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+[    {        "title": "2009.04894v3.Charge_Spin_Interconversion_in_Epitaxial_Pt_Probed_by_Spin_Orbit_Torques_in_a_Magnetic_Insulator.pdf",        "content": "Charge-Spin Interconversion in Epitaxial Pt Probed by Spin-Orbit Torques in a\nMagnetic Insulator\nPeng Li,1, 2Lauren J. Riddiford,1, 2Chong Bi,1, 3Jacob J. Wisser,1, 2Xiao-Qi Sun,1, 4Arturas Vailionis,5, 6Michael\nJ. Veit,1, 2Aaron Altman,1, 2Xiang Li,1, 3Mahendra DC,1, 3Shan X. Wang,1, 3Y. Suzuki,1, 2and Satoru Emori7\n1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA\n2Department of Applied Physics, Stanford University, Stanford, CA 94305, USA\n3Department of Material Science, Stanford University, Stanford, CA 94305, USA\n4Department of Physics, Stanford University, Stanford, CA 94305, USA\n5Stanford Nano Shared Facilities, Stanford University, Stanford, CA 94305, USA\n6Department of Physics, Kaunas University of Technology,\nStudentu Street 50, LT-51368 Kaunas, Lithuania\n7Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n(Dated: June 3, 2021)\nWe measure spin-orbit torques (SOTs) in a unique model system of all-epitaxial ferrite/Pt bilayers\nto gain insights into charge-spin interconversion in Pt. With negligible electronic conduction in the\ninsulating ferrite, the crystalline Pt \flm acts as the sole source of charge-to-spin conversion. A\nsmall \feld-like SOT independent of Pt thickness suggests a weak Rashba-Edelstein e\u000bect at the\nferrite/Pt interface. By contrast, we observe a sizable damping-like SOT that depends on the Pt\nthickness, from which we deduce the dominance of an extrinsic spin-Hall e\u000bect (skew scattering)\nand Dyakonov-Perel spin relaxation in the crystalline Pt \flm. Furthermore, our results point to\na large internal spin-Hall ratio of \u00190.8 in epitaxial Pt. Our experimental work takes an essential\nstep towards understanding the mechanisms of charge-spin interconversion and SOTs in Pt-based\nheterostructures, which are crucial for power-e\u000ecient spintronic devices.\nI. INTRODUCTION\nSpin-orbit torques (SOTs)1,2have been recognized\nas a viable means to manipulate magnetization in\nthin-\flm heterostructures. A prototypical SOT-driven\nmedium consists of a ferro(ferri)magnetic metal (FM)\ninterfaced with a nonmagnetic heavy metal (HM) with\nstrong spin-orbit coupling (e.g., Pt). In a conventional\npicture of SOTs in such a bilayer, an in-plane charge\ncurrent through the HM (or its surface) generates non{\nequilibrium spin accumulation via the spin-Hall e\u000bect\n(or Rashba-Edelstein e\u000bect)1{4. This charge-to-spin\nconversion then results in SOTs1,2,5,6, typically classi\fed\ninto (1) a \\damping-like\" torque that either enhances\nor counteracts damping in the magnetic layer and (2) a\n\\\feld-like\" torque that acts similarly to a torque from an\nexternal magnetic \feld.\nAlthough SOTs are often attributed to charge-to-spin\nconversion e\u000bects in the HM, recent studies point to\nother e\u000bects that impact SOTs in metallic FM/HM\nbilayers7{26. For example, current shunted through\nthe FM can generate additional SOTs through spin-\ndependent scattering within the FM or across the\nFM/HM interface7{16,27. Roughness at the interfaces\nof FM/HM bilayers, which are typically disordered\n(i.e., polycrystalline or amorphous), may also contribute\nto SOTs17{19. Even with atomically sharp FM/HM\ninterfaces, SOTs may be intrinsically impacted by spin-\nmemory loss20{24and proximity-induced magnetism25,26\ndue to orbital hybridization.\nThese possible complications in FM/HM bilayers make\nit di\u000ecult to elucidate the fundamental mechanisms of\nSOTs and, more generally, the underlying charge-to-spin conversion phenomena. These factors also impede\nreconciling the wide spread of reported spin transport\nparameters { particularly for the often-used HM of Pt,\nwith its spin di\u000busion length in the range \u00181-10 nm and\nits spin-Hall ratio \u00180.01-120,23,28{43.\nHere, we demonstrate a clean ferrimagnetic-\ninsulator/heavy-metal (FI/HM) model system where\nSOTs originate solely in the HM layer, permitting\na simpler analysis of charge-to-spin conversion\nmechanisms. Speci\fcally, we investigate SOTs at\nroom temperature in FI/HM bilayers where the FI\nis an epitaxial spinel ferrite \flm of MgAl 0:5Fe1:5O4\n(MAFO)44and the HM is an epitaxial \flm of Pt, whose\nhigh crystallinity is enabled by its excellent lattice\nmatch to the spinel45. The insulating nature of MAFO\nremoves all complications from electronic spin transport\nin the magnetic layer7{16,27, and the Pt layer with a\nsharp crystalline interface minimizes roughness-induced\nmechanisms17{19. Spin-memory loss and proximity-\ninduced magnetism are also expected to be signi\fcantly\nweaker in FI/HM46{49compared to FM/HM20{26due to\nweaker interfacial hybridization22.\nWe leverage the low damping of MAFO44to\nquantify both the damping-like and \feld-like SOTs\nin a straightforward manner through dc-biased spin-\ntorque ferromagnetic resonance (ST-FMR)50{54. We\nobserve a large damping-like SOT due to the spin-\nHall e\u000bect in the bulk of Pt1,3, along with an order-\nof-magnitude smaller \feld-like SOT attributed to the\ninterfacial Rashba-Edelstein e\u000bect4,55. Modeling the\nPt thickness dependence of the damping-like SOT\nand spin-pumping damping indicates that the skew\nscattering1,3,37,56and Dyakonov-Perel57,58mechanismsarXiv:2009.04894v3  [cond-mat.mes-hall]  1 Jun 20212\nprimarily govern charge-to-spin conversion and spin\nrelaxation, respectively, in epitaxial Pt. This\ncombination of mechanisms is distinct from the intrinsic\nspin-Hall e\u000bect and Elliott-Yafet spin relaxation often\nfound in Pt-based systems38,39,41,42,59. Our modeling\nresults point to a large internal spin-Hall ratio of\n\u00190.8 in Pt, while a small spin-mixing conductance of\n\u00191\u00021014\n\u00001m\u00002primarily limits the e\u000eciency of the\ndamping-like SOT in the MAFO/Pt bilayer. Our work\ndemonstrates a unique material system and experimental\napproach to uncover the mechanisms of charge-spin\ninterconversion in Pt, with minimal spurious in\ruence\nfrom the adjacent magnetic layer.\nII. FILM GROWTH AND STRUCTURAL\nPROPERTIES\nMAFO is a low-damping FI with a Curie temperature\nof\u0019400 K, which can be grown epitaxially on spinel\nMgAl 2O4(MAO) substrates44. We \frst deposit epitaxial\nMAFO \flms on (001)-oriented single-crystal MAO by\npulsed laser ablation. A sintered ceramic target of\nstoichiometric MgAl 0:5Fe1:5O4is ablated in 10 mTorr\nof O 2at a \ruence of \u00192 J/cm2, repetition rate of\n1 Hz, target-to-substrate separation of \u001975 mm, and\nsubstrate temperature of 450\u000eC. No post-annealing at\na higher temperature is performed. All MAFO \flms\nare grown to be 13 nm thick, which is within the\noptimal thickness range that ensures coherently strained\ngrowth and low Gilbert damping44,60. Broadband\nferromagnetic resonance (FMR) measurements con\frm\na Gilbert damping parameter of \u000b\u00190:0017 for these\nMAFO \flms, similar to prior reports44,49,60. Then, 3-19\nnm thick Pt layers are sputtered onto the MAFO \flms\nin 3 mTorr of Ar at room temperature. To avoid surface\ndamage, we used a low dc power of 15 W.\nX{ray di\u000braction (XRD) measurements indicate\nepitaxy and high crystallinity of our MAFO/Pt samples.\nFigure 1(a) shows symmetrical scans for MAFO/Pt and\nMAFO samples. Strong Pt(111) and MAFO(004) Bragg\npeaks indicate a high degree of out-of-plane epitaxy. The\nvisible Laue oscillations around the Pt(111) peak for\nthe MAFO/Pt bilayers further indicate high structural\nquality of the Pt \flm. The degree of crystallinity of the\nPt layer is determined by performing a rocking curve\nmeasurement around the Pt(111) peak. The narrow\nrocking curve width of \u00190.4\u000e(Fig. 1(b)) indicates a\nuniform out-of-plane orientation of Pt crystals with an\nonly small mosaic spread.\nThe in-plane orientation of MAFO/Pt is investigated\nby measuring asymmetrical (113) Bragg peaks for Pt,\nMAFO, and MAO layers. The MAFO layer is fully\ncoherently strained to the MAO substrate as indicated\nin the previous study44. As can be seen from Fig. 1(c),\nthe MAFO layer and MAO substrate exhibit four-fold\nsymmetry that is expected from their cubic structures.\nThe Pt(113) peak exhibits twelve maxima indicating a\n30 40 50\n-3 -2 -1 0 1 2 3\nMAFO (004)Pt (111)\nMAO (004)\nMAFOIntensity ( a.u.)\n2θ(deg)\n0 120 240 360\n90o\n30o\nφ(deg)Intensity ( a.u.)MAFO(MAO)\nPtPt (111) twins aMAO=8.083 Å aPt=3.912 Å\nPeaks at =\n0, 120, 240\nPeaks at =\n30, 150, 270\nPeaks at =\n60, 180, 300\nPeaks at =\n90, 210, 330(c)(a) (b)\n(d)\nIntensity ( a.u.)\nΔω(deg)FWHM≈0.4o\nMAFO/Pt -5MAFO/Pt -3FIG. 1. XRD analysis of samples. (a) XRD 2 \u0012=!scans of\nMAFO (13 nm)/Pt (5 nm), MAFO (13 nm)/Pt (3 nm), and\nMAFO (13 nm). (b) Rocking curve scan about the Pt (111)\npeak for the MAFO (13 nm)/Pt (5 nm) sample shown in (a),\nwith FWHM \u00190:4\u000e. (c) XRD \u001escans on the (113) plane of\nthe MAFO (13 nm)/Pt (5 nm) sample. Pink: MAFO. Green:\nMAO. (d) Lattice matching relationship between the Pt and\nMAFO (MAO) unit cells.\nrather complex epitaxial relationship. Careful analysis\nof the Pt in-plane orientation on MAFO reveals a\ntwinning pattern of the Pt domains, which is presented\nin Fig. 1(d). One can distinguish four Pt domains that\nmatch MAFO epitaxially and produce in total twelve\nPt(113) peaks as shown in Fig. 1(c).\nIt should be noted that the epitaxial growth of Pt on\nMAFO is in contrast to polycrystalline or amorphous\nPt on iron garnets33,61,62. Further, X-ray re\rectivity\nindicates a small roughness of <0.2 nm at the MAFO/Pt\ninterface. Our structural characterization thus con\frms\nthat MAFO/Pt is a high-quality model system with a\nhighly crystalline structure and sharp interface.\nIII. RESULTS AND DISCUSSION\nA. DC-Biased Spin-Torque Ferromagnetic\nResonance\nThe MAFO/Pt bilayers are lithographically patterned\nand ion-milled to 60 \u0016m\u000210\u0016m strips with the edges\nparallel to the in-plane h110iaxes of MAFO. They are\nthen contacted by Ti (5 nm)/Au (120 nm) ground-signal-\nground electrodes to allow input of a microwave current\nfor our ST-FMR measurements at room temperature,\nas illustrated in Fig. 2(a). We have veri\fed that the\nmagnetic properties of MAFO/Pt are unchanged by the\nlithographic patterning process (see Appendix A).\nThe microwave current in Pt induces SOTs and a3\nFIG. 2. ST-FMR measurement setup. (a) MAFO/Pt stack\netched to a 60 \u0016m\u000210\u0016m strip. Magnetization, external\n\feld, rf \feld, and SOTs are shown as the arrows. The\nground-signal-ground Au electrode connects MAFO/Pt to\nthe external circuit. (b) FMR spectrum at 4 GHz. Red\ncurve: symmetric Lorentzian contribution. Green curve:\nantisymmetric Lorentzian contribution. Blue curve: total \ft.\nclassical Oersted \feld torque on the magnetization in\nthe MAFO layer. ST-FMR spectra are obtained from\nthe recti\fed voltage due to magnetoresistance and spin-\npumping signals63,64with \feld modulation65. Each\nintegrated ST-FMR spectrum (e.g., Fig. 2(c)) can be\n\ft with a superposition of symmetric and antisymmetric\nLorentzians to extract the half-width-at-half-maximum\nlinewidth \u0001 Hand resonance \feld Hres.\nWe use an additional dc bias current to directly extract\nthe damping-like and \feld-like SOTs50{54in MAFO/Pt.\nThis dc bias approach circumvents ambiguities of the\noft-used symmetric/antisymmetric Lorentzian ST-FMR\nlineshape analysis (e.g., where the symmetric Lorentzian\ncan contain voltage signals from spin pumping and\nthermoelectric e\u000bects63,64,66,67) and instead probes both\nSOTs in a direct manner. In particular, the dc damping-\nlike SOT modi\fes the e\u000bective damping ( /linewidth\n\u0001H) linearly with the dc bias current density Jdc; the\ndc \feld-like torque shifts the resonance \feld Hreslinearly\nwithJdc. Since all of the current \rows in the Pt layer,\nthe classical Oersted \feld HOeis easily determined from\nHOe=Jdc=tPt=2, wheretPtis the Pt thickness, and\nsubtracted from dHres=dJdcto extract the \feld-like SOT.\nFigure 3(a,b) shows the e\u000bect of Jdcon \u0001Hand\nHres. The linear dependence on current indicates\nthat Joule heating contributions68are minimal in these\nmeasurements. By reversing the magnetization direction\n(external magnetic \feld direction), we observe a reversal\nin the slope for \u0001 H(orHres) versusJdcconsistent with\nthe symmetry of the SOTs1,2.\nFrom the linear slope of linewidth \u0001 HversusJdc\n(Fig. 3(a)), the damping-like SOT e\u000eciency \u0012DLis\nreadily quanti\fed with50,52\nj\u0012DLj=2jej\n~(Hres+Me\u000b=2)\u00160MstM\njsin\u001ej\f\f\f\fd\u000be\u000b\ndJdc\f\f\f\f;(1)\nwhere\u000be\u000b=j\rj\u0001H=(2\u0019f),j\rj=(2\u0019) = 29 GHz/T is\n0 5 10 15 200.000.050.100.15\n0 5 10 15 200.0000.0080.016\n-3 -2 -1 0 1 2 30.81.21.6Jdc(1010 A/m)Linewidth µ 0ΔH (mT)(a)\n+H\n-H\nPt thickness, tPt(nm)ƟDL(c)\nƟFL\nPt thickness ,tPt(nm)(d)\n-3 -2 -1 0 1 2 3-0.4-0.20.00.20.4(b)\n-H\n+Hµ0ΔHres(mT)\nJdc(1010 A/m)FIG. 3. Measurement of SOT e\u000eciencies. (a) Dependence of\nlinewidth \u0001 Hon dc bias current density Jdcfor MAFO (13\nnm)/Pt (5 nm). Linewidths and linear \fts under positive\n(blue boxes and line) and negative (red dots and line)\nmagnetic \felds are shown. (b) Resonance \feld change \u0001 Hres\nas a function of Jdcfor the MAFO (13 nm)/Pt (5 nm).\nResonance \felds and linear \fts under positive (purple dots\nand line) and negative (green dots and line) magnetic \felds\nare shown. The Oersted \feld contributions are shown as\npurple (positive) and green (negative) dashed lines. (c,d) Pt\nthickness dependence of (c) \u0012DLand (d)\u0012FLfor MAFO/Pt.\nNote the di\u000berent vertical scales for \u0012DLand\u0012FL. The error\nbars in (c) and (d) are derived from the linear \fts of linewidth\nand resonance \feld change vs. Jdc.\nthe gyromagnetic ratio of MAFO44,fis the microwave\nfrequency (e.g., f= 4 GHz in Figs. 2 and 3), tM=\n13 nm is the MAFO thickness, and \u001e= 45\u000eor 225\u000e\nis the in-plane magnetization orientation with respect\nto the current axis ( x-axis in Fig. 2(a)). In applying\nEq. 1, we account for the sample-to-sample variation\nin the saturation magnetization Ms= 90\u000095 kA/m\n(determined by SQUID magnetometry) and the e\u000bective\nmagnetization \u00160Me\u000b= 1:2\u00001:5 T (determined by \ftting\nthe frequency dependence of resonance \feld44). The large\ne\u000bective magnetization of epitaxial MAFO arises due to\nsigni\fcant magnetoelastic easy-plane anisotropy44.\nThetPtdependence of \u0012DLis summarized in Fig. 3(c).\nThe increase in \u0012DLwithtPtup to\u00195 nm (Fig. 3(c))\nsuggests that the spin-Hall e\u000bect in the Pt bulk is the\ndominant source of the damping-like SOT6,38. The\ndecrease in \u0012DLat highertPtmight seem surprising, but\na similar trend has been observed in prior experiments38.\nWe also quantify the \feld-like SOT e\u000eciency \u0012FL\nfrom the linear shift of HreswithJdc(Fig. 3(b)) and\nsubtracting the Oersted \feld contribution19,52\nj\u0012FLj=2jej\n~\u00160MstM\njsin\u001ej\u0012\f\f\f\fdHres\ndJdc\f\f\f\f\u0000tPt\n2jsin\u001ej\u0013\n;(2)\nwhere the term proportional to tPtaccounts for the4\nOersted \feld. As shown in Fig. 3(d), the constant value\nof\u0012FLwith Pt thickness implies that the \feld-like SOT\narises from the MAFO/Pt interface, e.g., via the Rashba-\nEdelestein e\u000bect4,55,69. However, this \feld-like SOT is\nweak, i.e., similar in magnitude to the Oersted \feld\n(Fig. 3(b)). Indeed, we \fnd that \u0012FL\u00180:01 is about\nan order of magnitude smaller than \u0012DL.\nBased on the dominance of the strongly tPt-dependent\ndamping-like SOT over the tPt-independent \feld-like\nSOT, we conclude that charge-spin interconversion\nprocesses in the bulk of Pt dominate over those at the\nMAFO/Pt interface. It has been proposed that a \feld-\nlike SOT could arise from the bulk of Pt in the presence\nof an imaginary part of the spin-mixing conductance,\nIm[G\"#]70. A substantial Im[ G\"#] would manifest in a\nshift in the gyromagnetic ratio (or g-factor) in MAFO\nwith the addition of a Pt overlayer71. Since such a shift\nis not observed, we rule out this scenario of a \feld-like\nSOT of \\bulk\" origin. In other words, the damping-like\ntorque is the predominant type of SOT that arises from\nthe bulk of Pt. Therefore, in the following sections, we\nuse the damping-like SOT as a measure of charge-to-spin\nconversion in Pt.\nB. Modeling the Pt-Thickness Dependence of the\nSpin-Pumping Damping and Damping-Like\nSpin-Orbit Torque\nWe employ a model similar to the one used by Berger\net al.23to assess charge-spin interconversion mechanisms\nin Pt. This model estimates key parameters that\ngovern charge-spin interconversion by \ftting the tPt\ndependence of two experimentally measured quantities:\nthe Gilbert damping parameter \u000band the damping-like\nSOT conductivity \u001bDL.\nWe have measured the damping parameter \u000bby\ncoplanar-waveguide-based FMR and ST-FMR, which\nyield consistent results for unpatterned and patterned\nMAFO/Pt (see Appendix A). As can be seen in\nFig. 4(b,c), MAFO/Pt bilayers exhibit higher \u000bthan\nthe bare MAFO \flms with tPt= 0. In Sec. III C, we\nattribute this damping enhancement to spin pumping71,\ni.e., due to the loss of spin angular momentum pumped\nfrom the resonantly excited MAFO layer to the adjacent\nspin sink layer of Pt. In Sec. III D, we further consider\nan additional contribution to the enhancement of \u000bdue\nto spin-memory loss or two-magnon scattering.\nTo parameterize the strength of the damping-like\nSOT, we employ the \\SOT conductivity,\" \u001bDL =\n\u0012DL=\u001aPt. Normalizing \u0012DLby the Pt resistivity \u001aPtmakes\nexplicit the relationship between the SOT and electronic\ntransport. We also remark that \u001bDLis equivalent to the\nSOT e\u000eciency per unit electric \feld \u0018E\nDLin Refs. 38, 42.\nThetPtdependence of \u001aPt(\ft curve in Fig. 4(a)) is\ninterpolated by using the empirical model outlined in\nAppendix D.\nIn contrast to Ref. 23 that studies FM/Pt bilayerswhere electronic spin transport in the FM can generally\nyield additional e\u000bects that impact SOTs, our MAFO/Pt\nsystem restricts the source of SOTs to Pt. We\nare therefore able to examine the spin-Hall e\u000bect\nof Pt without any complications from an electrically\nconductive FM.\nTo model our experimental results, we consider two\ntypes of spin-Hall e\u000bect1,3:\n•the intrinsic mechanism, where the internal\nspin-Hall ratio \u0012SH{ i.e., the charge-to-spin\nconversion e\u000eciency within the Pt layer itself { is\nproportional to \u001aPt, with a constant internal spin-\nHall conductivity \u001bSH=\u0012SH/\u001aPt;\n•the skew scattering mechanism, where \u0012SHis\nindependent of \u001aPt.\nWe also consider two mechanisms of spin relaxation that\ngovern the spin di\u000busion length \u0015sin Pt35,57,58:\n•Elliott-Yafet (EY) spin relaxation, where spins\ndepolarize during scattering such that \u0015sscales\ninversely with \u001aPt, i.e.,\u0015s=\u0015bulk\ns\u001abulk\nPt=\u001aPt;\n•Dyakonov-Perel (DP) spin relaxation, where spins\ndepolarize between scattering events such that \u0015sis\nindependent of \u001aPt(as outlined by Boone et al.35).\nThus, we model four combinations of the above-\nlisted charge-to-spin conversion and spin relaxation\nmechanisms, as shown in Fig. 4(b-e).\nSimilar to Ref. 23, we self-consistently \ft \u000bvs.tPt\n(Fig. 4(b,c)) and \u001bDLvs.tPt(Fig. 4(d,e)) by using\nstandard spin di\u000busion models6,35,71, as elaborated in\nAppendix E, with four free parameters:\n•spin di\u000busion length \u0015sin the case of DP spin\nrelaxation, or its bulk-limit value \u0015bulk\nsin the case\nof EY spin relaxation;\n•internal spin-Hall ratio \u0012SHof Pt in the case of skew\nscattering, or its bulk-limit value \u0012bulk\nSH=\u001bSH\u001abulk\nPt\nin the case of intrinsic spin-Hall e\u000bect;\n•real part of the spin-mixing conductance G\"#at the\nMAFO/Pt interface, neglecting the imaginary part\nas justi\fed in Sec. III A;\n•e\u000bective damping enhancement \u000bSML due to\ninterfacial spin-memory loss or two-magnon\nscattering, as discussed in detail in Sec. III D.\nA key assumption here is that the spin-pumping damping\nand damping-like SOT share the same values of \u0015s,\nG\"#, and\u000bSML. This is justi\fed by the enforcement\nof Onsager reciprocity on the charge-spin interconversion\nprocesses of spin pumping and SOT23,72. We also assume\na negligible interfacial contribution to the spin-Hall e\u000bect\nin Pt73, which would yield a \fnite value of \u001bDLwhentPt\nis extrapolated to zero. Indeed, as shown in Fig. 4, the5\ntPtdependence of \u001bDLis adequately modeled without\nincorporating the interfacial spin-Hall e\u000bect.\nFor simplicity, we \frst proceed by setting \u000bSML= 0 in\nSec. III C. This is a reasonable assumption considering\nthat interfacial spin-memory loss is likely much weaker\nin MAFO/Pt than in all-metallic FM/Pt systems20{26.\nNevertheless, we also discuss the consequence of \u000bSML>\n0 in Sec. III D.\nC. Mechanisms and Parameters for Charge-Spin\nInterconversion in Pt: Zero Spin-Memory Loss\nOur modeling results under the assumption of zero\nspin-memory loss are summarized in Fig. 4 and Table I.\nWe \fnd that the combination of skew scattering and\nDP spin relaxation (solid green curves in Fig. 4(c,e))\nbest reproduces the tPtdependence of both \u000band\u001bDL.\nAlthough this observation does not necessarily rule out\nthe coexistence of other mechanisms23,43,57,58, it suggests\nthe dominance of the skew scattering + DP combination\nin the epitaxial Pt \flm. Skew scattering in highly\ncrystalline Pt is consistent with what is expected for\n\\superclean\" Pt, in contrast to the intrinsic spin-Hall\ne\u000bect that is dominant in \\moderately dirty\" Pt37.\nThe dominance of DP spin relaxation { i.e.,\nspin depolarization (dephasing) from precession about\ne\u000bective spin-orbit \felds { is perhaps surprising, since\nit is usually thought to be inactive in centrosymmetric\nmetals (e.g., Pt). Indeed, in the context of spin transport\nin Pt, it is typical to assume EY spin relaxation where\nspins depolarize when their carriers (e.g., electrons)\nare scattered38,39,41,42,59. However, a recent quantum\ntransport study indicates the dominance of DP spin\nrelaxation in crystalline Pt57, which is in line with our\nconclusion here. Possible origins of the DP mechanism\ninclude symmetry breaking between the substrate and\nthe surface of the crystalline Pt \flm74and strong spin\nmixing caused by the distinct band structure (large spin\nBerry curvature) of Pt58. DP spin relaxation may also\nbe more pronounced when proximity-induced magnetism\nin Pt is negligible58, as is likely the case for Pt interfaced\nwith the insulating MAFO75. We also note that DP\nspin relaxation has been previously used to model the\nangular dependence of spin-Hall magnetoresistance76,77\nin MAFO/Pt49. The combination of skew scattering and\nDP spin relaxation, though not reported in prior SOT\nexperiments, is reasonable for MAFO/Pt.\nWe now discuss the parameters quanti\fed with our\nmodel, as summarized in the \\skew scatt.+DP\" row\nin Table I. The value of G\"#\u00191\u00021014\n\u00001m\u00002\nis comparable to those previously reported for FI/Pt\ninterfaces33,49,78,79, and\u0015s\u00193 nm is in the intermediate\nregime of\u0015s\u00181\u000010 nm in prior reports on Pt20,23,28{43.\nWe \fnd a large internal spin-Hall ratio of \u0012SH\u0019\n0:8. While a few studies have alluded to \u0012SHon\nthe order of unity in transition metals23,33,42,80,81, our\nexperimental study is the \frst to derive such a large\n0246a (10-3)intrinsic\n𝜃SH∝𝜌Pt\nskew scatt .\n𝜃SH=constEY\n𝜆s∝1/𝜌Pt\nDP\n𝜆s=const\n0 5 10 15 20 intrinsic\n skew scatt.\nPt thickness, tPt (nm)\n0 5 10 15 200246\n intrinsic\n skew scatt.sDL = qDL/rPt (105 W-1m-1)\nPt thickness, tPt (nm)\n(d)\n02468rPt (10-7 Wm)(a)\n(b)\nEY DP(e)(c)FIG. 4. Pt thickness dependence of: (a) resistivity \u001aPt,\nwith the solid curve showing the \ft obtained with the model\ndescribed in Appendix D; (b,c) Gilbert damping parameter \u000b,\nwith the black horizontal dashed line indicating the average\ndamping parameter of uncapped MAFO; (d,e) damping-like\nSOT conductivity \u001bDL. Modeling results based on Elliott-\nYafet (EY) spin relaxation are shown in (b,d), whereas those\nbased on Dyakonov-Perel (DP) spin relaxation are shown in\n(c,e). The dotted curves are based on the intrinsic spin-Hall\ne\u000bect, and the solid curves are based on skew scattering. The\nmodeling results in (b-e) are obtained by assuming zero spin-\nmemory loss and two-magnon scattering (i.e., \u000bSML = 0).\nIn (b-e), the error bars are comparable to or smaller than\nthe symbol size and are derived from the linear \fts of FMR\nlinewidth vs. frequency (b,c) and dc bias current density\n(d,e).\nvalue in Pt without uncertainties from a conductive\nFM23,42,80,81or microwave calibration23,33,80,81. Our\n\fnding of\u0012SHapproaching unity is also distinct from\npreviously reported spin-Hall ratios <0:1 in all-epitaxial\nFM/Pt59,82{85. This discrepancy may be partially\nexplained by the conductive FM reducing the apparent\ncharge-to-spin conversion e\u000eciency, or by the indirect\nnature of the measurements in these reports. With direct\nSOT measurements on the model-system MAFO/Pt\nbilayers, our study points to the possibility of a strong\nmodel \u000bSMLG\"#(\n\u00001m\u00002)\u0015s(nm)\u0012SH\nintrinsic + EY 0 2 :5\u0002101421 0.21\nskew scatt. + EY 0 1 :1\u000210144.7 1.2\nintrinsic + DP 0 1 :8\u000210145.7 0.25\nskew scatt. + DP 0 1 :3\u000210143.3 0.83\nTABLE I. Parameters for the modeled curves in Fig. 4. For\ncharge-to-spin conversion = intrinsic (for spin relaxation =\nEY),\u0012SH(\u0015s) is the value at \u001aPt=\u001abulk\nPt= 1:1\u000210\u00007\nm.6\nspin-Hall e\u000bect in highly crystalline Pt in the skew-\nscattering regime, where the charge-to-spin conversion\ne\u000eciency could be greater than the limit set by the\nintrinsic spin-Hall e\u000bect1,3,37,42.\nD. Mechanisms and Parameters for Charge-Spin\nInterconversion in Pt: Finite Spin-Memory Loss\nA natural question at this point is how \fnite spin-\nmemory loss at the MAFO/Pt interface impacts the\nparameters quanti\fed in our modeling. Moreover, while\nbare MAFO exhibits negligible two-magnon scattering44,\nan overlayer (Pt in this case) on top of MAFO may\ngive rise to two-magnon scattering at the interface86.\nBoth spin-memory loss and two-magnon scattering would\nhave the same consequence in that they enhance the\napparent damping parameter, \u000b, independent of tPt23,87.\nWe therefore model spin-memory loss and two-magnon\nscattering with a phenomenological parameter, \u000bSML.\nFigure 5 and Table II summarize our modeling results\nincorporating \fnite spin-memory loss or two-magnon\nscattering (i.e., \u000bSML>0). Finite \u000bSML does not\nimprove the \ft quality in \u001bDLvs.tPtof the EY models\n(Fig. 5(a,b)). By contrast, the \ft quality is improved for\nthe DP models with increasing \u000bSML, particularly in \u000b\nvs.tPt(Fig. 5(c,d)). We therefore focus on the results\nfor the DP models.\nAs shown in Table II, increasing \u000bSML signi\fcantly\ndecreasesG\"#, consistent with the reduced share of spin\npumping in the damping enhancement. To compensate\nfor the smaller G\"#, the internal spin-Hall ratio \u0012SH\nmust increase to reproduce the tPtdependence of \u001bDL\n(Ref. 87). In the \\skew scattering + DP\" model, shown\nto be most plausible in Sec. III C, \u0012SHincreases to values\nexceeding unity with \fnite \u000bSML. At a su\u000eciently large\n\u000bSMLof&0:002, the \\intrinsic + DP\" model appears to\nbecomes plausible (see Fig. 5(c)), but this scenario also\nyields\u0012SH>1.\nIn both of the above DP scenarios, substantial spin-\nmemory loss or two-magnon scattering apparently leads\nto an unphysically large value of internal spin-Hall ratio\nin Pt exceeding unity. It is then reasonable to conclude\nthat spin-memory loss and two-magnon scattering is\nnegligibly small in epitaxial MAFO/Pt. This is in\nstark contrast to the large spin-memory loss deduced for\nall-metallic FM/Pt bilayers23. The small spin-memory\nloss in MAFO/Pt also suggests fundamentally di\u000berent\nspin-transport mechanisms between FM/Pt and FI/Pt\nsystems, which could be exploited for more e\u000ecient SOT\ndevices in the future. Our \fnding motivates further\nstudies to test whether the negligible spin-memory loss\nis due to the crystalline growth or due to the absence of\nproximity-induced magnetism.E. Implications of the Large Internal Spin-Hall\nRatio in Pt\nFrom our analysis in Sec. III C, we have arrived\nat a large internal spin-Hall ratio of \u0012SH\u00190:8 in\nepitaxial Pt. Yet, the observed spin-torque e\u000eciency\nof\u0012DL.0:15 implies an interfacial spin transparency\nratio\u0012DL=\u0012SHof.0:2. In other words, at most only\n20% of the spin accumulation generated by the spin-Hall\ne\u000bect in Pt is transferred to the magnetic MAFO layer\nas the damping-like SOT. The origin of this ine\u000ecient\nspin transfer, according to the spin di\u000busion model\nemployed here, is the small spin-mixing conductance\nofG\"#\u00191\u00021014\n\u00001m\u00002, which is several times\nlower than G\"#computationally predicted for FM/Pt\ninterfaces88{90. The small G\"#results in a substantial\nspin back\row87,91that prevents e\u000ecient transmission of\nspin angular momentum across the MAFO/Pt interface.\nWe emphasize that spin-memory loss is likely negligible\nat the MAFO/Pt interface (see Sec. III D) and hence not\nresponsible for the ine\u000ecient spin transfer.\nThere may be an opportunity to enhance the spin\ntransparency { and hence the SOT e\u000eciency { by\nengineering the interface. One possible approach is\nto use an ultrathin insertion layer of NiO, which has\nbeen reported to increase the spin transparency ratio\nto essentially unity in FM/Pt systems91. However,\nit remains to be explored whether the ultrathin NiO\ninsertion layer can increase the spin transparency without\ncausing substantial interfacial spin scattering86in FI/Pt\nbilayers. An increased spin transparency (via enhanced\nG\"#) also leads to higher spin-pumping damping71,92,\nwhich may not be desirable for applications driven by\nprecessional switching or auto-oscillations.\nAnother striking implication of the large internal spin-\nHall ratio is a large maximum spin-Hall conductivity\n\u001bSH=\u0012SH=\u001abulk\nPt of\u00198\u0002106\n\u00001m\u00001, which is at\nleast an order of magnitude greater than \u001bSH\u0018104\u0000\n105\n\u00001m\u00001typically predicted from band structure\ncalculations93{97. It should be noted, however, that these\ncalculations are for the intrinsic spin-Hall e\u000bect, whereas\nour experimental data are best captured by the extrinsic\nspin-Hall e\u000bect of skew scattering. We thus speculate\nthat this di\u000berence in mechanism could account for the\ndiscrepancy in \u001bSHderived from our experimental work\nand from prior calculations.\nFinally, we comment on remaining open fundamental\nquestions. Comparing MAFO/epitaxial-Pt and\nMAFO/ polycrystalline -Pt could reveal the critical\nrole of crystallinity in charge-spin interconversion, spin\nrelaxation, and the internal spin-Hall ratio in Pt. This\ncomparison study is precluded here due to the di\u000eculty\nin growing polycrystalline Pt on MAFO; Pt has a strong\ntendency to be epitaxial on MAFO due to the excellent\nlattice match, even when Pt is sputter-deposited with\nthe substrate at room temperature. Moreover, while the\nepitaxial Pt \flm on MAFO is single-crystalline in the\nsense that its out-of-plane crytallographic orientation7\nintrinsic + EY skew scattering + EY intrinsic + DP skew scattering + DP\n0246a (10-3)\n0246a (10-3)\n0 5 10 15 200246\n aSML = 0.002\n aSML = 0.001\n aSML = 0sDL = qDL/rPt (105 W-1m-1)\nPt thickness, tPt (nm)\n0 5 10 15 200246\n aSML = 0.002\n aSML = 0.001\n aSML = 0sDL = qDL/rPt (105 W-1m-1)\nPt thickness, tPt (nm)\n0246a (10-3)\n0 5 10 15 200246\n aSML = 0.002\n aSML = 0.001\n aSML = 0sDL = qDL/rPt (105 W-1m-1)\nPt thickness, tPt (nm)\n0246a (10-3)\n0 5 10 15 200246\n aSML = 0.002\n aSML = 0.001\n aSML = 0sDL = qDL/rPt (105 W-1m-1)\nPt thickness, tPt (nm) (a) (b) (c) (d)\naSML= 0.001aSML= 0.002\nFIG. 5. Pt thickness dependence of the Gilbert damping parameter \u000band the damping-like SOT conductivity \u001bDL, taking into\naccount di\u000berent strengths of spin-memory loss or two-magnon scattering (parameterized by \u000bSML), for the four combinations\nof charge-to-spin conversion and spin relaxation mechanisms: (a) intrinsic spin-Hall e\u000bect + Elliott-Yafet (EY), (b) skew\nscattering + EY, (c) intrinsic spin-Hall e\u000bect + Dyakonov-Perel (DP), and (d) skew scattering + DP. The error bars are\ncomparable to or smaller than the symbol size; they are derived from the linear \fts of FMR linewidth vs. frequency (for \u000b)\nand ST-FMR linewidth vs. dc bias current density (for \u001bDL).\nintrinsic + EY skew scatt. + EY intrinsic + DP skew scatt. + DP\n\u000bSMLG\"#(\n\u00001m\u00002)\u0015s(nm)\u0012SHG\"#(\n\u00001m\u00002)\u0015s(nm)\u0012SHG\"#(\n\u00001m\u00002)\u0015s(nm)\u0012SHG\"#(\n\u00001m\u00002)\u0015s(nm)\u0012SH\n0 2:5\u0002101421 0.21 1:1\u000210144.7 1.2 1:8\u000210145.7 0.25 1:3\u000210143.3 0.83\n0.001 1:5\u0002101423 0.40 0:7\u000210144.7 2.7 1:0\u000210146.2 0.53 0:9\u000210143.6 1.5\n0.002 0:6\u0002101426 1.3 0:4\u000210145.0 7.5 0:6\u000210147.1 1.2 0:5\u000210143.8 4.1\nTABLE II. Parameters for the modeled curves in Fig. 5. For charge-to-spin conversion = intrinsic (for spin relaxation = EY),\n\u0012SH(\u0015s) is the value at \u001aPt=\u001abulk\nPt= 1:1\u000210\u00007\nm.\nis exclusively (111), it is yet unclear how the twin\ndomains (discussed in Sec. II) in\ruence charge-spin\ninterconversion in Pt. Determining the impact of\nmicrostructure on spin-Hall and related e\u000bects in Pt\nremains a subject of future work.\nFurthermore, we acknowledge the possibility that\nthe model employed in our present study (outlined in\nSec. III B and Appendix E) is incomplete. For instance,\nwe have assumed that the damping-like SOT and spin-\npumping damping are reciprocal phenomena with shared\nG\"#and\u0015s. This commonly made assumption23{ with\nprior studies suggesting that such reciprocity holds46,52\n{ is necessary for constraining the \ftting of the limited\nnumber of experimental data points. Further studies are\nrequired for con\frming whether the damping-like SOT\nand spin-pumping damping can be captured by the same\nvalues ofG\"#and\u0015s.\nIV. SUMMARY\nWe have measured SOTs in a low-damping, epitaxial\ninsulating spinel ferrite (MgAl 0:5Fe1:5O4, MAFO)\ninterfaced with epitaxial Pt. This model-system bilayer\nenables a unique opportunity to examine charge-spin\ninterconversion mechanisms in highly crystalline Pt,while eliminating complications from electronic transport\nin (or hybridization with) a magnetic metal. Our key\n\fndings are as follow.\n1. Charge-to-spin conversion in Pt appears to be\nprimarily a bulk e\u000bect, rather than an interfacial\ne\u000bect. A sizable damping-like SOT, which depends\nstrongly on the Pt thickness, arises from the spin-\nHall e\u000bect within Pt. An order-of-magnitude\nsmaller \feld-like SOT, independent of the Pt\nthickness, is attributed to the Rashba-Edelstein\ne\u000bect at the MAFO/Pt interface.\n2. In crystalline Pt, the extrinsic spin-Hall e\u000bect\nof skew scattering and the Dyakonov-Perel spin\nrelaxation mechanism likely dominate. This is\nin contrast to the combination of the intrinsic\nspin-Hall e\u000bect and Elliott-Yafet spin relaxation\ntypically reported for Pt-based systems.\n3. The internal spin-Hall ratio deduced for crystalline\nPt is large, i.e., \u0012SH\u00190:8. While a\nsimilar magnitude has been suggested before from\nexperiments on all-metallic FM/Pt bilayers, greater\ncon\fdence may be placed in our result owing to\nthe cleanliness of the MAFO/Pt system, the direct\nnature of the SOT measurement method, and8\nthe self-consistent modeling of the SOT and spin-\npumping damping.\n4. Spin-memory loss appears to be minimal in\nthe epitaxial MAFO/Pt system. Modeled\nscenarios with substantial spin-memory loss yield\nunphysically large internal spin-Hall ratios that\nexceed unity.\n5. The factor limiting the damping-like SOT e\u000eciency\nin the MAFO/Pt bilayer, despite the apparently\nlarge\u0012SH, is the small spin-mixing conductance\nG\"#. Enhancing G\"#while keeping spin-memory\nloss minimal could increase the SOT e\u000eciency.\nOverall, our work demonstrates the utility of\nepitaxial insulating-ferrite-based heterostructures\nfor understanding spin-transport phenomena in the\nwidely-used spin-Hall metal of Pt, as well as for\nengineering materials for e\u000ecient spintronic devices.\nACKNOWLEDGMENTS\nThis work was funded by the Vannevar Bush Faculty\nFellowship of the Department of Defense under Contract\nNo. N00014-15-1-0045. L.J.R. acknowledges support\nfrom the Air Force O\u000ece of Scienti\fc Research under\nGrant No. FA 9550-20-1-0293. J.J.W. acknowledges\nsupport from the U.S. Department of Energy, Director,\nO\u000ece of Science, O\u000ece of Basic Energy Sciences,\nDivision of Materials Sciences and Engineering under\nContract No. DESC0008505. S.X.W. acknowledges\nfunding from NSF Center for Energy E\u000ecient Electronics\nScience (E3S) and ASCENT, one of six centers in\nJUMP, a Semiconductor Research Corporation (SRC)\nprogram sponsored by DARPA. L.J.R. and M.J.V.\nacknowledge the National Science Foundation Graduate\nFellowships. X.-Q.S. acknowledges support from DOE\nO\u000ece of Science, O\u000ece of High Energy Physics under\nGrant NO. DE-SC0019380. S.E. acknowledges support\nfrom the National Science Foundation, Award No.\nDMR-2003914. Part of this work was performed at\nthe Stanford Nano Shared Facilities (SNSF)/Stanford\nNanofabrication Facility (SNF), supported by the\nNational Science Foundation under award ECCS-\n1542152. S.E. acknowledges Makoto Kohda, Xin\nFan, and Vivek Amin for fruitful discussions. P.L.\nacknowledges Wei Zhang for valuable suggestions in\nbuilding the ST-FMR system.Appendix A: E\u000bect of Sample Processing on the\nMagnetic Properties of MAFO\nWe have used both broadband FMR (i.e., with\nunpatterned \flms placed on a coplanar waveguide, see\nRef. 44 for details) and ST-FMR (i.e., with microwave\ncurrent injected through patterned 10- \u0016m-wide strips)\nto measure the frequency dependence of FMR linewidth\nand resonance \feld. Thus, it is important to con\frm the\nconsistency of measurements between the two techniques.\nFigure 6(a) plots the linewidth vs. frequency data\nfor a bare MAFO \flm (13 nm) that we started with,\nthe MAFO (13 nm)/Pt (5 nm) \flm after Pt deposition,\nand the ST-FMR pattern with MAFO (13 nm)/Pt\n(5 nm) after the microfabrication processes. The\ndamping parameters of the MAFO/Pt unpatterned \flm\nand patterned strip are essentially identical, con\frming\nthe consistency of the broadband FMR and ST-FMR\nmeasurements.\nWe also show in Fig. 6(b) that the frequency\ndependence of resonance \feld is unaltered before and\nafter microfabrication. The \ft using the Kittel\nequation44indicates negligible ( \u001c5%) di\u000berence in the\ne\u000bective magnetization (dominated by mangnetoelastic\neasy-plane anisotropy) and gyromagnetic ratio for the\nunpatterned \flm and patterned strip. The results\nin Fig. 6 therefore con\frm that the microfabrication\nprocesses have little to no e\u000bect on the essential magnetic\nproperties of MAFO/Pt.\n0 5 10 15 20 25 30024f(GHz)HWHM Linewidth (mT)\nMAFO /Pt Pattern\nα=0.00506\nMAFO /Pt Film\nα=0.00514\nMAFO Film\nα=0.00149\n200 400102030\nField (mT)00f(GHz)(a) (b)\nFIG. 6. Frequency dependence of (a) linewidth and (b)\nresonance \feld a bare MAFO \flm (13 nm), unpatterned\nMAFO (13 nm)/Pt (5 nm) \flm, and patterned MAFO (13\nnm)/Pt (5 nm) ST-FMR strip.\nAppendix B: Microwave Power Dependence of the\nSpin-Torque Ferromagnetic Resonance Signal\nFigure 7 shows the dependence of the ST-FMR signal\namplitude on the microwave power. The ST-FMR\nvoltage increases linearly with the microwave power,\nindicating that all the measurements are done in the\nlinear regime in this present study.9\nField\nFIG. 7. (a) Exemplary ST-FMR spectra at di\u000berent\nmicrowave powers. (b) ST-FMR amplitude vs. microwave\npower.\nAppendix C: Frequency Dependence of the\nSpin-Orbit Torque E\u000eciencies\nFIG. 8. Dependence of SOTs in MAFO (13 nm)/ Pt (5 nm).\n(a) Damping-like torque e\u000eciency \u0012DL. (b) Field-like torque\ne\u000eciency\u0012FL.\nWe have carried out a frequency dependence study of\ndamping-like and \feld-like SOT e\u000eciencies. The dc-\nbiased ST-FMR method is used to extract each data\npoint. Figure 8 shows that both the damping-like and\n\feld-like SOT e\u000eciencies are nearly constant across the\nfrequency range of 3-8 GHz. This veri\fes that the SOT\ne\u000eciencies are independent of the microwave frequency.\nAppendix D: Model for the Pt Thickness\nDependence of Resistivity\nWe model the Pt thickness dependence of resistivity\n\u001aPtby using an approach similar to that reported by\nBerger et al.23. This model takes into account theconductivity \u001bas a function of position along the \flm\nthickness axis z, expressed as the sum of bulk and\ninterfacial contributions,\n\u001b(z) =1\u0000exp\u0000\n\u0000z\nL\u0001\n\u001abulk\nPt+exp\u0000\n\u0000z\nL\u0001\n\u001aint; (D1)\nwhere\u001abulk\nPt= 1:1\u000210\u00007\nm is the bulk resistivity of Pt,\n\u001aintis the interfacial resistivity, and Lis an empirical\ncharacteristic length scale capturing the decay of the\ninterfacial contribution to resistivity. The resistivity of\nthe Pt \flm with thickness tPtis then given by,\n\u001aPt(tPt) =\u00121\ntPtZtPt\n0\u001b(z)dz\u0013\u00001\n=\u001abulk\nPt\n1 +L\ntPt\u0010\n\u001abulk\nPt\n\u001aint\u00001\u0011\u0000\n1\u0000exp\u0000\n\u0000tPt\nL\u0001\u0001:(D2)\nThe \ft curve for the experimentally measured tPt-\ndependence of \u001aPt(Fig. 4(a)) is obtained with Eq. D2\nwith\u001aint= 1:3\u000210\u00006\nm andL= 10:nm.\nAppendix E: Equations for the Pt Thickness\nDependence of \u000band\u001bDL\nWe \ft thetPtdependence of \u000bwith35\n\u000b(tPt) =\u000b0+\u000bSML+\u000bSP\n=\u000b0+\u000bSML+g\u0016B~\n2e2MstM\u00141\nG\"#+ 2\u001aPt\u0015scoth\u0012tPt\n\u0015s\u0013\u0015\u00001\n;\n(E1)\nwhere\u000b0= 0:0017 is the mean value for the\n\fve bare MAFO \flms ( tPt = 0) prior to Pt\ndeposition for the MAFO/Pt bilayers, \u000bSPis the\nspin-pumping contribution to Gilbert damping, \u000bSML\nis the phenomenological parameter capturing the tPt-\nindependent enhancement of damping (from interfacial\nspin-memory loss or two-magnon scattering), g= 2:05 is\nthe spectroscopic g-factor44,Ms= 93 kA/m is the mean\nvalue of the saturation magnetization of MAFO used in\nthis study, and tM= 13 nm is the thickness of MAFO.\nThetPtdependence of \u001bDLis \ft with6\n\u001bDL(tPt) =\u0012DL\n\u001aPt=\u0012SH\n\u001aPt8\n>><\n>>:(1\u0000e\u0000tPt=\u0015s)2\n(1 +e\u00002tPt=\u0015s)G0\nG0+ tanh2\u0012tPt\n\u0015s\u00139\n>>=\n>>;\u000bSP\n\u000bSML+\u000bSP; (E2)\nwhereG0=G\"#2\u001aPt\u0015stanh(tPt=\u0015s). We also remark that \u001aPtis dependent on tPtas given by Eq. D2.10\n0246a (10-3)\n0 5 10 15 200246sDL = qDL/rPt (105 W-1m-1)\nPt thickness, tPt (nm)intrinsic + EY\naSML= 0\n0 5 10 15 200246sDL = qDL/rPt (105 W-1m-1)\nPt thickness, tPt (nm)\n(a)\n(e)(a)\n(e)\n0246a (10-3)(a)(e)skew scattering + EY\naSML= 0\n0246a (10-3)\n0 5 10 15 200246sDL = qDL/rPt (105 W-1m-1)\nPt thickness, tPt (nm)\n0246a (10-3)\n(a)(e)\n(a)(e)\n(a)(e)(a)(e)\n0 5 10 15 200246sDL = qDL/rPt (105 W-1m-1)\nPt thickness, tPt (nm)\n(a)(e)intrinsic + DP\naSML= 0skew scattering + DP\naSML= 0\nFIG. 9. Exemplary \ft results for the Pt thickness dependence of the Gilbert damping parameter \u000band the damping-like SOT\nconductivity \u001bDLwith several values of G\"#. The solid curves (parameterized by the values in bold font in Table III) are the\ncurves shown in Fig. 4. The error bars are comparable to or smaller than the symbol size; they are derived from the linear \fts\nof FMR linewidth vs. frequency (for \u000b) and ST-FMR linewidth vs. dc bias current density (for \u001bDL).\nintrinsic + EY skew scatt. + EY intrinsic + DP skew scatt. + DP\nG\"#(\n\u00001m\u00002)\u0015s(nm)\u0012SHG\"#(\n\u00001m\u00002)\u0015s(nm)\u0012SHG\"#(\n\u00001m\u00002)\u0015s(nm)\u0012SHG\"#(\n\u00001m\u00002)\u0015s(nm)\u0012SH\n(a) 1.5\u0002101416 0.24 1.0\u000210140.1 48 1.2\u000210145.7 0.32 1.1\u000210143.2 0.96\n(b) 2.0\u0002101419 0.22 1.1\u000210144.7 1.2 1.6\u000210145.6 0.27 1.2\u000210143.3 0.89\n(c) 2.5\u0002101421 0.21 1.2\u000210145.7 0.96 1.8\u000210145.7 0.25 1.3\u000210143.3 0.83\n(d) 3.0\u0002101422 0.20 1.3\u000210146.7 0.80 2.0\u000210145.8 0.23 1.4\u000210143.4 0.77\n(e) 5.0\u0002101424 0.19 1.4\u000210146.1 0.68 2.5\u000210146.1 0.21 1.5\u000210143.5 0.72\nTABLE III. Parameters for the modeled curves in Fig. 9. 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Adv. 5, eaav8575 (2019).\n98We used the MATLAB function of nlinmulti\ft, which is a\nwrapper for nlin\ft that allows for simultaneous \ftting of\nmultiple data sets with shared parameters, available\nat: https://www :mathworks:com/matlabcentral/\nfileexchange/40613-multiple-curve-fitting-with-\ncommon-parameters-using-nlinfit ."    },    {        "title": "1001.0391v4.Electrically_stabilized_magnetic_vortex_and_antivortex_states_in_magnetic_dielectrics.pdf",        "content": "Electrically stabilized magnetic vortex and antivortex states \n in magnetic dielectrics \n \nA.P. Pyatakov*, G.A. Meshkov \n \nPhysics Department, M.V. Lomonosov MSU, Leninskie gori, Moscow, 119991, Russia \n \nThe micromagnetic distribution in a dielectric nanoparticle is theoretically considered. It is shown that the existence of \ninhomogeneous magnetoelectric interaction in magnetic dielectrics provides the possibility to stabilize the vortex and \nantivortex state in magnetic nanoparticle. The estimation of the critical voltage necessary for vortex/antivortex \nnucleation in bismuth ferrite and iron garnet nanoparticles yields a value of ±150 V. This system can be considered as \nelectrically switchable two state-logic magnetic element.  \n \n *) pyatakov@physics.msu.ru \n  \nNumerous micromagnetic structures observed in magnetic media are the result of \ncompetition between only a few types of interact ions that include magnetostatic and exchange \nenergy. These two provide us with magnetic vortex structures in ferromagnetic nanodiscs or \nnanodots [1-6]. An equally fundamental though much less known is antivortex state that is a topological counterpart of the vortex . The realization of an tivortex state is a challenging task since it \nis unfavorable for magnetostatic reasons due to the formation of the magnetic charges at the edge of \nthe particle. There are only a few reports in which the antivortex was observed either in a complicated four ferromagnetic rings cross-junction structure [7] or as a metastable state of \nasymmetric cross-like nanostructures [8].  \nMeanwhile there is an additional interaction that should be taken in to account when we \nconsider micromagnetic structure under influence of electric field or the structures in magnetic \nferroelectrics (multiferroics). It is proportional to  spatial derivatives of magnetic order parameter \nvector. This spin flexoelectricity is the variety of magnetoelectric coupling related to the magnetic \ninhomogeneties and it is described by P\niMj∇kMn coupling term, where P and M, are electric and \nmagnetic order parameters, resp ectively [9-18]. This type of interaction is responsible for \nmagnetically induced electric polarization in spiral multiferroics [19] and electric field driven \nmagnetic domain wall motion in ferr imagnetic iron garnet films [20]. \n In this short Letter we will show that the gradient electric field produced by point electrode \n(e.g. cantilever tip of atomic force microscope) can stabilize in magnetic dielectric nanoparticle \neither vortex or antivortex state depending on the electric polarity of the tip.  \n \na) \n   b)\n  \nFig. 1 Magnetic dielectric nanoparticle subjected to the electric field from the cantilever tip (the electric polarity \ndepends on the sign of magnetoelectric constant in (1)) \na) electrically induced magnetic vortex state b) electrically induced magnetic antivortex state.  \nMagnetoelectric materials, mostly being ferrimagnets or weak ferromagnets, have rather \nmoderate value of spontaneous magnetization. This results in large exchange length \n22s exch MA l= , where A is the exchange stiffness, MS is saturation magnetization, and large \nmagnetostatic lengths 22s MS M AK l=  (the maximum size of single domain particle, where K is \nanisotropy constant) compared to conventional ferromagnets. The typical values for room \ntemperature magnetoelectric iron garnet films are:  m lexchμ5,0= , m lMSμ5=  and room temperature \nmultiferroic bismuth ferrite: m lexchμ5.1= , m lMSμ240= . Thus the magnetic dielectric particles of \nnanometric size should be in homogeneous magnetic state since exchange interaction is the \npredominant one. \nThe only thing that can compete against the exchange interaction in magnetic dielectrics on \nthis nanometer scale is the spin flexoelectricity that for isotropic media can be represented in the \nform [10,13]:  \n() ()( )n n n nP ∇⋅−⋅∇⋅⋅⋅=γMEF     ( 1 )  \nwhere γ is magnetoelectric constant, P is electric polarization, n is unit vector of magnetic \norder parameter.  \nThe linear charge density corresponding to magnetic vortex can be found as in [13]:  \ne LQπγχ2±= ,        ( 2 )  \nwhere eχ is electric susceptibility of the medium, signs “±” correspond to the vortex and antivortex \nstate, respectively.  \nThe magnetoelectric energy thus can be represented in the form of electrostatic energy of the \ncharge (2) in the electric potential φ of the point electrode:  \n2ME eWq hϕπγχ ϕ== ⋅ ⋅ ,        (3) \nwhere q is the integral charge, h is the height of the particle.  \nThe exchange energy can be estimated as \n() h A h AVAk WExch2 22\n222ππ=Δ⎟⎠⎞⎜⎝⎛\nΔ==    ( 4 )  \nwhere k is the modulation wave vector, Δ is the lateral size of the particle.  \nThus the critical potential of the electrode  tip in which the nucl eation of the vortex \n(antivortex) occurs that correspond to the condition W ME  + W exch <0 can be found as \n2\nC\neAπϕγχ=⋅       ( 5 )  \nAssuming γ~10-6 cm erg/ (typical values for BiFeO 3 and iron garnet), dielectric \nsusceptibility χe~4 (it can be estimated using the value of BiFeO 3 dielectric constant ε=1+4πχe=50) \nand exchange stiffness A~3 *10-7 erg/cm (typical value for room temperature magnets) we obtain \nφc=±150V.  \nMore rigorous approach should take into account the finite size of the vortex core in which \nthe spin come out of the plane. In the case of dielectric nanoparticle its diameter is determined by the balance between the exchange energy and the inhomogeneous magnetoelectric energy. The \nestimation (5) remains valid provided that the critic al potential corresponds to the lowest voltage at \nwhich the vortex/antivortex state appears at the perimeter of the particle, the charge (3) is distributed on the surface of the cylinder wrapping the vortex core and the parameter Δ in (4) stands for the core \ndiameter. Thus the electric field can stabilize the vortex state for one sign of the charge and antivortex \nstate for another (see figure). This possibility besi des fundamental interest can be considered as a \nprototypical example of electrically switchable magnetic system with two logic states. The use of \nferroelectric media as a material of the tip electrode will make it possible to implement the memory cell based on this principle.   \n \nAuthors are grateful to A. Zvezdin, D. Khomskii and M. 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Fraerman, \nO.G.Udalov, Anti-vortex state in cross-like nanomagnets, Phys. Rev. B, 81, 094436 (2010) \n9. V. G. Bar’yakhtar, V. A. L’vov,  and D. A. Yablonskii, JETP Lett. 37, 673 (1983) \n10. Yu. F. Popov, A. Zvezdin, G. Vorob’ev, A. Ka domtseva, V. Murashev, and D. Rakov, JETP Lett. 57, 69 (1993) \n11. A. Sparavigna, A. Strigazzi, and A. Zvezdin , Phys. Rev. B 50, 2953 (1994) \n12. A.A. Khalfina, M.A. Shamtsutdinov, Ferroelectrics, 279, 19 (2002) \n13. M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006). \n14. Logginov A.S., Meshkov G.A., Nikolaev A.V., Pyatakov A.  P., Shust V. A., Zhdanov A.G., Zvezdin A.K., J. Magn. \nMagn. Mater., 310, 2569 (2007) \n15. I. Dzyaloshinskii, EPL, 83, 67001 (2008) \n16. D. Mills, I.E.Dzyaloshinskii, Phys.Rev. 78, 184422 (2008) \n17. A.A. Mukhin, A.K. Zvezdin, JETP Lett. 89, 328 (2009) \n18. A.P. Pyatakov and A.K. Zvezdin, Flexomagnetoelectric interaction in multiferroics, The European Physical Journal \nB - Condensed Matter and Complex Systems, Volume 71, Number 3, p. 419 (2009) \n19. S.-W. Cheong, M. Mostovoy, Nature Materials, 6, 13 (2007) \n20. A.S. Logginov, G.A. Meshkov, A.V. Nikolaev, E.P. Ni kolaeva, A.P. Pyatakov, A.K. Zvezdin Room temperature \nmagnetoelectric control of micromagnetic structure in iron garnet films, Applied Physics Letters ,  v.93, p.182510 (2008) "    },    {        "title": "1702.02022v1.Giant_spin_nonlinear_response_theory_of_magnetic_nanoparticle_hyperthermia__a_field_dependence_study.pdf",        "content": "Giant-spin nonlinear response theory of magnetic nanoparticle hyperthermia: a \feld\ndependence study\nM. S. Carri~ ao,\u0003V. R. R. Aquino, and A. F. Bakuzis\nInstituto de F\u0013 \u0010sica, Universidade Federal de Goi\u0013 as, 74690-900, Goi^ ania-GO, Brazil\nG. T. Landi\nDepartamento de Ci^ encias Naturais e Humanas,\nUniversidade Federal do ABC, 09210-580, Santo Andr\u0013 e-SP, Brazil\nE. L. Verde\nInstituto de Ci^ encias Exatas e da Terra, Universidade Federal de Mato Grosso, 3500, Pontal do Araguaia-MT, Brazil\nM. H. Sousa\nFaculdade de Ceil^ andia, Universidade de Bras\u0013 \u0010lia, 72220-140, Bras\u0013 \u0010lia-DF, Brazil\n(Dated: February 8, 2017)\nUnderstanding high-\feld amplitude electromagnetic heat loss phenomena is of great importance,\nin particular in the biomedical \feld, since the heat-delivery treatment plans might rely on analytical\nmodels that are only valid at low \feld amplitudes. Here, we develop a nonlinear response model valid\nfor single-domain nanoparticles of larger particle sizes and higher \feld amplitudes in comparison\nto linear response theory. A nonlinear magnetization expression and a generalized heat loss power\nequation are obtained and compared with the exact solution of the stochastic Landau-Lifshitz-\nGilbert equation assuming the giant-spin hypothesis. The model is valid within the hyperthermia\ntherapeutic window and predicts a shift of optimum particle size and distinct heat loss \feld amplitude\nexponents. Experimental hyperthermia data with distinct ferrite-based nanoparticles, as well as\nthird harmonic magnetization data supports the nonlinear model, which also has implications for\nmagnetic particle imaging and magnetic thermometry.\nKeywords: magnetic nanoparticles, hyperthermia, cancer, nonlinear, alternating magnetic \feld\nI. INTRODUCTION\nThe response of nanomaterials to alternating electro-\nmagnetic \felds is of great importance nowadays in the\nbiomedical \feld, where new approaches to treat diseases\nare under development. One of the most innovative and\nimportant applications is related to heat delivery through\nthe interaction of nanomaterials with electromagnetic\n\felds. This heat delivery method can be used to release\ndrugs[1], activate biological processes[2{4] and even treat\ntumors[5{9]. Indeed, using Maxwell's equations and the\n\frst law of thermodynamics one \fnds that the heat loss\nper unit volume per cycle is given by\n1\nV\f\ncycle\u000eQ=\u0002\n~E\u0001~Jdt\u0000\f\ncycle~P\u0001d~E\u0000\f\ncycle\u00160~M\u0001d~H;(1)\nwhereVis the nanomaterial volume, Qis the heat loss, ~E\nthe electric \feld, ~Jthe free volumetric density current, ~P\nthe electric polarization, \u00160the vacuum magnetic perme-\nability,~Mthe magnetization and ~Hthe magnetic \feld.\nThe \frst term in equation (1) correspond to the \\free-\ncurrent\" loss, whereas the last two describe dielectric and\nmagnetic losses.\nMoreover the \\free-current\" loss term has an important\nimpact on the biomedical application since it is related\nto a biological constraint. This term states that the fre-quency (f) and magnitude of the alternating magnetic\n\felds need to be lower than a critical value in order to\ninhibit possibly harmful ionic currents in the patient's\nbody[7]. For instance, for a frequency of 100 kHz the\nmaximum \feld amplitude is in the order of 20 :8 kA=m\n(261 Oe) for a single air-core coil radius of 0 :035 m\n(expected dimension for breast cancer application[10]).\nNote that this value is higher than the one usually re-\nported (order of 4 :9 kA=m) only because the estima-\ntion of Atkinson used a coil radius of 0 :150 m. Since\nthe free current loss is proportional to the square of the\ndistance from the coil axis, an estimation of the criti-\ncal \feld for a given coil radius ( r) might be obtained\nfromHf < (0:150=r)\u00034:85\u0002105kA=(m\u0002s). Figure\n1 shows the biological critical \feld as a function of \feld\nfrequency in the usual therapeutic hyperthermia range\nusing Atkinson's criteria[7, 10, 11], which indicates that\nthe higher the frequency the lower is this \feld (the pa-\nrameters used to generate the curve are presented in the\n\fgure captions).\nOn the other hand, the last terms of Eq. (1), which\nrepresent hysteretic losses, has been the subject of ana-\nlytical models within the Linear Response Theory (LRT)\nand was used to estimate optimal particle size, un-\nderstand particle-particle interaction e\u000bects and maxi-\nmum heat generation for hyperthermia[12{15]. Curi-\nously, most LRT studies from the literature do not dis-arXiv:1702.02022v1  [cond-mat.mes-hall]  7 Feb 20172\ncuss a fundamental limitation of the model, namely, that\nit is only valid at the low particle size range and low \feld\namplitudes.\nIn Fig.1 we show the range of validity of the LRT,\nwhich, as can be seen, is far below the typical \felds\nused for hyperthermia studies. There are several sug-\ngestions for identifying this limit. For example, Car-\nrey et al.[16] found that the hysteresis area for a lon-\ngitudinal case (\feld applied parallel to the easy axis -\nsee Fig. 5(g) of Ref. [16]) deviates from the LRT for\n\u0018\u00140:2 (\u0018=\u00160MSVH=kBT- whereMSis the satura-\ntion magnetization, kBis the Boltzmann constant and\nTis the temperature), which suggests that this model\ncan only be applied for particles below a critical size.\nAlternatively, Verde et al. suggested that deviations oc-\ncurs for \felds H < 0:02HK(HKis the anisotropy \feld,\nthat for uniaxial case is HK= 2K=\u0016 0MSwithKthe\nanisotropy constant)[17, 18]. It is important to empha-\nsize that experimentally, it is possible to determine if\none is still in the linear regime or not, by verifying if the\nthe heating e\u000eciency (also known as speci\fc loss power\n- SLP) scales quadratically with the \feld. Throughout\nthis manuscript, when discussing the theoretical models,\nlow magnetic \felds mean values within the LRT range.\nIn addition, in Fig. 1 we also include an estimation of the\nrange of validity of the nonlinear response model (NLRT)\ndeveloped in this work, which will be shown later in the\nmanuscript to be H < 0:14HK. This corresponds to a 7-\nfold increase in the range of \feld validity in comparison to\nthe LRT de\fnition used above. The result suggests that\nthe model may be useful for biomedical applications, in\nparticular for magnetic hyperthermia.\nIn the subject of heat loss, the term \\nonlinear\" has\nbeen used in a variety of ways. For instance, nonlin-\near dielectric e\u000bects have been related to the correla-\ntion of distinct relaxation times[19]. In this case, a\nsuperposition of Debye processes is used, which pre-\ndict heat loss scaling with the square of \feld ampli-\ntude. Conversely, for relaxor ferroelectric materials a\nnonlinear polarization term is included in the dynamic\nresponse equation[20]. Such approach allowed the au-\nthors to investigate the third harmonics of the relaxor.\nOn the other hand, in magnetic materials, nonlinear\nresponse is investigated using the stochastic Landau-\nLifshitz-Gilbert (SLLG) equation[17, 18, 21, 22]. In\nthis case, thermal \ructuations are addressed using the\nBrown's approach[23], where the giant spin hypothesis\nallow one to use the Fokker-Planck equation to the mag-\nnetic moment orientational distribution function. One\ncan then show that this leads to an in\fnite hierarchy of\nequations, which can be solved numerically to \fnd the\nmagnetic moment response of the nanoparticle [24{27].\nThe method is valid for any \feld amplitude, but due to\nits mathematical complexity, it does not yield simple an-\nalytical expressions that could be useful in the applied\n\feld.Indeed, the \feld and frequency-dependence of heat loss\nin magnetic materials have been attracting the attention\nfor a long time due to technological applications [28].\nIn general, the loss in magnetic materials can show sev-\neral contributions, spanning from eddy currents (that\nscales with f2H2), anomalous eddy current con- tribu-\ntions (due to complex domain wall dynamics which scales\nwithf3=2H3=2) up to multidomain magnetichys- tere-\nsis contribution. The later term can be explained us-\ning the Rayleigh correction to the magnetic permeability\nand reveals a power loss scaling with fH3. Curiously,\nthis type of behavior had been reported before in mag-\nnetic nanoparticle hyperthermia experiments [29]. The\nauthors suggest that this can be explained by the exis-\ntence of large particles in the sample [29]. So, multi-\ndomain particles could be relevant to heat generation\nthrough domain wall motion loss. However, for most\nused magnetic \ruid samples, multidomain particles are\nnot expected. For example, in magnetite nanoparticles\nthe single-domain limit is around 80 nm [30]. Moreover,\non the theoretical point of view, Carrey et al. inves-\ntigated the SLP \feld exponent using numerical simula-\ntions of the SLLG equation (see Fig. 7 of Ref. [16]). The\nauthors found theoretically that this exponent is size de-\npendent and showed values below or higher than 2. This\ntype of behavior was found experimentally by Verde et\nal. [17]. However, in both works, no simple analytical\nexpression was used to explain this behavior.\nHere we show that through a modi\fcation of Bloch's\nequation, which is linear with respect to the magneti-\nzation, one is able to obtain a heat loss expression valid\nbeyond the LRT. Indeed, di\u000berent from other works from\nthe literature, we demonstrate that even the linear fre-\nquency term has higher order \feld contributions. Also,\nour model introduces a nonlinear frequency term which\nadequately describes the magnetic response within the\nhyperthermia therapeutic window. The validity of the\nmodel is explicitly tested by comparing it with numeri-\ncal simulations of the SLLG approach. In addition, we\nincluded experimental magnetic hyperthermia data that\nsupports our theoretical \fndings. Twelve powder sam-\nples were studied, including cobalt-ferrite, copper-ferrite,\nnickel-ferrite, maghemite and manganese-ferrite (doped\nwith Zn or Co and also undoped) based nanoparticles.\nThe analytical nonlinear response model is believed to\nbe useful not only for improving our understanding of\nmagnetic losses, but also may impact other related ar-\neas, which could bene\ft from analytical expressions, as\nfor example, magnetic particle imaging[31, 32] and mag-\nnetic nanothermometry[33, 34].\nThe article is organized as follows: In section II we dis-\ncuss several models from the literature. In particular, we\npresent the linear response theory (LRT), the nonlinear\nFerguson-Krishnan model (FK) (usually applied in mag-\nnetic particle imaging), the perturbation method devel-\noped by Raikher and Stepanov (RS model), and \fnally3\nthe stochastic Landau-Lifshitz-Gilbert model, which is\nexpected to be the exact solution of the magnetic re-\nsponse of the nanoparticle at alternating \feld conditions.\nAll the models are critically compared showing the ne-\ncessity of developing a simple nonlinear analytical model.\nThe SLLG model is than compared with the proposed\nnonlinear response model (NLRT) developed in section\nIII. Section IV we present the experimental procedure,\ni.e. the synthesis and characterization of magnetic \ru-\nids. In section V we discuss all the theoretical and ex-\nperimental results. Here we focus on magnetic nanopar-\nticle hyperthermia, but also compare our model with the\nthird-harmonic magnetic particle imaging data from the\nliterature. Finally, in section VI we summarize our \fnd-\nings.\n1002 003 004 005 006 007 008 00050100150200250N\nLRT (0.14 HK)L\nRT (0.02 HK)Biologically Safe F\nrequency (kHz)Applied Field (Oe)B\niologically HarmfulAtkinson's Criterion0\n48121620Applied Field (kA/m)\nFigure 1. (Color online) The \fgure shows the calculated bi-\nological critical \feld according to Atkinson's criteria scaled\nfor a coil radius of 0 :035 m (expected for small tumors in\nbreast [10]), which results in a Hf < 20:78\u0002105kA=(m\u0002s) .\nHuman treatment can only occur below this \feld. The LRT\nlimit is calculated assuming H < 0:02HKfor a particle diam-\neter of 15 nm, MS= 270 emu=cm3,Kef= 8\u0002104erg=cm3,\nT= 300 K,\u000b= 0:05 and\u001a= 5 g=cm3. The nonlinear re-\nsponse critical \feld for our model (NLRT) corresponds to the\nsolid line.\nII. MODELS REVIEW\nAll the models discussed in this manuscript are valid\nwithin the single-domain range. Also, they assume the\ngiant-spin hypothesis of Brown[23], i.e. coherent spin\nrotation. Here we will consider the case of uniaxial mag-\nnetic nanoparticle, where the energy is given by\nE=KVsin2\u0012\u0000\u00160MSVHcos(\u0012\u0000'): (2)\nThe \frst term is the uniaxial anisotropy energy, while\nthe other is the Zeeman interaction. \u0012represents the\nangle between the magnetic moment of the nanoparticleand the easy axis direction, while \u0012\u0000'corresponds to\nthe angle between the magnetic dipole and the applied\n\feld. It is common to name the longitudinal case as\n'= 0, which is the case where the \feld is applied in the\nanisotropy axis direction.\nThe simplest quasi-static magnetization model in\nthe literature, named Langevin model, neglects the\nanisotropy term, which can only be done if the ratio of\nthis anisotropy contribution to the thermal energy is very\nlow. In this case the magnetization can be calculated\nfrom\nM\nMS=hcos\u0012i=\u0019\u0001\n0cos\u0012e\u0018cos\u0012sin\u0012d\u0012\n\u0019\u0001\n0e\u0018cos\u0012sin\u0012d\u0012=L(\u0018): (3)\nL(\u0018) =coth(\u0018)\u00001=\u0018is the Langevin function, whose\nseries expansion to \ffth order gives\nM=MS\u0012\u0018\n3\u0000\u00183\n45+2\u00185\n945\u0000:::\u0013\n(4)\n=\u001fLA;1H+\u001fLA;3H3+\u001fLA;5H5+:::\nThe \frst term is the initial (linear) susceptibility, the\nsecond the cubic, and there on.\nA. Linear Response Theory\nThe \frst linear response model to describe heat loss\nwas probably described by Debye in the context of rigid\nelectric dipoles[12]. Here we focus in the magnetic case.\nLet us \frst start by assuming that a magnetic particle\nis subjected to a harmonic \feld H(t) =<e\b\nH0ei!t\t\n=\nH0cos!t, with the magnetic susceptibility \u001f=\u001f0\u0000i\u001f00,\nwhere\u001f0and\u001f00corresponds to the real and imaginary\nlinear susceptibility terms, respectively. So, the magne-\ntization term can be written as M(t) =<ef\u001fH(t)g=\nH0(\u001f0cos!t+\u001f00sin!t), where!= 2\u0019fwithfthe\n\feld frequency. Therefore, de\fning the heating e\u000eciency\n(SLP) as the frequency times the hysteresis loss divided\nby the particle density ( \u001a) one \fnds\nSLP =f\n\u001aVp\f\ncycle\u000eQ=\u0000f\n\u001a\u00160\f\nMdH (5)\n=\u0019f\n\u001a\u00160H2\n0\u001f00:\nThis equation represents the heat loss of the magnetic\nmaterial. So, one now needs an expression for the imag-\ninary susceptibility term. If the projection of the mag-\nnetization, M(t), in the \feld direction satis\fes the Bloch\nequation, i.e. \u001c(dM=dt) +M=\u001fH(t), where\u001cis the\nmagnetization relaxation time, one can show the linear\nsusceptibility term is \u001f=\u001f0=(1 +i!\u001c), revealing that\n\u001f00=\u001f0!\u001c\n1 + (!\u001c)2: (6)4\n\u001f0is the equilibrium susceptibility, which within the\nLangevin model is equal to \u001fLA;1. However, depending\non the model this term would be di\u000berent. The relax-\nation of the magnetization for an uniaxial nanoparticle\nis\u001c=\u001c0e\u001b=\u001b1=2with\u001b=KV=kBT, that is valid when\n\u001b>2 [35]. Here Vis the particle volume, Tis the tem-\nperature,kBin Boltzmann's constant and Kthe mag-\nnetic anisotropy. \u001c0=p\u0019MS(1 +\u000b2)=(\r02K\u000b) (about\n10\u000010\u000010\u00008s), with\r0the electron gyromagnetic ratio\nand\u000bthe dimensionless damping factor. For the \feld\napplied in the anisotropy direction one \fnds for the re-\nlaxation in the limit of high anisotropy\n\u001ch=2\u001c0h\n(1\u0000h)e\u0000\u001b(1\u0000h)2+ (1 +h)e\u0000\u001b(1+h)2i\u00001\n\u001b1=2(1\u0000h2):\n(7)\nThe \feld term his given in reduced units, i.e. h=\nH0=HK. This expression returns to the later in the ab-\nsence of an applied \feld. The \frst one to describe this\nheat loss for magnetic \ruids was Rosensweig[14]. The\nmodel above predicts a loss proportional to the square\nof the applied \feld. However, this is only true exper-\nimentally at low \feld amplitudes as found in several\ncases dealing with magnetic nanoparticles [17, 18, 29].\nNote that the same issue occurs in the electric case for\ndielectrics[19] or relaxor ferroelectrics[20]. In addition,\nthe LRT model predicts elliptical magnetic hysteresis\ncurves, which have been observed at low \feld amplitudes\n(less than 4 kA =m) by Eggeman et al. [36] and Tomitaka\net al. [37]. However, this is not consistent with \fndings\nat higher \feld amplitudes, as for instance in magnetic\nparticle imaging where a nonlinear response plays a cru-\ncial role [31, 32].\nB. Ferguson-Krishnan Approach\nIn an attempt to include nonlinear phenomena in the\ndescription, Ferguson and Krishnan[38] proposed a gen-\neralization of linear magnetization, using the Langevin\nfunction:\nM(t) =MS\u00121\n1 + (!\u001c)2L(\u0018cos(!t)) (8)\n+!\u001c\n1 + (!\u001c)2L(\u0018sin(!t))\u0013\n:\nThis approach assumes that the frequency response of\nhigher \feld order (quasi-static) terms are the same as\nthe linear dynamic susceptibility term and neglects the\nquasi-static contribution from the magnetic anisotropy\nenergy term. This expression is usually used to obtain the\nnth-order harmonic magnetization, which represents an\nimportant quantity in magnetic particle imaging [31, 32].\nThe harmonic calculation will be discussed in detail in\nsection III.C. Raikher-Stepanov Perturbation Method\nUsing perturbation theory, Raikher and Stepanov[39],\nincluded the anisotropy term and showed that the mag-\nnetization could be written as M(t) =<(\u001f1H0ei!t+\n\u001f3H3\n0e3i!t+:::). However, di\u000berent from the FK model\nabove, the frequency dependence of the cubic term was\nfound to be di\u000berent from the linear term. The authors\nfound that the cubic susceptibility could be written as\n\u001f3=\u00001\n4\u001f(0)\n3(1 +S2\n2)(1\u0000i!\u001c)\n45(1 +i!\u001c)(1 + 3i!\u001c); (9)\nwhere\u001f(0)\n3=\u001e\u00163\n0M4\nSV3=(kBT)3,\u001eis the particle vol-\nume fraction of the assembly and S2=1\n21\u0001\n0(3x2\u0000\n1)exp(\u001bx2)dx=1\u0001\n0exp(\u001bx2)dx. So, the real and imaginary\nsusceptibility terms are given by\n\u001f0\n3=1\n180\u001f(0)\n3(1 +S2\n2)(7!2\u001c2\u00001)\n(1 +!2\u001c2)(1 + 9!2\u001c2); (10)\n\u001f00\n3=\u00001\n180\u001f(0)\n3(1 +S2\n2)!\u001c(3!2\u001c2\u00005)\n(1 +!2\u001c2)(1 + 9!2\u001c2): (11)\nUsing up to the cubic term the magnetization of the\nnanoparticle in the RS model gives\nM(t) = (\u001f0\n1cos(!t) +\u001f00\n1sin(!t))H0 (12)\n+ (\u001f0\n3cos(3!t) +\u001f00\n3sin(3!t))H3\n0;\nwhere\u001f0\n1=\u001f(0)\n1(1 + 2S2)=(1 + (!\u001c)2),\u001f00\n1=!\u001c\u001f(0)\n1(1 +\n2S2)=(1 + (!\u001c)2), with\u001f(0)\n1=\u001e\u00160M2\nSV=(kBT) and\u001f0\n3\nand\u001f00\n3are given by Eq. 10 and Eq. 11. Note that those\nexpressions are valid for an ensemble and low \feld ampli-\ntudes. In order to obtain the equivalent expressions for\nthe nanoparticle one only need to neglect the particle vol-\nume fraction in the equilibrium susceptibilities. Because\nof the perturbation approach this model is expected to\nbe valid only at very low \feld amplitudes.\nD. Stochastic Landau-Lifshitz-Gilbert Model\nThe model that is expected to correctly describe the\nmagnetization response of a single-domain nanoparti-\ncle at any \feld amplitude and frequency range is the\nSLLG model. In this case, the magnetic moment of the\nnanoparticle is assumed to be described by the Landau-\nLifshitz-Gilbert equation\nd~M\ndt=\u0000\r~M\u0002~He\u000b\u0000\u000b\r\nMS~M\u0002\u0010\n~M\u0002~He\u000b\u0011\n;(13)5\nwhere\n~He\u000b(t) =~H(t) +~Hani+~Hth(t): (14)\nThe e\u000bective \feld has three contributions: the applied\nexternal \feld, the anisotropy \feld and the thermal \ruc-\ntuation \feld. So the Landau-Lifshitz-Gilbert equation\nfor a magnetic dipole is augmented with a Gaussian\nwhite noise thermal \feld ~Hthwhose Cartesian coordi-\nnates satisfy the statistical properties: h~Hi\nth(t)i= 0 and\nh~Hi\nth(t)~Hi\nth(s)i= 2 (kBT\u000b=V )\u000eij\u000e(t\u0000s). The Kronecker\nand Dirac deltas indicates that the thermal \feld is both\nspatial and temporally uncorrelated. In principle, one\ncould use the equation above and do numerical simu-\nlations. However, the approach of Brown was to con-\nnect the SLLG equation to the Fokker-Planck equation\nof the magnetic moment orientational distribution func-\ntion [23], which can be used to obtain the nanoparticle\nmagnetic moment response.\nIn this work we focus in the longitudinal case. The\n\frst authors to study in detail this problem analytically\nwas Dejardin and Kamilkov [24]. Later, others used the\nsame approach to describe dynamic magnetic hysteresis\n[25{27]. Here, the magnetic moment orientational distri-\nbution function f(z;t) can be shown to obey the Fokker-\nPlanck equation\n2\u001cN@f\n@t=@\n@z\u0014\u0000\n1\u0000z2\u0001\u0012@f\n@z\u0000f(z;t)he\u000b(z;t)\u0013\u0015\n;(15)\nwith\u001cN=\u0016\u0000\n1 +\u000b2\u0001\n2\r0\u000bkBTthe free di\u000busion time, z= cos\u0012\nand\u0012the angle between the magnetic dipole and the ap-\nplied \feld. The magnetic anisotropy is assumed uniaxial.\nSo, the ratio of the particle energy to thermal energy can\nbe written as\nUe\u000b\nkBT=\u0000\u001bz2\u00002h\u001bz; (16)\nwhere the \feld term h=H=HK. Therefore, the e\u000bective\n\feld is\nhe\u000b=\u00001\nkBT@Ue\u000b\n@z= 2\u001b(h+z): (17)\nThe Fokker-Planck equation is then used to obtain\nthe time evolution of the lth-order moment pl(t) =hPli,\nwhich is shown to be described by\n2\u001cNdpl\ndt=l(l+ 1)\n2l+ 1(A1+A2)\u0000l(l+ 1)pl; (18)\nwith\nA1= 2\u001bh(pl\u00001+pl+1); (19)\nand\nA2=2\u001b\u0014l\u00001\n2l\u00001pl\u00002+2l+ 1\n(2l\u00001)(2l+ 3)pl (20)\n\u0000l+ 2\n2l+ 3pl+2\u0015\n:This equation shows that each moment depends on others\nin a nonlinear fashion. This in\fnite hierarchy may be\nsolved numerically using fast sparse solvers [17, 22, 25,\n27] and discarding several periods of the external \feld.\nAlternatively, one could also expand the pl(t) in a Fourier\nseries as\npl(t) =1X\nk=\u00001Fl\nk(!)eik!t; (21)\nwith allpl(t) real, which implies that Fl\n\u0000k= (Fl\nk)\u0003, where\nthe asterisks refer to the complex conjugate [24]. This\nwill then lead to a hierarchy of algebraic equations for\nthe Fourier amplitudes, which also need to be solved\nnumerically[24].\nE. Magnetization Loops\nWe are now in condition to compare the hysteresis\nloops of each model, namely the linear response theory\nusing the \feld-independent relaxation time (LRT) and\nalso the \feld-dependent relaxation time of Eq. (7) (LRT\n\u001ch), the FK model, the RS model and the exact solution\nfor the SLLG model. In Fig. 2(a) we show the magne-\ntization curves of all those models. It is clear that the\nLRT model, independent of the relaxation time equation\nused, shows an elliptical loop. The RS model showed a\nsimilar behavior. The only model that shows a signi\fcant\ndi\u000berence from LRT is the FK model. However, it also\nshows an elliptical hysteresis, which is distinct from the\nLRT model because of the Langevin equilibrium suscep-\ntibility. So, di\u000berent from the other models, it does not\ntake into account the anisotropy term. Nevertheless, for\nthe parameters used in this simulation, it is shown that\nnone of the models above represent well the exact solu-\ntion given by the SLLG magnetization hysteresis loop.\nAlthough improvements were obtained in each model, in\ngeneral they are not yet satisfactory. Motivated by this\nfact, we decided to work out a nonlinear response model,\nfrom now on named NLRT model. This model is able\nto better represent the magnetization response at higher\n\feld conditions, not only in comparison to the LRT, but\nalso far better than the FK or RS models.\nIII. THEORETICAL MODEL\nIn this section we present our nonlinear response\nmodel. Firstly, we include the magnetic anisotropy en-\nergy term in the longitudinal case, which allow us to ob-\ntain any quasi-static (equilibrium) susceptibility terms.\nThose expressions will be named \u001fQS;n, i.e. the nth-\norder quasi-static (QS) coe\u000ecient obtained in the low-\nfrequency limit ( !!0). In the next subsection we in-\ntroduce our dynamic model, where a new expression for6\n-0.10- 0.050 .000 .050 .10-1.0-0.50.00.51.00\n5 1 01 52 02 5110 LRT \nLRT τh  \nFK \nRS \nSLLGM(t) / MsH\n(t) / HkaD\nimensionless anisotropy - σn = 5n\n = 3 \n χQS,n/χLA,nn\n = 1b\nFigure 2. (Color online) (a) Dynamic hysteresis curves for the\nLRT, LRT considering \feld dependence on relaxation time \u001c,\nFerguson-Krishnan approach, Raikher Stepanov method and\nnumerical solution of SLLG for \u001b= 6 and!\u001c0= 10\u00003.\n(b) Longitudinal to Langevin susceptibilities ratio for n= 1,\nn= 3 andn= 5.the heat loss and the particle magnetization is obtained.\nThe last subsection is related to the cubic harmonic cal-\nculation, which is an important parameter for magnetic\nparticle imaging application.\nA. Quasi-static longitudinal case\nFor an uniaxial magnetic nanoparticle in the longitu-\ndinal case, the average magnetization is obtained from\nM\nMS=hcos\u0012i=\u0019\u0001\n0cos\u0012e\u001bcos2\u0012+\u0018cos\u0012sin\u0012d\u0012\n\u0019\u0001\n0e\u001bcos2\u0012+\u0018cos\u0012sin\u0012d\u0012:(22)\nFor\u001b >0, one can show that the longitudinal magneti-\nzation is [40]\nM=MS0\nBBB@2isinh (\u0018)p\u001b\u0019e\u001b+\u00182\n4\u001b\nerf\u0014\ni\u0012p\u001b+\u0018\n2p\u001b\u0013\u0015\n+ erf\u0014\ni\u0012p\u001b\u0000\u0018\n2p\u001b\u0013\u0015\u0000\u0018\n2\u001b1\nCCCA: (23)\nExpanding the longitudinal magnetization in a Taylor series:\nM=MS\u0014ie\u001b\np\u001b\u0019erf(ip\u001b)\u00001\n2\u001b\u0015\n\u0018+MS\"\ne\u001b(6\u001be\u001b+ip\u001b\u0019(2\u001b+ 3)erf(ip\u001b))\n12\u0019(\u001berf(ip\u0019))2#\n\u00183+::: (24)\n=\u001fQS;1H+\u001fQS;3H3+:::;\nwhere erf(iz) = (2i=p\u0019)\u0001z\n0eu2duand\u001fQS;3<0.\nNote that all \u001fQS;n are real. In Fig. 2(b) we show\nthe ratio of the \u001fQS;n=\u001fLA;n up to the \ffth-order (n=5).The longitudinal linear susceptibility ( \u001fQS;1) calculation\ndemonstrate that in the absence of (or very low) mag-\nnetic anisotropy the susceptibility approaches the ex-7\npected Langevin result. On the other hand, in the high\nanisotropy limit, the linear ratio approaches 3, which in-\ndicates that the longitudinal result tends to the Ising re-\nsult, as expected in this case. Other ratios are also shown\nin the \fgure. Therefore, we can conclude that in general\nit is of great importance to include the anisotropy term\nwhen investigating the magnetic response of nanoparti-\ncles.\nB. Nonlinear Response Model\nAs in LRT model, let us assume that a magnetic par-\nticle is subjected to a harmonic \feld and that the pro-\njection of the magnetization, M(t), in the \feld direction\nsatis\fes the Bloch equation, i.e.\n\u001c\u0012dM\ndt\u0013\n+M=f(t); (25)\nwhere\u001cis the relaxation time and f(t) is a function of the\nalternating \feld. Hence, it will be periodic, i.e. f(t) =\nf(t+ 2\u0019=!). Also, in general one might represent f(t) =\n\u001f1H(t)+\u001f3H(t)3+:::, where\u001fnis the nth-order magnetic\nsusceptibility. The LRT, discussed before, corresponds\nto considering just the \frst term in f(t). The nonlinear\nresponse under Bloch's assumption may be computed as\nfollows. In general f(t) is a function of H(t) so it may\nbe expanded in a cosine series as f(t) =1P\nn=1cncos(n!t)\nfor certain coe\u000ecients cn, that can be easily identi\fed by\nexpandingf(t) in terms of cos(n!t) (another alternative\nway to obtain those coe\u000ecients is using the integrating\nfactor method directly to Bloch's equation). The steady-\nstate solution of the Bloch equation is therefore\nM(t) =1X\nn=1cncosn!t+ (n!\u001c) sinn!t\n1 + (n!\u001c)2: (26)\nIn this approach, the corresponding SLP is\nSLP =\u0019f\n\u001aH0!\u001c\n1 + (!\u001c)2c1 (27)\n=\u0019f\n\u001a!\u001c\n1 + (!\u001c)2\u0012\n\u001f1H2\n0+3\n4\u001f3H4\n0+5\n8\u001f5H6\n0+:::\u0013\n:\nThis means that one only needs to worry with the co-\ne\u000ecientc1(H0). This comes from the fact that in the\nheat loss integral only the terms obtained from n= 1\nis nonzero. Note that the \frst term corresponds to the\nusual Debye model, if one assumes that \u001f1=\u001fQS;1, i.e\nthat\u001f1is the quasi-static limit linear coe\u000ecient. Also, it\nmight be important to mention that the existence of the\nhigher order \feld dependent terms indicate a correction\nnot reported before in the literature. As for instance, if\none uses the magnetization equation of the RS model,\nonly the the quadratic \feld term appears. The same ap-\nproach can also be used in the dielectric loss case. Forexample, the electric \feld dependence dielectric loss of\nglycerol (see inset of Fig. 3 of Ref [19]).\nAccording to equation (26), the Bloch solution for the\nmagnetization M(t) up to cubic terms in the \feld is\nM(t) =\u0012\n\u001f1H0+3\n4\u001f3H3\n0\u0013cos(!t) +!\u001csin(!t)\n1 + (!\u001c)2(28)\n+\u001f3H3\n0\n4cos(3!t) + 3!\u001csin(3!t)\n1 + (3!\u001c)2;\nwhere\u001fnare the nth-order magnetic susceptibility coef-\n\fcients. In the equation above is clear that higher-order\nterms are also relevant to the magnetization dynamics.\nAs for instance, this nonlinearity e\u000bect can be identi-\n\fed even for the \frst harmonic contribution, which shows\nhigher \feld order terms.\nIn addition, if !\u001c\u001c1 one may write the magnetiza-\ntion (considering higher-order terms in f(t)) asM(t) =\n\u001f1H0cos(!t) +\u001f3H3\n0cos(!t)3+:::. For the sake of ar-\ngument, if one assumes that the nth-order susceptibility\nterms are equal to the quasi-static terms ( !!0) and\nthat the nanoparticle is at the superparamagnetic regime,\nthanM(t)=MS=L(\u0018cos(!t)) +O(!\u001c). Note that the\n\frst term of this equation has been used systematically in\nboth, magnetic particle imaging (MPI)[31] and magnetic\nnanothermometry (MNT)[33, 34]. In MNT the magne-\ntization expression was shown to be useful only at the\nlow frequency range[33], which is easily explained by our\nmodel due to the range of validity of the later expres-\nsion. Moreover, in MPI the magnetization is similar, but\nnot identical to our model, and di\u000bers mainly due to the\ntermn!\u001c and that the later assumes quasi-static sus-\nceptibility terms and superparamagnetic particle. As a\nconsequence, our model give di\u000berent higher-order har-\nmonic magnetization terms and might represent better\nthe experimental MPI data[38]. Curiously, our model\ngives a similar expression as Ref. [38] for the heat loss\nif we assume that \u001fn=\u001fQS;n. However, this approx-\nimation does not represent correctly the magnetization\ndynamics.\nFurther, Eq. (27) shows that the Bloch equation pre-\ndicts the same frequency dependence as the LRT, which\nwill result in elliptical-like hysteresis curves that are in\ndisagreement with experiment. The reason for this dis-\ncrepancy is that Bloch's equation is linear, whereas the\nunderlying physical phenomena is not, as discussed be-\nfore in section II-D. One way to circumvent this is to\nassume that the coe\u000ecients \u001fndepend explicitly on !.\nThe exact form of this dependence is problem speci\fc,\nbut it must be such that when !!0, one recovers the\nequilibrium nonlinear susceptibilities. The heuristic im-\nprovement approach, also used by others[39], is able to\nbetter represent the magnetization dynamics.\nSo, to correct for the aforementioned de\fciency of the\nBloch approach, we replace \u001fnwith a frequency depen-\ndent function and compare the approximation with exact8\nresults, which are obtained for the longitudinal case using\nthe SLLG model[17, 18, 22, 35]. In this strategy we wrote\n\u001fn=\u001fQS;ngn, wheregnis a function of the frequency.\nThe quasi-static susceptibility coe\u000ecients were obtained\nfrom the series expansion of the quasi-static longitudinal\nsolution[40]. Also, from our assumption is obvious that\none should have gn(!\u001c!0) = 1. Moreover, for the \frst\nterm we should have g1= 1, which corresponds to the\nLRT result. For the cubic term we found that\n\u001f3=\u001fQS;33\u0000(!\u001c)2\n3 (1 + (!\u001c)2): (29)\nSimilarly as the RS model, the magnetization can be\nwritten in the same functional form as Eq. (12). How-\never, now the real susceptibility terms are\n\u001f0\n1=\u001fQS;1\n1 + (!\u001c)2+1\n4H2\n0\u001fQS;33\u0000(!\u001c)2\n(1 + (!\u001c)2)2; (30)\n\u001f0\n3=1\n12\u001fQS;33\u0000(!\u001c)2\n(1 + (!\u001c)2)(1 + (3!\u001c)2); (31)\nwhile the imaginary terms are \u001f00\n1=!\u001c\u001f0\n1and\u001f00\n3=\n3!\u001c\u001f0\n3. Those results indicate that the susceptibility\nterms are distinct from the RS model (see Eqs. 10 and\n11), even though the quasi-static susceptibility coe\u000e-\ncients give the same result. Also, the linear susceptibility\nterm shows a nonlinear \feld and frequency contribution,\nwhich was absent in other models.\nSo, returning to the heat loss integral (Eq. (5)) and\nusing the cubic magnetization (Eq. (28)) with this cor-\nrection (Eq. (29)), the new expression for SLP is now\ngiven by\nSLP =\u00160\u0019f\n\u001aH2\n0\u0014\u001fQS;1!\u001c\n1 + (!\u001c)2(32)\n+1\n4H2\n0\u001fQS;3!\u001c\u0000\n3\u0000(!\u001c)2\u0001\n(1 + (!\u001c)2)2#\n:\nMoreover, in section V, besides discussing the magnetic\nnanoparticle hyperthermia, the cubic harmonic magnetic\nparticle imaging (MPI) experimental signal data ob-\ntained in Ref. [38] will also be compared with the the-\noretical calculations using the FK model and the NLRT\nmodel (see section V for details). We will show a bet-\nter agreement with experimental data using the nonlin-\near response theoretical model developed in this work. In\naddition, because we also investigate soft-magnetic nano-\nmaterials (low \u001b), the empirical uniaxial relaxation time\nexpression, valid for any anisotropy value, has been con-\nsidered [35]\n\u001c=\u001c0(e\u001b\u00001)\u0014\n2\u0000\u001b+2\u001b3=2\np\u0019(1 +\u001b)\u0015\u00001\n: (33)IV. EXPERIMENTAL PROCEDURE\nManganese-ferrite samples were synthesized by hy-\ndrothermal route and separated for the hyperther-\nmia analysis after characterization by x-ray di\u000braction\n(XRD) and vibrating sample magnetometer (VSM). All\nchemical reagents (FeCl 3:6H2O, MnCl 2:4H2O, ZnCl 2,\nCoCl 2:6H2O) citric acid trisodium salt - Na 3C6H5O7,\nmethylamine - CH 3NH2, and acetone - CH 3COCH 3)\nwere purchased with analytical quality and used with-\nout any further puri\fcation. In a typical ap-\nproach, Mn 0:75[(Zn or Co)] 0:25Fe2O4magnetic nanopar-\nticles were prepared as follows: adequate amounts\nof 1:0 mol=L metal stock solutions were diluted with\n40:0 mL of distilled water to form a precursor solution\ncontaining 10 :0 mmol of Fe3+, 3:75 mmol of Mn2+, and\n1:25 mmol of Zn2+or Co2+. Thus, 120 mmol of methy-\nlamine at 40% (w/w) were quickly poured into the stock\nsolution under vigorous stirring for 10 min and then\ntransferred into a 120 mL Te\ron-sealed autoclave and\nheated up to 160\u000eC for 6 h. After cooling to room tem-\nperature, the precipitate was separated by magnetic de-\ncantation, washed with H 2O three times and re-dispersed\nin 50:0 mL of water. Then, 4 :0 mmol of citric acid\ntrisodium salt was added into the solution which was\nheated up to 80\u000eC for 60 min. After adjusting the pH of\nslurry to 7.0 and washing with acetone three times, the\nprecipitate was re-dispersed in 50 :0 mL of water to form\na magnetic sol, after evaporating residual acetone. Thus,\na size-sorting process was done by adding 1 g of NaCl to\nthe as-prepared magnetic sol[41]. 5 min afterwards un-\nder a permanent magnet (NdFeB), salt adding induced a\nphase transition and formed an upper (botton) sol phase\nwith populations of smaller (larger) nanoparticles. Once\nseparated, precipitate of each phase was washed twice\nwith a mixture water/acetone 1:10 (volume/volume) and,\nafter evaporating residual acetone, nanoparticles were re-\ndispersed in water. This procedure was repeated several\ntimes. Powders were obtained from evaporation of sols\nat 55\u000eC for 8 h. Details about cobalt-ferrite samples can\nbe found in Ref. [18] and copper-ferrite and nickel-ferrite\nsamples can be found in Ref. [17].\nAfter the size-sorting process powder samples were an-\nalyzed by XRD (Shimadzu 6000) to separate samples\nwith similar sizes. The previous analysis was performed\nusing the well-known Scherrer equation, which is given by\nDXRD =\u0014\u0015=\f cos , where\u0014= 0:89 is the Scherrer con-\nstant,\u0015= 0:15406 nm is the X-ray wavelength, \fis the\nline broadening in radians obtained from the square root\nof the di\u000berence between the square of the experimental\nwidth of the most intense peak to the square of silicon\nwidth (calibration material), and  is the Bragg angle of\nthe most intense peak (311). This procedure allowed us\nto select three distinct samples of similar sizes contain-\ning MnFe 2O4, Mn 0:75Zn0:25Fe2O4, or Mn 0:75Co0:25Fe2O49\nnanoparticles. All the nanoparticles were surface-coated\nwith citric acid, which guarantee stability at phisiologi-\ncal conditions. The samples were also characterized by\nVSM (ADE Magnetics, model EV9, room temperature\nmeasurements, \feld up to 2T). Table I summarizes the\nrelevant characterization properties of the nanoparticles.\nFinally, magnetic hyperthermia data was obtained in\ntwo systems, one home-made which operates at 500kHz,\nand another one from nanoTherics. In particular, the\nlater system operates in a broad frequency range, span-\nning from 110 up to 980kHz. While details about\nthe home-made hyperthermia system has been described\nelsewhere [17, 18]. The calorimetric method used to ob-\ntain the experimental SLP of the sample used the equa-\ntion\nSLP =C\nmNP\u0014dT\ndt\u0015\nmax; (34)\nwhereCis the heat capacity of the sample (here assumed\nas the heat capacity of the liquid carrier due to the low\nconcentration of particles), mNPis the mass of magnetic\nnanoparticles in unit of grams (obtained from the anal-\nysis of the magnetisation curves of the colloid samples),\nTis the temperature of the sample measured with a \f-\nbre optic thermometer. Note that in the SLP calculation\nwe use the value of the maximum rate of temperature\nincrease ([dT=dt ]max), as discussed previously by others\n[17, 42]. This method is believed to better estimate SLP\nthan the most common initial-slope procedure that can\nunderestimate this value [43].\nV. RESULTS AND DISCUSSION\nA. Theoretical results\nSeveral experimental results show the existence of an\noptimal particle size for hyperthermia [16, 18, 30]. This\nis also contemplated in Eq. (6), which predicts that this\noptimal size should occur when !\u001c= 1. This, however,\nis only true at low \feld amplitudes. Increasing \feld am-\nplitude one notice a shift of maximum size towards larger\nparticles in a noninteracting system. This can be easily\nmodelled within LRT using the \feld dependent magne-\ntization relaxation time [24]. Indeed, such drift becomes\nclear when h >0:04 (see discussion of Fig. 3(f) below).\nFurther, numerical dynamic hysteresis simulations using\nthe SLLG model or Kinetic Monte Carlo method [16{\n18, 44] show that, as the \feld amplitude increases, the\noptimal size shifts towards larger particles. It may even\ndisappear, depending (also) on the magnetic anisotropy\nof the nanoparticle [16{18, 44]. Most of the results above\nconsider a noninteracting system. However, in colloids,\nor real in vivo situation, agglomerate formation plays a\nkey role. In this case, it has been shown within LRT, that\nthe opposite e\u000bect occurs, i.e. increasing the strength of\n-0.10-0.050.000.050.10-0.6-0.4-0.20.00.20.40.6-\n0.10-0.050.000.050.10-0.4-0.20.00.20.40.6-\n0.10-0.050.000.050.10-0.10-0.050.000.050.100.154\n8 1 21 60204060801001201400\n.0000.0020.0040.0060.0080.010020406080100120140160-0.10-0.050.000.050.10-0.8-0.40.00.40.84\n8 1 21 602040600\n5 01 001 502 00-8-6-4-202 SLLG \nLRT \nNLRT \n M (t) / MSH\n (t) / HK   s = 4w\nt0 = 10-3a \nSLLG \nLRT \nNLRT \n  s = 10w\nt0 = 10-3 \n  \n H\n (t) / HKM (t) / MSc \nSLLG \nLRT \nNLRT \n  s = 12w\nt0 = 10-3H\n (t) / HKM (t) / MS \n dw\nt0 = 10-30.14 HK0\n.09 HKSLP (W/g)D\nimensionless anisotropy - σ \n  \nLRT \nNLRT0\n.04 HKe \nSLLG \nLRT \nNLRTw\nt0 = 10-3s = 10 SLP (W/g)(\nH0/HK)2s = 6h   s = 6w\nt0 = 10-3 SLLG \nLRT \nNLRT \n   \nH (t) / HKM (t) / MSb \nLRT \nNLRT \nLRT τh \nNLRT τh w\nt0 = 10-30.09 HKSLP (W/g)D\nimensionless anisotropy - σ f d\n = 11 nm      \nwt0 = 10-3 \nNLRT χ3' \nNLRT χ3'' \nRS χ3' \nRS χ3''  \nCubic susceptibility (a.u.)T\nemperature (K)gR\nSN\nLRTFigure 3. (Color online) Dynamic hysteresis curves for the\nLRT (dashed line), NLRT (solid line) and the exact solution\n(dash-dot line) using the SLLG equation for \feld H0= 0:1HK\nand!\u001c0= 10\u00003. In (a)\u001b= 4, (b)\u001b= 6, (c)\u001b= 10 and\n(d)\u001b= 12. (e) SLP as function of \u001bfor the LRT and NLRT\nwith distinct \feld amplitudes. (f) SLP as function of \u001bfor\nthe LRT and NLRT with and without the \feld dependence on\nthe relaxation time. (g) Real and imaginary susceptibilities as\nfunction of temperature for the RS and the NLRT models. (h)\nSLP as a function the square of the \feld for the LRT (dash),\nNLRT (solid) and exact solution using the SLLG (points) for\n\u001bvalues of 6 and 10.\nparticle interaction shift the optimal diameter to lower\nsizes [15]. The same was found including particle-particle\ninteraction using a mean \feld approach to the SLLG\nmodel at the low \feld regime [27]. Anyway, a valuable\nanalytical nonlinear response theoretical model (NLRT)\nshould be able to explain at least some of the features\ndiscussed above.\nA comparison between the hysteresis curves obtained\nfrom the LRT, our NLRT model and the numerical solu-\ntion of the SLLG model is shown in Fig.3(a)-(d), for dis-\ntinct\u001bvalues considering !\u001c0= 10\u00003andH0=HK= 0:1.\nIt is found that the inclusion of the corrected cubic term10\nleads to a good agreement with the numerical simula-\ntions, adequately describing the deviations from the lin-\near response. Note that the agreement is far better than\nany other model discussed previously (see Fig. 2(a)).\nThe LRT model is shown as a dash line, the exact result\nusing the SLLG equation is shown as dash-dot line, while\nNLRT (considering Eq. (29)) is shown as solid line. It is\nvery surprising that, with such a simple assumption, an\ninteresting nonlinear e\u000bect is obtained able to represent\nfar better the magnetization dynamics. Indeed, we found\nthat the present model works very well close to this limit\nof anisotropy value ( H0\u0018=0:14HK). It also has a slight\nfrequency dependence which can be monitored by non-\nphysical results in the magnetization curve or kinks in the\nSLP versus \u001bcurves increasing the \feld. At higher \felds\nwe observe deviations from the exact solution that might\nbe only addressed if higher-order terms are determined.\nNevertheless, as shown in Fig. 1 (see the NLRT line),\nthe range of validity of the model is almost completely\nwithin the hyperthermia therapeutic window. This sug-\ngests that this model might be applicable for real clinical\nsituations.\nFigure 3(e) shows the SLP as function of \u001bfor the LRT\n(dashed line) and the NLRT (solid line) for distinct \feld\namplitudes. For simplicity, we are not considering the\n\feld dependence on the relaxation time. One can clearly\nobserve a shift of the maximum of SLP towards higher\nparticle sizes in the nonlinear case. Also a decrease of\nthe maximum SLP value for the NLRT case. The phe-\nnomenon is strictly related to the nonlinear e\u000bect intro-\nduced in the model and not due to the \feld e\u000bect from\nthe relaxation. This result is in accordance with numeri-\ncal simulations from the literature [18, 44]. On the other\nhand, Fig. 3(f) also shows SLP as function of \u001bin both\ncases, but now investigating the \feld e\u000bect on the relax-\nation time for H= 0:09HK. Similar behavior as before\nis observed. Nevertheless, in comparison with the LRT,\nthe NLRT-\u001c(H) shows a larger size shift. As for instance,\nthe optimum anisotropy term change from \u001bopt= 8:1 for\nLRT to\u001bopt= 9:0 for NLRT- \u001c(H), which corresponds to\na shift in optimal diameter of the order of 4%.\nAs discussed in section II, there are other nonlinear\nmodels. In particular, cubic susceptibility expressions\nusing the RS model had been suggested to represent\nexperimental data of noninteracting magnetic nanopar-\nticles [45]. Figure 3(g) shows the cubic susceptibility\nterms, imaginary and real, for the RS model and the\nNLRT model as a function of temperature. Here the pa-\nrameters used were d= 11 nm, MS= 270 emu =cm3,\nKef= 8\u0002104erg=cm3,\u000b= 0:05 and\u001a= 5 g=cm3. As\nfound in Ref. [39] the real cubic term in the RS model\nshows a signi\fcant variation as a function of temperature,\nin particular in the range below 60 K, where a quite high\npositive cubic value is found theoretically. It is curious to\nnotice that experimentally such e\u000bect has not been ob-\nserved in Ref. [45] for noninteracting nanoparticle sam-ples. In fact, discrepancies between the RS model and\ndata of Ref. [45] had been attributed to polydispersity\nand particle-particle interaction e\u000bects. Note that the\ninclusion of such e\u000bects could be responsible for some\nof those di\u000berences between theory and data. However,\nthere might be another explanation. As we have just\nshown, the NLRT model represents far better the mag-\nnetization response. Di\u000berently from the RS model, the\nreal cubic susceptibility from NLRT does not show such\nstrong positive contribution at low temperatures. As a\nconsequence it might represent better experimental data.\nAnother point that could be commented about the im-\nprovement in the NLRT model in comparison to others is\nthe SLP calculation. Note that in the RS model the SLP\ncalculation, using the magnetization expression of Eq.\n(12), provides the same result as the LRT. So, although\nthe magnetization equations are not the same, the hys-\nteresis area is the same as the LRT case. Again, this is\nin contradiction with several experimental results. From\nthe experimental point of view, after obtaining the SLP\ndata of the samples as a function of the applied alternat-\ning \feld, it is common to try to describe the heating e\u000e-\nciency in terms of a \feld exponent, i.e. one might try to\n\ft the data with an allometric expression as SLP =aH\u0017;\nwhereais a constant and \u0017the \feld exponent. If this\nexponent is equal to 2 one might argue that the sample\nis within the linear response regime.\nFigure 3(h) shows the SLP as a function of the\nquadratic \feld for distinct \u001bvalues considering the LRT\n(dash), NLRT (solid) and SLLG (points). Both sit-\nuations shows that depending on the particle size or\nanisotropy, deviations from the expected quadratic \feld\ndependence of the LRT are found. At the low barrier\nregime (\u001b < \u001bopt), i.e. for particle sizes lower than the\noptimum value, the \feld dependence exponent is lower\nthan 2. While at the high barrier regime, an exponent\nhigher than 2 is observed. The same behavior is found\nfrom SLLG, as expected since NLRT model is based on\nthe assumption that SLLG is the exact result. However,\nbecause in the NLRT only the cubic term was introduced,\ndeviations between both models are expected for higher\n\felds. The nonlinear regime has been studied experi-\nmentally before on Ref. [17], where the transition to the\nnonlinear regime was explained using the SLLG model,\nthough without any analytical expression. The expla-\nnation for such behavior may be understood using Eq.\n(28). Note that \u001fQS;3<0, so when !\u001c <p\n3\u00191:7 the\nhigh-order contribution term lowers the linear SLP \feld\ndependence term. The consequence of this is an apparent\n\feld exponent lower than 2. On the other hand, when\n!\u001c >p\n3 the higher order SLP term changes sign, which\nnow adds a value to the \frst order term. In this case\nexponents larger than 2 might appear if the \feld is high\nenough.11\nSample DXRDMSHcoer\u0017\n(nm) (emu/cm3) (Oe)\nMnFe 2O4 11.3 293 21 2.2\nMn0:75Zn0:25Fe2O411.1 302 0.4 1.6\nMn0:75Co0:25Fe2O411.4 309 77 2.6\nCoFe 2O4 9.1 272 152 3.9\n\r\u0000Fe2O3 9.3 209 2.7 2.0\nCuFe 2O4 9.4 124 0.5 1.2\nCoFe 2O4 3.4 103 1.4 1.9\nCoFe 2O4 12.9 253 261 2.5\nCoFe 2O4 13.6 281 299 5.5\nNiFe 2O4 5.3 153 0.3 1.5\nNiFe 2O4 7.9 151 0.4 2.1\nNiFe 2O4 12.8 185 4.4 2.3\nTable I. Characterization parameters of the samples. DXRD\ncrystalline size, MSsaturation magnetization and Hcoercoer-\ncive \feld.\u0017is the apparent SLP \feld exponent from allomet-\nric \ft.\nB. Magnetic hyperthermia evidence\nEvidence of nonlinear behavior the SLP \feld depen-\ndence can be found in distinct ferrite-based powder sam-\nples. Table I summarizes the parameters obtained from\nsample analysis. Four sets of samples were studied. The\n\frst set is composed of three samples: manganese-ferrite\nbased nanoparticles undoped, doped with zinc and doped\nwith cobalt. Since samples were produced using the\nsame method, have (approximately) the same magneti-\nzation, and the same diameter, this set allow the study\nof anisotropy in\ruence over SLP versus Hbehavior. The\nsecond set is composed by other three samples: cobalt-\nferrite, maghemite and copper-ferrite. These samples\nhave very di\u000berent magnetization and anisotropy, but the\nsame diameter (some results published in Ref. [18]). The\nthird set is composed by other three samples of cobalt-\nferrite, which have a high anisotropy, with di\u000berent diam-\neters. And, the last set is composed by three samples of\nnickel-ferrite, which have a lower anisotropy than cobalt-\nferrite, with di\u000berent diameters.\nMagnetic hyperthermia experimental data around\n500 kHz is shown in Figs. 4(a), 4(c), 4(e) and 4(g) for\npowder samples, where we present the SLP as a func-\ntion of the applied \feld for distinct ferrite-based samples.\nMost of the applied \felds are above the therapeutical\nvalues (see Fig.1), but are necessary to experimentally\nobserve deviations from LRT. Symbols represent experi-\nmental data, while the lines are the \ft of the data using\nthe allometric function. Firstly, notice that soft-like ma-\nterials heat more e\u000eciently at low \feld amplitudes, in\naggreement with what was found before experimentally\nand theoretically [15, 17, 18]. This property, although not\ndiscussed in this work, is relevant for in vivo applications[5]. Table I shows the apparent \feld exponents obtained\nfrom this type of phenomenological approach for all the\nsamples, as well Figs. 4(b), 4(d), 4(f) and 4(h), com-\npared with 2 (gray dashed line which represents LRT).\nThe result indicates deviation from linear behavior and\nthe samples shows distinct exponents values, depending\n(probably) on sample anisotropy. The same behavior has\nbeen observed with other ferrites [17]. This behavior is in\naccordance with our previous theoretical analysis. How-\never, a direct comparison between experimental data and\ntheoretical analysis is compromised by the fact that sam-\nple are solid, allows a random anisotropy axis nanoparti-\ncle con\fguration that decreases the equilibrium suscepti-\nbility values lowering the SLP [17]. So, the nanoparticles\nat this highly packed con\fguration are at strong interact-\ning conditions, which may a\u000bect the magnetic anisotropy\n[15, 17]. In this case, one can not use the longitudinal\ncalculation developed in this work for the powder sam-\nples, since the quasi-static susceptibility values are now\ndi\u000berent. Nevertheless, powder con\fguration inhibit fric-\ntional loss contributions due to the Brownian relaxation\nmechanism [46{48] and a similar behavior for SLP (with\ndistinct absolute values) is also expected.\nThe NLRT model developed here is valid for H\u0014\n0:14HK, where magnetization relaxation mechanisms\nplays a role in the spin reorientation by overcoming the\nbarrier energy. Increasing the \feld value one need to\nuse directly the SLLG model, which due to the complex-\nity of the problem does not reveal any simple analytical\nequation. Nevertheless, a simple approach for qualitative\nanalysis under high \feld conditions ( H >HK) might be\nachieved using the Stoner-Wohlfarth (SW) model [13].\nC. Magnetic particle imaging evidence\nBesides magnetic hyperthermia, the present model\nmight be useful for magnetic particle imaging (MPI) too.\nMPI is a nonionizing imaging technique, introduced in\n2005 by Gleich and Weizenecker [31], which is capable\nof imaging magnetic tracers through the nonlinear mag-\nnetic response of magnetic nanoparticles. In MPI a DC\nplus an AC \feld are applied to the magnetic material\nin such a way to create a free \feld point volume where\nthe nanoparticles can respond to the ac \feld excitation.\nThe magnetic response signal can then be measured us-\ning detector coils. The received voltage by the detector\ncoil is\nu=\u0000\u00160\u0002\nVS0(x)@M(x;t)\n@tdV; (35)\nS0is the coil sensitivity (assumed to be \u00160S0=\n2:25 mT=A) and the integration is over the magnetic\nmaterial. The MPI third harmonic magnetization sig-\nnal per unit volume emf 3!0is de\fned as the module of12\n4060801001201401601800.00.51.01.52.02.53.03.54.04.5 MnFe2O4 \nMn0.75Zn0.25Fe2O4 \nMn0.75Co0.25Fe2O4 H\n (Oe)SLP (W/g)f\n = 522 kHz \nd @ 11 nma4\n060801001201400.00.10.20.30.40.50.60.70.8 CoFe2O4 \nγ-Fe2O3 \nCuFe2O4 H\n (Oe)SLP (W/g)f\n = 500 kHz \nd @ 9 nmc2\n04060801001201400.00.20.40.60.8 13.6 nm \n  9.1 nm \n12.9 nm \n  3.4 nm H\n (Oe)SLP (W/g)f\n = 500 kHz \nCoFe2O4e2\n04060801001201400.00.20.40.60.81.0g \n12.8 nm \n  7.9 nm \n  5.3 nm H\n (Oe)SLP (W/g)f\n = 500 kHz \nNiFe2O401234567d\n @ 11 nmb Field exponent  - - -  LRTM\nnFe2O4M\nn0.75Zn0.25Fe2O4M\nn0.75Co0.25Fe2O40\n1234567d\n @ 9 nmd Field exponent  - - -  LRTC\noFe2O4g\n-Fe2O3C\nuFe2O40\n1234567C\noFe2O4f Field exponent  - - -  LRT1\n3.6 nm9\n.1 nm1\n2.9 nm3\n.4 nm0\n1234567hN\niFe2O4 Field exponent  - - -  LRT1\n2.8 n7\n.9 nm5\n.3 nm\nFigure 4. (Color online) (a) SLP as function of the mag-\nnetic \feld for distinct manganese-ferrite nanoparticles around\n11 nm in powder con\fguration at f= 522 kHz. (b) Appar-\nent SLP \feld exponent \u0017obtained for manganese-ferrite in\npowder con\fguration. (c) SLP as function of the magnetic\n\feld for distinct ferrite nanoparticles around 9 nm in powder\ncon\fguration at f= 500 kHz. (d) Apparent SLP \feld expo-\nnent\u0017obtained for distinct ferrite nanoparticles in powder\ncon\fguration. (e) SLP as function of the magnetic \feld for\ncobalt-ferrite nanoparticles with distinct sizes in powder con-\n\fguration at f= 500 kHz. (f) Apparent SLP \feld exponent \u0017\nobtained for cobalt-ferrite in powder con\fguration. (g) SLP\nas function of the magnetic \feld for nickel-ferrite nanoparti-\ncles with distinct sizes in powder con\fguration at f= 500\nkHz. (h) Apparent SLP \feld exponent \u0017obtained for nickel-\nferrite in powder con\fguration. Symbols are data and lines\nrepresent the best \ft using the allometric function.\nthe Discrete Fourier Transform given by\nemf 3!0=\u00160S0jDFT [u3]j; (36)\nwhere\nDFT [u3] =N\u00001X\nk=0f[k]e\u0000i6\u0019\nNk: (37)\n01 2 3 4 5 6 7 0.11101001000 \nData of Ref. 36 \nFK model  \nNLRT model \n  D\nimensionless anisotropy - σThird harmonic MPI signal - emf (mV/mgFe)Figure 5. (Color online) MPI third harmonic signal of\nmagnetite-based magnetic \ruids containing nanoparticles of\ndi\u000berent sizes as a function of \u001b. The \fgure shows the exper-\nimental data (circles) from Ref. [38], calculations using the\nFK model of Ref. [38] (squares) and the NLRT (triangles)\ncalculation.\nThe function f[k] is obtained using f[t] =@M(t)\n@tand\nthe time discretization as t=k\nNf0, wheref0is the exci-\ntation \feld frequency and Ncorresponds to the number\nof intervals discretized within one period. In this work\nN= 40. In NLRT model the complete magnetization ex-\npression is unknown, so we only use the terms up to the\nthird harmonic. On the other hand, for the FK model,\none can expand the Langevin expression up to any order.\nFigure 5 we shows the experimental MPI data of\nthe third harmonic magnetization signal of magnetite\nnanoparticles of distinct sizes performed at 250 kHz (see\nRef. [38] for details). Spheres correspond to experimen-\ntal data, while squares are related to the FK model of\nRef. [38]. Note the logarithmic scale and that we are\npresenting the data in terms of \u001b. Here we assumed the\nbulk magnetic anisotropy value, although is well known\nthat the anisotropy is size dependent [40, 49{51]. Never-\ntheless, size dispersity was taken into account. The calcu-\nlations used a relaxation time valid for any \u001b[15, 35] and\nparameters from Table 1 of Ref. [38]. Triangles corre-\nspond to our polydisperse calculation taking the Discrete\nFourier Transform and using Eq. (35) in units of V/g,\ni.e. taking into account in the calculation of emf 3!0the\namount of magnetic material in mass per unit volume.\nNote that our model represents better the MPI exper-\nimental data. Indeed from 10 data points NLRT is in\nbetter agreement with 80% of the data. Better theoreti-\ncal results might be achievable if the anisotropy of each\nsample is known, or even more if one is able to take into\naccount possible particle-particle interaction e\u000bects due\nto agglomerate formation [52]. So, it might be fair to say\nthat, both hyperthermia and MPI experiments seem to\nbe more adequately described by the NLRT model.13\nFinally, it might be relevant to comment that there\nis a huge interest of not only deliver heat using mag-\nnetic nanoparticle hyperthermia, but also, monitor non-\ninvasively heat delivery using magnetic nanoparticles. In\norder to be successful in such area, analytical expressions,\nas the ones derived in this work, that better represent the\nnon-linear response of magnetic nanoparticles, are highly\nneeded. The authors believe that the model developed\nhere might indicate a useful approach towards this im-\nportant clinical goal.\nVI. CONCLUSION\nIn conclusion, a nonlinear response model of magnetic\nnanoparticles valid for single-domain nanoparticles was\ndeveloped. The model is valid beyond the linear response\ntheory, and showed good agreement with dynamic hys-\nteresis simulations using the stochastic Landau-Lifshitz-\nGilbert approach and experimental hyperthermia data\nfor \feld amplitudes as high as 10% of the magnetic\nanisotropy \feld. In particular, a generalized expression\nfor the magnetization and the heat loss e\u000eciency (SLP)\nwere obtained. The model showed many features found\nexperimentally in magnetic hyperthermia and MPI stud-\nies. 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Andersenb \n \na Departamento de Electrocerámica, Instituto de Cerámica y Vidrio, CSIC, \n28049 Madrid, Spain  \nb Departamento de F ísica de Materiales, Universidad Complutense de Madrid, \n28040 Madrid, Spain  \n*c.granados.miralles@icv.csic.es  \n \nAbstract  \nPermanent magnets are integral components in many of the modern \ntechnologies  that are critical for  the transition to a sustainable society . However, \nmost  of the high -performance  (BH max > 100 kJ/m3) permanent magnets that are \ncurrently employed contain rare -earth elements (REE), which have long been \nclassified as  critical materials with a high supply risk  and concerns regarding \npollution in their mining . Therefore, suitable  REE -lean /free magnets must be \ndeveloped in order to ensure  the sustainability of clean energy generation  and \nelectric mobility . The REE -free h exagonal ferrites (or hexaferrites ) are the most \nused permanent magnet s across all applications , with an 85  wt.% pie of the \npermanent magnet market . They are the dominant  lower -grade  option \n(BH max < 25 kJ/m3) due to  their  relatively good hard magnetic properties , high \nCurie temperature (>700  K), low cost  and good  chemical stability.  In recent years , \nthe hexaferrites have also emerged as  candidates for substituting REE -based \npermanent magnets in applications requiring  intermediate magnetic performance \n(25–100 kJ/m3), due to considerable performance improvements  achieved \nthrough  chemical tuning,  nanostructuring  and compaction /sintering  \noptimization . This  chapter reviews the state -of-the-art sintering strategies  being  \ninvestigated with the aim of manufacturing hexaferrite magnets with optimized \nmagnetic properties , identifying key challenges and highlighting  the natural future \nsteps to be followed.  \n \nKeywords:  permanent magnets, hard ferrites, hexaferrites, ceramic magnets, \nrare-earth -free magnets , SrFe 12O19, BarFe 12O19  \n   \n \n1. Introduction  \n1.1 Classification of magnetic materials  \nThe magnetism of magnetic materials arises at the atomic scale  and is \ninfluenced by characteristics spanning several orders of magnitude  (see Figure 1a). \nIn the atoms of most compounds, the electrons exist in pairs with opposite spins \nthat cancel out each other’s magnetic moment. However, some elements or ions \nhave unpaired electrons, whose  spin and orbital motion cause them to exhibit a \nmagnetic field  giving the atom a magnetic moment . The organization  of these \nmagnetic atoms in the atomic structure of the material determines its magnetic \nproperties.  Figure 1b shows a schematic illustration of the main types of magnetic \nordering.  In paramagnetic materials, the atomic magnetic moments are randomly \noriented leading to no net magnetization and a relatively  weak attraction to an \nexternal magnetic field. Antiferromagnetic materials are magnetically ordered, but \nalso exhibit zero net magnetization due to an antiparallel organization of equal \natomic magnetic moments. However, in ferro - or ferri -magnetic materia ls (below \ntheir Curie temperature, Tc, which is the  critical temperature above which thermal \nfluctuations lead to the material being paramagnetic ), the magnetic atoms are \norganized in a way that leads to a net magnetization along a c ertain direction \n(magne tic easy axis) in the structure, and it is these types of materials that are used \nfor permanent magnets  (PMs) . \n \nFigure 1. (a) Illustration of the multiscale origin of the magnetism in magnetic materials. (b) Schematic \nillustration of  main  magnetic ordering types  and the  resulting  net zero field magnetization . (c) Size \ndependency of the coercivity. (d) Hysteresis curve of an ideal  permanent magnet . \nThe ferro/ferri -magnetic materials are generally categorized as either ‘soft’ or \n‘hard’ depending on their resistance to demagnetization. This is evaluated in terms \nof the coercive field (or coercivity, Hc), which is the external magneti c field \nrequired to reset the magnetization of the material. Magnetically soft materials are \neasily (de)magnetized by an external magnetic field ( typically defined as \nHc < 10 kA/m) and their magnetization is therefore often temporary, while hard \n(or perman ent) magnetic materials have a high resistance to demagnetization \n(Hc > 400 kA/m) and once magnetized they can therefore sustain a magnetic field \n \n \nindefinitely.[1] The coercivity of a material is determined in part by the intrinsic \nmagnetocrystalline anisotropy of the crystal structure as well as by microstr uctural \n(extrinsic) effects such as crystallite size or  structural defects, which influence the \nformation (nucleation and growth) of magnetic domains in the material.  For most \nmagnetic materials, Hc is found to increase as the crystallite  size is reduced, \nreaching a maximum value at the critical single -domain size (see Figure 1c). \nAnother key property of a magnetic material is its remanence field ( Br or Mr), \nwhich is th e spontaneous magnetic flux density  or magnetization  exhibited by the \nmaterial in zero external field conditions. Figure 1d shows a schematic illustration \nof the external magnetic field ( H)-dependent flux density ( B) and magnetization \n(M) curves, commonly called hysteresis curves, of an ideal permanent magnetic \nmaterial. As illustrated, it is the combination of these two param eters, i.e., the \ncoercivity (magnetic stability) and remanence (spontaneous magnetization), that \nultimately determines the magnetic strength of the magnet. This magnetic strength \nis quantified  by the so -called maximum energy product ( BH max), defined by the  \narea of the largest possible rectangle that fit s under the BH curve in the second \nquadrant, which measures the potential energy stored in the stray field of the \nmagnet .[2] \nFigure 1d shows the magnetic hysteresis of an ideal permanent magnet, in \nwhich all magnetic spins are perfectly aligned  (and therefore , Mr = Ms) but in real \nmagnets, the remanence value is smaller than the saturation ( i.e., Mr < Ms). It \nfollows that , as the Mr value approaches Ms, the loop turns more square d, and in \nturn, BH max is maximized. Hence, the squaren ess and magnetic alignment is of ten \nmeasured in terms of in Mr/Ms ratio ,[3] which is another of the key parameters to \nbe improved  for permanent  magnets.  \n1.2 Materials for permanent magnets : Current status  \nMagnetic materials have the unique ability to directly interconvert between \nelectrical and mechanical energy. A moving magnet can induce an electric current \nto generate electrical energy, and oppositely , an electric current can be used to \ngenerate a magnet ic field and exert a magnetic force. These electromagnetic \nproperties underpin the operation of electric generators and motors, making \nmagnetic materials critical for the transition towards an environmentally friendly \nand sustainable future.[2] As a result, the worldwide permanent magnet market is \nexpected  to reach  $39.71  Billion by 2030, according to the 8.6 % compound annual \ngrowth rate (CAGR) forecast  in the last Grand View Research report.[4] \nFigure 2a illustrates  the relative performance in terms of BH max and Hc for the \nmost important families of  commercial  PM materials , including AlNiCo alloys, \nhard ferrites  ceramics , Nd 2Fe14B and SmCo 5. The high -performance \n(BH max > 100 kJ/m3) permanent magnet market is currently dominated by the \nrare earth element (REE) -containing  materials  Nd 2Fe14B (strongest magnet) and \nSmCo 5 (best high temperature performance) due to their superior energy \nproduct s,[1,5] which is a critical parameter for the performance in applications \nwhere miniaturization is a major driving force ( e.g., electric vehicle motors, \ndirect -drive generators, electro -acoustic devices, accessory electric motors, mobile \nphones, sensors, portable electronics, etc.). Unfortunately, the use of REE -based \nmaterials  entails various problems. The compounds rely  on scarce REE  such as \nneodymium , samarium or dysprosium, which are classified as critical raw \nmaterials, not only owing to their supply risk and price volatility, but also to the \nharmful environmental impact of their extraction.[6] China has been the \nundisputed leader in REE mining and production for the last 40 years,[7] and \ndespite other countries attem pting to gain ground, today China still accounts for \nmore than 60% of the world REE production.[8] Consequently, o ver the last 20  \n \nyears geopolitical circumstances have often  led to erratic price fluctuations . \nFurthermore, the c obalt used in Sm Co 5 magnets is another problematic element . \nThe supply chains for the bulk part (>50%) of the cobalt used in advanced \nmaterials can be traced b ack to the cobalt mines in the Democratic Republic of the \nCongo, where artisanal miners (including thousands of children) work under \nextremely hazardous conditions.[9] As a consequence, the development of  \nREE -poor or REE -free alternatives has long been an important research topic in \nthe PM field.  \nAlthough the undisputed strength of REE -magnets is necessary  for the \nhigh est-performance applications, there are many other applications  that are less \ndemanding in terms of magnetic strength, where a compromise (see Figure 2b) \nmust be made between other factors such as price, stability, processability, etc.[10] \nAt this end of the spectrum, hard ferrite magnets have long been the material  of \nchoice fo r lower grade applications (<25  kJ/m3). However, as illustrated by the \narrow in Figure 2a, a considerable performance gap exists in the in termediate \nperformance range between the cheaper AlNiCo and hard ferrite PMs and REE \nPMs. Consequently, for many applications it is often necessary to use an expensive \nand excessively strong REE magnet , in lack of an intermediate alternative . Here, a \nmodes t performance improvement of lower grade magnets would be sufficient to \nreplace REE PMs while remaining within a weight range suitable for the \napplication.  \n \nFigure 2: (a) Diagram of BH max vs coercive field for the main families of commercial ly available  hard  \nmagnetic materials . (b) Radar plots of key extrinsic properties of sintered Nd 2Fe14B, sintered SmCo 5, \nanisotropic AlNiCo and sintered hexaferrite magnets.  Figures based on v alues from [11]. \n \n \nIn this context, hexaferrites have long been considered  good candidates for \nreplacing REE magnets in the intermediate performance range, due to their \nreasonably good performanc e, high Curie temperature (>700  K) and excellent \nchemical stability , which all comes at a fraction of the cost of REE magnets.[12,13]  In \nfact, hard ferrites are the most produced magnetic material, despite their moderate \nperformance compared to REE magnets.[14] In 2013 they were reported to account \nfor 85 % of the total PM market by  manufactured  mass, although they only \nrepresented 50  % of the market by sales.[15] \nWhile recent studies have  demonstr ated new approaches to improve magnetic \nproperties  of hard hexaferrite powders  (e.g. nanostructuring,[16–19] chemical \nsubstitution,  [20–23] exchange spring  composites [19,24,25] ), manufacturing dense \nsintered pellets of sufficient structural integrit y without degrading the optimized \nproperties  has proven  a key challenge.  In practice , this prevents  the replacement  \nof expensive and unsustainable REE  PMs  in a range of  applications , and is  the \nreason why  hard ferrites still generate  great scientific  inter est.[26] The present \nchapter aims at summarizing  the most relevant  recent  achievements and progress \nin the field, as well as key challenges encountered during the fabrication and \nsintering of dense ferrite magnets.  \n2. Hard ferrites : M-type hexaferrites  \n2.1 Crystal and magnetic structure  \nThe so -called hexaferrites, hexagonal ferrites  or simply hard ferrites , are a \nfamily of ternary or quaternary iron oxides with  hexagonal crystal lattice  of long  \nunit cell  c-axis (≈23–84 Å).[26] Of the materials in the  hexaferrite  family , the \nM-type  hexaferrites  have been widely  used for application  as permanent magnets . \nWith chemical formula MFe12O19 (M = Sr2+ or Ba2+), the Sr and Ba M -type ferrites \n(SrM and BaM)  are isostructural and exhibit  very similar magnetic characteristics . \nThe compounds have  a large uniaxial magnetocrystalline anisotropy and a \nmagnetic easy axis along the crystallographic c-direction.  This strong intrinsic  \nanisotropy results in  a high Hc, making them very resistant towards  \ndemagnetization ( i.e. magnetically hard) and therefore attractive  as PM  material s. \n \nFigure 3. Crystal and magnetic structure of  Sr (Ba) hexaferrite . Black and red spheres represent Sr2+ \n(Ba2+) and O2– ions. Colored polyhedra illustrate the 5 different crystallographic sites of Fe3+ and arrows \nsymbolize the Fe3+ magnetic spins.  \n \n \nFigure 3 illustrates the crystal and m agnetic structures of M -type hexaferrites. \nThey display a hexagonal magnetoplumbite structure (space group P63/mmc ) with \nvery anisotropic unit cell ( a ≈ 5.9 Å, c ≈ 23 Å). Fe3+ ions occupy interstitial \npositions in a hexagonal  close -packed structure of O2– and Sr2+ (Ba2+) ions .[26–28] \nWith 2 formula units per unit cell (64 atoms) , SrM has a crystallographic density \nof 5.3  g/cm3 (5.1 g/cm3 for BaM ).[29,30]  The crystal structure may  also be described  \nin terms of stacking of simpler structural blocks (cu bic S and hexagonal R blocks)  \nwhich are in turn stacked onto similar blocks rotated 180° about  the c-axis (S* and \nR* blocks, respectively).[28] \n2.2 Magnetic properties  \nTable 1 compares  the intrinsic magnetic properties of SrM and BaM with that  \nof other important magnetic compoun ds. The theoretical  magnetic moments (at \n0 K) of the hexaferrite crystal structure s can be calculated from the ferrimagnetic \nordering of the magnetic Fe3+ ions in the structure (see arrows in Figure 3), yielding \nvalues of 20.6 µB/molecule for SrM and 20  µB/molecule  for Ba M.[26,31]  This results \nin fairly good  saturation  magnetization , Ms, and magnetic induction, Bs, values.  \nThe Curie temperature, TC, of the M -type hexaferrites is more than 100  °C above \nthat of the much used REE -based  Nd 2Fe14B hard phase.  \nThe large  uniaxial  anisotropy of the hexagonal lattice of SrM and BaM \n(c/a = 3.9) causes a large magnetocrystalline anisotr opy along the c-axis, which \nyields relatively high anisotropy constants, K1 (see Table 1)[32–34] and a large \ntheoretical maximum Hc of 594 kA/m .[26] For a hypothetical fully -dense and \nperfectly -oriented hexaferrite magnet, a theoretical maximum BH max of 45 kJ/m3 \nhas been estimated.[1] \n \nTable 1. Intrinsic magnetic parameters  at room temperature  (RT)  for some representative soft and hard \nmagnetic phases . Data extracted  from [34] unless otherwise stated.  \n Ms (Am2/kg)  Bs (T) TC (K) K1 (MJ/m3) \nFe 0.65Co 0.35 240 2.45  1210  0.018  \nFe 217 2.15 1044  0.048  \nAlNiCo5  [1] 159 1.40  1210  0.68*  \nCoFe 2O4 [11] 75 0.5 793 0.27  \nBaFe 12O19 72 0.48  740 0.33  \nSrFe 12O19 72 0.48  746 0.35  \nNd 2Fe 14B 165 1.61 588 4.9 \nSmCo 5 100 1.07 1020  17.2 \nSm 2Co 17 118 1.25 1190  4.2 \n             *shape anisotropy  \n3. Sintered hard ferrite permanent magnets  \nTowards the effective implementation of permanent magnets into a device, the \nmaterial in powder form has to be compacted into dense, mechanically stable  and \nmagnetically -oriented  pieces ( i.e., magnets ). This conforming /densification \nprocess  (called sintering) generall y involves applying elevated pressures and/or \ntemperatures to the material in powder shape .[35] As for most other materials , the \nmechanical properties of the sintered piece  relies on a high density. However,  the \nimportance of achieving a highly dense magnet is enhance d for PMs, since the \nmagnetic performance ( BH max) is measured per volume unit, and hence , it is \ndirectly proportional to the density.   \n \nThe high sintering  temperatures  often end up undesirably  altering the \nfunctional properties  of the starting material and therefore, great efforts are \ndedicated to both (i) adapting the sintering methods to the specific material of \ninterest and (ii) developing  novel sintering strategies that lower the working  \ntemperature s, aiming at minimizing the damage.[35] In the particular case of \nhexaferrites, a common problem is the format ion of hematite (α-Fe2O3) as a side \nphase.  This iron oxide is very prone to appear,  as a result of  its high stability , and \ncauses a decrease of saturation magnetization, due to the antiferromagnetic nature \nof the phase. Fortunately, it has been shown that α-Fe2O3 can be avoided when the \nstarting MFe12O19 powders have the right M:Fe stoichiometry, yielding Ms values \napproaching the expected ≈70 Am2/kg.[26] In contrast , limiting the grain growth to \ncircumvent the detrimental impact on Hc has proven more challenging.[36] Thus, \nM-type ferrites in powder form often present coercivities far below the theoretical \nvalue , and the situation worsens for sintered pieces (see Table 7 in ref. [26] for an \nextensive sample record) . Owing to this, sintered hexaferrite magnets are generally \ninferior to the theoretic ally achievable  45 kJ/m3,[1] although specific studies have \nmanaged to come fairly close to this value.  \nAnother important aspect i n the sintering of PMs is the magnetic alignment of \nthe constituent particles and domains in the material. The magnetic particles may \n(or may not) be magnetically aligned, resulting in anisotropic (isotropic  if not \naligned ) magnets. The greater the magneti c alignment, the more the Mr value \napproaches Ms, yielding a more square -shaped  MH curve ( as illustrated by the \nblack curve in  Figure 1c), there by maximizing BH max. Thus, the BH max of \nmass -produced  isotropic M-ferrite magnets is around 10  kJ/m3, while the \nanisotropic kind ranges from 33 to 42  kJ/m3.[37–40]* The magnetic alignment has \nbeen traditionally carried out by application of an external magnetic field,[26,41]  \nalthough recently patented methods have succeeded in suppressing the external \nfield by taking advanta ge of the shape of the particles .[42,43]  Notably, t he M-type \nferrite s are prone to form platelet -shaped particle s, with magnetizati on direction \nparallel to the platelet normal vector (see Figure 4a). As illustrated in  the figure, \nthe platelet shape  of the particles  favors magnetic (and crystallogr aphic) alignment \nupon compaction.  \n \nFigure 4. MFe12O19 particles, displaying typica l hexagonal platelet shape with the easy  axis of  \nmagnetization normal to the plate let plane (and parallel to the crystallographic c-axis). This shape \nfavors magnetic (and crystallographic) alignment upon application of uniaxial pressure. Adapted  with \npermission from [13] \n \n* Ferrite magnets with higher BH max values are available commercially (up to 44  kJ/m3), but in those \ncases the material is doped with e.g., La or Co.[39,40]  \n \n \nDuring the last decades, different sinte ring strategies have been investigated \naiming at maximizing both the magnetic alignment (boosting Br and Mr/Ms) and \nthe Hc on the sintered material. Lately, efforts have also been devoted to making \nthe processes greener and increasing recycling rates. The following sections intend \nto offer an overview of the pros and cons of each of the alternative s. \n3.1 Conventional sintering  \nHexaferrites were first developed  as a PM material by researchers at the Philips \nResearch Laboratories in 1950s. In 1952, Went et al.  prepared a Ba-ferrite  magnet \nwith a good  Hc value (≈240 kA/m) , although a limited Br derived from its isotropic \nnature  (0.21  T) yielded a modest BH max of 6.8  kJ/m3.[44,45]  Two years later, Stuijts \net al.  developed a conventional sintering  (CS)  strategy  to produce anisotropic BaM \nmagnets with BH max up to 28  kJ/m3,[41] which is essentially the method used \nnowadays to make sintered ferrite magnets  industrially . In brief, a sludge of Ba M \npowders and water is compacted while being held it in an  external magnetic field, \nproducing a consolidated piece (still  poor in  density ) which is subsequently \nsintered at temperatures above 1100  °C to promote densification . Stuijts et al.  \nexplored  sintering temperatures between  1250  and 1340  °C and noted  that \nincreasing the temperature maximizes the density and the magnetic alignment \n(and therefore Br), but at the cost of decreasing  Hc, as a consequence of the grain \ngrowth promoted by the elevated temperatu res. This problem , encountered \nalready in 1954, h as been subject  of extensive  research  since . \nAs mentioned earlier, structural characteristics such as crystallite size, size \ndistribution and crystallite morphology can largely affect the coercivity of ferrite \nmagnets.  In particular, highest Hc values are attained for crystallite size s close to \nthe critical single -domain size  defined earlier .[33,45,46]  The difficulty  not only lies in \nbeing able to produce particles of a specific size in a controlled manner, but  it \nbegins with determining what this critical size is for a specific material . For \nisotropic SrM crystallites , the critical sing le-domain size  has been estimated to be \naround 620–740 nm.[16,47]  However, the experimentally reported \ncrystallite /particle  single -domain sizes  of SrM span  from  30 nm all the way up to  \n830 nm.[47] This is due to the high influence of particle morphology in the attained \ncoercivity, as well as to  the different characterization methods used to determine \nthe reported size  (i.e. particle vs. crystallites, number vs. volume weighted, etc.). A \nstudy by Gjørup et al.  showed that a much smaller critical single -domain size is \nobtained for highly anisotropic crystallites , and therefore not only the overall size, \nbut also the aspect ratio of anisotropic SrM crystallites  should be considered  when \ntrying to maximize Hc.[47] \nNotably, r educing the size of the starting powders  does not necessarily yield to \nbetter coercivities, as the grain growth upon sintering seems to be even greater \nwhen dealing with  materials of smaller particle size s.[48–50] El Shater et al . sintered \nnanometric BaM ( 100–200 nm) at 1000 and 1300  °C, producing average particle \nsizes of  0.537  and 16.35  µm, respectively , with the consequent drop  in coercivity \n(from 271 to 56 kA/m ) and the Mr gain.[51] Therefore, the choice of sintering \ntemperature must be a compromise  between minimizing grain growth (to \nmaximize Hc) and maximizing densification (and in turn, Mr). \nA common approach for limiting grain growth has been the use of sintering \nadditives . Kools  proposed a mechanism through with SiO 2 would prevent the \ngrowth of SrM grains during sintering and proved the effect for a range of SiO 2 \nconcentrations (0.36 –1.44 wt.% ).[52,53]  Beseničar et al.  reported that, besides \nlimiting the growth, SiO 2 induces some ordering of the SrM particles, resulting in  \nvery anisotropic magnets  with high relative density (97%) and satisfactory \nmagnetic properties ( Br ≈ 0.39  T, Hc ≈ 340 kA/m).[54] Kobayashi et al.  determined  \n \nthe optimal SiO 2 concentration to be between 1 and 1.8 wt.% , showing a \ndetrimental effect on Hc for greater SiO 2 additions.[55] Guzmán -Mínguez  et al.  \nreported the appearance of ≈20 wt.%  α-Fe2O3 as a secondary phase for SiO 2 \nconcentrations>1  wt.% .[56] \nCaO has been reported to favor densification,  and therefore, it has also been \nexplored as a sintering additive  for hexaferrites , in this case with the aim of  \nboosting Mr, although at the expense of aggravating the grain growth  effect .[46,55,57]  \nIn this context, the combined use of both additives  has also been investigated . Lee \net al . reported a decent BH max of 29.4  kJ/m3 when adding  0.6 wt.%  SiO 2 and \n0.7 wt.%  CaO, but neither remanence nor coercivity were terrific ( Br = 0.36  T, \nHc = 281 kA/m).[58] Töpfer et al.  fabricated a very dense SrM magnet (98%) with a \nnotable Br value of 0.42  T by incorporating 0.25  wt.%  of SiO 2 and 0.25  wt.%  CaO , \nalthoug h a moderate  coercivity value of 282  kA/m only allowed for a \nBH max = 32.6  kJ/m3.[59] Huang et al.  tested the combined add ition  of CaCO 3, SiO 2 \nand Co 3O4 (1.1, 0.4 and 0.3  wt.% , respectively), managing a remarkable BH max of \n38.7  kJ/m3, owing to an exceptional  remanence (0.44  T) and despite a modest \ncoerci vity (264  kA/m).[60] \nSlightly superior magnetic parameters ( Br = 0.44  T, Hc = 328 kA/m , \nBH max = 37.6  kJ/m3) have been obtained by from a two -step sintering (TSS) \nmethod adapted to SrM by  Du et al ..[61] Here, the powders were cold -pressed as \nusual  CS, but the subseq uent thermal cycle used for sintering was slightly more \nelaborate : after a first high temperature step, in which the maximum temperature \n(1200  °C) is maintained for only 10  min, a longer (2  h) heating step at 1000  °C \nprovides for full densification of the SrM magnet.[61] The scanning electron \nmicrograph (SEM) in Fig. 6(e) from ref. [61] illustrates the confined grain size, the \nhigh density and high degree of alignment justifying the good magnetic \nperformance. A more recent work by Guzmán -Mínguez et al.[62] combined a TSS \napproach with the addition of 0.2% PVA and 0.6% SiO 2, realizing great control of \nthe grain growth at 1250  °C (see Figure 5) although the obtained magn etic \nproperties were not as good  as the  ones  previously reported by Du  et al. . \n \nFigure 5. SEM images of SrM pellet  sintered at 1250  °C by (a) conventional sintering and (b ) two -step \nsintering. Reprinted from [62], Copyright 2021, with permission from Elsevier.  \n3.2 Spark plasma sintering  \nIn the 1990s , a new commercial apparatus based on  resistive sintering , called \nspark plasma s intering (SPS)  was developed by Sumitomo Heavy Industries Ltd. \n(Japan ).[63] The SPS method is based on the use of an electrical current and a \nuniaxial mechani cal pressure  under low atmos pheric pressure,  to simultaneously \nheat and compact a powder sample.[64] The starting powders are typically loaded \nin a graphite die,  which is placed between two electrodes in a water -cooled vacuum \nchamber . A uniaxial pressure is applied to the die while passing a DC electrical \ncurrent through, which heats up the sample due to  the Joule effect  (see Fig. 1 in \nref. [65] for a typical SPS setup ). The inventors of the syst em claimed  the g eneration \n \n \nof plasma to take place, thus leading to the technique ’s name. However, a lthough \nit is generally accepted that plasma may be generated between particles due to \nelectrical discharges, there is no conclusive experimental evidence of such \noccurrence.[64] Therefore, SPS is sometimes referred to by alternative names, such \nas field-assisted sintering t echnique (FAST).  The simultaneous application of \ntemperature and pressure can also be obtained by conventional hot pressing  (HP) . \nHowever, in SPS and HP , heat  is produced and transmitted to the material in \ndifferent ways. In conventional heating  the powders are sintered by heating the \nentire container using external heating elements in a furnace. This leads to slow \nheating rates , long sintering times and waste of energy in heating up all the \ncomponents . The SPS method , however,  has allowed increasing the heating rates, \nlowering the working temperatures and reducing the dwell times.[66,67]  These \nbenefits make SPS a good alternative when the goal is to limi t the grain g rowth \nduring  sintering ,[67] and potentially improve the obtained Hc (and BH max) values of  \nsintered hexaferrite  magnets . \nNumerous investigations focusing on sintering hexagonal ferrites by SPS have \nbeen  published in the last two decades.  Obara et al.  prepared fully -dense SrM \nmagnets by SPS at 1100  °C and 50  MPa for only 5  min.[65] A fairly competitive Hc \nof 325  kA/m was obtained by doping with La2O3 (1 wt.%) and Co 3O4 (0.1 wt.%) . \nAlthough  the measured hysteresis loops were rather squared, the remanence value \n(0.32 T) was not sufficient to  guarantee a noteworthy energy product \n(BH max = 18.3  kJ/m3). Mazaleyrat et al.  sintered BaM nano powders  with sizes \nbelow 100  nm and managed to hold grain growth and produce a Hc of \n390 kA/m,[68] which even surpasses the value reported for the La and Co -doped \nmaterial described above. Unfortunately, a deficient density (88%) degraded the \nBH max down to 8.8 kJ/m3.Ovtar et al.  sintered the same batch of 90  nm BaM \nnanoparticles by both CS and SPS , producing much smaller sizes through the \nsecond  method.[69] Additionally, they realized that secondary phase s (Fe 3O4, \nα-Fe2O3) tend to form on the surface of the BaM SPS pellets, and tested different \nmaterials for the protective discs separating the sample from the graphite die (BN, \nAu, α-Al2O3) concluding th at α-Al2O3 was the one performing best. The resulting \ndensity was rather low 82% but the coercivity was adequate  (350  kA/m).  Stingatiu \net al.  attempted  downsizing a µm-sized SrM material by a ball-milling step prior \nto consolidation through SPS.[70] The resulting density was sat isfactory (90%) but \nunfortunately, ball -milling was seen to amorphize  the surface of the SrM, which \ntrigger ed formation of secondary phases during SPS, this having a detrimental \neffect on the magnetic properties ( BH max < 10 kJ/m3). \nSaura -Múzquiz et al.  prepared nm -sized  SrM powders by hydrothermal \nsynthesis (HT)  with hexagonal plate -like particles  (such as those in Figure 4) with \nvery small sizes ; in some cases, th e platelets  were  as thin as a single unit cell \n(i.e. <3 nm).[17] These HT -synthesized SrM powders were consolidated by SPS \nyielding appropriate Hc values of 301  kA/m. More importantly, the highly \nanisotropic shape of the particles provided for a pronounced  magnetic alignment \nof the sintered SrM magnets , inherently occurring as a result of simultaneous \napplication of elevated temperature and uniaxial pressure, just as illustrated in \nFigure 4. Here, an Mr/Ms ratio  of 0.89 was reached without applying an external \nmagnetic field neither before nor during sintering , yielding a BH max value of \n26 kJ/m3. Figure 6a shows the magnetic hysteresis of the HT powders and the \ncorresponding SPS pellet, evidencing the squareness of the latter. Achieving  \nmagnetic alignment without a magnetic field is very  conve nient from an industrial \npoint of view, because it allows a full step to be remove d from the manufacturing \nprocess  (i.e., the magnetization), which simplifies the procedure, reduces costs and \nincreases energy efficiency.[42] Figure 6b displays the powder X -ray diffraction \n(PXRD) data measured on both SrM  powders and SPS pellet. Despite the very \ndissimilar appearance, Rietveld analysis demonstrates that both PXRD patterns \nare consistent with  pure -phase SrFe 12O19 although with notable differences in  \n \ncrystallite size and orientation.  The highly anisotropic s hape of the powders is \nvisible from the sharpness of the hkl-reflections describing the crystallite on the \nplatelet plane, such as (110) or (220), compared to the large broadening of those \nassociated to the platelet thickness, e.g., (008), all this in agre ement with much \nsmaller sizes along the c-axis than on the ab-plane (i.e., thin platelet s). Regardless \nof the difference in peak broadening, Bragg reflections of all orientations are \npresent in the PXRD pattern measured for the SrM powders, demonstrating a  \nrandom orientation of the crystallites. However, the very intense hh0 reflections \nare absent for the PXRD pattern recorded for the SPS pellet, while 00l reflections \n(as well as others with high contribution from the c-crystallographic direction) are \nsyste matically intensified, thus indicating a marked preferred orientation of the \nplate lets. As explained before, for M -type platelet -shaped  particles, \ncrystallite/particle alignment goes together with magnetic alignment. The \ncrystallographic alignment was furt her studied based on pole figure measurements \n(Figure 6c), a slightly more comple x diffraction measurement enabling \nquantification of the degree of orientation ( Figure 6d). \n \nFigure 6. (a) Magnetic hysteresis loop of HT-synthesized SrM nanoparticles and corresponding SPS \npellet. (b) PXRD data along  with Rietveld model of the same samples. (c)  X-ray pole figure \nmeasurements and (d) oriented volume fraction of SPS  pellet . Reproduced from Ref. [17] with \npermission from the Royal Society of Chemistry.  \nOptimization of both the HT synthesis route[71] and the SPS protocol[18,72]  as well \nas correlation of structural and magnetic properties, allowed reaching \nMr/Ms ratios as high as 0.95, altho ugh at the cost of reducing Hc down to 133  kA/m, \nwith which the BH max improvement was only moderate (29  kJ/m3).[73] However, \nperforming  a thermal treatment at 850  °C after  SPS was enough to reach a \nBH max = 36 kJ/m3, value on the order of the highest -grade commercially available \nferrite magnets,[37–40] while avoiding the use of  an external magnetic field.  \nApplying this SPS protocol to SrM powders produced by synthesis methods other \nthan HT  did not yield  such  outs tanding magnetic properties, due to an inferior \nparticle or ientation degree and, hence , a poorer magnetic alignment.[72,74]  A newer \nstudy by Saura -Múzquiz et al.  confirmed that the degree of magnetic alignment \nusing this preparation method could be tuned by modifying the aspect ratio of the \ninitial powder s, reaching almost fully -aligned pellets  (Mr/Ms = 0.9) wi th densities \nabove 90%.[74] Higher alignment leads to higher squareness and thus greater  \nMr/Ms ratio and BH max, but it is accompanied by a reduction in Hc due to the \ninversely proportional relationship that exists between magnetization and coercive \nfield. Nonetheless, by reducing the degree of alignment they were able to obtain \nSrM magnets with a large Hc of 412  kA/m, proving the potential of SPS to overcom e \nthe reduction of Hc due t o excessive crystallite growth.  \n \n \nRecently, Vijayan et al.  reported the use of SPS not only for densification of \nferrite powders, but for the direct synthesis of aligned SrM magnets.[75–77] In th is \nstudy, SrM  is synthesized directly during the SPS process, using a precursor \npowder of antiferromagnetic six -line ferrihydrite (FeOOH) platelets mixed with \nSrCO 3. A low SPS temperature of ≈750 °C was sufficient to drive the reaction \nbetween the six -line phase and SrCO 3 to produce S rFe 12O19, while the  anisotropic \nshape of the hydrothermally synthesized six -line phase ensured the alignment of \nthe resulting SrM particles. Following this synthesis method, they were able to \nproduce a dense SrM magnet with a BH max of 33(4)  kJ/m3, a Mr/Ms of 0.93 and a \nHc of 247  kA/m.  \n3.3 Microwave sintering  \nIn the field of  hexafe rrite  research , microwaves (MWs) have mainly  been used \nfor synthesis purposes although a few sintering attempts using MWs have also \nbeen reported.[78–80] In all of them, powders are initially cold -pressed  followed by \na MW treatment , using frequencies in the GHz range , to sinter the piece.[35] In \n1999, Binner et al . used MWs to sinter ferrite nanoparticles reporting a limited \ngrain growth for non -agglomerated starting powders, although they failed to avoid \ncracks in the final sintered pieces.[78] Ten years later, Yang et al . succeed ed in \nmaking 97  % dense BaM magnets by MWs sin tering.[79] They also managed to \nprevent the app earance of -Fe2O3 in the final material, although they did not \nsucceed in preventing grain growth, in turn producing a rather low Hc (<50 kA/m).  \nRecently, Kanagesan et al.  tested fast heating rates (50  °C/min) and short dwell \ntimes (10 min) to MW sinter s ome Sr -ferrite powders synthesized by sol -gel.[80] \nThe MW sintering at 1150  °C yielded a 95  % SrM ceramic magnet with a fairly high  \nHc of 445  kA/m. H owever, the Ms value (50  Am2/kg) was not outstanding, \nalthough the sample seems relatively pure from powder diffraction data. The Mr \nvalue is also rather low ( ≈30 Am2/kg), which is expected from the poor alignment \nof the SrFe 12O19 particles observed in the  corresponding SEM micrograph (see Fig. \n2 in ref. [80]). \n3.4 Cold sintering  process  \nIn 2016 , Guo et al . reported a new sintering strategy  named  cold sintering \nprocess (CSP) , with which they were able to attain  high densification degrees for a \nwide range of inorganic materials at temperatures ≤ 200 °C, fabricating materials \nwith functional properties comparable to those made by conventional \nhigh -temperature approaches .[81] For CSP, the ceramic powders are mixe d with a \nsmall amount of aqueous solution which partially dissolves the particle edges and \nfacilitates diffusion and mass transport , aiding the sintering process, which in turn \noccurs at lower temperatures.  Sintering at low temperatures is very attractive in \ngeneral, as it reduces the energy demands, making the process greener and more \ncost-efficient. This  is especially interesting for M -ferrite  magnet s, as lower working \ntemperatures are expected to minimize grain growth.  The exact role of the solvent \ndurin g CSP is still under discussion, but it is believed to induce the formation of an \namorphous phase at the grain boundaries  which ease s sintering  and may also \nrestrict grain growth.[81] \nTo the best of our knowledge, there is only one research group which has tested \nCSP on hard ferrites. In 2021, Serrano et al.  patented a CSP method that allows \nfabricati on of  dense SrM magnets with magnetic properties in the order of  \nmedium -grade commercia l ferrite magnets.[43] In the CSP method devel oped by \nSerrano, SrM powders are mixed with glacial acetic acid and the wet mixture is  \n \nsubject ed to a uniaxial pressure ( ≈400 MPa) while heat ed at 190  °C.[82] After CSP, \nrelative densities of about 85% are obtained, which can be driven up to 92% by \nsubsequently treating the sintered piece at 110 0 °C for 2 h . This last sintering step  \nalso ha s a beneficial effect on the magnetic properties (see Figure 7A). In \nparticular, Ms at 5 T increases from 49.2 to 73.7  Am2/kg and Hc goes from 119 to \n223 kA/m. For the final product, a Mr/Ms ratio of 0.68 was obtained.  The density \nobtained by conventional sintering at 1100  °C for 4 h (no solvent, no hot \ncompression) was only 77% and the magnetic properties slightly inferior (see \nFigure 7B). Conventional sintering at 1300  °C yielded higher density (97%) but \nvery poor magnetic properties ( Hc = 48 kA/m, Mr/Ms = 0.33), due to  the dramatic \ngrain growth caused by the high temperature (see bottom FE -SEM micrograph \nfrom Figure 7B). \n \nFigure 7. Magnetic hysteresis and FE -SEM corresponding to SrFe 12O19 magnets fabricated by A) CSP  at \n190 °C, CSP followed by annealing  at 1100  °C, B) conventional  sintering at 1100 and 1300  °C. Reprinted \nfrom [82] with permission from Elsevier.  \nFurther investigations have been carried out using different organic solvents \n(i.e., oleic acid, oleylamine)  and widening the pressure and temperature ranges \nexplored (up to 270  °C and 670  MPa).[83] In all cases, the average grain size of the \nCSP ceramic magnet was about 1  µm (similar to the starting SrM powders) while \nsimilar conventional sintering processes typically yield average grain sizes above \n3 μm.[62] \nWith the aim of further improving the density and magnetic properties of CSP \nmagnets,  the addition of a small amount (10 wt.% ) of nanometric SrM to the \noriginal micrometric SrM powders was tested, moderately increasing  Hc \n(239 kA/m ) and Mr/Ms (0.73), although the density value continued at 92%.[84] \nThese numbers are competitive in the context of commercial SrM magnets. As an \nexample, the Hitachi’s NMF -7C series display values of Hc = 220–260 kA/m and \nMs = 68 Am2/kg).[40]  \n \n \n4. Summary  and perspective  \nIn the present chapter, the main sinterin g approaches applied to manufacturing \nhard ferrite ceramic magnets have been  reviewed. Table 2 summarizes the \nproperties of top SrFe 12O19 magnets fabricated by the various discussed sintering \nstrategies . Conventional sintering (CS) continues to be the quintessential \nindustrial method  for M -type hexaferrite PM fabrication , owing to its technical \nsimplicity and  the relatively  good result ing p roperties . However, this approach is \nhighly inefficient, as most of the energy employed is irreversibly dissipated as \nheat.[85] Therefore, the search  for more energy -efficient methods continues to be \nan active field of research.  \n \nTable 2. Magnetic parameters and relative density for top representatives of SrFe 12O19 magnets \nmanufactured following the different sintering approaches descr ibed in the present chapter, i.e., \nconventional sintering (CS),[61] spark plasma sintering (SPS),[18] cold sintering process  (CSP),[84] and \nmicrowave sinterin g (MWs).[80] \n Ms (Am2/kg)  Mr/M s Mr (Am2/kg)  Hc (kA/m)  BH max (kJ/m3) ρrel (%) \nCS ≈68 ≈1 68 328 37 ≥99%  \nSPS     73    0.93   225 36 >95%  \nCSP     73    0.73   239 –    92%  \nMWs     50 ≈0.62   445 –    95%  \n*Approximate values ( ≈) are graphically estimated from the article figures.  \nMultiple studies have demonstrated that spark plasma sintering (SPS) allow s \nproduction of PMs with  much higher  Mr/Ms ratios than CS. However,  the increase \nin texture comes at a cost of reduction in  Hc values , which therefore  still need to be \nimproved . As a result, magnets made using SPS end up displaying a similar \nperformance ( BH max) to the best CS examples. Additionally, technical challenges \nhinder the replacement of CS by  SPS in the industrial production of magnetic \nferrite s, since current SPS machines only allow producing relatively small pieces \nwith very few specific shapes (typically cylindrical pellets).  \nOnly a few attempts  have so far been made  to densify SrM by the relatively new \ncold sintering process (CSP) and theref ore, there is still much to explore and \noptimize. However, the CSP has already allowed  preparation of  hexaferrite \nmagnets with  magnetic properties comparable to medium -high grade commercial \nferrites , while lowering the sintering temperature . This reduces the energy \nconsumption by about 9 kWh /kg, which  leads to  energy savings of ≈29% compared \nto the sintering methods employed industrially  at present . \nThe results obtained by microwave sintering (MWs) have been very satisfactory \nin terms of both density and Hc, but the  resulting  Ms and Mr/Ms values are still \ninsufficient  to be commercially competitive . As with CSP, reports are scarce and \nfurther explorat ion is required.  \nSintering has undergone significant innovation over the last decade,[35] with the \nintroduction of a number of new sintering technologies, such as flash \nsintering,[86,87]  and various modified SPS methodologies, like flash SPS (FSPS),[88] \ndeformable punch SPS (DP -SPS),[89] or cool -SPS.[90] As a result, there are more \nalternatives available for sintering ferrites with enhanced magnetic characteristics \nand microstructure. To our knowledge, none of the  just mentioned  have yet been \nexamined on hard hexagonal ferrites, leaving lots of room for additional study in \nthis area.  \n   \n \nAcknowledgments  \nC.G. -M. acknowledges financial support from grant RYC2021 –031181 -I funded \nby MCIN/AEI/10.13039/501100011033 and by the “European U nion \nNextGenerationEU/PRTR”.  M.S. -M. acknowledges the financial support from the \nComunidad de Madrid, Spain, through an “Atracción de Talento Investigador” \nfellowship (2020 -T2/IND -20581). H.L.A acknowledges the financial support from \nThe Spanish Ministry o f Universities (Ministerio de Universidades) and the \nEuropean Union —NextGenerationEU through a Maria Zambrano —attraction of \ninternational talent fellowship grant . \nReferences  \n[1] Coey, J. M. D. Hard Magnetic Mate rials: A Perspective. IEEE Trans. \nMagn.  47, 4671 –4681 (2011).  \n[2] Jimenez -Villacorta, F. & Lewis, L. H. Advanced Permanent Magnetic \nMaterials. in Nanomagnetism  (ed. 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Control of \nbarium ferrite decomposition during spark plasma sintering: Towards \nnanostructured samples with anisotropic magnetic properties. J. Eur. \nCeram. Soc.  34, 337 –346 (2014).  \n[70]  Stingaciu, M., Topole, M., McGuiness, P. & Christensen, M. Magnetic \nproperties of ball -milled SrFe 12O19 particles consolidated by Spark -Plasma \nSintering. Sci. Rep.  5, 14112 (2015).  \n[71] Granados -Miralles, C. et al.  Unraveling structural and magnetic \ninformation during growth of nanocrystalline SrFe 12O19. J. Mater. Chem. \nC 4, 10903 –10913 (2016).  \n[72]  Eikeland, A. Z., Stingaciu, M., Mamakhel, A. H., Saura -Múzquiz, M. & \nChristensen, M. Enhancement of magnetic propertie s through \nmorphology control of SrFe 12O19 nanocrystallites. Sci. Rep.  8, 7325 (2018).  \n[73] Stingaciu, M., Eikeland, A. Z., Gjørup, F. H., Deledda, S. & Christensen, \nM. Optimization of magnetic properties in fast consolidated SrFe 12O19 \nnanocrystallites. RSC  Adv.  9, 12968 –12976 (2019).  \n[74]  Saura -Múzquiz, M. et al.  Elucidating the relationship between \nnanoparticle morphology, nuclear/magnetic texture and magnetic \nperformance of sintered SrFe 12 O 19 magnets. Nanoscale  12, 9481 –9494 \n(2020).   \n \n[75] Vijayan, H., K nudsen, C. G., Mørch, M. I. & Christensen, M. Ultrathin \nnanoplatelets of six -line ferrihydrite enhances the magnetic properties of \nhexaferrite. Mater. Chem. Front.  5, 3699 –3709 (2021).  \n[76]  Thomas -Hunt, J. et al.  Alignment of strontium hexaferrite, by cold  \ncompaction of anisotropic non -magnetically interacting crystallites. Dalt. \nTrans.  51, 3884 –3893 (2022).  \n[77] Vijayan, H. et al.  High -Performance Hexaferrite Ceramic Magnets Made \nfrom Nanoplatelets of Ferrihydrite by High -Temperature Calcination for \nPerman ent Magnet Applications. ACS Appl. Nano Mater.  6, 8156 –8167 \n(2023).  \n[78]  Binner, J. G. P., Caro, I. & Firkins, J. Microwave Sintering of Nanometer \nand Micrometer Ferrite Powders. J. Microw. Power Electromagn. Energy  \n34, 131 –136 (1999).  \n[79]  Yang, Q., Zhang, H., Liu, Y. & Wen, Q. Microstructure and magnetic \nproperties of microwa ve sintered M -type barium ferrite for application in \nLTCC devices. Mater. Lett.  63, 406 –408 (2009).  \n[80]  Kanagesan, S. et al.  Effect of microwave sintering on microstructural and \nmagnetic properties of strontium hexaferrite using sol –gel technique. J. \nMater. Sci. Mater. Electron.  24, 3881 –3884 (2013).  \n[81] Guo, J. et al.  Cold Sintering: A Paradigm Shift for Processing and \nIntegr ation of Ceramics. Angew. Chemie Int. Ed.  55, 11457 –11461 (2016).  \n[82]  Serrano, A. et al.  Hexaferrite -based permanent magnets with upper \nmagnetic properties by cold sintering process via a non -aqueous solvent. \nActa Mater.  219, 117262 (2021).  \n[83]  Serrano, A. et al.  Effect of organic solvent on the cold sintering processing \nof SrFe12O19 platelet -based permanent magnets. J. Eur. Ceram. Soc.  42, \n1014 –1022 (2022).  \n[84]  García -Martín, E. et al.  Dense strontium hexaferrite -based permanent \nmagnet composites assist ed by cold sintering process. J. Alloys Compd.  \n917, 165531 (2022).  \n[85]  Grasso, S. et al.  A review of cold sintering processes. Adv. Appl. Ceram.  \n119, 115 –143 (2020).  \n[86]  Yu, M., Grasso, S., Mckinnon, R., Saunders, T. & Reece, M. J. Review of \nflash sinter ing: materials, mechanisms and modelling. Adv. Appl. Ceram.  \n116, 24–60 (2017).  \n[87]  Biesuz, M. & Sglavo, V. M. Flash sintering of ceramics. J. Eur. Ceram. Soc.  \n39, 115 –143 (2019).  \n[88]  Grasso, S. et al.  Flash Spark Plasma Sintering (FSPS) of α and β SiC. J. \nAm. Ceram. Soc.  99, 1534 –1543 (2016).  \n[89]  Muche, D. N. F., Drazin, J. W., Mardinly, J., Dey, S. & Castro, R. H. R. \nColossal grain boundary strengthening in ultrafine nanocrystalline oxides. \nMater. Le tt. 186 , 298 –300 (2017).  \n[90]  Herisson de Beauvoir, T., Sangregorio, A., Cornu, I., Elissalde, C. & Josse, \nM. Cool -SPS: an opportunity for low temperature sintering of \nthermodynamically fragile materials. J. Mater. Chem. C  6, 2229 –2233 \n(2018).  \n "    },    {        "title": "1103.3666v1.Are_magnetite__Fe3O4__films_on_MgAl2O4_auxetic_.pdf",        "content": "arXiv:1103.3666v1  [cond-mat.mtrl-sci]  18 Mar 2011Are magnetite (Fe 3O4) films on MgAl 2O4auxetic?\nM. Ziese∗\nAbteilung Supraleitung und Magnetismus, Universit¨ at Lei pzig, D-04103 Leipzig, Germany\n(Dated: June 20, 2018)\nMagnetite (Fe 3O4) films were fabricated on MgAl 2O4(001) single crystal substrates by pulsed\nlaser deposition. In-plane and out-of-plane lattice const ants were determined by X-ray diffraction.\nThe apparent Poisson’s ratio was determined as the negative ratio of the out-of-plane to in-plane\nstrains. The results show that (i) the determination of Pois son’s ratio by this method is only reliable\nfor fully strained films and (ii) Poisson’s ratio ν100≃0.3 along the /angbracketleft100/angbracketrightdirection is positive for\nthis archetypal ferrite. Fe 3O4films grown on MgAl 2O4(001) are not auxetic.\nPACS numbers: 62.20.dj, 75.47.Lx, 75.70.Ak, 81.40.Jj\nAuxetic materials are substances with a negative Pois-\nson ratio, i.e. these expand in the transverse direction\nwhen stretched and contract when compressed [1]. This\nbehavior is often found in foams and organic material\nsuch as cartilage, where it is related to the microstruc-\nture and the appearance of so-called re-entrant cells [2].\nAlthoughcounterintuitive,auxeticbehaviordoesnotvio-\nlate thermodynamic principles, since the non-negativity\nof the elastic constants restricts the possible values for\nPoisson’s ratio νin isotropic materials to the range of\nvalues−1< ν <0.5. In anisotropic media the situa-\ntion is more complicated, since Poisson’s ratio depends\nin an intricate way on the values of the elastic moduli\nCijas well as on the directions of the longitudinal stress\nand transverse strain. For cubic materials this has been\nstudied in detail [3–9]. In case of Poisson’s ratio in the\n/angbracketleft100/angbracketrightdirection, which is of interest here, one obtains\nν100=C12\nC11+C12. (1)\nWith the stability criteria, see e.g. [9, 10], C11>0,\n−C11/2< C12< C11andC44>0, one again obtains\n−1< ν100<0.5 as for isotropic materials. In general,\nhowever, Poisson’s ratio can have smaller or larger val-\nues than these for general directions in the crystal [5–9],\nespecially for directions close to the /angbracketleft111/angbracketrightdirection [8].\nRecentlymolecularauxeticbehaviorinepitaxialcobalt\nferrite films grown on SrTiO 3(001) substrates as deter-\nmined by X-ray diffraction measurements of the in-plane\nand out-of-plane strains was reported [11]. This observa-\ntion was interpreted in terms of the atomic spinel struc-\nture, but not related to the specific microstructure of the\nfilms. Since variousferrites crystallize in the spinel struc-\nture, especiallythe parent compound magnetite (Fe 3O4),\nit is tempting to search for auxetic behavior in magnetite\nfilms.\nMagnetite films were grown by pulsed laser depo-\nsition from stoichiometric polycrystalline targets onto\nMgAl2O4(001) substrates. Deposition conditions were a\nsubstrate temperature Tsubbetween 370 and 530◦C and\nan oxygen partial pressure between 7 ×10−7mbar and\n1.5×10−5mbar. An excimer laser (Lambda Physik) op-erating at a wavelength of 248 nm (KrF), a repetition\nrate of 10 Hz and a fluence of about 1 .5 J/cm2was used\nfor the ablation. The thickness of the films was measured\nwith a Dektak surface profile measuring system. Struc-\ntural characterization of the films was made by X-ray\ndiffraction (XRD) with a Philips X’pert system using Cu\nKαradiation. Both θ−2θscans as well as reciprocal\nspace maps of the (226) reflections were recorded. Mag-\nnetic characterization of the films was made by SQUID\nmagnetometry (Quantum Design model MPMS-7). For\nfurther details on film growth and characterization see\n[12].\nTwo series of films with thicknesses between 25 nm\nand 240 nm (series I) as well as between 10 nm and\n160 nm (series II) were studied. The bulk lattice con-\nstant of the cubic phase of magnetite at room tempera-\nture isab= 0.8398 nm, the lattice constant of MgAl 2O4\nisas= 0.8084 nm, respectively. The lattice mismatch\nis−3.7% and, accordingly, the films are expected to be\nunder compressive in-plane strain. Out-of-plane lattice\nconstants a⊥were determined from θ−2θscans, in-plane\nlattice constants a/bardblfrom reciprocal space maps of the\n(226) reflection. Out-of-plane and in-plane strains were\ncalculated from\nǫ⊥=a⊥−ab\nab(2)\nǫ/bardbl=a/bardbl−ab\nab, (3)\nand the degree of relaxation was defined as\nR=a/bardbl−as\nab−as. (4)\nThe results of the X-ray analysis are presented in ta-\nble I and are illustrated in Fig. 1. Although for all films\nthe volume is smaller than the bulk volume, the in-plane\nstrain is negative and compressive, whereas the out-of-\nplane strain is positive and tensile. The thinnest films of\neach series are fully strained, but thicker films show con-\nsiderable strain relaxation approaching almost 100%, see\nFig. 1(b) (left scale). The films of series II were grown\nunder identical conditions for substrate temperature and2\nTABLE I. Results of the X-ray analysis. Parallel a/bardbland perpendicular a⊥lattice constants were determined from reciprocal\nspace maps and θ−2θscans, respectively. In-plane strain ǫ/bardbl, out-of-plane strain ǫ⊥, degree of relaxation R, volume change\n∆V/V= (a2\n/bardbla⊥−a3\nb)/a3\nband apparent Poisson ratio ν∗=−ǫ⊥/ǫ/bardblwere calculated from the lattice constants. Film thickness t\nand oxygen pressure pO2during deposition are also shown.\nt(nm)Ts(◦C)pO2(mbar)a/bardbl(nm)a⊥(nm)R(%)ǫ/bardbl(%)ǫ⊥(%)∆V\nV(%)ν∗\nSeries I\n25 530 3 ×10−6– – – – – – –\n35 500 3 ×10−60.8084 0.8494 0 -3.74 1.14 -6.29 0.31\n80 450 1 .5×10−50.8272 0.8431 0.60 -1.50 0.39 -2.60 0.26\n90 450 7 ×10−70.8231 0.8503 0.47 -1.99 1.25 -2.74 0.63\n155 450 3 ×10−60.8369 0.8452 0.91 -0.35 0.64 -0.05 1.86\n165 370 1 .5×10−50.8354 0.8444 0.86 -0.52 0.55 -0.50 1.05\n240 370 3 ×10−5– – - – – – –\nSeries II\n10 450 3 ×10−60.8084 0.8498 0 -3.74 1.19 -6.25 0.32\n20 450 3 ×10−60.8136 0.8475 0.17 -3.12 0.92 -5.28 0.29\n30 450 3 ×10−60.8310 0.8480 0.72 -1.05 0.98 -1.13 0.93\n80 450 3 ×10−60.8341 0.8462 0.82 -0.68 0.76 -0.60 1.12\n160 450 3 ×10−60.8316 0.8428 0.74 -0.98 0.36 -1.59 0.37\noxygen partial pressure, whereas both parameters were\nvaried for the films of series I. This might explain that\nthe thickness at which strain relaxation sets in, is signifi-\ncantlydifferent. However,thestrainstateiscontrolledby\nmany factors such as fluence, growth rate and substrate\nquality that are not at all times exactly reproduced.\nThe apparent Poisson ratio was defined as [11]\nν∗=−ǫ⊥\nǫ/bardbl. (5)\nThis is presented in table I and shown in Fig. 1(b) (right\nscale). The apparent Poisson ratio is positive for all film\nthicknesses studied here. For the fully strained films a\nconventional value around 0 .3 is obtained, whereas in re-\nlaxed films unphysically large values were found. This\nis reasonable, since the strain state in a relaxed film is\ncertainly inhomogeneous such that the definition of the\nPoisson ratio via averaged strain values is not valid. Lit-\neraturevalues forthe elastic moduli with C11,C12= 270,\n108 GPa [13], 268, 106 GPa [14], 260, 148 GPa [15], and\n312, 184 GPa [16] yield Poisson ratios ν100= 0.29, 0.28,\n0.36, and 0 .37. These values are in good agreement with\nthe value obtained here for the fully strained films.\nOne problem in the growth of oxide films lies in the\nstabilization of the correct oxygen stoichiometry. Oxy-\ngen stoichiometry was indirectly assessed in the films by\nthe measurement of the Verwey temperature, i.e. the\ntransition temperature of the transformation from the\nhigh temperature cubic into the low temperature mono-\nclinic phase. The Verwey temperature sensitively de-\npends on the oxygen stoichiometry [17]. Fig. 2 shows (a)\nthe normalized magnetic moment mand (b) its deriva-\ntivedm/dTof the fully strained and strain relaxed films\nfrom series II with thicknesses of 20 nm and 160 nm, re-\nspectively. The thick film showed a single transition at\nthe Verwey temperature of 114 K, whereas the thin film0 30 60 90 120 150 180 0.0 0.2 0.4 0.6 0.8 -4 -2 02\n0.0 0.5 1.0 1.5 2.0  R (%) \nFilm thickness (nm) (b) ε⊥\n  strain ε (%)  \n ε||\n(a) \n ν*\nFIG. 1. (Color online) (a) In-plane ǫ/bardbland out-of-plane ǫ⊥\nstrain as a function of film thickness. (b) Degree of relax-\nationR(red triangles, left axis) and apparent Poisson’s ratio\nν∗(black squares, right axis) as a function of film thickness.\nSolid symbols: series I, open symbols: series II.\nshowed a double transition (most clearly seen by the two\npeaks in the derivative dm/dT) with Verwey tempera-\ntures of 122.8 and 105.4 K. Except for the thickest film\nall films of series II showed a double transition, whereas\nall films from series I showeda single transition. The cor-3\n0.88 0.92 0.96 1.00 50 100 150 200 \nt (nm) = \n 160 \n 20 \n(b) \n  Temperature  T (K) \n m\n(a) \n024 \n dm/dT (10 -3 /K) \n0 50 100 150 200 250 105 110 115 120 \n series I \n series II \n(c) \n  TV (K) \nThickness (nm) \nFIG. 2. (Color online) (a) Magnetic moment of the 20 nm and\n160 nm magnetite films of series II normalized to the value at\n130 K.(b)Temperature derivativeofthenormalized magneti c\nmoment. The maxima indicate the Verwey temperature TV.\n(c) Verwey temperature TVvs. film thickness.\nresponding Verwey temperatures are shown in Fig. 2(c)\nas a function of thickness. There are two trends: with\ndecreasing thickness the Verwey transition temperature\nof most of the films increases to a value near 123 K which\nisclosetothebulk valueof125K;ifadoubletransitionis\npresent, the lower transition temperature decreases with\ndecreasing film thickness. The latter behavior was re-\nported before and was related to strain effects [18–21].\nThe present data indicate that – even if the films ap-\npear fully strained in X-raydiffractometry measurements\n– some films might be in an inhomogeneous strain state\nwith amagnetitelayerwith highVerweytemperaturead-\njacent to the substrate and another layer with lower Ver-\nwey temperature and probably different microstructure\non top of the first layer. Without further structural in-\nvestigations it is impossible to determine whether strain,\nmicrostructural effects or deoxygenation is the main fac-\ntor determining the value of the Verwey temperature and\nits dependence on the film thickness. Overall, however,\nthe Verwey temperature of the thin films is close to the\nbulk valueindicatinganearlyidealoxygenstoichiometry.\nThe results presented here show that magnetite filmsgrown on MgAl 2O4are not auxetic, at least not along\nthe/angbracketleft100/angbracketrightdirection. Poisson’s ratio ν∗= 0.3 deter-\nmined from the in-plane and out-of-plane strain ratio\nof a fully strained film is in good agreement with the\nvalue derived from measurements of the elastic mod-\nuli. Since both magnetite and cobalt ferrite crystallize\nin the inverse spinel structure, these data cast doubt on\nthe claim of molecular auxetic behavior in cobalt ferrite\nfilms grown on SrTiO 3(001) [11]. Indeed, using the ex-\nperimentally measured elastic moduli of cobalt ferrite,\nC11= 257 GPa, C12= 150 GPa [14], or the calculated\nvaluesC11= 240−282 GPa, C12= 137−168 GPa [22],\na positive Poisson’s ratio ν100= 0.37 is obtained that is\nincompatiblewith auxeticbehavior. Asimilarconclusion\nagainstauxeticbehaviourwasreachedin[23]forNiFe 2O4\nfilms on MgAl 2O4. The data presented here clearly show\nthat Poisson’s ratio can be reliably determined from X-\nray measurements on fully strained films.\nThis work was supported by the DFG within SFB 762\n“Functionality of Oxide Interfaces”. I thank H. C. Sem-\nmelhack for the X-ray diffractometry and R. H¨ ohne for\nthe magnetization measurements.\n∗ziese@physik.uni-leipzig.de\n[1] R. S. Lakes, Science 235, 1038 (1987).\n[2] R. Lakes, J. Materials Science 26, 2287 (1991).\n[3] J. Turley and G. Sines, J. Phys. D: Appl. Phys. 4, 264\n(1971).\n[4] D. J. Gunton and G. A. Saunders, Proc. R. Soc. 343, 68\n(1975).\n[5] S. P. Tokmakova, phys. stat. sol. (b) 242, 721 (2005).\n[6] T. C. T. Ting and D. M. Barnett, J. Appl. Mech. ASME.\n72, 929 (2005).\n[7] K. W. Wojciechowski, Comp. Methods Sci. Technol. 11,\n73 (2005).\n[8] A. N. Norris, Proc. R. Soc. A 462, 3385 (2006).\n[9] T. Paszkiewicz and S. Wolski, phys. stat. sol. (b) 244,\n966 (2007).\n[10] L. D. Landau and E. M. Lifshitz, Elastizit¨ atstheorie (Ver-\nlag Harri Deutsch, 1991).\n[11] M. Valant, A.-K. Axelsson, F. Aguesse, and N. M. Al-\nford, Adv. Func. Mater. 20, 644 (2010).\n[12] A. Bollero, M. Ziese, R. H¨ ohne, H. C. Semmelhack,\nU. K¨ ohler, A. Setzer, and P. Esquinazi, J. Magn. Magn.\nMater.285, 279 (2005).\n[13] P. K. Baltzer, Phys. Rev. 108, 580 (1957).\n[14] Z. Li, E. S. Fisher, J. Z. Liu, and M. V. Nevitt, J. Mater\nScience26, 2621 (1991).\n[15] H. J. Reichmann and S. D. Jacobsen, Am. Mineralogist\n89, 1061 (2004).\n[16] D. Chicot, F. Roudet, A. Zaoui, G. Louis, and V. Lep-\ningle, Mater. Chem. Phys. 119, 75 (2010).\n[17] Z. Kakol and J. M. Honig, Phys. Rev. B 40, 9090 (1989).\n[18] R. A. Lindley, S. P. Sena, H. J. Blythe, and G. Gehring,\nJ. Phys. IV France 7, C1 (1997).\n[19] S. P. Sena, R. A. Lindley, H. J. Blythe, C. Sauer, M. Al-\nKafarji, and G. A. Gehring, J. Magn. Magn. Mater. 176,4\n111 (1997).\n[20] S. K. Arora, R. G. S. Sofin, I. V. Shvets, and M. Luys-\nberg, J. Appl. Phys. 100, 073908 (2006).\n[21] J. Orna, P. A. Algarabel, L. Morell´ on, J. A. Pardo, J. M.\nde Teresa, R. L. Ant´ on, F. Bartolom´ e, L. M. Garc´ ıa,\nJ. Bartolom´ e, J. C. Cezar, and A. Wildes, Phys. Rev. B81, 144420 (2010).\n[22] D. Fritsch and C. Ederer, Phys. Rev. B 82, 104117\n(2010).\n[23] M. N. Iliev, D. Mazumdar, J. X. Ma, A. Gupta,\nF. Rigato, and J. Fontcuberta, Phys. Rev. B 83, 014108\n(2011)."    },    {        "title": "1005.3169v1.Micromagnetic_simulations_of_spinel_ferrite_particles.pdf",        "content": "arXiv:1005.3169v1  [cond-mat.mtrl-sci]  18 May 2010Micromagnetic simulations of spinel ferrite particles\nChristine C. Dantas and Adriana M. Gama\nDivis˜ ao de Materiais (AMR), Instituto de Aeron´ autica e Es pa¸ co (IAE), Departamento\nde Ciˆ encia e Tecnologia Aeroespacial (DCTA), Brazil\nAbstract\nThis paper presents the results of simulations of the magnetization fieldac\nresponse (at 2 to 12 GHz) of various submicron ferrite particles (c ylindrical\ndots). The ferrites in the present simulations have the spinel stru cture, ex-\npressed here by M 1−nZnnFe2O4(where M stands for a divalent metal), and\nthe parameters chosen were the following: (a) for n= 0: M = {Fe, Mn,\nCo, Ni, Mg, Cu }; (b) for n= 0.1: M ={Fe, Mg}(mixed ferrites). These\nruns represent full 3D micromagnetic (one-particle) ferrite simula tions. We\nfind evidences of confined spin waves in all simulations, as well as a com plex\nbehavior nearby the main resonance peak in the case of the M = {Mg, Cu\n}ferrites. A comparison of the n= 0 andn= 0.1 cases for fixed M reveals a\nsignificant change in the spectra in M = Mg ferrites, but only a minor ch ange\nin the M = Fe case. An additional larger scale simulation of a 3 by 3 partic le\narray was performed using similar conditions of the Fe 3O4(magnetite; n= 0,\nM = Fe) one-particle simulation. We find that the main resonance peak of\nthe Fe 3O4one-particle simulation is disfigured in the corresponding 3 by 3\nparticle simulation, indicating the extent to which dipolar interactions are\nable to affect the main resonance peak in that magnetic compound.\nKeywords:\nPACS:75.78.Cd, 75.30.Ds, 75.47.Lx, 75.75.Jn, 76.50.+g\nEmail address: ccdantas@iae.cta.br; adriana-gama@uol.com.br (Christine C.\nDantas and Adriana M. Gama)\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials May 30, 20181. Introduction\nFerrites(ferromagneticoxides) present convenient dielectric an dmagnetic\nproperties for microwave and millimeter-wave applications, consider ing their\nrelatively large magnetic losses and resistivities [1]. It is well known th at\nseveral physical properties at sub-micron scales, such as the siz e and shape\nof the particles in the system, their composition and concentration , including\ntheir interactions, are important factors that shape the charac teristics of the\nmagnetic material in a sensitive manner [2, 3, 4]. The main interactions\namong these particles are the dipolar (or long range interactions) a nd the\nspin exchange interactions. The interplay between these interact ions often\nleadtonovelandcomplexmagneticphenomena. Therefore, inorde rtodesign\nmaterials appropriate to specific applications, a thorough underst anding of\nthese phenomena is needed.\nMicromagnetism addresses the study of magnetism at sub-micron s cales\nin the continuum approximation, and its main theoretical equation is t he\nso-called Landau-Lifshitz-Gilbert equation (LLG) [5, 6, 7, 8]. It des cribes\nthe magnetization vector field dynamics (the local precessional mo tion of the\nmagnetizationvector field), including a phenomenological damping te rm, un-\nder an “effective” magnetic field, representing various interaction s amongst\nthe spins. Due to the fact that this is highly nonlinear vector partial dif-\nferential equation, it is generally solved by numerical methods (ana lytical\nsolutions can only be found in very few cases [8, 9, 10]).\nDue to the advance of computer capabilities, micromagetic simulation s\nhave been carried out with increasing validity, elucidating several co mplex\nmagnetic phenomena, but still with many open questions [11]. In part icular,\nstudies of the dynamics of confined spin waves in patterned arrays of mag-\nnetic particles in thin films [12, 13, 14, 15, 16, 17] is of great interest and\nis the subject of the present investigation, in which ferrite particle s are the\nconstituent elements. Although the literature on micromagnetic sim ulations\nof ferromagnetic or permalloy particles is quite vast (see, e.g., Ref. [16] and\nreferences therein), possibly due to the fact that such a magnet ic material\nis able to support a reasonable range of magnetic structures (spe cially the\nvortex structure, relevant to magnetic recording systems), th e literature spe-\ncific on ferrimagnetic or ferrite particle simulations is still somewhat s carce.\nAn inspection of the OOMMF citation list on June 2009 [18] revealed mor e\nthan 750 papers that have used that simulator, in which only a few of them\nfocused on ferrimagnetic particles/films (e.g., Refs. [19, 20, 21]) or bilayers\n2(e.g., Ref. [22]). Zero-field absorption spectra of magnetite cubic p articles\nhave been reported in Ref. [23]. This motivates our project to sys temat-\nically investigate the magnetization field acresponse of various magnetic\ncompounds (apart from permalloy) according to several physical properties,\nsuch as size and shape of the particles, their composition and conce ntration,\ninter-particle interactions, etc.\nThe main purpose of the present work is to study the absorption sp ectra\nand magnetization dynamics of full 3D micromagnetic simulations repr esent-\ning submicron spinel ferrite (e.g., [1], [24] and references therein) p articles\n(cylindrical dots). Several of these ferrites, already studied fo r many decades,\nare now being explored in recent advances in nanotechnology, spec ially in\nspintronics (e.g., Ref. [25] and references therein). It is well know n that\nthe saturation magnetization of several ferrites can be increase d by a proper\ncombinationwiththenon-magneticzincferrite. Wehaveattempted torepro-\nduce qualitatively the effect of an addition of zinc content, in order t o have a\npicture of its possible contributions to the resulting spectrum. In t hat case,\nwe have focused on a small addition of zinc, which lies in the linear part o f\nthe the relation between saturation magnetization and zinc conten t [1]. This\nfirst exploration is intended as a basis for a future systematic nume rical work\nexploring several material properties of ferrites of various type s.\nAnother relevant analysis in the present work resulted from the pe rfor-\nmance of an additional larger scale simulation consisting of a 3 by 3 par ticle\narray, in a similar fashion to our previous work with permalloy particles [26].\nIt was performed using analogous conditions of the Fe 3O4one-particle sim-\nulation. That larger simulation was performed with the aim of indicating\nthe extent to which dipolar interactions are able to affect the spect rum char-\nacteristics in a ferrite patterned film, that is, one formed of closely spaced\ndots.\nThis paper is organized as follows. A summary of the time domain mi-\ncromagnetic simulation setups are given in Sec. II. In Sec. III, we d escribe\nthe absorption spectra of the simulations and the equilibrium magnet ization\nfields and discuss the results, concluding in Sec. IV.\n32. Materials and Methods\n2.1. The Fundamental Equation of Micromagnetism and Spin Wa ve Phe-\nnomena\nThe Landau-Lifshitz-Gilbert equation is a vector partial differentia l equa-\ntion for the magnetization vector /vectorM, defined as the sum of Nindividual\nmagnetic moments /vector µj(j= 1,...,N), specified in a elementary volume dV\nat a position vector /vector rwithin a magnetic system (particle). Being a contin-\nuum limit expression, it assumes that the direction of /vectorMvaries continuously\nwith position [7]. The LLG equation describes the movement of the mag -\nnetization field /vectorM(/vector r,t) under the action of a external magnetic field ( /vectorHext)\nas precession movement of /vectorMaround an effective magnetic field ( /vectorHeff), de-\nfined as /vectorHeff≡ −µ−1\n0∂Eeff\n∂/vectorM. It is assumed that Eeffembeds the energy\nof an effective magnetic field, which is in turn generally expressed by t he\nsum of four fields composing spin interactions of distinct origins, nam ely:\nEeff=Eexch+Eanis+Emag+EZee(respectively: the exchange energy, the\nanisotropyenergy, themagnetostaticordipolarenergyandtheZ eemanorex-\nternal magnetic field energy). The resulting equilibrium state of tha t system\nminimizes the total energy. Other physical parameters of the LLG equa-\ntion are: the saturation magnetization, Ms(determined by the temperature,\nhere fixed throughout), the gyromagnetic ratio, γ, and a phenomenological\n(Gilbert) damping constant, α. The resulting dynamics is that in which the\nmagnetization vector precesses around the /vectorHefffield, loosing energy accord-\ning to the damping term, eventually leading to an alignment of /vectorMwith/vectorHeff.\nThe LLG equation is therefore written as:\nd/vectorM(/vector r,t)\ndt=−γ/vectorM(/vector r,t)×/vectorHeff−γα\nMs/vectorM(/vector r,t)×/bracketleftBig\n/vectorM(/vector r,t)×/vectorHeff/bracketrightBig\n.(1)\nThe magnetization dynamics allows uniform and non-uniform (spatially\nvarying) precession movement within the system. An oscillating magn etic\nfieldHacat a frequency ω0, applied perpendicularly to the magnetization\nfield leads to a coupling of /vectorMandHac, in which the energy will be absorbed\nby the system from the acfield. The acfield couples to uniform (leading\nto main resonance peak) and to nonuniform (spin wave) modes[27, 28, 29].\nIn the latter case, one notices that exchange and dipolar interact ions may\ncontribute to the energy of these modes. According to the Kittel’s model\n4[30], additional resonances will be found at frequencies ωp=ω0+Dk2\np, where\nDis a function of the exchange interaction between adjacent spins, andkpis\nthe (quantized) wave vector corresponding to a given spin wave ex citation.\nIt has been noted (e.g., Ref. [31]) that the resonant peaks associa ted with\nthe exchange interactions are found at the left of the uniform res onance\npeak, and that dipolar interactions would be generally independent o f the\nsize of the system, leading to an interparticle dipolar coupling field (se e, e.g.,\nRefs. [12, 26]). At a reasonably high signal level, they may exist signifi cant\ncoupling between the uniform precession mode and spin wave modes, causing\nalterations on the main resonance line (e.g., broadening and lowering) . A\nreview of confined spin waves can be found in Demokritov et al. [32].\n2.2. Simulations Setup\nThe present work follows the general procedure described in our p revious\nwork [26], where a similar analysis of permalloy cylindrical dots has been\nperformed, based in the method outlined by Jung et al. [13]. We have\nused the freely available integrator OOMMF (Object Oriented Microm ag-\nnetic Framework)[18] in order to numerically integrate the LLG equa tion\nand evolve the magnetization field of the ferrite particles. The part icles were\ncircular dots with finite thickness, that is, cylindrical dots of 0 .5µm of di-\nameter and 85 nm of thickness. The chosen spinel ferrites for the present\nsimulations (M 1−nZnnFe2O4; M, a divalent metal), were the following: (a)\nforn= 0: M = {Fe, Mn, Co, Ni, Mg, Cu }; (b) for n= 0.1: M ={Fe,\nMg}. Our simulations were performed in a method suitable for a qualitative\ncomparison with Brillouin light scattering spectroscopy measuremen ts [33],\nnoting that formally the strength of the Brillouin cross-section diffe rs quan-\ntitatively from the amplitude of the absorption spectrum. We descr ibe the\ndetails of the simulations as follows.\nIn order to study the absorption as a function of frequency, an e xter-\nnal magnetic field in the plane of the particle was applied, formed by tw o\ncomponents: a static ( dc) magnetic field ( Bdc≡µ0Hdc) of 100 mT in the y\ndirection, andavarying ( ac)magneticfield( Bac≡µ0Hac)ofsmall amplitude\n(1 mT) in the xdirection:\nBac= (1−e−λt)Bac,0cos(ωt), (2)\nwith the acfield frequency ( f=ω/(2π)) ranging from 2 to 12 GHz, in steps\nof 0.2 GHz (that is, 51 different OOMMF frequency runs were performed for\n5each ferrite simulation). We have discretized the time domain of the a pplied\nBacfieldatintervalsof0 .005ns, whichwereusedasinputsinthe“fieldrange”\nrecord of OOMMF (stepped linearly by the simulator). The simulations\nwere run up to 5 ns, resulting in 1000 outputs (dumps) for each of t hese 51\nfrequency runs. An additional simulation involving a 3 by 3 array of pa rticles\nwas performed using the same parameters and conditions of the Fe 3O4one-\nparticle ferrite simulation, but running a smaller set of selected freq uencies\naround the resonance peak, due to the high computational deman d of this\nsimulation.\nWelistinTable1themainglobalparametersoftheOOMMF, whichwere\nfixed for all sets of simulations. These global parameters were also adopted\nfor the larger scale simulation (the 3 ×3 array). Note that the exchange\nstiffnesshasafixedvalueinallsimulations(oftheorder ∼10−11J/m). Inthe\nnext section we address in more detail this parameter in context of exchange\nlength effects. Table 2 lists particular data of the simulations, specifi cally the\nvalue of the saturation magnetization and anisotropy constant ad opted for\neach ferrite particle. We have extracted the data from Fig. 4.9 of R ef. [1],\nwhich presents the experimental values of the saturation magnet ization of\nmixed ferrites (in Bohr magnetons) according to the Zinc content ( nvalue).\nData was also extracted from Refs. [22] and [34] (c.f. Table A.1). The\nsimulations were executed on a 3 GHz Intel Pentium PC running Kurum in\nLinux, taken an average of ∼28 hours of computation for each set of one-\nparticle simulations, whereas the3 ×3 array simulation tookabouttwo weeks\nto be run.\n2.3. Exchange stiffness considerations\nIn numerical micromagnetism, it is important to observe the restric tion\nthat, in order to obtain accurate results, the value of the comput ational cell\nsize should not exceed the exchange length (see, e.g. Ref. [10]), d efined as\nlex=/radicalbig\n2A/(µ0M2\ns), whereAis exchange stiffness of the material. Notice the\nstronger dependence of lextoMsthan toA(e.g., a 10% smaller value for A\nwithMsfixed implies a ∼5% decrease in the resulting lex, whereas the same\n10% reduction in MswithAfixed implies a ∼23% increase in lex). The cell\nsize here adopted is 5 nm (enough for a meshing of O(100) magnetization\ncells along the particle’s diameter), and therefore materials with lexabove\nthat limit are in accordance with the present numerical requirement s.\nBased on magnetoresistive measurements, Smith et al. [35] obtaine d the\nexchange stiffness forPermalloy within10%error( ANiFe= 1.05×10−11J/m).\n6This method improves on previous estimates based on spin-wave fer romag-\nnetic resonance spectra, which can give discrepancies of a factor of 2 around\nthe value ANiFe∼1×10−11J/m. From measurements of the domain width,\nLivingston [36] found for Fe-Nd-B magnets AFe−Nd−B= 1.1×10−11J/m,\nbut this method depends on the measurement of the anisotropy co nstantK1\n(the quote value for AFe−Nd−Bwas increased by a ∼1.5 factor given a new\nmeasurement of K1, as mentioned in a note added in proof). Estimates for A\ncan also be obtained by a formula that includes the exchange integra lJand\nother parameters (see, e.g., [37] for an estimate of the exchange stiffness of a\nnanocrystalline Ni0.5Zn0.5ferrite, although uncertainties are not quoted).\nGiven the experimental uncertainties, we have decided to adopt an ad-\nequate order of magnitude value for A(such that the resulting lexis above\nthe computational cell of 5 nm), namely A∼10−11J/m. In particular, we\nhave fixed A= 1.2×10−11J/m, as usually quoted for magnetite (Ref. [19],\nbut see e.g. Ref. [23] for a quoted value larger by a factor ∼3). Notice that,\nby the use of a global relation for ferrimagnetic polycrystals [38], na mely,\nA(T) = (kTc/a)(1−T/Tc)2, one can alternatively infer the A(T) value for\nthe ferrites from the lattice constant ( a) and Curie temperature ( Tc) with\n∼<20% precision. Using this relation and data collected from literature\n(see Refs. [39], [40], [41], and [42]), we find that the resulting values of\nA(T) at room temperature for all ferrites in the present work are well within\n∼20%of the presently adoptedvalue of A= 1.2×10−11J/m; in other words,\nreasonably within current experimental uncertainties.\nWith the adopted value for A, we see that lex∼8.2 nm for the (M = Fe,\nn = 0.1) ferrite (the highest Msof the set) and lex∼33.8 nm for the (M =\nCu, n = 0) ferrite (the lowest Msof the set). Hence the latter range for lex\nis above the cell size, in accordance with the numerical requirement s. Notice\nthat, for the most critical cases (namely, M = Fe ferrites, with the highest\nMsvalues), one could ask how much an error in the corresponding Avalue is\nallowed for in order to still be in accordance with the numerical requir ements\n(considering that the Msvalue is correct). It results that a factor of ∼1/3\n(namely, a decrease in ∼33% inA) would result in lex<∼5 nm for the M =\nFe ferrites. We conclude that the adopted value for Ais acceptable for the\npresent simulations.\nHowever, it is important to understand how sensitive our simulations are\nto variations in Ato the point that the final results could change appreciably.\n7We will investigate the effect of lower values of A1in supplementary simula-\ntions to be discussed in the next section along with the main simulations\n2.4. Calculation of the Spectra\nFor the spectra computation, we have also followed the general pr ocedure\noutlined in Ref. [13] (see also [26]). In order to obtain the absorptio n spectra\nof the ferrite particles, we proceeded as follows, for each simulatio n. The first\n1 ns of the averaged magnetization vector in the xdirection, /an}bracketle{t/vectorM/an}bracketri}htx(t≤1 ns),\nhas been excluded, and the Fourier transform of the remaining time domain\ndata,/an}bracketle{t/vectorM/an}bracketri}htx(1< t≤5 ns), has been calculated. The amplitude of the\nmaximum Fourier peak at each frequency run was then selected for providing\nthe absorpion at the respective frequency, hence building up the s pectrum of\neach ferrite particle.\n3. Results\n3.1. Main Micromagnetic Simulations\nIn this section we outline the main results found in the present work. A\nmore detailed (qualitative) analysis will be offered in the next section.\nFig. 1 shows the resulting spline fit absorption spectra of the ferrit e one-\nparticle simulations for n= 0. It is observed that the ferrites with higher\nsaturation magnetization ( Ms) have their main resonance peaks at higher\nfrequencies. It is also possible to notice in each spectrum the prese nce of\nsmall amplitude absorption peaks at the left of the main resonance p eak;\nthese small peaks appear to increase in amplitude for the ferrites w ith lower\nMs. It is already pronounced in the M = Ni case, and results in a “double-\nlike” peak in the case of the M = {Mg, Cu}ferrites (which have very similar\nMs). As already mentioned, resonant peaks associated with the exch ange\ninteractions are found at the left of the uniform (main) resonance peak (e.g.,\nRef. [31]). Hence, the small peaks found in the spectra are probab ly confined\nspin-wave excitations of the magnetization field of the particles.\nFig. 2showsthespectrumofthesimulationrepresentingaFe (1−n)ZnnFe2O4\n(n= 0.1) mixed ferrite compared with that of n= 0 (Fe 3O4). It can be seen\n1Clearly, for a fixed Ms, lower values of Aare of more interest than higher values, since\nthe latter are “safe” with respect to the numerical requirements forlex, according to our\nconsiderations.\n8that the main peak in the mixed ferrite is slightly moved to higher frequ en-\ncies. Fig. 3 shows the comparative result for the Mg(1−n)ZnnFe2O4mixed\nferrite. This is a more complex case. Clearly, peak A ( n= 0 case), which\nis quite pronounced, lowers in amplitude significantly in the correspon ding\nn= 0.1 case (peak C), whereas peak B appears to be modified into peak D,\nwhich is at a higher amplitude and frequency.\nIn Figs. 4 and 5 (top panel), the simulation output “snapshots” of t he\nmagnetization vector field related to the peaks of interest of the M ={Cu,\nMg}ferrites are shown. At each peak, the snapshots were chosen (r estricted\ntot >3 ns) at two points of the acfield cycle ( ωt=π/2 for the snapshot at\nright, and ωt= 3π/2 for the one at left of those figures). The varying pixel\ntonalities of the particle’s snapshots correspond to different value s of thex\ncomponent of the magnetization field, which was subsampled to show an\narrow for the average of 9 vectors per cell element. Both simulatio ns show\nsimilar results due a close Msvalue for these ferrites. It is clear that the\npronouncedpeakattheleftofthespectra(inboththesecases) isofadifferent\nnature from the one at right: in the former peak, the magnetizatio n field in\nthe center of the particle is mostly static and aligned with the direction of the\ndcfield; the response of themagnetization field is limited tosmall oscillatio ns\nof the (nonuniform) magnetization near the edges . The corresponding peak\nat right present instead a quasi-uniform behaviour .\nIn both panels of Fig. 5, which refer to a comparison between the n= 0\nandn= 0.1 cases for fixed M = Mg, one is able to contrast the snapshots\nof each of the peaks of interest as a function of the zinc content a ddition.\nClearly, the peaks at left (A, C) show a different magnetization field b e-\nhaviour than the right ones (B, D), as already pointed out in previou sly.\nSince the zinc addition is here implemented in a qualitative manner, this\nresult must be interpreted as a general trend.\nFig. 6 shows the snapshots of the magnetization vector field at the main\nresonance peak of each of the one-particle ferrite simulations of F ig. 1 (n=\n0). In the case of the M = {Mg,Cu}ferrites, the snapshots were selected\nfrom the peaks at the rightof their spectra (see Figs. 4 and 5 for peaks\nB and D), given that they show a similar nature with respect to the ma in\nresonance peaks of the other ferrites, as already pointed out. I t is observed\nasystematic change in the magnetization field response as a fu nction of the\nsaturation magnetization of the ferrite (which increases to the left in Fig. 6).\nThis systematic change is revealed in terms of a higher overall amplitu de of\nresponse for higher Msas well as an increasingly important presence of small\n9oscillations about a nonuniform static magnetization distribution. Fig . 7 is\na similar figure to the previous one, but presents instead the snaps hots of\nthe simulations with addition of zinc content ( n= 0.1 runs) as compared to\ntheirn= 0 counterparts. We will discuss these results in more detail in the\nnext section.\nFig. 8 shows the absorption spectrum of the Fe 3O43×3 particles (ar-\nray) simulation superposed to the corresponding single particle simu lation.\nClearly, the main resonance peak of the Fe 3O4one-particle simulation is dis-\nfigured in the corresponding 3 by 3 particle simulation. There are now 3\nresolved peaks approximately within the region of the single main peak of\nthe one-particle simulation counterpart, and these peaks increas e in ampli-\ntude for higher frequencies, but never reach the same amplitude o f the main\npeak of one-particle run. This figure should provide some indication o f the\nextent to which dipolar interactions are able to affect the main reson ance\npeak in that magnetic compound. This result is compatible with that of Ref.\n[43] for cylindrical Permalloy 3 ×3 dot arrays, in which the fundamental\nmode is found to be split into three modes.\nFig. 9 shows the corresponding snapshots of the magnetization ve ctor\nfield at each of the three peaks identified in the previous figure, con cerning\nthe 3 by 3 array simulation. Snapshots number 3 (panel at the right of\nthat figure) should be compared with that of Fig. 6, M = Fe one-part icle\nsimulation. There are several issues to be observed in the 3 by 3 arr ay\nsimulation, which will be addressed in more detail in the next section.\n3.2. Supplementary Micromagnetic Simulations\nAs mentioned on Sec. 2.3, we report on additional simulations perfor med\nin order to evaluate the impact of smaller values of the exchange stiff ness\nconstant, A, on our results. As explained in that section, it is interesting to\nanalyse that impact for the well-known M = Fe ferrite (magnetite). I n other\nwords, we have artifically lowered the Avalue for that one-particle ferrite\nmodel by a factor 1 /3 (simulation labeled “S1”) and by 20% (“S2”), see Ta-\nble 3. Notice that the S1 run brings lex∼5 nm (cell size), and therefore is\nexpected to bring noticeable change in the results (all other param eters re-\nmained fixed). Indeed, as Fig. 10 shows, there is a decrease in the a mplitude\nof the main peak as Adecreases. Otherwise, the resonance frequency and\nother minor modes at the left of the main peak show little variation. Th is\nsuggeststhat, inadditiontothepreliminaryconsiderations already expressed\nin Sec. 2.3, our results are qualitatively robust.\n10In order to verify the sensitiveness of the appearance of the “th ree peaks”\nfoundinthe3 ×3particlessimulationwithrespecttoachangeinsomespecific\nparameter, wehaverunadditional3 ×3simulationswiththesameparameters\nof the original one, except for a change in some parameter of our c hoice. Due\nto the fact that these array simulations are computationally deman ding, so\nthat a fine-grain covering of the parameter space is prohibitive at t his time,\nwe have limited our analysis to a small set of additional simulations in ord er\nto infer possible trends. Also, we have limited the simulations to the 5 .0 -\n7.0 GHz frequency range, in steps of 0 .2 GHz. Table 3 lists the parameter\nchanged in these simulations (labelled “S3” to “S7”).\nFig. 11 shows the resulting spectra of the additional 3 ×3 simulations.\nThe “Reference” spectrum is thatresulted fromtheoriginal 3 ×3run, that is,\nthe same as shown in Fig. 8 (the spectrum with “three peaks”, as ind icated).\nWe have also presented a re-analysis of that original simulation by se lecting\nthe last 2 ns of the remaining time domain data, 3 < t≤5 ns, for the\ncalculation of the Fourier transform (instead of the 1 < t≤5 ns data; see\nSec. 2.4). This selects a clearer steady state condition. We see tha t (top\npanel of Fig. 11) the form of the spectrum is practically unchanged , except\nin amplitude, which is decreased.\nA lower value of A(“S3”) also produces a smaller amplitude spectrum,\nwith the overall form maintained (except perhaps for the first, sm aller peak\nat left), which is the same result as the one-particle cases (see Fig. 10). A\nhigher value for Ms(“S4”) results in significant distortion of the reference\nspectrum, namely: adecreaseinamplitudeofthepeaks, andthefir st, smaller\npeak at left is not seen in the range of frequencies simulated. An anis otropy\nconstant K1 set to zero (“S5”; Fig. 11, bottom panel) shows no significant\nchange in the spectrum. On the other hand, the spectrum resultin g from a\nlarger damping constant (factor of 10) misses entirely the three peaks . This\nis interesting in the light of our previous work, where the same dampin g\nconstant was used, and no splitting of the main peak was found for t he\nPermalloy 3 ×3 particles run (see discussions in [26]), although a splitting\nwas indicated in Ref. [43]. Our present analysis thus confirm that a lar ger\ndamping parameter possibly explains the difference in the previous re sults.\nFinally, the spectrum of the “S7” run (model B3 of [26]), where the d ots\n“touch” each other, shows a different spectrum as compared to t he reference\nsimulation (which in turn has 0 .122µmof interparticle spacing). However, a\nsplitting of the main resonance mode is also visible.\n114. Discussion\nThe elucidation of peculiar features in the absorption spectra of fe rrite\nparticles must take into account recent interpretations on the na ture and role\nof the spin-wave modes. Modes with nodal planes parallel to the mag neti-\nzation are associated with high frequency modes, whereas modes w ith nodal\nplanes perpendicular to the magnetization can exhibit frequencies lo wer and\nhigher than the quasi-uniform mode, and their presence mainly depe nds on\nthe number of nodes and the equilibrium between the dipolar and exch ange\ninteraction effects (see, e.g., Ref. [31]). In the present work, we a re inter-\nested in the overall qualitative magnetization field acresponse of various\nferrite cylindrical dots according to the micromagnetic numerical p redictions\nfor these systems, in order to have a basis for a more detailed subs equent\ninvestigation. For definiteness, we list here three possible collective acre-\nsponses of the magnetization field according to the following criteria [12]:\n•“Quasi-uniform” behavior ( QU): the motion of each arrow is approxi-\nmately the same to that of its neighbors, except for the regions ar ound\nthe edge of the particle;\n•“Spin-wave” behavior ( SW): the arrows exhibit small oscillations about\na nonuniform static magnetization distribution;\n•“Edge-like” behavior ( ED): the magnetization field in the center of the\nparticle is mostly static and aligned with the direction of the dcfield;\nthe response of the magnetization field is limited to small oscillations\nof the (nonuniform) magnetization near the edges – these modes m ay\nbe influenced by the dipolar field coming from another particle placed\nnearby.\nTheobservedcharacteristics oftheabsorptionspectraaslinked toavisual\ninspection of the magnetization fields at resonant peaks of interes t can be\nclassified under those criteria, a subject to which we address now.\nA general trend ( n= 0 cases) can be seen by comparing the spectra\nof Fig. 1 with the corresponding snapshots of Fig. 6. The nature of the\nresonance peaks based on the appearance of the snapshots can be inferred,\nwhich we list below:\n•A remarkable feature is that the main resonance peak in all these sim -\nulations seem to be of a similar nature and follow a systematic pattern ,\n12namely: an increase in amplitude response (in the central body of th e\nparticle) and the presence of small oscillations about a nonuniform\nstatic magnetization distribution (SW) – both effects as a function o f\na larger saturation magnetization. In the most extreme case, M = F e,\none can see clearly the presence of nonuniformity in the magnetic fie ld\noscillation.\n•The anisotropy constant K1appears to play a minor (but noticeable)\nrole to affect the above-mentioned trend. For example, let us comp are\nthe M = Co ferrite with its two “neighbours” (in terms of Msvalue),\nnamely: M = Mn and M = Ni (c.f. Fig. 6). If one focuses on the pixel\ntonalities (a measure of the magnetization amplitude), the M = Co\nferrite shows a smaller extent in tonality of the central region (whe rein\nthe amplitude of the magnetization field is larger) as compared with\nthose of the M = Mn (which has a close, but larger value of Ms) and\nM = Ni (lower Ms) ferrites. It would be expected from the above-\nmentioned trend (an increase in amplitude magnetization response in\nthe central body of the particle) that such an extent in tonality fo r the\nM= Co ferrite would be ofintermediate size (between theM = Mnand\nM = Ni ones). The fact that this is not observed points to a relatively\n“easier” alignment of the magnetization with the external field in the\nM = Co ferrite case. A more systematic study fixing Msand varying\nK1, however, was not performed, and more study is needed to confir m\nthese general trends.\nIt is specially interesting to observe the snapshots in the cases of t he\n“double-peaks” seen in the compounds M = {Cu, Mg}(Figs. 4 and 5, top\npanel), where the pronounced peak at right of the spectrum (in bo th cases)\npresents an “edge-like” behaviour ( ED). Such materials will probably show\ncomplex spectra in a properly manufactured patterned film, in which “edge-\nlike” modes may be significantly influenced by the dipolar field arising fro m\nanother particle placed nearby (depending on the interparticle spa cing and\npossibly other factors). We observe that these “edge-like” effec ts aresmaller\nfor higher saturation magnetizations . For instance, a comparison between\nthen= 0 and n= 0.1 case (M = Mg; Figs. 5 and 7 ) clearly shows this\neffect (note that the n= 0.1 ferrite has a larger value of Msthan that of\nn= 0, for a fixed M).\nA comparison between Figs. 2, 5 and 7 allow us to qualitatively infer how\npeaks of interest possibly morph from one to another as the zinc co ntent\n13is added in the ferrite particle. For the case M = Fe, the nonuniformity\n(SW), as expressed by the pixel tonality distribution, tends to increase in\namplitude for n= 0.1 (or larger Ms), specially in the central body of the\nparticle. Similarly, for the M = Mg case, “edge-like” effects ( ED)decrease\nin amplitude and extent for n= 0.1 (compare snapshots related to peaks A\nand C of Fig. 7). For the quasi-uniform modes (peaks B and D of Fig. 7 ) a\nmore uniform magnetization field oscillation ( QU) in the central the body of\nthe particle is found for the n= 0.1 case. These effects qualitatively explain\nthe observed transformation of the corresponding spectra.\nA note is necessary at this point. The n= 0.1 simulations were intended\nas preliminar test-cases for a more systematic subsequent work. In Ref. [44],\nfor example, an experimental study on the overall energy absorp tion behav-\nior of Mn-Zn mixed ferrites in the frequency range of 8 to 12 GHz for various\nchemical compositions is presented. A clear understanding of the b ehaviour\nshown in that work from the point of view of the magnetic absorption dy-\nnamics would be desirable. It would be interesting to analyse and comp are\nthe corresponding numerical predictions with experimental result s in order\nto allow for predictions and guidance for specific applications2. The present\ntest-cases clearly show that such an analysis is feasible.\nTheresultsfortheabsorptionspectrumoftheFe 3O43×3particles(array)\nsimulation is very interesting and follows a previous investigation that we\nhave performed on similar permalloy arrays [26]. As already mentioned , the\nmain resonance peak of the Fe 3O4one-particle simulation is disfigured in\nthe corresponding 3 by 3 particle simulation, resulting in 3 resolved pe aks\napproximately within the region of the original single main peak. Such a\nfeatureis notobserved in the permalloy ( Ms= 8.0×105[A/m]) 3 by 3 array\nsimulation of our previous work (c.f. Fig. 4, the simulation A0 – one-pa rticle\nrun – compared with A1 – 3 by 3 array run – of that paper, Ref. [26]).\nA reasonable explanation for not finding the splitting of the main mode in\nour previous work is due to a larger damping factor used in that work , as\nsuggested here by our supplementary simulations analysis (Sect. 3 .2). A\nsimilar splitting was found and discussed in Ref. [43]. There are, howe ver,\nsimilarities between the present and previous results, which we will ad dress\n2Although the literature on ferrites is extremely vast, it has proven somewhat difficult\nin the course of this work to find adequate papers to which the pres ent results could be\ndirectly compared.\n14now.\n•The new peaks in the 3 by 3 array simulation never reach the same\namplitude of the main peak of one-particle run, and this is also true in\nthe permalloy simulation of our previous work, although only one peak\nhad been observed in that case. As already noted in that paper, th e\ndecrease in amplitude of the emerging response is probably due to th e\naveraging out of the magnetization field over the array particles, w hich\nshow several modes not present in the one-particle case.\n•A very interesting effect (c.f. Fig. 9), which we attribute to dipolar\ninteractions, which was found in our previous work and is confirmed\nhere, is the following. Representing the 3 by 3 array as a matrix, A,\nonecan observe that themagnetization field of elements in the A1,jand\nA3,jrows (j= 1,2,3) evolve in symmetric opposition to each other.\nThe central row A2,jappears to have its magnetization field evolving\nindependently of the other two.\n•The central dot in the 3 by 3 simulation has a particularly unique\nbehaviour compared to the others in the array. Comparing snapsh ot\nnumber 3 (panel at the right of Fig. 9) with that of Fig. 6, which\nrepresents the same M = Fe ferrite dot, but completely isolated, we\nnotice that the presence of small oscillations about a nonuniform st atic\nmagnetization distribution ( SW), seen in the latter simulation, is only\npresent in that central dot of the 3 by 3 array. In other words, this\neffect is apparently attenuated in the other dots of the array, wh ich do\ngain instead a more “edge-like” behaviour ( ED).\n•Several dots in the snapshots corresponding to peaks 1 and 2 pre sent\nEDbehaviour, and this effect appears to be more intense in the dots\nof snapshot number 2 than of number 1, which is consistent with the\ncorresponding amplitude of the peaks in the spectrum (c.f. Fig. 8).\nThe effects outlined above must be of a dipolar nature, but the exac t\nprediction of the resulting behavior of the magnetization field (anti- ) “syn-\nchronism” as a function of the array symmetry or mutual dispositio n of the\nparticles still needs elucidation (see Ref. [26] for several examples and dis-\ncussion).\n155. Conclusion\nIn the present paper we have reported on a set of 3D micromagnet ic\nsimulations of cylindrical dots supposed here to represent submicr on spinel\nferrite particles excited by an external periodic magnetic field. We h ave anal-\nysed the resulting absorption spectra and the magnetization field b ehavior\nat modes of interest, limited to the timespan covered in the simulation s (5\nns). We have identified the nature of confined spin waves and small o scilla-\ntions of the (nonuniform) magnetization in the absoption spectra t hrough an\ninspection of the magnetization field at extreme amplitudes of the cy cle. A\nqualitative analysis of the magnetization field behaviour for all simulat ions\nwas given.\nThe absorption spectra of ferrite particles may present complex b ehavior\nnearby the main resonance peaks, specially in the cases of M = {Mg, Cu}\nferrites. It is inferred that a significant change in the absorption s pectrum\ncan be achieved as the zinc content is added in M = Mg ferrites, but th is is\nunlikely in the M = Fe case, at least for a change from n= 0 ton= 0.1.\nA study of a larger scale simulation of a 3 by 3 particle array with similar\nconditions of the M = Fe one-particle one shows that the resonance peak of\nthe one-particle ferrite simulation is replaced by a “triple” peak or ot herwise\ndisfigured in agreement with Ref. [43]. We confirm our previous result that\nthereisindeed amagnetizationfield(anti-) “synchronism” effect int hearray.\nThis study permitted us to infer the extent which dipolar interaction s are\nable to affect the main resonance peaks in such ferrite particles.\nWe aim to perform additional numerical studies to analyse the role of\nconfined spin oscillations in various ferrimagnetic particles and array s with\ndifferent physical conditions in a future work.\n6. Acknowledgments\nWe thank the referee for useful suggestions. We would also like to t hank\nthe attention and technical support of Dr. Michael J. Donahue in t he initial\nphases of this project. We also wish to acknowledge the support of Dr.\nMirabel C. Rezende and FINEP/Brazil.\n16Table 1: Main parameters set to the OOMMF simulator, fixed for all sim ulations in the\npresent work.\nSimulation Parameter/Option Parameter Value/Option\nExchange stiffness [J/m] 1 .2×10−11\nAnisotropy Type cubic\nFirst Anisotropy Direction (x,y,z) (1 1 1)\nSecond Anisotropy Direction (x,y,z) (1 0 0)\nDamping constant 0 .005\nGyromagnetic ratio [m/(A.s)] 2 .21×105\nParticle thickness [nm] 85 .0\nParticle diameter [ µm]10.5\nCell size [nm] 5 .0\nDemagnetization algorithm type magnetization constant in each cell\nSaturation magnetization [A/m] see Table 2\n1There is a small difference in the case of the 3 ×3 array simulation. Due to constraints\ninthe drawingofthe array(bitmap imageto beused asinput forthe s imulator), individual\nparticles turned out to have 0 .552µmof diameter each, with 0 .122µmof interparticle\nspacing, therefore fitting in an exact square of 1 .9µmby 1.9µm. It is necessary that the\ninput bitmap size is set as an integer multiple of the cell size.\n17Table 2: Particular parameter values for each ferrite simulation\nSimulation (M, n)Ms[×105A/m] K1 [ ×104J/m3] Note\n(Fe,0.0) 5 .00 −1.10 single particle\n(Fe,0.1) 5 .48 −1.10 single particle\n(Fe,0.0) 5 .00 −1.10 3 by 3 array of particles\n(Mn,0.0) 4 .14 −0.28 single particle\n(Co,0.0) 3 .98 27 .00 single particle\n(Ni,0.0) 2 .70 −0.69 single particle\n(Mg,0.0) 1 .39 −1.50 single particle\n(Mg,0.1) 2 .14 −1.50 single particle\n(Cu,0.0) 1 .35 −0.60 single particle\n18Table 3: Supplementary simulation data\nSimulation Parameter that has been modified Note\nS1 A= 0.40×10−11[J/m] single particle\nS2 A= 0.96×10−11[J/m] single particle\nS3 A= 0.40×10−11[J/m] 3 by 3 array of particles\nS4 Ms= 7.5×105[A/m] 3 by 3 array of particles\nS5 K1 = 0 [J/m3] 3 by 3 array of particles\nS6 damping const. = 0 .05 3 by 3 array of particles\nS7 Model B3 of Ref. [26] 3 by 3 array of particles\n19Figure 1: Absorption spectra of single ferrite particle simulations. C urves were arbitrarily\ndislocated for better comparison. Eachferrite spectrum is labelled by its divalent metal M\n(all cases here with n= 0), and organized in order of saturation magnetization, such tha t\nthe upper curve is from the highest Ms. The spectral curves were obtained from spline fits\nof the discrete simulation results (performed at frequencies from 2 to 12 GHz, sampled at\nintervals of 0 .2 GHz). Possible spin-wave excitations to the left of main resonance peaks\ncan be seen.\n20Figure2: The spectrumofthe simulationrepresentingaFe (1−n)ZnnFe2O4(n= 0.1)ferrite\ncompared with that of n= 0 (Fe 3O4).\n21Figure 3: Same of previous figure, but for the Mg(1−n)ZnnFe2O4ferrite. Labels A, B, C\nand D mark peaks of interest.\n22Figure 4: “Snapshots” of the magnetization vector field for the Cu Fe2O4ferrite, at two\nresonance peaks, as indicated. At each peak, the snapshots wer e chosen (restricted to\nt >3 ns) at two points of the acfield cycle ( ωt=π/2 for the snapshot at right, and\nωt= 3π/2 for the one at left). The varying pixel tonalities of the particle’s sn apshots\ncorrespond to different values of the xcomponent of the magnetization field (subsampled\nto show an arrow for the average of 9 vectors per cell element).\n23Figure 5: Same as the previous figure, but for the Mg(1−n)ZnnFe2O4ferrite.Top panel:\nn= 0.Bottom panel: n= 0.1.24Figure 6: “Snapshots” of the magnetization vector field for the fe rrites presented in Fig. 1,\ntaken at their main resonance peaks, labelled by its divalent metal M(all cases here with\nn= 0). Saturation magnetization increases to the left. In the case o f the M = {Mg,Cu}\nferrites, the snapshots were selected from the peaks at right of their spectra (see Figs. 4\nand 5, and the corresponding explanation in the text). As previous ly, the snapshots were\ntaken at two points of the acfield cycle ( ωt=π/2 for upper snapshot, and ωt= 3π/2 for\nthe lower one).\n25Figure7: Sameofpreviousfigure,butnowsnapshotsreferforth esimulationswithaddition\nof zinc content ( n= 0.1 runs) as compared to their n= 0 counterparts. Labels A, B, C\nand D refer to corresponding peaks in Figs. 3 or 5.\n26Figure 8: Comparison between the absorption spectrum of the Fe 3O4single particle sim-\nulation (thin line) with that of the 3 ×3 particles (array) simulation (thick line). Filled\nsquares mark the selected frequencies performed for the 3 ×3 simulation (the associated\ncurve is a spline fit to the corresponding data).\n27Figure 9: “Snapshots” of the magnetization vector field for the 3 ×3 particles simulation\n(n= 0, M = Fe), obtained at the resonancepeaks indicated in the previo usfigure (labelled\nby 1, 2 and 3). The snapshots were taken at two points of the acfield cycle ( ωt=π/2\nfor upper snapshot, and ωt= 3π/2 for the lower one). Snapshots number 3 (panel at the\nright) should be compared with that of Fig. 6, M = Fe one-particle simu lation.\n282 4 6 8 10 12\nFreq. [GHz]010203040506070Absorption [arb. units]Reference\nS1\nS2\nFigure 10: Comparison of the absorption spectra of a single ferrite particle (M = Fe;\nmagnetite), labeled as “Reference” in the figure, with additional sim ulations “S1” and\n“S2”. The latter runs had their stiffness parameter Aartifically lowered the by a factor\n1/3 (“S1”) and by 20% (“S2”) in relation to the reference model.\n292 3 4 5 6 7 8\nFrequency [GHz]010203040Absorption [arb. units]Reference\nReference - cut after 3ns\nS3\nS4\n3 x 3\n2 3 4 5 6 7 8\nFrequency [GHz]010203040Absorption [arb. units]Reference\nS5\nS6\nS7\n3 x 3\nFigure 11: Comparison of the absorption spectra of a 3 ×3 ferrite (M = Fe; magnetite)\nparticles simulation, labeled as “Reference” in the figure, with additio nal simulations “S3”\nto “S7”, as listed in Table 3 and explained in the text.\n30References\n[1] R. F. Soohoo, Theory and Application of Ferrites, Prentice-Hall, Inc.,\n1960.\n[2] T. Hyeon, Chemical synthesis of magnetic nanoparticles, Chem. Com-\nmun. (2003) 927–934.\n[3] M. Blanco-Mantec´ on, K. O’Grady, Interaction and size effects in mag-\nnetic nanoparticles, Journal of Magnetism and Magnetic Materials 2 96\n(2006) 124–133. doi:10.1016/j.jmmm.2004.11.580 .\n[4] R. 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Harris, Computational stud y of cop-\nper ferrite (CuFe 2O4), Journal of Applied Physics 99 (8) (2006)08M909.\ndoi:10.1063/1.2170048 .\n34[43] G. Gubbiotti, M. Madami, S. Tacchi, G. Carlotti, T. Okuno,\nNormal mode splitting in interacting arrays of cylindrical permal-\nloy dots, Journal of Applied Physics 99 (8) (2006) 080000–+.\ndoi:10.1063/1.2150806 .\n[44] A. M. Gama, M. C. Rezende, The relationship between Mn-Zn fer rites\nwith different iron ion contents and the absorption energy in X-band ,\nProceedings of the IEEE 01 (2005) 322–325.\n35"    },    {        "title": "1711.09980v2.Cavity_Magnon_Polaritons_with_Lithium_Ferrite_and_3D_Microwave_Resonators_at_milli_Kelvin_Temperatures.pdf",        "content": "Cavity Magnon Polaritons with Lithium Ferrite and 3D Microwave\nResonators at milli-Kelvin Temperatures\nMaxim Goryachev,1Stuart Watt,2Jeremy Bourhill,1Mikhail Kostylev,2and Michael E. Tobar1,a)\n1)ARC Centre of Excellence for Engineered Quantum Systems, School of Physics, University of Western Australia,\n35 Stirling Highway, Crawley WA 6009, Australia\n2)Magnetisation Dynamics and Spintronics Group, School of Physics, University of Western Australia,\n35 Stirling Highway, Crawley WA 6009, Australia\n(Dated: 24 September 2018)\nSingle crystal Lithium Ferrite (LiFe) spheres of sub-mm dimension are examined at mK temperatures, mi-\ncrowave frequencies and variable DC magnetic \feld, for use in hybrid quantum systems and condensed matter\nand fundamental physics experiments. Strong coupling regimes of the photon-magnon interaction (cavity\nmagnon polariton quasi-particles) were observed with coupling strength of up to 250 MHz at 9.5 GHz (2.6%)\nwith magnon linewidths of order 4 MHz (with potential improvement to sub-MHz values). We show that\nthe photon-magnon coupling can be signi\fcantly improved and exceed that of the widely used Yttrium Iron\nGarnet crystal, due to the small unit cell of LiFe, allowing twice more spins per unit volume. Magnon mode\nsoftening was observed at low DC \felds and combined with the normal Zeeman e\u000bect creates magnon spin\nwave modes that are insensitive to \frst order order magnetic \feld \ructuations. This e\u000bect is observed in the\nKittel mode at 5.5 GHz (and another higher order mode at 6.5 GHz) with a DC magnetic \feld close to 0.19\nTesla. We show that if the cavity is tuned close to this frequency, the magnon polariton particles exhibit an\nenhanced range of strong coupling and insensitivity to magnetic \feld \ructuations with both \frst order and\nsecond order insensitivity to magnetic \feld as a function of frequency (double magic point clock transition),\nwhich could potentially be exploited in cavity QED experiments.\nI. INTRODUCTION\nRecently, applications of low loss magnonic systems in\nphysics have drawn considerable attention. Modern ex-\nperimental demonstrations range from Quantum Electro-\ndynamics (QED)1,2to fundamental physics3{5. The main\nfocus of all these experiments is on the rare-earth iron\ngarnets, particularly, Yttrium Iron Garnet (YIG), a ma-\nterial that combines good photon and magnon properties\nover the entire microwave spectrum as well as the possi-\nbility to enhance its performance for particular applica-\ntions with chemical substitution. For room temperature\napplications, it has been widely used as a tunable element\nin many common microwave devices such as \flters and\noscillators6, as a material for nonlinear studies7,8, and\nas a platform for prospective magnon based information\nprocessing devices9,10. Additionally, YIG has been con-\nsidered as a material of choice for many implementations\nof hybrid quantum systems11{18. On the other hand, it\nsu\u000bers from high losses in the optical frequency band as\nwell as very high refractive index, which makes optical\ncoupling ine\u000ecient and very hard to implement19. Also,\nthe performance of YIG for many applications may su\u000ber\nfrom coupling to higher order magnon modes due to pos-\nsible crystal imperfections, especially at very low temper-\natures. Thus, it is interesting to compare this material to\nsimilar ferrimagnetic single crystals that have very high\nspin density and narrow magnon linewidths.\nOne particular candidate is the single-phase crys-\ntal Lithium Ferrite (LiFe) of empirical formula\na)Electronic mail: michael.tobar@uwa.edu.auLi0:5Fe2:5O420{24, which belongs to the cubic spinel fer-\nrites family with properties competitive with YIG23,24.\nLiFe belongs to the class of soft magnets with square\nmagnetic hysteresis loops, large magnetization and\nstrong anisotropy. Some additional properties such as\nstrong Dzyaloshinskii-Moriya interaction lead to propos-\nals for novel magnonic devices based on relativistic band\nengineering of LiFe crystals25. Besides microwave ap-\nplications, the material is used for coatings of anodes\nfor lithium ion batteries26, Terrahertz devices27and gas\nsensing28.\nExperiments that couple magnon spin wave modes\nto photonic cavities to create cavity magnon polariton\n(CMP) two-level systems have recently been a fertile area\nof investigation. For example CMP bistability29, CMP\nexceptional points30and CMP manipulation of distant\nspin currents31have recently been observed. In this work,\nwe couple a 3D lumped photonic resonant cavity to a\nLiFe sphere to create CMPs and illustrate the potential\nfor QED at microwave frequencies and mK temperatures.\nFurthermore, we observe in the dispersive regime of the\ncavity, magnon spin-wave modes, which are \frst order in-\nsensitive to magnetic \feld close to 5.5 GHz in frequency.\nWe show that if the cavity mode can be tuned to this\nmagnetic \feld insensitive point of the magnon, where\ndf=dB = 0, we create a two-mode CMP with enhanced\ncoupling range and reduced curvature (point of in\rection,\nwith bothdfCMP=dB = 0 andd2fCMP=dB2= 0), similar\nto a double magic point atomic clock transition32. Previ-\nously, solid state clock transitions have been observed in\nnuclear, electron and hyper\fne solid state spin systems,\nsuch as NV in diamond33,34, Eu3+in YSO35, Bismuth in\nSi36{40as well as molecular spin systems41,42.arXiv:1711.09980v2  [cond-mat.str-el]  22 Mar 20182\nII. CAVITY MAGNON POLARITONS WITH A\nLITHIUM FERRITE SPHERE\nThe LiFe specimens were highly polished 0 :46 mm di-\nameter spheres glued onto a microwave ceramic post with\nspeci\fc orientation. The post as well with the external\nDC \feld are oriented along the (110) crystallographic\ndirection of the sphere.The spheres exhibit ferrimagnetic\nresonance linewidth \u0001 Hof 1 Oe and the saturation mag-\nnetization\u00160Msof 0:37 T at room temperature. The re-\nfractive index of LiFe thin crystal is measured to be \u00182:6\nat 800 nm and\u00182:3 at 1550 nm27, and permittivity at\nmicrowave frequencies is approximately 15, close to that\nof YIG43,44. In this work, we investigate LiFe properties\nusing cavity methods at 20 mK and very low photon ex-\ncitation number. For this purpose, a cavity containing a\nLiFe crystal is attached to the lowest temperature stage\nof a Dilution Refrigerator inside a superconducting mag-\nnet and characterised as a function of the DC magnetic\n\feld using continuous wave excitation. The experimental\nsetup is described in detail in previous works45,46.\nThe material is characterised using a re-entrant mi-\ncrowave cavity47,48. Such resonators are typically cylin-\ndric, with a metallic post attached to one of the con-\nducting faces which stops just short of the opposite face,\nforming a small gap. The re-entrant mode has the elec-\ntrical \feld con\fned in this gap and the magnetic \feld\naround the post, and thus the metallic rod forms a 3D\nlumped element LC resonator. The re-entrant mode res-\nonance frequency can be calculated as a resonance of a\ncorresponding LC-circuit, thus the resonance frequency\nis proportional to a square root of the gap distance47,48.\nIn the double-post structure as shown in Fig. 1, low and\nhigh frequency re-entrant resonances have symmetric (co-\ndirectional currents and electric \felds for both posts) and\nantisymmetric (contra-directional currents and electric\n\felds for both posts) structure respectively. As a result,\nthe symmetric resonance expels magnetic \feld from the\nspace between the posts and is henceforth referred to as\nthe dark mode, whilst the anti-symmetric mode focuses\nthe \feld and will be referred to as the bright mode. This\nproperty has been used to enhance the spin-photon cou-\npling by placing crystal samples of sub-wavelength size\nbetween the posts46,49. The position of the crystal inside\nthe cavity and orientation of the external \feld is shown\nin Fig. 1. The other advantage of this method is the abil-\nity to spatially separate the magnetic properties of test\ncrystals from electric ones. This property is achieved due\nto the fact that for the re-entrant type structure, most\nof the electric \feld is concentrated in the gap, while the\nmagnetic \feld spreads around the posts leading to their\nstrong spatial separation. For this reason, dielectric prop-\nerties of LiFe crystals need not be considered.\nThe cavity was measured at temperatures near 20 mK\nin a dilution refrigerator as a function of DC magnetic\nusing a superconducting magnet. The excitation signals\nwere attenuated at 4 K and 20 mK stages down to the\nlevel of a few photons. The transmitted photons were\ngapLiFe\n\u0000\u0000\u0000~H\u0000\u0000\u0000\n2 min max\"\" \"#(B)\n(C)(A)FIG. 1: (A) Double post reentrant cavity (without top\nlid) with a LiFe sphere in external magnetic \feld, with\ncross-section in (B). This cavity supports a dark \"\"and\na bright mode#\"as shown in (C).\n(A)\n(B)\nFIG. 2: Frequency response of the LiFe sphere-cavity\nsystem as a function of external DC \feld for the HF\ncase: (A) close to the bright mode #\", (B) near dark\nmode\"\".\nampli\fed by a low noise 4 K ampli\fer separated from\nthe cavity by a circulator. The cavity of 10 mm diameter\nand 3.8 mm height was made of oxygen free Copper with\n2 mm diameter posts separated by 1.5mm from the cavity\ncentre. With these dimensions, the \flling factors (ratio\nof the magnetic energy stored in the LiFe sphere and\nthe total magnetic energy in the cavity) for the bright\nand dark mode were estimated to be 0.1% and 10\u00004%\nrespectively. The former may be greatly increased by3\n(A)\n(B)\nFIG. 3: Frequency response of the LiFe sphere-cavity\nsystem as a function of external DC \feld for the LF\ncase: (A) close to the bright mode, (B) close to the dark\nmode\"\", which has been \ftted with a two-mode model,\nwith the coupled (uncoupled) solutions in dashed purple\n(white) and shows two mode crossings. The lower\ncrossing in (B) is not in the strongly coupled regime due\nto excess magnon losses at low magnetic \feld, while the\nupper crossing is strongly coupled.\nminimising the cavity height and separation between the\nposts46without change of resonant frequency.\nThe photon-magnon coupling experiment with the\n0:46 mm diameter sphere was repeated twice using di\u000ber-\nent values of the gap between the top of the posts and the\ncavity lid. The system responses for the cases of larger\ngaps and smaller gap as a function of external magnetic\n\feld and frequency are shown in Fig. 2 and 3 respectively.\nA run with larger gaps gives higher resonance frequencies\nand is referred to as Higher Frequency (HF) experiment,\naccordingly, the smaller gap case results in Lower Fre-\nquencies (LF). In each case, the system is characterised\nin the vicinity of the dark and bright modes with corre-\nsponding frequencies f\"\"andf#\".\nNear each avoided level crossing the system may be\napproximated by the simplest linearly coupled two har-\nmonic oscillator model:\nH=h =\u0017phaya+\u0017m(B)byb+g(ayb+bya);(1)\nwhereay(a) andby(b) are creation (annihilation) opera-tors for photonic and magnon modes with correspond-\ning resonance frequencies of \u0017phand\u0017m(B), and the\nmagnon resonance frequency is controlled via the exter-\nnal magnetic \feld ( B) due to the Zeeman e\u000bect \u0017m=\ng\u0016B(B+B0)=~. Here\u0016Bis the Bohr magneton, g is\nthe e\u000bective Land\u0013 e g-factor and B0is an e\u000bective mag-\nnetic \feld bias. The summary of the model parameters\nfor two modes of two experiments are given in Table I.\nFor all experiments with the 0.46 mm (110) sample, the\nmagnon linewidth is estimated to be on the order of\n9 MHz. The maximum observed coupling is 250 MHz\nat 9.5 GHz (i.e. g=!= 2:6%), which would be improved\nfurther by increasing the \flling factor. If the \flling factor\ncould be increased to even 3%, a value which was previ-\nously demonstrated with a YIG sphere inside the same\ntype of cavity46, one could increase the coupling to 1.3\nGHz or 13.7% of the resonant frequency. In the case of\nthe YIG sphere experiment, magnon-photon coupling of\n2 GHz at 20.6 GHz or 9.7%46was measured, and was\nachieved using a sphere of 0.8 mm diameter hence a 5.3\ntimes larger volume than the LiFe described here.\nIII. MAGNETISATION PROPERTIES OF LITHIUM\nFERRITE AND COMPARISON TO YIG\nFor several particular applications, the spin density or\nthe total number of spins is the important parameter for\ncomparison. Thus, it is interesting to compare LiFe to\nthe most popular magnonic material YIG. Unlike YIG,\nLiFe exhibits spinel structure with one Fe3+ion occu-\npying the tetrahedral sites compared to 1.5 ions occupy-\ning the octahedral sites and oriented antiparallel, which\nmeans there are, \u0001 n= 0:5, dominant octahedral ions in\neach unit cell. Both types are Fe3+, which carry 5 Bohr\nmagnetons and thus determines the e\u000bective spin density\nto beNe\u000b=Ms=\u0016B\u00193:2\u00021022cm\u00003. With a mag-\nnetic moment of a unit cell equal to 10 \u0016Band a typical\nMsvalue of 0:175 T, the e\u000bective spin density of YIG\nmay be estimated to be 1 :5\u00021022cm\u00003, two times less.\nCompared to YIG, the larger spin density of LiFe is due\nits unit cell occupying a smaller volume, which is veri\fed\nvia the considerably stronger normalised photon-magnon\ncoupling as discussed above.\nBesides the main uniform magnetisation precession\nmode, the sphere under cryogenic conditions exhibit a\nnumber of higher order magnon modes as seen for both\nbright and dark modes in both Fig. 2 and 3. These ef-\nfects are also seen for all tests of YIG at low tempera-\ntures and typically attributed to arising anisotropies. It\nis found that the frequencies of these modes cannot be\npredicted by the same theory50as successfully applied to\nYIG spheres46due to the strong anisotropy.\nHere one has to mention that there has been only one\nrecent experiment where FMR measurements for LiFe\nwere taken over a broad range of frequencies51. There-\nfore, not much detail can be found in the literature on\nthe \feld dependence of the FMR frequency for this type4\nTABLE I: Parameters of avoided level crossing between\nthe uniform magnon made to cavity resonances.\nMode\u0017ph(GHz) gB0(mT)g(MHz)\nHF\"\" 6.67 \u00182 6.4 9\nHF#\" 9.5 \u00182 10 250\nLF\"\" 5.96 \u00182 8.2 68\nLF#\" 8.7 \u00182 9 240\nof ferromagnetic spinel. What is known, is that our\nLiFe material should possess a cubic magneto-crystalline\nanisotropy52,53with (111) being an easy magnetisation\naxis (and (100) being the axis along which the resonance\nlinewidth is minimal54). However, the shapes of the hys-\nteresis loops for the spheres (Fig. 4 (A)) taken for (110)\nand (111) directions are qualitatively the same, and both\nare characteristic for hysteresis measurements taken at\nan angle with an easy axis. This may suggest that in\naddition to the magnetocrystalline cubic anisotropy, our\nsamples may also possess a uni-axial anisotropy poten-\ntially induced during sphere grinding, as the axis of this\nanisotropy is not along (110) or (111). A more detailed\nexperimental analysis of the nature of the spheres mag-\nnetic anisotropy is beyond the scope of this paper. Im-\nportantly, we show there is enough evidence that the\nspheres possess a signi\fcant anisotropy \feld. This ev-\nidence is not only the shape of the hysteresis loops, but\nalso a fact that linear \fts to sections of the lines in Fig. 4\n(B) which correspond to the magnetically saturated state\nof the spheres deliver an vertical o\u000bset of fb= 440 MHz\nfrom zero \feld for the (110) 0.58mm sphere at 20 mK\n(see Table II for other cases).\nGiven this evidence, the non-monotonicity of the\ncurves in Fig. 2 (B) and Fig. 3 (B) can be explained\nas mode softening55. This is seen in their magnetic \feld\ndependence resulting in a magnon magnetic \feld insen-\nsitivity (dB=df = 0) atBs= 0:191 T as seen in as turn-\nover points in the magnon resonance dependence on the\nexternal \feld. This value of the turn-over point is asso-\nciated with the properties of the hysteresis loop of LiFe.\nTo identify the softening \feld, the sample hysteresis loop\nwas measured at 3 K as well as at room temperature and\nplotted in Fig. 4 (A). As it is seen from this \fgure, the\nmagnetic hysteresis loop measurement demonstrates that\nBscorresponds to the saturating \feld of the material.\nIn Silber et al55, it is shown that if a sphere is mag-\nnetized at an angle to the easy axis, the frequency vs\n\feld dependence can be separated into three sections.\nAbove the saturating \feld, all spins in the material are\nperfectly co-aligned and also co-aligned with the external\n\feld. This results in dynamics obeying Kittel equation\nfor the ferromagnetic resonance frequency (Field Range\n3). Below this \feld, two regimes can be identi\fed. The\n\frst regime is in the closest vicinity of the saturating \feld\n(Field Range 2). Here the material is in a single-domain\nstate, as above the saturating \feld, but the magnetisa-\ntion vector is not collinear to the applied \feld - the vector\nis closer to the easy axis than the \feld. With an increasein the \feld, the vector rotates closer to the \feld. This\nleads to an increase in the FMR frequency. Importantly,\nin contrast to the fully saturated state, the dependence\nis not a straight line, with the curvature diminishing\nwith an approach to the magnetic saturation (at which\nthe static magnetization vector becomes aligned to the\n\feld). At smaller \felds (Field Range 1), the sphere is bro-\nken into magnetic domains with magnetization vectors in\nthem aligned along the easy axis (or multiple easy axes\nfor cubic anisotropy). As shown theoretically by Smith\nand Beljers56, in this regime, the resonance decreases to\na \fnite value at the critical \feld, and then increases fol-\nlowing the single-domain theory from Silber et al.55.\nThis explanation is consistent with the variation in the\nresonance linewidth with the applied \feld seen in Figs. 2\n(B) and 3 (B). The FMR responses for Ranges 2 and 3\nare characterised by a relatively narrow resonance line.\nThis is coherent with the single-domain state of static\nmagnetization for these ranges. For Range 1 the reso-\nnance linewidth monotonically broadens as the magnetic\n\feld decreases. This is consistent with a more devel-\noped domain structure for lower magnetic \feld values.\nAt room temperatures, parameter Bswas measured to\nbe 0:154 T giving the expected temperature dependence\nfor this value. Finally, it is also seen from Fig. 2 (B) that\nthe higher order magnon mode exhibits the same soften-\ning phenomenon at the same value of external magnetic\n\feld.\nTo characterise the softening phenomenon, LiFe sam-\nples of di\u000berent size and orientation with respect to the\napplied \feld were measured. The results are summarised\nin table II and shown in Fig. 4 (B), which compares pa-\nrameters of the magnon resonance frequency dependen-\ncies on the external magnetic \feld. Here g \u0000, g+,Bs,fs,\nfb,\u000eare the e\u000bective g-factor below the saturation mag-\nnetisation, g-factor above the saturation magnetisation,\nthe softening \feld (identi\fed as saturation magnetisa-\ntion), frequency of the magnetic \feld insensitive point\nwheredf=dB = 0, zero \feld frequency bias for a linear\n\ft to the magnetically saturated state, and an estima-\ntion for the magnon linewidth at B > B srespectively.\nFirstly, for the (110) 0.46 mm diameter sphere, the satu-\nration magnetisation \feld decreases by going from 20 mK\nto 4 K, which is consistent with the room temperature\nto 20 mK comparison. Secondly, the test reveals a de-\npendence on sphere size, with the turnover point's \feld\nvalue decreasing for larger spheres. Thirdly, the sphere\nwith (111) orientation demonstrates signi\fcantly di\u000ber-\nent minimum frequency and e\u000bective g-factors. From the\ntable it follows that the gyromagnetic ratio (or e\u000bective\ng-factor) depends on orientation, with the value for (110)\nsamples exceeding previously measured values at room\ntemperature51.\nTable II also compares estimations of magnon\nlinewidths for di\u000berent samples. It suggests that the big-\nger samples exhibit some reduction in linewidths with the\nlowest value of 4 MHz at 9 GHz, possibly due to reduction\nof the contribution of surface magnon scattering to the5\n(B)(A)\nRange 1Range 2\nRange 3\nFIG. 4: (A) Magnetisation loops for LeFe spheres at\n300 K and 3 K with the external \feld scaled to the\nsaturation \feld. (B) Curve \fts to resonance peaks at\nlow frequencies.\nTABLE II: Parameters of the gyromagnetic curve for\nthe uniform precession magnon mode.\nSample g\u0000g+Bsfsfb\u000e\n(mT) (GHz) (MHz) (MHz)\n(110) 0.46mm, 20 mK -0.81 2.02 194 5.56 90 9\n(110) 0.46mm, 4 K -0.72 2.01 192 5.57 142 9\n(110) 0.58mm, 20 mK -0.71 2.01 180 5.55 440 4\n(111) 0.58mm, 20 mK -1.19 1.96 189 5.32 10 5\ntotal linewidth. Nevertheless, this value is larger than\nthe lowest demonstrated linewidths for state-of-the-art\nYIG spheres at mK temperatures, but within an order of\nmagnitude1,2. The observed minimum value of linewidth\nis in accordance with the mean value of 1.35 Oe at 5.1\nGHz (or equivalently 3.8 MHz) for a batch of selected\nlow defect concentration samples measured at 4.2 K57.\n0.180.190.200.210.220.10.20.51\nB Field(T)߿ࠀ.߾߿.߾9߿.߾8߾ࠀ.߾5.45.86.߾6.ࠀ5.6Frequency (GHz)ࠛagneࡂic Field  (T)f(1)\"\"f(0)\"\"f(2)\"\"ࠀ.߾߾.߿.߾5f+\u0001- f\u0000(GHz)Bs(A)(B)(C)(D)(c)(b)(a)ࠀgࠛagneࡂic Field  (T)5.45.86.߾6.ࠀ5.6߿ࠀ.߾߿.߾9߿.߾8߾ࠀ.߾ࠀࠀ.߾★★★f+\nf+f+\nf\u0000f\u0000f\u0000FIG. 5: Modelled interaction of mode crossing shown in\nFig.3(B), lowered from its LF \"\"frequency of f(0)\n\"\"= 5:96\nGHz (A), to f(1)\n\"\"= 5:76 GHz (B), and \fnally\nf(2)\n\"\"=fs= 5:56 GHz (C). The white and yellow curves\nrepresent the uncoupled magnon and cavity modes'\nmagnetic \feld dependence, respectively, whilst the dark\nlines represent the behaviour when the two are coupled\nat a rate of g= 68 MHz. The frequency di\u000berence\nbetween the higher frequency branch and lower\nfrequency branch (the transition frequency,\nfCMP =f+\u0000f\u0000) is then plotted for each of these three\ncases in (D). The red stars represent the location where\nthe photon and magnon maximally hybridise.\nTo the best of our knowledge, the lowest reported val-\nues is 0.274 Oe (equivalently 770 kHz) for a (100) sphere\nat 4 K around 10.8 GHz54. Such low values of magnon\nlinewidths make this ferrite material a strong alternative\nto YIG spheres taking into account the fact that LiFe\nmaterial research has not been developing with the same\npace as the YIG industry in the last 50 years. Also, uni-\nform precession modes for both samples exhibit strong\ncoupling of 50 MHz to the dark cavity mode at around\n8 GHz, and the second strongest higher order magnon\nmode is coupled at the rate of 11 MHz.\nIV. CAVITY MAGNON POLARITON WITH REDUCED\nMAGNETIC FIELD SENSITIVITY\nFig. 5 shows the frequency response between the dark\nmode and magnons in the (110) 0.46 mm, 20 mK LiFe\nsample as the cavity photon frequency is tuned to the\nturning point of the magnon mode at 5.56 GHz. In our\nexperiment the cavity dark mode is only tuned 400 MHz\naway (Fig. 5 (A)) and we show how the characteristics\nchange using a two-mode coupled model, when the cav-\nity frequency is also tuned 200 MHz away (Fig. 5 (B))\nand then to the point of exact tuning (Fig. 5 (C)). When\nthe cavity is tuned to the point of exact tuning, the CMP\ntransition frequency, fCMP =f+\u0000f\u0000, range is enhanced,\nresulting in a lower curvature CMP transition frequency\nversus B \feld characteristic and a broader range of mag-6\nd(f+\u0000f\u0000)\ndB(GHz/T)\n0.19 0.20 0.21 0.22-4-2024\nB Field (T)d2(f+-f-)\ndB2(x103GHz /T)d2(f+\u0000f\u0000)\ndB2⇥103(    GHz/T )20.19 0.20 0.21 0.22-20-1001020\nB Field (T)d(f+-f-)\ndB(GHz /T)\n0.18 0.19 0.20 0.21 0.22-20-100102030\nB Field (T)d(f+-f-)\ndB(GHz /T)\n60.19 0.20 0.21 0.22-20-100102030\nB Field (T)d(f+-f-)\ndB(GHz /T)\n0.18 0.19 0.20 0.21 0.22-20-100102030\nB Field (T)d(f+-f-)\ndB(GHz /T)\n0.18 0.19 0.20 0.21-4-20246\nB Field (T)d2(f+-f-)\ndB2(x103GHz /T)\n0.18 0.19 0.21 0.22-4-20246\n0.20\nB Field (T)d2(f+-f-)\ndB2(x103GHz /T)Bs\n★ ★ ★\n★★ ★(c)(b)(a)\n(c)\n(b)(a)(A)\n(B)\nFIG. 6: First (A) and second (B) derivatives of the\nfrequency of the CMP quasi particle transition with\nrespect to magnetic \feld for the same values of\ndetuning as shown in Fig. 5.\nnetic \feld where strong coupling is attained (Fig. 5 (D)).\nTo study the properties of the CMP transition as we\ntune to the magnon mode turn over point, we plot the\n\frst and second derivative of fCMP with respect to mag-\nnetic \feld as a function of photonic mode cavity tuning,\nwhich is illustrated in Fig.6. The \fgure clearly shows\nthe point of maximal hybridisation (centre of an avoided\nlevel crossing) occurs when dfCMP=dB = 0 for all cavity\nmode de-tunings. However, as the photonic mode cavity\nfrequency approaches the magnon mode turning point\nthe second order is reduced and \fnally reaches zero at\nthe maximal hybridisation (curve (c)). This is similar\nto a double magic point atomic clock transition32which\nsigni\fcantly reduces the e\u000bects of magnetic \feld biasing\n\ructuations and could potentially be exploited in cavity\nQED experiments.\nV. CONCLUSION\nIn conclusion, we investigated the properties of LiFe\nas a magnonic material coupled to a photonic cavity at\nultra-low temperatures. It is demonstrated that due to\nvery high spin density and relatively low magnon losses,\nLiFe spheres allow strong coupling regimes of magnon-\nphoton interactions similar to YIG (coupling strengths of\nhundreds of MHz), and to thus create cavity magnon po-\nlariton two level systems. 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We investigated the ME voltage coe \u000ecient as a complex quantity for all composite samples using\nthe dynamic piezomagnetic coe \u000ecient, qac=@\u0015ac=@H. The results reveal that tuning the magnetostrictive phase has a strong e \u000bect\non the real part of the ME voltage coe \u000ecient. Doping zinc into cobalt ferrite modified the magnetic properties of the magnetic\nphase, such as magnetic anisotropy and coercive field, and hence the ME properties. The highest ME coe \u000ecient value of 12.33\nmV\ncm:Oewas obtained for x =0.1 at the magnetic field of 755 Oe. In addition, the magnetic field at which the maximum value of the ME\ncoe\u000ecient was observed ( Hpeak) strongly depends on the value of Zn substitution. The results were interpreted using the magnetic\nfield dependence of the CF 2\u0000xZxO magnetostriction.\nKeywords: Multiferroics, Magnetoelectric effect, Piezoelectrics, Magnetostriction, Cobalt ferrite\n1. INTRODUCTION\nMultiferroics are a class of materials that have at least two\nferroic orders (ferroelectric, magnetic, and elastic). Magneto-\nelectric (ME) materials are a subset of multiferroics in which\nferroelectric and magnetic orders are coupled. The ME ma-\nterials are classified into single-phase and composite groups in\nwhich composites show higher ME coupling values than single-\nphase ones[1, 2, 3, 4, 5, 6, 7, 8]. Since in this class of materials\nmagnetization (polarization) can be controlled by an /a electric\nfield (magnetic field), they can be used in both electrical and\nmagnetic areas, and have a wide range of applications, includ-\ning magnetic field sensors, spintronics, data storage, actuators,\nphotovoltaic solar cells, etc[9, 10, 11, 12].\nThe ME voltage in composite magnetoelectric materials is\ngenerated by polarization change in the piezoelectric phase as a\nresult of dynamic mechanical deformation of the magnetostric-\ntive phase induced by the applied ac magnetic field in the pres-\nence of the dc magnetic field. In addition, the ME coupling\ncoe\u000ecient is a \u000bected by magnetostriction and piezoelectric be-\nhavior of magnetic and electric phases, respectively. As a re-\nsult, significant magnetostriction and piezoelectric coe \u000ecients\nare essential for obtaining a high ME voltage coe \u000ecient.\nMany studies have been conducted on piezoelectric-\nferrite composites like: (BiBa)(FeTiZn)O 3/CoFe 2O4[13],\nPFN-PT /(Co,Ni)Fe 2O4[14], PMN-PT /CoFe 2O4[15]\nBaTiO 3/CoFe 2O4[16, 17], PMN-PT /NiFe 2O4[18],\nPZT/Ni0:2Co0:8Fe2O4[19], PZT /CoFe 2O4[20],\n\u0003Corresponding author\nEmail address: torabishahbazmohammad@gmail.com (Mohammad\nTorabi-Shahbaz)PVDF /CoFe 2O4[21], and etc. Among them, Pb-based\nferroelectrics, such as PZT and PMN-PT, have been frequently\nutilized as the ferroelectric phase in ME composites due to\ntheir excellent piezoelectric characteristics. For the magnetic\nphase, high magnetostrictive materials such as cobalt ferrite\nare commonly used in these composites.\nIn this work, as piezoelectric material, we used\nthe solid solution of lead-iron niobate and lead\ntitanate;0.94Pb(Fe 1=2Nb1=2)O3-0.06PbTiO 3(PFN-PT)[22].\nThere is a ferroelectric-paraelectric (F-P) phase transition\nat approximately 125C for this compound with x =0.06 (\nmorphotropic phase boundary)[23]. It also has the highest\ndielectric constant and piezoelectric coe \u000ecient, d 33, among\nother concentrations[22]. Possessing a high d 33makes it a\nsuitable candidate for use in magnetoelectric composites as a\npiezoelectric phase(d 33=600pC\nNfor x=0.06)[22]. However, the\nME e \u000bect in a particulate composite of PFN-PT with magne-\ntostrictive materials has been rarely reported[14]. On the other\nhand, the magnetic properties of magnetostrictive materials\ncan be controlled by doping them. In this research, we doped\nzinc elements into cobalt ferrite to tune its magnetostriction\nproperties. This let us study the a \u000bect of manipulation of the\nmagnetic phase on magnetoelectric coe \u000ecient.\nIn essence, in this work we select PFN-PT as the ferroelectric\nphase and zinc-doped CoFe 2O4(CFZO) as the magnetic phase.\nThe structural, dielectric, magnetic, and ME voltage coe \u000e-\ncients (a complex quantity) are investigated for the composites\nwith di \u000berent contents of Zn in 0.6(PFN-PT)-0.4(CF 2\u0000xZxO).\nPreprint submitted to Journal of L ATEX Templates August 25, 2022arXiv:2208.11317v1  [cond-mat.mtrl-sci]  24 Aug 20222. EXPERIMENTAL DETAILS\nThe polycrystalline solid solution of 0.94Pb(Fe 1=2Nb1=2)O3-\n0.06PbTiO 3(PFN-PT) was synthesized using the solid-state re-\naction method. The high purity raw materials of PbO (with\n4Wt% excess of PbO to compensate for lead volatility during\nthe heat treatment), Fe 2O3, Nb 2O5, TiO 2, and one Wt% Li 2CO 3\n(to make a better structure of the perovskite phase and reduce\nelectrical conductivity) were blended and ground for 1.5 h by\nhand in an agate mortar. The obtained powder was calcined at\n900\u000eC for four hours, and then reground and annealed in the\nair in two stages, 1070 and 1100\u000eC for 4 and 6 h, respectively.\nCo(NO 3)2.6H 2O, Fe(NO 3)3.9H 2O, Zn(NO 3)2.6H 2O,\nC6H8O7.H2O, and deionized water were used to synthesize\nnanoparticles of CoFe 2\u0000xZnxO4(x=0,0.25,0.1,0.2,0.3) using\nstandard sol-gel method. Then the resultant powder was\ncalcined at 800\u000eC for four hours.\nFinally, the Zn-doped cobalt ferrite and PFN-PT were mixed\nto prepare (60Wt%) PFN-PT - (40 Wt%) CoFe 2\u0000xZnxO4\n(x=0,0.25,0.1,0.2,0.3) particulate composites. The resultant\npowders were ground and prepared as pellets pressed at a pres-\nsure of 300 MPa and sintered at 800\u000eC for four hours.\nThe X-ray di \u000braction (XRD) patterns of the samples were\nmeasured at room temperature by using an Empyrean di \u000brac-\ntometer with Cu - K \u000bradiation. The microstructure of sam-\nples was analyzed using Field Emission Scanning Electron Mi-\ncroscopy (FE-SEM). Magnetization measurements of the com-\nposites were carried out by using a vibrating sample magne-\ntometer (VSM) at room temperature. The relative permittivity\nof pure PFN-PT and other composites were measured at various\nfrequencies (10 - 650 kHz) as a function of temperature using\nan LCR meter (IM3570 Hioki). Polarization-electric field (P-E)\nloops of the samples at room temperature were performed by\nusing a homemade setup based on the Sawyer-Tower method.\nAfter polling the samples in an electric field of 10 kV /cm for\none hour, the transverse magnetoelectric voltage coe \u000ecient was\nmeasured at room temperature by using the lock-in amplifier\ntechnique.\n3. Results and discussion\nThe XRD patterns of all composites, as well as polycrys-\ntalline PFN-PT solid solution and cobalt ferrite (CFO) samples\n(for comparison), are shown in figure. 1. The XRD pattern\nof the composite shows the structures of perovskite and spinel\nassociated with PFN-PT and CFO, respectively. No impurity\nphase is observed in the XRD pattern of sintered composites\nwhich means that no significant chemical reactions occurred\nbetween the grains of ferrite and ferroelectric during the sin-\ntering.\nFigure 2 shows the FE-SEM images of the sintered pellets\nof the pure PFN-PT (a), as well as all composite samples (b-\nf). It can be seen from the images that the samples consist of\npolygonal micrometer grains (related to the PFN-PT) and small\nCFO particles that are distributed in the shape of agglomerates\ninside the ferroelectric matrix. FE-SEM images exhibit that\nFigure 1: XRD patterns of (PFN-PT)-(CF 2\u0000xZxO) composites (x =0, x=0.025,\nx=0.1, x =0.2, x =0.3) as well as pure PFN-PT and CFO.\nthe ferrite particles are uniformly spread within the ferroelec-\ntric phase, which is suitable for achieving high magnetoelectric\ncoupling. Figure 2 (g) shows the EDS mapping images of the\nsintered pellet of the sample PFN-PT /CoFe 1:7Zn0:3O4. The ele-\nments Pb, Nb, and Ti, are more prominent in areas of the main\nimage where ferroelectric grains are present. On the other hand,\nCo, Fe, and Zn elements are more distinguished in regions of\nthe main image where there are tiny grains of ferrite.\nFigure 3 illustrates the temperature dependence of relative\npermittivity ( \u000fr) and dielectric loss tangent (tan \u000e) at four fre-\nquencies (10, 100, 300, 650 kHz) in the temperature range\nof 25 - 250\u000eC for pure PFN-PT (a) and composites samples\n(b-f). The dielectric constant of samples increases with rising\ntemperature and reaches its maximum at Tmbefore decreasing\nwith further temperature increases. As clearly seen from fig-\nure. 3 (a), the dielectric constant of pure PFN-PT has a peak\nat 124.5\u000eC. At this temperature, the dielectric constant has its\nmaximum value and a broad peak, related to a phase transition\nfrom high-temperature paraelectric (cubic structure) to the low-\ntemperature ferroelectric phase (tetragonal structure). Figures\n3 (b-f) show that by adding the magnetic phase, the dielectric\nconstant (\u000fr) decreases and the loss tangent (tan \u000e) increases.\nThis is owing to the decreased resistance of the ferrite mag-\nnetic phase as opposed to PFN-PT. The peak of the paraelectric\nto ferroelectric transition is also found to be moved to higher\ntemperatures. The interdi \u000busion of ions between the magnetic\nand ferroelectric phases during the sintering process is respon-\nsible for the shift in transition temperature [24, 25]. As shown\nin figure 3 (a), Tmfor PFN-PT is frequency independent, but\nit moves toward higher temperatures as frequency increases for\ncomposite samples (see figure 3 (b-f)).\nAt temperatures above the transition temperature, the real di-\n2Figure 2: (a-f) FE-SEM images of the sintered pellets of the pure PFN-PT\nas well as (PFN-PT)-(CF 2\u0000xZxO) composites samples (with x =0, x=0.025,\nx=0.1, x =0.2, x =0.3). (g) EDS mapping images for the sample PFN-PT-\nCoFe 1:7Zn0:3O4.electric constant follows the modified Curie-Weiss equation.\n1\n\u000f0\u00001\n\u000f0m=(T\u0000Tm)\r\nC1(1)\nwhere\u000f0\nmis the highest dielectric permittivity at the transition\ntemperature, C1is the modified Curie-Weiss constant, and \ris\nthe slope of log(1 /\u000f0-1/\u000f0\nm) vs. log ( T\u0000Tm). The value of \r\nindicates the ferroelectric’s degree of relaxation. Its value is 1\nfor normal ferroelectrics, but for relaxor ferroelectrics, \rranges\nbetween 1 and 2, whereas ( \r=2) indicates the entirely di \u000buse\nphase transition (1 \u0014\r\u00142) [26, 27, 28]. The graphs of ln(1 /\u000f0-\n1/\u000f0\nm) vs. ln ( T\u0000Tm) at 10 kHz are shown in the insets of figure\n3 (be). When the experimental data are fitted to Equation 1, the\nvalues\r=1.63, 1.85, 1.86, 1.92,1.91 and 1.94 were obtained\nfor PFN-PT, x =0, 0.025, 0.1, 0.2, and 0.3 respectively.These\nresults, when compared to pure PFN-PT with a smaller gamma,\ndemonstrate typical relaxor behavior in the composite samples.\nBecause of composition fluctuations and structural disorder in\nthe cationic configuration on the sites of the crystal structure,\nthis behavior is linked to the emergence of polar nano regions\n(PNRs), which are responsible for this behavior[27].\nFigure 4 exhibits the ferroelectric hysteresis loops of all sam-\nples up to the electric field of 20 kV /cm at a frequency of 50 Hz.\nPure PFN-PT shows a coercive field (E c) of 3.75 kV /cm and\nremanent polarization (P r) of 16\u0016C/m2. E cand P rfor com-\nposite samples are plotted in the inset of figure 4 as a func-\ntion of Zn content. According to the graph, introducing the\nnon-ferroelectric phase of ferrite to the ferroelectric phase PFN-\nPT reduces remanent polarization while increasing the coercive\nfield. It is also noted that as the electric field increases due to the\nincreased conductivity and leakage current of the ferrite mag-\nnetic phase, the composite samples are not as well-saturated\nas the pure sample and the loops form becomes rounder. The\nEcvalue is nearly identical for all composite samples. A high\nEcindicates that a larger electric field is required to rotate the\norientation of the electric domains and make the sample’s total\npolarization zero. It is primarily due to porosity within the ma-\nterial and the space charges that generate electric domain wall\npinning at ferrite and ferroelectric grain boundaries[29, 30, 31].\nAlso, the quantity of remanent polarization in the composite\nsamples varied according to the amount of zinc impurity in the\nmagnetic phase. As the amount of zinc in the magnetic phase\nincreases, the remanent polarization of the composite drops.\nAccording to the dielectric properties data, this decrease in po-\nlarization can be attributed to the higher conductivity of sam-\nples with a higher zinc content. As conductivity increases due\nto electric leakage current, a lower quantity of surface charge\naccumulates on the electrodes, reducing electric polarization.\nThe magnetic hysteresis curves for all composite samples\nare shown in figure 5. Because the PFN-PT is paramagnetic\nat ambient temperature, it contributes only a minor amount to\nthe magnetic hysteresis curves, and the main contribution is re-\nlated to zinc-doped cobalt ferrite. The saturation magnetization\n(Ms) and coercive field ( Hc) value change with zinc impurity,\nas shown in the inset of figure 5.\nFor composite samples, the value of Msgrows and then\ndrops. The value of Msis determined by the distribution of\n3Figure 3: Temperature dependence of relative permittivity ( \u000f0) (solid lines) and the corresponding dielectric loss tangent (tan \u000e) (dashed lines) in frequency range of\n10 to 650 kHz for (PFN-PT)-(CF 2\u0000xZxO) composites samples with (a) x =0, (b) x =0.025, (c) x =0.1, (d) x =0.2, (e) x =0.3, as well as pure PFN-PT (f).Insets show\nfits of data above transition temperature to modified Curie-Weiss law.\nFigure 4: Room temperature polarization-electric field (P-E) loops of (PFN-\nPT)-(CF 2\u0000xZxO) composites samples, as well as pure PFN-PT, measured at a\nfrequency of 50 Hz. Inset shows the coercive field (Ec) and remnant polariza-\ntion (Pr) as a function of Zn content.zinc in the tetrahedral and octahedral sites. According to Neel’s\nmodel, the net magnetization of spinel ferrites is equal to the\ndi\u000berence in magnetizations in two sublattices AandB, which\nrepresent the tetrahedral and octahedral sites, respectively. The\nZn+2ion preferentially occupies tetrahedral sites. Because the\nZn+2ion is non-magnetic and has no magnetic moment, by\nzn doping the magnetization value in sublattice Adrops when\ncompared to the magnetization value in sublattice B, causing\nthe net magnetization to increase. When the Zncontent in-\ncreases, some Zn+2ions are placed in the octahedral site, caus-\ning magnetization to decrease further. The leading cause of\nthe decrease in magnetization is the weakening of the superex-\nchange interaction between the A and B sites, as well as the\ntilting of the moments[32].\nTheHcdecreases with increasing zinc impurity in the iron\nsite, as shown in the inset of figure 5. The Hcin spinel fer-\nrites is a \u000bected by parameters such as anisotropy constant,\nsaturation magnetization, superexchange interactions, and lat-\ntice defects[33]. The Hcis directly related to anisotropy.\nAnisotropy reduces as zinc impurity increases. As a result, the\ncoercive field decreases. It is significant to mention that dur-\ning the heating process in the furnace, atoms of ferroelectric\nand ferrimagnetic phases can penetrate each other, a \u000becting the\nmagnetic properties of ferrite.\nMagnetoelectric coupling in composites is primarily created\nby transmitted stress caused by magneto-mechanical interac-\n4Figure 5: Room temperature magnetization loops of (PFN-PT)-(CF 2\u0000xZxO)\ncomposites. Inset (i) shows the coercive field (H c) and saturation magnetization\n(Ms) as a function of Zn content. Inset (ii) shows a zoomed view of the m-h\nloop in the vicinity of the center.\ntion at the interface of the piezoelectric and magnetostrictive\nphases[34]. The magnetoelectric voltage coe \u000ecient is stated as\na complex quantity[14]:\n\u000b\u0003\nME=\u000bMEe\u0000i'=\u000b0\u0000i\u000b00(2)\nwhere'is the phase shift between the driving current for the\nac magnetic field and the output ME voltage from the sample.\nThe real and imaginary parts of the magnetoelectric coe \u000ecient\nare represented by \u000b0and\u000b00, respectively. This phase shift re-\nsults in the formation of an imaginary portion, which represents\nthe presence of energy loss during the measurement of \u000b\u0003\nMEfor\nME composites. A detailed explanation of ME measurement is\ndescribed in the supplemental material of Ref[14].\nTo explore the influence of zinc substitution on magnetoelec-\ntric characteristics, after polling samples up to the 10kV\ncm, the\ncomplex magnetoelectric coe \u000ecient was evaluated for compos-\nites. Figure 6 reveals the real and imaginary parts of ME volt-\nage coe \u000ecient for the composites as a function of DC magnetic\nfield ( HDC) from 0 to 8 kOe in a 0.77 Oe AC magnetic field\n(f=978 Hz).\nIn the absence of a direct magnetic field ( HDC), the real part\nof the ME coe \u000ecient has a non-zero value ( \u000bsel f\u0000biased ) in all\nsamples, which is the result of the interaction between the alter-\nnating magnetic field ( HAC) and remanent magnetization of the\nferrite phase. As seen in the figures, the magnetoelectric co-\ne\u000ecient grows nearly linearly with increasing magnetic field\n(HDC) for all samples until a specific magnetic field ( Hpeak)\nreaches its maximum value ( \u000bmax). The magnetoelectric co-\ne\u000ecient then decreases as the magnetic field increases until it\nreaches zero in a specific magnetic field. As the field strength\nincreases, the value of the real magnetoelectric part changes\nsign and becomes negative. The values of \u000bsel f\u0000biased ,\u000bmax, and\nHpeakfor all composite samples are presented in table 3.sample name \u000bsel f\u0000biased (mV\ncm:Oe)\u000bmax(mV\ncm:Oe)Hpeak(Oe)\nx=0 2.57 7.16 1465\nx=0:025 2.38 8.09 1360\nx=0:1 4.20 12.33 755\nx=0:2 4.79 11.2 388\nx=0:3 1.41 4.96 249\nTable 1: The values of \u000bsel f\u0000biased ,\u000bmax, and Hpeakfor all composites.\nAs shown in the figures, the values of \u000b0andHpeakstrongly\ndepend on the zinc content in the cobalt ferrite magnetic phase.\nAccording to figure 6, the value of \u000bmaxgrows with increas-\ning Zn content up to x =0.1 and thereafter drops. Similar to\ncoercivity, increasing Zn content causes Hpeakto shift toward\nlower fields. Sample x =0.1 has the optimal amount of Zn and\nthe most robust magnetoelectric coupling.\nAnother significant feature visible in the curves is hysteresis\nbehavior between the increasing and decreasing branches of the\n\u000b0. This hysteresis behavior reflects the magnetic hysteresis\nbehavior. As shown in the pictures, as the Zn amount increases,\nthe hysteresis curves become narrow, which is consistent with\nthe magnetic hysteresis behavior of the samples.\nIn general, the magnetoelectric voltage coe \u000ecient in com-\nposites depends on the magnetostriction ( \u0015) and piezomagnetic\ncoe\u000ecient (@\u0015\n@M) of the magnetostrictive phase [35, 36]. The ME\ne\u000bect is proven to be driven by dynamic piezomagnetic, qac,\nrather than quasi-static magnetostriction[37]. qacis described\nby:\nqac=@\u0015ac\n@HacjHdc (3)\nWhere qacis the dynamic magnetostriction induced by the vi-\nbration of magnetic moments by Hacwhile applying a constant\nDC magnetic field[37]. Because of dynamic magnetostriction\nin the magnetostrictive phase, \u000bsel f\u0000biased appears in the sam-\nples. The self-biased ME is due to the magnetic hysteretic re-\nsponse of the piezomagnetic phase[38]. As a result, even with-\nout a direct magnetic field, all samples exhibit \u000bsel f\u0000biased .\nThe variation in magnetostriction behavior among the com-\nposites in the presence of a magnetic field accounts for the vari-\nance in magnetoelectric behavior. Many investigations have\nbeen published on the magnetostriction curves of pure cobalt\nferrite and cobalt ferrite doped with zinc[39, 40, 41]. The mag-\nnetostriction of zn-doped cobalt ferrite increases initially and\ndecreases with increasing magnetic field. The magnetostriction\nof a zn-doped cobalt ferrite polycrystal sample is a combina-\ntion of\u0015100<0 and\u0015111>0, where\u0015100and\u0015111correspond\nto magnetostriction in the easy and hard magnetic directions,\nrespectively. The dominant contribution of \u0015100drives the in-\ncreasing magnetostriction behavior in low fields. In contrast,\nthe decreasing behavior of magnetostriction in high fields is due\nto the higher contribution of \u0015111[42, 43, 44]. As the amount\nof the zinc element in the iron sites increases, the peak value\nof the magnetostriction curve decreases. Zinc substitution in\ncobalt ferrite causes part of magnetization vectors to change\n5their directions from the easy axis (100) to the hard axis (111).\nLess magnetocrystalline anisotropy is a consequence of these\nchanges in the direction of the magnetization vectors, which is\nin agreement with the results of the magnetic hysteresis curves.\nIt turns out that the Zn-substituted samples magnetostriction\npeak (x =0.025,0.1,0.2,0.3) occurs at lower magnetic fields than\nthe sample without Zn (x =0)[39, 41].\nThe value of \u000b0, as previously mentioned, is proportional to\nthe strain sensitivity (@\u0015\n@M). Indeed, the larger the slope of the\nmagnetostriction curve versus the magnetic field, the larger the\ncoupling coe \u000ecient. In this investigation, the sample with the\nhighest magnetoelectric coupling is X=0:1. In prior investiga-\ntions, the strain sensitivity of ferrite CoFe 1:9Zn0:1Oconfirmed\nthat this impurity had the highest strain sensitivity[39, 41].\nIn order to gain further insight, we can interpret the magnetic\nfield dependency of \u000b0by using the \u0015(H) trend. As was already\ndiscussed,\u0015dctends to grow as the magnetic field increases. In\nthe case of high fields, the behavior of \u000b0is controlled by the\ntype of the magnetic phase and the percentage of zinc impurity.\nWhen the DC magnetic field is strong enough to rotate the mag-\nnetic domains ( Hpeak), the ac magnetic field can cause the max-\nimum length change in the magnetic phase, yielding the highest\nvalue of\u000b0. With further increasing the magnetic field, the value\nof\u000b0becomes zero (@\u0015\n@H=0) and then the sign of \u000b0changes\nfrom positive to negative, which is due to the negative sign of\n(@\u0015\n@H) for zinc-doped cobalt ferrite [39, 41]. There have rarely\nbeen reports on negative ME voltage coe \u000ecients[45, 46, 47].\nFigure 6 also includes the samples’ imaginary part of the\nME coe \u000ecient (\u000b00). ME responses have rarely been studied\nin terms of their imaginary part[48, 49, 50]. The imaginary part\nof the ME coe \u000ecient represents the energy dissipation in the\nsystem. As can be seen from the figure, the \u000b00has a peak close\nto the Hpeak. As mentioned earlier, \u000bmaxis obtained when the\nmagnetic domains collectively rotate, producing the highest en-\nergy dissipation in the magnetic phase. It implies that the peak\nof\u000b00should occur in the vicinity of Hpeak, where the peak of\nthe real part occurs. This is true for all samples, as evidenced\nby the curves.\n4. Conclusion\nTo sum up, the composite 0.94PbFe 0:5Nb0:5O3-0.06PbTiO 3\nand CoFe 2\u0000xZnxO4were prepared for x =0, 0.025, 0.1, 0.2, 0.3\nand a weight percentage of 60-40 (60 percent ferroelectric and\n40 percent ferrite). The XRD characterization revealed the for-\nmation of perovskite and spinel structures for ferroelectric and\nferrite phases, respectively. In addition, FE-SEM and EDS\nmapping images revealed a homogeneous and suitable distribu-\ntion of the magnetic phase within the ferroelectric matrix. The\nferroelectric to paraelectric phase transition occurs at 124.5C in\nthe dielectric constant curve for the pure PFN-PT. The dielec-\ntric constant is reduced by adding the ferrite phase. The elec-\ntric hysteresis loop for the pure PFN-PT is well-saturated in the\n20kV\ncmfield. The remanent polarization is reduced by including\nthe non-ferroelectric ferrite phase. Furthermore, the value of\nthe coercive field increases with the addition of ferrite, which\nis caused by the domain wall pinning. The magnetic hysteresis\nFigure 6: Real and imaginary part of transverse ME voltage coe \u000ecient (\u000b\u0003\n31) as\na function of DC magnetic field for (PFN-PT)-(CF 2\u0000xZxO) composites (x =0,\nx=0.025, x =0.1, x =0.2, x =0.3), h ac=0.77 Oe, and f =978 Hz.\n6curves revealed that the coercive field value reduces as the zinc\nimpurity increases. For each composite, the real and imaginary\nparts of the magnetoelectric coe \u000ecient were measured. All the\ncomposite samples possess \u000bsel f\u0000biased . The highest ME voltage\ncoe\u000ecient value of 12.33mV\ncm:Oeat DC magnetic field of 755 Oe\nwas achieved for x =0.1 sample. The measurements revealed\nthat the magnetoelectric coe \u000ecient is highly dependent on the\nzinc content. 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Within this model the \nkinetics of evolution of the sizes of precipitates of cons tant chemical compositi on under irradiation is \ninvestigated. The results obtained are compared to the experimental data [I. Monnet et al., J. Nucl. \nMater. 335 (2004) 311] for the ferritic ODS steel ЕМ10+MgO under electron irradiation. \n \nKeywords: semicoherent boundary; second phase precip itate; inequilibrium point  defects; ODS steel. \n \n1. Introduction \nThere are several mechanisms of irradiation infl uence on stability of heterophase precipitates\n1 in alloys \n(see e.g. [1]). Radiation disordering violates the phase stability of st ochiometric precipitates. Ballistic \neffects lead to the forced transpor t of substance between precipitate and matrix. If pr ecipitates contain \nsinks of point defects, then the radiation-induced se gregation of alloy’s compone nts in their vicinities \ntakes place (the inverse Kirkendall effect). Irra diation can cause polymorphous transformations \n(martensite transitions, amorphization, etc.). Migr ation of inequilibrium point defects between a \nprecipitate and a matrix can lead to change of a chemical compositi on of the precipita te. Irradiation-\nproduced continuous defects of crys talline structure (dis location loops, pores, etc.) can be nucleation \ncentres for secondary precipitates. However, some recent experimental data have no e xplanation in the framework of existing theoretical \nmodels. In the paper [2] the complex analysis of the effect of different types of irradiation on stability \n                                                \n \n1 Hereinafter we consider the heterophase precipitates whic h are different from the matrix alloy in their chemical \ncomposition.  2of precipitates in several types of ODS ferritic marten sitic steels is carried out. It has been shown, that \nneutron irradiation in Phénix re actor leads to dissolution of Y 2O3–based precipitates at high \ntemperatures and irradiation doses. To reveal th e mechanisms of precipitate dissolution, different \nirradiations by several types of ch arged particles were performed. It  turned out that irradiation by \nhelium ions with energy 1 MeV (the  region of domination of electr on losses, where the point defects \nare not produced) has no effect on the precipitates in  the steels studied. Irradiation by argon ions with \nenergy 300 keV, which produce cascades of point defect s (PD), leads to dissoluti on of the precipitates. \nIrradiation by 1 MeV and 1.2 MeV electrons, which produce isolated PD, leads to considerable \ndecrease of the precipitates based on MgO and Y 2O3 respectively. Thus it wa s demonstrated, that these \nare isolated PD which are basica lly responsible for precipitate di ssolution. Electron i rradiation with \nenergy 1 MeV does not affect Y 2O3-based precipitates, and 1.2 MeV el ectron irradiation leads to their \ndissolution. The displacement threshol d energy for yttrium ion in the Y 2O3 lattice is eV 57=dE  [2]. \nDuring electron-yttrium collision, th e maximum energy transferred is eV 49=tE  by 1 MeV electron \nand eV 64=tE  by 1.2 MeV electron [3]. Thus, the necessary  condition of precip itate dissolution is \nproduction of PD inside the precipita te. Therefore, the mechanism of radiation-induced segregation [4] \nin this case does not play a crucial role.  The analysis [2] of experimental  data on dissolution of MgO-based precipitates under 1 MeV electron \nirradiation shows, that the ballis tic mechanism of dissolu tion [5, 6] yields about  10 % of the observable \nvelocity of precipitate di ssolution. Besides, the observable velocity  of precipitate di ssolution increases \nwith temperature increase, whereas in both cases of radiation disordering relaxation and back-diffusion \nof recoil atoms one expects for an  opposite temperature effect.  \nThis paper presents a microscopic model of subs tance transport between a heterophase precipitate \n(below – a precipitate) and a substi tution solid solution (matrix) due to thermally activated migration \nacross the interphase boundary of inequilibrium PD produced by irradiation. Consider a precipitate \nwith concentration of one of its  components, conventionally desi gnated A (below – an impurity) \nconst c\np\nA= , located in a matrix with impurity concentration m\nAc, such that m\nAp\nAc c> . Irradiation \nproduces inequilibrium PD in the precipitate and in the matrix. Since concentration of the impurity \ninterstitials in the precipitate is greater than in the matrix, their diffusion fl ux through the interphase \nboundary is directed from the precipita te to the matrix. If concentrati on of the impurity in lattice sites \nof the matrix near the boundary exceeds its solubil ity limit in the solid solution, then there is an \nopposite flux of the impurity from the matrix to  the precipitate by a vacancy mechanism. The \ncompetition of these two fluxes determines the kinetics of evolution of the precipitate size. \n \n  32. A model of homogeneous semicoherent interphase  boundary \n \nOwing to the misfit of crystalline lattices of the adjacent phases the shear stresses occur at the interphase boundary. These stresses increase with in creasing of the precipitate size and at reaching of \nsome threshold value they are compensated by form ation of a network of mi sfit dislocations at the \nboundary surface. Such a boundary is conventionally  called semicoherent. The cores of misfit \ndislocations are sinks for inequili brium PD. At the same time, the c oherent regions of the boundary are \ntransparent for PD. Therefore, the flux of PD of the type n (\nin= – interstitial, vn= – vacancy), \naveraged over the bounda ry surface, in a phase ϕ ( p=ϕ  – precipitate, m=ϕ  – matrix) near the \nboundary can be represented as a sum of the absorbed ()panrjϕ and transited ()ptnrj  components: \n() () ()ptn pan p n rj rj rj +=ϕ ϕ. (1) \nBelow in the sections 2.1 and 2.2 within the framew ork of the model of homogeneous semicoherent \ninterphase boundary we find an explicit form for ()panrjϕ and ()ptnrj , and in the section 2.3 we derive a \nkinetic equilibrium impurity concen tration under irradiation conditions. \nThe stationary concentration profil es of PD with diffusion constants ϕ\nnD in an effective homogeneous \nabsorbing medium with a volume concentration of sinks ()2ϕκ , which is supposed to be identical for \nvacancies and interstitials, are set up during the characteristic times ()[]12~−ϕϕ\nκκτnD , which are much \nless than one second already at a room temperature. Therefore, one can consider the PD diffusion to be \nstationary. In further description th e mutual recombination of interstit ials and vacancies is neglected. \nThis assumption is valid for not very high PD con centrations, when the characteristic recombination \nlength is greater than the mean distance between PD sinks [1]. \n \n2.1 Absorption of PD at the interphase  boundary \n \nConsider a semicoherent interphase boundary (bel ow – a boundary) containi ng a network of misfit \ndislocations with a mean distance L between the dislocati on cores with radius dr. For a square network \nof misfit dislocations their surface de nsity (the length of the dislocati on line per surface unity) is given \nby the expression: \nLd2=σ . (2) \nTo estimate the value of ()panrjϕ we employ the following model. E ach misfit dislocation is assigned \nits region of influence in the form of a half of cyli nder with radius L/2 in both phases (see Fig. 1).   4 \nFig. 1. Regions of influence and core s of misfit dislocations  in the vicinity of an interphase boundary. \n \nIn this way, the whole surface of the interphase boundary appears to be divided into the regions of \ninfluence. Suppose that a region of influence is free from other sinks and sources of PD, and the space \nout of a region of influence is a homogeneous e ffective absorbing medium with PD concentration \nvolncϕ. This assumption is valid for small values of L: \n{}abs rec p llr L ,, min<< , (3) \nwhere pr is a precipitate radius, and recl and abslare respectively recombinati on and absorption lengths \nfor PD. \nThen the diffusion of PD in the region of infl uence is governed by the following diffusion problem: \n0 div=ϕ\nnj , ϕϕ ϕ\nn n n cD j∇−= ; (4) \n()0ϕ ϕ\nn d n c rc= , (5) \n()voln n c Lcϕ ϕ=2/ , (6) \nwhere ϕ\nnD is a PD diffusion constant, 0ϕ\nnc is a thermodynamic equilibrium PD concentration. \nThe equation (4) with the boundary conditions (5), (6) has the following solution: \n()()()\n().2 lnln 0 0ϕ ϕ ϕ ϕ\nn\ndd\nn voln n crLrrc c rc + −=  (7) \nThe total PD flux per unity of  length of the dislocation lin e is given by the expression: \n()\n()dn voln n\nSn nrLc cDdsj J2 ln0ϕ ϕϕ\nϕϕ ϕ π−==∫\n∈. (8) \nThen the average PD flux ab sorbed at the interphase boundary (a unit normal vect or is directed to the \nboundary) is given by the expression:  5()()\n()dn voln n\nd n panrLLc cDJ rj2 ln20ϕ ϕϕ\nϕ ϕ πσ−== . (9) \nNow we employ the approach of an  effective homogeneous absorbin g boundary. For th is purpose one \nneeds to express the flux of abso rbed PD (9) in terms of their mean concentration at the boundary. \nFrom (7) one finds the mean concentr ation of PD inside the region of influence (and, therefore, at the \nboundary): \n()()\n()()012\n0\n2\n212/\n2ln21211\n2/ϕ ϕ ϕϕ\nϕ\nn\ndd\nn volnL\nrn\np n crL\nLrc c\nLrdrrc\nrcd+\n⎪⎭⎪⎬⎫\n⎪⎩⎪⎨⎧\n⎥\n⎦⎤\n⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n⎟\n⎠⎞⎜\n⎝⎛−−−= =− ∫\n. (10) \nThen from (9) and (10) it is straightforward to find a relation between the flux of PD, absorbed at the \nboundary, and their mean con centration at the boundary: \n() ()[]0ϕ ϕϕ ϕαn p n n pan c rc rj − = , (11) \nwhere \n() ()[]{}2\n2121 2 ln2\nLr rL LD\nd dn\nn−−=ϕ\nϕ πα . (12) \nFrom (11) and (12) it follows that  there is a critical value of the distance between misfit dislocations \nd c r L 2= , at which the quantity ()[]0ϕ ϕ\nn p n c rc−  turns to zero, i.e. the boundary becomes completely \nincoherent. The case of 0 →ϕαn  is realized at ∞→L  and corresponds to a coherent boundary. \n \n2.2 Transition of PD across the interphase  boundary \n \nWithin this model the PD concentration profiles ar e supposed to be discontinuous at the interphase \nboundary. Therefore, PD transition across the boundary is  considered as a reve rsible surface chemical \nreaction. Thus, a frequency of transitions of interstitials of  each type across the boundary is proportional to their \npartial concentration in the corres ponding phase. Therefore, the flux of the interstitials, transited across \nthe boundary, is given by the following expression (herei nafter the normal unit vector is considered to \nbe guided from the precipitate into the matrix): \n() () ()[ ] ∑\n∈− =\npApm\niAm\niA pp\niAp\niA p it rc rc rj β β~ ~, (13) \nwhere the summation is taken over the precipitate’s components. \nThe kinetic coefficients in (13) depend on te mperature according to the Arrhenius’s law:  6()\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−∝Tkr G\nBp\niAiAϕβ ϕβ exp~, (14) \nwhere ()pr G\niAϕβ is an activation energy of transition of an  interstitial of type A in the corresponding \nphase, Bk is the Boltzmann’s constant, Tis a temperature. \nWithin this model the exchange of places between a vacancy and an  atom A, being on the different \nsides of the boundary, is repres ented in the form of a reve rsible chemical reaction: \nm p\nAp\nAm\nA vl vl +↔+ , (15) \nwhere m\nAl is an atom A in the regular site of the matrix lattice; p\nAv is a vacancy in the A sublattice of \nthe precipitate; p\nAl is an atom A in the regular site of the precipitate lattice; mv is a vacancy in the \nmatrix. Hereinafter it is considered, that atoms in the precipitate can occupy sites only in their native \nsublattices. In the spirit of the th eory of velocities of chemical reactions, the flux of the vacancies, \ntransited across the boundary, is gi ven by the following expression: \n() ()() ()[ ] ∑\n∈− =\npAp\nA pm\nvm\nvA pm\nA pp\nvAp\nvA p vt crc rcrc rj β β~ ~, (16) \nwhere the summation is taken over the precipitate’s components. \nThe kinetic coefficients in (16) depend on te mperature according to the Arrhenius’s law: \n()\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−∝Tkr G\nBp\nvAvAϕβ ϕβ exp~, (17) \nwhere ()pr G\nvAϕβ is an activation energy of transition of a vacancy to the place of an atom A in the \ncorresponding phase. \nThe total atomic flux, transited across the boundary, is given by the following expression: \n()()()p vt p it pt rj rj rj −= . (18) \nFrom the requirement of conservation of a chemical  composition of the precipitate, the partial flux of \natoms A, transited across the boundary, is related to  the total atomic flux (18) by the expression: \n()()0ωp\nA pt p At crj rj= , (19) \nwhere 0ω is a mean atomic volume. \nConsider, that partial concentrat ions of interstitials and vacanci es in the precipitate, and also \ninterstitials in the matrix are related to  their total concentrations as follows: \nm\nim\nAm\nAm\niAp\nnp\nAp\nAp\nnA cc x cvin cc x c0 0 ;, , ω ω = = = , (20) \nwhere ϕ\nAx is a dimensionless constant, accounting for a po ssible deviation of PD partial concentrations \nfrom the relation 0ωϕϕϕ\nA n nA c cc= . \nThen from (13), (16), (18), taki ng into account (20), one obtains:  7() ()() [ ]()()() [ ]pp\nvp\nvA pm\nim\niA pm\nA pm\nvm\nvA pp\nip\niAp\nA p At rc rc rc rc rc c rj β βω β βω + − + =0 0 , (21) \nwhere for brevity of records the kinetic coefficients are renormalized as follows: \n.~,~,~,~\n0ωββββ ββββm\nvAm\nvAm\nAm\niAm\niAp\nAp\nAp\nvAp\nvAp\nAp\niAp\niA x cx x = = = =  \nFrom comparison of (19) and (21) one obtains: \n()()()()() [ ]()p\nA pm\nA pp\nvp\nvA pm\nim\niA pm\nvm\nvA pp\nip\niA pt crcrc rc rc rc rj β β β β + − + = , (22) \nwhere A is some2 component of the precipitate. \nFrom comparison of (13), (16), (18), (22) it follows that \n()()()()p\nA pm\nA pm\nim\niA pp\nip\niA p it crcrc rc rj β β− = , (23) \n()()()()pm\nvm\nvAp\nA pm\nA pp\nvp\nvA p vt rc crcrc rj β β − = . (24) \n \n2.3. A local kinetic equilibrium at the interphase boundary \n \nThe state of the local kinetic equilibrium at the interphase boundary is determined by the requirement that the atomic flux transited across the interphase boundary turns to zero: \n() 0=peq\ntrj . (25) \nTaking into account (25), one can fi nd from (22) the relation between  the equilibrium impurity and PD \nconcentrations at the boundary: \n() () ()\n() ()peqp\nvp\nvA peqm\nim\niApeqm\nvm\nvA peqp\nip\niA\np\nApeqm\nA\nr c r cr c r c\ncr c\nβ ββ β\n++= . (26) \nThe equilibrium PD concentrations at both sides of the boundary are governed by the corresponding \ndiffusion equations:  \n()p\nip p\nip p\ni c D Kj2div κ−= , p\nip\nip\ni c D j∇−= , (27) \n()()0 2divp\nvp\nvp p\nvp p\nv cc D Kj − −= κ , p\nvp\nvp\nv c D j∇−= , (28) \n()m\nim m\nim m\ni c D K j2div κ−= , m\nim\nim\ni c D j∇−= , (29) \n()()0 2divm\nvm\nvm m\nvm m\nv c c D K j − −= κ , m\nvm\nvm\nv c D j∇−= , (30) \nwhere pK and mKare the volume PD generation rates in the pr ecipitate and in the matrix respectively. \nThe first set of boundary conditions for the diffusion e quations (27) – (30) is given by the expressions \n(1), (11), (23), (24) in the following explicit form: \n                                                 \n2 The kinetics of evolution of the size of a multicomponent preci pitate is usually considered to be determined by the \nvelocity of migration of its slowest component. In this way it is possible to make an unambiguous choice of the component \nA.  8()()()()()p\nA pm\nA pm\nim\niA pp\nip\niAp\ni pp\ni crcrc rc rj β βα − += , (31) \n() ()[]() () ()pm\nvm\nvAp\nA pm\nA pp\nvp\nvAp\nv pp\nvp\nv pp\nv rc crcrc c rc rj β β α − +− =0, (32) \n() () ( )()()pp\nip\niA pm\nip\nA pm\nAm\niAm\ni pm\ni rc rccrc rj β βα + +−= , (33) \n() ()[]()()()pm\nvm\nvAp\nA pm\nA pp\nvp\nvAm\nv pm\nvm\nv pm\nv rc crcrc c rc rj β β α − +− −=0. (34) \nThe second set of boundary conditions represents  the requirement of finitness of the PD \nconcentrations: \n()∞<0p\nic , (35) \n()∞<0p\nvc , (36) \n()∞<∞m\nic , (37) \n()∞<∞m\nvc . (38) \nIn the state of the local kineti c equilibrium the impurity and PD concentrations at the boundary are \nsubject to the relation (26). Then the equations ( 27) – (30) with the boundary conditions (31) – (34) \nand (35) – (38) repres ent a diffusion problem.  \nAs a result of its solution one can find a value of th e equilibrium impurity concentration at the \nboundary: \n()\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−− = 14122\n102\n21\naaa\naa\ncr c\np\nApeqm\nA, (39) \nwhere \n()()() ()()\n()⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡ +− − +⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− − =−\n− −\n21\n1\n21 0\n2 coth coth\nmm\npm\npp p\nppp\np\nip\ni\npp p\npp\nvp\nv p\nvp\nvA\nm\nim\niArKr rK\nDr rDcDDa\nκκκκ\nκακκββ,(40) \n()()() ()\n()()\n(), cothcoth coth\n0\n21\n11\n21 0 1\n1\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−+\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− −+\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n− +⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ +− −⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛++ =\n−\n−− − −\nm\nvm\nvAmm\npm\np\nip\ni\npp p\np m\nim\niApp p\nppp\np\nip\niAp\ni\npp p\npp\nvp\nvm\np m\nim\ni\np\nvp\nvA\ncrK\nDr rDr rK\nDr rDc rD Da\nβ\nκκ ακκβκκ\nκβακκ καβ\n, (41) \n()()()\n()\n()() . cothcoth\n1\n21\n1 01 1\n2 0\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−++\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡ ++⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+++⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+++ − =\n−−\n−− −\np p\nip\niAp\ni\npp p\nmm\npm\nm\nvm\nvA m\np m\nim\ni m\nvm\nvAm\nvm\nvA m\np m\nim\ni\np pp p\npp\np\nip\niA\nrDrrK\nDrDcDrDr rK\nDa\nβακκ\nκκ βκαββκακκ\nκβ\n (42) \nIf irradiation is absent ( )0 ,0==p mK K , the equilibrium impurity concentration at the boundary is as \nfollows:  9() ()\n()pp\nvp\nvApm\nvm\nvA\np\nApm\nA\nrcrc\ncrc\n00 0\nββ= . (43) \n \n3. Kinetics of evolution of  the precipitate size \n \nLet the impurity diffusion in the matr ix is governed by the usual equation: \nm\nAm\nAm\nAm\nAm\nAc D j jtc∇−= −=∂∂, div , (44) \nwhere m\nAD is the impurity diffusion constant in the matrix. \nConsider, that prior to irradiation the system precipi tate-matrix was in its equilibrium state. Thus, the \ninitial condition is \n()00,m\nAm\nA c rc= , (45) \nwhere 0m\nAc is a thermodynamic equilibrium impurity concentration in the matrix. \nThe impurity flux transited across the boundary is given by the expression (21). In  the first order in a \ndeviation of the impurity concentrat ion at the boundary from its equ ilibrium value (39), the expression \n(21) looks like: \n() () ( ) () [ ]()() [ ]trc r cr c r c trjtrjpm\nA peqm\nA peqp\nvp\nvA peqm\nim\niA p At pm\nA , , ,0 − + = = β βω . (46) \nConsider, that in the matrix each precipitat e occupies a spherical region with radius R, such, that the \nimpurity flux at the boundary of this region is zero. Such a region is conventi onally called an influence \nregion. Thus, the second boundary condition looks like: \n() 0 ,=tRjm\nA . (47) \nTaking into account the initial (45) and the boundary  conditions (46), (47), the impurity diffusion \nequation (44) has the solution: \n() () ()[] ()[]()()()[] ∑∞\n=−−⋅ − + +=\n120\n00exp1 ,\nnn\nn npn n p m\nA peqm\nA peqp\nvp\nvA peqm\nim\niAm\nAm\nA t\nyryry\nrrc r cr c r c ctrc λ\nλβ βω , (48) \nwhere \n()∫=R\nrn n\npdxxy y2 2. (49) \nHere the eigenfunctions of the Sturm-Liouville problem, corresponding to the given diffusion problem, \nare as follows: \n() xDRDRDxDxym\nAn\nm\nAn\nm\nAn\nm\nAn\nnλ λ λ λcos arctantan sin ⋅\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+ = , (50)  10and the corresponding eigenvalues nλ are roots of the equation \n() ()[ ]() n rRD Dr c r crDRDp m\nAn\nm\nApeqp\nvp\nvA peqm\nim\niA\np m\nAn\nm\nAnπλ β βωλ λ−− =\n⎥⎥⎥\n⎦⎤\n⎢⎢⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ ++ −⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−\n−1\n0 1arctan arctan . (51) \nThe velocity of change of the pr ecipitate size is determined by th e atomic flux tran sited across its \nboundary: \n()ptprjdtdr\n0ω−= . (52) \nThe atomic flux transited across the boundary is given by the expression (22). In the first order in a \ndeviation of the impurity concentrat ion at the boundary from its equ ilibrium value (39), the expression \n(22) looks like: \n() ( ) ( )[ ]()() [ ]p\nA pm\nA peqm\nA peqp\nvp\nvA peqm\nim\niA pt ctrc r cr c r c trj , , − + = β β . (53) \nTaking into account the above-stated, the expr ession (52) gets the next explicit form: \n() ()[ ]()[]\n() ()[]()()[] .1 exp1\n122\n00\n0\n⎪⎭⎪⎬⎫\n⎪⎩⎪⎨⎧\n−−−⋅ ⋅ +×− +=\n∑∞\n=nn\nn np n\npeqp\nvp\nvA peqm\nim\niAp\nAm\nA peqm\nA peqp\nvp\nvA peqm\nim\niA p\nt\nyryr c r ccc r cr c r c\ndtdr\nλ\nλβ βωβ βω\n (54) \nThe characteristic time for the impurity diffusion front  to reach the boundary of the influence region is \nas follows: \n()m\nA p R D rR2−=τ . (55) \nIn the case when the observat ion time is much less than Rτ, one can consider the impurity diffusion to \noccur in an infinite matrix and the second boundary condition becomes: \n()0,m\nAm\nA ct c=∞ . (56) \nThe solution of the equation (44) with the initial condition (45) and the boundary conditions (46), (56) \nhas a more obvious look: \n()() ()[ ]()[]\n() ()[]()\n()() () () ()\n() ().\n2erfcexp2erfc ,\n012\n01\n01010\n0 0\n⎪⎭⎪⎬⎫\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ +++−×\n⎥⎥⎥\n⎦⎤\n⎢⎢⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ +++⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ ++−⎪⎩⎪⎨⎧\n−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−⋅\n+ +− ++=\n−− −−\ntDDr c r cr\ntDrrtDDr c r crDr c r crrrtDrr\nrr\nDr c r c rDc r cr c r cctrc\nm\nA m\nApeqp\nvp\nvA peqm\nim\niA\npm\nApm\nA m\nApeqp\nvp\nvA peqm\nim\niA\np m\nApeqp\nvp\nvA peqm\nim\niA\np pm\nAp p\nm\nA peqp\nvp\nvA peqm\nim\niA pm\nAm\nA peqm\nA peqp\nvp\nvA peqm\nim\niA m\nAm\nA\nβ βωβ βωβ βωβ βωβ βω\n (57) \nAt the same time the expression (54) also becomes simpler:  11() ()[ ]()[] ()( )[ ]\n() ()[]()\n() () () (). erfc exp11\n012\n010100\n0\n⎪⎭⎪⎬⎫\n⎥⎥⎥\n⎦⎤\n⎢⎢⎢\n⎣⎡\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ ++\n⎥⎥⎥\n⎦⎤\n⎢⎢⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ ++ −⎪⎩⎪⎨⎧\n×\n+ +++−− +=\n− −−\ntDDr c r cr tDDr c r crDr c r c rDr c r c\ncc r cr c r c\ndtdr\nm\nA m\nApeqp\nvp\nvA peqm\nim\niA\npm\nA m\nApeqp\nvp\nvA peqm\nim\niA\npm\nA peqp\nvp\nvA peqm\nim\niA pm\nApeqp\nvp\nvA peqm\nim\niA\np\nAm\nA peqm\nA peqp\nvp\nvA peqm\nim\niA p\nβ βωβ βωβ βωβ βω β βω\n (58) \nIt follows from the expression (58)  that the kinetics of evolution of the precipitate radius has the \ncharacteristic time scale \n() ()[]()m\nAm\nA peqp\nvp\nvA peqm\nim\niA p r D Dr c r c r\np2\n01−−+ += β βω τ , (59) \nduring which the impurity concentration at the boundary saturates to its equilibrium value ()peqm\nAr c . \nAt the initial stage of the precipitate evolution, when \npr tτ<< , the velocity of dissolution is determined \nby the transboundary kinetics of  migration of the impurity: \n() ()[ ]()[]\np\nAm\nA peqm\nA peqp\nvp\nvA peqm\nim\niA p\ncc r cr c r c\ndtdr0\n0− +−≈β βω . (60) \nAt the later stage of the precipitate evolution, when \npr tτ>> , the velocity of dissolution is determined \nby the volume diffusion of the impurity \n() ()[ ]()[]\n() ()[]()m\nA peqp\nvp\nvA peqm\nim\niA pp\nAm\nA peqm\nA peqp\nvp\nvA peqm\nim\niA p\nDr c r c r cc r cr c r c\ndtdr\nβ βωβ βω\n+ +− +−≈\n00\n01. (61) \n \n4. Comparison of the model results to experimental data \n \nTo compare the results of this model to the expe rimental data [2] on di ssolution of the MgO-based \nprecipitates in the ferritic mart ensitic steel EM10 under electron i rradiation with energy 1 MeV it is \nreasonable to make some additional simplifications. Since in the absence of irradiation both magnesium and oxygen are practical ly insolvable in the ferritic matrix [7], one can put \n.00 0=p\nAm\nAc c  (62) \nSince the melting temperature of MgO essentially exceed s that of a steel, in the absence of irradiation \nthe equilibrium concentrations of vacan cies should satisfy the inequality 0 0m\nvp\nv c c<< . Then from (62), \ntaking into account (43), it follows, that \n,0=m\nvAβ  (63) \ni.e. the flux of vacancies from the matr ix into the precipitate is negligible.  12Under the experimental conditions [2 ] considered, the rate of PD pr oduction both in the precipitate and \nin the matrix is of the order of 10-3 dpa/s, and the temperature is C 550°≤T . Therefore, in the both \nphases the nonequilibrium vacancies produced by i rradiation dominate over the equilibrium ones: \n()p\nvp p p\nv D K c2 0κ<< , (64) \n()m\nvm m m\nv D K c2 0κ<< . (65) \nTherefore, in the expression for the impurity di ffusion constant in the matrix we neglect the \ncontribution from the equilibrium vacancies and l eave only the contribution from the nonequilibrium \nPD, assuming lack of their recombination (see e.g. [1]): \n()2\n0 2m m m\nA K D κω= . (66) \nSince the maximal dose p\nmaxΦ  reached for the precipitate (see Fig.  2 below) is much less than the dose \ndpa100~0Rp\nRKτω=Φ , at which the diffusion front of th e impurity reaches the boundary of the \ninfluence region w ith the radius nm 10~3R  of the order of the radius of the focused electron beam, \nfor comparison with the experimental data we use the expression (58).  \nTaking into account the assumptions (62) – (66) ma de, the results of numerical integration of the \nequation (58) with the initial condi tions from Table 1 and the values of the model parametres from \nTable 2, together with the experimental data [2 ] on dissolution of the MgO-based precipitates in the \nEM10 steel under 1 MeV electron irra diation at several temperatur es, are given in Fig. 2.  \n \n0 2 4 6 8 10 12 14 16 180102030405060708090100 Radius decrease (nm)\nDose for the precipitate (dpa)      Experiment:  Theory:\n3000C           \n4000C           \n5000C           \n5500C           \n \nFig. 2. Decrease of the precipitate radius vs. the dose absorbed by it. Points represent the experimental \ndata [2]. Lines represent the numer ical solution of the equation (58) with the initial conditions from \nTable 1 and the parametres from Table 2.       13Table 1. The initial precipitate sizes \nTemperature, C° 300 400 500 550\nInitial radius, nm 100 125 150 120\n \nTable 2. Values of the parametr es entering the expression (58). \ndpa/s,0ωpK \nnm, 1pκ \ndpa/s,0ωmK\nnm, 1mκ nm,L \nnm,dr\nnm,m\niAm\niDβ \nnm,p\niAp\niDβ \nnm,p\nvAp\nvDβ \n310− 55 3106−⋅ 10 5,10 6,0\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛⋅T2400exp3,0 ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛⋅T2400exp3,0 ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛⋅T3000exp3 \n \nFrom Fig. 2 one can see, that at the chosen values  of the model parametres the theory gives a good fit \nto the experimental data at low temper atures and a qualitative fit at 550ºC. \n \n5. Discussion \nThe model of homogeneous semicoherent interphase  boundary describes the processes of substance \ntransport between a precipitate and a matrix a lloy due to thermoactivated migration across the \ninterphase boundary of irradiation- produced inequilibrium point defects.  It allows to investigate the \nkinetics of substance redistributi on and change of the precipitate si ze in alloys under irradiation.  \nIn section 2 of the paper it is shown, that within  the model irradiation leads to modification of the \nvalue of the impurity equilibrium concentration (39)  in the matrix near the boundary. Expression (39) \nincludes 11 parametres: ϕϕαn nD, characterising an absorptive ability  of the boundary and identical for \nall types of PD in bot h phases; 4 parametres ϕϕβn nAD, characterising a tran sparency of the boundary \nfor PD from both sides, 2 parametres ϕκ, characterising a volume density of PD sinks in the phases, 2 \nparametres ϕK, characterising volume PD production rates un der the influence of irradiation and 2 \nparametres 0ϕ\nvc, being thermodynamic equilibrium concentrations of vacancies in both phases.  \nIn section 3 the kinetics of substance redistributi on between the precipitate and the matrix is studied \nand the expression (54) for the velocity of the precipitate radius change  under the influence of \nirradiation is found. The expressi on (54) and its subsequent simp lifications include an additional \nparametre m\nAD, being the impurity diffusion constant in the matrix.  \nIn section 4 the results of numerical solution of the equation for the velocity of the precipitate radius \nchange under the influence of irradi ation in the simplified form (58) are compared to the experimental \ndata [2] on dissolution of the Mg O-based precipitates in the fe rritic ODS steel EM10 under 1 MeV \nelectron irradiation. With the values of initial cond itions from Table 1 and the model parametres from  14Table 2, the theory has a good agreement with the e xperiment at temperatures 300 ºC and 400 ºC and a \nqualitative agreement at 550 ºC.  \n6. Summary \n• The model of homogeneous semicoherent in terphase boundary provides a microscopic \ndescription for the processes of absorption a nd thermoactivated migration of irradiation-\nproduced inequilibrium point defects at a semicoherent boundary between a heterophase \nprecipitate and a substi tution solid solution. \n• The main result of the model is the expre ssion for the atomic flux, transited across the \nboundary. According to the model, the flux incr eases with increasing the production rate of \npoint defects and the temperature. \n• The model allows a good fit to the experiment al data [2] on dissolution of the MgO-based \nprecipitates in the ferritic ODS steel  EM10 under 1 MeV electron irradiation. \n \nAcknowledgements \nThe author is grateful to Prof. A.  Bakai, Dr. N. Lazarev and Dr. A. Turkin for introducing him into the \nphysics of radiation effects in so lids and numerous discussions.  \n \nReferences \n[1] Frank V. Nolfi Jr. (Ed.), Phase Transformations  During Irradiation, Applied Science Publishers, \nLondon, 1983. \n[2] I. Monnet, P. Dubuisson, Y. Serruys, M.O. Ru ault, O. Kaitasov, B. Jouffrey, Microstructural \ninvestigation of the stabili ty under irradiation of oxide  dispersion strengthened ferritic steels, J. Nucl. \nMater. 335 (2004) 311–321. [3] O.S. Oen, Cross Sections for Atomic Displ acements in Solids by Fast Electrons, Report #4897, \nOak Ridge National Laboratory, Oak Ridge, 1973. [4] H. Wiedersich, P. R. Okamoto, N. Q. Lam,  A theory of radiation- induced segregation in \nconcentrated alloys, J. Nucl. Mater. 83 (1979) 98-108; A. S. Bakai, A. A. Tu rkin, Radiation-modified \nphase diagrams of binary alloys, in: R. E. Stoller, A. S. Kumar, D. S. Gelles (Eds.), Effects of \nRadiation on Materials: 15th International Symposium, vol. I, ASTM STP 1125, American Society for \nTesting and Materials, Phila delphia, PA, 1992, pp. 709-730. \n[5] R. S. Nelson, J. A. Hudson, D. J. Mazey, The stab ility of precipitates in an  irradiation environment, \nJ. Nucl. Mater. 44 (1972) 318-330.  15[6] H. J. Frost, K. C. Russell, R ecoil dissolution and particle stabilit y under irradiation, J. Nucl. Mater. \n104 (1981) 1427-1432; D.S. Gelles, F.A. Garner, An e xperimental method to determine the role of \nhelium in neutron-induced microstructural evol ution, J. Nucl. Mater. 85&86 (1979) 689-694. \n[7] M. Hansen, K. Anderko, Constitution of Bi nary Alloys, second ed., vol. II, McGraw-Hill Book \nCompany, New York, 1958. "    },    {        "title": "1309.5218v2.High_values_of_Ferroelectric_polarization_and_magnetic_susceptibility_in_BFO_nanorods.pdf",        "content": "High values of Ferroelectric polari zation and magnetic susceptibility \nin BFO nanorods \nNabanita Dutta1, S.K.Bandyopadhyay1*, Subhasis Rana1, Pintu Sen1, A.K.Himanshu1 and \nP.K.Chakraborty2 \n1. Variable Energy Cyclotron Centre, 1/ AF, Bidhan Nagar, Kolkata-700 064, India. \n2. Department of Physics, Burdwa n University, Burdwan 713104, India. \n*Corresponding author. E-mail: skband@vecc.gov.in \n \nAbstract:  \n Remarkably high values of polarization as well as a significant magnetic susceptibility \nhave been observed in multiferroic Bismuth Ferri te (BFO) in the form of nanorods protruding \nout. These were developed on porous Anodised Al umina (AAO) templates using wet chemical \ntechnique. Diameters of nanorods are in the ra nge of 20-100 nm. The high values of polarization \nand magnetic susceptibility are attributed to th e BFO nanorod structures giving rise to the \ndirectionality. There is no leak age current in P-E loop examined at various frequencies. \nMagnetocapacitance measurements reflect a significant enhancement in magnetoelectric \ncoupling also.   \nKeywords: A. Bismuth Ferrite; A. Nanorod; E.Polarizat ion; E. Magnetic susceptibility  \n \n1. Introduction:   \nBismuth ferrite (BFO) is a quite interest ing multiferroic showing coexistence of \nferroelectric as well as antiferromagnetic propertie s [1,2]. BFO exists in nature as a partially \ncovalent oxide with a rhombohedrally distorted pe rovskite structure belonging to the space group \nofR3cwhich allows for a ferroelectricity below 1083 K and G type antiferromagnetism with a \nvery high Neel temperature TN of 643 K [3]. However, so metimes it exhibits weak \nferromagnetism as a result of size manifestation that leads to a relative ordering. This type of \nweak ferromagnetism in antiferromagnets is explained through Dzialoshinsky-Moriya \ninteraction, exchange bias, strain  manipulation etc [4,5]. BFO seem s to be a promising candidate \nfor magnetic storage or other devices in spintr onics applications [6]. The presence of 6s2 lone \npair of electrons in bismuth acc ounts for the polarization for ferroelectricity. However, leakage \ncurrents due to oxygen vacancies or impurities of BFO lead to a major shortcoming. Recent \napproaches on it are being focused on single/poly crystals and substrate- free nanostructures as \nwell as low-dimensional nanostructures, like na noparticles, nanowires/rods fiber etc. In particular, the nanostructured forms of this materi al are expected to deli ver enhanced properties \nof magnetization and polarization  necessitating the drive for synthesizing in nanostructured \nform. However, each synthesis protocol has go t some advantages and limitations. We have \nchosen Anodized Alumina (AAO) templates to gr ow BFO nanowires. In this communication, we \nreport the development, characterization of BF O nanorods by template assisted wet chemical \ntechnique and their magnetic suscep tibility and pola rization studies.  \n \n2. Experimental details:   \nCommercially available AAO templates of 60µ m thickness and 13mm dia with pore size \ndistribution ranges from 20-200nm have been employed. 0.1M aqueous solutions of Bi(NO 3)3 \nand Fe(NO 3)3 were prepared with stoichiometric amount s of the nitrates using methoxymethanol \nas solvent. pH of the solution was adjusted to  2-3. Since the filling takes place with capillary \naction, it gets hindered due to ve ry small diameter of the nanopores  leading to inad equate filling \nof pores.  The filling of pores was achieved by the directional flow of the ions in the template \nadopting the controlled vacuum technique.  The templates with pores containing solution were \ndried in air and sintered for 3 hours at 7500C to get the required phase without unwanted grain \ngrowth. The templates are very much fragile and get bent during sintering. Hence we took \nspecial care keeping them sandwiche d between two thin ceramic plates  to receive in intact form.  \nThe most important part of the process is  the etching to obser ve nanowires/rods by \nelectron microscopy. Templates were half etched so  that the rods would be visible. They were \nnot self supporting. We attempted controlled et ching process with 1M NaOH and the bundle of \nnanowires emerged. Weight of BFO nanorod developed was measur ed using a very sensitive \nmicrobalance (with resolution of 1µgm) by subtracting the weight  of AAO and it was in the \nrange 80-100µgms. for various templates.  \nThe nanowires and nanorods were examin ed by FEI Scanning Electron Microscope \n(SEM) with a resolution of 6nm and provided with Energy Dispersive X-ray (EDX) Analysis. \nTransmission electron microscopy (TEM) was ta ken by high resolution TEM (Model: FEI T20 \nwith applied voltage of 200KV). Since the nano wires/ rods are grown in AAO templates, the \nsample preparation for TEM analysis is the cr ucial matter and it was subjected to extreme care \nthrough a number of steps. The AAO templates c ontaining BFO nano wires half etched with \nNaOH were washed thoroughly to remove NaOH before TEM studi es. Physical properties of \nsynthesized BFO nano wires basically  ferroelectric measurements were carried out to investigate \nferroelectric behavior. Ferroel ectric polarization was measured  through P-E loop analyzer \n(Model: Precession, Radiant Technology)  at different electric fields  and frequency. Up to a field \nof 60KV/cm was employed withou t any break down. Magnetic susc eptibility was measured from \nroom temperature down to 25K  at a field of 1300 gauss. \n 3. Results and Discussions: \n \n3.1. Electron Microscopy:  \nWe have received two kinds of  1D nanostructure in the fo rm of nanowires as well as \nnanorods as apparent from SEM similar to earlier  groups [7, 8]. Cross sectional view is showing \ndevelopment of nanorods in Fig. 1a- a representati ve case. It demonstrat es the structures of \nseveral nanorods (around 20 in a region of 5 µm x 5 µm) with high aspect ratio protruding out of \nthe pores after partial etching. There were almo st similar number of nanorods embedded in the \npores. We noticed around 400 pores in 5 µm x 5 µm . TEM reveals nanorods of high density with \nintact structure in Fig. 1b. A clea r SAED pattern (inset of Fig. 1b) with a pr ominent ring signifies \ndevelopment of polycrystalline BFO.   \n \n3.2. Polarization:  \nFig. 2 depicts P-E loops at various frequencies from 90-400 Hz. The hysteresis loops are \nquite appreciable characteristic of a ferroelectric material with the loop width decreasing with \nincreasing applied frequencies. We have not attain ed saturation up to the applied voltage field of \n60KV/cm. Measurement has been done for blank AAO template too and there is no signature of \nany hysteresis loop. The gap in the plot is  seen due to improper depolarization.   \nThe relative displacements of Bi and O ion induced by the repulsion of Bi(III) 6s2 lone \npair of electrons result in a net polarization. Such polarization is an example of orientational \npolarization which is affected by distribution of the polarizati on domains. Relatively low value \nof polarization is a characteristic  of nanoparticles. However, it is quite high in epitaxial BFO \nfilms and BFO single crystals [9], where the proj ection of polarization takes the orientation along \nthe easy axis of growth of crystals. The net pol arization matches along the direction of growth \nbasically along (111)  plane in case of  BFO [10].                                                         \nMost significantly, the obser ved polarization of BFO na norods is remarkably high \n(~0.04µC/cm2) considering the weight of the na norods (80 micrograms). The observed \npolarization value comes in the same  order as that of the single crys tal. The value is substantially \nlarge as compared to our earlier polarization values obtained in case of agglomerated BFO \nnanoparticles [11]. This high value of polarization can be attributed to the di rectional growth of \nBFO as nanorod structure. We have employed a moderate sintering time to avoid the grain \ngrowth. Moreover, preheated furnace was employed to get rid of the loss of Bismuth. Bismuth to \nIron ratio was nearly 1:1 as verified through E DX studies. There is no leakage current (indicated \nby decrease in polarization with the electric fiel d) associated with our nanorods. The leakage \ncurrent is observed mainly in grain boundary  region due to the accumulation of defects as \noxygen vacancies as well as metallic impurities like Bi3+, iron oxide etc. Organized growths of \nBFO nanorods facilitate minimization of gr ain boundaries as well as absence of oxygen \nvacancies, metallic impurities leading to th e disappearance of the leakage current.   \n3.3. Magnetic measurements:   \nMagnetic susceptibility versus temperature of BFO nanorods is shown in Fig. 3. BFO \nnanorods show a substantially hi gh magnetic susceptibility of th e order of 2.5emu/mole at low \ntemperature under the application of magnetic field of 1300 Gauss.   This result corresponds to \nthe susceptibility obtained in single crystal repo rted earlier [12]. High su sceptibility of BFO in \nsuch nanostructured form is attributed to the directional growth.  \nUsually a weak ferromagnetism is observed in bulk BFO, which gets enhanced in case of \nsingle crystal due to the matc hing of magnetization direction and easy axis of growth. The \ngrowth directionality of BFO na norods in this case can be compar ed to the directionality in \nsignal crystal leading to the high value of suscep tibility. The negative susceptibility at room \ntemperature is due to the diama gnetic contribution of AAO template , which is large compared to \nthe weak ferromagnetic susceptibility of BFO nanorods at that  temperature.   \n \n3.4. Magneto capacitance:   \nMagnetocapacitance gives an idea about the magne toelectric coupling in an indirect way   \n[13]. The capacitance per unit field has been plotted against the magnetic field as depicted in Fig. \n4. Magnetocapacitance has been normalized with respect to the magnetic field to attain \nquantitative information and it appears to be quite high.  Generally magnetocapacitance increases \nwith the increasing value of the field and this si gnature is also evident in our measurement. In \ncase of epitaxially grown thin films, if the polar ization direction is in plane with the magnetic \nfield, the film experiences maximum effect of magnetic field, but if  it is out of plane, the net \neffect is zero. Similarly in this case also it becomes large due to the directional growth of the \nnano rod depending upon their orientation with re spect to the magnetic field. The increasing \ncapacitance with the increase of applied field suggests an indication of magneto electric coupling. \nMagnetocapacitance value is also high by a factor of five compared to that of BFO nanoparticles \nobserved by us earlier [11] i ndicating a larger magnetoelectric coupling in case of nanorods. \nThus, the nanostructure has not only enhanced th e magnetic susceptibility  and polarization, but it \nhas also led to the enhancement of the magnetoel ectric coupling.  A smooth pattern is not noticed \nhere due to the unequ al interaction of na norods. Each individual nanorod responds to the \nmagnetic field differently as per their area and this leads to a random effect in \nmagnetocapacitance.  \n                                  \n4. Conclusion: We have studied magnetic susceptibility as  a function of temeperature, ferroelectric \npolarization and magnetocapacitanc e of BFO nanorods developed on AAO templates at various \nfields and frequencies. The va lues of polarization (~0.04µC/cm2) are remarkably high compared \nto the bulk considering the low wt. of 80 µgms . Furthermore, the magnetic susceptibility is \nsignificantly high. This high value of polarization and magnetic su sceptibility as well  can be \nattributed to the directionality achieved in th e nanorod structure. The pattern of the growth \nprohibits development of grain boundaries paving the pathway for leakage current free material.  \nThe magnetoelectric coupling in BFO nanor ods as obtained thro ugh magnetocapacitance \nmeasurements is also quite high as compared to  that of BFO nanoparticles. All measurements \nconsistently establish the enhancement of multiferroic properties in the nanostructure, which is \nvery important from the applications point of view.  \n \nAcknowledgements: Authors gratefully acknowledge Dr. P.K.Mukhopadhyay and Mr. Sakti \nNath Das of S.N.Bose Center for basic Scien ces for SEM studies and Mr. Pulak Kumar Roy of \nSaha Instititute of Nuclear Physics for TEM studies.  \n \nReferences: \n [1] J. Wang, J.B. Neaton, H. Zheng, V. Naga rajan, S.B. Ogale, B. Liu, D. Viehland, V. \nVaithyanathan, D.G. Schlom, U. V. Waghmare, N.A. Spaldin, K. M. Rabe, M. Wuttig and  R. \nRamesh, Science , 299  (2003)  1719-1722. \n[2] M. M. Kumar, V.R. Palkar and S.V. Suryanarayana, Appl. Phys. Lett.  76 (2000) 2764-2766. \n[3] G. A. Smolenskii and I. E. Chupis, Sov. Phys. Usp.   25 (1982) 475–493.  \n[4] Dzyaloshinesky, J. Phys. Chem. Solids  4 (1958)  241-255.   \n[5] Toru Moriya, Phys. Review  120 (1960)  91–98.   \n[6]    J. F. 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Rev  B 7 \n(2003) 18040-1846(R).  \n     \n \n      \n \n     \n \n  \n   \n \n \nFig. 1(a) SEM of nanorod protruding from pores and (b) TEM of BFO nanorods and \ncorresponding SAED pattern shown in the inset.  \n \n \n      \n \n  Fig.  2. Ferroelectric hyste resis loop of BFO nanorod. \n    \n \n  \n (a) \n (b)\n-80 -60 -40 -20 0 20 40 60 80-0.05-0.04-0.03-0.02-0.010.000.010.020.030.040.05 \n Field (kV/cm)Polarization (µC/cm2) 400Hz\n 300Hz\n 200Hz\n 90Hz\n0 50 100 150 200 250 300-1.0-0.50.00.51.01.52.02.5Susceptibility (emu/mole)\nTemperature (K) \n   \n \n \\     \n \n   Fig. 3. Magnetic susceptibility versus temp of BFO nanorod. \n \n  \n \n      0 3000 6000 9000 12000 150000.0020.0040.0060.0080.010MC (F/Gauss)\nMagnetic Field (Gauss) \n   \n \n      \n \n \nFig. 4. Magnetocapacitan ce plot of BFO nanorod.\n\n "    },    {        "title": "2205.06233v1.Structural__Dielectric__and_Electrical_Transport_Properties_of_Al3__Substituted_Nanocrystalline_Ni_Cu_Spinel_Ferrites_Prepared_Through_the_Sol_Gel_Route.pdf",        "content": "1 | P a g e \n Structural, dielectric, and electrical transport properties of Al3+ substituted nanocrystalline \nNi-Cu spinel ferrites prepared through the sol-gel route \nM. M. Rahmana, N. Hasanb, M. A. Hoquec, M. B. Hossend, M. Arifuzzamane,* \naDepartment of Industrial and Production Engineering, Bangladesh University of Textiles, Dhaka-\n1208, Bangladesh \nbDepartment of Electrical and Computer Engineering, North South University, Dhaka-1229, \nBangladesh \ncBangladesh Council of Scientific and Industrial Research, Dhaka-1205, Bangladesh \ndDepartment of Physics, Chittagong University of Engineering and Technology, Chattogram-\n4349, Bangladesh \neDepartment of Mathematics and Physics, North South University, Dhaka-1229, Bangladesh \n \n*Corresponding author: \nM. Arifuzzaman (md.arifuzaman01@northsouth.edu) \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 2 | P a g e \n Abstract \n \nIn this study, a series of nanocrystalline ferrites of Ni 0.7Cu0.30AlxFe2-xO4 (x=0.00 to 0.10 with a \nstep of 0.02) has been synthesized through the sol-gel auto combustion technique. The structural, \nmorphological, dielectric, and electrical properties of the Ni-Cu spinel ferrite nanoparticles are \nanalyzed due to the substitution of Al3+ content. The crystalline and structural characteristics of \nthe prepared nanoparticles (NPs) have been studied employing the x-ray diffraction (XRD) spectra \nand FTIR analysis. The extracted XRD patterns assure the single-phase cubic spinel structure of \nall samples with homogeneity and no impurity, which indicates the yielding of high crystalline \nNPs. The average crystallite size of the synthesized ferrite nanoparticles is found in the range  \n(55.63–70.74 nm) and the average grain size varies from 59.00 to 65.00 nm. FTIR study also \nconfirms the formation of spinel structures in the prepared Ni-Cu ferrite nanoparticles. A slight \ndecrease of average grain size with increment of Al3+ content is observed in the surface \nmorphological study carried out by the field emission scanning electron microscopy (FESEM). \nThe studied materials are found in semi-spherical shapes, showing the multi-domain grains \nseparated by grain boundaries with some agglomerations. The chemical composition study for the \nsynthesized NI-Cu spinel ferrites using energy dispersive x-ray (EDX) ensures the presence of \neach component in appropriate proportions in each sample. The dielectric dispersion nature of all \ninvestigated materials is reflected in the current study up to the frequency of 10 kHz. Because of \nthe high resistive grains, the value of ′ is higher at low frequencies, resulting in space charge \npolarization. However, the effect of cation distributions on A and B sites on the grain dependent \nspace charge polarization nature is reflected in the dielectric constant. The sample with x = 0.1 \ndemonstrates that the space charge polarization has increased, resulting in a higher dielectric \nconstant value. The impedance spectroscopy confirms the non-Debye relaxation phenomena of the \nsynthesized nanomaterials. The contribution of grains and grain boundaries is resolved through the \nmodulus study of the materials, which reconfirms their dielectric relaxation. The trend in variation \nof AC resistivity suggests the normal behavior of the materials with varying frequencies, which is 3 | P a g e \n explained by the hopping mechanism.   \nKeywords:  Spinel nano-ferrites; Sol-gel process; XRD; FTIR; FESEM; Impedance spectroscopy; \nDielectric dispersion; AC resistivity. \n \n1. Introduction:  \nMagnetic ferrite nanoparticles in the spinel phase have been considered as the influential class \nof materials, which are employed in various high-frequency device applications1. The cubic spinel \nstructure has the chemical formula of AB 2X4, where the anions X are occupied by O atom as of \nmetal oxides forming the cubic close-packed lattice, tetrahedral interstices fill the A site as the \n‘network formers’ and octahedral interstices occupy the B site as the ‘modifiers’, called the Ferro-\nspinel and semiconductor in nature 2–5.  Most spinel ferrites belong to the space group of Fd3m \n(No. 227, Z = 8), which provide the highest symmetrical face-centered cubic (FCC) spinel \nstructure. A spinel unit supercell's crystal is formed by 8 A-sites and 16 B-sites cations. Based on \nthe distribution of divalent metal ions and trivalent ferric ions over A and B sites, spinel ferrites \nare of three classes; normal spinel, inverse spinel, and mixed spinel 6. \nMagnetically soft spinel ferrites are used in a large spectrum of biomedical and industrial \napplications including medical treatments, such as magnetic resonance imaging, antenna \nfabrication, computer memories, energy storage in supercapacitors, high-density information \nstorage, high-frequency transformers, hyperthermia treatment, multi-layered chip inductors, water \npurification methods, sensing of nucleic acid, separation of DNA and RNA, gene therapy and \ndelivery, ferrofluids and so on 7–13. The advancement of electronic devices is now moved to \nintegrated circuits-based technology, where highly efficient transistors are increasing gradually in \naccordance with Moore’s law which requires nano-level engineering and fabrication.  Thus, in \ncontrast to bulk materials, researchers are now focusing on nanocrystalline ferrites’ for utilizing \nthem in the advancement of nano-technological devices. The physical and chemical characteristics \nof ferrite nanomaterials mostly depend on their scale size, shape, or morphology. The structural 4 | P a g e \n parameters such as crystal size and lattice parameters are somehow linked to the electrical and \nmagnetic properties of ferrite nanoparticles. Therefore, the controlling of several factors such as \nthe particle size, surface-to-volume ratio, magnetic anisotropy eventually improves the electronic \nproperties of magnetic nanoparticles in the spinel phase, owing to their transitions from bulk to \nnano-shape. \nResearchers are continuously paying their efforts to employ an easy and efficient method for \nyielding the nanocrystalline ferrites to tailor their structural, dielectric, electric, and magnetic \nproperties under favorable environmental conditions. Various techniques have been deployed to \nsynthesize nanostructured ferrite materials till now viz. sol-gel auto combustion, co-precipitation, \nhigh-energy milling, hydrothermal synthesis, precursor method, mechanochemical route, and \nmicrowave hydrothermal 7–9,14–17. Among these, the sol-gel route appears to be a prominent method \nfor preparing ferrite nanoparticles, as it is eco-friendly, less expensive, and effective without the \ninvolvement of expensive equipment to maintain a good stoichiometry during the synthesis \nprocess. The sol-gel is a wet chemical method, which is widely used due to its potential advantages \nsuch as enhanced control over homogeneity, elemental composition, and powder morphology with \na uniform narrow particle size distribution at relatively low temperature 7,16–18.  \nResearchers attempted sporadically to study the structural, electrical, morphological, \nphotocatalytic, and magnetodielectric properties of Ni-Cu series ferrite NPs demonstrating various \neffects of doping on the properties of the materials crystal. Doping is an effective method to \nameliorate the applications to a broad range by achieving excellent optoelectronic properties. \nInvestigations are still continued with selecting different atoms as dopants or substitutions in A \nand B- sites to tailor the physical, structural and electromagnetic properties of Ni-based mixed \nspinel ferrite nanoparticles 19–28. Munir et al.29 conducted an experiment with a noble \nnanocomposite CuFe 2O4/Bi2O3 by introducing Bi 2O3 nano-petals into the porous CuFe 2O4 and \nobserved a significant increase in the photocatalytic activity in effect of photo-degradation activity. \nHowever, the investigated nanocomposite has remarked with an excellent magnetic separation at 5 | P a g e \n room temperature for the reduced recombination and improved separation of electron-hole pairs. \nCarbon coated highly active magnetically recyclable hollow nano-catalysts have been synthesized \nby Shokouhimehr et al.30, where the authors projected that the prepared nanocomposite can be used \nas a general platform for loading other noble metal catalyst nanoparticles, resulting in high yields \n(up to 99 percent) in selective nitroarenes reduction and Suzuki cross-coupling reactions. \nFurthermore, magnetic properties revealed that the catalysts could be easily separated using a \nsuitable magnetic field and recycled five times in a row. Moreover, Rahman et al.31 thoroughly \ninvestigated the photocatalytic efficiency and recycling stability of rGO supported cerium \nsubstituted nickel ferrite nanoparticles under visible light illumination. According to their findings, \nNiCeyFe2-yO4/rGO (NCFOG) nanocomposite outperformed NiCe yFe2-yO4 nanoparticles by two \ntimes in photocatalytic efficiency and recycling stability, which is attributed to the formation of \nNCFOG heterojunction that enables in the separation of photo-induced charge carriers while \nmaintaining a strong redox ability. Recently, M. Arifuzzaman et al.  17 studied Cu substituted Ni-\nCd ferrite NPs and reported the decrease of average crystallite size and saturation magnetization \nof Ni0.7-xCuxCd0.3Fe2O4 up to x=0.2. Besides, V. A. Bharati et al. 32 reported the influence of \nparallel doping of Al3+ and Cr3+ on the structural, morphological, magnetic, and MÖssbauer \nproperties of Ni ferrite NPs and justified their suitability in HF device applications. In 33, K. Bashir \net al. revealed the electrical and dielectric properties of Ni-Cu ferrite NPs with the doping of Cr3+, \nmaking them the potential for HF applications and photocatalytic activity. Le-Zhong Li et al.  34 \nexamined the Al3+ substituted Ni-Zn-Co ferrites and observed a decrease in saturation \nmagnetization at >0.10. They reported about the metal-semiconductor transition behavior of Ni-\nZn-Co ferrites as an effect of varying temperature and the increase of dc resistivity with Al content \nwas found. The structural and magneto-optical properties of Ni ferrite NPs were propounded in 35, \nwhere the authors calculated the electronic bandgap of 1.5 eV and observed a decrease in saturation \nmagnetization and T c with Al3+ content. In addition, density functional theory (DFT) based \nsimulation was employed in estimating the electronic structure of CuO NPs with the optimized 6 | P a g e \n geometric crystal calculation, which showed the variation of energy band gap with Al content in \nthe samples 36. The effect of doping materials on the characteristics of the different spinel ferrite \nnanoparticles are also available in the literature, Zn ferrite 37,38, Ga ferrite 39, Co ferrite 40–42, Fe \nferrite 43, Mg ferrite 25,44, and Ni-Zn ferrite 45,46.  \nHowever, as per literature survey, no study has been found yet on the structural, dielectric, and \nelectrical properties of Al3+ substituted nanocrystalline Ni-Cu spinel ferrites. Therefore, it is \nimportant to perceive the role of Al substitution on nano-crystallinity and the physical \ncharacteristics of Ni-Cu ferrite NPs. Henceforth, the present study aims to explore the influence of \nAl3+ substitution on the structural, dielectric dispersion and electrical conductivity properties of \nthe synthesized nanocrystalline Ni 0.70Cu0.30AlxFe2-xO4 (x=0.00 to 0.10 with a step of 0.02) ferrites \nthrough the sol-gel process.  \n2. Experimental details  \nMaterials: \nTo synthesize the studied nanocrystalline Ni-Cu spinel ferrites, analytical-grade reagents- \nnickel (II) nitrate Ni(NO 3)2.6H2O (98%), copper (II) nitrate Cu(NO 3)2.3H2O (95-103%), ferric (III) \nnitrate Fe(NO 3)3.9H2O (98%), and aluminum (III) nitrate Al(NO 3)3.9H2O (98%) were used in this \nexperiment which purchased from the Research-Lab Fine Chem. \n2.1 Synthesis of Ni-Cu-Al nanoparticles:  \nDerivatives of Ni 0.70Cu0.30AlxFe2-xO4 (0 ≤ x ≤ 0.1) nanoparticles with a step of 0.02, were \nsynthesized by the sol-gel process. In this process, metal materials of Ni(NO 3)2.6H2O, \nCu(NO 3)2.3H2O, Fe(NO 3)3.9H2O, and Al(NO 3)3.9H2O were taken as raw materials and dissolved \nthem in ethanol and mixed in certain proportions with a magnetic steer to make a homogeneous \nsolution. The pH of the mixture was kept at 7 using the liquid NH 4OH solution and the sol was \ncontinued to heat up to a temperature of 70°C until turning it into a form of dry gel. In an electric 7 | P a g e \n oven, the dried gel was heated at 200ºC for 5 hours, during which a self-ignition process occurred \nand the compositions gradually became fluffy-loose powder. To obtain the resulting ingredients in \na highly crystalline form, the derived powder was annealed at 700°C for another 5 hours to \neliminate any impurity present in the samples. The powder was further homogenized by grinding \nit in a hand-milling process in a mortar. A hydraulic press of 65 MPa was then applied to the \nsamples for 2 minutes to condense and turned them into disk-shaped samples. The prepared \nsamples were 12 mm in diameter and 2.3 mm in thickness. Powder samples were finally sent for \nfurther study on dielectric and electrical measurements. \n2.2 Characterization and property measurements  \nThe structural parameters of the yielded nanocrystalline ferrites were determined through \nthe powder x-ray diffractometer (XRD) analysis using the model PW3040, with CuK α radiation \nof λ = 1.5418Å. The lattice parameter, crystal size (D), and the displacement density were retrieved \nby using the XRD data. The theoretical density (ρ th), micro-strain (ε ms), lattice strain (ε ls), and \nstacking faults in the crystal structure were also determined. The lattice parameter ( 𝑎) and \ncrystallite size (D) were measured by the following relations [20]: \n𝑎=𝑑ඥ(ℎଶ+𝑘ଶ+𝑙ଶ)   (1) \n𝐷=.ଽఒ\nఉೖ௦ఏ                          (2) \nwhere, λ, β hkl, θ, and d hkl, respectively, indicate the wavelength of the X-ray, the full width at half \nmaximum (FWHM) at the most prominent peak (311), the Bragg’s angle, and the distance between \nadjacent planes.  Fourier transform infrared (FTIR) spectroscopy was performed to investigate the \nabout spinel phase in structure in all of the prepared samples. The morphology of the studied \nmaterials has been interrogated by the Field Emission Scanning Electron Microscopy (FESEM) \n(JEOL-JSM 7600F model). The Wynne Kerr Impedance Analyzer (model:6500B) was used to \ndetermine the complex dielectric (ε*), AC resistivity (ρ AC), complex electric modulus (M*), and \ncomplex impedance (Z*) of nanocrystalline ferrite samples. 8 | P a g e \n 3. Result and Discussion:  \n3.1 Structural analysis: \nXRD patterns of Al3+ substituted Ni-Cu ferrites annealed at 700°C, are illustrated in  \nFig. 1, where the peaks are resulted due to diffractions from the planes of (111), (220), (311), \n(222), (400), (422), (511), and (440). The peaks are shaped well-defined with a homogeneous \ndistribution of nanoparticles, which attest to their highly crystalline nature with no impurity. Such \npeaks indicate the cubic single-phase formations of the spinel materials 32–34. The peak diffracted \nfrom the plane (311) is found as the high intensity, which was used to determine the average \ncrystallite size of the materials (see Table 1) using Debye-Scherer’s equation. The lattice constant \n(a0) values are calculated by the Nelson-Riley technique and unit cell volumes (V) of the \ncompositions are listed in Table 1. The decreasing trend of lattice constant and cell volume with \nincreasing Al3+ content is observed, which is due to the replacement of larger ionic (0.67 Å) cations \nby that with smaller radius (0.51 Å). As Al3+ is replacing to the place of Fe3+ in the investigated \nferrites, the unit cell becomes shrinkage, as a result, both a0 and V decrease linearly with Al3+ \ncontent, well satisfied by Vegard's law 47,48. As appeared in Table 1, the average crystallite size \ndecreases with Al3+ content, which might be because of the ionic radius difference between Al3+ \nand Fe3+ ions, offering redistribution of cations in A and B sites which ultimately cause the increase \nin stress and strain of the samples. Lattice spacing is determined by the following equation: \n𝑑=ఒ\nଶ௦ఏ          (3) \nwhere d is the inter-spacing distance between crystal planes and the value of n is taken as 1, which \nrepresents the order of diffraction.  9 | P a g e \n  \nFig. 1. XRD spectra for the synthesized Ni 0.7Cu0.3AlxFe2-xO4 ferrite nanoparticles annealed at \n700°C \nThe sharp diffraction peaks from XRD confirms the higher crystallinity of the ferrites. The \npercentage of crystallinity for the prepared nanoparticles is measured by the following \nequation49,50: \n% 𝐶𝑟𝑦𝑠𝑡𝑎𝑙𝑙𝑖𝑛𝑖𝑡𝑦 = ௨ௗ ௧ ௬௦௧  ௦\n  ௧  ௦×100                     (4) \nThe theoretical density ( ρth) is calculated by the following relation 51: \n𝜌௧=଼ெೢ\nேೌబయ                       (5) \nwhere M w and N a indicate the molecular weight of the compositions and Avogadro’s number, \nrespectively. The experimental density ( ρex) is calculated by the following equation: \n𝜌௫=ெ\nగమ                      (6) \nwhere M, r, and 𝑙 represent the mass, radius, and height of the synthesized samples in tabloid \nshapes, respectively.  \nThe experimental density and the theoretical density of the samples annealed at 700 °C are \nlisted in Table 1. The porosity is found to increases as presented in Table 1, which is due to the \ndiscontinuity of the grain size, resulting in the decrease of density. The P (%) is calculated by the \nfollowing relation: \n10 | P a g e \n 𝑃 (%) = ఘ ି ఘೣ\nఘ×100 %            (7) \nThe porosity is nothing but a relation between inter-granular and intra-granular porosity, which is \nshown by the following equation: \nP (%) = P inter + Pintra                         (8)  \nThe total displacement length per unit volume of the crystal structure can be referred to as \nthe dislocation density (δ) and the way to reduce it is to anneal the samples at high temperatures, \nwhich in turn increases their grain size [36]. This annealing is also considered as the regulator of \nthe strength and flexibility of the crystal structure. The visible parallel lines and random lines in \nthe crystal may indicate the displacements, which means these lines may result due to the \ndisplacement. Displacement density and particle size follow an inverse relationship with giving an \nerror called linearity error. The dislocation density is calculated by the following equation: \n𝛿=ଵ\nమ                                  (9) \nThe length due to the deformation of an object is closely related to the pressure applied, known as \nthe lattice strain (ε ls). The defects caused by imperfections in the crystal structure compel atoms to \ndeviate slightly from their normal position 52. These structural flaws include interstitial and/or \nimpurity atoms that cause lattice strain, which can be determined by the following relation: \n𝜀௦=ఉ\nସ ௧ఏ                    (10) \nwhere θ represents the angle of diffraction and β indicates the full width at half maximum. The \nstacking faults are induced in the atomic planes of the crystal because of the interruption of the \nlayered arrangement in a normal lattice structure. The stacking fault [SF] is determined by the \nfollowing equation: \n𝑆𝐹=ଶగమ\nସହ√(ଷ௧ఏ)       (11) \nVarious defects in the crystal structure such as displacement, plastic deformation, point defects, \nand domain boundary defects are considered to be the key factors of the deformation in the \nstructure and it is assumed that this deformation occurs in one part out of nearly one million parts 11 | P a g e \n of the material defined as the micro strain (ε ms). A notable feature of the micro strain is that it \nmaximizes the peak and the following equations are introduced to comprehend it 53: \n𝜀௦=ఉ௦ఏ\nସ                         (12) \nThe ionic radii of A and B sublattices are calculated by the following relations 9,54: \n𝑟=√3𝑎(𝑢−0.25)− 𝑟             (13) \n𝑟=𝑎(0.625− 𝑢)−𝑟                (14) \nwhere r o and u represent the radius of oxygen (1.32 Å) and oxygen parameter with the value of ଷ\n଼, \nrespectively. The distance between the centers of adjacent ions is defined as the hoping length and \nthe lengths for A-A sites, B-B sites, and A-B sites are calculated using the following equations, \nrespectively 54,55: \n𝐿ି=√ଷ\nସ                                    (15) \n𝐿ି=√ଵଵ\n଼                                  (16)  \n𝐿ି=\nଶ√ଶ                                      (17) \nwhere a 0 represents the lattice constant. \n \nFig. 2.  Variation in lattice constant and crystallite size with Al3+ content. 0.00 0.02 0.04 0.06 0.08 0.10545760636669 Crystallite Size\n Lattice Parameter\n \nAl3+ ContentCrystallite Size (D)\n0.8290.8300.8310.8320.8330.834\n Lattice Parameter (ao)12 | P a g e \n FTIR Study \nTo confirm the structure of the spinel phase in all of the prepared samples, Fourier transform \ninfrared (FTIR) spectroscopy was utilized in this study. Fig. 3 depicts the FTIR spectra of \nnanocrystalline Ni 0.70Cu0.30AlxFe2-xO4 ferrites taken in the frequency region of 450-4000 cm-1. The \nv1 and v 2 are two fundamental strong absorption bands can be observed in effect of the metal-\noxygen (M-O) bonds at the tetrahedral and octahedral sites. The entity of high frequency v 1 band \nfound in the range of 585-615 cm-1 which is formed by the internal stretching vibration of the M-\nO bonds at tetrahedral sites whereas the low frequency v 2 band around 400 cm-1 corresponds to \nthat of octahedral site 56. The formation of spinel structures in the prepared Ni-Cu ferrite \nnanoparticles is ascertained by the observed bands. The bands observed in this investigation is \nconsistent with previous findings. 57,58. The absorption peaks, however, are induced by the \ntetrahedral site of the metal's intrinsic stretching vibration. Moreover, the stretching vibration of \nM-O at both sites is influenced by changes in the lattice parameter. The tetrahedral stretching \nfrequency band (v 1) shifts to higher frequency regions as Al3+ doping increases, as observed from \nthe FTIR spectra. As illustrated from the Fig. 3, the observed band shifting with changing of Al3+ \nconcentrations might be due cations distribution followed by lighter Al3+ substitution over the \ntetrahedral and octahedral sites 56,58,59. 13 | P a g e \n  \nFig 3. Room temperature FTIR spectra of the synthesized Ni-Cu spinel ferrite nanoparticle with \nAl3+ concentrations \n \n3.2 FESEM and EDX Studies \nFig. 4 shows the FESEM micrographs of nanocrystalline Ni 0.70Cu0.30AlxFe2-xO4 annealed at \n700 °C. As depicted in all figures of Fig. 4 (A-F), the grains are found in semi-spherical shapes \nwith a uniform and even distribution in multi-domains separated by grain boundaries. The average \ngrain size of the synthesized ferrite nanoparticles is measured by 54: \n𝐺=ଵ.ହ\nே          (18) \nwhere L, X, respectively, indicate the total length in cm and the magnification of the micrographs, \nand N is the number of intercepts. Fig. 5 illustrates the EDX analysis of Ni 0.70Cu0.30AlxFe2-xO4, \nwhich ensures the presence of each and every component in each sample with appropriate \nproportions. The expected sum of each of the observed compositions is found 100%, which \nconfirms the accuracy of the sol-gel analysis technique and manifests its novelty. \n 500 1000 1500 2000 2500 3000 3500 4000\n Transmittance (%)\nWavenumber (cm-1)x=0.00x=0.02x=0.04x=0.06x=0.08x=0.10\n59159559159161458714 | P a g e \n Table 1. Structural parameters of nanocrystalline Ni 0.7Cu0.3AlxFe2-xO4 varying Al3+ content. \nParameters x = 0.00 x = 0.02 x = 0.04 x = 0.06 x = 0.08 x=0.10 \nd (nm) 0.2510 0.2502 0.2508 0.2514 0.2501 0.2504 \na0 (nm) 0.8324 0.8296 0.8316 0.8337 0.8294 0.8304 \nD (nm) 70.74 60.51 64.71 55.63 64.73 62.78 \nV (nm3) 0.5768 0.5711 0.5752 0.5795 0.5705 0.5727 \n% Crystallinity  87.10 93.83 94.92 97.29 94.10 94.38 \nδ (×1014) (lines/m2) 1.998 2.731 2.338 3.230 2.387 2.538 \nεLS (×10-3) 1.597 1.860 1.744 2.033 1.738 1.795 \nSF 0.446 0.445 0.446 0.446 0.445 0.445 \nεms (×10-4) (line-2/m-4) 4.9 5.729 5.357 6.230 5.355 5.220 \nρex (kg/m3) (×103) 3.832 3.832 3.832 3.832 3.832 3.832 \nρth (kg/m3) (×103) 5.256 5.297 5.247 5.195 5.264 5.230 \nP (%) 27.093 27.657 26.968 26.237 27.204 26.730 \nGa (nm) 59 64 65 62 61 62 \nrA (nm) 0.0482 0.0476 0.0481 0.0485 0.0476 0.0478 \nrB (nm) 0.0761 0.754 0.0759 0.0764 0.0753 0.0756 \nLA-A (nm) 0.3605 0.3605 0.3605 0.3605 0.3605 0.3605 \nLA-B (nm) 0.3451 0.3451 0.3451 0.3451 0.3451 0.3451 \nLB-B (nm) 0.2943 0.2943 0.2943 0.2943 0.2943 0.2943 \n \n3.3 Dielectric Property \nFig.6 (A, B) demonstrates the variation in real (ε′) and imaginary (ε′′) parts of complex \ndielectric constant of nanocrystalline Ni 0.70Cu0.30AlxFe2-xO4 annealed at 700 °C with increasing \nfrequency. The dielectric property of ferrites is contingent on different factors such as preparation \nmethod, chemical composition, grain size, electronic di-polarity, and so on. The ε′, ε′′, and \ndielectric loss tangent ( tan δE) are calculated by the following relations: 15 | P a g e \n 𝜀ᇱ=௧\nఌ           (19),       𝜀ᇱᇱ=𝜀ᇱ𝑡𝑎𝑛𝛿ா           (20), and 𝑡𝑎𝑛𝛿ா=ଵ\nఠఌఌᇲఘ            (21) \nwhere C is the capacitance, ω=2πf, f represents the applied field frequency, εo represents the free-\nspace permittivity, t is the thickness and A is the area of the contact surface of the tabloids.  16 | P a g e \n  \nFig. 4. FESEM micro-graphs and corresponding histogram analysis of the synthesized \nNi0.7Cu0.3AlxFe2-xO4 NPs; ((A) x= 0.00, (B) x=0.02, (C) x=0.04, (D) x=0.06, (E) x=0.08, and (F) \nx=0.10)). \nA \n B \n C \nD E F 17 | P a g e \n  \nFig. 5. Energy dispersive spectra (EDX) of the prepared Al3+ substituted nanocrystalline \nNi0.7Cu0.3AlxFe2-xO4. \nFig. 6(A) illustrates that ε′ decreases with the frequency up to 105 Hz and thereafter remains almost \nconstant with showing a very low value. On the contrary, the imaginary part (ε′′) reveals higher \nvalues at low frequency regime and decreases vigorously with frequency as observed in Fig. 6(B). \nThe observed dispersive dielectric nature of these investigated materials can be described by the \nx= 0.00 \n x= 0.02 \nx= 0.04 \n x= 0.06 \nx= 0.08 \n x= 0.10 18 | P a g e \n Maxwell–Wagner interfacial theory of polarization supported with the Koop’s phenomenological \ntheory 54,55,60. The grain boundaries are more active at low frequencies, whereas at high frequencies \ngrains are more contributing. At the low frequency, the value of ε′ is higher because of the high \nresistive grains, which gives the space charge polarization 47,48. The decrease of real dielectric \nconstant (ε′) with increasing frequency is found in Fig. 6A, because the grains come into action at \nhigher frequencies and the hopping electrons cannot follow the applied electric field, causing the \npolarization to decrease, and the value of 𝜀′ appears to be very low, becoming almost constant 61.  \n \nFig. 6. Extracted initial permeability (A) real and (B) Imaginary part for the investigated \nnanocrystalline Ni 0.7Cu0.3AlxFe2-xO4. \nThe sample with x = 0.1 shows the maximum value of 𝜀′ because of the redistribution of \nFe3+ at both A- and B-sites in Ni 0.70Cu0.30AlxFe2-xO4. The substitution of Fe3+ by Al3+ in the \ncompositions results the transfer of Al3+ at A-sites and replaces some Fe3+ at B-sites, which causes \nthe enhancement of Fe3+ ions in the grain and assembles them in the grain boundary 48,61. \nConsequently, the space charge polarization is increased and caused a higher value of the dielectric \nconstant. Due to the generation of heat in dielectric materials by the high flow of electricity which \nis dissipated and considered as the material’s loss that is characterized as the imaginary part (ε′′) \nof dielectric constant 54. From Fig. 6(B), it is observed that the value of ε′′ increases significantly \nwith increasing Al3+ content in ferrites. The decrease of ε′′ with frequency is occurred due to the \nA \n B 19 | P a g e \n high resistive effect of the grain boundaries. The electrons reverse their direction of motion \nfrequently at higher frequencies and the hopping electrons can no longer follow the applied electric \nfield, the probability of charge transport at the grain boundary decreases, resulting in the decrease \nof polarization, giving the low value of ε′′ 17,55,60,61. \n \nFig. 7. Dielectric loss tangent vs frequency plot of the synthesized Al3+ Ni-Cu NPs. \nFig. 7 shows the variation of dielectric loss tangent ( tan δE) of the synthesized samples \nannealed at 700 °C with varying frequencies. Due to impurities and imperfections, the polarization \nlags behind the applied voltage, causing tanδ E to form there 54,61. The highest value of tanδE is \nfound under the relaxation condition of ωτ = 1, where ω = 2πf max, and τ= 1/2P and fmax, τ represents \nthe peak frequency and the relaxation time, respectively and both of which are closely related to \nthe hopping or jumping probability. Electron sharing between Fe3+ and Fe2+ requires very little \nenergy and the maximum peak is achieved when the hopping frequency between them is well-\nmatched with the applied electric field. Koop’s theory explains how tanδE of the investigated \nmaterials decreases with frequency, in a very simple, smooth, and neat way 62,63. It is noted that at \nlower conductive grain boundaries, tanδE exhibits the maximum value as more electrons are \navailable to be conductive at the low-frequency region. There is energy loss that occurred during \nthe electrons sharing between Fe3+ and Fe2+, therefore high energy is required 47,64,65. \n20 | P a g e \n The role of microstructure is important in determining the tanδE. H. Jia et. al. showed that \nthe grain boundaries and porosity between polycrystalline crystals affect the ε′ and ε′′ 66. The inter-\nrelation among porosity, grain boundaries, and dielectric loss is defined by the following relation: \n𝑡𝑎𝑛𝛿ா=(1−𝑃)𝑡𝑎𝑛𝛿+𝐶𝑃                     (22) \nwhere C m is the material-dependent constant, P represents the porosity and tanδo is the dielectric \nloss of material with full densification. Uniform density and lower porosity reduce the ε′ and ε′′, \nrespectively and the intrinsic and extrinsic fault are responsible for the dielectric loss.  \n3.4 AC Resistivity  \n The variation in ac resistivity ( ρac) of the investigated samples with frequencies (annealed \nat 700 °C) is depicted in Fig. 8. The variation in ρac of the investigated ferrite nanoparticles is \nexplained based on the hopping mechanism. The ρac is calculated by the following equation 54: \n𝜌=ଵ\nఌబఌᇱఠ௧ ಶ              (23) \nwhere ω defines as the angular frequency. According to the hopping mechanism, electrons jump \nfrom one state to another, which prefer to be distributed over the sites in the lattice. In Fig. 8, it is \nanticipated that at lower frequencies the ρ ac of the investigated ferrites has higher values and \ndepletes with increasing frequency. After a certain frequency, it gets almost saturation with \nshowing a very small value. This variation of ρac with frequency can be described by the frequency \ndependency of grains and grain boundaries. The conductivity mechanism illustrates the particles \nability to be highly electrically conductive 67,68. \nThe high-resistive boundary separating the grains are more active at lower frequencies, which \nimpedes the movement of free charges and thus the hopping of electrons between Fe2+ and Fe3+ is \nless, which in turns result the higher values of ρ ac54. To increase the hopping of electrons between \nFe2+ and Fe3+, it must be operated at higher frequencies, which plays a critical role in reducing the \nρac value. The main reason for the low values of ρac is that the hopping of electrons almost stops \nafter a certain frequency range. As depicted in Fig 8, the maximum value of ρac is found for the  21 | P a g e \n  \nFig.  8. AC resistivity of the synthesized Al3+ substituted Ni-Cu nanoparticles. \nmother sample, Ni 0.70Cu0.30Fe2O4. With the increase of Al3+ concentration in Ni–Cu ferrites, the \nAC conductivity increases; therefore Ni 0.70Cu0.30Al0.1Fe1.9O4 shows the minimum value at the low-\nfrequency region, and the conduction takes place through highly resistive grain boundaries, while \nat high frequencies conduction occurs through low resistive grains 65–67. \nThe high-resistive boundary separating the grains are more active at lower frequencies, which \nimpedes the movement of free charges and thus the hopping of electrons between Fe2+ and Fe3+ is \nless, which in turns result the higher values of ρ ac54. To increase the hopping of electrons between \nFe2+ and Fe3+, it must be operated at higher frequencies, which plays a critical role in reducing the \nρac value. The main reason for the low values of ρac is that the hopping of electrons almost stops \nafter a certain frequency range. As depicted in Fig 8, the maximum value of ρac is found for the \nmother sample, Ni 0.70Cu0.30Fe2O4. With the increase of Al3+ concentration in Ni–Cu ferrites, the \nAC conductivity increases; therefore Ni 0.70Cu0.30Al0.1Fe1.9O4 shows the minimum value at the low-\nfrequency region, and the conduction takes place through highly resistive grain boundaries, while \nat high frequencies conduction occurs through low resistive grains 65–67. \n \n \n22 | P a g e \n 3.5 Complex Electric Modulus \nThe electric relaxation mechanism in the materials can be explained through the \nspectroscopy of electric modulus (M*), which is resolved into two components 63 as given in the \nfollowing: \nM*=ଵ\nఌ∗=ଵ\nఌᇱିఌ=ఌᇲ\nఌᇲమାఌᇲᇲమ−𝑖ఌᇲᇲ\nఌᇲమାఌᇲᇲమ= 𝑀′+𝑖𝑀′′                  (24)  \nwhere 𝑀ᇱ=ఌᇲ\nఌᇲమାఌᇲᇲమ is the real and 𝑀ᇱᇱ=ఌᇲᇲ\nఌᇲమାఌᇲᇲమ is the imaginary part of the electric modulus. \nFrom the above equations, both the real (M′) and imaginary (M′′) parts of modulus are found to be \nfrequency-dependent, which plays a key role in investigating the relaxation mechanism of the \nmaterials. From the Fig. 9(A), it is perceived that M′ responds very well to higher frequencies with \nexhibiting the highest value for x=0.00. It indicates the lower value of ε′ at high frequencies. The \ninadequacy of the restorative force and the release of space charge polarization near the grain \nboundary helps to attain its saturation. This phenomenon occurs at higher frequencies and at the \nsame time ensures frequency independency in the electrical properties of the materials47,63,64. \n \nFig. 9. Electric modulus behavior for the synthesized Al3+ substituted Ni-Cu nanoparticles. \nTo illustrate the peaking behavior, one has to look at the variation of M′′ as shown in Fig. \n9(B). The hopping mechanism is used to illustrate the peaking behavior better as it more accurately \nexplains the transition of the charge carriers. In the figure above, it is clear and understand that \nA \n B 23 | P a g e \n charge carriers contributing to the hopping process cover long distances at low frequencies. On the \nother hand, charge carriers are able to cover short distances at higher frequencies, which indicates \nthe relaxation in the polarization process 69,70. \n                      \nFig. 10. M\" vs M' plot for the investigated Al3+ substituted Ni-Cu NPs. \nThe relaxation of the material is distinguished by the cole-cole plot (M′′ vs M′) of the \nelectric modulus as presented in Fig. 10. The grain and grain boundary are thought to be \nresponsible for this separation 62,69. A clear non-Debye type relaxation is found by looking closely \nat the non-overlapping semicircular pattern in Fig. 10. Nanoparticles annealed at 700 °C show two \nidentical non- overlapping semicircular patterns 47. \n3.6 Complex Impedance Analysis \nTo study the electrical behavior of the material, the impedance spectroscopy was employed in this \nstudy for the synthesized nanoparticles. This is a long-established method to distinguish the \nimpedance contributions of the materials’ grains, grain boundaries and electrodes. The complex \nimpedance (Z*) includes both the resistive and reactive components of the impedance as follow: \nZ*=Z' - jZ\"                (25) \n24 | P a g e \n where the resistive part is designated as the real part Z ' which is the horizontal component of the \ncomplex impedance denoted as Z ' = |Z*|cos𝜃 and the imaginary part is designated as the reactive \n(capacitive) part expressed as Z \" = |Z*|sin𝜃. However, these two components are combined \nimpedance effect of resistance and capacitance due to grain and grain boundary which are \nembroiled to dielectric and electric modulus parameters following the relation: \ntan𝛿 = ఌᇲᇲ\nఌᇲ = ᇲᇲ\nᇲ=ெᇲᇲ\nெᇲ                  (26) \nThe variation in real part of complex impedance (Z′) of the investigated ferrites annealed at 700 \n°C is illustrated in Fig. 11(A) with varying frequencies. The higher values of Z′ of synthesized \nferrites are revealed at lower frequencies with dispersed behavior and drop sharply up to 1 KHz \nand thereafter it remains to constant in high frequency.  \n \nFig. 11. Impedance analyzer extracted plots for the synthesized Al doped Ni-Cu spinel nano-\nferrites. \nBesides, in Fig. 11(B), the variation in imaginary part (Z′′) of the complex impedance for the \nsynthesized Ni 0.7Cu0.3AlxFe2-xO4 nano-ferrites is illustrated. As observed in Fig 11(B), the \nmaterials show higher values at the lower frequency likewise the real part (Z′) and decrease rapidly \nwith increasing frequency (up to 10 KHz) as the conductivity of the ferrites increases. However, \nat higher frequencies ( ≥100KHz), it appears with frequency-independent behavior of small \nA \n B 25 | P a g e \n constant values in effect of the reduction in polarization 54,61,71,72. Both of the Fig. 11 shows the \nsimilar trend to the dielectric nature of the materials. For all compositions, the impedance curves \nare appeared to merge at higher frequencies indicating the predominance contribution of low \nresistive grains. Moreover, the space-charge polarization is considered important only when the \nmaterials are resolved into grains and grain boundaries 61,68. The curves tend to converge at higher \nfrequencies owing to a decrease in space charge polarization and this behavior elucidates the \nincreasing tendency of ac conductivity with frequency, confirming the semiconducting behavior \nof the prepared nanocrystalline spinel ferrites 47,71. \nThe Nyquist impedance plot (also known as cole-cole plot) of the prepared Ni 0.7Cu0.3AlxFe2-xO4 \nferrites annealed at 700 °C is shown in Fig. 12 which reveals the contribution of grain and grain \nboundary resistance as the plot is combined response of RC circuit by parallelly connected resistor \nand capacitor. The heterostructure nature of synthesized materials along with characteristic nature \nof complex impedance spectra by determining the existence of multiple electrical responses (due \nto grain resistance R g, grain boundary resistance R gb and electrode effects) can be easily \ndetermined by observing the semicircular arcs appeared in the cole-cole plot 47,54,72. By looking at \nFig. 12, the two semicircular arcs are clearly visible which are formed with its center placed below \nthe real axis, which manifests the single-phase of Al3+ substituted nanocrystalline Ni-Cu materials. \nThe diameter of the semicircle arcs is found to decrease with increasing Al3+ concentration, which \nis actually caused by the resistance of grain boundaries. However, at lower frequencies, the R g \ndominates the appearance of the first semicircle, whereas at higher frequencies, the R gb dominates \nthe appearance of the second semicircle. The difference in relaxation time is considered as the \nmain catalyst behind the separation of semicircles arcs 61. 26 | P a g e \n  \nFig. 12. Nyquist impedance plot of the prepared Ni-Cu ferrites annealed at 700 °C \n \nConclusion \nThe sol gel method was used to synthesize a series of high crystalline nanomaterials of \nNi0.7Cu0.30AlxFe2-xO4. The single-phase cubic spinel structure of the investigated materials was \nconfirmed through XRD study with no impurity. The surface morphology was studied through the \nFESEM measurements, which illustrated the distribution of semi-spherical grains separated by the \ngrain boundaries with a homogenous distribution of particles on the surface. The structural \nparameters were determined using the XRD and FESEM data. The electrical and dielectric \nproperties were carried out by using the impedance analyzer supported with the modulus and \nimpedance spectroscopy. Both the average crystallite size and the average grain size of the studied \nmaterials are found in the nano-scale range (55.63–70.74 nm) and (59.00– 65.00 nm), respectively. \nThe dielectric dispersion nature of the materials was revealed through the dielectric study of the \nmaterials. The electrical response of the materials was inspected by means of impedance and \nmodulus spectroscopy, which well resolved the contribution of grains and grain boundaries in the \nelectrical properties of the investigated Al3+ substituted Ni-Cu ferrite nanoparticles. The relaxation \nphenomena in the materials was justified through the cole-cole analysis of both impedance and \n27 | P a g e \n modulus spectra. A little substitution of Al3+ is found to be influential in the structural, dielectric \nand electrical properties of Ni-Cu spinel ferrites prepared by the cost-effective sol gel method. \nAcknowledgment \nThe authors are grateful to the center of excellence of the Department of Mathematics and \nPhysics at North South University (NSU), Dhaka 1229, Bangladesh. This research is funded by \nthe NSU research grant CTRG-20/SEPS/13.  \nData Availability \nThe authors are currently using all related data for the purpose of further research. 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Frequency and temperature-\ndependence of dielectric permittivity and electric modulus studies of the solid solution Ca \n0.85 Er 0.1 Ti 1−x Co 4x/3 O 3 (0 ≤ x ≤ 0.1). RSC Advances  8, 17139–17150 (2018). 32 | P a g e \n 63. Kaiser, M. Electrical conductivity and complex electric modulus of titanium doped \nnickel–zinc ferrites. Physica B: Condensed Matter  407, 606–613 (2012). \n64. Nitika & Rana, A. Study of anneal-tuned dielectric properties, AC conductivity, complex \nimpedance, and modulus of cobalt ferrite nanoparticles. Materials Today: Proceedings  45, \n5444–5448 (2021). \n65. Butt, K. Y. et al. The study of structural, magnetic and dielectric properties of spinel \nferrites for microwave absorption applications. Applied Physics A  127, 714 (2021). \n66. Jia, H. et al. Monodomain MgCuZn ferrite with equivalent permeability and permittivity \nfor broad frequency band applications. Ceramics International  43, 5974–5978 (2017). \n67. Santhosh Kumar, M. V. et al. 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Journal of Materials \nScience: Materials in Electronics  32, 2832–2844 (2021). \n72. Dabbebi, T. et al. Investigations of microstructural and impedance spectroscopic \nproperties of Mg0.5Co0.5Fe1.6Al0.4O4 ferrite prepared using sol–gel method. Journal of \nMaterials Science: Materials in Electronics  32, 12521–12534 (2021). \n  "    },    {        "title": "1407.4657v1.Magnetic_Properties_of_Bismuth_Ferrite_Nanopowder_Obtained_by_Mechanochemical_Synthesis.pdf",        "content": " 1 Magnetic Properties of Bismuth Ferrite Nanopowder O btained \nby Mechanochemical Synthesis \nIzabela Szafraniak-Wiza a* , Bartłomiej Andrzejewski b, Bożena Hilczer b \na Institute of Materials Science and Engineering, \nPozna ń University of Technology, \nM. Skłodowska-Curie Sq. 5, PL-60965 Pozna ń, Poland \n* corresponding author: izabela.szafraniak-wiza@put.poznan.pl  \nb Institute of Molecular Physics \nPolish Academy of Sciences \nSmoluchowskiego 17, PL-60179 Pozna ń, Poland \n \nAbstract — Multiferroic bismuth ferrite (BiFeO 3) \nnanopowders have been obtained in room temperature by \nmechanical synthesis. Depending on the post-synthes is \nprocessing the nanopowders have exhibited differenc es in \nthe mean sizes, presence of amorphous layer and/or \nsecondary phases. Extended magnetic study performed  for \nfresh, annealed and hot-pressed nanopowders have \nrevealed substantial improvement of the magnetic \nproperties in the as-prepared powder.  \n \nPACS numbers:  81.07.Bc Nanocrystalline materials; 81.07.Wx \nNanopowders; 81.20.Ev Powder processing; 75.85.+t \nMagnetoelectric effects, multiferroics; 75. Magneti c properties and \nmaterials  \nKeywords-component; multiferroics, bismuth ferrite,  magnetic \nproperties, nanopowder \n \nI.  INTRODUCTION \nMultiferroics exhibit at least two primary ferroic orders: \nferroelectric, ferromagnetic, ferroelastic or ferro torroic in  \na single homogeneous phase and these order paramete rs can \nbe mutually coupled [1]. Especially interesting are  \nferroelectromagnets (or magnetoelectric multiferroi cs) having \nmagnetization and dielectric polarization, which ca n be \nmodulated and activated by an external electric fie ld and \nmagnetic field, respectively. For this reason, mult iferroic \nmaterials are being considered for a host of potent ial \napplications like magnetic recording media, informa tion \nstorage, spintronics, and sensors (for review of ph ysics and \napplications of multiferroics see ref. [2]). Betwee n \nmultiferroics bismuth ferrite (BiFeO 3) attacked the most \nscientific interest because its antiferromagnetic a nd \nferroelectric properties are observed at room tempe rature and \nthe phase transitions occur at very high temperatur es i.e.  \nTN~640 K and TC~1100 K [2]. At room temperature BiFeO 3 \nhas rhombohedrally distorted cubic perovskite cell ( R3c ) and \nthe antiferromagnetic properties are related to G-t ype ordering \nwith a cycloid modulation (62 nm) apparent down to 5 K. Recently it has been shown that BFeO 3 nanoparticles \nexhibit strong size-dependent magnetic properties. That effect \nhas been correlated with: (1) increased suppression  of the \nknown spiral spin structure with decreasing nanopar ticle size, \n(2) uncompensated spins with spin pinning and strai n \nanisotropies at the surface [3] and (3) presence of  impurities \n(like γ-Fe 2O3) and/or oxygen vacancies [4]. In the current \nwork, we studied magnetic properties of BiFeO 3 nanopowders \nwith different sizes obtained by room temperature \nmechanochemical synthesis [5, 6], high-temperature annealing \nand bulk ceramics made from this powder by hot-pres sing \nmethod.  \n \nII.  EXPERIMENTAL \nBismuth ferrite nanopowder was synthesized by \nmechanochemical route. Details of synthesis were pu blished in \nprevious paper [6]. Commercially available oxides ( Bi 2O3 and \nFe 2O3 purchased from Aldrich, 99% purity) in stoichiomet ric \nratio were milled in a SPEX 8000 Mixer Mill for 120  h.  \nThe thermal treatment was performed for 1h in norma l \natmosphere at 500 oC. The ceramic samples were prepared by \nhot isostatic pressing of nanopowder at 200 MPa and  800 oC \nfor 2h. Magnetic properties were studied by Oxford \nInstruments Ltd. MagLab 2000 System in temperature range \nbetween 2÷350 K. The temperature dependent measurem ents \nwere carried out in two different procedures: zero field cooled \n(ZFC) and field cooled (FC) one. In the case of ZFC  \nmeasurements the sample was first cooled in absence  of \nexternal magnetic field and next the field was appl ied when \nthe sample reached desired temperature. For FC meas urements \nthe sample was cooled in applied magnetic field (1 mT÷1 T).  \nIn both cases data were acquired during the heating  cycle.  \n \nIII.  RESULTS AND DISCUSSION \nEarlier XRD studies [6] confirm that the nanopowder  obtained \nafter 120 h milling exhibits rhombohedrally distort ed  2 perovskite structure. The mean grain size estimated  using \nScherrer formula is 19÷26 nm. Transmission electron  \nmicroscope studies (TEM) indicate that nanograins o f this \npowder are composed of crystalline core BFO and amo rphous \nshell [6]. Thermal annealing (1h at 500 °C) causes \ncrystallization of the shell and his leads to the i ncrease of the \nmean grain size to 26÷42 nm. This sample contains a lso \nsubstantial amount of Bi 2Fe 4O9 parasitic phase formed during \nhigh temperature processing. Hot pressing method yi elds \ndense, bulk bismuth ferrite ceramic.  \nThese three different BiFeO 3 samples were used for \nmagnetometric measurements. ZFC and FC temperature \ndependences of magnetic susceptibility χ(T) were measured \nfor all samples and for various applied magnetic fi elds.  \nThe origin of the difference between ZFC and FC \nsusceptibilities in as-prepared nanopowder is domai n wall \npinning and/or reorientations of weak ferromagnetic  domains \nin bismuth ferrite phase. A spin-glass phase as a s ource of this \ndifference should be rather excluded because in BFO  \ncompounds there is a lack of memory, which is commo n effect \nfor spin-glass systems [7]. In ZFC susceptibility a nomalous, \nfield-dependent maximum about 8 K (for the field 1 T) is \nobserved whereas FC susceptibility decreases monoto nously \nwith temperature (see Fig. 1).  \n0 50 100 150 200 250 300 350 400 0123456\n7 8 9 10 11 12 13 0.0 0.3 0.6 0.9 1.2 \nTF \nBiFeO 3 \nas prepared powder \nµ0H=1 T \n  \nχ [m3kg -1 ] x 10 -7 \nT [K] ZFC FC \n  \n Almeida-Thouless line \nµ0H [T] \nT [K] \n \nFigure 1.  ZFC and FC magnetic susceptibilities for as-prepare d nanopowder. \nThe insert shows freezing temperature dependence on  magnetic field fitted \nwith Almeida-Thouless model. \nThe low temperature maximum is probably related to spin \nclusters in amorphous shell of nanograins [8]. Acco rding this \napproach, at high temperature the spins are in a pa ramagnetic \nstate. At lower temperature the ordered spin cluste rs are \nformed, which freeze into random directions below \ntemperature TF. It is assumed that the freezing temperature TF \ncorresponds to the maximum of ZFC susceptibility.  \nThe validity of the above explanation can be tested  using the \nAlmeida-Thouless relation between the freezing temp erature \nand the applied magnetic field [8]:  \n( )\n( ).THTHH/\nFF23\n001\n− =  (1) In eq. (1) H0 denotes magnetic field for which TF→0 and TF(0) \nis the freezing temperature for H=0. The best fit using eq. (1) \nto the data is obtained for the parameters: µ0H0=3.5 T and \nTF(0)=11 K (see the inset to Fig. 1).  \nThe annealed sample shows no low temperature anomal y \nbut it exhibits substantial difference between ZFC and FC \nsusceptibilities (Fig. 2a). The absence of maximum in ZFC \nsusceptibility well corresponds to the crystallizat ion of \namorphous shell and thus disappearance of a spin-gl ass phase \nafter thermal annealing. The thermal processing als o leads to \nthe increase of the mean grain size to 26÷42 nm. In  these \nnanoparticles with improved crystal structure we ex pect \nenhanced energy barriers for magnetic moment reorie ntations \nor pinning of domain walls leading to pronounced \nirreversibility effects. The anomaly observed about  250 K is \nvery common for orthoferrites [2]. However, the cal cined \nsample contains Bi 2Fe 4O9 parasitic phase which undergoes  \na transition from paramagnetic to antiferromagnetic  state near \nTN=264±3 K [9] which can be also produce this anomaly .  \nFor the hot-pressed sample, the ZFC and FC suscepti bilities \n(Fig. 2b) are almost identical, with pronounced ano maly about \n250 K. This indicates weak pinning of magnetic doma in or \ndomain walls in hot-pressed sample.  \n0 50 100 150 200 250 300 350 400 0.0 2.0 4.0 6.0 8.0 10.0  \n \nχ [m3kg -1 ] x 10 -7 \nT [K] FC \nZFC annealed \nµ0H= 1T a)\n \n0 50 100 150 200 250 300 350 1.15 1.20 1.25 1.30 1.35 1.40 \n  \nhot-pressed \nµ0H=1 T \nZFC FC \nχ [m3kg -1 ] x 10 -7 \nT [K] b)\n \nFigure 2.  ZFC and FC magnetic susceptibilities for calcined B FO \nnanopowder (a) and hot-pressed ceramics (b). The ap plied magnetic field was \nequal to 1 T.  3 The magnetic hysteresis loops presented in Fig. 3 h ave been \nrecorded at temperatures 4 K, 50 K and 300 K for al l samples.  \n-2 -1 0 1 2-0.8 -0.4 0.0 0.4 0.8 \n0 100 200 300 510 15 20 25 30    as prepared \n 4 K \n 50 K \n 300 K \n M [Am 2kg -1 ]\nµ0H [T] a)\n  \n T [K] µ0Hc [mT ]\n \n-2 -1 0 1 2-0.8 -0.4 0.0 0.4 0.8 \n0 100 200 300 020 40 60 80    annealed \n 4 K \n 50 K \n 300 K\n M [Am 2kg -1 ]\nµ0H [T] b)\n T [K]  µ0Hc [mT] \n \n-2 -1 0 1 2-0.2 -0.1 0.0 0.1 0.2 \n0 100 200 300 510 15 20    hot pressed \n 4 K \n 50 K \n 300 K M [Am 2kg -1 ]\nµ0H [T] c)\n  \n T [K]  µ 0Hc [mT] \n \nFigure 3.  Hysteresis loops recorded at different temperatures  for a) as-\nprepared b) annealed nanopowders and c) hot-pressed  ceramics made from \nnanopowder. Insets show temperature dependence of c oercive field for each \nsample. The hysteresis loops for nanopowders are not well-s aturated \nwhich may be related either to antiferromagnetic st ate or to the \nspin disorder and spin pinning at the nanograin sur faces [3].  \nThe maximum of magnetization, MS=0.710 Am 2/kg  is \nobserved at 4 K and 2 T for the as-prepared nanopow ders \ncomposed of shell-core nanograins (see Fig. 3a). Th is value is, \ndepending on temperature range, 2÷4 times bigger th an MS for \nthe thermal processed ceramics. The increased magne tization \nis due to size effect and presence of amorphous she ll.  \nAll recorded loops are superposition of ferromagnet ic non-\nlinear and antiferromagnetic or paramagnetic linear  \ncontributions. The nonlinear, ferromagnetic contrib ution is \nmost pronounced in low temperatures and for as-prep ared \nBFO sample (Fig. 3a). This ferromagnetic behavior c an be \nrelated to suppression of spin cycloid leading to t he lack of \nspin compensation or to surface spins in BFO nanogr ains.  \nThe complex relation of coercive field Hc(T) on \ntemperature found in the BFO samples has different origins. \nFor the as-prepared BFO sample (inset to Fig. 3a) t he \ncoercivity Hc(T) exhibits non-monotonic behaviour with the \nhighest value over 25 mT at the temperature 4 K and  \nminimum at 50 K. This dependence is a superposition  of \nstrong pinning of spin clusters below freezing temp erature TF \nand pinning of weak ferromagnetic domains prevailin g at \nhigher temperatures. The temperature dependence of \ncoercivity for calcined BFO sample (inset to Fig. 3 b) is also \nunusual because Hc(T) linearly increases with temperature. \nThe increase of coercivity with temperature sometim es takes \nplace in granular systems if intergranular coupling  decreases \nwith temperature [10]. In this case, the coercivity  Hc(T) is \ndescribed by a phenomenological formula: \n.JN H Hs eff a k ex c − =0 0 µαα µ  (2) \nwhere: αk is the structure-dependent Kronmuller parameter, \naex  parameter represents deleterious influence of the intergrain \ncoupling, Neff  is the magnetostatic parameter, J is the \nmagnetization and Ha denotes the anisotropy field. For the \nnanopowders the intergranular coupling is obviously  very \nweek. This coupling is temperature dependent and ca n be \nfurther suppressed at higher temperatures. For dens e, hot-\npressed ceramic the intergranular coupling is very strong and \nthe coercivity reflects only decrease of anisotropy  field Ha(T) \nwith temperature (see inset to Fig. 3c). \n \nIV.  CONCLUSIONS \nThe studies of the magnetic properties of BiFeO 3 nanopowder \nobtained by mechanochemical synthesis and its ceram ics have \nshown the improvement of magnetization. The improve ment \nin the magnetization of nanosized particles we woul d like to \nrelate to the suppression of cycloidal order, i.e. , incomplete \nrotation of the spins along the direction of the wa ve vector and \nalso to an increase in spin canting, due to the lat tice strain \nwhich gives rise to weak ferromagnetism. The presen ce of the \ncore-shell structure in the BiFeO 3 grains obtained by  4 mechanochemical synthesis may be important from the  \napplication point of view because of possibility to  improve \nmagnetic properties of nanomaterials.  \n \nV.  ACKNOWLEDGEMENT \nAuthors would like to thank Dr. B. Malic (Institut Jozef \nStefan, Ljubljana, Slovenia) for preparation of the  hot-pressed \nceramics.  \n \nREFERENCES \n[1]  H. Schmid, J Phys: Condens Matter 20 (2008) 43 3 DOI:10.1088/0953-\n8984/20/43/434201 \n[2]  G. Catalan, J. F. Scott, Adv Mater 21 (2009)  2463 DOI: \n10.1002/adma.200802849  \n[3]  R. Mazumder, P. Sujatha Dev, D. Bhattacharya, P. Choudhury, A. 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Shen, J.P. Li u, Appl Phys Lett 88 \n(2006) 042504 DOI: dx.doi.org/10.1063/1.2167795  \n "    },    {        "title": "1807.09013v1.Enhanced_magnon_spin_transport_in_NiFe__2_O__4__thin_films_on_a_lattice_matched_substrate.pdf",        "content": "Enhanced magnon spin transport in NiFe 2O4thin films on a\nlattice-matched substrate\nJ. Shan,1,a)A. V. Singh,2L. Liang,1L. Cornelissen,1A. Gupta,2B. J. van Wees,1and T. Kuschel1, 3\n1)Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,\nThe Netherlands\n2)Center for Materials for Information Technology, The University of Alabama, Tuscaloosa, AL 35487,\nUSA\n3)Center for Spinelectronic Materials and Devices, Department of Physics, Bielefeld University, Universit ¨atsstraße 25,\n33615 Bielefeld, Germany\n(Dated: 25 July 2018)\nWe investigate magnon spin transport in epitaxial nickel ferrite (NiFe 2O4, NFO) films grown on magnesium gallate\nspinel (MgGa 2O4, MGO) substrates, which have a lattice mismatch with NFO as small as 0.78%, resulting in the\nreduction of antiphase boundary defects and thus in improved magnetic properties in the NFO films. In the nonlocal\ntransport experiments, enhanced signals are observed for both electrically and thermally excited magnons, and the\nmagnon relaxation length ( \u0015m) of NFO is found to be around 2.5 \u0016m at room temperature. Moreover, at both room\nand low temperatures, we present distinct features from the nonlocal spin Seebeck signals which arise from magnon-\npolaron formation. Our results demonstrate excellent magnon transport properties (magnon spin conductivity, \u0015m\nand spin mixing conductance at the interface between Pt) of NFO films grown on a lattice-matched substrate that are\ncomparable with those of yttrium iron garnet.\nMagnons, the collective excitation of spins, are playing the\ncentral role in the field of insulator spintronics.1Magnons in\nmagnetic materials can interact with conduction electrons in\nadjacent heavy metals, transferring spin angular momentum\nand thus allowing for magnonic spin current excitation and de-\ntection using electrical methods.2–9Besides, magnons can be\ndriven thermally, known as the spin Seebeck effect (SSE).10–14\nBoth magnons generated by a spin voltage bias and a tem-\nperature gradient can be transported for a certain distance in\nthe order of a few to tens of micrometers, as reported recently\nin ferrimagnetic3,15and even in antiferromagnetic materials,16\nmaking magnons promising as novel information carriers.\nNickel ferrite (NFO) is a ferrimagnetic insulator with in-\nverse spinel structure. It is widely used in high-frequency\nsystems and as inductors in conventional applications.17Re-\ncently, NFO and other spinel ferrites were explored for spin-\ntronics applications, where effects like spin Hall magne-\ntoresistance (SMR),18–22SSE23–30and nonlocal magnon spin\ntransport15were reported. In most of these studies, large mag-\nnetic fields of a few teslas are required to align the magnetiza-\ntion of the ferrites, possibly due to the presence of antiphase\nboundaries.31\nHowever, it was recently shown that the NFO films grown\non nearly-lattice-matched substrates with similar spinel struc-\ntures, such as MgGa 2O4and CoGa 2O4, exhibited supe-\nrior magnetic properties due to the elimination of antiphase\nboundaries, leading to, for instance, a larger saturation mag-\nnetization (MS), smaller coercive fields and a lower Gilbert\ndamping constant, compared to the NFO films grown on the\ntypically used MgAl 2O4(MAO) substrate.32An enhanced\nlongitudinal SSE effect was reported on such NFO films.33\nIt can be expected that the nonlocal transport properties of\na)j.shan@rug.nlmagnon spin are also elevated in these NFO films, as we dis-\ncuss in this paper.\nWe studied two NFO films on MGO (100) substrates, with\nthicknesses of 40 nm and 450 nm, respectively. NFO films\nwere grown by pulsed laser deposition, in the same way as\ndescribed in Refs.32,33. Prior to further processes, the 450-\nnm-thick sample was characterized by superconducting quan-\ntum interference device (SQUID) magnetometry, exhibiting\nan in-plane coercive field lower than 5 mT (see Fig. 1(b)). Af-\nterwards, multiple devices were fabricated on both samples.\nFigure 1(a) shows schematically the typical geometry of a de-\nvice, where two identical Pt strips are patterned in parallel\nwith a center-to-center spacing d, ranging from 0.3 to 25 \u0016m\nfor all devices. The lengths and widths of the Pt strips are\ndesigned to be different for shorter- and longer- ddevices, as\nsummarized in Table I. In Geometry I, Pt strips are 100 nm in\nwidth, allowing for fabrication of devices with narrow spac-\nings. In Geometry II, Pt strips are wider and longer, permitting\nlarger injection currents which yield larger signal-to-noise ra-\ntio, so that small signals can be resolved. For all devices, Pt is\nsputtered with a thickness of 8 nm, showing a conductivity of\naround 3\u0002106S/m. Contacts consisting of Ti (5 nm)/Au (60\nnm) were patterned in the final step of device fabrication.\nElectrical measurements were performed with a standard\nlock-in technique, where a low-frequency ac current, I=p\n2I0sin(2\u0019ft), was used as the input to the device, and\nvoltage outputs were detected at the same ( 1f) or double fre-\nquency ( 2f), representing the linear and quadratic effects, re-\nspectively. In this study, typically I0is 100\u0016A andfis set\nTABLE I. Sample details of Geometry I and II.\nGeometry Pt length ( \u0016m) Pt width ( \u0016m) distances ( \u0016m)\nI 10 0.1 0.3 - 2\nII 20 0.5 2 - 25arXiv:1807.09013v1  [cond-mat.mes-hall]  24 Jul 20182\n(e)\n(b)\nV1f (nV)\n α (deg)\n-180-135-90-450 4590135180-1000-50005001000\n α (deg)V2f (nV)-180-135-90-450 4590135180020040060080010001200\nVEI\nVTG-180-135-90-450 4590135180494.9495.0495.1495.2495.3495.4\n α (deg)V1f (mV)(c)\n(d)\n-180-135-90-45045901351808090100  \n \n α (deg)V2f (/uni03BCV)\nNFO (100) B\nα\nVNL+\n-\nd\nVL+-\nI \nMGO (100) (a)\n(f)d=350 nm local\nVSMR\nVSSEd=350 nm localM (emu/cm3)\n-150-100-50050100150\n-100 -50 0 50 100\nmagnetic /f_ield (mT)\nLL\nNLNL\nFIG. 1. (a) Schematic geometry of local and nonlocal measurements. An electric current Iis applied at one Pt strip, and voltages can be\ndetected at the same strip (locally) or at the other one (nonlocally). An in-plane magnetic field is applied at an angle denoted by \u000b. (b) In-\nplane magnetization of the 450 nm-thick NFO film obtained from SQUID at room temperature. (c)-(f), Room-temperature local and nonlocal\nmeasurements shown in first and second harmonic signals, with I= 100\u0016A. They are measured on the 40-nm-thick NFO sample with an\nexternal magnetic field of 300 mT under angular sweep. Only for (e), a background of 910 nV is subtracted.\nto be around 13 Hz. For the local detection VL, as shown in\nFig. 1(a),V1f\nLdetects the resistance and magnetoresistance\n(MR) effect of the Pt strip, and V2f\nLincorporates the current-\ninduced local SSE.34,35For the nonlocal detection VNL,V1f\nNL\nrepresents the nonlocal signals from magnons that are in-\njected electrically via SHE,3,4andV2f\nNLstands for the nonlo-\ncal SSE.3,15,36–40The conductance of the NFO thin films was\nchecked by measuring resistances between random pairs of\nelectrically detached contacts, which yielded values over G \n,\nconfirming the insulating nature of the NFO films.\nWe first perform angular-dependent measurements at room\ntemperature for both local and nonlocal configurations, with\nresults plotted in Figs. 1(c)-(f). The sample was rotated in-\nplane with a constant magnetic field applied. The strength of\nthe field is 300 mT, large enough to saturate the NFO mag-\nnetization along the field direction. A strong MR effect, with\n\u0001R=R\u00190:1%, was observed from the local V1f\nLsignal (see\nFig. 1(c)). This MR effect was checked to be magnetic-field\nindependent in the range from 100 to 400 mT, indicating that\nthe observed MR effect is the SMR effect which is sensitive\nto the NFO magnetization that is saturated in this range, in-\nstead of the Hanle MR effect41which depends on the exter-\nnal magnetic field. This is in marked contrast to the previous\nobservations from sputtered NFO thin films grown on MAO,\nwhere only the Hanle MR effect was observed at fields above\n1 T.15The SMR ratios for both 40- and 450-nm thick sam-\nples exhibit similar values, ranging between 0.07% to 0.1%,around 3 to 4 times larger than those for Pt/yttrium iron garnet\n(YIG) systems with similar Pt thickness.7,36,42It is also more\nthan twice as large as the SMR reported from Pt/NFO sys-\ntems with the NFO layer grown by chemical vapor deposition\non MAO substrates.19Using the average SMR ratio of 0.08%\nand the spin Hall angle of Pt of 0.11,7,36we estimated the real\npart of the spin mixing conductance ( Gr) for Pt/NFO systems\nto be 5:7\u00021014S/m2with the SMR equation,43being more\nthan 3 times larger than that of the Pt/YIG systems determined\nwith the same method.7\nFigures 1(e) and 1(f) plot typical results from the non-\nlocal measurements in V1f\nNLandV2f\nNL, showing respectively\ncos2(\u000b)andcos(\u000b)dependences, same as observed previ-\nously in YIG or NFO films with Pt or Ta electrodes.3,15,36,44,45\nFor the magnon transport process represented by V1f\nNL, both\nthe magnon excitation and detection efficiencies are governed\nbycos(\u000b), which in total yields a cos2(\u000b)behavior. For V2f\nNL,\non the other hand, the thermal magnon excitation is indepen-\ndent of\u000bbut the detection process is, thus showing a cos(\u000b)\ndependence. Their amplitudes, denoted as VEIandVTGre-\nspectively, can be obtained from sinusoidal fittings.\nNext, we present VEIandVTGfor all devices as a function of\ndon both the 40- and 450-nm-thick samples to investigate the\nmagnon relaxation properties, as shown in Fig. 2. For both\nVEIandVTG, discontinuities are found between Geometry I\n(d\u00142\u0016m, filled with yellow color) and II ( d\u00152\u0016m),\neven though the data from Geometry II are carefully normal-3\n0 2 4 6 8 10 12 140.11101001000 tNFO= 40 nm\n tNFO= 450 nm\n distance (/uni03BCm)VEI (nV)\nλm=2.8±0.4 /uni03BCm\nλm=2.5±0.5 /uni03BCm(a)\n0 5 10 15 20 250.11101001000\n VTG (nV)(b)\ndistance (/uni03BCm)λm=2.2±0.2 /uni03BCm\nλm=2.3±0.1 /uni03BCm= 40 nm\n= 450 nm tNFO\n tNFO\nFIG. 2. Distance dependence of (a) VEIand (b)VTGmeasured at\nB=200 mT on both NFO samples at room temperature, normalized\ntoI=100\u0016A. The datapoints filled with yellow color are obtained\nfrom devices in Geometry I, while the rest belongs to Geometry II.\nThe datapoints from Geometry II are normalized to Geometry I as de-\nscribed in Ref.36for better comparison. Dashed lines are exponential\nfittings with the formula V=Aexp(\u0000d=\u0015m), with the coefficient\nAbeing different for each fitting. The extracted \u0015mfrom each fitting\nis indicated nearby. The dotted orange lines in (b) are 1=d2fittings\nfor long-dresults.\nized to Geometry I as was done for Pt/YIG nonlocal devices to\nlink the data between the two geometries.36However, this nor-\nmalization method is based on the assumption of noninvasive\ncontacts and does not account for the additional spin absorp-\ntion that was induced by widening the Pt contact width. This\nnormalization method works well for Pt/YIG systems but be-\ncomes less satisfactory for Pt/NFO systems as we study here,\nwhich is expected in view of a larger Grvalue.\nForVEI, the datapoints at d >15\u0016m (d > 12\u0016m for 450\nnm NFO) are not plotted as the signal amplitudes become\nmuch smaller than the noise level. For shorter distances ( d<1\n\u0016m), the signals on both samples are even comparable to those\nmeasured on thin YIG films with similar device geometry3,36,\nthough a fairer comparison should be made with the same\nthickness of the magnetic insulators. We can also make a com-\nparison between the VEIsignals from the 40-nm-thick NFOfilm studied here and the 44-nm-thick sputtered NFO film on\nMAO substrate studied in Ref.15. We found that for the same\ndevice geometry ( d=350 nm) and Pt thickness, the VEIsig-\nnal amplitudes obtained here is around 100 times larger than\nfound in Ref.15, showing the superior quality of the NFO films\nstudied in this paper.\nTo extract\u0015mfor these NFO samples at room temperature,\nwe performed exponential fittings as shown in Fig. 2(a) by the\ndashed lines. We limit the fit to the datapoints in the expo-\nnential regime where d > 2\u0016m. Both datasets yield \u0015m\u0019\n2.5\u0016m for the two NFO samples with different thicknesses.\nWorthnotingly, the VEIsignals for the 450 nm NFO are in gen-\neral smaller than those for the 40 nm NFO sample, except for\none datapoint at the shortest distance. However, one would\nexpect the opposite, as increasing the NFO thickness from 40\nto 450 nm enlarges the magnon conductance without intro-\nducing extra relaxation channel vertically, given that 450 nm\nis still much smaller than \u0015m\u00192.5\u0016m. This puzzle is similar\nas for Pt/YIG systems36and the reason is not yet clear to us.\nNow we move to the thermally generated nonlocal SSE\nsignalsVTGas shown in Fig. 2(b). According to the bulk-\ngenerated SSE picture,7,36,40,46at a certain distance ( drev)VTG\nshould reverse sign, where in short distances VTGhas the same\nsign as the local SSE signal, and further away the sign al-\nters.drevis influenced by the thickness of the magnetic insu-\nlator and interfacial spin transparency at the contacts.36With\nour measurement configuration (the polarities of local and\nnonlocal measurement configurations are opposite as shown\nin Fig. 1(a)), the VTGmeasured from all devices are in fact\nopposite in the sign compared to the local SSE signals (see\nFig. 1(d)), meaning drevis positioned closer than the shortest\ndwe investigated. Only an upturn is observable for VTGof\n450 nm NFO sample at short- drange. Compared to Pt/YIG\nsystems, where drevis about 1.6 times of the YIG thickness,\nfor Pt/NFO systems the sign-reversal takes place much closer\nto the heater, possibly because of the Pt/NFO interface being\nmore transparent for a larger Gr.\nExponential fittings can also be carried out for VTGon both\nsamples. Note that only the datapoints in the exponential\nregime can be used to extract \u0015m, which typical starts at\nd=\u0015mand extends to a few \u0015m.38Further than the exponen-\ntial regime, VTGstarts to decay geometrically as 1=d2, dom-\ninated by the temperature gradient present near the detector.\nBased on the \u0015mthat we extracted from the decay of the elec-\ntrically injected magnon signals, we identify 2\u0014d\u00148\u0016m\nas the exponential regime and obtained \u0015mto be around 2.2\nor 2.3\u0016m from the decay of VTG. The consistency between\nthe\u0015mfound from magnon signals excited electrically and\nthermally illustrate again the same transport nature of the\nmagnons generated in both methods.\nOwing to the excellent quality of the NFO films, we are able\nto study its magnetoelastic coupling by means of the nonlocal\nSSE. It was observed in YIG that for both the local and non-\nlocal SSE signals, spike structures arose at certain magnetic\nfields, at which the magnon and phonon dispersions became\ntangent to each other, resulting in maximal magnetoelastic\ninteraction and the formation of magnon-polarons.40,47–49At\nthese conditions, the spin Seebeck signals have extra contri-4\n-800-600-400-2000200400600800VTG (nV)\n-6 -5 -4 -3-810-800-790-780 \n3 4 5790800810820830 (a)\n6\n-6 -4 -2 0 2 4 6-600-400-2000200400600 (b)\nVTG (nV)\nmagnetic /f_ield (T)T=293 K\n-6 -5 -4 -3-570-560-550-540\n3 4 5 6540550560570 T=150 K\nFIG. 3. Magnetic field sweep measurements of VTGat (a)T= 150\nK and (b)T= 293 K, on the 450 nm NFO sample ( d=1\u0016m) from\nthe nonlocal spin Seebeck measurements with \u000b= 0\u000e. Insets show\nclose-ups of the resonant dips.\nbutions from the magnon-polarons, provided that the magnon\nand phonon impurity scattering potentials are different.47,48It\nwas found that for YIG films, the acoustic quality is higher\nthan the magnetic one, with peaks observed in local SSE\nand nonlocal SSE ( d < d rev) measurements and dips ob-\nserved for nonlocal SSE where d > d rev.40,47This effect is\nexplained as several parameters such as \u0015m, the bulk spin\nSeebeck coefficient, the magnon spin and heat conductivities\nare all modified by the emergence of magnon-polarons.40,48\nSo far, this resonant enhancement/suppression of SSE caused\nby magnetoelastic coupling has only been clearly observed in\nYIG; besides, a bimodal structure was found in the SSE of a\nNi0:65Zn0:35Al0:8Fe1:2O4thin film and was speculated to be\nrelated to magnon-phonon interactions.50\nHere we present distinctive magnon-polaron features in the\nnonlocal SSE measurements on our NFO films. Figure 3\nshows field-sweep data of VTGperformed on one device of the\n450-nm-thick NFO sample at T=150 K and 293 K. At both\ntemperatures, asymmetric dip structures of VTGare clearly\nvisible, around\u00064.2 T atT=150 K and shift to \u00064.0 T at\nT=293 K. The change of the characteristic magnetic field\nof 0.2 T for a temperature decrease of about 150 K is com-\nparable to Pt/YIG systems.40The sign of the anomalies is in\naccordance with the previous observation reported in Ref.40,considering that the spacing between the Pt strips ( d=1\u0016m)\nis further than drev. This implies that the studied NFO film\nmay also have a higher acoustic than magnetic quality like\nYIG, though a careful study which measures the anomalies\nfrom the local SSE is needed.\nThe magnetic fields where the anomalies occur can be eval-\nuated by the phonon and magnon dispersions. In our experi-\nments, limited by the maximal applied magnetic field ( \u00160H\u0019\n7T), we could only probe the first anomaly which involves\ntransverse acoustic (TA) phonons with a lower sound veloc-\nity. The TA phonons follow the dispersion relation !=vTk,\nwherevTis the TA phonon sound velocity. vTis related to the\nelastic constant C44and material density \u001abyC44=\u001av2\nT,50,51\nand is determined to be 3968 m/s for NFO using C44=82.3\nGPa and\u001a= 5230 kg/m3.33,52\nWe assume that magnons in NFO can be also described\nby a parabolic dispersion relation like for YIG ( !=p\n(Dexk2+\r\u00160H)(Dexk2+\r(\u00160H+MS))), whereDexis\nthe exchange stiffness, \u00160the vacuum permeability and \rthe\ngyromagnetic ratio. From Fig. 1(b) we obtain MSof our\nNFO sample to be 160 emu/cm3at room temperature, which\nequals 201 mT. To the best of our knowledge, the Dexof NFO\nis not experimentally reported. In our experiment, from the\npeak positions observed at room temperature ( \u00160HTA=\u00064:0\nT) we can determine the only unknown parameter Dexto be\n5.5\u000210\u00006m2/s with both phonon and magnon dispersions.\nThis value is close to the Dexwhich can be estimated from\nthe exchange integrals among Ni2+, Fe3+(octahedral site)\nand Fe3+(tetrahedral site).53,54Using the parameters given\nin Ref.55,Dexof NFO can be estimated to be 6.4 \u000210\u00006m2/s,\nwithin 17% difference of our experimental value.\nAnomalies were also observed in the 450 nm NFO sample\nfor electrically excited magnons in the field-sweep measure-\nments ofVEIatT=150 K, albeit with a lower signal-to-noise\nratio. For the 40 nm NFO sample, however, no clear anoma-\nlies were identified in the measured range ( \u00160H\u00146:6T) for\nVEIorVTG.\nIn summary, we have studied the magnon spin transport\nproperties of epitaxial NFO films grown on MGO substrates\nin a nonlocal geometry. We obtained large nonlocal signals\nfor both electrically and thermally excited magnons at short\ncontact spacings, comparable to that of YIG. From the relax-\nation regime \u0015mwas found to be around 2.5 \u0016m. Further, we\nobserved anomalous features as a result of magnon-polarons\nformation in the field-dependent SSE measurements at both\n150 and 293 K, from which the exchange stiffness constant\nof NFO can be determined. Our results demonstrate the im-\nproved quality of NFO grown on a lattice-matched substrate,\nshowing NFO to be a potential alternative to YIG for spin-\ntronic applications.\nWe thank the helpful discussions with Gerrit Bauer,\nMatthias Althammer and Koichi Oyanagi, and would like to\nacknowledge M. de Roosz, H. Adema, T. Schouten and J. G.\nHolstein for technical assistance. 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The oleic acid  coated  CFO  nanoparticles are  stable in non-\npolar organic media, such as hexane but are  not well dispersible in water . The sur face of these \nsnano particles has  been further  modified by citric acid using ligand e xchange proces s, which \nmakes CFO nanoparticle s more stable colloidal solution  in water . Citric acid coated CFO \nnanoparticles exhibits high dispersibility in water, high zeta potential , very low coercivity and \nmoderate saturation magnetization. Biocompatibility of these CFO nanoparticles is \ndemonstrated through cytotoxicity test in L929 cell line.  \nKEYWORDS:  CoFe 2O4, Biomedical a pplications, Ligand exchange,  Superparamagnetism , \nNanoparticles . \n*Authors to whom all correspondence should be addressed.  \nE-mail: nkhare@physics.iitd.ernet.in   \n \n doi: 10.1016/j.jmmm.2015.12.017   \n2 \n 1. Introduction  \nIn recent years m agnetic nanoparticles have been extensively explored for  their potentiality \nin many biomedical applications such as for targeted  drug delivery [ 1], as contrast enhancement \nagents  in magnetic resonance imaging (MRI) [ 2], and in hyperthermia treatments as heat \nmediators [ 3].The main advantage of magnetic nanopart icles for biomedical applications is its \nlarger surface area for easy ligand attachment, better tissue diffusion and reduced dipole -dipole \ninteraction . The magnetic properties of the  nanoparticles can be tuned by controlling its  size \n[4], composition [ 5], shape [ 6] and strain/defects [ 7]. By carefully reducing its size below to a \ncritical diameter, the magnetic nanoparticles can be turned to superparamagnetic nanoparticles.  \nIron oxide magnetic  nanoparticles such as Fe3O4 [2] and γ−Fe2O3 [8] have been widely  \nexplored for biomedical applications . The saturation magnetization and hysteresis losses of \nthese iro n oxide nanoparticles are small  compared to pure metals (Co, Fe, or CoFe  etc.), but \nthe metallic nanoparticles are highly toxic and very sensitive to oxidation and hence are not \nuseful for biomedical applications. Another alternative can be spinel ferrites such as MFe 2O4 \n(M  Co, Mn, Ni) [ 9, 10, 11 ]. Among these ferrites , CoFe 2O4 is interesting due to its large curie \ntemperature, high effective anisotropy and moderate saturation magnetization  [12]. CFO has \nan inverse spinel structure with general formula AB 2O4 (A = Fe and B = Co, Fe) where half of \nthe Fe3+ occupies the octahedral sites and the other half Fe3+ occupies the tetrahedral sites \nwhereas all the Co2+ occupy the octahedral sites.  \n  For biomedical applications the magnetic nanoparticles should be of small sizes with \nnarrow size distribution. These nanoparticles should be coated with some organic or inorganic \nmaterial  which  ensure their biocompatibility , nontoxicity and colloidal stability in biophase . \nSeveral techniques  such as m icroemulsion  [13], coprecipitation  [14], ball milling  [15], sol−gel \n[16], thermal  decomposition  [17], sonochemical  [18] and electrosynthesis [9] method have   \n3 \n been employed for the  synthesis of magnetic nanoparticles but all these synthesis methods often  \nproduce larger size nanoparticles with wide  particle size distribution . \nIn the present work , we have synthesized uniform size (~6nm)  CoFe 2O4 magnetic  \nnanoparticles using hydrothermal techniques with oleic acid  as surfactant.  These oleic acid \ncoated CFO nanoparticles are  not dispersible in water and i n order to make these nanoparticles \nwater dispersible, the surface of these oleic acid coated nanoparticles was modified with citric \nacid using ligand exchange method.  It is found that these citric acid coated CoFe 2O4 \nnanoparticles makes a good colloidal solution in water in a wide range of pH.  The \nbiocompatibility of citric acid coated CFO nanoparticles was  studied  with mouse fibroblast \nL929 cells lines, using a  MTT cytotoxicity assay.   \n2. Experimental  \nCFO nanoparticles were synthesized by hydrothermal method  [7], using ferric  and \ncobalt nitrate as precursors and oleic acid as surfactant . A solution of 2 mmol  of ferric nitrate \nand 1 mmol of cobalt nitrate in 20 ml water was added with  a 10 m mol NaOH  solution, ethanol \nand 12 ml oleic acid. The resultant solution was mixed thoroughly using a magnetic stirrer and \nwas put into  a Teflon lined stainless steel autoclave.  The a utoclave was  placed into a preheated \noven at 180o C for 16 h ours. After cooling , the particles were washed several times in hexane \nand ethanol. A permanent magnet was used for t he separation of the nano particles  from the \nliquid . These nanoparticles is named as OA -CFO.  \nFor synthesizing  citric acid coated CFO nanoparticles from the oleic acid coated CFO \nnanoparticles, ligand e xchange method was used. The OA -CFO nanoparticles were kept in a \nsolution of toluene, citric acid and dimethyl sulfoxide (DMSO) and stirred thoroughly for 30 \nhours.  Modified nanoparticles were collected , washed in ethanol and dried at 60 oC. These \ncitric acid coated nanoparticles is named as CA -CFO in the  subsequent discussion.   \n4 \n Structural properties of CFO samples were investigated using Rigaku Ultima IV X -ray \ndiffractometer (XRD) equipped with Cu Kα ( λ=1.542 Å) radiation source and the morphology \nof the samples were characterized by using a JEOL JEM -2200 -FS Transmission electron \nmicroscope (TEM). Magnetic measurements of the CFO nanoparticles  were performed at room \ntemperature using Quantum Design Evercool -II Physical pr operty measurement system,  in the  \nmagnetic  field range of -4 to 4 Tesla . For the magnetic measurements samples were prepared \nby adding a known amount of CFO nanoparticles in DI  water and then drop casted onto a glass  \nsubstra te of dimensions 0.4cm × 0.4 cm .  FTIR studies of  OA-CFO and CA -CFO nanoparticles \nwere carried out using Thermo Scientific™ Nicolet™ iS™ 50 FT -IR Spectrometer.  Zeta \npotential ( 𝜁) and hydro dynamic diameter  of CA-CFO nanoparticles were studied using \nmalvern zetasizer nano zs90.  \nIn vitro cell viability studies of CA -CFO nanoparticles were carried out with mouse \nfibroblast L929 cells lines , using a 3-(4 5-dimethylthiazol -2-yl)-2 5-dipheny ltetrazolium \nbromide ( MTT)  cytotoxicity assay. The MTT cytotoxicity assay is a colorimetric assay which \nmeasures the cellular metabolic activity based on mitochondrial NADPH dependent \ndehydrogenase enzymes  [19]. These enzymes reduce the MTT dye to form formazan crystals \nin viable cells. DMSO dissolves these crystals to give a purple colored solution, which can then \nbe quantified s pectrophotometrically at 540 nm , using microplate spectrophotometer \n(PowerWave XS2,  BioTek Instruments, USA) . \n3. Results and discussion  \nXRD patterns of OA-CFO  and CA -CFO nanoparticles  are shown in Fig. 1.  The observed \npeaks  at 2θ = 30.12º, 35.44º, 43.08º, 53.52 º, 57.04º and 62.58 º corresponds  to (220), (311), \n(400), (422), (511) and (440 ) planes  of CoFe 2O4 (JCPDS No. 22 -1086) . This confi rms the  \nformation of single phase cubic spinel structure of CoFe 2O4 nanoparticles . The XRD patterns  \n5 \n of OA -CFO and CA -CFO nanoparticles are similar because these nanoparticles have same \ncrystalline core of CoFe 2O4. \nScherer formula is used to determine the average crystallite size for CFO nanoparticles, \nwhich is given as  [20]; \nt = 0.9λ\nβ cosθ      (1) \nwhere β represen ts the full width at half maximum  of the XRD peak, θ is the Bragg’s \nangle, λ (1.542 Å) is the wavelength of X -ray, and  t is the average crystallite size. The average \ncrystallite sizes for OA-CFO  and CA-CFO  nanoparticles are found as    ̴ 5.9 nm and 5.4 nm \nrespectively.   \nTEM images of OA-CFO nanoparticles and CA -CFO nanoparticles are shown in Fig. \n2. The histogram of the particle size distribution  are also shown as inset in Fig. 2a and 2b. The \nsize distribution of the CFO nanoparticles reveals that the maximum number of the particles \nhas a diameter in the size range of 4 nm to 7 nm with a log normal peak appearing at 5.8 and \n5.6 nm for OA-CFO and CA -CFO nanoparticles.  These results are in go od agreem ent with the \ncrystallite size  obtained from XRD analysis.  \nFig. 3 shows the field dependence of magnetization  for OA-CFO and CA -CFO \nnanoparticles at room temperature. OA-CFO nanoparticles are found to have a slightly high \nsaturation magnetization  as compared to CA-CFO  nanoparticle . For OA -CFO nanoparticles  \nthe value of the saturation magnetization (M s) and coercivity (H c) are found as ~47 emu/gm  \nand ~11 Oe respectively, and for CA-CFO  nanoparticles, the values of the M s and H c are found \nas ~42 emu/gm  and ~13 Oe respectively. The  small value s of coercivity  of CFO  nanoparticles \nindicates that these nanoparticles are  near the superparamagnetic limit  which is the ideal regime \nfor several  biomedical applications .  \n6 \n The lower value of saturation magnet ization (M s) of these CFO nanoparticles compared \nto the Ms value  of bulk cobalt ferrite ( ~ 90 emu/gm ) can be attributed to a “nanoscale size \neffect ” according to which the magnetic moments present near the surface of the nanoparticles \nbeha ves differently from that  present in the core of the particles and a much higher spin disorder \nis present on the surface of the particles which  leads to the reduction of M s of CFO \nnanoparticles  [21].   \nFTIR spectr um of OA-CFO and CA -CFO nanoparticles are  shown in Fig.  4. The \ndifference in FTIR spectra of OA -CFO and CA -CFO nanoparticles is due to the presence of \ndifferent coating on CFO for both of these two samples. The presence of oleic acid  on as \nsynthesized  OA-CFO  nanoparticles  was confirmed by two CH 3 stretching at 2920 cm-1 and \n2850  cm-1 present in FTIR spectra of the sample . The t wo bands appears near 1538 and 1410 \ncm-1, which are charact eristic bands of the asymmetric  and the symmetric stretch of  (COO) . It \nis evident  that oleic acid  was chemisorbed onto the surface of CFO nanoparticles via its \ncarboxylate group  [22].  \nThe CA -CFO nanoparticles shows three strong abs orption peaks at  3275 , 1575, and \n1405 cm-1 corresponding  to the stretching ba nd of hydroxyl group ( -OH),  antisymmetric 𝜐as \n(COO)  and symmetric 𝜐s (COO)  stretching band of the carboxy l group, respectively  [23]. This \nconfirms that the surface of the CA-CFO nanoparticles was covered with carboxylate species  \nof citric acid . An intense peak at ∼590 cm−1 is observed, which is  attributed to the stretching \nof the metal ion at  the tetrahedral A -site, M A↔O [24].  \nThe CA -CFO nanoparticles are well dispersible in water in a wide range of pH. The \nhydrodynamic diameter (D H) determined by Dynamic Light Scattering (DLS) and zeta \npotenti al (ζ) values for CA -CFO nanoparticles at different pH values ranging from 2.2 to 10.8, \nare shown in Fig. 5. As the ζ is related to the surface charge present on the nanoparticles, the  \n7 \n large/small value of |𝜁| indicates the more/less electrostatic repulsion between the \nnanoparticles. In the case of magnetic nanoparticles this electrostatic repulsion opposes the \nmagnetic attraction acting between the nanoparticles. Initially increasing the pH from 2.2 the \n|ζ| ap proaches toward “0” and at pH value ~3.5, |ζ| is minimum. At this point the magnetic \nattraction exceeds the electrostatic repulsion which leads to more agglomeration of CA -CFO \nnanoparticles. Due to this agglomeration the D H shows its maximum value at pH ~3 .5. At \nhigher pH values (> 3.5) the |ζ| starts increasing, which increases the electrostatic repulsion \nbetween the nanoparticles that leads to a decrease in D H. At pH ~7, ζ is sufficiently negative ( -\n22.3 mV), which indicates that negative charges are pres ent on the surface of CA -CFO \nnanoparticles in a larger amount. Smaller value of D H for CA -CFO nanoparticles indicates that \ngood electrostatic repulsive forces are acting between the particles that opposes the \nagglomeration of these nanoparticles and increa ses the dispersion and colloidal stability in \nwater. All these observations suggests that the surface modification of OA -CFO nanoparticles \nby citric acid allows us to obtain a good dispersion in water with more colloidal stability.  \nThe OA -CFO nanoparticles  are not dispersible in water so these nanoparticle cannot be \nused in biomedical applications but the high colloidal stability of CA -CFO nanoparticles at \nneutral pH values make the CA -CFO magnetic nanoparticles a suitable candidate for \nbiomedical applicati ons. The biocompatibility  of citric acid  coated CFO  nanoparticles  was \nstudied  on the L9 29 (mouse fibroblast) cells  line, using an MTT cytotoxicity assay. Cells were \nseeded at a density of 10 × 103 cells per well in 96 well plate.  The plate was incubated a t 37 \n°C for 24 h in a CO 2 incubator. CA-CFO nanoparticles  suspension was added to each well so \nthat final concentration range was from 100 µg/ml to 1000 µg/ml and inc ubated for another 24 \nh. The media was replaced with fresh media and 10 μL  of 5% MTT solution, filtered through \nsterile 0.22 µm filter was added to each well and incubated for 4 h. Cell viability was calculated  \n8 \n relative to negative control ( phosphate buffered saline ) and a positive control (1% Triton X -\n100) using the following  relation : \n   Cell Viability (%) = Sample 540nm −Positive  Control 540nm\nNegative  Control 540nm −Positive  Control 540nm × 100 \nFig. 6 shows the  surviving fraction of L929 cells incubated during 24 hours with \ndifferent concentrations of CA -CFO nanoparticles and evaluated by the MTT assay. The \nsynthesized  CA-CFO  nanoparticles were found to biocompatible at relatively high \nconcentrations (100 µg/ml to 1000 µg/ml), with cell viability almost 100 %, i.e. the survival \nfraction of treated cells is similar to the controls cell under the experimental conditions . This \nhigh cell viability of CA -CFO nanoparticles is attributed to the  coating of citric acid  on these \nnanoparticles through ligand exchange method. As th e zeta potential of CA-CFO  nanoparticles \nwas found to be large (-18.8 mV ), these CA-CFO nanoparticles  will have long c irculatory \neffect in biophase . \n4. Conclusion  \nWe have successfully synthesized oleic acid capped, uniform size (~6 nm) CFO \nnanoparticles by hydrothermal method and demonstrated the conversion of these nanoparticles \nto a citric acid (CA) coated CFO nanoparticles  using ligand e xchange method. The CA-CFO  \nnanoparticle s exhibits very s mall coercivity,  moderate saturation magnetization and a high \ndegree of dispersibility in water . At pH ~ 7, z eta potential  of CA-CFO  nano particles was higher \n(~22 mV) , which  gives CA -CFO nanoparticles colloidal stability  in water. 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Jun, Y.M. Huh, J.S. Choi, J. H. Lee, H. -T. Song, S. Kim, S. Yoon, K. -S. Kim, J. -\nS. Shin, J. -S. Suh, and J. Cheon, J. Am. Chem. Soc. , 127  (2005), 5732.  \n[22] L. Zhang, R. He, and H. C. Gu, Appl. Surf. Sci. , 253 (2006) , 2611 . \n[23] A. Goodarzi, Y. Sahoo, M. T. Swihart, and P. N. Prasad, MRS Proc. , 789 (2004 ) N6.6. 1.  \n12 \n [24]  M. V. Limaye, S. B. Singh, S. K. Date, D. Kothari, V. R. Reddy, A. Gupta, V. Sathe, R. \nJ. Choudhary, and S. K. Kulkarni, J. Phys. Ch em. B , 113 (2009) 9070 . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n13 \n Figure Captions  \nFig. 1  X-ray diffraction patterns of OA-CFO and CA -CFO nanoparticles.  \nFig. 2.  TEM images of (a) OA -CFO nanoparticles and (b) CA -CFO nanoparticles. The inset \nin the figure shows  distribution of particle size and its log normal di stribution fit. \nFig. 3.  Magnetic hysteresis loops for OA -CFO and CA -CFO nanoparticles at room \ntemperature.  \nFig. 4. FTIR spectra of OA -CFO and CA -CFO nanoparticles . Arrows marked OA -CFO \nspectra corresponds to 2990 cm-1, 2850 cm-1, 1538 cm-1, 1410 cm-1 and 590 cm-1 (from \nleft to right) whereas the arrows marked to CA -CFO spectra corresponds to 3275 cm-1, \n1575 cm-1, 1405 cm-1 and 590 cm-1  (from left to right).  \nFig. 5. Zeta potential and hydrodynamic diameter  as a function of pH for CA -CFO \nnanoparticles.  \nFig. 6. Cytotoxicity profiles of CA -CFO nanoparticles for 24 h  on L929 cell line at different \nconcentrations (100, 400, 600, 8 00 and 1000 µg/mL)  \n \n \n \n \n \n \n  \n14 \n  \n \n \nFig. 1  X-ray diffraction patterns of OA -CFO and CA -CFO nanoparticles.  \n \n \n \n \n \n \n \n \n15 \n  \n \n \n \nFig. 2.  TEM images of (a) OA-CFO nanoparticles and (b) CA -CFO nanoparticles. The inset \nin the figure shows distribution of particle size and its log normal distribution fit.  \n \n \n \n \n \n \n \n \n \n16 \n  \n \n \nFig. 3.  Magnetic hysteresis loops for OA -CFO and CA -CFO nanoparticles at room \ntemperature.  \n \n \n \n \n \n \n \n17 \n  \n \n \nFig. 4.  FTIR spectra of OA -CFO and CA -CFO nanoparticles.  Arrows marked OA -CFO \nspectra corresponds to 2990 cm-1, 2850 cm-1, 1538 cm-1, 1410 cm-1 and 590  cm-1 (from left to \nright) whereas the arrows marked to CA -CFO  spectra corresponds to 3275 cm-1, 1575 cm-1, \n1405 cm-1 and 590 cm-1  (from left to right).  \n \n \n \n \n \n \n18 \n  \n \n \n \n \nFig. 5. Zeta potential and hydrodynamic diameter as a function of pH for CA -CFO \nnanoparticles.  \n \n \n \n \n \n19 \n  \n \n \n \n \nFig. 6. Cytotoxicity profiles of CA -CFO nanoparticles for 24 h on L929 cell line at different \nconcentrations (100, 400, 600, 8 00 and 1000 µg/mL).  \n \n"    },    {        "title": "1005.2244v1.Short_Range_B_site_Ordering_in_Inverse_Spinel_Ferrite_NiFe2O4.pdf",        "content": "arXiv:1005.2244v1  [cond-mat.mtrl-sci]  13 May 2010Short-Range B-site Ordering in Inverse Spinel Ferrite NiFe 2O4\nV. G. Ivanov1, M. V. Abrashev1, M. N. Iliev2, M. M. Gospodinov3, J. Meen2, M. I. Aroyo4\n1Faculty of Physics, University of Sofia, 1164 Sofia, Bulgaria\n2Texas Center for Superconductivity and Department of Physi cs, University of Houston, Texas 77204-5002, USA\n3Institute of Solid State Physics, Bulgarian Academy of Scie nces, 1184 Sofia, Bulgaria\n4Departamento de F´ ısica de la Materia Condensada,\nUniversidad del Pas Vasco, 48080 Bilbao, Spain\n(Dated: May 14, 2010)\nThe Raman spectra of single crystals of NiFe 2O4were studied in various scattering configurations\nin close comparison with the corresponding spectra of Ni 0.7Zn0.3Fe2O4and Fe 3O4. The number\nof experimentally observed Raman modes exceeds significant ly that expected for a normal spinel\nstructure and the polarization properties of most of the Ram an lines provide evidence for a micro-\nscopic symmetry lower than that given by the Fd¯3mspace group. We argue that the experimental\nresults can be explained by considering the short range 1:1 o rdering of Ni2+and Fe3+at the B-sites\nof inverse spinel structure, most probably of tetragonal P4122/P4322 symmetry.\nPACS numbers: 78.30.-j, 63.20.D-, 75.47.Lx\nI. INTRODUCTION\nThe spinel ferrites with general formula AFe 2O4have\ninteresting physical properties and are of technological\nimportance.1In particular, NiFe 2O4is of increased in-\nterest as this material, in the form of bulk, powder, thin\nfilm or nanoparticles, finds or promises numerous ap-\nplications in magnetic storage systems,2magnetic res-\nonance imaging,3, spintronics,4,5etc. At present it is\naccepted that NiFe 2O4crystallizes with inverse spinel\nstructure,6–9described by the face-centered cubic (FCC)\nspace group Fd¯3m(No.227, Z=8). In this structure the\ntetrahedral A-sites (8a) are occupied by half of the Fe3+\ncations, whereas the rest of the Fe3+and Ni2+cations\nare distributed over the octahedral B-sites (16d). A\nfundamental question however arises whether Fe3+and\nNi2+are spread in a random fashion among the B-\nsites or exhibit specific short-range order on a spatial\nscale, which is below the detection limit of the standard\ndiffraction techniques. This issue can be addressed ef-\nfectively by polarization Raman spectroscopy since the\nnumber, the frequencies and the polarization selection\nrules of the Raman-active vibrational modes are highly\nsensitive to the atomic short-range order. For the nor-\nmal spinel structure only five Raman allowed phonons\n(A1g+Eg+ 3F2g) are expected and this is the case\nfor number of materials such as Co 3O4,10CdCr2Se4,11\nand Fe 3O4above the Verwey transition temperature,12\nto mention a few. Another group of spinels, such as\nNiFe2O4,13,14NiAl2O4,15CoFe2O4,16,17exhibit much\nricher Raman spectra with a number of additional peaks.\nThe latter materials are with inverse or partly inverse\nspinel structure and it is reasonable to assume that the\nappearanceofadditional Raman lines is somehowrelated\nto the presence of non-equivalent atoms at the B-sites,\nwhich may have the following consequences for the Ra-\nman spectra: (i) The random distribution ofinequivalent\nB′and B′′atoms destroys the translation symmetry, in\nparticular of the oxygen sublattice, and activates other-wise forbidden phonon modes. One expects in this case\nadditional broad Raman structures which roughly repro-\nduce the smeared one-phonon density of states;18(ii) A\nshort-range ordering of B′and B′′atoms may result in\nformationofdomainsofsymmetrylowerthan Fd¯3mwith\nnew sets of Raman-allowed phonons. The coexistence of\ntwin variants of these local structures leads to a super-\nposition of spectra corresponding to different scattering\nconfigurations.\nIn this work we report results of a polarized Raman\nstudy of NiFe 2O4. The analysis is done in close compar-\nison with lattice dynamics calculations for spinel struc-\ntures with either full disorder or ordering of Ni2+and\nFe3+at the B-sites as well with the corresponding spec-\ntra of Ni 0.7Zn0.3Fe2O4and Fe 3O4. A conclusion is made\nthat at a microscopic level the structure of NiFe 2O4can\nbe considered as a mixture of twin variants of a structure\nwith Fe3+and Ni2+ordered over the B-sites.\nII. SAMPLES AND METHODS\nThe first step in the growth of NiFe 2O4and\nNi0.7Zn0.3Fe2O4single crystals was sintering of polycrys-\ntalline samples by solid-state reaction of stoichiometric\namounts of NiO, Fe 2O3, and ZnO annealed for 48 h at\n1150◦C in oxygen atmosphere. As a next step the high\ntemperature solution growth method was applied using\nPbO-PbF 2-B2O3flux with ratio of the components of\n0.50 : 0.48 : 0.02 for NiFe 2O4and of 0.67 : 0.32 : 0.01 for\nNi0.7Zn0.3Fe2O4, respectively. The flux was mixed with\nNiFe2O4powder in a 10 : 1 ratio or with Ni 0.7Zn0.3Fe2O4\nin a 7 : 1 ratio and annealed in a 500 ml platinum\ncrucible at 1225◦C in air for 48 h. After annealing\nthe temperature was decreased to 950◦C at a rate of\n0.5◦C/h for NiFe 2O4and to 1000◦at a rate of 1◦C/h for\nNi0.7Zn0.3Fe2O4. The flux was decanted and the crystals\nof up to 5 mm in size were removed from the bottom of\nthe Pt crucible. These crystals were of octahedral shape2\nwith large (111) and smaller (100) and (110) facets. The\nFe3O4samplewasnaturalpolycrystallinemagnetite with\ntypical grain size of 300 µm, large enough for obtaining\npolarized Raman spectra in exact scattering configura-\ntionfromproperlyorientedmicrocrystalsurfaces. Theel-\nemental content has been confirmed by x-ray wavelength\ndispersive spectrometry (WDS)using a JEOL JXA8600\nelectron microprobe analysor.\nThe polarized Raman spectra were measured from\n(100) cubic surfaces with a triple T64000 spectrometer\nequipped with microscope. The spectra obtained with\n633 nm, 515 nm, 488 nm, or 458 nm excitation were\npractically the same.\nIII. RESULTS AND DISCUSSION\nA. Polarized Raman spectra of NiFe 2O4, Fe3O4,\nand Ni 0.7Zn0.3Fe2O4\nFromsymmetryconsiderationsoneexpectsforthe nor-\nmal spinel Fd¯3mstructure five ( A1g+Eg+3F2g) Raman\nactive modes, which could be identified by measuring the\nRaman spectra in severalexact scatteringconfigurations,\ne.g.XX,XY,X′X′, andX′Y′. The first and second\nletters in these notations correspond, respectively, to the\npolarization of incident and scattered light, where X,Y,\nX′, andY′denotethe [100] c, [010]c, [110]c, and[1¯10]ccu-\nbic directions. Asit followsfrom TableI, oneexpects two\nRaman lines ( A1g+Eg) in theXXspectrum, five lines\n(A1g+Eg+ 3F2g) in the X′X′spectrum, three (3 F2g)\nlines in the XYspectrum, and only one ( Eg) line in the\nX′Y′spectrum.\nInFig.1arecomparedtheexperimentalRamanspectra\nof NiFe 2O4and the closely related spinel Fe 3O4obtained\nat room temperature with 488 nm excitation. The spec-\ntra of NiFe 2O4taken with 633 nm excitation are shown\nin more detail in Figure 2. The selection rules for the\nRaman bands of Fe 3O4follow strictly those for the nor-\nmal spinel structure (see Table I). This is to be expected\nsincethe experimental temperatureiswell abovethe Ver-\nwey transition temperature of magnetite and, therefore,\nthe charge is smeared uniformly among the Fe B-sites.\nIn contrast to Fe 3O4, the number of experimentally ob-\nserved Raman lines in the spectra of NiFe 2O4exceeds\nsignificantly the number expected for a normal spinel\nstructure and the polarization rules are strictly followed\nfor only a few lines, namely those at 213 cm−1(F2g),\n333 cm−1(Eg), and 705 cm−1(A1g). It is remarkable,\nhowever, that these spectra are practically identical to\nthosereportedearlierforNiFe 2O4crystals,thin filmsand\nnanocrystalline samples from different sources (see e.g.\nRefs. 13,14). Much richer than expected Raman spectra\nhave been also reported for other compounds, such as\nNiAl2O4, with nominally inverse spinel structures.15\nThe larger number of Raman active modes in inverse\nspinels have been discussed before and has tentatively\nbeen explained in terms of defect-induced lattice distor-200\r 300\r 400\r 500\r 600\r 700\r 800\r\n200\r 300\r 400\r 500\r 600\r 700\r 800\r200\r 300\r 400\r 500\r 600\r 700\r 800\r\n200\r 300\r 400\r 500\r 600\r 700\r 800\rNiFe\r2\rO\r4\r\nFe\r3\rO\r4\rE\rg\rA\r1g\rXX\r\nNiFe\r2\rO\r4\r\nFe\r3\rO\r4\rF\r2g\r F\r2g\r A\r1g\rX'X'\rNiFe\r2\rO\r4\r\nFe\r3\rO\r4\rF\r2g\r F\r2g\rXY\rINTENSITY (arb. units)\r\n RAMAN SHIFT (cm\r-1\r)\rNiFe\r2\rO\r4\r\nFe\r3\rO\r4\rE\rg\r488 nm\r\n300 K\r\nX'Y'\r\nFIG. 1: (Color online) Polarized Raman spectra of NiFe 2O4\nand Fe 3O4as obtained at room temperature with 488 nm\nexcitation.\ntions due to deviation from stoichiometry and/or coexis-\ntenceof’normal’and’inverse’domains.13,15Itseemsthat\nthe deviation from the stoichiometry has little effect on\ntheRamanspectra. In Figure3arecomparedthe Raman\nspectraofNiFe 2O4and Ni 0.75Zn0.25Fe2O4. The Zn+2ion\nhas a larger ionic radius than Ni+2and is expected to in-\ncrease the structural disorder of the oxygen sublattice.\nIf the extra lines in the Raman spectra of NiFe 2O4were\ninduced by a disorder, one would expect their relative\nintensity to increase in the Zn-substituted samples. In\ncontrast, upon Zn substitution for Ni these lines broaden\nand decrease in intensity. Therefore, the additional lines\nin the Raman spectrum of NiFe 2O4indicate to a short-\nrange order of Fe and Ni cations rather than a random\ndistribution over the octahedral B-positions.3\n100\r 200\r 300\r 400\r 500\r 600\r 700\r 800\r633 nm\r\n300 K\rNiFe\r2\rO\r4\r\nX'X'\r\nXX\r\nX'Y'\r\nXY\r\n704 -\r- 658\r - 656\r- 593\r469 -\r482 -\r - 487\r449 -\r\n- 571\r- 487\r- 381\r- 273\r\n- 301\r\n- 333\r- 235\r- 140\r\n213 -\r - 213\rSCATTERING INTENSITY [arb. units]\r\nRAMAN SHIFT [cm-1]\r\nFIG. 2: Polarized Raman spectra of NiFe 2O4as obtained at\nroom temperature with 633 nm excitation.\n200\r 300\r 400\r 500\r 600\r 700\r\n200\r 300\r 400\r 500\r 600\r 700\r200\r 300\r 400\r 500\r 600\r 700\r\n100\r 200\r 300\r 400\r 500\r 600\r 700\r 800\r \r\nNi \r0.7\rZn \r0.3\rFe\r2\rO\r4\r (x=0.3)\rNiFe\r2\rO\r4\r (x=0)\rNi\r1-x\rZn\rx\rFe\r2\rO\r4\r633 nm\r\n300 K\r\nX'X'\r \r\nNi \r0.7\rZn \r0.3\rFe\r2\rO\r4\r (x=0.3)\rNiFe\r2\rO\r4\r (x=0)\rXX\rNi \r0.7\rZn \r0.3\rFe\r2\rO\r4\r (x=0.3)\rNiFe\r2\rO\r4\r (x=0)\r\nXY\rX'Y'\rSCATTERING INTENSITY [arb. units]\r \r\nNi \r0.7\rZn \r0.3\rFe\r2\rO\r4\r (x=0.3)\rNiFe\r2\rO\r4\r (x=0)\r\nRAMAN SHIFT [cm-1]\r\nFIG. 3: Polarized Raman spectra of NiFe 2O4and\nNi0.7Zn0.3Fe2O4as obtained at room temperature with\n633 nm excitation.B. Raman Phonons in B-Site Ordered Phases in\nInverse Spinels\n1. Symmetry aspects of the 1:1 ordering at the B-sites of\ninverse spinel structure\nThesymmetryaspectsofthe1:1orderingattheB-sites\nof inverse spinel structure have been discussed in detail\nby Haas.21It has been shown that there are two possible\ntypes of such ordering, α-type and β-type, illustrated in\nFig.4 and shortly described below.\nTheα-type order is characterized by ...-B′′-B′-B′′-B′-\n... chains along the [110] and [1 ¯10] cubic directions. The\nspace group is P4122 (#91) with a tetragonal unit cell\ntwo times smaller than the face-centered spinel unit cell\nwith lattice parameters /vector at=1\n2(/vector ac+/vectorbc),/vectorbt=1\n2(/vector ac−\n/vectorbc), /vector ct=/vector cc. The same type of ordering can be described\nby theP4322 (#95) space group, which is enantiomor-\nphic toP4122. The atomic site symmetries and the clas-\nsification of the normal modes of vibration in the two\nspace groups are equivalent. For this reason our fur-\nther analysis will be done almost exclusively in the con-\ntext ofP4122 group. From symmetry considerations the\nFd¯3m−P4122/P4322 disorder-ordertransition is of first\norder. Therefore, in the phase diagram one expects two-\nphase region (miscibility gap) where the cubic and the\ntetragonal phases coexist. If upon cooling α-type order-\ning does take place, the tetragonal axis may be aligned\nalong each of the three equivalent /vector ac,/vectorbcor/vector cccubic direc-\ntions. This implies that at a microscopic level six types\nof tetragonal domains, three types for P4122 and three\nforP4322, with mutually orthogonal 4-fold axes may co-\nexist at room temperature (Fig.4). The enantiomorphic\nP4122/P4322 pairs of domains with the same orientation\nof the 4-fold axis will be referred by a common number,\nI, II and III for the cubic X,YandZdirections re-\nspectively. In the case of domains of relatively small size\n(≤50 lattice constants), their presence remains below\nthe detection limits of standard diffraction techniques.\nTheβ-type order is characterized by ...-B′-B′-B′-B′-\n... chains along the [110] direction and ...-B′′-B′′-B′′-B′′-\n... chains along the [1 ¯11] cubic directions (Fig.4). The\nspace group is Imma(#74) with an orthorhombic unit\ncell two times smaller than the face-centered spinel unit\ncell (lattice parameters /vector ao=1\n2(/vector ac+/vectorbc),/vectorbo=1\n2(/vector ac−\n/vectorbc), /vector co=/vector cc). Here againthe phase transition from spinel\nto orthorhombic structure is of first-order type and at a\nmicroscopic level three pairs (IV, V and VI) of mutually\northogonaldomains may coexist within the frameworkof\nan averaged spinel structure.\nThe experimentally confirmed averaged cubic struc-\nture of NiFe 2O4will be compatible with tetragonal\nP4122/P4322 and/or orthorhombic Immastructure(s)\nif the twin variants of these structures are uniformly ori-\nented with respect to the cubic axes as shown in Fig.4.\nThis means that the experimental Raman spectrum in a\ngiven cubic scattering configuration will be a superposi-4\n/c97-type /c98-typeab\ncab\nc\nNi2+\nocta Fe3+\ntetra Fe3+\nocta O2-XYX’Y’\nI(IV)\nII(VI) III(V)\nP4 221 Immb\nFIG. 4: (Color online) α-type (P4122/P4322) and β-type\n(Imma) ordering in inverse spinel structure. The orienta-\ntion of the twin variants, I, II, III for the tetragonal struc ture\nand IV, V, VI for the orthorhombic structure, with respect to\nthe cubic directions are also shown.\ntion of spectra obtained simultaneously from tetragonal\nP4122 twin variants in three different scattering configu-\nrations, corresponding to types (I), (II), or (III) orienta-\ntion. The number of twin variants and, hence, the scat-\ntering configurations is doubled to six for the orthorhom-\nbicImmastructure, accounting that aoandboparame-\nters are interchangeable.\nDue to lower symmetry (compared to that of an ideal\nspinel) the number of Raman allowed modes in B-site-\nordered structures increases. Their classification is given\nin Table II. The polarizationselection rulesfor the B-site\norderedP4122 andImmastructures, averaged over all\ntwin variants, are summarized in Tables III and IV.\nThe tetragonal structure gives rise to 84 normal modes\n(accounting for the mode degeneracy), which is twice the\nnumber of normal modes in the cubic structure. This is\ndue to the fact that the primitive cell of P4122 has two\ntimes bigger volume than the primitive cell of the FCC\nstructure. By means of the group-subgrouprelationshalf\nof the normal modes in the tetragonal structure can be\nmapped onto Γ-point modes of the cubic structure. For\nthe Raman-active modes the correspondence is:\nA1g→A1 (1)\nEg/Eu→A1+B2 (2)\nF1g/F1u→A2+E (3)\nF2g/F2u→B1+E (4)\nThe rest of the Γ-point modes of the tetragonal struc-\nture originate from a zone-folding of the FCC Brillouin\nzone, which maps X∗, the star of zone-boundary X-\npoint of Fd¯3m, onto Γ-point of P4122. Among theRaman-activemodesofthe tetragonalstructuresuchare:\n5A1+5B1+8B2+10E. From a physical point of view\nthe new spectral features in P4122 can be considered as\na result of splitting of the degenerated Raman modes of\nFd¯3minto doublets, as well as activation of the IR F1u,\nΓ-point silent F1gandF2u, and zone-boundary modes.\nThe primitive cell volumes of the orthorhombic Imma\nstructure and FCC structure are equal, and no zone-\nfolding takes place. All Raman-active modes of the or-\nthorhombicstructurecanbe mapped ontoΓ-pointmodes\nofFd¯3m:\nA1g→ Ag (5)\nEg→Ag+B1g (6)\nF2g→Ag+B2g+B3g (7)\nF1g→B1g+B2g+B3g (8)\nTherefore, the extra Raman bands in Immashould con-\nsist of a doublet originating from the Egmode, 3 triplets\noriginating from the F2gmodes, and a triplet resulting\nfrom activation of the silent F1gmode of the FCC struc-\nture.\n2. Lattice dynamics calculations of Γ-point Raman phonons\nof inverse spinel NiFe 2O4\nTheoretical results for the lattice dynamics of NiFe 2O4\nwith either disorderly distributed or ordered Ni2+and\nFe3+over the octahedral B-sites were obtained by means\nof a shell model (SM) using the General Utility Lattice\nProgram (GULP).19\nIn order to reduce the number of adjustable model pa-\nrameters some approximations were applied. First, a va-\nlence shell was considered for the O atoms only while\nNi and Fe were treated as rigid ions. Second, the van\nder Waals attractive interaction was considered to act\nonly between O shells, while it was neglected for the\nNi (Fe) core - O shell pairs. These assumptions are\njustified by the much higher polarizability of the O−2\nion compared to that of the transition-metal ions. The\nrigid-ion approximation for transition-metals is a com-\nmon approximation in the shell-model calculations on\ntransition-metal oxides.20. The short-range interatomic\ninteractions were modelled by a Buckingham potential:\nV(r) =Aexp(−r/ρ)−C/r6, where a non-zero van der\nWaals constant Cwas retained for the O shell - O shell\npairs only.\nThe starting model parameters were taken from the\nwidelyutilizedparametersetofLewisandCatlow20, with\na formal charge of +3 assigned to the Fe ions in both\nA- and B-positions. As a next step the Buckingham A\nparameters for the Ni+2core - O shell and Fe+3core -\nO shell pairs were optimized in order to reproduce the\nexperimentally observed lattice parameters for NiFe 2O4,\nthe cubic face-centered lattice constant ac= 8.337˚A\nand the fractional oxygen position u= 0.831. It was as-\nsumed at this stage that even if a cation ordering takes5\nplace in the B-positions, on a macroscopic scale (above\nthe detection limit of the diffraction techniques) the ma-\nterial can be described in a cubic approximation. For\nthis reason the fit was performed by setting equal par-\ntial occupancies of 0.5 for the Fe+3and Ni+2ions in the\nB-position of the ideal cubic Fd¯3mstructure. This is\nequivalent to introduction of an “average” cation in B-\nposition having charge, mass and short-range potential\nparameters, which are arithmetic means between those\ncorresponding to Ni+2and Fe+3. The as obtained shell-\nmodel parameters are summarized in Table V. Finally,\nthe set of fitted parameters was used to calculate the Γ-\npoint normal modes for the average-atom cubic Fd¯3m\nstructure, which are assumed to mimic the positions of\nthe main Raman bands in the case of complete cation\ndisorder at the B-sites. The same parameters were uti-\nlized to optimize the lattice parameters of the ordered α-\nandβ-type structures and to calculate the corresponding\nΓ-point modes. The calculated normal mode frequencies\nfor the three structural models are listed in Table VI.\n3. Raman spectroscopy evidence for 1:1 ordering at the\nB-sites of NiFe 2O4\nUnlike standard X-ray and neutron diffraction tech-\nniques, which are most sensitive to the long range order,\nthe Raman scattering is more sensitive to the local short\nrange order, which may differ from the averaged long-\nrange one. As discussed above, the 1:1 ordering at the\nB-sites gives rise to structures of tetragonal P4122(#91)\nor orthorhombic Immb(#74) symmetry.\nAs it follows from the polarization selection rules,\nthe fully symmetrical modes A1g(Fd¯3m),A1(P4122) or\nAg(Imma) canbe identified bytheirstrongerintensityin\ntheXXandX′X′spectra compared to that in the XY\nandX′Y′spectra. Such are the Raman peaks at 140,\n235, 381, 449, 487, 571, 593, and 704 cm−1. Their num-\nber exceeds the expected single A1gmode for Fd¯3mor\nfiveAgmodes for Immastructure. Therefore, it is plau-\nsible to accept that at least part of these peaks originate\nfromα-type ordering at the octahedral sites, correspond-\ning to local P4122 structure. Indeed, these experimental\nfrequencies show closest match with the following calcu-\nlated frequencies ofthe A1modes in the P4122structure:\n168, 253, 395, 498, 573, and 694 cm−1. In the same time,\nthe calculations show a large frequency gap between 387\nand 605 cm−1in theAgchannel of the Immastructure,\nwhich makes the β-type ordering unlikely from a spec-\ntroscopic point of view.\nIt is instructive to comment on the experimentally ob-\nserved splitting of the intense band around 580 cm−1\ninto two components at 571 and 593 cm−1. Accord-\ning to our calculations there are two closely separated\nmodes in the P4122 structure at 573 and 574 cm−1of\nA1andB2symmetry respectively. The selection rules\nfor the tetragonal structure (see Table III) predict the\nB2modes to appear in the same scattering configurationFIG. 5: (Color online) A z= 3/8 cross-section of the Fd¯3m\nunit cell. Two FCC zone-boundary normal modes, which\nsplit into A1−B2pairs upon α-type 1:1 arrangement at B-\npositions: 571 - 593 cm−1(a) and 449 - 487 cm−1(b).\nasA1modes. Thus the experimentally observed compo-\nnents can be assigned to an A1−B2pair. The inspection\nof the atomic displacement pattern in these modes shows\nthat they can be mapped to a doubly degenerated X-\npoint zone-boundary mode of the cubic structure, which\nbecomes a Raman-active Γ-point mode in the P4122\nstructure due to zone folding (see Fig. 5 (a)). Similarly,\nthe 449 and 487 cm−1bands can be ascribed to another\nA1−B2pairoriginatingfromthe FCCzone-boundary X-\npoint(see Fig. 5 (b)). In a cubic structure the B′and B′′\nsites are equivalent and the double degeneracy of each of\nthese modes results from the fact that depending on the\nchoice of the BO 2chains the oxygenatoms can vibrate in\ntwo independent directions, e.g. [110] cand [011] c, while\nthe vibration in the third symmetry-equivalent direction\n[101]cis a linear superposition of the other two vibra-\ntions.\nAdditionalpiecesofevidenceforthepresenceoftetrag-\nonalα-type ordering could be drown from an analysis\nof the other scattering configurations. The mode at\n213 cm−1is active in X′X′andXYconfigurations and\ncould be assigned to F2g,B1+E, orAg+B2g+B3gvi-\nbrationsin the cubic, tetragonaland orthorhombicstruc-\ntures respectively. Again, the best correspondence is\nfound with the E-mode at 208 cm−1for theP4122 struc-\nture. However, taking into account the unavoidable un-\ncertainty of calculations, the B2gmode at 229 cm−1of\ntheImmastructure is also a like candidate for that spec-\ntral feature.\nThe cubic Egchannel includes the XX,X′X′, and\nX′Y′scattering configurations with a lowest expected\nintensity in the X′X′configuration. Similar selection\nrules are expected for the A1+B2modes in the P4122\nstructure and the Ag+B1gmodes in the the Imma\nstructure. The experimentally detected modes at 273,\n301, and 333 cm−1follow closely these selection rules.\nThe multiplicity of the observed frequencies suggests the\npresence of ordered structures since one single Egmode\nis expected for the cubic structure. Most likely, the ex-\nperimentally observed spectral bands correspond to the\nA1andB2modes of the P4122 structure, whose calcu-6\nlated frequencies fall in the range 295 - 353 cm−1(see\nTable VI). Since the frequency splitting between some\nof the modes (305-306 and 350-353 cm−1) is compara-\nble to the spectrometer resolution, experimentally they\nmay appear as single spectral features leading to only\nthree observable frequencies. It is worth mentioning that\naccording to our calculations no AgorB1gmodes are ex-\npected in this frequency range for the Immastructure.\nFinally, the band at 656-658 cm−1is active in all scat-\ntering configurations. According to Table IV such a be-\nhavior is expected for the Agmodes of the orthorhom-\nbicImmastructure. One plausible explanation of this\nfeature is the calculated Agmode at 659 cm−1. Alterna-\ntively, a doublet of closely separated modes of B2sym-\nmetry at 660.7 cm−1and ofEsymmetry at 665.4 cm−1\nis predicted for the tetragonal P4122 structure. Due to\ntheir similar frequencies the two modes may be indistin-\nguishable experimentally, and should appear in all scat-\ntering configuration ( Table III).\nIV. SUMMARY AND CONCLUSIONS\nWepresentdetailedpolarizationRamanmeasurements\nof the inverse spinel NiFe 2O4. The number and the po-\nlarization selection rules for the observed Raman bands\ndo not support the model of stochastic distribution of\nNi2+and Fe3+cations among the octahedral B-sites.\nByusingsymmetryanalisysandshell-modellatticedy-namics calculations we examined the experimental data\nagainst two models of B-site 1:1 ordering, α-type of\ntetragonal P4122/P4322 symmetry and β-type of or-\nthorhombic Immasymmetry. All experimental Raman\nbandsareconsistentbysymmetryandfrequencywiththe\ncalculated normal modes of the α-type structure, The\nβ-type ordering can explain only part of the observed\nbands.\nOn the basis of the above arguments we can con-\nclude that Ni2+and Fe3+exibit 1:1 ordering at the oc-\ntahedral sites of NiFe 2O4, most probably of tetragonal\nP4122/P4322 symmetry. However, the Immastructure\ncan not be definitively ruled out due to the good corre-\nspondence of some calculated frequencies to experimen-\ntally observed bands. It is possible that the two types of\nordering coexist with prevalence of the tetragonal struc-\ntures. Macroscopically the material exhibits cubic sym-\nmetryduetothepresenceofrandomlyorientedtwinvari-\nants of the ordered structures.\nAcknowledgments\nThis work is supported in part by the State of Texas\nthrough The Texas Center for Superconductivity at the\nUniversity of Houston and partly by the contracts # DO\n02-167/2008 and TK-X-1712/2007 of the Bulgarian Na-\ntional Scientific Research Fund.\n1V. A. M. Brabers, Handbook of Magnetic Materials (Else-\nvier Science B.V., Amsterdam, 1995).\n2H. D. Han, H. L. Luo, and Z. Yang, J. Magn. Magn. Mater.\n161, 376 (1996).\n3C. H. Cunningham, T. Arai, P. C. Yang, M. V. McConnell,\nJ. M. Pauly, and S. M. Connolly, Magn. Reson. Med. 53,\n999 (2005).\n4P. Seneor, A. Fert, J.-L. 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Matter 19186217 (2007).\n16W. H. Wang and X. Ren, J. Crystal Growth 289, 605\n(2006).\n17M. N. Iliev, unpublished results.\n18M. N. Iliev, M. V. Abrashev, V. N. Popov, and V. G. Had-\njiev, Phys. Rev. B 67, 212301 (2003).\n19G. D. Gale, J. Chem Soc. Faraday Trans. 93, 629 (1997).\n20G. V. Lewis and C. R. A. Catlow, J. Phys. C: Solid State\nPhys.18, 1149 (1985).\n21C. Haas, J. Phys. Chem. Solids 26, 1225 (1965).7\nTABLE I: Raman polarization selection rules for the spinel\nFd¯3mstructure.\nModeXXX′X′XYX′Y′\nA1ga2a200\nEg4b2b203b2\nF2g0d2d208\nTABLE II: Atomic site symmetries and corresponding Γ-point modes for the Fd¯3m,P4122, andImmastructures of NiFe 2O4.\nFor the ordered P4122 andImmastructures only Raman modes are shown.\nFd¯3m(cubic) P4122 (tetragonal) Imma(orthorhombic)\nWickoff Γ-point Wickoff Raman Wickoff Raman\nAtom index modes Atom index modes Atom index modes\nFe(1) 8a F2g+F1u Fe(1) 4c A1+2B1+B2+3EFe(1) 4e Ag+B2g+B3g\nNi/Fe(2) 16d A2u+Eu+F2u+2F1uNi 4a A1+B1+2B2+3ENi 4c —\nFe(2) 4b A1+B1+2B2+3EFe(2) 4b —\nO 32e A1g+A2u+Eg+Eu+O(1) 8d 3 A1+3B1+3B2+6EO(1) 8h 2 Ag+B1g+B2g+2B3g\nF2u+2F2g+2F1u+F1gO(2) 8d 3 A1+3B1+3B2+6EO(2) 8i 2 Ag+B1g+2B2g+B3g\nTOTAL Raman: A1g+Eg+3F2gTOTAL: 9 A1+10B1+11B2+21ETOTAL: 5 Ag+2B1g+4B2g+4B3g\n1 acoustic + 4 IR: 5 F1u\ninactive: 2 A2u+2Eu+2F2u+F1g\nTABLE III: Polarization selection rules for the scattering fromP4122 structures averaged over the three twin variants with\nI-,II,- and III-type orientation with respect to the cubic a xes.\nModeRaman tensor XX(cubic)XY(cubic) X′X′(cubic) X′Y′(cubic)\nA1\na\na\nb\n2\n3a2+1\n3b201\n3a2+1\n6(a+b)21\n6(a−b)2\nB1\nc\n−c\n 01\n3c2 1\n3c20\nB2\nd\nd\n2\n3d201\n6d2 1\n2d2\n\ne\ne\n\nE 02\n3e2 2\n3e20\n−e\n−e\n9\nTABLE IV: Polarization selection rules for the scattering f romImmastructures averaged over the six twin variants with\nI(IV)-,II(V),- and III(VI)-type orientation with respect to the cubic axes.\nMode Tensor XX(cubic) XY(cubic) X′X′(cubic) X′Y′(cubic)\nA1\na\nb\nc\n1\n6(a+b)2+1\n3c21\n12(a−b)21\n6a2+1\n6b2+1\n24(a+b+2c)21\n24(a+b−2c)2\nB1g\nd\nd\n2\n3d201\n6d2 1\n2d2\nB2g\ne\ne\n1\n3e2 1\n3e2\nB3g\nf\nf\n 01\n3f2 1\n3f20\nTABLE V: Shell-model parameters for NiFe 2O4.\nCore-shell\nAtom Core charge Shell charge spring constant\nX Y k(eV/˚A2)\nNi +2 — —\nFe +3 — —\nO 0.513 -2.513 72.53\nAtomic pair A(eV) ρ(˚A)C(eV×˚A6)\nNi core - O shell 681.9 0.337 0\nFe core - O shell 986.1 0.337 0\nO shell - O shell 22764.0 0.149 27.87910\nTABLE VI: Calculated frequencies of the Raman-active\nmodes in NiFe 2O4for the three structural models: B-site\ndisorder of a macroscopic Fd¯3msymmetry, α-type ordering\n(symmetry P4122) and β-type ordering (symmetry Imma)\n.\nFd¯3m P4122 Imma\nAgEgF2gA1B1B2EAgB1gB2gB3g\n168148155147\n171205207\n235235227 229\n255253248 246 248\n253 252\n263\n300 295 305293\n329306314\n350 353328 336\n395381374367387\n415 407 405\n438 438\n465 465 464\n469\n498498496490\n516514\n573 574556 588\n618 612 615605 613\n635649661665659 643\n687 694 718 689\n744"    },    {        "title": "1311.6217v2.Structural_and_Dielectric_Characterization_on_Multiferroic_xNi0_9Zn0_1Fe2O4__1_x_PbZr0_52Ti0_48O3_Particulate_Composite.pdf",        "content": "1 \n Structural and Dielectric Characterization o n Multiferroic \nxNi 0.9Zn 0.1Fe2O4/(1-x)PbZr 0.52Ti0.48O3 Particulate Composite  \n \nRishikesh Pandey, Braj Raj Meena and Akhilesh Kumar Singh *  \nSchool of Materials Science & Technology, Indian Institute of Technology  (Banaras Hindu \nUniversity ) Varanasi - 221005 , India  \n \nABSTRACT  \nWe have carried out the powder x-ray diffraction and dielectric studies on  multiferroic \nparticulate composite xNi0.9Zn0.1Fe2O4/(1-x)PbZr 0.52Ti0.48O3 with x= 0.15, 0.30, 0.45, 0.60, \n0.75 and 0.90 to explore the structural and ferroelectric properties.  A conventional double \nsintering method was used to prepare the xNi0.9Zn0.1Fe2O4/(1-x)PbZr 0.52Ti0.48O3 composite s. \nThe structure of one of the component Ni 0.9Zn0.1Fe2O4 is spinel cubic with space  \ngroup      , while the other component PbZr 0.52Ti0.48O3 is selected around the morphotropic \nphase boundary region in which the tetragonal and monoclinic phases with space group \nP4mm and C m coexist respectively.  We have carried out Rietveld refinement of the s tructure \nto check  the formation of ideal composite s with separate  ferroelectric and ferrite phases. Even \nthough the structural characterization does not reveal the formation of any new phase due to \nreaction between the two components of the composite during sintering , the tetragonality of \nthe PbZr 0.52Ti0.48O3 continuously decreases with increasing the ferrite fraction while the \nlattice parameter of ferrite phase increases with increasing fraction of the ferroelectric phase. \nSimilarly , the dielec tric study reveals clear  shift in the  ferroelectric to paraelectric phase \ntransition  temperature of PbZr 0.52Ti0.48O3 during composite formation suggesting that part of \nNi2+, Zn2+/ Fe3+ ions are diffusing at the B-site of PbZr 0.52Ti0.48O3 replacing Ti4+, which in \nturn decrease s its transition temperature.  Scanning electron micrograph of sintered pellet  \nsurface confirms  the presence of two  types of particle  morphology  in the particulate \ncomposite , corresponding to ferrite and ferroelectric phases.   2 \n  \n* Corresponding author  \nE-mail address: akhilesh_bhu@yahoo.com, aksingh.mst@itbhu.ac.in  \n1. INTRODUCTION:   \nIn recent  years , multifunctional materials such as multiferroics have stimulated great  \ninterest of researchers due to the potential applications for devices such as sensors, actuators , \ntransducers  and memor ies [1-2]. In multiferroics , more  than one ferroic properties  like \nferroelectric ity (FE), ferromagneti sm (FM)  etc. are present  simultaneously  and are coupl ed. \nFor example, presence of coupled ferroelectric and ferromagnetic responses provides an extra \ndegree of freedom in which data could be written electrically and read magnetically and vice \nversa  for memory applications  [3]. Usually presence of f erroelectricit y in perovskites require s \nthe empty d -orbital s at B-site cation  which soften  the orbital  overlapping between B -site \ncation s and O anions in the ABO 3 perovskite  structure  [4]. However , ferromagnetism requires  \nunpaired electrons in the d -orbit als of the ions at B -site. That is the reason  why these \nmaterials are rarely found in nature  and difficult to synthesize in lab oratory  [5]. Single phase \nmultiferroics  such as BiFeO 3, BiMnO 3, etc. exhibit  very low  magnetoelectric  (ME) \ncoefficient and low resistivity  limiting their applications  [6-7]. Most of them exhibit \nmagnetoelectric coupling  much below room temperature and therefore  are not suitable for  \npotential device application s [3, 8-9]. In view of this, the c omposite multiferroics in which \ntwo phases are coupled via strain may be a good substitute for single phase multiferroics  [2]. \nThe composite multiferroics may  exhibit  high ME- coefficient which have not been observed \nin the single phase ME materials.  The appearance of magnetoelectric coupling in composite s \nis relatively new concept introduced in 1972 by Suc htelen  [10]. Subsequently large number \nof papers appeared , reporting significant ME -coefficient  (~ 100 V/cm -Oe) in ferroelectric/  \nferromagnetic composites  [11]. In the recent years, large numbers of particulate composites  3 \n and piezoelectric -magnetostrictive heterostrucures  have  been explored such as  \nNi0.9Co0.1Fe2O4/Pb(ZrxTi1-x)O3 [12], BiFe 0.5Cr0.5O3/NiFe 2O4 [13], Ni0.93Co0.02Mn 0.05Fe1.95O4/ \nPb(Zr xTi1-x)O3 [14], Pb(Zr 0.52Ti0.48)O3/NiFe 2O4 [15], BaTiO 3/(Ni 0.3Zn0.7)Fe 2.1O4 [16], \nPb(Zr 0.52Ti0.48)O3/NiFe 1.9Mn 0.1O4 [17], BaTiO 3/NiFe 1.98O4 [18], CoFe 2O4/PbZr 0.52Ti0.48O3 \n[19] etc. Significant  improvement in ME -coefficient (~ 25 mV/cm -Oe) is reported in the \ncomposites prepared by NiFe 2O4 (NFO) and Pb(Zr xTi1-x)O3 (PZT) using spark plasma \nsintering method  [20]. Particularly the  composite heterostructure  of Terfenol -D with PVDF  \nprepared in the form of multilayer thin film which exhibit very high ME -coefficient  (> 1 \nV/cm-Oe) termed a s giant magnetoelectric coefficient  (GME)  [21]. There are several papers \nin the literature which report the magnetoelectric investigations on particulate composites \nusing various types of ferroelectric phases such as BaTiO 3, PZT etc. and magnetic phases \nsuch as nickel f errite, cobalt f errite, nickel zinc ferrite etc . assuming that the two components \nform ideal composite without reacting/  modifying each other. However, at high sintering \ntemperature during composite formation it is very unlikely that the magnetic and ferroelectric \nphases remain intact without diffusion of ions from one component to other. To the best of \nour knowledge the pos sible modifications in the ferroelectric and magnetic components \nduring particulate composite formation has not been investigated into detail which may have \nvery crucial impact on the ME response of the composite. In the present work we have \nlooked into th is aspect by investigating several compositions of xNi 0.9Zn0.1Fe2O4/(1-\nx)PbZr 0.52Ti0.48O3 (NZFO/PZT ) composite.  \nMultiferroic composite of NZFO and PZT has been prepared choosing the suitable \ncomposition of perovskite PZT in the MPB region and NZFO using conventional double \nsintering route. PZT is a well -known piezoelectri c material and the composition \nPb(Zr0.52Ti0.48)O3 chosen is in the MPB region in which both tetragonal (P4 mm) and \nmonoclinic (C m) phases coexists showing maximum piezoelectric response [2 2]. The 4 \n transition temperature for PZT with x=0.52 is ~  380 0C [23]. Nickel zinc ferrite \n(Ni 0.9Zn0.1Fe2O4) shows ferrimagnetic to paramagnetic phase transition at T N ~ 530 0C [24]. \nOur study reveals that both the ferrite and ferroelectric phase s get modified during composite \nformation at high sintering temperature which in turn may deteriorate the ME response of the \ncomposite. We observed a systematic variation in the lattice parameter of the ferrite and \nferroelectric phases with changing their phase fraction  in the composite  which suggest that \nthe ions from the two components are diffusing  into each other thereby modifying the lattice \nparameter s. The transition temperature (T C) of PZT also shows a systematic shift  to lower \ntemperatures  for variou s compositions of the composite and the ferroelectric to paraelectric \ntransition becomes more diffuse d with increasing the ferrite fraction in the composite. To \nprevent the reaction between two components and to  improve the ME response of such \nparticulate composites, the sintering temperature needs to be  brought down by using \nnanocrystalline powders  [25] or by reducing the sintering time by adopting the methods like \nhot pressing  [26] or spark plasma sintering  [20].  \n2. EXPERIMENTAL  DETAILS :  \nSamples used in the present work were prepared by  conventional  double sintering \nsolid state route. To prepare Ni 0.9Zn0.1Fe2O4, stoichiometric amounts of AR grade  NiO \n(obtained by thermal decomposition of  NiCO 3.2Ni(OH) 2 (QUALIGENS ) at 550 0C), ZnO  \n(QUALIGEN S, 99% ) and Fe 2O3 (HIMEDIA , 99% ) were mixed in agate mortar  and then ball \nmilled  for 4 h. The powder mixture was calcined at 800 0C for 6 h. Similarly, PbZr 0.52Ti0.48O3 \n(PZT) was prepared by conventional solid state route using AR grade PbCO 3 (HIMEDIA, \n99.9%), ZrO 2 (HIMEDIA, 99%) and TiO 2 (HIMEDIA, 99%). Stoichiometric amounts of \nthese ingredients were ball milled in acetone  (as mixing media)  for 6 h . The mixed powder \nwas dried and then calcined at 800 0C for 6 h . Powder x-ray diffraction  (XRD)  patterns were \nrecorded using an 18 kW rotating  Cu-target  based RIGAKU  (Japan) powder x-ray \ndiffractometer fitted with a graphite monochromator  in the diffracted beam.  To prepare \ndifferent compositions of NZFO /PZT composites , the mixture of calcined powders of PZT \nand NZFO were first ball -milled in acetone for 6 h and then dried. 2% polyvinyl alcohol 5 \n (PVA) solution in water , which acts as binder was mixed with the  powders . The powder \nmixture was pressed in the form of pellets  of diameter 12 mm and thickness  ~ (1.5- 2.0) mm \nusing a  stainless - steel die  and uniaxial  hydraulic press  at an optim ized load of 65 kN.  Before \nsintering, the green pellets were kept at 500  °C for 10 h to burn out the binder. The pellets \nwere finally sintered at 1150 0C for 6 h  in PbO  atmosphere  in sealed Alumina crucible . For \ndielectric measurements, the flat surface of sintered pellets was gently polished with 0.25 μm \ndiamond paste for about 2 min and then washed with acetone. Isopropyl alcohol was then \napplied for removing the moisture, if any, on the pellet surfaces. Fir ed on  silver paste was \nsubsequently applied on both the surface s of the pellet s. It was first  dried around 120 0C in an \noven and then cured by firing at 500 0C for about 5 min.  Dielectric measurements were \ncarried out using a Novocontrol, Alpha -A high perf ormance frequency analyzer . For the high \ntemperature dielectric measurements, the temperature of the sample was controlled by using a \nEurotherm programmable temperature controller with an accuracy of ±1 0C. The \nmeasurements were carried out during heating the sample at a rate of 1 0C per min.  \nFULLPROF program (Rodriguez - Carvajal ) [27] was used for Rietveld refi nement of the  \ncrystal  structure. Pseudo -Voight  function was used to define the peak profiles and sixth -order \npolynomial was used to fit the background.  In the spinel cubic phase of space group F d  m \n(Space group # 227), Zn2+ and Fe I3+ ions occupy the tetrahedral 8(a) sites at (1/8, 1/8, 1/8), \nFeII3+ and Ni2+ ions occupy  octahedral sites 16(d) at (1/2, 1/2, 1/2) and O2- ions occupy 32(e) \nsites at ( -x+1/4, -x+1/4, x) as listed in  the “International Table for Crystallography, Vol. A \n(2005)” . In the tetragonal phase with P 4mm space group, the Pb2+ ion occupies 1(a) sites at \n(0,0, z), Ti4+/Zr4+ and O I2- occupy 1(b) sites at (1/2, 1/2, z), and O II2- occupy 2(c) sites at (1/2, \n0, z). In the monoclinic phase with space group C m, Pb2+, Ti4+/ Zr4+ and O I2- occupy  2(a) sites \nat (x, 0, z) and O II2- occupy  4(b) sites at (x, y, z).  The microstructur e of the sintered pellets  \nwas studied using  scanning electron microscope  (SEM) (ZEISS SUPRA 40).  Before the \nmicrostructural study , sintered pellets were sputter coated with  Pd/Au alloy .  \n3. RESULTS & DISCUSSION:  \n3.1 Crystal Structure:   \nThe XRD patterns  of calcined powder s of NZFO  and PZT  are shown in Fig.1 (a) and \n(b), respectively . All the peaks  shown in Fig.1 (a) are indexed with the  cubic spinel structure  \nwith space group F d  m and no impurity phase  is present . Similarly, for PZT all the peaks 6 \n shown in Fig.1 (b) correspond to perovskite structure. The indices shown on the peaks in \nFig.1(b) correspond to pseudocubic cell. For this composition of PZT, both the tetragonal \n(P4mm) and monoclinic (C m) phases are reporte d to coexist [ 22] which is evident from the \nbroad triplet character of the (200) pseudocubic profile. The Rietveld refinement of the \nstructure by us confirms the coexistence of the monoclinic and tetragonal structures. Fig.2 \nshows the XRD profile for different compositions of NZFO /PZT composites  prepared by \nsintering the mixture of the calcined powders of NZFO and PZT in different proportions . \nPeaks marked with “p” and “n” denote  the reflections corresponding to PZT and NZFO  \nphases , respectively.  We can se e that as we increase the fr action  of NZFO, the intensity of \npeaks corresponding to PZT decreases while the peak intensity corresponding to NZFO \nincreases.  For all the patterns shown in Fig.2, except the peaks corresponding to the PZT and \nNZFO, there  is no additional peak. This suggests that during composite formation any new \nphase due to possible reactions between the two components is not present.  Phase fractions \nof the PZT and NZFO were calculated using the intensity of the m ost intense peak of the \nXRD pattern of  the two phases  using equation  (i) and (ii) , \n                                                 \n                                                                (i)              \n                                                              \n                                                                (ii)                      \nAs shown in Fig.3(a) and Fig.3 (b), we obtained  good agreement between this calculated \nphase fractions (%) with weight fraction of the PZT and NZFO used t o prepare various \ncompositions . This again suggests that the particulate composite formation during sintering of \nthe mixture is not leading to any new phase.   \nWe could successfully refine the structure of the various compositions of the \ncomposites using Rietveld me thod. The Rietveld fit for the XRD pattern of  NZFO/ PZT 7 \n composite with x=0.60 is shown in Fig.4.  Using  the cubic space group F d  m for NZFO , and \ncoexisting  tetragonal (P4mm) and monoclinic (Cm) space group s for PZT , very good  fit \nbetween  observed  and calculated profiles  is obtained . The refined structural parameters  for \npure NZFO, PZT and  NZFO/PZT composit e with x=0.60 are listed in table I.  The isotropic \nthermal parameters (Biso) for the Pb2+ ion was found to be  very high (>3  Å2). In view of this , \nwe considered anisotropic thermal parameters (β) for these  ions in tetragonal  and monoclinic  \nstructure s. For the cubic phase , Biso were used  for all the ions.  The refined structural \nparameters for pure PZT are in good agreement with that reported by Ragini et al [22].  \nVariation of lattice parameter of NZFO and tetragonal phase of PZT obtained after \nRietveld refinement of the structures for various compositions of composite is shown in \nFig.5(a) and (b) respectively. As can be seen from Fig.5(a) the lat tice parameter of NZFO \nsystematically increases with increasing the fraction of PZT in the composite. Similarly the \n‘a’ and ‘c’ parameters of the tetragonal phase of PZT come closer to each other with \nincreasing the fraction of the NZFO in the composite. T he tetragonality  (c/a) continuously \ndecreases with increasing the fraction of NZFO and structure becomes pseudocubic for \nx=0.85.  These modifications in the lattice parameters of NZFO and PZT are not possible \nunless the ions from the two components diffuse to each other. Thus the solid state sintering \nroute of the composite formation is not leading to the ideal composite where the t wo \ncomponents should not react with each other  [28]. Most important impact of the modification \nin the lattice parameter will correspond to the ferroelectric phase where reduced tetragonality \nwill decrease the magnitude of the polarization of the unit cell which in turn will reduce  the \npiezoelectric response of the ferroelectric phase. Since the ME effect in particulate composite \nresults from the coupling between polarization and magnetization mediated by strain, the \nreduced piezoelectric response will decre ase the value of ME coefficient also.  \n 8 \n 3.2 Microstructure studies : \nThe SEM image for NZFO/ PZT composite  with x=0.60 is shown in Fig. 6. \nAppearance of two types of morphology could be seen clearly in the SEM image confirm ing \nthe formation of composite. The smaller grains  with elongated oval shape are identified  as \nNZFO . The rock shaped bigger grains  correspond to PZT. The average grain size for NZFO \nand PZT is calculated to be ~ 0.09 μm and 1.0 μm , respectively. As can be seen from the \nmicrograph, there is u niform distribution of the two components .  \n3.3 Dielectric Studies : \nThe room temperature permittivity  (ε/) of the NZFO/PZT composite in the frequency \nrange of (1  kHz- 106 kHz) for different composite compositions  is shown in Fig. 7. It is \nevident from Fig. 7 that with  increasing the amount  of NZFO, the permittivity decreases.  This \nis due to decreas ing the fraction of  the ferroelectric  phase  PZT.  Another reas on for decrease \nin the dielectric constant with increasing ferrite content may be attributed to the  Verwey type  \nelectron exchange polarization [29].  \nFig.8 shows the temperature variation of permittivity f or NZFO/ PZT composite with \nx=0.60  at various frequencies . With increasing frequency, the peak in the temperature \nvariation of the permittivity shifts to the higher temperature side along with the significant \nfrequency dispersion.  This suggests the relaxor  nature of the ferroelectric phase transition in \nPZT after composite formation. However, the pure PZT behaves like a normal ferroelectric \nand does not exhibit relaxor features  [30]. The appearance of relaxor features must be li nked \nwith the modification of the PZT phase by diffusion of ions from NZFO during composite \nformation.   It is well known that the relaxor nature of ferroelectric phase transition i s \nassociated with the local ionic size/charge disorder in perovskite solid solutions [ 31]. Relaxor \nnature of ferroelectric phase transition in the NZFO/PZT composite must be resulting from 9 \n the introduction of more disorder at the B -site cations in the ABO 3 perovskite structure due to \ndiffusion of Ni2+/Zn2+/Fe3+ ions from the NZFO to the Zr4+/Ti4+ site.  This may lead to the  \nMaxwell -Wagner type  interfacial polarization [32] as discussed in Koop’s phenomenological \ntheory  [33].   \nFig.9 shows the temperature variation of permittivity at a frequency of 10  kHz for the \nNZFO/PZT composite with x=0, 0.15, 0.30, 0.45  and 0.60. For pure PZT with x=0 the \npermittivity shows a peak at 380 0C which is in well agreement with the value of T C reported \nin literature [ 23] for Pb(Zr 0.52Ti0.48)O3. With increasing the value of ‘x’ i.e. the NZFO fraction \nthe permittivity peak systematically shifts to the lower temperature side and the nature of \nphase transition becomes very diffused. This suggests t hat the relaxor feature of the phase \ntransition become more pronounced with increasing NZFO concentration in the composite. \nAs discussed earlier, the appearance of relaxor features is linked with the diffusion of \nNi2+/Zn2+/Fe3+ ions from NZFO to the Zr4+/Ti4+ site of the PZT . It is expected that at higher \n‘x’ more Ni2+/Zn2+/Fe3+ ions are available to diffuse to the Zr4+/Ti4+ site of the PZT which in \nturn strengthen the relaxor nature of the ferroelectric phase transition.  To elaborate the \nstrengthening of  the relaxor character with increasing NZFO concentration in the composite, \nwe have plotted the full width at half maximum (FWHM) of the permittivity peaks at 10 kHz \nfor various compositions  in Fig.10 . It is evident from Fig.10 that the FWHM increases \ncontinuously with increasing NZFO concentration , indicating enhanced diffuseness of the \nferroelectric phase transition. Fig.11 shows the composition dependence of T C determined at  \n1 kHz frequency for NZFO/PZT composite. As compared to pure PZT  with T C=380  0C, the \nTC decreases drastically for x=0.15 (~ 350 0C) and then decreases linearly with increasing \nNZFO concentration. The significant change in T C of PZT after composite formation clearly \nsuggests that the ferrite and ferroelectric phases react with each other. This is in contrast to \nthe proposition of the Chougule  et al. [34] that the two phases do not react.  10 \n Chougule et al. [34] have reported that the transition temperature for the composition \nwith x=0 (for pure PbZr0.52Ti0.48O3) is 420 0C, which is incorrect , and not consistent with the \nPZT phase diagram [ 23]. Our study on the composition with x=0 shows that the T C ~ 380 0C \nwhich is consistent  with the  reported phase diagram of PZT for pure PZT composition in the \nMPB region [23,35]. Similarly , for the other compositions with x=0.15, 0.30 and 0.45, the \nChougule et al . have reported that TC is 380  0C, 375  0C and 370 0C. In contrast , our studies \nclearly reveal that the T C for x=0  is 380  0C, decreases drastically to ~  350 0C for x= 0.15, and \nthen shows li near decrease for x=0.30, 0.45 and 0.60 as 348  0C, 345  0C and 342 0C, \nrespectively . Also, the reported value of ε/ (= 6160)  by Chougule et al. [34]  for pure NZFO is  \nextremely high  and comparable to ferroelectric (PZT) rich  composit ion with x=0.15 (ε/= \n6446) . Such high value of ε/ for NZFO is not expected . The reported value of ε/ in literatures \nis as high as ~ 850 (at 1 kHz)  at the sintering temperature of 800 0C which falls down  to ~ \n250 with increasing the sintering temperature above 1000 0C  [36-37]. The value of ε/ (~ 200) \nmeasured at room temperature  and 1 kHz by us ( see Fig.7) , for pure NZFO  sintered at 1150 \n0C is nearly comparable to that reported by Chen et al  [36].  \n4. CONCLUSIONS:  \nThe dielectric and structural characterization of NZFO /PZT composites by us reveals \nthat at the XRD level no secondary phases appear after composites formation. However, the \nlattice parameters of both the ferrite and ferroelectric phases are modified suggesting that the \nions from the two phases diffuse to each oth er during sintering. The tetragonality of \nferroelectric phase continuously decreases with increasing the ferrite fraction in the \ncomposite . To get the maximum piezoelectric response of the ferroelectric phase and \nconsequently better ME -coefficient in compo site, a composition with higher tetragonality \nmay be chosen, the structure of which latter may get modified to MPB phase.  The transition \ntemperature of the ferroelectric phase (PZT) also decreases significantly after composites 11 \n formation and the nature of the phase transition becomes relaxor like. The SEM studies \nreveal the two types of particle morphology in the particulate composite, corresponding to \nferrite and ferroelectric phases.  \nACKNOWLEDGEMENT S: \n RP acknowledges University  Grant Commission (UGC), India for financial  support.  \nAuthors are thankful to Professor Dhananjai Pandey, School of  Materials Science and \nTechnology, IIT (BHU) Varanasi, India  for extending laboratory facilities  and support . \nREFERENCES:  \n1. R. Ramesh  and N. A. Spaldin, Nature Mater. 6 (2007) 21.  \n2. W. Eerenstein, N. D. Mathur and J. F. Scott, Nature 442 (2006) 759.  \n3. N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha and S -W. Cheong, Nature 429  \n(2004)  392. \n4. R. E. Cohen, Nature 358 (1992)  136. \n5. N. A. Hill, J . Phys. Chem. B, 104 (2000) 6694.  \n6. N. Wang, J. Cheng, A. Pyatakov, A. K. Zvezdin, J. F. Li, L. E. Cross and D. Viehland, \nPhys. Rev. B 72 (2005) 104434.  \n7. V. G. Prokhorov, G. G. Kaminsky, J. M. Kim, Y. J. Yoo, Y. P. Lee, V. L. \nSvetchnikov, G. 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G. Chen, X. G. Tang, Q . X. Liu, Y. P. Jiang, C. B. Ma  and R. Li , J. Appl. Phys. 113 \n(2013) 214110.  \n37. A. Verma , O. P. Thakur, C. Prakash, T. C. Goel and R. G. Mendiratt, Mater. Sci. Eng. \nB 116 (2005) 1.  \n \n \n \n \n 14 \n FIGURE CAPTIONS:  \nFig.1Powder XRD p attern  of NZFO and PZT ceramics calcined at 800 0C for 6 h.  \nFig.2Powder XRD profile of xNZFO /(1-x)PZT particulate  composites  for the composition  \nwith x= 0, 0.15, 0.30, 0.45, 0.60, 0.75, 0.90 and 1.0  sintered at 11 50 0C for 6 h . \nFig.3Variation of  phase fractions  (%) for PZT (P PZT) and NZFO  (PNZFO) in xNZFO/(1 -x) PZT  \nparticulate composite .   \nFig.4Observed (dots), calculated (continuous line), and difference (bottom line) profiles \nobtained after the Rietveld refinement of the structure of 0.60NZFO/ 0.40PZT \nparticulate composite sintered at  1150  0C using coexisting [Cubic (F d  m)+ \nTetragonal (P4 mm)+ Monoclinic (C m)] phases . The vertical tick marks above the \ndifference plot show the positions of the Bragg peaks.  \nFig.5 Variation of  the (a) lattice parameter of NZFO in the term of  doped PZT (%) in NZFO , \nand (b)  lattice parameters and tetragonality of  PZT (tetragonal phase ) in the term of \ndoped NZFO  (%) in PZT , in xNZFO/(1 -x)PZT particulate composite.  \nFig.6 SEM image of  the sintered pellet surface  of xNZFO/(1 -x)PZT  particulate composite  for \nthe composition  with x=0.60  sintered at 1150 0C for 6 h . \nFig.7Room temperature  permittivity  (ε/) of xNZFO/(1 -x)PZT particulate  composite  for the \ncomposition s with x=0, 0.15, 0.30, 0.45, 0.60, 0.75, 0.90 and 1.0  measured  in the \nfrequency range 1  kHz- 106 kHz. \nFig.8Frequency  dependent  permittivity (ε/) of xNZFO/(1 -x)PZT  particulate  composite  for the \ncomposition  with x=0.60  measured in the frequency range  5 kHz -102 kHz . \nFig.9Temperature variation  of permittivity  (ε/) of xNZFO /(1-x)PZT particulate  composite  at \n10 kHz for the composition s with x= 0, 0.15, 0.30, 0.45 and 0.60 . 15 \n Fig.10 Full width at half maximum (FWHM) variation of permittivity  (ε/) peaks  at 10 kHz for \nthe compositions  with x= 0, 0.15, 0.30, 0.45 and 0.60  of xNZFO/(1 -x)PZT particulate  \ncomposites .  \nFig.11Variation of the transition temperature  (TC) for the different composition s of \nxNZFO /(1-x)PZT  particulate  composite with x= 0, 0.15, 0.30, 0.45 and 0.60  at 1 \nkHz. \n \n \n \n \n \n \n \n \n \n \n \n \n \n 16 \n  \n \n17 \n  \n \n18 \n  \n \n19 \n  \n \n20 \n  \n \n21 \n  \n \n \n \n \n \n \n \n \n \n22 \n  \n \n23 \n  \n \n24 \n  \n \n25 \n  \n \n \n \n26 \n  \n \n27 \n TABLE. 1  \n \n \n"    },    {        "title": "2105.13943v1.Anisotropic_Magnon_Spin_Transport_in_Ultra_thin_Spinel_Ferrite_Thin_Films____Evidence_for_Anisotropy_in_Exchange_Stiffness.pdf",        "content": "Anisotropic Magnon Spin Transport in Ultra-thin Spinel Ferrite Thin Films –\nEvidence for Anisotropy in Exchange Stiffness\nRuofan Li,1Peng Li,2Di Yi,2,3Lauren Riddiford,2,4Yahong Chai,5,6\nYuri Suzuki,2,4Daniel C. Ralph,7,8,\u0003and Tianxiang Nan5,6,7,y\n1Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853, USA\n2Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA\n3State Key Lab of New Ceramics and Fine Processing,\nSchool of Materials Science and Engineering, Tsinghua University, Beijing 100084, China\n4Department of Applied Physics, Stanford University, Stanford, CA, 94305, USA\n5School of Integrated Circuits and Beijing National Research Center for Information\nScience and Technology (BNRist), Tsinghua University, Beijing 100084, China\n6Institute of Microelectronics, Tsinghua University, Beijing 100084, China\n7Laboratory of Atomic and Solid-State Physics, Cornell University, Ithaca, NY 14853, USA\n8Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY 14853, USA\nWe report measurements of magnon spin transport in a spinel ferrite, magnesium aluminum\nferrite MgAl 0:5Fe1:5O4(MAFO), which has a substantial in-plane four-fold magnetic anisotropy.\nWe observe spin diffusion lengths >0:8\u0016mat room temperature in 6 nm films, with spin diffusion\nlength 30% longer along the easy axes compared to the hard axes. The sign of this difference is\nopposite to the effects just of anisotropy in the magnetic energy for a uniform magnetic state. We\nsuggest instead that accounting for anisotropy in exchange stiffness is necessary to explain these\nresults.\nUsing magnons, the quanta of spin waves, for informa-\ntion transmission offers the potential for low energy dis-\nsipation compared to traditional electronic transport[1].\nMagnon spin transport has been demonstrated experi-\nmentally in both insulating ferrimagnets [2–14] and anti-\nferromagnets [15–17]. The magnon spin diffusion length,\nthe characteristic propagation length, has been stud-\nied under various conditions of temperature [11, 18–20],\nmagnon chemical potential [21–23], and external mag-\nnetic field [11, 20, 24]. Previous measurements on ferri-\nmagnetic insulators have focused on yttrium iron garnet\n(YIG), either with thick films (>100 nm) which are fully\nrelaxed relative to the substrate [3, 5, 20] or with thinner\nfilms [25]. In either case, YIG has very weak magnetic\nanisotropy, andnoanisotropieshavebeenobservedinthe\nspin transport.\nHere we report measurements of magnon transport in\na low-loss spinel material that has been recently stabi-\nlized in epitaxial thin-��lm form, magnesium aluminum\nferrite MgAl 0:5Fe1:5O4(MAFO) [26–30]. When grown\nepitaxially on an MgAl 2O4substrate, MAFO possesses\na substantial in-plane cubic anisotropy ( \u001813mT with\neasy axes in the <110> directions), while maintain-\ning low magnetic damping into the regime of ultrathin\nfilms. We report that magnon spin transport depends\nstrongly on the propagation direction relative to the\nanisotropy axes, with a spin diffusion length 30% greater\nfor magnon propagation along the easy axes compared\nto the hard axes. We argue that this difference has\nthe wrong sign to be explained taking into account only\nthe usual magnetic anisotropy energy which applies to\nspatially-uniform states, but require also a consideration\nof anisotropy in the exchange stiffness for nonuniformstates. Our results suggest that spin transport measure-\nments can be used as a sensitive probe of the exchange\nstiffness and that manipulation of this stiffness (e.g., via\nstrain) provides an alternative strategy for controlling\nmagnon spin diffusion.\nWe employ a measurement geometry commonly used\nfor measuring magnon spin transport – parallel Pt wires\nwith different separation distances deposited on top of\nthe magnetic insulator to be investigated (Fig. 1(a))\n[3, 5, 7, 10, 11, 15, 17]. The Pt wires have widths 200\nnm, lengths 10 \u0016m, and spacings that range from 0.4 to\n3.2\u0016m. A charge current passing through one of the Pt\nwires results in the excitation of magnons in the mag-\nnetic film below the wire by the spin Hall effect (SHE)\n[31] and, because of a thermal gradient arising from Joule\nheating, also by the spin Seebeck effect (SSE) [32]. The\nexcited magnons diffuse through the film to the other Pt\nwire where they are detected by means of a voltage sig-\nnal generated by the inverse spin Hall effect (ISHE) [33].\nTo measure magnon spin transport in different directions\nrelativetotheunderlyingcrystallinefilmwemeasuresep-\narate devices on the same chip with different orientations\nof the Pt wires (Fig. 1(b)).\nThe magnetic insulators we probe are (001)-oriented 6-\nnm-thick MAFO thin films that are epitaxially grown on\nMgAl 2O4(MAO) substrates [26] (see Supplemental Ma-\nterial[34]forsamplegrowthandfabrication). Tetragonal\ndistortion due to epitaxial strain acting on MAFO (3%\nlattice mismatch) induces an in-plane 4-fold magnetic\nanisotropy with an effective field strength of 13 mT as\nshown by ferromagnetic resonance (FMR) measurements\non a MAFO film of similar thickness (5 nm) grown under\nthe same conditions (Fig. 1(c)). The angular dependencearXiv:2105.13943v1  [cond-mat.mtrl-sci]  28 May 20212\nof the anisotropy field is consistent with cubic symmetry,\nand the easy axes are along <110>. The Gilbert damp-\ning in MAFO as measured by FMR remains small and\nisotropic with respect to the angle of in-plane magnetic\nfield relative to the crystal axes [26].\ndVIMAOMAFOPtPt𝜙M(a)\n[100][1!10][110][010](b)[100][010](c)\nFIG. 1. Measurement geometry and magnetic anisotropy of\nMAFO (a) Schematic layout of the experimental setup (not\nto scale). (b) Schematics for measuring magnon transport\nalong different directions relative to the crystal axes of the\nMAFO film (c) FMR resonance field at 9 GHz as a function\nof the angle ( \u001eH) between applied in-plane magnetic field and\nthe [100] crystal axis for a MAFO thin film of 5 nm thickness\ngrown under the same conditions as the 6 nm film. The solid\nline represents a cos(4\u001eH)fit to the measured data.\nFor the magnon spin-transport measurements, we pass\na low-frequency current (5.9 Hz) through one Pt wire,\nexciting magnons in the MAFO film via both the SHE\nand SSE. All the measurements were performed at room\ntemperature, using different angles and magnitudes of in-\nplane applied magnetic field, and different spacings be-\ntweentheinjectorand detectorPtwires. Thecomponent\nof the nonlocal voltage ( VNL) detected in the distant Pt\nwire that originates from SHE ( VSHE) has a linear depen-\ndence on the current ( I), while the component from SSE\n(VSSE), which arises due to a temperature gradient from\nJoule heating, varies quadratically with I. These two\nkinds of nonlocal voltages can therefore be distinguished\nby detecting the first ( V1!=R1!I) and second-harmonic\n(V2!=R2!I2) responses using lock-in amplifiers. De-\npending on their origin, the nonlocal resistances can be\nthen written as\nR1!=RSHE +R0;1! (1)\nR2!=RSSE+R0;2! (2)whereRSHEandRSSErepresent the nonlocal resistances\nthat we wish to measure arising from the SHE and SSE,\nwhileR0;1!andR0;2!are offset resistances due to in-\nductive and capacitative couplings in the sample and the\nmeasurement setup.\nTo subtract out the constant-impedance parts ( R0;1!\nandR0;2!), we collected V1!andV2!as a function of the\nin-plane magnetic angle \u001efor a field magnitude of 75 mT,\nas shown in Fig. 2(a). \u001eis defined as the complement of\nthe angle between the magnetization of MAFO and the\napplied current axis (see Fig. 1(a)). For the electrically-\ngenerated nonlocal signal VSHE, both the injection and\nthe detection of the magnons have a cos(\u001e)dependence\ncoming from SHE and ISHE respectively, resulting in a\ntotal dependence approximately /cos2(\u001e)(Fig. 2(a)).\nIn the case of the thermally-generated nonlocal signal\nVSSE, themagnoninjectionisgeneratedbyJouleheating,\nwhich has no angular dependence, while the detection of\nthe magnons through ISHE varies with angle \u001e, which\ngives rise to an approximately cos(\u001e)angular dependence\n(Fig. 2(c)).\n(a)\n(d)(b)\n(c)\nFIG. 2. (a) First-harmonic RSHEand (c) second-harmonic\nRSSEnonlocal signals as functions of the magnetic-field angle\n\u001efor a field magnitude of 75 mT and samples with d= 0.6, 1,\nand 1.4\u0016m, with the Pt wires oriented along the [100] direc-\ntion. Solid lines are the fits to cos2(\u001e)andcos(\u001e)dependence\nfor first and second-harmonic signals respectively. Magnitude\nof (b) first-harmonic and (d) second-harmonic nonlocal sig-\nnals as functions of distance between the Pt wires oriented\nalong the [100] and [1\u001610]directions. The solid lines are fits to\nEq. 3.\nAlthough generally magnon conductance decreases\nwith decreasing thickness in magnetic insulator films due\nto increased damping [35], nevertheless even in 6 nm of\nMAFO we observe long-range magnon transport across\n3.2\u0016m gaps. The spin diffusion length can be extracted\nfrom the decay of RSHEandRSSEas a function of the\nseparation ( d) between the Pt wires (injector and detec-\ntor). This decay can be well fitted to a magnon diffusion3\nmodel [3]:\nRNL=C\n\u0015exp (d=\u0015)\n1\u0000exp (2d=\u0015)(3)\nwhereRNLcould be either RSHEandRSSE,Cis a\ndistance-independent constant, and \u0015is an effective spin\ndiffusion length in the direction perpendicular to the Pt\nwires.\nWe have investigated the effects of anisotropy by com-\nparing spin diffusion lengths for the orientation of the\nwires along [100], [110], [010] and [1\u001610]axes. Fig. 2(b)\nshows the first-harmonic non-local resistances for the\n[100] and [1\u001610]wire orientations plotted as a function\nof the wire spacing. The dots in the plots correspond to\nthe experimental data while the solid lines show the fits\nto Eq. (3). A similar decay of the nonlocal resistance vs.\nspacing was also observed for the second-harmonic signal\nasshowninFig.2(d). Thespindiffusionlengths, \u00151!and\n\u00152!, extracted from the fits for first and second-harmonic\nsignals are shown in Fig. 3 for the different crystal-axis\norientations.\n[100] [110] [010] [110]\nCrystal axes0.60.70.80.91 (m)\n1\n2\n0.81.01.2\n2 (m)\nFIG. 3. Spin diffusion lengths extracted from the decay of\nthe first (blue circles) and second-harmonic (red triangles)\nnonlocal signals for wires along corresponding axes. Error\nbars represent the standard deviations of the fits.\nFor both\u00151!and\u00152!we observe significantly larger\n(> 30%) spin diffusion lengths along the <110> family\nof axes (easy axes) compared to the <100> axes (hard\naxes). We also find that the extracted values of \u00152!are\nslightly larger than \u00151!, which has also been observed\npreviously for YIG thin films [20]. This difference can be\nexplained as due to the different mechanisms by which\nthe nonequilibrium magnon distributions are generated\nforthetwosignals. Furthermore, duetoalateralthermal\ngradient near the Pt injector bar, the second-harmonic\nvoltage can have contributions from both local and non-\nlocal SSE signals, while for the first-harmonic signal the\nSHE excites the magnons only locally [5]. The angular\ndependencies of \u00151!and\u00152!correspond to the same 4-\nfold symmetry as the in-plane magnetic anisotropy, con-\nsistent with the cubic symmetry of MAFO (Fig. 1(c)).If one assumes that the primary cause of anisotropic\nmagnon transport in MAFO is simply the anisotropy in\nthe magnetic energy for a uniform magnetic state, it is\nsurprising that the spin diffusion length is longer in the\ndirection of the magnetic easy axis, rather than the re-\nverse. The magnetic anisotropy energy for a uniform\nmagnetic state should cause the same qualitative behav-\nior as an increased applied magnetic field along the easy\naxis. Both previous measurements on YIG [20, 24], and\nour own measurements on MAFO (Fig. 4) show that\nthe magnitude of the nonlocal spin signal decreases as a\nfunction of the increasing magnitude of an applied mag-\nnetic field, corresponding to a decreased spin diffusion\nlength with increasing magnetic field. This behavior has\nbeen ascribed within the context of the SSE to the influ-\nence of the magnetic field increasing the energy of long-\nwavelength magnons [24]. Quantitatively, the effect of a\nmagnetic field is also far too weak to explain the scale of\nthe effect that we measure. Figure 4 shows that a 140\nmT magnetic field decreases the spin signal by only 18%,\nindicating that an in-plane cubic anisotropy of 13 mT\ncould not generate the 30% difference we observe.\nFIG. 4. The ratio of first harmonic nonlocal signal with its\nmaximum value as a function of applied magnetic field at\n\u001e= 0for samples with d = 1 \u0016mand Pt wires oriented along\n(a) hard axis (b) easy axis. The inset in each plot shows the\norientation of the Pt wires and applied magnetic field with\nrespect to the crystal axis.\nThe sign of the effect we observe is also surpris-\ning within the usual theoretical framework for model-\ning the energies and group velocities of long-wavelength\nmagnons. The only type of anisotropy that is ordinarily\nconsidered is the anisotropy energy for a uniform mag-\nnetic state, accounted for in terms of an anisotropy field\n2\u00160Han. For a 4-fold in-plane magnetic anisotropy, the\ndispersion curve for long-wavelength magnons (taking\ninto account both exchange and magnetic dipole contri-\nbutions) takes the form [36–38]\n!(k;\u001ek;\u001eH) =g\u0016B\n~p\nB1B2; (4)4\nwith\nB1=B+\u00160Meff(1\u0000Pk) +Dk2+1\n4\u00160Han(3 + cos(4\u001eH))\n(5)\nB2=B+\u00160MeffPksin2(\u001ek) +Dk2+\u00160Hancos(4\u001eH);\n(6)\nand where k=j~kjis the magnitude of the wavevector,\n\u001ekis the angle of the wavevector relative to an easy axis,\n\u001eHis the angle of the average magnetization relative to\nthe [100] direction, B= 0.075 T in our angle-dependent\nmeasurements, Meffis the saturation magnetization, D\nis the exchange stiffness, and Pk= 1\u0000[(1\u0000e\u0000kd)=(kd)]\nwithdas the film thickness. For our MAFO samples,\n\u00160Meff= 1:5T,2\u00160Han= 13mT,d= 6nm. For oxide\nferrimagnets, a typical value of the exchange stiffness is\nD= 5\u000210\u000017T m2. The effect of the anisotropy field\nis to increase the energy of magnons with small values of\nkfor\u001eknear the easy axis, but to cause little change in\nthe energy of magnons with larger kdue to the increas-\ning importance of the exchange and dipole terms. As\na result, the group velocity, vg=d!(k;\u001ek;\u001eH)=dkis al-\nways decreased by an increase in the magnetic anisotropy\nenergy. A larger value of Hanwill also decrease the ther-\nmalmagnonpopulation. Botheffectsshoulddecreasethe\nspin diffusion length in the direction of a magnetic easy\naxis.\nWe therefore draw the conclusion, based on both the\nsign and magnitude of the effect we observe, that the\nanisotropic nonlocal spin signal must be caused by crys-\ntalline anisotropies which are different from simply the\nmagnetic anisotropy energy for a uniform magnetic state.\nWe considered whether the scattering time for spin relax-\nation might depend on the orientation of ~kwith respect\nto the anisotropy axes. But if this were the case, we\nwould expect the scattering to also depend on the ori-\nentation of ~kwith respect to the an applied magnetic\nfield. We do not observe deviations from the behavior\nVSHE/cos(\u001e)andVSSE/cos2(\u001e)and therefore con-\nclude that scattering time is not ~korientation dependent\n(See Supplemental Material [34] for residuals in the an-\ngular fits).\nWe suggest, instead, that the anisotropy of our sig-\nnal is dominated by anisotropies in the exchange ener-\ngies associated with the MAFO crystal structure. In-\nstead of assuming an isotropic exchange stiffness as in\nEqs. (4-6), we can model the exchange stiffness Das a\nfunction of the orientation ~krelative to the crystal axes\nfor the long wavelength spin waves that contribute most\nto the non-local measurements; more specifically Dis\nlarger for~kalong the magnetic easy axis so as to increase\ngroup velocity in those directions. The possibility of an\nanisotropic exchange stiffness has been considered pre-\nviously [39–41]. For non-relativistic exchange processes,\nthe spin stiffness should not depend on the orientation of\nthe magnetization with respect to the crystal axes ( \u001eH),but it can depend on the orientation of the wavevector\nrelative to the crystal axes ( \u001ek). This is the symmetry\nrequired to explain our results without significant devia-\ntions from the observed dependence on the angle of mag-\nnetic field ( VSHE/cos(\u001e)andVSSE/cos2(\u001e)).\nThe existence of anisotropy in exchange stiffness can\nalso help to explain the differences in the magnitude of\nthe spin transport signal extrapolated to small spacings\ndbetween source and detector wires – the fact that the\nspinsignalsinthelimitofsmall dbecomelargerfortrans-\nport along the hard axis compared to the easy axis. The\nanisotropiesinboththeexchangestiffnessandtheenergy\nof the uniform magnetic state have the sign to increase\nthe energies of long-wavelength magnons with ~kalong\nthe easy axes, so the population of those magnons will\nbe decreased relative to magnons with ~kalong the hard\naxes.\nWe are not aware of previous observations of\nanisotropy in exchange stiffness by broadband ferro-\nmagnetic resonance (FMR) or Brillouin light scattering\n(BLS), the two most common techniques for making di-\nrect measurements of exchange stiffness in thin-film sam-\nples [42–45]. Broadband FMR measures the exchange\nstiffness in only one direction ( ~kperpendicular to the\nplane of the thin film) since it requires measuring spin-\nwave standing waves within the film thickness. To sep-\narate the effects of anisotropy in exchange stiffness from\na simple magnetic anisotropy energy using BLS would\nlikely require measurements as a function of the magni-\ntude of~k.\nIn summary, we have measured magnon-mediated\nspin transport in epitaxially-grown ultrathin (6 nm)\nMgAl 0:5Fe1:5O4thin films. The small isotropic Gilbert\ndamping parameter ( \u00180.0015) of these films, their soft\nmagnetism (in-plane coercive field < 0.5 mT), and\nlow processing temperature ( \u0018450/uni2103) make MAFO an\nparticularly attractive platform for the study of magnon\ntransport and integrated magnonic devices. Unlike\nprevious studies of YIG samples, tetragonally-strained\nepitaxial MAFO posseses substantial in-plane cubic\nmagnetic anisotropy. We find also a strong anisotropy\nin magnon-mediated spin transport, with spin diffusion\nlengths 30% larger along the easy axes as compared\nto that along the hard axes. The sign of this effect is\nopposite to what would be expected due simply to the\nmagnetic anisotropy energy of a uniform magnetic state,\nso we suggest that the anisotropy in spin transport is\ndominated instead by anisotropy in exchange stiffness.\nAn exchange stiffness that is larger for ~klong the\nmagnetic easy axis can explain not only the longer spin\ndiffusion lengths for transport along the easy axes but\nalso larger nonlocal spin signals in the limit of small\nspacing that we observe for transport along the hard\naxes. Nonlocal spin wave transport measurements might\ntherefore serve as a sensitive probe of exchange-stiffness\nanisotropy in thin-film samples. Since crystalline5\nanisotropies can be tuned by strain, we also suggest\nthat strain-mediated manipulation of exchange stiffness\nmight provide a strategy for modulating spin transport\nin magnetic thin films.\nWe thank Andrei Slavin and Vasyl Tyberkevych for\na critique of an initial draft of this paper, and Satoru\nEmori, Jiamian Hu, Pu Yu, Dingfu Shao and Evgeny\nTsymbal for helpful discussions. Research at Cornell\nwas supported by the Cornell Center for Materials Re-\nsearch with funding from the NSF MRSEC program\n(Grant No. DMR-1719875). This work was performed\nin part at the Cornell NanoScale Facility, a member\nof the National Nanotechnology Coordinated Infrastruc-\nture, which is supported by the NSF (Grant No. NNCI-\n2025233). Research at Tsinghua was supported by the\nNational Natural Science Foundation (52073158), and\nthe Beijing Advanced Innovation Center for Future Chip\n(ICFC). 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Prellier2, and J F Scott3 \n \n \n1Department of Physics and Institute of Functi onal Nano Materials, University of Puerto \nRico, PR 00931-3343, USA  \n2Laboratoire CRISMAT, CNRS UMR 6508, EN SICAEN, 6 Bd du Maréchal Juin, F-\n14050 Caen Cedex, France \n3Department of Earth science, University of Cambridge, Cambridge CB2, 3EQ,UK \n \nE-mail: jsco99@esc.cam.ac.uk and rkatiyar@speclab.upr.edu  \n \nReceived 7 October 2008 \n \n \nAbstract \nLow-temperature magnetic properties of epitaxial BiFeO 3 (BFO) thin films grown on \n(111) SrTiO 3 substrates have been studied. Zero-field-cooled (ZFC) and field-cooled \n(FC) magnetization curves show a large discrepancy beginning at a characteristic \ntemperature fT that is dependent on the magnetic-field strength. )(HfT  varies according \nto the well known de Almeida–Thouless line 3/2HfT∝ suggesting an acentric long \nrange spin-glass behavior and mean-field system.   \n \n \n* PACS numbers :  (i) 78.30.Hv, (ii) 63.20.Dj, (iii) 75.50.Lk, (iv) 5.40.Gb,   \n* Keywords :  (i) BiFeO 3 epitaxial films, (ii) multiferroics, (iii) long range magnetic \norder, (iv) spin-glass-like behavior   2BiFeO 3 (BFO) is unusual or perhaps unique in that it exhibits magnetism and \nferroelectricity at room temperature.  Ferroel ectromagnetic materials, i.e., multiferroics, \nexhibit ferroelectric (or antiferro-electric) properties in combination with ferromagnetic (or antiferromagnetic) properties.[1,2] BFO is a rhombohedrally distorted ferroelectric \nperovskite (T\nc ≈ 1100K) with space group R3c,[3,4] which permits coupling between \nmagnetism and ferroelectricity.  However, BiFeO 3 shows G-type antiferromagnetism up \nto 643K (T N),[3-5] in which  all  neighboring magnetic spins are oriented antiparallel to \neach other, and addition, the axis along which the spins are aligned precesses throughout \nthe crystal, resulting in a modulated spiral sp in structure with a long periodicity of ~ \n620Å.[4,5]  This cycloidal spin modulation is thought to cause linear magnetoelectric \ncoupling to average spatially to zero in si ngle-crystals; however, this modulated spin \nstructure was once thought considered to be absent in constrained films.  As a result, \nweak ferromagnetism has been suggested experimentally[6-9] and also predicted \ntheoretically [10] in thin films. For example, Wang and co-workers[6] fabricated an \nepitaxially constrained BFO film that exhi bited pronounced thickness dependence of \nferromagnetism.  More recent studies,[8,11] however, show clearly that the compressive \nepitaxial strain does not enhance the magnetization in BFO films, and Lebeugle et al. , for \nexample,[11] note that just 1% mol of paramagnetic Fe3+ (probably due to presence of \nBi25FeO 39) can account for all the low-temperature magnetic enhancement in their single \ncrystals, and that removing such impurities with HNO 3 removes virtually all traces of \nferromagnetism in their samples.   \nA survey of the literature re veals a dramatic change in the magnetic properties of \nBFO at temperatures below 200 K.[8,9,13-15] Recently we inferred a spin glass \ntransition below 120K, which follows mean field theory[16] and is similar in some \nrespects to those in other orthoferrites.  Latter we observed magnons in BFO by inelastic \nlight scattering techniques,[17] showing the spin wave behaviour near the transition \ntemperatures 140K and 201K and the enhancement the Raman intensity of the \nmagnon.[18-20] Cazayous et al.[21] also reported strong magnon anomalies at the same \ntemperatures. Spin reorientation (SR) transitio ns in orthoferrites have been extensively \nstudied.[22,23] For example, in ErFeO 3 Koshizuka and Ushioda[22] observed two one–\nmagnon branches by inelastic scattering technique showing the frequency dependence  3near the transition temperature and the enhancement the magnon intensity. More recently \nRedfern et al.[24]noted that transition near 200K shows strong magnetoelastic coupling \nin the Hz regime whereas anomalies near 140 K show strong elastic coupling in the MHz regime in data from Carpenter et al.[24]  \nThe conventional wisdom (Young et al.[25]) is that short-range Ising-like spin \nglasses cannot have a de  Almeida-Thouless line (AT-line), and that the critical exponent \n0.10.8±=νz . However, Fischer and Hertz[26] point out that acentric (non- \ncentrosymmetric) ferroelectric magnetic spin glasses cannot be Ising-like and probably \nviolate other predictions of standard spin-glass theory. The original spin glass model of \nKarkpatrick and Sherrington[27] described magnetic systems within a mean field theory, \nfor which the critical exponent z ν = 2 characterizes the frequency dependence of a \ncharacteristic freezing temperature. Recent work has generally applied Ising statistics to \nsuch spin glasses, But BiFeO 3 is a unique case of an asymmetric spin glass exhibiting \ncritical exponent 4.1=νz similar to the mean-field prediction of 2.0.  \nIn the present work we report magnetic and phonon properties of the pseudo-\ncubic [111] c-oriented rhombohedral BFO thin films that have been known to possess \ngiant spontaneous polarizations ( Ps) along the [111] c axis[28] with the easy \nmagnetization plane perpendicular to this pseudo-cubic [111] c axis (or equivalently \nperpendicular to hexagonal [001] h).[29]  We observed a splitting in the ZFC and FC \nmagnetization curves at a characteristic temperature, irrT.  This splitting temperature was \nstrongly dependent on the applied magnetic field, typical of a spin-glass-like transition.  \nBFO thin films were grown on (111) STO substrates by employing pulsed laser \ndeposition (PLD) method.[30] The average thic kness of these films, as estimated using \nfield-emission scanning electron microscopy, was 300±3 nm.  To examine the structure \nof the PLD-grown BFO film on a STO (111) substrate, theta-2-theta ( θ-2θ) x-ray \ndiffraction (XRD) and Ф-scan experiments were carried out, and their results are shown \nin Fig. 1.  The pattern reveals purely [111] c-oriented rhombohedral BFO reflections.  The \ndegree of in-plane orientation was assessed by examining XRD Ф-scan spectra.  As \npresented in the inset of Fig. 1, the peaks for (022) reflection of the [111] c-oriented \ndomain occur at the same azimuthal Ф angle as those for STO (022) reflection and are \n120o apart from each other.  This clearly indicates the presence of threefold symmetry  4along the [111] c direction and a coherent epitaxial growth of the BFO film with R3c \nsymmetry on a STO (111) substrate. A superconducting quantum-interference-device-\nbased magnetometer (Quantum design MPMS-5) was used for the magnetization measurements which were carried out by cooling the sample to a desired temperature in \nthe presence or absence of an applied magnetic field.   \nFigure 2 displays the ZFC and FC magnetization curves of the epitaxial BFO film \nhaving rhombohedral R3c symmetry. The substrate effect from STO(111) has been \nsubstracted from the magnetization data. During the measurements, an external magnetic \nfield of 10 kOe (1\nT) was applied parallel to the out-of-plane [111] c direction. The \nmagnetization induced along the in-plane direction which is perpendicular to [001] h (i.e., \n[111] c) was measured because the magnetization easy axis of R3c BFO is parallel to \n[110] h which is vertical  to [001] h.[29]  As shown in Fig. 2, the ZFC and FC \nmagnetizations gradually increase with decreasing temperature, which is presumably \ncaused by local clustering of spins.[12 ]  The most prominent feature of Fig. 2 is that there \nis a large discrepancy between the ZFC and FC curves in the film beginning at ~74.7 K, \nwhich increases with decreasing temperature.  The observed splitting in the ZFC and FC \ncurves at low temperatures is a hallmark of sp in-glass-like transition.  In addition to this, \nwe have also observed a sharp cusp at around 50 K in the ZFC curve, which can be \nattributed to a typical blocking process of an assembly of superparamagnetic spin \nmoments.[31] On the contrary, these moments are aligned parallel to the applied field \nduring the FC measurement, leading to a large discrepancy between the FC and ZFC \ncurves below the freezing temperature.  Other researchers have very recently reported \naging within the inferred spin-glass temperature range.  Shvartsman et al.[31]  confirm \nnon-ergodic behavior of the low-field magnetization at low T. They suggest that that this \nmight be a reentrant phenomenon, since the system being primarily antiferromagnetic \nreveals a spin spiral counteracting the formation of weak ferromagnetism due due to \nglobal spin canting. However, they exclude a generic spin glass phase, “since only \ncumulative relaxation is found af ter isothermal aging below T g instead of classic hole \nburning and rejuvenation.” \nThe ZFC and FC magnetization characteristics of the epitaxial BFO film were \nfurther examined by applying the bias magnetic field with various strengths. The results  5are presented in Fig. 3(a) with the field strength of 0.5, 1.0, 1.5, 3.0, 5.0, 7.0 and 10.0 \nkOe in ascending order.  The splitting temperature )(HTirr  (irreversibility in ZFC) \ngradually decreases with increasing field st rength.  The splitting is accompanied by the \nobservation that the cusp maximum becomes smeared out with decreasing field strength \n[Fig. 3(a)]. This indicates that the magnetic energy at a high-field becomes sufficient to \novercome the energy barrier between possibl e equilibrium orientations of the magnetic \nmoments, thereby decreasing )(HTirr . This observation also su pports the spin-glass-like \nbehavior of the present BFO film, which arises from the spin reorientation in the easy \nmagnetization plane.  Similar behavior was also reported by Park et. al. [14] in their ZFC \nand FC curves of BFO nanoparticles, suggesting a spin-glass-like transition at low \ntemperatures.  \nTo test the validity of the present spin-glass model of the [111] c-oriented BFO \nfilm, we have examined the field-dependent freezing temperature, ,)(HfT  using the \nAlmeida-Thouless(AT) equation,[12,32] namely, ,}] / {1[2/3)0( )(f f AT T T AH H−= where \n)(HfT is equal to  )(HirrT which  corresponds to the onset of the irreversible behavior \nunder the field H.  The data fitting by the AT equation for higher magnetic fields (1.5~10 \nkOe) yields Tf (0) = 140 K, as shown in Fig 3(b).  Note that this freezing temperature \nagrees within a small uncertainty with the te mperature [18-20] at which the magnon cross \nsection diverges T = 140.3K.  From the present data, however, it is very difficult to \nassign an exact spin-glass temperature in this  cross-over regime be cause sharp transitions \nat higher fields, i. e. the presence of Almeida–Thouless line stability does not favor the \nshort range Ising-type spin configuration,[25] whereas the Heisenberg-type ferromagnet \nwith non-commuting spin operators is likely to  be more appropriate at lower fields.[26]  \nIn Ref [16], we reported that the critical exponent describing the slowing down of the \nglassy dynamics 4.1≈υz , which is much closer to the value excepted in a mean field \nsystem (where 0.2≈υz ) then in the classical short range Ising magnetic spin glass   \n( 107−≈υz ). La 0.5Mn 0.5FeO 3 is a good example of such a non- standard spin glass with \n0.1≈υz .[33] Fischer and Hertz[26] have emphasized that no published theories apply to \nspin glasses lacking an inversion center, and fu rther, that such glasses can not possibly be \nIsing–like. On this basis it can be suggested th at the spin glass transition is coupled with a  6long-range order parameter (strain) responsible for its mean field behavior and that the \nsymmetry is acentric.                                                                           \nThe behavior of magnetization with temp erature is in good agreement with the \nelectromagnon description of BFO.[18-20] These authors inferred spin reorientation at \n140 and 200K which are temperatures very close to the predicted transition temperature \nin this work. In Ref [18] we see both sigma (FM-magnon) and gamma (AFM-magnon) \nmodes at 18.3 cm-1 and 26.4 cm-1(80K) which also reveal the presence of spin glass \nbehavior in BiFeO 3. They are very similar to those two branches in orthoferrites such as \nErFeO 3.[22]  However, at exactly 201K there is an abrupt change in the frequency and \nintensity of the sigma mode and the gamma mode disappears.  This suggests a spin \nreorientation, as is common in orthoferrites.  Therefore there appear to be subtle and \nunpredicted magnetic changes going on in the region of 201K, far below T N = 640K and \nthese may influence the spin-glass behavior we see initially on cooling at 140K.  To \nrephrase this important point:  Spin-glass phases are usually not so far below the Neel \ntemperature in magnetic materials where they occur at all; previously this cast doubt on \nspin glass behaviour in BiFeO 3 at cryogenic temperatures, since T N > 630K.  However, \nour recent discovery of low-temperature spin-reorientation transitions makes it more \nplausible. \nIn conclusion, we examined the ZFC and FC magnetization curves of the [111] c-\noriented epitaxial BFO film with R3c symmetry. The two curves showed a discrepancy \nbeginning at a characteristic temperature, )(HfT , which did depend on the applied field, \nrevealing spin-glass-like behavior.  Howeve r, it is known [34] that such an H2/3 \ndependence is not in itself proof for a spin-glass state, which can also arise from \nsuperparamagnetic behaviour.  Therefore, other data, including the frequency dependence \nof the temperature peak in susceptibility a nd the Vogel-Fulcher dependence,[16] as well \nas aging phenomena,[31]  are helpful in inferring a glassy state.  Note also in this context \nthat antiferromagnetic ordering and spin-glass phenomena may coexist both \nexperimentally and in mean field models.[35] \nIn general it is not easy to prove the existence of a spin glass:  An AT-line can \noccur in superparamagnets; aging and rejuvenation (which we will show for bismuth \nferrite in a separate paper) can occur in any ferroic system with domain pinning; and  7frequency dependent susceptibilities can occur in relaxors.  Therefore numerous separate \nexperiments are required, but we believe the AT-line shown here will be very helpful in \nthis regard. \n \nWe gratefully acknowledge the financial support from the DOD grants W911NF \n–06–1–0030 and W911NF-06-1-0183. The magnetic measurements were supported by \nNoE FAME (FP6-500159-1), the STREP Ma CoMuFi (NMP3-CT-2006- 033221) from \nthe European Community and by the CNRS, France.  \n \n \n* REFERENCES  \n \n[1] Hill N A 2000 J. Phys. Chem. B 104 6694  \n[2] Eerenstein W, Mathur N D and Scott J F 2006  Nature  442 759  \n[3] Kubel F and Schmid H 1990 Acta Cryst.  B 46 698  \n[4] Fischer P, Po łomska M, Sosnowska I , and Szyma ński M 1980 J. Phys. C : Solid St. \nPhys. 13 1931  \n[5] Przenioslo R, Palewicz A, Regulski M, Sosnowska I, Ibberson R M, and Knight K S \n2006 J. Phys: Condens. 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Phys. 100 033908  \n[14] Park T J, Papaefthymiou G C, Viescas A J, Moodenbaugh A R, and Wong S S 2007 \nNano Lett . 7 766    \n[15] Xu X, Qian T, Zhang G, Zhang T, Li G, Wang W, and Li X 2007 Chem. Lett.  36 112    \n[16] Singh M K, Prellier W, Singh M P, Katiyar R S, and Scott J F 2008 Phys. Rev . B. 77  \n  144403; Chukalkin Yu G and Goshchitskii B N 2003 Phys. Stat. Sol.  A 200 R9 \n[17] Cottam M G and Lockwood D J 1986 Light Scattering in Magnetic Solids  (Wiley,      \n  New York)  \n[18] Singh M K, Katiyar R S, and Scott J F 2008 J. Phys: Condens. Matter 20 252203  \n[19] Scott J F, Singh M K, and Katiyar R S 2008 J. Phys: Condens. Matter  20  322203  \n[20] Scott J F, Singh M K, and Katiyar R S 2008, J. Phys: Condens. Matter 20 425223  \n[21] Cazayous M, Gallais Y, Sacuto A, De Sousa R, Lebeugle D and Colson D 2008    \n  Phys. Rev. Lett.  101 037601  \n[22].Koshizuka N and Ushioda S 1980 Phys. Rev . B 22 11 \n[23] White R M, Nemanich R J, and Herring C 1982 Phys. Rev.  B 25 1822 \n[24] Redfern S A T, Wang C, Catalan G, Hong J W, and Scott J F 2008  J. Phys:  \n  Condens. Matter  20 at press; Carpenter M C et al. 2008 at press \n[25] Young A P 2008 Bull. Am. Phys. Soc . 53, paper L5.5; Mattsson J, Jonsson T,   \n  Nordblad P, Katori H A, and Ito A 1995 Phys. Rev. Lett.  74, 4305 (1995); Young A P   \n  and Katzgraber H G 2004 Phys. Rev. Lett.  93 207203   \n[26] Fisher K H and Hertz J A 1986 Spin Glasses  (Cambridge University Press,  \n  Cambridge, U. K.).  \n[27] Kirkpatrick S and Sherrington D 1978 Phys. Rev . B 17 4384 \n[28] Bai F, Wang J, Wuttig M, Li J F, Wang N, Pyatakov A P, Zvezdin A K, Cross L E,  \n   and Viehland D 2005 Appl. Phys. Lett.  86 032511  \n[29] Ruette B, Zvyagin S, Pyatakov A P, Bu sh A, Li J F, Belotelov V I, Zvezdin A K, \nand Viehland D 2004 Phys. Rev . B 69 064114  \n[30] Lee D, Kim M G, Ryu S, Jang H M, and Lee S G 2005 Appl. Phys. Lett . 86 222903  \n[31] Martinez B, Obradors X, Balcells L, Rouanet A, and Monty C 1998 Phys. Rev. Lett . \n80 181; Shvartsman V V, Haumont R, and Kleemann W 2008 (at press)  9[32] de Almeida J R L and Thouless D J 1978 J. Phys.  A 11 983  \n[33] De K, Thakur M, Manna A and Giri S 2006 J. Appl. Phys.  99 013908 \n[34] Wenger L E and Mydosh J A 1984 Phys. Rev . B 29 4156 \n[35] Wong P-Z, Vonmolnar S, Palstra T T M, Mydosh J A, Yoshizawa H, Shapiro S M,  \n   and Ito A 1985 Phys. Rev. Lett . 55 2043 \n  10 \n \n \n* Figure Captions  \n \nFIG. 1.  Theta-2-theta ( θ-2θ) XRD pattern of a PLD-grown BiFeO 3 thin film on a SrTiO 3 \n(111) substrate with the intensity profile in  logarithmic scale.  The inset presents Ф-scan \ndiffraction patterns on (022) planes.  \n \nFIG. 2.  Temperature dependence of the dc magnetization (ZFC and FC) of the [111] c-\noriented BiFeO 3 thin film measured under the applied magnetic field of 10 kOe along the \nout-of-plane [111] c direction.  \n \nFIG. 3.  (a) Temperature dependence of the dc magnetization (ZFC and FC) of the [111] c-\noriented BiFeO 3 thin film measured at various strengths of the applied magnetic field \n(0.5, 1.0, 1.5, 5.0, and 10.0 kOe in ascending order). (b) Experimentally observed values \nof irrT fitted with the Almeida-T houless (AT) line equation.   \n \n \n \n \n   \n \n  \n \n  \n \n  \n \n  \n  11 \n \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n  \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n2θ (degree)I. (a.u.) BFO(111) \nSTO(111) -60 0 60 120 180\n  \nΦ (degree) Intensity (A.U.)  BFO (022) \nSTO (022) \n40 50 60 70 80 90\n  \nBFO(222) \nSTO(222)  12 \n \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n  \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n 13 \n \n \n  \n \n \n \n \n \n  \n \n  \n \n  \n \n \nFigure:2  14 \n \n \n  \n \n  \n \n  \nFigure:3(b) \nFigure:3(a) "    },    {        "title": "1103.5839v1.Analysis_of_volume_distribution_of_power_loss_in_ferrite_cores.pdf",        "content": "arXiv:1103.5839v1  [cond-mat.mtrl-sci]  30 Mar 2011Analysis of volume distribution of power loss in ferrite cor es\nM. LoBue,1V. Loyau,1and F. Mazaleyrat1\nSATIE, ENS de Cachan, CNRS, UniverSud, 61 av du President Wil son,\nF-94230 Cachan, France\n(Dated: 26 September 2021)\nWe present a technique to estimate the inhomogeneities of magnetic loss across the\nsection of ferrite cores under AC excitation. The technique is base d on two distinct\ncalorimetric methods that we presented elsewhere. Both the meth od are based on\nthe measurement of the rate of increase of the sample temperatu re under adiabatic\ncondition. The temperature ramp is recorded either measuring the sample bulk\nresistivity, or using a platinum probe pasted on the sample surface. As an example\nwe apply the procedure to an industrial sample of Mn-Zn ferrite und er controlled\nsinusoidalexcitationwithapeakinductionof50mTintherangebetwe en100kHzand\n2MHz. The results are discussed by comparison with simulations of th e dissipation\nfield profile through the sample, calculated using a FEM code.\nPACS numbers: 75.50.-y, 75.50.Gg,\n1I. INTRODUCTION\nSoft sintered Mn-Zn ferrites are widely used in high frequency applic ations based on\nswitching circuits. Their relatively high resistivity and permeability mak e them optimal for\nthe 30kHz-1MHz bandwidth. Miniaturization and the increase of wor king frequencies make\nthermal and loss managements the keys for future design of elect ronic switching devices. It\nis thus necessary to establish reliable methods to measure, unders tand and predict the losses\nin soft ferrites at medium and high frequencies.\nLosses are commonly measured using the flux-metric method1–3. However, when excita-\ntion frequencies are above 10 kHz, measurements can be dramatic ally affected by spurious\nphase contributions due to stray capacitances and inductances o f the circuits and probes. A\npossible alternative is the calorimetric method; in adiabatic conditions , the power dissipated\nin the magnetic material is directly related to the heating rate, redu cing the measurement\nto that of a temperature ramp. We discussed in detail this techniqu e in a recent paper4\nwhere an experimental set-up using a platinum probe as temperatu re sensor was presented.\nA further improvement of the technique has been proposed5by measuring the temperature\nfrom the bulk DC resistivity of the ferrite core5. Whereas the temperature deduced from\nbulk resistivity represents a measurement averaged over the who le sample volume (as in the\ncase of flux-metric measurements), the platinum probe measurem ent gives information on\ndissipation taking place in a thin strip of the material just near its sur face. Consequently\nthe comparison between the two measurements can be used to est imate the loss inhomo-\ngeneity, and thus, the relevance of skin effect for a given sample ge ometry and excitation\nfrequency. It will be shown that, global low level electromagnetic p arameters, obtained by\nimpedance spectroscopy, can be used as local input parameters f or the FEM computation of\nmagnetic loss in linear regime. In this way we shall calculate the volume o r surface averaged\nloss values, corresponding to the two types of calorimetric measur ements and compare them\nwith experimental results.\nII. EXPERIMENTAL AND COMPUTATION DETAILS\nPower losses in magnetic materials under AC excitation are associate d with all the ir-\nreversible phenomena taking place during the core magnetisation (i.e . Barkhausen jumps,\n2eddy currents, domain wall resonance, spin dumping, magnetoaco ustic emission, etc.). Un-\nder adiabatic conditions4, all the dissipated power is converted into heat and induces a tem-\nperature increase of the sample. Thus the temperature field thro ugh the core T(x,y,z,t)\ncan be described by means of the linearised heat equation giving:\n∂T(x,y,z,t)\n∂t=1\nρmcpps(x,y,z) (1)\nwhereps(x,y,z) is the local heat source density (in Wm−3),cpis the specific heat (in\nJK−1kg−1) of the sample and ρmits density (in kgm−3). Now, if V0is the volume of the\nsample, the magnetic loss per cycle measured using the flux-metric m ethod, at a given\nexcitation frequency f, corresponds to the average loss over the sample volume, namely:\n/angbracketleftW/angbracketright=cp\nfV0/integraldisplay\nV∂T(x,y,z,t)\n∂tdV. (2)\nWhereas a measure of the average temperature of the whole samp le will result in a correct\nextrapolation of /angbracketleftW/angbracketright, any local measurement will be related with dissipations taking place\njust in a small volume near the position of the temperature probe. I n the latter case /angbracketleftW/angbracketright\ncan be deduced properly only when volume losses are uniformly distrib uted through the\nsample.\nTwo different techniques where used to measure the temperature ramp: the first using\na platinum probe pasted (Pt500) on the sample4and therefore sensible to the temperature\nchange in a narrow strip just near the surface, and a second tech nique based on the mea-\nsurement of previously calibrated bulk resistivity of the ferrite cor e5which returns a global\naverageofthetemperatureramp. Thelatterleads /angbracketleftW/angbracketrightnotwithstanding theinhomogeneities\nof the loss field due to the geometry of the sample and to the skin effe ct. In both cases,\nthe sample was a square cross section bar (I rod: 0 .64×0.64×2.54 cm3) made of 3E27\nFerroxcube Mn-Zn ferrite. Complex impedance spectra of the per meability µ=µ′+iµ′′and\nof the electrical conductivity σ=σ′+iσ′′=σ′+iǫω(whereǫis the electrical permittivity\nandω= 2πfis the angular frequency) where performed with a HP 4195 impedanc e analyser\nbetween 100 kHz and 10 MHz. The losses where measured in a vacuum chamber between\n100 kHz and 2 MHz as described in ref.4,5from 10 second long temperature recordings.\nIn order to validate the interpretation of the measurements perf ormed with the two\ntechniques, an approach based on the solution of Maxwell equation s was used. For this\n3(a)\n (b)\nFIG. 1. Total loss field wtot(x,y) calculated from Eq. (4) and (5) for f= 100kHz (a) and f=2MHz\n(b). The dashed rectangle indicates the region used to calcu late the surface loss as measured by the\nplatinum probe. The thickness of the rectangle is defined as t he product of the heat propagation\nspeed in the material and the measure duration (10 seconds). Grey scales indicate the local loss\nin mJkg−1where the loss varies from 0 .04mJkg−1(dark regions) to 0 .05mJkg−1(white regions)\nin (a) , and from 0 .45mJkg−1to 0.85mJkg−1in (b) . The ratio between the maximum and the\nminimum values is 1 .25 atf= 100kHz (a) and 2 at f=2MHz (b)\npurpose, the code FEMM6was used taking measurements of complex magnetic permeability\nand conductivity as intrinsic material parameters. All the following a nalysis is based on\nthree strong assumptions: ( i) the peak induction values are sufficiently low to consider the\nmagnetic behaviour aslinear. Thus theconcept ofreal andimaginar y magnetic permeability\nand their relationship with energy losses is valid. This assumption is con firmed by the fact\nthat allthe lossmeasurements presented inthispaper scale rough lyasthesquare of thepeak\ninduction. Thismeansthattheterm µ′′/|µ|2remainsconstantintheinvestigatedinduction\nrange (from 3mT to 50mT); ( ii) all the measurements of complex magnetic permeability are\nperformed on a toroidal ferrite sample with section small enough to make eddy current loss\nand skin effect negligible; ( iii) all the measurements of complex conductivity are performed\non bar shaped samples which section is sufficiently reduced to neglect skin effect.\nFrom (ii) the measured µis a parameter related to the local phase lag between HandB\ndue to all dissipative phenomena but eddy current losses. On the ot her hand, the complex\nelectrical conductivity σresults from the mixed electrical properties of the grain and of the\ngrain boundaries7. Its real part σ′, is related to ohmic and dielectric loss effect whereas the\nimaginary part σ′′, is due to the effect of mixed permittivity.\n4FIG. 2. Comparison between the bulk measurements of the aver age total loss and the calculated\nones. Peak inductions are: 3mT, 10mT, 30mT, 40mT and 50mT\nFIG. 3. Comparison between bulk and surface loss measuremen ts and the same for computations\n(over the whole volume and the one delimited by the dashed lin e in Fig.1. The peak induction is\nof 50mT\nGiven the geometry of the problem, with the magnetic field always per pendicular (along\nzaxis) to the core section, the calculation of the loss field reduces to a two dimensional\nproblem in the xyplane. Now, under sinusoidal excitation when Hz(x,y,t) =H0(x,y)eiωt,\nsubstituting the measured σandµin the diffusion equation for the magnetic field we have,\n1\nµ▽2Hz(x,y,t) =iωσH0(x,y). (3)\n5From this equation and from the fact that j=▽×Hwe can calculate the eddy-current\nloss field across the core section,\npec(x,y) =1\n2ρ′|j(x,y)|2(4)\nwhereρ′=Re(1/σ). This expression includes all the eddy current related losses. The\neffect of ohmic and dielectric dissipation in both, the grains and their b oundaries, are de-\nscribed by the real part of the resistivity ρ′. On the other hand, the complex part of the\npermeability µ′′, measured on the very small sample, will account for all the magnet isation\nprocess related dissipation phenomena ( excess losses, spin dampin g, etc.). So we can write\nthe magnetic loss field as:\npmag(x,y) =1\n2ωµ′′|Hz(x,y)|2=1\n2ωµ′′\n|µ|2|Bz(x,y)|2(5)\nThe total power loss will be the sum of the two terms of Eq. (4) and ( 5), and the loss per\ncycle and unit mass will be wtot(x,y) =ptot(x,y)/(fρm). By averaging over the core section\nand multiplying by the length, we obtain /angbracketleftW/angbracketrightv=ℓ/integraltexta/2\n−a/2/integraltexta/2\n−a/2w(x,y)dxdycorresponding to\nthe bulk (resistance) measurement, whereas /angbracketleftW/angbracketrights=ℓ/integraltexta/2\n0/integraltextd\n0w(x,y)dxdyyields the value\ncorresponding tosurface(PT500)measurement. Thethickness disdefined astheproductof\nthe heat diffusion velocity in the material and the durationof the mea surement (10 seconds).\nIII. RESULTS AND DISCUSSION\nIn order to determine the magnetic field Hz(x,y) across the section area we must specify\nthe induction conditions under which losses have been both calculate d and measured. Here\nwe consider the case where a sinusoidal magnetic induction of avera ge value /angbracketleftB0/angbracketrightis forced\nacross the core section area S. This implies that the magnetic field verifies:\n/angbracketleftB0/angbracketright=1\nS/integraldisplay\nSµH0(x,y)dS (6)\nThe magnetic field Hz(x,y) has been determined solving Eq.(3) coupled with Eq.(6) with\na two dimensional finite element code in a frequency range between 1 00kHz and 2MHz on\nthe sample geometry. Calculation results are shown in Fig. 1 where th e loss field wtot(x,y)\nis plotted for f= 100kHz in Fig.1(a) and f= 2MHz in Fig.1(b).\n6In Fig.2 comparison between the bulk measurements of loss and the c alculated average\ntotal loss /angbracketleftW/angbracketrightvis shown for peak inductions ranging from 3mT to 50mT. The agreeme nt be-\ntween measurements andcalculationsisparticularlygoodforfrequ encies above300kHz. The\nover estimation of computed losses may be due to the compensation error in the measure-\nment ofµ′′performed with the impedance analyser. Indeed, at low frequency the resistive\npartoftheimpedance isnearlyzero, soeven asmall absoluteerror canyieldastrong system-\natic error. Fig.3 shows the comparison between the surface measu red loss and corresponding\ncomputation /angbracketleftW/angbracketrights. Calculations and measurements are in good agreement. With the sa m-\nple geometry studied here, as the average magnetic path is not muc h different from the\nextremal paths in the sample, all the loss inhomogeneities can be asc ribed to the skin effect.\nSo, eithermeasurements andfiniteelements calculations, showtha tskineffect isnegligiblein\na frequency regime below 600-700kHz. Above these frequencies s kin effect becomes relevant,\nand a largepart of thedissipation phenomena is concentrated near the surface of the sample.\nIn conclusion, it has been shown that the calorimetric loss measurem ent method, exploiting\nbulk and surface temperature recordings, is able to give indirect inf ormation on the eddy\ncurrent losses in Mn-Zn ferrites. Using FEM computation –for which material parameters\ninput where obtained by impedance spectroscopy– it is possible to co mpute the loss fields\nand to obtain very good agreement with measurements without intr oducing any fitting pa-\nrameter. Hence, this model could be integrated in magnetic compon ents design software in\norder take into account the magnetic losses including eddy current s for the dimensioning.\nREFERENCES\n1T. Sato, and Y. Sasaki, IEEE Trans. Magn. 23, 2593 (1987)\n2H. Saotome, and Y. Sakaki, IEEE Trans. Magn. 33, 728 (1997)\n3F. Fiorillo, M. Co¨ ısson, C. Beatrice, and M. Pasquale, J. Appl. Phys. 105, 07A517 (2009)\n4V. Loyau, M. LoBue, F. Mazaleyrat, Rev. Sci. Instrum. 80, 024703 (2009)\n5V. Loyau, M. LoBue, F. Mazaleyrat, IEEE Trans. Magn. 46, 529 (2010)\n6Finite Element Method Magnetics, D. Meeker, freeware, http://ww w.femm.info\n7J. B. Goodenough, IEEE Trans. Magn. 38, 3398 (2002)\n7"    },    {        "title": "1910.06643v1.Achievements_and_New_Challenges_for_CERN_s_Digital_LLRF_Family.pdf",        "content": "ACHIEVEMENTS AND NEW CHALLENGES FOR CERN’S DIGITAL LLRF\nFAMILY\nM. E. Angoletta†, S. Albright, A. Findlay, V . R. Myklebust, M. Jaussi, J. C. Molendijk, N. Pittet,\nCERN, Geneva, Switzerland\nAbstract\nAn innovative digital Low-Level RF (LLRF) family has\nbeen developed at CERN and deployed on several circular\nmachines. Operation of CERN’s PS Booster (PSB), Low\nEnergy Ion Ring (LEIR) and Extra Low ENergy Antipro-\nton (ELENA) ring all reaped great benefit from the flexi-\nbility and processing power of this new family. Beam and\ncavity feedback loops have been implemented, as well as\nbunch shaping, longitudinal blowup and bunch splitting.\nFor ELENA, longitudinal diagnostics such as bunched\nbeam intensity and bunch length measurements have also\nbeen deployed. During Long Shutdown 2 (LS2) the ferrite-\nbased High-Level RF (HLRF) systems of the Antiproton\nDecelerator (AD) and of the four-ring PSB will be replaced\nwith Finemet-based HLRF. This will require a new LLRF\nsystem for the AD and deep upgrades to the existing PSB\nLLRF systems. This paper gives an overview of the main\nresults achieved by the digital LLRF family so far and of\nthe challenges the LLRF team will take on during LS2.\nCERN VXS DIGITAL LLRF OVERVIEW\nCERN’s VXS Digital Low-Level RF (LLRF) family is\nthe second generation digital LLRF for small synchrotrons\ndeveloped at CERN. The first generation, now obsolete,\noperated successfully the Low Energy Ion Ring (LEIR)\nmachine [1] and allowed carrying out machine tests in the\nProton Synchrotron Booster PSB [2] for over 10 years.\nThe VXS digital LLRF family is based on the VXS bus\nand on CERN-designed custom hardware [3]. This in-\ncludes: a) FMC-DSP carrier boards hosting up to two, high\npin count FPGA Mezzanine Card (FMC) daughtercards;  b)\nrear transition modules hosting the power supplies of the\ncarrier board and providing timings and interlocks input-\noutput capabilities ; c) VXS switch modules allowing the\nvarious carrier boards in a crate to communicate ; d) three \ntypes of FMC daughtercards providing the functions of\nMaster Direct Digital Synthesizer (DDS), Slave DDS and\nDigital Down Converter. Figure 1 shows a FMC-DSP car-\nrier board hosting two FMC daughtercards. The Field Pro-\ngrammable Gate Arrays (FPGAs) and the Digital Signal\nProcessor (DSP) are also shown.\nFigure 1: FMC-DSP carrier board.The VXS LLRF family typically runs synchrotrons with\nlow-frequency cavities (up to 20 MHz) hence it operates\nvia direct sampling and baseband I,Q processing.\nInitially based on a sweeping frequency clock, the LLRF\nis now being moved to fixed frequency operation [4]. This\nallows seamless handling of the frequency swing and im-\nproves the signal-to-noise ratio in the analogue-to-digital\nand in the digital-to-analogue conversion.\nThe family was successfully deployed on several CERN\nmachines [5-7] and on the synchrotron of the MedAustron\ncomplex for hadron therapy [8]. It was also adopted by\nCERN’s Beam Instrumentation group to implement orbit\nmeasurement systems [9, 10] and used for longitudinal di-\nagnostics by CERN’s Radiofrequency (RF) group [11]. Ta-\nble 1 summarises the main deployment milestones,\nachieved and planned, together with the staff involved. The\nused clocking scheme (fixed vs. sweeping) is also indi-\ncated. Finally, studies to develop the 3rd generation LLRF\nfamily will start in 2023, after the post-Long Shutdown 2\n(LS2) machines restart.\nTable 1: VXS digital LLRF family deployment milestones.\nKeys: MA - MedAustron; RF - CERN BE Radiofrequency\ngroup; BI - CERN BE Beam Instrumentation group.\nWhen What Who\n2014MedAustron LLRF (sweeping clock)MA,\nRF\nPSB 4 rings LLRF (sweeping clock) RF\n2016AD orbit BI\nLEIR LLRF upgrade to 2nd genera-\ntion LLRF (sweeping clock)RF\n2017ELENA orbit BI\nELENA LLRF\n(fixed frequency clock)RF\n2018LEIR orbit BI\nLEIR LLRF upgrade to fixed fre-\nquency clockRF\nELENA LLRF upgrade to include\nsome longitudinal diagnosticsRF\n2019 Long Shutdown 2\n2020PSB LLRF upgrade to full Finemet\nHLRF operation and Linac4 injection\n(fixed frequency clock)RF\nELENA LLRF upgrade to include\nfull longitudinal diagnosticsRF\n2021 AD LLRF and longitudinal diagnos-\ntics (fixed frequency clock)RF\n~2022 Beam loops implementation in PS\nLLRF (sweeping clock)RF\n≥2023 Start studies for 3rd generation LLRF RF\nPS BOOSTER\nThe Machine\nCERN’s PSB is a Large Hadron Collider (LHC) injector,\nwhich accelerates protons with four superposed rings and\nsupplies beam to an experimental area. It is being upgraded\n[12] as part of CERN’s LS2 activities.On the contrary, the PSB operated with a single-tone\nmodulation at high harmonic. Table 2 shows a comparison\nbetween the two methods, which will both be available af-\nter LS2.\nAnother achievement was operating with three harmon-\nics (h = 1+2+3) instead of two ( h = 1+2).\nFigure 2: Post-LS2 LLRF for one PSB ring. Keys: MDDS – Master Direct Digital Synthesiser, ADC – Analogue-to-\nDigital Converter, DAC – Digital-to-Analogue Converter, TPU  – Transverse Pick-Up, CTRV – timing receiver module,\nMEN A20 – Master VME board, RTM – Rear Transition Module.\nThe LLRF\nThe PSB was successfully equipped in 2014 with one in-\ndependent LLRF system per ring, based on the VXS LLRF\nfamily [5, 13]. These will be upgraded during LS2 to com-\nply with new operational specifications. In particular, each\nLLRF will operate the three new Finemet-based HLRF\nsystems per ring, that deliver up to 8 kV each in the (1 –\n20) MHz bandwidth. Figure 2 shows the layout and func-\ntionalities of the LLRF system for one ring [14].\nSelected Achievements\nMany new features were tested and reliability runs car-\nried out over the years, particularly in view of post-LS2\noperation [5, 13, 14]. Here two achievements are selected.\nTable 2: Comparison of PSB longitudinal blowup methods:\nsingle tone modulation at high harmonic and phase noise.\nHigh hPlusesEasy to track changing synchrotron\nfrequency fS\nFaster\nMinusesMinimum of 5 D parameters space\nRequires control of high harmonicPhase noisePlusesNo need for high h control\nSmaller parameters space\nTargets specific fS amplitudes\nMinusesMore difficult to track changing fS.\nSlower.\nA novel longitudinal blowup method by phase noise at\nthe accelerating harmonic was validated in the PSB [15].\nThis method was so far used only in larger CERN machines\nsuch as the Super Proton Synchrotron (SPS) and the LHC.This was enabled by the test Finemet HLRF system in-\nstalled in PSB Ring 4, that allowed operation also at har-\nmonic h=3. Beam tests showed that triple harmonic opera-\ntion improved the brightness of the LHC25 beam and re-\nduced the vertical emittance of the Multi Turn Extraction\n(MTE) beam. Figure 3 depicts the simulated bunch profile\nfor operation at triple harmonic, at standard double har-\nmonic and at double harmonic with extra voltage on h=2\n(double harmonic overloaded). The capability to control\nharmonic h=3 in voltage and phase is therefore a strong\nwish for the post-LS2 LLRF system.\nFigure 3: Simulated bunch profile for different harmonics\nused. Traces: triple harmonic [blue], double harmonic\noverloaded [orange], double harmonic standard [green].\nNew Challenges\nPost-LS2 challenges include operation with a new Btrain\nsystem, ring synchronisation for Linac4 injection and con-\ntrol of the Finemet-based HLRF systems. The last one is a\nmajor change as it includes implementing in the FPGA ser-\nvoloops at 16 harmonics for each HLRF system. This ap-\nproach has been the model for other accelerators [16].\nLEIR\nThe Machine\nCERN’s LEIR is an LHC injector that accumulates and \naccelerates ions; particles accelerated so far include O4+, \nAr11+, Xe39+ and Pb54+.The method allowed improving both the reproducibility \nand the beam transmission through the machine. Figure 5 \nshows the progress in LEIR’s extracted intensity in the last \nthree Pb54+ runs; the LEIR LLRF was instrumental in ob-\ntaining this result.\nFigure 4: LEIR LLRF schematic view. Keys: MDDS – Master Direct Digital Synthesiser; ADC – Analogue-to-Digital\nConverter; DAC – Digital-to-Analogue Converter; TPU – Transverse Pick-Up; CTRV – timing receiver module; MEN\nA20 – Master VME board; RTM – Rear Transition Module; CCI – Cavity Control Interface.\nThe LLRF\nLEIR began commissioning in 2005 equipped with the \n1st generation LLRF system [1]. It was upgraded to the 2nd \nLLRF generation with sweeping frequency clock operation \nin 2016 [7]. Finally, it was upgraded to fixed frequency \nclock operation in 2018. Figure 4 shows the LEIR LLRF \nlayout and functionalities implemented. In particular, the \nLLRF operates in double-harmonic mode (acceleration and \nshaping) using either of the two HLRF systems installed in \nLEIR. Both HLRF systems can be also driven in parallel \nfor machine development sessions. More details on the sys-\ntem operation are available elsewhere [7].\nSelected Achievements\nThe LEIR LLRF allowed many machine improvements \n[7]. Here three achievements are outlined. \nFirst, a novel method for capturing ion beams [17] was \nroutinely used for operation during the 2018 Pb54+ run. This \nconsists of a programmable modulation of the frequency \nduring the capture, embedded in the LLRF. \nFigure 5: Progress in LEIR’s extracted intensity for Pb54+\nion operation from 2015 to 2018.Second, a NOMINAL scheme operating at harmonics \nh=3+6 instead of h=2+4 was deployed in the 2018 run. \nWith this novel scheme, originally unplanned, three \nbunches were successfully accelerated, synchronised, ex-\ntracted (see Figure 6) and routinely sent through the ion \naccelerator chain to the LHC. This scheme will be the \nbackup operational mode in case of problems with post-\nLS2 SPS momentum slip stacking operation.\nFigure 6: Pb54+ NOMINAL bunches at extraction under the\nkickers. Traces: three bunches [brown], standard two\nbunches [black], extraction kickers [blue, green, violet].\nFinally, tests were made with triple harmonic operation \nat h=2+4+6 and by using two cavities in parallel. Inspired \nby the PSB experience, this method showed a transmission \nslightly higher than with standard double harmonic. It will \nbe further studied in future runs.\nNew Challenges\nOperation with the new Btrain system will be required \nafter LS2. The automatic generation of LLRF parameters \nand voltage functions will be integrated within the LEIR \ncontrols infrastructure. \nELENA\nThe Machine\nThe Extra Low ENergy Antiproton (ELENA) ring [18]\ndecelerates antiprotons injected from the Antiproton De-\ncelerator (AD). For setting up it can also accelerate/decel-\nerate H- ions and protons from a source. Its commissioning,\nstill under way, began in December 2016. In 2018 ELENA\nstarted delivering antiprotons to GBAR [19].\nThe LLRF\nFigure 7 shows a schematic view of the ELENA LLRF\nlayout and functionalities. In particular, the LLRF controls\na Finemet-based HLRF capable of delivering up to 500 V.\nMore details on the system are given elsewhere [6, 11].\nFigure 7: ELENA LLRF layout in 2018. Keys: MDDS –\nMaster Direct Digital Synthesiser, ADC – Analogue-to-\nDigital Converter, DAC – Digital-to-Analogue Converter,\nCTRV – timing receiver module, MEN A20 – Master\nVME board, RTM – Rear Transition Module, CCI – Cavity\nControl Interface, LPU/TPU – Longitudinal/Transverse\nPick-Up.\nSelected Achievements\nELENA’s LLRF allowed to carry out bunch-to-bucket\ntransfer from the AD, to decelerate antiprotons as well as\nH- ions and to synchronise at extraction the bunch(es). [6,\n11]. Here previously unpublished results are shown.\nThe H- beam was routinely accelerated and decelerated\nto setup the machine. Figure 8 shows the cycle and several\nLLRF signals such as the magnetic field, the phase and ra-\ndial loop contributions in units of Hz, the radial loop posi-\ntion in units of mm measured by the LLRF and received by\nthe orbit system over optical fibre.\nFigure 9 shows the bunch length measurement for an an-\ntiproton cycle as an example of the longitudinal diagnostics\ndeveloped for ELENA. As expected, the bunch length in-\ncreases during the acceleration and decreases on the extrac-\ntion plateau, also thanks to the bunched-beam cooling. Thediscontinuity in the bunch length on the first ramp is due to\na phase jump on the signal changeover between high-fre-\nquency and low-frequency LPU.\nFigure 8: Accelerated and decelerated H- beam .\nFigure 9: Antiprotons bunch length measurement. Traces:\nbunch length [pink], magnetic field [blue], voltage [green].\nNew Challenges\nThe longitudinal diagnostics will be expanded with the\nObsBox, a custom processing module based on a server\nPC, with high data rate optical interfaces and large storage\ncapabilities [20]. The resulting diagnostics will be exported\nfirst to the AD, then to other machines. The combination of\nsignals in the LPU will be corrected as a function of fre-\nquency. Finally, the control of the LLRF parameters will be\nintegrated within the RF cycle editor, to automatically\nadapt the parameters to different cycles.\nAD\nThe Machine\nCERN’s AD has been providing antiprotons to experi-\nments since July 2000. It is now being upgraded and con-\nsolidated [21].\nThe LLRF\nThe AD will restart after LS2 equipped with a custom-\nised copy of ELENA’s LLRF as its new digital LLRF sys-\ntem. This will provide beam and cavity control as well as\nlongitudinal diagnostics. Figure 10 shows the system lay-\nout and its functionalities. In particular, the LLRF will con-\ntrol a Finemet-based HLRF delivering up to 3500 V [20].\nChallenges\nThe AD will profit from ELENA’s LLRF features and\ncommissioning experience. The AD commissioning will\nhowever have to compete for manpower with the PSB\ncommissioning and the restart of other LHC injectors.\nFigure 10: Post-LS2 AD LLRF and longitudinal diagnostics. Keys: DDS –Direct Digital Synthesiser, ADC – Analogue-\nto-Digital Converter, DAC – Digital-to-Analogue Converter, SFP – Small Form-factor Pluggable Transceiver,  LPU/TPU\n– Longitudinal/Transverse Pick-Up, CTRV – timing receiver module, MEN A20 – Master VME board, RTM – Rear\nTransition Module, ObsBox – custom processing module, SPEC – Simple PCI Express Carrier module.\nCONCLUSION AND FUTURE WORK\nDeveloping CERN’s VXS LLRF family was a large RF\ngroup manpower investment, well repaid by synergies\namongst the various machines this LLRF operates. Its flex-\nibility and processing power has allowed to implement fea-\ntures not originally planned and to fully harness the wide-\nband characteristics of the Finemet HLRF. The RF group\nsuccessfully used this family on ELENA for longitudinal\ndiagnostics, which will be expanded and used by other ma-\nchines. Finally, studies for the 3rd generation LLRF family\nwill start in 2023, after the post-LS2 machines restart.\nACKNOWLEDGEMENTS\nWe are grateful to our colleagues in the operation and\nbeam physics groups for their support. The fruitful collab-\noration with HLRF colleagues is also acknowledged.\nREFERENCES\n[1] M. E. Angoletta et al., “CERN’s LEIR Digital LLRF: Sys-\ntem Overview and Operational Experience”, in Proc. IPAC’\n10, Kyoto, Japan, May 2010, pp. 1464-1466.\n[2] M. E. Angoletta et al., “CERN’s PS Booster LLRF Renova-\ntion: Plans and Initial Beam Tests”, in Proc. IPAC ’10 ,\nKyoto, Japan, May 2010, pp. 1461-1463.\n[3] M. E. Angoletta et al., “A Leading-Edge Hardware Family\nfor Diagnostics Applications and Low-Level RF in CERN’S\nELENA Ring”, in Proc. IBIC ’13 , Oxford, UK, September\n2013, pp. 575 – 578.\n[4] J. C. Molendijk, https://public.cells.es/work-\nshops/www.llrf2017.org/pdf/Orales/O-22.pdf\n[5] M. E. Angoletta et al., “Operational Experience with the\nNew Digital Low-Level RF System for CERN’s PS\nBooster”, in Proc. IPAC ’17 , Copenhagen, Denmark, May\n2017, pp. 4058 – 4061\n[6] M.E. Angoletta et al, “Initial Beam Results of CERN\nELENA’s Digital Low-Level RF System”, in Proc. IPAC\n’17, Copenhagen, Denmark, May 2017, pp. 4054 – 4057.\n[7] M. E. Angoletta at al., “The new LEIR Digital Low-Level RF\nsystem”, in Proc. IPAC ’17 , Copenhagen, Denmark, May\n2017, pp. 4062 – 4065.\n[8]https://www.medaustron.at/[9] R. Marco-Hernandez et al., “A New Orbit System for eth\nCERN Antiproton Decelerator”, in Proc. IBIC ’15 , Mel-\nbourne, Australia, September 2015, pp. 465 – 469.\n[10] O. Marqversen et al., “The Orbit Measurement System for\nthe CERN Extra Low Energy Antiproton Ring”, in Proc.\nIBIC ’15 , Grand Rapids, USA, August 2017, pp. 206 – 208.\n[11] M. E. Angoletta. M. Jaussi, J. C. Molendijk, “The New Dig-\nital Low-Level RF System for CERN’s Extra Low ENergy\nAntiproton machine”, in Proc. IP AC’19 , Melbourne, Aus-\ntralia, May 2019, pp. 3962 – 3965.\n[12] K. Hanke et al., “Status and Plans for the Upgrade of the\nCERN PS Booster”, in Proc. IP AC ’15 , Richmond, Virginia,\nUS, May 2015, pp. 3905 – 3907.\n[13] M. E. Angoletta et al., “Control and Operation of a Wide-\nband R System in CERN’s PS Booster” in Proc. IPAC ’17 ,\nCopenhagen, Denmark, May 2017, pp. 4050 – 4053.\n[14] M. E. Angoletta et al., “Upgrade of CERN’s PSB Digital\nLow-Level RF System”, in Proc. IPAC’19 , Melbourne, Aus-\ntralia, May 2019, pp. 3958 – 3961.\n[15] S. Albright, “Time Varying RF Phase Noise for Longitudinal\nEmittance Blow-up”, in Proc. IP AC’19 , Melbourne, Aus-\ntralia, May 2019, pp. 3954 – 3957.\n[16] F. Tamura et al. , “Multiharmonic Vector RF V oltage Control\nfor Wideband Cavities Driven by Vacuum Tube Amplifiers\nIn a Rapid Cycling Synchrotron”, Phy. Rev. Accell. Beams ,\nvol. 22, no. 9, p. 092001, Sept 2019.\n[17] S. Albright, M. E. Angoletta, “Frequency Modulated Cap-\nture of Cooled Coasting Ion Beams”, in Proc. IP AC’19 , Mel-\nbourne, Australia, May 2019, pp. 2356 – 2358.\n[18] T. Eriksson et al., “ELENA – From Installation to Commis-\nsioning”, in Proc. IP AC ’17 , Copenhagen, Denmark, May\n2017, pp. 3327 – 3330.\n[19] Mansoulié, B. & on behalf of the GBAR Collaboration Hy-\nperfine Interact (2019) 240: 11.\nhttps://doi.org/10.1007/s10751-018-1550-y\n[20] M. E. Angoletta, S. Albright, M. Jaussi, V . R. Myklebust, J.\nC. Molendijk, “New Low-Level RF and Longitudinal Diag-\nnostics for CERN’s AD”, in Proc. IP AC’19 , Melbourne,\nAustralia, May 2019, pp. 3966 – 3969.\n[21] T. Eriksson et al., “Upgrades and Consolidation of the\nCERN AD for Operation During the Next Decades”, in\nProc . IP AC’13 , Shanghai, China, May 2013, pp. 2654 –\n2656.\n"    },    {        "title": "1203.3924v1.Structural__magnetic__and_optical_properties_of_zinc__and_copper__substituted_nickel_ferrite_nanocrystals.pdf",        "content": "1 \n Structural,  magnetic , and optical properties of zinc - and copper - \nsubstituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani a; V. Daadmehr  a,*; A. T. Rezakhani b; R. Hosseini  \nAkbarnejad  a; S. Gholipour  a \na Magnet& Superconducting  Lab , Department  of Physics, Alzhara University, Tehran 19938 , Iran  \nb Department  of Physics , Sharif University of Technology, Tehran, Iran  \nAbstract  \nFerrite nanocrystals are interesting material due to their rich physical properties. Here we add \nnonm agnetic  dopants Zn and Cu to nic kel ferrite nanocrystals,  Ni1-xMxFe2O4 (0≤x≤1 , M{Cu, Zn }), \nand characterize how relevant properties of the samples are modified accordingly. Basically, these \ndopings cause a rearrangement of Fe+3 ions into the two preexisting octahedral and tetrahedral sites. In \nfact, this, we show, induces pert inent magnetic properties of the doped samples. In the case of the Cu -\ndoping, the Jahn -Teller effect also emerges, which we identify through the FTIR Spectroscopy  of the \nsamples.  Moreover, we show an increase in the lattice parameters of the doped samples,  as well a \nsuperparamagnetic behavior for the doped samples is shown, while the Jahn -Teller effect precludes a \nsimilar behavior in the CuFe 2O4 nanocrystals.  The influences of Zn and Cu substitutions are \ninvestigated on the optical properties of nickel ferr ite nanocrystals by photoluminescence \nmeasurement  at room temperature .    \n \n Keywords: ferrites 75.50.Gg, sol -gel processing 81.20.Fw, nan ocrystalline magnetic materials \n75.50.Tt, superparamagnetic 75.20. -g, magnetic anisotropy 75.30.Gw , Jahn-Teller effect  71.70.Ej  \n \n \n 2 \n  \n1. Introduction  \n     Spinel ferrites , with common  formula of MFe 2O4 (M: a divalent metal ion) , have wide \ntechnological  applications , e.g., in multilayer chip inductor (MLCI) , ferrofluids, high -speed \ndigital tape or recording disks, rod antenna , and humidity sensor  [1-9]. Ferrite nanocrystals \nare also of interest in various applications, such as inter -body drug delivery [10-12], \nbioseparation , and magnetic refrigeration systems  [13], in particular due to their specific \nproperties, such as  superp aramagnetism . In addition, among ferrospinels  zinc ferrites  are used \nin gas sensing [ 14, 15], catalytic application [1 6], photocatalyst [ 17, 18], and absorbent \nmaterials [ 19]. \n     The unit cell of spinel ferrite s is com posed of 32 oxygen atoms in cubic cl osed- packed \narrangement distributed in tetrahedral („ A‟) and octahedral sites („B‟). C hemical and \nstructural properties of spinel ferrite nanocrystals are affected by their compositions and \nsynthesis methods, and corresponding electric and magnetic proper ties depends on cation \nsubstitutions.  \n     Doping ferrite nanocrystals with various metals, such as chromium, copper, manganese, \nand zinc are usually used to improve some of their electric or magnetic properties  [20-22]. For \nexample, Zn/Ni ferrites have a pplications as soft magnetic materials with high frequency (due \nto high electrical resistivity and low eddy -current loss  [8]).  Along that line, (Cu, Zn)/Ni \nferrites offer a further improvement as softer magnetic materials [ 23]. \n     In this work, we study  effects of Zn /Ni and Cu /Ni substitutions  on nickel ferrite  \nnanocrystals synthesized  through the sol -gel method . Specifically, we characterize structural \nand magnetic properties of the doped samples at room temperature. We also exhibit the \nemergence of the  Jahn-Teller effect in the case of the CuFe 2O4 nanocrystals , and argue that \nthe modification of the structural properties is in principle due to the occupation of the A sites \nvs. B sites. In this exhaustive characterization, various techniques are employed  such as X -ray 3 \n diffraction (refined with the Rietveld method with the MAUD software ), Field -Emission  \nScannin g Electron Microscope (FE -SEM), Fourier Transform Infra -Red (FTIR) spectr oscopy,  \nVibrating Sample Magnetometer ( VSM), and Photoluminescence (PL) , an d UV -Vis \nspectrometer  at room temperature . We thus compare the effect of dopings not only regarding  \nstructural and magnetic properties , but also in the sense of optical properties.  In addition , here \nthe Jahn -Teller effect has is studied through the investi gation of the structural and magnetic \nproperties.  \n2. Experimental  details  \n2.1. Preparation method  \n    The sol -gel method is widely used in synthesis of ferrite nanocrystals because of its high  \nreaction rate, low preparation temperature , and production of small particles . Hence, in our \nexperiment, magnetic nanocrystalline Ni 1-xZnxFe2O4 with x=0, 0.3, 0.5, 0.7, 1 and              \nNi1-xCuxFe2O4 with x=0, 0.5, 1 were synthesized by this method. Citric acid C 6H8O7, ferric  \nnitrate Fe(NO 3)3.9H 2O (98%), nickel nit rate Ni(NO 3)2.6H 2O (99%) , zinc nitrate \nZn(NO 3)2.6H 2O (99%) and copper nitrate Cu(NO 3)2.3H 2O (99%) were produced by Merck® \ncompany and were used as raw materials. The stoichiometric amount of nitrate s and acid \ncitric were dissolved separately in deionized w ater to make  0.5M solutions. The mole ratio of \nmetal nitrates to citric acid was taken as 1:1. To obtain smaller nanocrystals, we added \nethylenediamine to the solution until its pH became  1. Next, the sol was heated continuously \nat 70 °C under st irring to f orm a br own dried gel. This gel was fire d at 135 °C in oven for 24h, \nand was then ground into powder. To obtain various Cu+2 and Zn+2 substitutions, we  calcined  \nthe powder at 300 -600°C for 4h. \n \n 4 \n 2.2. Measurements and c haracterization s  \n   The X-ray diffract ion patterns (XRD) of the synthesized nanocrystals were obtained using a \nPhilips® PW1800 X -ray diffracto meter  with Cu Kα radiation ( = 1.54056Å) operated at 40kV \nand 30 mA. The refinement method of Rietveld was applied with the “ Mate rial Analysis Using \nDiffraction” (MAUD) program ( v.2.056 ). Fourier Transform Infra -Red (FTIR) spectra of \nsamples were detected by a BRUKER®TENSOR27 FTIR spectrometer with transmission \nfrom 4000 to 400cm-1 using KBr pellets.  The average grain size and morphology of the \nsamples wer e observed by a Hitachi® S4160 Field Emission Scanning Electron Microscopy \n(FE-SEM). Photoluminescence (PL) measurement was performed by a VARIAN® CARY \nECLIPSE spectrometer (using a diode laser with the wavelength  of 430nm ). The optical \nabsorption spectra were recorded by a Perkin -Elmer® Lambada35 UV -Vis spectrometer.  The \nmagnetic properties of the samples were measured by Meghnatis Daghigh Kavir Co.® \nVibrating Sample Magnetometer (VSM) at room temperature. Note that  during such \nmeasurements t he maximum app lied magnetic field was 9 kOe.  \n3. Results and discussion  \n3.1. Structural studies  \n    The X -ray diffraction patterns of the synthesized ferrite nanocrystals have been  shown in \nFigs. 1 and 2. The existence of the (220), (311), (400), (422), (511) and (440) ma jor lattice \nplanes in  the XRD patterns confirm s the formation of spinel cubic structure with the Fd3m \nspace group , which is  consistent with the powder diffraction file of JCPDS. Also the presence \nof the (111), (222), (331), (533), (622), (444), (642) and ( 731) minor lattice planes in the \nXRD patterns agree s well with the powder diffraction of spinel cubic JCPDS file . All samples \nare considered to be single -phase spinel structure . In addition, as Fig.  2 shows , the XRD \npatterns of the Ni0.5Cu0.5Fe2O4 and CuFe 2O4 nanocrystals confirm that  almost  all Cu atoms 5 \n have been placed in the  cubic spinel structure . However, note that in some previous studies, \nthere are reports of excess phases, e.g., the CuO phase [24] or tetrahedral CuFe 2O4 phase [25].  \nSpecifically, i t has been reported in Ref. [24] that both cubic spinel structure and CuO phase \nwere observed in the Ni0.5Cu0.5Fe2O4 nanocrystals  (with pH 3 -4) when the calcina tion process  \nlasted for 3 h at 600 °C, while only the single cubic spinel structure was formed when the \ncalcination temperature was up to 1000 C. In the case of  our experiment,  the Ni0.5Cu0.5Fe2O4 \nnanocrystals  (with pH=1)  were calcined at 375 C for 4h, and crystallize d in the single cubic \nspinel structure with  almost  no excess phases.  \n    The average cr ystallite sizes D are calculated from the characteristics of the (311) XRD -\npeaks through the Scher rer formula in the range of 1.4 -19.6nm (Table s 1 and 2 ). Note that, \nsimilar to the case of Zn/Ni ferrites prepared by the combustion -reaction method [2 6], her e an \nincrease in the Zn and Cu contents does in fact increase the particle size. The existence of \nbroad peaks in the XRD pattern of our synthesized nickel ferrite nanocrystals can be \nattributed to the small crystallite size of them (see the inset of Fig. 1 ). \n    The X -ray diffraction data of the synthesized ferrite nanocrystals has been  refined by using \nthe MAUD software and Rietveld‟s method for the structural analysis, cation distribution and \nlattice parameter calculations. The obtained lattice parameters  for the nanocrystals are listed \nin Table s 1 and 2. It is seen that the lattice parameters increase for larger  Zn and Cu contents. \nThis increase can be attributed to larger ionic radiuses of Cu2+ (0.72Å)  and Zn2+ (0.82Å)  \nrelative to  Ni2+ (0.69Å) ; this is c onsistent with Refs. [2 4,27]. In addition,  because the ionic \nradius of Zn2+ is larger than the ionic radius of Cu2+, the Zn2+ substitution leads to larger \nexpansion of the lattice; thus the lattice parameter increase s more in comparison to the Cu2+ \nsubstit ution in the synthesized nanocrystals. Also, the shift of  the (311) XRD  peak to small \ndiffraction angle with the increase in the Zn and Cu content s may be attributed to this point \nthat the sample with higher Zn and Cu content s has a larger lattice paramete r. These shifts are 6 \n larger  for the Ni 1-xZnxFe2O4 nanocrystals because of larger ionic radius of Zn2+ in comparison \nto Cu2+.  \n    To ensure that spine l structure for the synthesized nanocrystals has been formed, as well to \ninvestigate how the ferromagnetic ions ( Fe3+ ,Ni2+) and no nmagnetic transition metal ions \n(Cu2+ ,Zn2+) occupy tetrahedral and octahedral sites, we refined the XRD data by employing \nthe MAUD software. Table 3 shows the before - and after -refinement values, indicating \nrelatively high accuracy  in synthesizing the nanocrystals without formation of extra phases.  \n    Based on the XRD data refinement, the formation of single -phase spinel cubic structure \nwith Fd3m space group has been confirmed  in all samples. A dditionally, these results indicate \nthat the synthesized nickel ferrite (NiFe 2O4) and copper ferrite (CuFe 2O4) nanocrystals have \ninverse spinel structure in which half of the Fe3+ ions spatially  fill the  tetrahedral sites and the \nrest occupy the octahedral sites with the Ni2+ ions in nickel fe rrite and the Cu2+ ions in copper \nferrite nanocrystals . Generally, an inverse spinel ferrite can be represented by th e formula \n[Fe3+]tet[A2+, Fe3+]octO42- (A= Ni, Cu ,..), where the \"tet\" and \"oct\" indices represent the \ntetrahedral and octahedral sites, res pectively. Likewise, these results specify that the \nsynthesized zinc ferrite (ZnFe 2O4) nanocrystals ha ve a normal spinel structure in which all \nZn2+ ions fill tetrahedral sites , hence the Fe3+ ions are forced to occupy all of  the octahedral \nsites. As a res ult, this compound can be displayed by the formula [Zn2+]tet[Fe 23+]oct O42-.  \nAlso these results indicate  that the synthesized Zn/Ni ferrite (0 <x<1) and Cu/Ni ferrite \n(Ni 0.5Cu0.5F2O4) nanocrystals have a mixed spinel structure where  the M2+ (M= Zn, Cu) \ncations are substituted for the Fe3+ ions in the tetrahedral sites , and these Fe3+ ions  move  to \nthe octahedral sites, as shown by the gray color in Table 3. In other words , in these \nnanocrystals the M2+ (M= Zn, Cu) ions occupy the tetrahedral sites with the Fe3+ ions  and the \nwhile the rest  of  the Fe3+ ions fill the octahedral sites with the Ni2+ ions. Thus, these samples \ncan be represented by the formula [M x2+ ,Fe1-x3+]tet [Ni 1-x2+ ,Fe1+x3+]oct O42-. All of these results 7 \n for the structural analysis and cati on distribution calculations for synthesized Zn/Ni ferrites \nare in agreement with the structure and cation distribution for bulk spinel ferrites of Refs. \n[28,29]. Table 3 represents the occupancy of the Zn2+ and Cu2+ (for the Ni0.5Cu0.5F2O4 \nnanocrystals ) contents in the tetrahedral sites, formation of the inverse spinel structure  for \nnickel and copper ferrites,  formation of normal spinel structure for zinc ferrites , and formation \nof mixed spinel structure for the other nanocrystals . Also the cation distribu tion of  the \nprepared nanocrystals are summarized in Table s 1 and 2, and Fig. 3 represents the  XRD \npatterns  for the ZnFe 2O4 nanocrystals .  \n    The pattern fitness can be checked for XRD data. The re are several parameters for the \ncalculation of pattern fitne ss. The goodness of fit (S) is described by S= R wp /Rexp,  where R wp \nis the weighted residual error and R exp is the expected error. Refinement has been continued \nuntil a convergence  was reached for a value of S close to 1 . which confirms the goodness of \nrefinement. These parameters are listed in Table 4.  \n    For Cu doped samples, we have also tested  a mixed scenario that put s the copper into both \nsublattices (for example the cation distribution for CuFe 2O4 nanocrystals  can be displayed by \nthe formula [Cu y2+, Fe 1-y3+]tet [Cu 1-y2+, Fe 1+y3+]oct , where 0≤y≤1 with step 0.1 ) and thus refined \nthe distribution of the copper in the tetrahedral and octahedral sites and also the distribution \nand the positions of the other ions by using the MAUD software. With refinement of the XRD \ndata for our Cu doped samples,  the MAUD software obtained that for 0<y≤1, it is unable to \nsolve the refinement with  the supplied initial parameters , because the correlation matrix from \nthe Choleski decomposition has negative diagonal. Thus we found that for our synthesized \nNi0.5Cu0.5Fe2O4 nanocrystals, all copper ions go into the tetrahedral sites so that  the cation \ndistribution for this sample can be displayed by the formula [Cu 0.52+, Fe 0.53+]tet [Ni 0.52+, \nFe1.53+]oct. Also for our synthesized CuFe 2O4 nanocrystals , all copper ions go i nto the \noctahedral sites thus the cation distribution can be displayed by the formula [Fe3+]tet [Cu2+ , 8 \n Fe3+]oct. Note that the FTIR results are consistent with the MAUD analysis for the cation \ndistribution.  In addition, this  cation distribution for Cu -doped samples explains well thier  \nmagnetic properties.                                                \n \n3.2. FTIR study  \n     The FTIR spectra of the synthesized nanocrystals,  shown in Fig s. 4 (a) a nd (b ), confirm the \nformation of the spinel structures  suggeste d by the MAUD analysis. The bands around 3415 \nand 1621cm-1 are due to the O-H stretching vibration of the free or absorbed water . This \nindicates existence of hydroxyl groups in our synthesized ferrites , which is apparently seen in \nearlier experiments too  [30, 31]. The band at 1380cm-1 is attributed to the C=O stretching \nvibration of the carboxylate group (CO2-), and the band around 1099cm-1 is related  to the NO - \nstretching vibrati on due to nitrate group  [32, 33]. These bands disappear when Cu and Zn \ncontent s (and thus calcina tion temperature) increased.   \n     In the range of 800 -400cm-1, two main absorption bands were observed in the range of \n604-542cm-1 and 463-416cm-1. These bands are attributed to the vibration of the tetrahedral \nand octahedral metal -oxygen (M-O) bonds in the lattices of the synthesized nanocrystals , \nrespectively. These FTIR frequency bands for various Zn and Cu content s have been listed in \nTables  5 and 6 . It is understood from  these tables that the value of 1 and 2 are found to \ndecreas e and increase , respectively, with the increase in the Zn and Cu content s except for the  \nCuFe 2O4 sample.  \nThese results can be explained  as fllows. The addition of Zn2+ and Cu2+ (from x=0 to x=0.5 \nfor Ni/Cu ferrites) ions in the tetrahedral sites with larg er radius and greater atomic weight, \nmakes the Fe3+ ions migrate to the octahedral sites , and consequently decreases the tetrahedral \nvibration frequency. Similarly , migration of the Fe3+ ions to the octahedral site increases the \noctahedral vibration freque ncy. For the CuFe 2O4 nanocrystals, the increase of the tetrahedral 9 \n vibration frequency and the decrease of the octahedral vibration frequency may be due to the \npresence of half of the Fe3+ ions in the A sites and presence of the Cu2+ ions with larger radiu s \nand greater atomic weight in the B sites.  There is a weak splitting around octahedral vibration \nfrequency ( 2) of the CuFe 2O4 nanocrystals that can be attributed to the existence of Cu2+ \nions, a Jahn -Teller ion [3 4, 35], in the octahedral sites , and causes some  lattice distortion . This \ndistortion normally takes the form of str etching the bonds to the ligand  lying along the z axis, \nbut sometimes happens as a shortening of these bonds as well . This affects the octahedral Cu -\nO vibration frequency in the CuFe2O4 nanocrystals.  Note that all of these results are also \nconsistent with the MAUD analysis for the cation distribution.                                              \n3.3. Morphological study  \n    The FE -SEM images of the CuFe 2O4, Ni0.5Zn0.5Fe2O4 and ZnFe 2O4 nanocrystals have been \nshown in Figs.  5(a)-(c). The average grain size of the CuFe 2O4, Ni0.5Zn0.5Fe2O4 and ZnFe 2O4 \nnanocrystals are 15, 20, and 30nm, respectively.  Note that t he average grain size of the \nsamples obtained from SEM images is larger than nan ocrystals size as calculated using the \nXRD measurements , which simply indicates that each grain is formed by aggregation of a \nnumber of nanocrystals.  The samples are spherical and uniform , and cohesion of grains is due \nto the magnetic attraction.  \n3.4. Magn etic study  \n    The magnetic properties of the synthesized nanocrystals are analyzed using a \nMagnetometer (VSM) at room temperature. Figures 6(a) and (b) show the M-H curves  of the \nprepared Ni 1-xMxFe2O4 nanocrystals. The saturation magnetization (M s) and co ercivity (H c) \nvalues have been directly extracted from these curves and have been listed for various Zn and \nCu contents in Tables 1 and 2. These Tables indicate that Ms increases while increasing the \nZn/Ni and Cu/Ni contents up to x=0.5 (after which Ms decreases); i.e., the maximum value of 10 \n Ms is found in the Ni0.5M0.5Fe2O4 (M=Zn, Cu) nanocrystals. The latter increase is higher in \nthe Zn/Ni substitution against the Cu/Ni substitution.  \n    The saturation magnetization of the synthesized nickel ferrite and zi nc ferrite nanocrystals \nhave been  obtained at room temperature from the hysteresis loops as 6.4 and 1.7 emu/g,  \nrespectively (Table s 1 and 2). These values are less than the corresponding values for bulk \nnickel and zinc ferrites (50 and 5emu/g, respectively)  [36, 37]. This observation can be \nattributed to the presence of small nanoparticles having core-shell morphology, containing a \nspin-glass -like surface layer and ferrimagnetically lined -up core spins. As a result , the canting \nand disorder of the surface la yer spins may be due to broken superexchang e bonds , and unlike  \nlocal symmetry for tho se atoms near the surface layer  lead to a smaller M s in these \nnanocrystals. These effects are especially dominant  in the case of ferrites due to the presence \nof superexcha nge interactions . In smaller ferrite nanoparticles  with larger surface/volume \nratio, these effects are strongest . The M S of the synthesized NiFe 2O4 nanocrystals (with 1.4 nm \ncrystallite size ) is very smaller than the value for the bulk.   \n    According to Ne el‟s ferrimagnetic theory, in the s pinel structure  the cations on different \nsublattices (tetrahedral and octahedral sites) have oppositely aligned magnetic moments [ 31, \n38]. Hence,  the magnetic moment per formula unit (nB) in the µB units is: \nnB = M oct – Mtet                                  (1) \nwhere, Moct and M tet are the magnetic moments of the octahedral and tetrahedral  sites, \nrespectively.  The magnetic behavior of our synthesized nanocrystals measured by VSM  can \nbe attributed to the competition of ferr omagnetic ions such as Fe3+, Ni2+ and the Cu2+ and Zn2+ \nions as non -magnetic transition metal ions in the occupancy of the tetrahedral and octahedral \nsites. This can be explained by  the obtained cation distribution from the MAUD analysis of \nXRD patterns  (summarized in Tables 1 and 2 ), and Neel‟s theory with the knowledge of ionic \nmagnetic moment of  0, 1, 2, and 5 µB for Zn2+, Cu2+, Ni2+ and Fe3+ ions, respectively.  11 \n     It is evident from Table s 1 and 2  that the saturation magnetization of the synthesized \nnanocrystals has been enhance d with the increase in Zn and Cu content s up to x=0.5 . This is \nconsistent with the obtained cation distribution the MAUD analysis of the XRD patterns. The \noccupancy of Zn2+ and Cu2+ substituted ions in the tetrahedral sites cause s transfer  of Fe3+ \nions from there to the octahedral sites. This  increase s the concentration of Fe3+ ions in \noctahedral sites and so increase s the magnetization in the octahedral sublattice . This in turn \nenhances the magnetization of the nanocrystals.  Thes e results agree well with the bulk sample  \nin that the maximum M s is found in the Ni0.5Zn0.5Fe2O4 for Zn/Ni bulk ferrites. Table s 1 and 2 \nindicate  that the increase in M s for Zn/Ni ferrite is more than Cu/Ni ferrite nanocrystals . This \ncan be explained by th e total spin moment of the tetrahedral site and octahedral site \ninteractions. Due to placement  of Zn2+ and Cu2+ ions in the tetrahedral sites, the magnetization \nof the B sublattice is expected to be identical for Ni0.5Zn0.5Fe2O4 and Ni 0.5Cu0.5Fe2O4 \nnanocry stals, while for the A sublattice it increase  with substitution of higher magnetic \nmoment of Cu2+ (1µB) in comparison to Zn2+ (0µB). Consequently , because  of the \nantiferromagnetic coupling, the result is a decrease in the net overall magnetic moment (Eq.1)  \nof the Ni0.5Cu0.5Fe2O4 relative to the Ni0.5Zn0.5Fe2O4 nanocrystals. In addition,  higher \ncalcination temperature for  the Ni0.5Zn0.5Fe2O4 (450C) in contrast  to the Ni0.5Cu0.5Fe2O4 \n(375C) nanocrystals implies  larger grain size and higher crystalline quali ty in the ferrites, \nthereby lead s to higher M s value.  \n    It is observed from Table s 1 and 2 that  with the increase  in the Zn and Cu content s from \nx=0.5 to x= 1, MS shows a decrease.  This reduction is remarkable for  the Zn/Ni substitution. \nThe decrease of MS for the Ni0.3Zn0.7Fe2O4 nanocrystals is referred to  as spin canting  in the \nsurface layer of nanoparticles.  When the Zn content increase s to x=0.7, the occupation of Zn2+ \nions in the tetrahedral sites decrease s the number of spins occupation in the A sit es, thus the 12 \n A-B superexchange interactions are weakened and the spines of the octahedral sites ca nnot \nhold collinearity to the little tetrahedral spins . Therefore,  the M s value decreases.   \n    Based on the MAUD analysis  and the VSM measurements in Fig. 6 (a), the small  Ms of \nZnFe 2O4 can be attributed to the absen ce of the Ni2+ ions as ferromag netic ions and the \noccupancy of Zn2+ ions with zero magnetic moment in all tetrahedral sites , i.e., replacing the \ncomplete Fe3+ ions in the octahedral sites . Consequ ently , the interactions between Fe3+ ions in \nthe octahedral sites are too weak to be important (the B -B interaction is weak in ferrites), \nhence  the magnetization of ZnFe 2O4 nanocrystals  is small . Also, with the increase in the Cu \ncontent from x=0.5 to x=1,  a small  reduction is observed in the saturation magnetization. To \nexplain this behavior,  we can a rgue that based on the MAUD analysis , the Cu2+ ions move \nfrom the tetrahedral to octahedral sites, which have smaller ionic magnetic moment in \ncontrast with t he Ni2+ ions as ment ioned above , and the half of Fe3+ ions fill the A sites, so \nreduce the super exchange interactions between octahedral and tetrahedral sites, which cause \nthe decrease in the M s value. But since the Jahn -Teller effect occurs, this decrease  is small.  \nThe existence of Cu2+ ion as a Jahn -Teller ion in the octahedral sites of the CuFe 2O4 \nnanocrystals causes a lattice distortion , which has the effect of removing the electronic and \norbital degeneracies of Cu2+ cations. It  is anticipated t hat this  effect in turn  create s large \nstrains in t he copper ferrite lattice , and as a result modified  magnetic properties. Hence , this \ncauses a reduction  in saturation magnetization from x=0.5 to x= 1 to be smaller.  Note that t he \nJahn-Teller theorem does not predic t the direction of the distortion ; it only marks the \nexistence of unstable lattice geometry.  \n     Since the coercivity (H c) of a magnetic material is a measure of its magneto -crystalline \nanisotropy, the  small nano particles , which have  close to zero coerciv ity and no remanence  \nbecome single domain  with little anisotropy energy.  This is a characteristic of \nsuperparamagnetic nanocrystals.  The coercivity (H c) of the synthesized nanocrystals has been  13 \n obtained from Fig s.6 (a) and (b), and is listed in Table s 1 and 2. Note that the H c value s of all \nprepared nanocrystals  are close to zero , except CuFe 2O4 nanocrystals  (Tables 1 and 2) . Thus  \nthese  nanocrystals show superparamagnetic behavior  at room temperature . Such properties \nmake these materials  favorable for wide engineering applications such as drug delivery  [39], \nbioseparation [40] and magnetic refrigeration systems  [41].  \n     The H c of the samples increase s with the  Zn/Ni and Cu/Ni substitution s. This increase  is \nsmall  for Zn/Ni ferrite nanocry stals, while it is significant for Cu/Ni ferrite nanocrystals. This \nincrease can be attributed to higher  magneto -crystalline anisotropy of Cu2+ in comparison to \nNi2+ and Zn2+ ions (hence  the Jahn-Teller effect ). Although the  CuFe 2O4 nanocrystals have \naverage crystallite si ze smaller than 10nm , they have large coercivity,  as a result of complete \nsubstitution of the Cu2+ to the Ni2+ ions in the octahedral sites  and the occurrence of the Jahn-\nTeller effect . The existence of Cu2+ as a Jahn -Teller ion in the octahedral sites of the CuFe 2O4 \nnanocrystals causes the lattice distortion which is anticipated to create large strains in  the \ncopper ferrite lattice. This  increase s the anisotropy  and coercivity of this sample , which  agrees \nwell with the FTIR analysis of the CuFe 2O4 nanocrys tals (see Table 6 ). Likewise, this  effect \nshows its small  influence on the coercivity in Ni 0.5Cu0.5Fe2O4 as a result of that the Cu2+ ions \nfill the tetrahedral sites . Thus the Jahn-Teller effect is much less noticeable in tetrahedral \ncomplexes . \n3.5. Optica l properties      \n    Photoluminescence spectroscopy is measured at room temperature to detect the optical \nproperties of all synthesized nanocrystals for various substitutions of the Zn2+ and Cu2+ ions \n(Figs.  7(a) and 7 (b)). Five clear emission bands have b een observed in the entire PL spectra \nfor these nanocrystals. The emission bands , around 541.94 and 518.93 and 460.89nm are \nconstant with the increase in the Zn and Cu content , and are a ttributed to  the 3d5 3d3 4s2 14 \n transitions of Fe3+ ions. We remark that  in Ref. [ 42] similar transitions at 530 and 470nm have \nbeen reported for ZnFe 2O4 thin films.   \n    In addition,  the intensity of the emission bands at 541.94  and 518.93 nm vary with the \nincrease in the Zn/Ni substitution. The  dependence of these transition  intensit ies of Fe3+ ions \non the Zn content (x) is depicted in Fig. 8. It is evident that these intensities  increase  with the \nincrease in the Zn content except for x=0.5. This may be explained  as follow s: i) Based on the \nMAUD analysis, with the increase in the Zn2+ concentration and leaving of the Ni2+ cations in \nthe structure, Zn2+ ions occupy the tetrahedral sites and transfer the Fe3+ to the octahedral \nsites. ii) These changes cause a decrease in the structural isotropy of the synthesized \nnanocrystals except for the Ni0.5Zn0.5Fe2O4 nanocrystals . iii) The nanocrystals  have biggest \nsaturation magnetization among the  synthesized nanocrystals and  have a structural isotropy in \nthe tetrahedral sites . iv) The occupation of half of the tetrahedral sites by the Zn2+ ions and the \nrest by the  Fe3+ ions causes  the decrease in transitions of the Fe3+ ions in  the tetrahedral sites.  \n     The broad emission bands around 660 and 484nm are attributed  to the indirect and direct \nband gaps  for our synthesized nanocrystals , respectively.  The values of the indirect and direct \nband gaps are listed for various substitutions of the Zn and Cu content in Table s 5 and 6 . In \nparticular, there are two minor peaks detected by the spectrometer around 660nm and 484nm, \npresenting, respective ly, the indirect and direct band gaps (E=hc/ λ). Admittedly the accuracy \nof this estimation is not high.  The minor peaks around 660nm for Ni 1-xCuxFe2O4 nanocrystals \nare displayed next to  Fig. 7 (b). However, to confirm our results, we also performed a UV -Vis \nexperiment for all samples. Figure 9 show s the optical absorption spectra for our  Ni1-\nxZnxFe2O4 nanocrystals in the UV region. The optical band gap E g can be determined by the \nequation αEp=A(Ep‒Eg)q [43], where  Ep is the photon energy, A is a constant that depends on \nthe transition probability an d q depends on the nature of the transition and is theoretically  \nequal to 2 and 1/2 for allowed indirect and direct electronic transition, respectively. The 15 \n connection  of the linear fits of (α Ep)1/q versus Ep plots for q=1/2 and 2 (shown in Fig. 10) on \nthe Ep-axis determines the direct and indirect band gaps, respectively.  The obtained indirect \nand direct band gaps from the UV-Vis spectra are consistent with these results from  the PL \nmeasurements.  \nNote that t he value of the indirect band gap increase s with the increase in the Zn content, \nwhile  it is almost constant with the Cu content . The values of the direct band gap are constant \nfor various substitutions of Zn and Cu  content. The value of the indirect band gap is affected  \nby various factors such as crys tallite size, structural parameter, and presence of impurities . \nThe increase in the indirect band gap for our synthesized Ni1-xZnxFe2O4 nanocrystals may be \nattributed to the more increase in the structural parameter (lattice constant) with the increase \nin the Zn concentration in comparison to the Cu content  (Tables 1 and 2) .  \n    The obtained indirect band gap values for  our synthesized Zn/Ni nanocrystals (1.85 –\n1.90eV) are higher than the determinant values for the Zn/Ni  ferrite nanocomposite s (1.50 –\n1.66eV)  in Ref . [44], which  is due to presence of impurities . Note that t he obtained values of \nthe indirect and direct band gap s for our prepared ZnFe 2O4 nanocrystals are hi gher than  \nestimated values of the direct and indirect band gaps for zinc ferrite nanowires  [45], which are \n2.23 and 1.73eV, respectively  and are similar to  the ZnFe 2O4 films [46].  \n4. Conclusion      \n   We have investigated experimentally how properties of spin el nickel ferrites are affected \nbecause of doping by Cu and Zn. We have prepared these  samples through the sol -gel \nmethod, a nd performed various structural , magnetic, and optical measurements on them.  The \ncrystallite size of synthesized nanocrystals  was calculated using Scherrer ‟s formula in the \nrange of 1. 4-19.6nm. The X -ray diffraction me asurements, refined with the Rietveld method, \nhave indicated formation of a single -phase cubic spine l with the space group Fd3m, and \ncation distributions. In particular, the  analysis of  X-ray diffraction has showed that the 16 \n structural and magnetic properti es are induced by competition of ferromagnetic ions Ni+2 and \nFe+3 and nonmagnetic ions Zn+2 and Cu+2 ions in occupation of the tetrahedral  and octahedral  \nsites. Moreover it has been shown that, as in the bulk case, Zn+2 cations have preferred to \nsubstitute  for Fe+3 ions in the tetrahedral sites. Magnetic measurements through VSM \nindica ted an increase in the saturation  magnetization (MS) with the Zn/Ni and Cu/Ni content \nup to x=0.5  and a decrease for x>0.5. We have argued that this is due to an increase in t he \nconcentration of the Fe+3 ions in the octahedral sites.  This increase in M S is the highest for the \ncase of Ni0.5Zn0.5Fe2O4 nanocrystals.  All samples showed  superparamagnetism behavior  \nexcept CuFe 2O4 nanocrystals.  \n    We have also observed two broad emis sion bands in the PL spectra . 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Phys. 105 (2009)  \n07A512.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 19 \n \n \nFig. 1. XRD patterns of synthesized Ni 1-xZnxFe2O4 nanocrystals: (a) NiFe 2O4, (b) Ni 0.7Zn0.3Fe2O4 , (c) \nNi0.5Zn0.5Fe2O4, (d) Ni 0.3Zn0.7Fe2O4 , (e) ZnFe 2O4 (  : Bragg reflection positions)  \n \nFig. 2. XRD patterns of synthesized Ni 1-xCuxFe2O4 nanocrystals: (a) NiFe 2O4, (b) Ni 0.5Cu0.5Fe2O4, (c) CuFe 2O4   \n(  : Bragg reflection positions)  \nStructural, magnetic, and optical properties of zinc - and copper - substituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakhani; R. Hosseini Akbarnejad ; S. Gholipour  20 \n \n \nFig. 3. XRD pattern r efinement s using MAUD software for  ZnFe 2O4 nanocrystals ( : experimental data, upper \nsolid line: calculated pattern, lower solid line: subtracted pattern)  \n \n \nFig. 4(a). FTIR spectra of Ni 1-xZnxFe2O4 nanocrystals at room temperature: (a) NiFe 2O4, (b) Ni 0.7Zn0.3Fe2O4 , (c) \nNi0.5Zn0.5Fe2O4, (d) Ni 0.3Zn0.7Fe2O4 , (e) ZnFe 2O4 \nStructural, magnetic, and optical properties of zinc - and copper - substituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakhani; R. Hosseini Akbarnejad ; S. Gholipour  21 \n \n \nFig. 4(b). FTIR spectra of Ni 1-xCuxFe2O4 nanocrystals at room temperature: (a) NiFe 2O4, (b) Ni 0.5Cu0.5Fe2O4, (c) \nCuFe 2O4 \n \nFig. 5(a). FE -SEM image of synthesized CuFe 2O4 nanocrystals  \nStructural, magnetic, and optical properties of zinc - and copper - substituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakhani; R. Hosseini Akbarnejad ; S. Gholipour  22 \n \n \nFig. 5(b). FE-SEM image of synthesized Ni 0.5Zn0.5Fe2O4 nanocrystals  \n \nFig. 5(c). FE -SEM image  of synthesized ZnFe 2O4 nanocrystals  \n \n \n \n \n \nStructural, magnetic, and optical properties of zinc - and copper - substituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakhani; R. Hosseini Akbarnejad ; S. Gholipour  \n 23 \n \n \nFig. 6(a). Hysteresis curves of Ni 1-xZnxFe2O4 nanocrystals at room temperature: (a) NiFe 2O4, (b) Ni 0.7Zn0.3Fe2O4, \n(c) Ni 0.5Zn0.5Fe2O4, (d) Ni 0.3Zn0.7Fe2O4 , (e) ZnFe 2O4 \n \nFig. 6(b). Hysteresis curves of Ni 1-xCuxFe2O4 nanocrystals at room temperature: (a) NiFe 2O4, (b) Ni 0.5Cu0.5Fe2O4, \n(c) CuFe 2O4 \n \nStructural, magnetic, and optical properties of zinc - and copper - substituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakhani; R. Hosseini Akbarnejad ; S. Gholipour  24 \n \n \nFig.7 (a). Photoluminescence spectra of Ni 1-xZnxFe2O4 nanocrystals at room temperature: (a) NiFe 2O4, (b) Ni 0.7Zn0.3Fe2O4, \n(c) Ni 0.5Zn0.5Fe2O4, (d) Ni 0.3Zn0.7Fe2O4, (e) ZnFe 2O4 \n  \nFig. 7(b). Photoluminescence spectra of Ni 1-xCuxFe2O4 nanocrystals at room temperature: (a) NiFe 2O4,                (b) \nNi0.5Cu0.5Fe2O4, (c) CuFe 2O4 \n \nStructural, magnetic, and optical properties of zinc - and copper - substituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakhani; R. Hosseini Akbarnejad ; S. Gholipour  25 \n  \n \nFig. 8. Variation of the transi tion intensity of Fe3+ ions versus  the Zn content in Ni 1-xZnxFe2O4 nanocrystals  \n \n \n \nFig. 9.  UV -Vis spectra  of Ni 1-xZnxFe2O4 nanocrystals at room temperature: (a) NiFe 2O4, (b) Ni 0.7Zn0.3Fe2O4, (c) \nNi0.5Zn0.5Fe2O4, (d) Ni 0.3Zn0.7Fe2O4, (e) ZnFe 2O4 \n \n \nStructural, magnetic, and optical properties of zinc - and copper - substituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakhani; R. Hosseini Akbarnejad ; S. Gholipour  26 \n \n \n \nFig 10. Plots of ( αEp)2 and (αE p)1/2 versus E p for Ni 1-xZnxFe2O4 nanocrystals at room temperature: (a) NiFe 2O4, \n(b) Ni 0.7Zn0.3Fe2O4, (c) Ni 0.5Zn0.5Fe2O4, (d) Ni 0.3Zn0.7Fe2O4, (e) ZnFe 2O4 \n \n \n \nStructural, magnetic, and optical properties of zinc - and copper - substituted nicke l ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakhani; R. Hosseini Akbarnejad ; S. Gholipour  \n \n \n 27 \n  \n \n \n \n \n \n \nTable1. Crystalline size, lattice parameter, saturation magnetization, and coercivity of Ni1-xZnxFe2O4 nanocrystal s \n \n \n \n \n \n \nCoercivity  \n(Oe) Saturation \nmagnetization \n(emu/g)  Lattice \nparameter \n(Ǻ) Crystalline \nsize (nm)  Cation distribution  Composition  Cu \ncontent \n(x) \n0.6 6.4 8.340  1.4 (Fe 1.03+)tet[Ni 1.02+, Fe 1.03+]oct NiFe 2O4 0 \n2.3 16.2 8.355  4.6± 0.1 (Cu 0.52+, Fe 0.53+)tet[Ni 0.52+Fe1.53+]oct Ni0.5Cu0.5Fe2O4 0.5 \n168.2  14.9 8.370  5.8± 0.1 (Fe 1.03+)tet[Cu 1.02+, Fe 1.03+]oct CuFe 2O4 1 \nTable2. Crystalline size, lattice parameter, saturation magnetization , and coercivity of  Ni1-xCuxFe2O4 nanocrystals  \n \n \n \n \n \n \n \n \n \n \n \nStructural, mag netic, and optical properties of zinc - and copper - substituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakhani; R. Hosseini Akbarnejad ; S. Gholipour  \n \n \n \n Coercivity  \n(Oe) Saturation \nmagnetization \n(emu/g)  Lattice \nparameter \n(Ǻ) Crystalline size \n(nm)  Cation distribution  Composition  Zn  \nconte nt \n(x) \n0.6 6.4 8.340  1.4 (Fe 1.03+)tet[Ni 1.02+, Fe 1.03+]oct NiFe 2O4 0 \n0.7 25.6 8.370  6.2±0.1 (Zn 0.32+, Fe 0.73+)tet[Ni 0.72+Fe1.33+]oct Ni0.7Zn0.3Fe2O4 0.3 \n0.9 34.8 8.389  9.8 ± 0.1 (Zn 0.52+, Fe 0.53+)tet[Ni 0.52+Fe1.53+]oct Ni0.5Zn0.5Fe2O4 0.5 \n1.0 28.4 8.400 13.5±0.2 (Zn 0.72+, Fe 0.33+)tet[Ni 0.32+Fe1.73+]oct Ni0.3Zn0.7Fe2O4 0.7 \n3.3 1.7 8.430  19.6± 0.4 (Zn 1.02+)tet[Fe 1.03+, Fe 1.03+]oct ZnFe 2O4 1 28 \n  \nTable  3. Structural parameters before and after refinement fro m MAUD analysis for Ni\n1-xM\nxFe\n2O\n4 nanocrystals  \n \nStructural, magnetic, and optical properties of zinc - and copper - substituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakhani; R. Hosseini Akbarnejad ; S. Gholipour  Structural parameters after refinement  Structural parameters before refin ement  Composition  \nQuantit\ny Occupancy  Position   \n \nIons Quantit\ny Occupa\nncy Position   \n \nIons  \n \n  \nZ(Å)  \n  \nY(Å)   \nX(Å)   \nZ(Å)  \n  \nY(Å)  \n  \nX(Å)  \n8.00002  1.00002  2.310-7≈ 0 1.810-7≈ 0 1.510-7≈ 0 Fe3+\n(tet) 8 1 0 0 0 Fe3+\n(tet)  \n \nNiFe 2O4 \n 8.00002  0.50003  0.62536  0.62540  0.62511  Fe3+\n(oct) 8 1 0.625  0.625  0.625  Fe3+\n(oct) \n8.00003  0.50002  0.62513  0.62544  0.62541  Ni2+\n(oct) 8 0.5 0.625  0.625  0.625  Ni2+\n(oct) \n32.0002  1.00003  0.382504  0.382508  0.382502  O2- 32 1 0.3825  0.3825  0.3825  O2- \n         \n  \n   \n5.60003  0.700018  4.710-6≈ 0 4.910-6≈ 0 4.910-6≈ 0 Fe3+\n(tet) 5.6 0.7 0 0 0 Fe3+\n(tet)  \n \nNi0.7Zn 0.3Fe2O4 2.40003  0.300024  3.510-6≈ 0 5.510-6≈ 0 6.210-6≈ 0 Zn2+\n(tet) 2.4 0.3 0 0 0 Zn2+\n(tet) \n10.4001  0.6500002  0.6250193  0.6249957  0.6248848  Fe3+\n(oct) 10.4 0.65 0.625  0.625  0.625  Fe3+\n(oct) \n5.60004  0.3500001  0.624773  0.6250003  0.6248851  Ni2+\n(oct) 5.6 0.35 0.625  0.625  0.625  Ni2+\n(oct) \n32.0006  1.000018  0.382515  0.382522  0.382521  O2- 32 1 0.3825  0.3825  0.3825  O2- \n            \n4.00071  0.500008  4.210-6≈ 0 8.210-6≈ 0 4.410-6≈ 0 Fe3+\n(tet) 4 0.5 0 0 0 Fe3+\n(tet)  \n \nNi0.5Zn 0.5Fe2O4 4.00002  0.500004  5.910-6≈ 0 3.410-6≈ 0 9.410-6≈ 0 Zn2+\n(tet) 4 0.5 0 0 0 Zn2+\n(tet) \n11.9999  0.749999  0.6250024  0.625014  0.625183  Fe3+\n(oct) 12 0.75 0.625  0.625  0.625  Fe3+\n(oct) \n4.00002  0.249999  0.6250033  0.625434  0.624531  Ni2+\n(oct) 4 0.25 0.625  0.625  0.625  Ni2+\n(oct) \n32.0003  1.00001  0.3825001  0.3825001  0.3824999  O2- 32 1 0.3825  0.3825  0.3825  O2- \n            \n2.40003  0.300002  6.310-6≈ 0 7.610-6≈ 0 3.410-6≈ 0 Fe3+\n(tet) 2.4 0.3 0 0 0 Fe3+\n(tet)  \n \nNi0.3Zn 0.7Fe2O4 \n \n \n 5.58996  0.699993  6.310-6≈ 0 7.710-6≈ 0 2.210-6≈ 0 Zn2+\n(tet) 5.6 0.7 0 0 0 Zn2+\n(tet) \n13.6004  0.850004  0.625051  0.625001  0.624888  Fe3+\n(oct) 13.6 0.85 0.625  0.625  0.625  Fe3+\n(oct) \n2.40057  0.150003  0.624859  0.62513  0.625004  Ni2+\n(oct) 2.4 0.15 0.625  0.625  0.625  Ni2+\n(oct) \n32.0006  1.000002  0.3825055  0.382511  0.382488  O2- 32 1 0.3825  0.3825  0.3825  O2- \n            \n8.00065  1.00008  1.110-6≈ 0 1.110-6≈ 0 1.110-6≈ 0 Zn2+\n(tet) 8 1 0 0 0 Zn2+\n(tet)  \nZnFe 2O4 16.0001  1.00001  0.625302  0.65002  0.625001  Fe3+\n(oct) 16 1 0.625  0.625  0.625  Fe3+\n(oct) \n31.9999  0.999999  0.382505  0.382515  0.382504  O2- 32 1 0.3825  0.3825  0.3825  O2- \n            \n4.00001  0.50001  3.710-6≈ 0 3.110-6≈ 0 1.310-6≈ 0 Fe3+\n(tet) 4 0.5 0 0 0 Fe3+\n(tet)  \n \nNi0.5Cu 0.5Fe2O4 \n \n \n 4.00002 0.50002  3.110-6≈ 0 2.510-6≈ 0 1.110-6≈ 0 Cu2+\n(tet) 4 0.5 0 0 0 Cu2+\n(tet) \n11.9999  0.74999  0.6250024  0.625014  0.625183  Fe3+\n(oct) 12 0.75 0.625  0.625  0.625  Fe3+\n(oct) \n3.99999  0.24999  0.6250033  0.625434  0.625031  Ni2+\n(oct) 4 0.25 0.625  0.625  0.625  Ni2+\n(oct) \n32.0001  1.000003  0.3825012  0.3825001  0.382499  O2- 32 1 0.3825  0.3825  0.3825  O2- \n            \n8.0002  1.00002  1.810-7≈ 0 4.210-7≈ 0 1.610-7≈ 0 Fe3+\n(tet) 8 1 0 0 0 Fe3+\n(tet)  \n \nCuFe 2O4 \n 8.00001  0.50001  0.62566  0.62548  0.62541  Fe3+\n(oct) 8 0.5 0.625 0.625  0.625  Fe3+\n(oct) \n7.9999  0.49999  0.62532  0.62573  0.62513  Cu2+\n(oct) 8 0.5 0.625  0.625  0.625  Cu2+\n(oct) \n31.9999  0.999999  0.382505  0.382515  0.382504  O2- 32 1 0.3825  0.3825  0.3825  O2- 29 \n  \n \ncomposition  Rwp Rp Rexp S \nNiFe 2O4 0.118  0.110  0.095  1.24  \nNi0.7Zn0.3Fe2O4 0.131  0.103  0.103  1.27  \nNi0.5Zn0.5Fe2O4 0.113  0.108  0.098  1.15  \nNi0.3Zn0.7Fe2O4 0.114  0.108  0.101  1.13  \nZnFe 2O4 0.117  0.102  0.097  1.20  \nNi0.5Cu0.5Fe2O4 0.104  0.083  0.084  1.22  \nCuFe 2O4 0.089  0.069 0.073  1.21  \nTable 4. The parameters for the calculation of pattern fitness for Ni\n1-xM\nxFe\n2O\n4 nanocrystals  \n \nZn \ncontent \n(x)  \nComposition   \nFTIR frequency bands (cm-1) \n \n      A- site (tet)                     B -site (oct)  \n            1                                                      2  \nIndirect band \ngap (eV)   \nDirect band  \ngap (eV)  \n0 NiFe 2O4 604.33  425.66  1.85 2.55 \n0.3 Ni0.7Zn0.3Fe2O4 590.21  426.11  1.87 2.56 \n0.5 Ni0.5Zn0.5Fe2O4 578.28  426.60  1.87 2.56 \n0.7 Ni0.3Zn0.7Fe2O4 571.55  426.63  1.88 2.56 \n1 ZnFe 2O4 542.37  463.14  1.90 2.56 \nTable5. FTIR absorption band frequencies and optical parameter of Ni1-xZnxFe2O4 nanocrystals  \n \nCu  \ncontent  \n(x) Composition  FTIR frequency bands (cm-1) \n \n        A- site (tet)                         B-site (oct)  \n 1                                  2                                      2                     \nIndirect band \ngap (eV)   \nDirect band  \ngap (eV)  \n0 NiFe 2O4 604.33  425.66   1.85 2.55 \n0.5 Ni0.5Cu0.5Fe2O4 574.69  428.13   1.87 2.55 \n1 CuFe 2O4 585.86  422.22  416.61  1.87 2.55 \nTable 6. FTIR absorption band frequencies and optical parameter of Ni1-xCuxFe2O4 nanocrystals  \n \nStructural, magnetic, and optical properties of zinc - and copper - substituted nickel ferrite nanocrystals  \nF. Shahbaz Tehrani ; V. Daadmehr*; A. T. Rezakha ni; R. Hosseini Akbarnejad ; S. Gholipour  \n 30 \n  \nShahbaz Tehrani, Fatemeh  \nB. Sc.  \nAlzahra University, Tehran 19938, Iran  \nM. Sc. student  \nAlzahra University, Tehran 19938, Iran  \nE-mail: tehrani66@gmail.com  \nResearch intere sts: \nFerrits, Nanocatalysts, Magnetic Nanoparticles, Carbon Nanotubes  \n \n \n \n \n \nCorresponding author: Daadmehr, Vahid  \nDirector, Magnet & Superconducting Research Laboratory  \nAssociate Professor of Physics  \nDepartment of Physics, Alzahra University, Tehran 19938 , Iran  \nTel: (+98 21) 85692640 / (+98) 912608 9714  \nFax: (+98 21) 88047861  \nE-mail:  daadmehr@alzahra.ac.ir  \nWeb: http://  www.alzahra.ac.ir/ daadmehr/  \nResearch interests:  \nCondensed matter physics, Material science, nanoferrites, Carbon Nanostructures, High \nTemperature Superconductors: Electrical properties, Nanocrystals, Y -based cuprates.  \n \n \n \n \nRezakhani, Ali T.  \nAssistant Professor of Physics  \nDepartment o f Physics, Sharif University of Technology, Tehran, Iran  \nTel: (+98 21) 6616 4523  \nFax: ( +98 21) 6602 2711  \nE-mail:  rezakhani@sharif.edu  \nPersonal Page:  http://sharif.edu/~re zakhani  \nResearch interests:  \nQuantum Information Science, Quantum Computation, Dynamics of Open Quantum Systems, \nCondensed matter -Superconductivity.  \n \n \n \n31 \n  \nHosseini Akbarnejad, Razieh  \nB. Sc.  \nTehran University, Tehran 19938, Iran  \nM. Sc. student  \nAlzahra University, Tehran 19938, Iran  \nE-mail: rh.akbarnejad@gmail.com  \nResearch interests:  \nCarbon Nanotubes, Nanocatalysts, Magnetic Nanoparticles, Ferrites  \n \n \n \nGholipour, Somayyeh  \nB. Sc.  \nAlzahra University , Tehran 19938, Iran  \nM. Sc. student  \nAlzahra University, Tehran 19938, Iran  \nEmail: gholipour65@gmail.com  \nResearch interest:  \nFerrits, Nanocatalysts, High Temperature Superconductors: Electrical \nProperties, Nanocr ystals, Y -based cuprates.  \n \n \n \n \n \n \n \n \n"    },    {        "title": "1002.1087v2.Polarization_transformations_by_a_magneto_photonic_layered_structure_in_vicinity_of_ferromagnetic_resonance.pdf",        "content": " 1Polarization transformations by  a magneto-photonic layered \nstructure in vicinity of  ferromagnetic resonance \nVladimir R Tuz1,2, Mikhail Yu Vidil2, and Sergey L Prosvirnin1,2 \n1School of Radio Physics, Karazin Kharkiv National University, 4, Svobody Square, \nKharkiv 61077, Ukraine  \n2Institute of Radio Astronomy of National Academy of Sciences of Ukraine, 4, \nKrasnoznamennaya st., Kharkiv 61002, Ukraine \nE-mail:  Vladimir.R.Tuz@univer.kharkov.ua , vidil@rian.kharkov.ua  and \nprosvirn@rian.kharkov.ua  \nAbstract \nPolarization properties of a magnetophotonic layered structure are studied at the frequencies \nclose to the frequency of ferromagnetic resonance. The investigations are carried out taking \ninto account a great value of dissipative losses in biased ferrite layers in this frequency band. \nThe method is based on analysis of solution stability of ordinary differential equation system \nregarding the field inside a periodic stack of dielectric and ferrite layers. The \nelectromagnetic properties of the structure are ascertained by the analysis of the eigenvalues \nof the transfer matrix of the structure period. The frequency boundaries of the stopbands and \npassbands of the eigenwaves are determined. The frequency and angular dependences of the \nreflection and transmission coefficients of the stack are presented. Frequency dependences \nof a polarization rotation angle and ellipticity of the reflected and transmitted fields are \nanalyzed. An enhancement of polarization rotation due to the periodic stack is mentioned, in \ncomparison with the rotation due to some effective ferrite slab. \nKeywords: multilayer, ferrite, stopband, polarization, absorption  \n(Some figures in this article are in color only in the electronic version) \n \n1. Introduction \nElectromagnetic properties of materials that have artificially created periodic translation symmetry differ \nsignificantly from those of the homogeneous media. There is a direct analogy between the wave processes \nin such structures and the properties of the wave functions of electrons moving in the periodic potential of \na crystal lattice. The translation symmetry signifi cantly affects the spectrum of eigenwaves of such \nmaterials. There are alternating bandwidths in whi ch propagation of electromagnetic waves is either \npossible (passbands) or forbidden (stopbands). Due to these properties, lately, it is conventional to refer \nsuch one-, two- or three-dimensional periodic structures composed of substantial index contrast elements \nto photonic crystals (PCs). The forbidden bandwidths are called photonic band gaps (PBG) [1, 2]. Note \nthat in the one-dimensional case a PC is nothing more than a dielectric periodic layered structure. We use \nthis notation trough this paper to emphasize the contrast of such structures to 2D or 3D periodic PC [3]. \nThe PCs are now widely used in modern integrated optics and optoelectronics, laser and X-ray \ntechniques, microwave and optical communications. \nFrom the viewpoint of applications, it is obvious that not only design of PCs but also control of the \nposition and width of the band gap is of a great interest. One of the ways to realize the control is using \nmagnetic materials in fabrication of PCs to produce so-called magneto-photonic crystals (MPCs) [4-6]. \nIndeed, a biased external static magnetic field can a lter permittivity or permeability of MPC ingredients. \nIn addition to the possibility to control properties of the PCs, the MPCs manifest some unique magneto-\noptical properties accompanied by the Kerr and Farada y rotation enhancement arising from the effect of \nlight localization as a result of wave interference within the magnetic structure. Owing to these features, \nMPCs have already found several electronic applications as isolators and circulators and magneto-optic  2spatial light modulators. We call these multilayered one-dimensional structures as magneto-photonic \nlayered structures (MPLS), unlike 2D or 3D periodic MPCs. \nAccording to classical electromagnetics, the ferromagnetic properties of media are related to a spin \nmagnetic moment [7, 8]. In the presence of an external static magnetic field, the electron spin precess \naround the field with a frequency 0ω, that is called as the frequency of ferromagnetic resonance. The \nphenomenon of the ferromagnetic resonance plays a crucial role, because it determines largely the \nmagnitude of dissipative losses, Faraday rotation and the nonreciprocity of medium. Therefore, the \nproperties of a MPLS depend strongly on the proximity of an incident wave frequency to the frequency of \nthe ferromagnetic resonance. Thus a wave propaga ting through MPLS experiences its attenuation \ndepending on the ratio of the wave frequency ω to the resonant frequency0ω. In particular, the wave \ndecay will reach maximum at the frequency 0ωω=. In spite of the fact that the magnetic losses depend \non the frequency, in most studies of MPLSs the frequency of a propagating wave is chosen quite far from \nthe resonance frequency 0ω, i.e. a special situation of infinitesimally small dissipative losses is under \nconsideration [4-6].  \nIn addition to a strong dissipative attenuation, the resonance magnetic losses affect also the \npolarization characteristics of MPLS. For an observer looking in the direction of the biased magnetic \nfield, electron spins precess clockwise [7, 8]. Thus prec essing electron spins have different effect on the \nright-handed and left-handed circularly polarized wave  propagated along the biased field. Through the \npaper we use the optical definition of a circular polarization. We assume the wave has the right-handed \n(left-handed) circular polarization, if its vector of electric field rotates clockwise (anticlockwise) to an \nobserver looking opposite to the wave propagation direction. Only for a left-handed circularly polarized \nwave propagated along biased magnetic field, the resonance phenomenon occurs causing the difference of \neffective complex magnetic permeability related to the right-handed and left-handed polarized waves. It \nyields the magnetic rotation of the polarization plane of the linearly polarized wave, which has great \npractical interest.  \nIn most papers, the normal wave incidence is considered and the transversal or longitudinal \nmagneto-optic configuration of the biased external static magnetic field is chosen to study the \nelectromagnetic properties of MPLSs. In the transv ersal biased field configuration, the electromagnetic \nwave can be presented as TE− and TM−waves [7, 10-12, 14], and in the longitudinal one, as the right-\nhanded and left-handed circularly polarized waves [8, 10, 11]. In either case these modes are uncoupled ones and the solution of electromagnetic wave propagation problem is described via a \n22× transfer \nmatrix formulation. Generally when a MPLS consists of ferromagnetic layers with arbitrary orientation of \nthe anisotropy axes or the wave impinges obliquely, the modes are right-handed and left-handed elliptically \npolarized, and it needs to use a 44× transfer matrix [6, 13, 15-17]. \nThe essence of the transfer matrix method consists in derivation of the matrix to relate the tangential \nfield components at the beginning and the end of the structure period and thereafter treatment of this \nmatrix to study both eigenwaves and fields reflected from and transmitted through MPLS. As it is well \nknown, the transfer matrix of the period of the layered structure is a product of the transfer matrices of its \nseparate layers. After the transfer matrix of the period  is found, further investigation is based on the \nanalysis of the eigenvalues of this matrix since they are directly related to the properties of eigenwaves of \nthe periodic structure. \nBy definition, the eigenwaves of a periodic structure is non-trivial solutions of the homogeneous \nMaxwell equations that satisfy the conditions of the Floquet’s quasi-periodicity [18] \n(, ,) e x p ( i ) (, , )xyz L xyz L γ Ψ= Ψ +GG\n, (1) \nwhere ΨG\n is some linear function of the field components, L is the structure period. The condition (1) \nexpresses the intuitive notion about waves in a periodic structure: in the neighboring periods the field \ndiffers only by a certain phase factor γ. This parameter γ is a wavenumber related to a certain \neigenwave. They are named as the Bloch wavenumber and the Bloch wave, respectively. It is obvious that  3the Bloch wave exists in any section of an infinite periodic medium. For lossless structures, the parameter \nγ takes either purely real or purely imaginary values in the passbands and stopbands, respectively. \nWhen the material losses are taken into consideration, the formal solution of the dispersion equation \nleads to complex values of γ (// /iγγγ=+ ). In this case an exponential decay of the field follows from \nthe condition (1), which gives a contradiction to the definition of the Bloch waves in an infinite periodic \nstructure. Thus it is necessary to initially refuse th e imposition of the condition (1). Instead, the standard \nmethod of the theory of irregular waveguides [18] can be applied. In the context of this theory it is \nassumed that the eigenwaves of the irregular waveguide with impedance sidewalls are orthogonal in \nenergy terms. It means that every eigenwave propagates independently from the others inside the area free \nof sources. Thus the eigenwaves have clear physical meaning: it is the field that can be excited in the \nwaveguide outside the area occupied by the sources. The method is based on obtaining a system of \nordinary differential equations and further analysis of the stability of solutions of this system. In simple \nterms, the concept of stability is associated with the analysis of behavior of small deviations from the \ntrivial solution of the differential equation [19]. In the case of periodic media, the corresponding system \nof equations contains periodic coefficients, and, from a mathematical point of view, the transfer matrix \ndefines the fundamental solutions of the system on a dedicated interval. In particular, as in our present \nwork, the electromagnetic properties of the structure under study are found by the analysis of the \neigenvalues of the transfer matrix, as these eigenvalue s describe the stability of solutions of the system. \nIn the present paper, we focus on the study of electromagnetic properties of a finite one-\ndimensional periodic layered structure that consists of normally biased ferromagnetic and isotropic layers. \nA distinctive feature of this work is a study of the properties of MPLS in a frequency range of the \nferromagnetic resonance. In particular the polarization transformations are under investigation. Our goal \nis to derive the solution of the problem of electromagnetic wave propagation in the MPLS with \ndissipative losses typical over the frequency range close to the frequency of ferromagnetic resonance. In our theory treatment, we deny a prior choice of solution in the common form (1) and employ the general theory of differential equations in order to include other possible forms of solution into consideration. We \nalso study peculiarities of field transformations by MPLS in the case of oblique incidence of \nelectromagnetic wave. \n \n2. Problem formulation \nA periodic stack of N identical double-layer slabs placed along z-axis is investigated, see Fig. 1. We \nassume that the structure is isotropic and infinite in the x- and y- directions. Each slab consists of a ferrite \nlayer and a homogeneous isotropic dielectric layer. The ferrite layer is magnetized up to saturation by \nan external static magnetic field 0HG\n directed normally to the layer. Constitutive parameters of ferrite \nlayer are 1ε, 1ˆμ and its thickness is 1d. The dielectric layer has a thickness 2d, permittivity 2ε, and \npermeability 2μ. Thus the thickness of the slab of periodic structure is 12 Ld d=+. \nThe outer half-spaces 0z≤ and zN L≥  are homogeneous, isotropic and have constitutive parameters \n0ε, 0μ and 3ε, 3μ, respectively. Let us suppose that an incident field is a plane wave of a frequency ω \nand its direction of propagation, in the region 0z≤, is determined by angles 0θ and 0ϕ relative to z-axis \nand x-axis respectively (see Fig. 1). Time dependence is assumed exp( ) itω−  through the paper. \nWe use common expressions for constitutive parameters of normally biased ferrite taking into \naccount the magnetic losses [20] \n1 fεε=,      1\n11\n1i0\nˆ i0\n00T\nT\nLμα\nμα μ\nμ⎛⎞\n⎜⎟=−⎜⎟\n⎜⎟⎝⎠, (2)  4\n \nFigure 1.  (color online) A periodic stack of double-layer slabs composed of normally magnetized ferrite and \nisotropic dielectric layers  \nwhere // /\n11iTμχχ=+ + , /2 2 2 1\n00 (1 )m bD χω ω ωω−⎡⎤=− −⎣⎦, // 2 2 2 1\n0 (1 )mbb Dχω ωωω−⎡⎤=+ +⎣⎦, \n// /iα=Ω +Ω , /2 2 2 1\n0 (1 )m bD ωω ω ω−⎡⎤Ω= − +⎣⎦, // 2 1\n0 2mbDωωω−Ω= , 222 2 2 2 2\n00 (1 ) 4 D bb ωω ω ω⎡⎤=−+ +⎣⎦, \nand typical parameters in the microwave region are 10fε=, 11Lμ=, 0.05b= , 02 4GHzωπ= , \n2 5.6GHzmωπ= . The value mω corresponds to saturation magnetization of 2000 G [20]. The frequency \ndependences of the permeability parameters are presented in Fig. 2. Note, the values of ImTμ and Imα \nare so close to each other that the curves of their frequency dependences coincide in the figure. \n \nFigure 2.  (color online) Frequency dependences of permeability parameters of along to z-axis biased ferrite layer.  \n \n3. Formalism of ×44  transfer matrix  \nLet us consider the incidence of a plane monochromatic wave on a layered structure in which permittivity \nε and permeability ˆμ are scalar and tensor piecewise constant functions of the coordinate z \nrespectively. In Cartesian coordinates the system of Maxwell's equations for each layer  has a form  5()\n()\n()ˆ ii , i i ,\nˆ ii , i i,\nˆ ii i , i ii .yy\nyz x y zx\nxx\nxz y y zy\nxy yx z x y y xzHEkH k E kE k Hzz\nHEkH k E kE k Hzz\nkH kH k E kE kE k Hεμ\nεμ\nεμ∂∂−= − −=∂∂\n∂∂−= − −=∂∂\n−= − −=G\nG\nG (3) \nwhere 00sin cosxkkθϕ = , 00sin sinykkθϕ = , kcω=  is a free-space wave number. From six \ncomponents of the electromagnetic field EG\n and HG\n, only four ones are independent. Thus we can \neliminate the components zE and zH from the system (3) and derive a set of four first-order linear \ndifferential equations in the transversal field components inside a layer of the structure \n22 2\n22 2\n22 2\n22 200 i\n00 ii00\n00T\nx x xy x\nT\ny y yx y\nLL\nx x xy x\nLL\ny y yx yE E kk k k k\nE E kk k kkkHH kk k k k z\nHH kkk kkαε μ ε\nμε α ε\nμε μ\nεμ μ⎛⎞ +− ⎛⎞ ⎛⎞\n⎜⎟ ⎜⎟ ⎜⎟−+ − ∂ ⎜⎟ ⎜⎟ ⎜⎟=⎜⎟ ⎜⎟ ⎜⎟ −− + ∂⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟−⎝⎠ ⎝⎠ ⎝⎠, (4) \nwhere 1εε=, 0α≠, 1TTμμ= , 1LLμμ=  in the ferrite layers 1 mL z mL d≤≤+  and 2εε=, 0α=, \n2TLμμμ==  in the dielectric layers 1 (1 ) mL d z m L+≤≤ + , 0,1,..., 1mN= −. \nThe set of equations (4) can be abbreviated by using a matrix formulation \n() i () ()zk zzz∂=∂Ψ AΨ , (5) \nwhere ()zA  is a 44× matrix, and Ψ is a four-component vector with evident notations. \nFurther, we assume that the vector ()zΨ  is known in the plane 0zz= and find a solution of the \nCauchy problem  for equation (5) in the form \n00 () (, ) ( )zz z z=Ψ MΨ . (6) \nIn electromagnetic theory the matrix 0(, )zzM  is referred to as a transfer  matrix. The matrix 0(, )zzM  \nsatisfies the following relations [19] \n00\n11 2 2\n1\n12 21(,) ,\n(, ) ( , ) (, ) ,\n(, ) (,) ,zz\nzz z z zz\nzz zz−=\n=\n=MI\nMM M\nMM (7) \nand \n1\n00 0 0(, ) () ( )zz z z−= MM M , (8) \nwhere 0() (, 0 )zz≡ MM , and I is the 44× identity matrix. \nLet us consider the case of a periodical coefficient of the system (5), i.e.  \n() ( )zL z+= AA  (9) \nor in terms of the transfer matrix \n00 (, )( , )zL z L z z++= MM  (10) \nwhere L is period. \nTaking into account (7) and (8), a multiplicative identity is derived from (10) \n00 0() ( ) ( )zL z L+= MM M . (11) \nThe constant matrix 0()LM  is called as the monodromy  matrix [19]. Further, if it is assumed that \n0 ln ( ) LL≡Κ M ,      0 ( ) ( )exp( )zz z≡ − FM Κ, (12) \nand in view of 1\n0() e x p ( )LL−=− M Κ  and (11), we deduce the equality \n00 () () e x p [ () ] ( ) e x p ( ) ( )zL zL zL z z z+= + − + = − =FM Κ M Κ F . (13)  6Thus the matrix ()zF  is a periodic one. By using (8), the transfer matrix of the Cauchy problem for \nequation (5) with periodic coefficient is derived like this  \n1\n00 0( , ) ( )exp[ ( )] ( )zz z z z z−=− MF Κ F . (14) \nIn this case the corresponding general solution of (5) has a form \n1( ) ( )exp( ) ( ) exp( )zz z z z−==Ψ FΚcF c c Κc, (15) \nwhere c is some arbitrary constant matrix compiled from the set of fundamental solutions of equation (5) \nin 0z=, det 0≠c . Note, c can be linked to ()zΨ  via the substitutions ( ) ( ) zz= FF c\u0004 , 1−=ΚcΚc\u0004 . The \nformula (15) is a vector analogue of the Floquet theorem [19]. The use of the Floquet's theorem for \nsolutions of the system (5) with periodical coefficients can be more effective than a direct numerical \nsolution. Actually, according to (15), in order to find a solution suitable for any z, it suffices to define the \nfunction F in one period and to find the constant matrix Κ. Both latter tasks are provided by the \nknowledge of the matrix 0()zM  in one period 0zL≤≤; which is sufficient to solve the problem for \n00z= with the initial condition 00(,)zz= MI  on the interval [0, ]L. \nIn our case the structure period consists of two layers with thicknesses 1d and 2d (Fig. 1). \nTherefore the system is describ ed by the transfer matrices 11 (, 0 )d=MM  and 21 (, )Ld=MM  related to \nintervals 1 0zd≤≤  and 1dz L≤≤  respectively. As follows from expression (6) and relations (7), the \ntransfer matrix of the structure period is a product of the transfer matrices of layers 21=MMM\u0000  and the \nfield components referred to boundaries of the first double-layer period of the structure are related as \n21 2 1 () ( ) ( 0 ) ( 0 )Ld== =Ψ MΨ MMΨΨM\u0000 . (16) \nNote that M is the monodromy matrix related to equation (5) with periodical coefficients in a special \ncase of double-layer structure period. \nIt is obvious, that the field components referred to the outer boundaries of the whole structure \nconsisted of N periodic slabs are connected by transfer matrix M raised to the power N  \n() ( 0 )NNL=ΨΨ M . (17) \nIn order to search the fields reflected from and transmitted through structures with a large number \nof periods 1 N\u0015, for the raising of a matrix M to power N, the algorithm of the matrix polynomial \ntheory can be used [21, 22] \n4\n1NN\njj\njρ\n==∑ P M ,   1\njj−=PV E V , (18) \nwhere jρ are the eigenvalues of the transfer-matrix M, V is a matrix whose columns are the set of \nindependent eigenvectors of M, jE is the matrix with 1 in the (, )jj-locations and zeros elsewhere. \n \n4. Bloch eigenwaves theory \nOne of the approaches based on the Floquet’s theorem (15), is the method of Bloch waves that describes \nthe propagation conditions of eigenwaves of an infinite periodic structure. It is based on the fact that for \nany multiplicator  ρ there is a nontrivial solution of the periodic system (5), satisfying the condition [19] \n() ( )zL zρ+=ΨΨ , (19) \nwhere multiplicators are the eigenvalues of the monodromy matrix, i.e. they are the roots of the following \ncharacteristic polynomial \n[] det 0ρ−=IM . (20) \nAs follows from (12), the eigenvalues γ of the matrix ln L=KM  satisfy the equation \n[] det 0γ−=IK  (21) \nand the multiplicators ρ are related via the condition   711ln ln | | i(arg 2 )jj j j nLLγρρ ρ π⎡⎤ == ++⎣⎦,   ( 1 , 4 , 0 ,1 ,2 , . . . )jn== ± ± . (22) \nIn our case, ()zA  is the 44× matrix that satisfies the condition [] tr ( ) 0 z= A , where tr denotes the \nmatrix trace, i.e. the sum of all elements of matrix  main diagonal. Therefore, according to the Liouville-\nJacobi formula [19] \n[]\n0det exp tr ( ) 1L\ntd t = =∫A M . (23) \nThus the matrix M is unimodular, and, after the determinant calculating, equation (20) comes to the \nfollowing polynomial form \n432\n321 0 0 SSS Sρρρρ+++ + = ,  (24) \nwhere 0det 1 S==M , (23 4\n1\n111ii jj kk ij jk ki ik ji kj ii jk kj jj ki ik\nij i k jSm m m m m m m m m m m m m m m\n== += +=− + + − − −∑∑∑  \n) kk ij jimm m , ()34\n2\n11ii jj ij ji\nij iSm m m m\n== +=−∑∑ , 4\n3\n1ii\niSm\n==−∑ , and mαβ are the elements of the matrix M. If \ncoefficients of equation (24) are satisfy to equality 31SS=, the left part of the dispersion equation of this \ntype can be represented as the product of two quadratic polynomials [16, 17, 23] \n22\n1211 0 QQρρρρ⎡⎤ ⎡⎤+++ + =⎣⎦ ⎣⎦. (25) \nThe fact that the condition 31SS= is satisfied can be verified numerically. In this case the coefficients of \nthe equations (24) and (25) are related as the next \n12 1QQ S+=,   12 2 2QQ S+=. (26) \nExpressing 1Q and 2Q through the 1S and 2S, we obtain \n2\n11\n1,2 2 222SSQS⎛⎞=±+ −⎜⎟⎝⎠. (27) \nThus, the dispersion equation (25) is split into two independent parts. From a physical point of view it \nmeans that in the structure, there are two independent spectra of eigenwaves, each of them is \ncharacterized by its dispersion relation and wavenumber. \nFrom (27) the eigenvalues of the transfer matrix of one period M can be written in the form \n2\n22\n11 2 11\n1,2 2\n2\n22\n11 2 11\n3,4 22 121 ,44 4 4 22\n2 121 .44 4 4 22SS S SSS\nSS S SSSρ\nρ⎡⎤− ⎛⎞ ⎛⎞⎢⎥ =− − + ± + + − −⎜⎟ ⎜⎟⎢⎥ ⎝⎠ ⎝⎠⎣⎦\n⎡⎤− ⎛⎞ ⎛⎞⎢⎥ =− + + ± − + − −⎜⎟ ⎜⎟⎢⎥ ⎝⎠ ⎝⎠⎣⎦ (28) \nThe multiplicators are related to the propagation constants of the eigenwaves via the condition \nexp( i )jj L ργ±=± ; the sign choice for the j-th type of wave corresponds to the wave propagation \ndirection. \nTo analyze the stability of the obtained solutions, according to the Lyapunov theorem on the \nreducibility  [19], the changing of variables in equation (5)  \n0 () () e x p ( )zz z==− ΨFM Κ ϒ ϒ (29) \nleads equation (5) with the periodic matrix ()zA  to the equation with the constant matrix Κ \nz∂=∂Κϒϒ. (30)  8The general solution of this equation, as known, is given by \nexp( ) z=cΚϒ . (31) \nFrom all the solutions of this equation we select the trivial solution 0=ϒ , and ask the question about of \nits stability, namely: do small deviations from this solution at 0z= lead to small deviations for all 0z≥? \nThe answer varies depending on the form of the matrix Κ. From the general definition of the Lyapunov \nstability  it is implied that the solutions are stable if and only if the matrix exp( ) zΚ is bounded for all \n0z≥, and the solutions are asymptotically stable if the matrix exp( ) zΚ tends to zero for all 0z≥. Thus \nthe stability of the system (30) is completely determined by the form of the roots jγ of the characteristic \npolynomial of the matrix Κ (21), and these conditions are determined via the theorems of Lyapunov \n[19]. \nOn the basis of this theorem, using the relation (22), the following conditions of the solution \nstability of the periodic system (5) can be formulated. The solutions are stable if all the multiplicators jρ \nlie within the boundaries of the closed unit circle || 1jρ≤ (|| 2jQ≤). The multiplicators that lie on the \ncircle || 1jρ= (|| 2jQ=) possess simple elementary divisors. For the asymptotic stability of the solutions \nof the periodic system, it is necessary and sufficient that all the multiplicators lay inside the unit circle \n|| 1jρ< (|| 2jQ<). \nFrom the electromagnetic point of view, when considering the periodic structure, the regions of \nstability and instability of solutions of equation (5) correspond to regions where waves propagate or do \nnot propagate. Thus, in the first case || 2jQ<, the frequency range and basic-element parameters provide \nthe propagation of the j-th wave (passbands). In the second case || 2jQ>, the wave does not propagate \n(stopbands), and //exp( )jj L ργ±≡± . The band edges are the regimes where || 2jQ= ( 0jγ=).  \nNote that in a periodic structure without losses, multiplicators are usually located on a circle of unit \nradius or on the real axis. In some anisotropic structures the points || 1jρ= can be located at arbitrary \npoints of the unit circle on the complex plane of ρ. These degenerate points are of particular interest and \nare a subject of some special studies [24, 25] related to the problem of “slow light”. The advantage of the \ngeneral theoretical approach used here is its applicability for the analysis of fields in the periodic \nstructures under a degeneracy of their eigenwaves. \n \n5. Reflection and transmission coefficients \nTo find the reflection and transmission coefficients, we use the solution (17) of equation (5) that can be \nequivalently formulated as  \n1(0) ( ) ( ) ( )NNL NL−==ΨΨ ΨMT . (32) \nThe field vector at the input surface is made up of two parts that consist of the incident and \nreflected wave contributions \n(0)in ref=+ ΨΨ Ψ . (33) \nThe field at the output surface matches only a single transmitted wave field \n()tr NL=ΨΨ . (34) \nOn the other hand, the incident, reflected, and transmitted field can be written as follow \n0 () e x p ( i )in in inEr E k r=⋅G GGGG,   0 () e x p ( i )ref ref refEr E k r= ⋅G G GG G,   0 ( ) exp(i )tr tr trEr E k r= ⋅G GGG G. (35) \nThe fields (35) can be represented in terms of the linearly polarized waves. Thus the field components \nin the input and output half-spaces are (the factor exp[ ( )]xy itk xk yω−−−  is omitted) like this  90 0\n00\n00\n0 0\n00\n001exp(i ) exp( i ) ,\n1\nexp(i ) exp( i ) ,s ss\nx\nzz pppy\ns ss\ny\nzz pppxE Y ABkz kzE AB Y\nY H ABkz kzH AB Y⎧⎫⎛⎞ ⎧⎫ ⎧⎫ ⎧⎫ ⎪⎪=± ± − ⎜⎟ ⎨⎬ ⎨ ⎬ ⎨⎬ ⎨⎬⎜⎟⎩⎭ ⎩⎭ ⎩⎭ ⎪⎪⎝⎠⎩⎭\n⎧⎫⎛⎞⎧⎫ ⎧⎫ ⎧⎫⎪⎪=−⎜⎟ ⎨⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎜⎟⎩⎭ ⎩⎭ ⎩⎭⎪⎪⎝⎠⎩⎭∓ (36) \n()\n()33\n3\n3\n31\nexp[i ( )]\n1ss\nx\nz\nppyYCE\nkzN LEYC⎧⎫\n⎧⎫⎪⎪=± −⎨⎬⎨ ⎬\n⎩⎭⎪⎪⎩⎭,     3 3\n3\n33exp[i ( )]ss\ny\nzppxYC HkzN LHYC⎧⎫⎧⎫⎪⎪=−⎨⎬ ⎨ ⎬\n⎩⎭⎪⎪⎩⎭. (37) \nHere vA, vB and vC (,vp s= ) are the amplitudes of the incident, reflected and transmitted field, \nrespectively; coss\njj jYηθ−1= , 1(c o s)p\njj jYηθ−=  ( 0,3j= ) are the wave admittances of input and output \nhalf-spaces; coszj j jkk θ= , jjkk n= , j jj nεμ= , j jjημε= , 00 sin sinjj nn θθ= , and the term s is \nrelated to the perpendicular polarization (electric-field vector EG\n is perpendicular to the plane of incidence) \nand the term p is related to the parallel polarization (electric-field vector EG\n is parallel to the plane of \nincidence) of plane electromagnetic waves. \nThe substitution of (36), (37) at the interfaces 0z= and zN L=  into (33), (34) yields the following \nsystem of algebraic equations \n00 00\n11 22\n33 33\n33 44\n0 3 03 03 03,,\n,,ss pp\ns ss s p p p p s s p p\nsp sp\nsp sp\ns ss p p p s p\nss ps ps ppYY YYA B aC aC A B aC a CYY YY\naa aaAB C C A B C C\nY Y YY YY YY+= + −= − −\n−= + += + (38) \nwhere 14 3s s\njj jat t Y=+ , 23 3p p\njj jat t Y=− + , 1, 4j= , and tαβ are the elements of the transfer matrix T. \nThen, we assume that incident field is either p-type ( 0sA=), either s-type ( 0pA=), and the co-\npolarized reflection and transmission coefficients are determined by the expressions vv v vR BA=  and \nvv v vCAτ= , and the cross-polarized ones are ''vv v vR BA=  and ''vv v vCAτ= , respectively. From (38) \nthey are: \n()\n()\n()\n()03\n00 2 2 03\n03\n00 1 1 03,2 ,\n2, 2 ,\n,2 ,\n2, 2 ,ss ss s s\nsp ps ss pp ss\nsp s p p s sp s p\nsp ss sp\npp pp p p\npp ss ps sp pp\nps p s p s ps p s\npp ps psRb b b b Y Y b\nRY Y a b a b Y Y bRb b b b Y Y bRY Y a b a b Y Y bτ\nτ\nτ\nτ−− −− −\n−− −\n++ ++ +\n++ +=− Δ = − Δ\n=− Δ = Δ\n=− Δ = Δ\n=− Δ = − Δ (39) \nwhere pss p p ps sbb bb+− +−Δ= − , and 02 3ppp\nppbY a a±=± , 02 3pss\npsbY a a±=±, 01 4sss\nssbY aa±=±, 01 4spp\nspbY aa±=± .  \nThe polarization state of both reflected and transmitted fields can be obtained using standard \ndefinition [26]: 21 tan2 UUβ= , 30 sin2 UUη= , where β is the polarization azimuth, η is the \nellipticity angle (Fig. 3). jU are the Stokes parameters calculated from the components of the electric \nfield in the right-handed orthogonal frame.  10\n \nFigure 3.  (color online) Parameters of the polarization ellipse.  \n6. Numerical results and discussion. \nAt first we consider the case of normal incidence (00θ=) of a plane wave on a MPLS. For the sake of \nsimplicity, but without loss of generality, let us assume that the structure consists of ferrite layers \nseparated by air gaps, i.e. 22 1εμ==  is assigned. \nAs discussed above, the stability and instability domains of solutions of the system (5) correspond \nto the regions of problem parameters in which propagation of electromagnetic waves is permitted or \nforbidden respectively. Thus the values of 1,2Q defined by expression (27) determine the band spectrum \nof two pairs of eigenwaves. One pair of eigenwaves propagates in direction of z-axis and the other has the \nopposite direction of propagation. The passbands are determined by the condition 1|| 2Q≤ for one \neigenwave and 2|| 2Q≤ for another in each of these pairs (Fig. 4). These conditions are displayed in \nFig. 4a as a shaded area. Since the dispersion equation (25) consists of two independent factors, the \nbandwidth of these spectra can be mutually overlapped. One can see a significant difference between \nthese two solutions of the dispersion equation. This is because they correspond to two waves with \ndifferent polarizations. \n(a) (b) \nFigure 4.  (color online) Stability conditions (a) and  band spectrum (b) of magnetophotonic layered structure under \nnormal wave propagation (00θ=), 3\n12 51 0m dd−== × , 00ϕ=.  11As is well known, the eigenwaves of an unbounded ferrite medium propagating along the \nmagnetization direction are the left-handed (LCP) an d right-handed (RCP) circularly polarized waves \nwhich differ in the propagation constants Tkkεμα ε μ± ±Γ= ( ±) =  [7, 8]. Thus each of those waves \npropagates in the medium with different effective magnetic permeability μ±, where μ+ and μ− are \nrelated to the LCP and RCP waves, respectively. For clarity, the frequency dependences of μ± are shown \nin Fig. 5. In our case, Re( ) μ− is a positive value over the chosen frequency range, and the medium losses \nare infinitesimal for the RCP wave ( Im( ) 0.01 μ−∼ ). On the other hand, it is possible to select three \nspecific frequency ranges where μ+ acquires different properties. In the first range, located between \n1 GHz and 3 GHz, the effective permeability μ+ has a positive value of real part and a small imaginary \npart. In the second range, between 3 GHz and 5 GHz the real part of μ+ varies from positive values to \nnegative ones as frequency increases. This transition takes a place at the frequency of the ferromagnetic \nresonance (04f= GHz). In this range the medium losses are very significant. Finally, in the third \nfrequency range, located from 5 GHz to 10 GHz, the permeability μ+ has a negative real part and a small \nimaginary part. \n \nFigure 5.  (color online) Frequency dependences of effective permeability μ± related to the left-handed and right-\nhanded circularly polarized waves. \nThe properties of the ferrite medium affect the propagation conditions of the eigenwaves of the \nperiodic structure. On the basis of equations (22) and (28), the propagation constants jγ ( 1, 4j= ) of the \neigenwaves are obtained via the solutions of the dispersion equations (27) with 1Q and 2Q. These \npropagation constants obey the conditions 12γγ=− and 34γγ=−, where the sign defines the propagation \ndirection along or opposite to z-axis. Thus, it is defined that 1γ, 3γ and 2γ, 4γ correspond to the RCP \nwaves and LCP waves, respectively. In the case of the RCP wave propagating in the positive z-axis \ndirection, due to small dissipative losses related to μ−, the eigenwave spectrum has interleaved passbands \ngiven by 1|| 2Q<, 12Im Im 0γγ=≈ and stopbands given by 1|| 2Q>, 12Im Im 0γγ=−≠ . The band edges \nare the regimes that correspond to 1|| 2Q=. In the case of the LCP wave, starting with some frequency \nnear 3 GHz, it is no longer possible to select the alternation of the passband and stopband positions. As is  12clear from Fig. 4, the imaginary parts of 3γ and 4γ are significant above this frequency, and the condition \n2|| 2Q> holds almost over all selected frequency range. \nThe mentioned features of the eigenwaves of the infinite periodic structure composed of ferrite \nlayers appear particularly in the frequency dependences of the reflection and transmission coefficients of \nthe LCP and RCP waves of the bounded stack formed of N basic double-layer elements (see Fig. 6). \nThese frequency dependences have interleaved bands of  the reflection and transmission that corresponds \nto the stopbands and passbands of the eigenwaves. The finite number of the structure periods results as small-scale oscillations in the passbands. These oscillations are a consequence of the interference with \nwaves reflected from outside boundaries of the stack. The number of oscillations is \n1N− within each \npassband. Evidently, the dissipative losses reduce the average level of the reflection and transmission in \nthe whole frequency range and decrease the amplitude of the small-scale interference oscillations in the \npassbands. Also note that a finite number of stack slabs results in a partial transmission of the waves \nwithin the stopbands.  \n(a) (b) \nFigure 6.  (color online) Frequency dependences of transmission, reflection and absorption 221| | | |vvWR τ =− −  \ncoefficients of right-handed ( RCPv= ) (a) and left-handed ( LCPv= ) (b) circularly polarized waves under normal \nincidence (00θ=), 5N=, 3\n12 51 0m dd−== × , 00ϕ=. \nAll these properties are typical for both the LCP and RCP waves. But for the LCP wave some \ndistinctive features can be pointed out. In the frequency band near the frequency of ferromagnetic \nresonance 0ω almost all the energy of the LCP wave is absorbed. Note that this effect of the different \nabsorption of LCP and RCP waves is well known in optics as circular dichroism. At the higher frequency, \nthe real part of ferrite effective permeability μ+ is negative, which leads to a great imaginary value of the \npropagation constant +Γ. In this band the LCP wave is reflected except for a portion of the absorbed \nenergy.  \nAnalyzing the properties of the reflected and transm itted fields in terms of a linearly polarized wave \n(Fig. 7), we note that generally a transformation of a linearly polarized wave to an elliptically polarized \none appears at the MPLS output, and, over the whole selected frequency range the conditions || | |ssp pττ= , \n|| ||spp sττ=  and || | |ssp pR R= , || ||spp sR R=  are satisfied under normal wave incidence. The above \nmentioned peculiarities of the absorption and reflection of the LCP wave lead the degeneration of the  13elliptical polarization to the circular polarization of the transmitted field, i.e. the magnitudes of the co-\npolarized and cross-polarized components of the transmitted field are equal to each other, \n|| | | || | |ssp ps pp sττττ=== , and the ellipticity angle is 4ηπ=− . This condition is observed in the frequency \nrange starting from the frequency of a ferromagnetic resonance and up to the frequency where μ+ becomes \npositive. At some frequencies the elliptical polarization changes to a linear polarization, which corresponds to \n0η=. For the reflected field this situation appears at the frequencies of the band edges. Within the passbands \nthe conditions are also possible when both the reflected and transmitted fields are circularly polarized. One of \nthese cases is shown in Figs. 7c, 7d by dotted lines which corresponds to the frequency at 7.8 GHz. \n(a) (b) \n \n(c) (d) \nFigure 7.  (color online) Frequency dependences of reflection and transmission coefficients (a), ellipticity (b) and polarization \nellipse of transmitted (c) and reflected (d) fields under normal incidence (00θ=) of the linearly x-polarized wave, \n5N=, 3\n12 51 0m dd−== × , 00ϕ=. \nIt is interesting to further investigate the influence of the structure periodicity on the enhancement \nof the Faraday rotation as a result of the wave interference within the multilayer stack (see Fig. 8). For \nthis study three different structure compositions are cons idered. In the first case the structure consists of a \nsingle homogeneous ferrite layer with finite thickness. In the second structure configuration, there are two  14ferrite layers separated by an air gap. And in the third case, the structure is a periodical stack of four \nferrite layers separated by air gaps. In general, the total thickness of the ferrite layers in all the structure \nconfigurations remains unchanged. \nIn terms  of the linearly polarized wave the enhancement of the Faraday rotation can be estimated \nthrough the level of the amplitude of the co-polarized and cross-polarized components of the transmission and reflection coefficients, since these components are directly related to the definition of the polarization \nellipse (Fig. 3). As is seen in Fig. 8a, for the transmitted field, the enhancement of the Faraday effect is \nobserved in case when both the permeabilities \nμ+ and μ− are positive values. As pointed out above, \nunder normal wave incidence and in the frequency range where μ+ is negative, the transmitted field is \ncircularly polarized and the structure configuration difference occurs only in the level of the transmission \ncoefficient magnitude. \n(a) (b) \nFigure 8.  (color online) Enhancement of Faraday rotation depending on the thicknesses and number of ferrite layers for \ntransmitted (a) and reflected (b) fields, 00 0θϕ==. \nIf we consider the properties of the reflected field, the fact of the strong influence of the structure \nconfiguration on the magnitude of the co-polarized and cross-polarized reflection coefficients can be \nestablished (Fig. 8b). Thus, a choice of thicknesses and number of ferrite layers of the structure can \nproduce either enhancement or weakening of the Faraday rotation in the reflected field at the given \nfrequency. \nThe angle of incidence of the primary field significantly affects the relation between the level of the \nco-polarized and cross-polarized reflection and transmission. Three pairs of graphs are presented in \nFigure 8 to show the magnitudes of the reflected and transmitted fields. These pairs are related to the \ndifferent frequencies that correspond to the different values of μ+.  \nAt the frequency 2 GHz (Figs. 9a, 9b), the ferrite effective permeability μ+ is positive and \nmagnetic losses are vanishingly small. This frequency corresponds to the wavelength that is much greater \nthan the structure period length and the MPLS can be considered as a homogeneous gyrotropic layer of \nthe same length NL. It is clear that the angular dependences of the transmission and reflection coefficient \nrepresent this situation. Thus there are no significan t variations of the magnitudes of the reflection and \ntransmission coefficients, because the interference of waves inside the structure does not occur. Note that \nthe condition || ||pss pR R=  is satisfied for all angles of incidence but for a certain angles || ||pss pττ≠ .  15\n(a) (b) \n(c) (d) \n(e) (f) \nFigure 9.  (color online) Angular dependences of reflection and transmission coefficients of linearly polarized wave; \n5N=, 3\n12 51 0 dd m−== × , 00ϕ=.  16The frequency 4.5 GHz is close to the frequency of ferromagnetic resonance (Figs. 9c, 9d). Here \nμ+ is negative and the magnetic losses are significant. As mentioned above, under the normal incidence \n(00θ=), the transmitted field is circularly polarized. Oblique incidence leads to the elliptical polarization \nof the transmitted field. The ellipses related to orthogonally polarized waves have identical ellipticity \nparameters defined by the following conditions | | | |pps pττ= , | | | |ssp sττ= . \nThe last pair of the graphs corresponds to the frequency of 10 GHz (Figs. 9e, 9f). At this frequency \nμ+ is close to zero and magnetic losses are vanishingly small. In this case, starting from a certain cutoff \nangle (065, degreesθ≈ ), the wave is completely reflected from the MPLS. The polarization of this \nreflected field is linear and similar to the polarization of the primary incident field, i.e. | | | | 0ps spRR== . \nNote that at this frequency for all angles of inciden ce, the components of the reflected field are practically \nequal to each other for the orthogonally polarized waves, i.e. the conditions | | | |pps sR R≈ , | | | |pss pR R=  are \nsatisfied. \n \n7. Conclusion \nIn conclusion, we studied the electromagnetic properties of a magnetophotonic layered structure at the frequencies close to the frequency of ferromagnetic resonance. The research was carried out taking into account a great value of dissipative losses in the ferrite layers of the structure in the frequency range. This \ncircumstance requires generalizing the definition of eigenwaves of the periodic structure. It was stated \nthat the eigenwaves are orthogonal in the energy sense and every eigenwave propagates independently of \nthe others in the region which is free of sources. \nThe method of our study is based on deriving a system of ordinary differential equations on \ntransversal field components and further analysis of the solutions stability of this system using the general \ntheory of differential equations. Through the analysis of the eigenvalues of the transfer matrix of the \nstructure period, the boundaries of the stopbands and passbands of the eigenwaves were determined since \nthese eigenvalues describe the stability of the system solutions. We estimated the difference of the two \nkinds of eigenwaves in relation to the stability conditions of the system solution. In the assumption of \npropagation normal to structure layers, these two kinds of eigenwaves are the RCP and LCP waves. The \nparticular absorption and reflection of the LCP wave was studied. \nA dramatic effect of the biased ferrite material on the LCP wave propagation results in the \npolarization transformations of a linearly polarized plane wave impinged on the MPLS. Generally the \nreflected and transmitted fields have elliptical polarization. We ascertained that, under normal incidence, \nthe transmitted field has circular polarization in the frequency range of negative values of the biased \nferrite effective permeability related to the LCP wave. It is shown that in this frequency band, starting \nfrom some angle of incidence, the transmitted field becomes elliptically polarized. At the frequencies of \nthe stopband edges, the reflected field is linearly polarized. \nWe observed an enhancement of the Faraday rotation by the periodic stack in comparison with the \nrotation by some effective ferrite slab. The polarization rotation was examined for different configurations \nof the stack. We have shown that the rotation enhancement takes place in the transmitted field at the \nfrequencies corresponding to positive effective permeability of the ferrite for both RCP and LCP waves.  \nReferences \n1. Yablonovitch E 1993 J. Opt. Soc. Am . B. 10 283 \n2. Sakoda K 2001 Optical Properties of Photonic Crystals  (Springer) \n3. Yablonovitch E 2007 Optics & Photonics News 18 12 \n4. Inoue M, Arai K, Fujii T and Abe M 1998 J. Appl. Phys.  83 6768 \n5. Inoue M, Arai K, Fujii T and Abe M 1999 J. Appl. Phys.  85 5768 \n6. Sakaguchi S and Sugimoto N 1999 J. Opt. Soc. Am. A  16 2045  177. Mikaelyan  A L 1963 Theory and Application of Ferrites at Microwave Frequencies  \n(Gosenergoizdat, Moscow-Leningrad) [in Russian] \n8. Gurevich A G  1963  Ferrites at Microwave Frequencies (Heywood, London)  \n9. Kato H, Matsushita T, Takayama A, Egawa M, Nishimura K and Inoue M 2003 Opt. Commun . 219 \n271 \n10. Lyubchanskii I L, Dadoenkova N N, Lyubchanskii M I, Shapovalov E A and Rasing Th 2003 J. \nPhys. D: Appl. Phys.  36 R277 \n11. Inoue M, Fujikawa R, Baryshev A, Khanikaev A, Lim P B, Uchida H, Aktsipetrov O, Fedyanin A, \nMurzina T and Granovsky A 2006 J. Phys. D: Appl. Phys . 39 R151 \n12. Chernovtsev S V, Belozorov D P and Tarapov S I 2007 J. Phys. D: Appl. Phys.  40 295. \n13. Levy M and Jalali A A 2007 J. Opt. Soc. Am. B  24 1603 \n14. Belozorov D P, Khodzitsky M K and Tarapov S I 2009 J. Phys. D: Appl. Phys.  42 055003 \n15. Berreman D W 1972 J. Opt. Soc. Am.  62 502 \n16. Bulgakov A A and Kononenko V K 2001 Telecommunications and Radio Engineering 55 369 \n17. Vytovtov K A and Bulgakov A A 2006  Telecommunications and Radio Engineering  65 1307 \n18. Il'inskiy  A S and Slepyan  G Ya 1983 Oscillations and Waves in Electrodynamic Systems with \nLosses (Moscow University Press, Moscow) [in Russian] \n19. Jakubovich  V A and Starzhinskij V H 1975 Linear  Differential  Equations  with Periodic  \nCoefficients  (Wiley, New York), Vol. 1 \n20. Collin R E 1992 Foundations for Microwave Engineering  (Wiley-Interscience, New York) \n21. Dickey L J 1987 ACM SIGAPL APL Quote Quad . 18 96 \n22. Tuz V R and Kazanskiy V B 2009 J. Opt. Soc. Am. A  26 815 \n23. Tuz V R and Kazanskiy V B 2008 J. Opt. Soc. Am . A 25 2704  \n24. Figotin A and Vitebsky I 2006 Waves Rand. Compl. Media  16 293 \n25. Figotin A and Vitebsky I 2006 Phys. Rev. E  74 066613 \n26. Collett E 1993 Polarized Light: Fundamentals and Applications (New York: Dekker) \n "    },    {        "title": "1410.7969v2.Simultaneous_Multi_Harmonic_Imaging_of_Nanoparticles_in_Tissues_for_Increased_Selectivity.pdf",        "content": "Simultaneous Multi-Harmonic Imaging of\nNanoparticles in Tissues for Increased Selectivity\nAndrii Rogov,†Marie Irondelle,‡Fernanda Ramos-Gomes,¶Julia Bode,¶Davide\nStaedler,†,§Solène Passemard,§Sébastien Courvoisier,†Yasuaki Yamamoto,k\nFrançois Waharte,‡Daniel Ciepielewski,#Philippe Rideau,@Sandrine\nGerber-Lemaire,§Frauke Alves,¶Jean Salamero,‡Luigi Bonacina,\u0003,†and\nJean-Pierre Wolf†\nGAP-Biophotonics, University of Geneva, CNRS UMR144 Institut Curie, Paris,\nMax-Planck-Institute of Experimental Medicine, Göttingen, LSNP , EPF Lausanne, JEOL SAS,\nLaboratory of Synthesis and Natural Products, EPFL, Nikon AG - Instruments, and Nikon France\n& BeLux - Division Instruments\nE-mail: luigi.bonacina@unige.ch\nAbstract\nWe investigate the use of Bismuth Ferrite (BFO) nanoparticles for tumor tissue labelling\nin combination with infrared multi-photon excitation at 1250 nm. We report the efficient and\n\u0003To whom correspondence should be addressed\n†GAP-Biophotonics, Université de Genève, 22 Chemin de Pinchat, 1211 Genève, Switzerland\n‡Cell and Tissue Imaging Facility (PICT-IBiSA), CNRS UMR144 Institut Curie - Paris 75248, France\n¶Department of Molecular Biology of Neuronal Signals, Max-Planck-Institute of Experimental Medicine,\nHermann-Rein-Str. 3, 37075 Göttingen, Germany\n§Laboratory of Synthesis and Natural Products, Institute of Chemical Sciences and Engineering, École Polytech-\nnique Fédérale de Lausanne, Batochime, 1015 Lausanne, Switzerland\nkJEOL SAS Espace Claude Monet - 1,allee de Giverny 78290 CROISSY-SUR-SEINE(France)\n?Laboratory of Synthesis and Natural Products, Institute of Chemical Sciences and Engineering, École Polytech-\nnique Fédérale de Lausanne, Batochime, 1015 Lausanne, Switzerland\n#Nikon AG - Instruments, Im Hanselmaa 10, 8132 Egg / ZH, Switzerland\n@Nikon France & BeLux - Division Instruments 191, rue du marché Rollay - 94504 Champigny sur Marne, France\n1arXiv:1410.7969v2  [physics.optics]  3 Oct 2015simultaneous generation of second and third harmonic by the nanoparticles. On this basis,\nwe set up a novel imaging protocol based on the co-localization of the two harmonic signals\nand demonstrate its benefits in terms of increased selectivity against endogenous background\nsources in tissue samples. Finally, we discuss the use of BFO nanoparticles as mapping refer-\nence structures for correlative light-electron microscopy.\nKeywords: second harmonic generation, third harmonic generation, nanoparticles, tissue imag-\ning, multiphoton microscopy\nThe advent of multi-photon microscopy in the early nineties has revolutionized the field of op-\ntical imaging.1This technique has proven particularly beneficial for biological studies. Nowadays,\nthanks to the availability of compact ultrafast sources exceeding the traditional 700-1000 nm range\nof Ti:Sapphire oscillators and covering the spectral region up to 1300 nm, the way is paved for\nimproved performances in terms of imaging penetration and novel applications for tissue diagnos-\ntics and tumor invasion studies.2The tunability of these sources allows selecting the excitation\nwavelength for minimizing water absorption and scattering according to sample characteristics.3,4\nTraditional nanophotonics labelling approaches (quantum dots, plasmonic nanoparticles (NPs),\nup-conversion NPs) display fixed optical properties often in the UV-visible spectral region and\ncannot fully take advantage of this spectral extension. To circumvent wavelength limitations, a\nfew research groups in the last years have introduced a new nanotechnological approach based on\nmetal oxide nanocrystals with non-centrosymmetric lattice, harmonic nanoparticles (HNPs).5–7By\ntheir crystalline structure, HNPs present very efficient nonlinear c(2)response, and can be effec-\ntively imaged using second harmonic (SH) emission as contrast mechanism efficiently responding\nto excitation from the UV to the mid-IR.8Moreover their signal is not bleaching, blinking, nor\nsaturating because of the non-resonant character of the photo-interaction mechanism involved.9,10\nMulti-harmonic emission by HNPs has been already sparsely reported to date.8,11,12Extermann et\nal.firstly observed third harmonic (TH) generation in Fe(IO 3)3HNPs by exciting at 1500 nm with\na SH/TH intensity ratio of approximately 100.8\nIn this work, we use 100-120 nm diameter Bismuth Ferrite (BFO) HNPs with a PEG bio-\n2compatible coating prepared for further functionalization (see S.I. § 1.1). BFO HNPs have been\nrecently presented as the most promising candidates for translating HNPs to medical applications.\nIn fact, they present a very high second order nonlinear coefficient hdi=79 pm/V13(for comparison\nhdi\u00194 pm/V for BaTiO 3and KNbO 3HNPs)14and extremely good biocompatibility, in particular\nwhen PEG-coated. A thorough discussion about the effect of PEG-coated BFO HNPs on various\ncancer and healthy cell lines has been recently published.15In this reference, the effect of BFO\nHNPs in terms of cell viability, membrane permeability, lysosomal mass, intracellular localization\nand hemolytic potential are assessed by high throughput methods at different concentrations and\nfor several incubation times. Recently, BFO HNPs have been applied with success to novel in\nvitro applications in cancer research and regenerative medicine. Apart from polymer coating for\nincreased biocompatibility, the functionalisation of HNPs for specific targeting has been demon-\nstrated for BaTiO 3.16–18\nFor optical nonlinear imaging we employed a Nikon A1R multiphoton upright microscope\n(NIE-Nikon) coupled with an Insight Deepsee tunable laser oscillator (Spectra-Physics, 120 fs, 80\nMHz, 680 - 1300 nm). With respect to standard systems based on Ti:Sapphire oscillators, the mi-\ncroscope was optimized for infrared transmission using tailored reflection coatings and dedicated\ntransmission components. The nonlinear signals were epi-collected by two different long working\ndistance objectives (25 \u0002CFI75 APO N.A. 1.1 and 16 \u0002CFI75 N.A. 0.8) spectrally filtered by tai-\nlored pairs of dichroic mirrors and interference filters and acquired in parallel either by a standard\nphotomultiplier (600 - 655 nm) or a GaAsP photomultiplier (385 - 492 nm).\nThe top-left plot in Fig. 1 shows the SH emission at 625 nm generated by BFO HNPs deposited\non a substrate when excited at 1250 nm. The intensity differences among HNPs in the image reflect\ntheir size dispersion and crystal axis orientation with respect to the polarization of the excitation\nlaser.20When detecting at the TH frequency (416 nm), HNPs appear also very bright. We have ap-\nplied to this two color image the algorithm developed by Costes et al. to detect co-localized SH/TH\nevents.19This recognized computational method has the advantage of automatically determining\nthe signal thresholds for both detection channels with no user bias. Any event in the region above\n3Figure 1: Multi-harmonic emission by bare BFO HNPs deposited on a substrate. SH: second\nharmonic image. TH: third harmonic image. Scatter plot: TH vs.SH pixel intensity. The lines\nparallel to the axes indicate the threshold levels determined by Costes algorithm,19events with\nintensities >both thresholds are considered co-localized. The diagonal line is a linear fit to the\nwhole dataset. SH \\TH: co-localization image with co-localized pixels in white. Pixel size 0.79\nmm, PSF not oversampled.\n4the two thresholds ( TSHandTTH) indicated by the two thick lines parallel to the axes is assumed\nas co-localized and the associated pixels appear white in the SH \\TH plot. One can see that this\nregion encompasses the quasi-totality of the nanoparticles on the substrate. The diagonal line in\nthe scatter plot is automatically determined by the algorithm as a result of a linear regression of\nthe whole dataset. The dispersion of the data points around this reference line is not surprising:\na strict linear dependence between the two normalized harmonic signals is not expected, because\nof their different intensity dependence ( µI2andµI3, respectively for SH and TH) and of other\nfactors influencing differently the two emissions, e.g. orientation. More importantly, a constant\nratio between TH and SH by HNPs is not required for the co-localization procedure to work. By\nrunning a test of significance, we obtain a P-value of 1.0, indicating that all the events identified\nby the algorithm as co-localized are truly so from a statistical standpoint.21,22For a comparison,\nthe widely used (but less informative) Pearson coefficient obtained for this data set is 0.87. Im-\nportantly, such multi-harmonic emission is not limited to BFO HNPs although for this material the\nratio seems particularly favourable. In S.I. § 2.3, we report the results obtained using a different\nHNP material (KNbO 3) as additional demonstration of the procedure.\nThe normalized power dependence measured on a single BFO HNP for both SH and TH with\nthe characteristic I2andI3dependence is reported in Fig. 2a with the corresponding exponential fit\nconfirming their assignment (1.95 and 3.2 for SH and TH, respectively). As expected, the SH/TH\nratio changes according to excitation intensity and indeed Dai et al. have recently investigated this\ndependence in ZnO NPs for prospective applications in display technology.12By calibrating the\nspectral transmission of optics and response of the detectors, we could estimate the SH/TH BFO\nintensity ratio to a factor forty to one hundred at 1250 nm, depending on the intensity applied. A\ndetailed description of the procedures used for this estimate can be found in S.I. § 2.1. In S.I. § 2.2\nwe also present a calculation on the expected emission anisotropy as a function of particle size,\nindicating that within the particle dimension range used in this work, no destructive interference\neffects are expected for the epi-detected fraction of both SH and TH emission. In Fig. 2b, we\nshow how the nonlinear axial point spread functions (PSF) measured at the second and third order\n5~I 1.95\n~I 3.2FWHM = 1.4 umFWHM = 2 umPower dependance Z Point spead functionFigure 2: A. SH and TH power dependence measured on an individual BFO HNP. Markers: ex-\nperimental values. Thick lines: Infits yielding n=1.95 and 3.2 for SH and TH, respectively. B.\nNonlinear axial PSF at SH and TH obtained with a 1.1 N.A. objective and 1250 nm excitation.\nMarkers: experimental values. Thick lines: Gaussian fits yielding FWHM=1.97 mm and 1.42 mm\nfor SH and TH, respectively.\non a single sub-diffraction limited HNP are different. Because of higher nonlinearity order, TH\nPSF is narrower, leading to increased resolution, an aspect which might turn out to be particularly\nbeneficial when working at long wavelengths.23\nAlthough a large c(3)response is not surprising for large c(2)samples,24the simultaneous\ncollection of multiple harmonics by excitation-tunable nanometric systems can be very advanta-\ngeous for increasing selectivity in demanding applications, like ultra-sensitive detection in fluids\nas recently demonstrated.25For imaging, the use of >1100 nm excitation wavelengths ensures\nthat standard microscope collection optics and acquisition detectors can be efficiently employed\nat the TH frequency. The two harmonic signals are very well spectrally separated and character-\nized by narrow bandwidths, which make them easily distinguishable also when using conventional\nfluorescence filters for detection .\nTo investigate the advantages of multi-harmonic detection in relevant biomedical samples, we\nproceeded in imaging excised cancer tissues from a xenograft tumor model. Details of preparation\nare provided in S.I. § 1.3 and 1.4. Briefly, we analysed tumors developed in female nude mice after\nimplantation of human breast tumor cells MDA MB 231 either subcutaneously or orthotopically\nin the right abdominal mammary gland fat pad. Fresh breast tumor tissue sections were obtained\nusing a vibratome, followed either by incubation with BFO HNPs or in buffer as control.\n6SH, pixel intensity11 TH, pixel intensityFigure 3: Left. Representative multiphoton image of an unlabelled tissue section of an ortho-\ntopic breast tumor. Cyan: SH. Yellow: TH. Right. Scatter plot TH vs. SH. The algorithm\nindicates no positive events for co-localization (the diagonal line with negative slope points to\nanti-colocalization; the Pearson’s coefficients is -0.09). Pixel size 0.65 mm, PSF not oversampled.\nIn Fig. 3 we show a multi-harmonic image of an unlabelled section from the breast orthotopic\ntumor (negative control). It is known that strong endogenous sources of second (cyan, collagen)26\nand third harmonic (yellow, lipids)27exist in tissues. Both these tissue constituents are abundant\nin tumors2,28–30and are clearly present in the picture. Such harmonic background can affect the\nselective detection of HNPs. Some authors have already shown that HNPs can be imaged by their\nSH emission against collagen background (mammalian tendon).10,31However, it was also shown\nthat for individual HNPs the contrast was reduced,31even though instrumental sensitivity was\nsufficient for detecting single HNPs emission. Interestingly, the application of Costes’s algorithm\nfails on this image slice because of the almost total absence of co-localization events to perform\nthe computation. This finding is consistent with the distribution of pixels in the scatter plot, where\nelements yielding simultaneously high SH and TH are sorely missing. The calculation of Pearson’s\ncoefficient for this image yields a negative value (-0.09).\nThe first column of Fig. 4, displays the SH signal of a tissue section from the subcutaneous\nbreast tumor which was incubated with BFO HNPs. Collagen structures are clearly evident as\ndiagonal stripes, the presence of small bright spots (sometimes at the limit of pixel resolution)\n7is more pronounced than in the negative control of Fig. 3 and points to the presence of HNPs in\nthe sample. Likewise, TH image in the central row shows the presence of small spots, together\nwith other larger structures with different morphology with respect to collagen, which can be\nascribed to lipids.2,29The results of the co-localization procedure (SH \\TH, rightmost plot) enables\nhighlighting in white exclusively pixels which show a simultaneous multi-harmonic emission and\nthat, on the basis of the findings discussed in relation to Figs. 1 and 3, can be safely associated\nwith the presence of BFO HNPs. The second row of Fig. 4 provides a more quantitative analysis\nof this co-localization effect. The semi-log histograms of pixel intensity both at the SH and TH\ncovers the whole detection dynamic range. Co-localization bars (in blue) indicate the number of\noccurrences which can be simultaneously attributed to both harmonic channels. One can see that\nthe discrimination of HNPs signal against background emission of endogenous sources cannot\nbe based simply on intensity, as in both detection channels a relevant fraction of events at high\nintensity are not co-localizing. When comparing the intensity emitted by endogenous structures\nwith that of HNPs, one has to take into account that the former despite their lower nonlinear\nefficiency (e.g. <d>=0:94 pm/V for collagen32) are generally characterized by much larger\ndimensions (fulfilling or exceeding the focal volume) therefore the squared-volume dependence of\nharmonic signals can easily counteract the harmonic generation efficiency difference.\nWe cannot exclude that a few low intensity multi-harmonic positive events are not accounted\nfor by the algorithm as inevitable in the case of automatic image analysis.19On the other hand,\nfalse positive detections seem unlikely. Although we used relatively broadband fluorescence filters\nas compared to the narrow SH and TH emission bandwidths, we carefully ensured that no autoflu-\norescence is emitted upon two- and three-photon absorption when working at 1200 nm with the\nexcitation settings employed here (for a complete analysis refer to S.I. § 3). Clearly, a preliminary\nassessment of the anti co-localization of endogenous sources in a negative control sample (Fig. 3\nin the present case) is mandatory for the success of the approach.\nThe application of multi-harmonic correlation for increasing selectivity by background rejec-\ntion in optically congested environment like tissues can be performed in real-time, and therefore\n8Figure 4: Representative multiphoton images of subcutaneous breast tumor tissue labelled with\nBFO HNPs. First row. SH (cyan), TH (yellow), and SH \\TH merged image with co-localized\npixels in white. Second row. Semi-log SH and TH intensity histograms. Blue bars correspond to\nco-localized events. Scatter plot TH vs.SH. As in Fig 1, the lines parallel to the axes indicate the\nautomatically determined threshold values and the diagonal line a linear fit to the whole dataset.\nOnly the events above the thresholds (white pixels) are assumed as co-localized. Pixel size 0.65\nmm, PSF not oversampled.\n9opens the way to HNPs tracking protocols33,34with high selectivity and minimal image process-\ning requirements. Moreover, autofluorescence can be minimized when working in this wavelength\nrange, avoiding endogenous fluorophores like riboflavins and NADH which display two-photon\nabsorption bands peaked at wavelengths <1000 nm.23In addition, it is worth noting that reso-\nlution is generally not an issue for detecting HNPs. As demonstrated by the Beaurepaire group,\nit is advantageous when imaging thick samples by multiphoton excitation to employ objectives\nwith low-magnification and large field of view / N.A. to epi-collect efficiently multiple scattered\nphotons.35\nThe measurements presented were performed with peak pulse intensity at the sample going\nfrom 300 to 650 GW/cm2. Even by assuming the highest of these two values (calculated in the\nfemtosecond regime and corresponding to 16 mW average laser power) and considering the scan-\nning speed used for raster imaging, which corresponds to a pulse density of 1.2 \u0002103mm\u00002,36we\ncan calculate that these conditions are suitable for imaging of biological samples. Indeed these fig-\nures are sensibly lower than the photo-induced tissue damage threshold established by Supatto et\nal.on living drosophila embryos.36As a further assessment, the highly cited paper by Köning on\ncellular response to near infrared femtosecond excitation, sets a damage threshold on cell viability\nand reproduction at 1 TW/cm2.37Although we approach this intensity value in absolute terms,\nit should be noted that this estimate was derived at a shorter wavelength (730-800 nm instead of\n>1200 nm) implying a lower order multiphoton process for absorption by endogenous molecules\nand at a much longer dwell time (80 ms as compared to 4 ms here).\nTo further assess the actual penetration limits of the proposed HNP-based multi-harmonic\nimaging protocol, we have simulated the optical response of a HNP embedded at depth zin a\ntissue and calculated the theoretical epi-detected signals. This numerical analysis aims at deter-\nmining how the TH/SH ratio obtained at the uppermost surface of a three dimensional z-stack is\nmodified upon penetration in the tissue. Simulations are based on a free-ware C-code for light\npropagation38,39that we modified to account for nonlinear excitation. We assume that a 1200 nm\ninput beam is focused by a N.A. 0.8, 3 mm WD objective in two different kinds of tissue slabs (rat\n1010-810-710-610-510-410-310-210-1100Log10 (Normalized Signal)\n600 400 200 0\nPosition in tissue, z / µm1.0\n0.8\n0.6\n0.4\n0.2\n0.0Normalized TH/SH Ratioobjective\nLiver\n SH\n TH \n \nCartilage\n SH \n TH \n \nLiver\n TH/SH \n fit: exp(-z/ ζ); ζ=92 µm\n \nCartilage\n TH/SH  \n fit: exp(-z/ ζ); ζ=171 µmI 2\nSH\nI 3\nTHFigure 5: Results of Monte Carlo light transport simulations. Upper plot. Normalized detected\nSH and TH signals (semi-log scale) as a function of HNP position zfor two relevant tissue types:\nliver and cartilage. Lower plot. Normalized TH/SH ratio calculated from the same dataset. Right\npanels. Two dimensional slices of the simulation volume in the case of a HNP embedded at z=\n300mm in liver, reporting from top to bottom: the excitation intensity squared, the SH emission\npattern, the excitation intensity cubed, and the TH emission pattern. The upper side of each image\ncorrespond to the objective aperture, the dashed line indicates the position of tissue surface. The\nintensity logarithmic colormaps are conserved between focal intensity plots (spanning 5 orders of\nmagnitude) and TH, SH emission plots (seven orders of magnitude) allowing a direct comparison.\n11liver and ear cartilage) characterized by the sets of optical properties ( ma(l),ms(l),g(l)) reported\nin S.I. § 5.40The simulation volume is composed by 7003voxels of 7 mm side. The HNP placed\nat the focus is assumed to irradiate isotropically at the SH and TH with an efficiency depending\nrespectively on the local I2andI3excitation intensity. We then calculate the fraction of SH and\nTH photons emitted by the HNP reaching the objective and from these value, the TH/SH value.\nThis ratio is assumed to be 1 at the entrance surface. In the lower plot of Fig. 5, one can observe\nan exponential decrease of the TH/SH ratio as a function of depth z. The observed decrease is\nmore prominent for liver than for ear cartilage, consistently with the higher scattering and absorp-\ntion at the TH wavelength for the former tissue. This finding can be appreciated in the series of\nimages on the right of Fig. 5, reporting the normalized focal intensity and emission patterns for\na HNP embedded at 300 mm depth in a liver tissue. As sketched in the top panel, the upper side\nof each image corresponds to the objective lens and the red dotted lines the focusing beam. The\nhorizontal dashed line defines the position of tissue surface. Comparing the SH and TH emission\npatterns, one can see how the penetration of TH in the tissue is strongly limited by absorption as\ncompared to SH. As a general trend, we can deduce that for these two representative tissues, the\ninitial TH/SH ratio drops to its 1 =evalue within the 100-200 mm range. Based on the fact that\nthe co-localization algorithm does not require a fixed ratio and that we could roughly co-localize\nevents with TH/SH variations within one order of magnitude (see scatter plot in Fig. 1), for a given\nset of detection parameters, we can extrapolate a working range of approximately 300 mm in a\ncartilage-kind of tissue and approximately 200 mm in the optically more extreme liver case. These\nvalues are compatible with the requirements of many cell tracking applications.\nIn the light of previous results, an interesting development for the use of HNPs for tissue\nimaging is their application as reference mapping structures for correlating light (multi-photon)\nand electron microscopy (CLEM), facilitating the retrieval of specific regions of interest going\nfrom one technique to the other.41In this respect, BFO HNPs are particularly appealing, as they\nare electron dense and are expected to provide good contrast in electron-based imaging like TEM\nand SEM. Figure 6 shows a SEM representative image, including several cells imaged at different\n12voltages in backscattered electron (BE) mode. BE ( >50 eV) are preferable in this case, as they\nare known to be sensitive to the composition of the specimen and display brighter contrast for\nheavier elements. Overall, one can notice the good contrast provided by SEM for BFO HNPs. By\nchanging accelerating voltage one can modify the scattering region of the incident electron beam\n(S.I. § 6). While the 5 kV image reports details of cell’s surface and indicates the presence of\nindividual particles or small HNPs clusters, when SEM voltage is increased to 20 and 30 kV , larger\naggregates located below the cell surface appear as indicated by the arrows. The presence and\nposition of these aggregates correlate well with previous studies indicating that BFO HNPs initially\nadhere to the cell membrane (2 h) and are successively internalized with increasing exposure time\nand tend to concentrate in intracellular organelles, such as endosomes or lysosomes.15Finally,\npanel MP shows a maximal intensity projection image of the same R.O.I. imaged by multi-photon\nmicroscopy.\nIn this case, fluorescent labels were used to stain nuclei (DAPI, blue) and lipids such as cell\nmembranes (Nile Red, red). SH from HNPs appears in white. One can see that HNPs tend to\nlocalize at the membranes, in particular in region with higher density subcellular membrane com-\npartments where they are thicker (red intensity higher). Two vertical projections corresponding to\nthe sections indicated by the dashed lines are also reported. Within the optical resolution, they in-\ndicate that HNPs are mainly located inside the membranes or at cell upper surfaces confirming an\nalready observed tendency to avoid nuclear region and colocalize in lysosomes or at cell surface.15\nIn conclusion, we have shown that multi-harmonic emission of BFO HNPs can be easily de-\ntected by multi-photon microscopy when using excitation >1100 nm. Very advantageously, the\nintensity difference among second and third nonlinear response is not large. Using the right com-\nbination of detectors for the different spectral ranges (GaAsP and standard photomultipliers) the\ntwo signals can be acquired simultaneously using standard settings. Based on this result, we have\ndemonstrated that the co-localization of SH and TH allow identifying with high selectivity HNPs\nin a complex optical environment presenting endogenous sources of fluorescence and harmonic\n135/uni03BCm\nMP\nBE(20kV) BE(5kV)\nBE(30kV)BE(20kV) BE(5kV)\nBE(30kV)10 /uni03BCmFigure 6: Demonstration of CLEM imaging of MDA MB 231 cells labelled by BFO HNPs. BE:\nsecondary electrons SEM images at 5, 20 and 30 kV . MP: Multiphoton maximal intensity projec-\ntion image with two vertical cuts corresponding to the dashed lines. Red: Nile red fluorescence\nstaining lipids (membranes). Blue: DAPI fluorescence (nuclei). White: SH by HNPs. Arrows indi-\ncate throughout the different images two exemplary clusters appearing only at high SEM voltages\n(20 and 30 kV).\ngeneration, an excised xenograft tumor tissue in our experiment. The image processing necessary\nfor this approach relies on simple two-channel co-localization, allowing its real-time use in de-\nmanding imaging applications. Numerical simulations of light transport indicate that the proposed\nimaging protocol can be performed up to a thickness of a few hundreds micrometers, which are\nrelevant length scales for cell tracking applications. Moreover, by additional electron microscopy\nmeasurements, we have shown that BFO HNPs could prospectively serve as localization fiduciaries\nin advanced CLEM studies.42\nAcknowledgement\nThis research was partially supported by the European FP7 Research Project NAMDIATREAM\n(NMP4-LA-2010- 246479, http://www.namdiatream.eu), Fondation pour la Recherche Médicale\n( FRM n. DGE20111123020), and the Cancéropôle Ile de France (n. 2012-2-EML-04-IC-1).\n14246479, http://www.namdiatream.eu). 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Cell2011 ,146,\n148–163.\n19"    },    {        "title": "1808.08427v1.Photocatalytic_activity__optical_and_ferroelectric_properties_of_Bi0_8Nd0_2FeO3_nanoparticles_synthesized_by_sol_gel_and_hydrothermal_methods.pdf",        "content": "1 \nPhotocatalytic activity, optical and ferroelectric \nproperties of Bi\n0.8\nNd\n0.2\nFeO\n3\n nanoparticles \nsynthesized by sol-gel and hydrothermal methods \nHamed Maleki\n* \nFaculty of Physics, Shahid Bahonar University of Kerman, Kerman, Iran \ne-mail address: hamed.maleki@uk.ac.ir  \n \nKeywords\n: Multiferroics; photocatalytic activity; bismuth ferrite; sol-gel; Neodymium \ndoping; hydrothermal.  \nAbstract: In this study, the effects of synthesis method and dopant Neodymium ion on the \nferroelectric properties and photocatalytic activity of bismuth ferrite were studied. BiFeO\n3\n \n(BFO) and Bi\n0.8\nNd\n0.2\nFeO\n3\n (BNFO) nanoparticles were prepared through a facile sol-gel \ncombustion (SG) and hydrothermal (HT) methods. The as-prepared products were \ncharacterized by X-ray powder diffraction (XRD), Furrier transform infrared spectroscopy \n(FTIR) and transmission electron microscope (TEM) images. Both nanophotocatalysts have \nsimilar crystal structures, but the SG products have semi-spherical morphology. On the other \nhand, HT samples have rod-like morphology. TEM results indicated that the morphology of \nproducts was not affected by the doping process. The thermal, optical and magnetic properties \nof nanoparticles were investigated by thermogravitometry and differential thermal analysis \n(TG/DTA), UV-vis spectroscopy, and vibrating sample magnetometer (VSM). The \nferroelectric properties of BNFO nanoparticles were improved compared to the undoped \nbismuth ferrite. The photocatalytic activity of as-synthesized nanoparticles was also evaluated \nby the degradation of methyl orange (MO) under visible light irradiation. The photocatalytic \nactivity of nanoparticles prepared via sol-gel method exhibited a higher photocatalytic activity \ncompared to powders obtained by hydrothermal method. Also substitution of Nd into the BFO \nstructure increased the photocatalytic activity of products. \n \n  2 \n1. Introduction \nMultiferroic materials have simultaneously the properties of ferroelectricity, \nferromagnetism and also in some cases ferroelasticity [1,2].  These materials have attracted \nattention because of their potential applications in data storage, spintronic devices and sensors \nand photovoltaics and so on [3–8]. Bismuth ferrite is the only known multiferroics that has \nrhombohedrally-distorted perovskite structure with the space group R\n3\nC [4] and shows both \nferroelectricity and ferromagnetism at room temperature (RT) [9,10].  BiFeO\n3\n (BFO) has the \nferroelectric order below the Curie temperate T\nC\n~1103K and antiferromagnetic behavior below \nthe Neel temperature T\nN\n~643K [11,12].  \nIn recent years, BFO has received new attention due to its narrow optical band-gap (2.19 \neV) and excellent chemical stability, which allows the photocatalytic activity under visible \nlight [3,13–18].  BFO shows different catalytic activities such as oxidation of organic \ncompounds and degradation of pollutants [19–21].  In addition, BFO nanoparticles are \nmagnetic semiconductor materials which could be separable in aqueous organic media [14]. \nHowever, there are some difficulties for different applications of BFO. Weak ferroelectricity, \nremanent polarization, high leakage current density, poor ferroelectric reliability and \ninhomogeneous weak magnetization are some of these challenges [22–24].  \nMany theoretical and experimental studies of bismuth ferrite have been investigated to \nexpand the applications and solve these problems hindering practical usage of BFO. In order \nto overcome such problems and improve photocatalytic activity of BFO, many modification \nhave been investigated which includes doping rare earth elements instead of bismuth into the \nBFO structure [15,25–34]. In addition, it was investigated that the substitution of rare earth \n(RE) elements into the bismuth ferrite, can alter the photocatalytic activity [35].  \nIn recent years, there has been many reports on preparation methods for BFO nanoparticles \nwith a focus on photocatalytic applications, which includes sol-gel route [36,37], co-\nprecipitation [38] and hydrothermal reaction [39–41]. However, as the morphology of \nnanoparticles affects the physical properties of bismuth ferrite, we are interested to study the \neffect of synthesis process on the photocatalytic activity of pure and Nd-doped bismuth ferrite, \nas well as other physical properties of BFO and BNFO nanoparticles. Although several \nresearches on the structural, optical and multiferroic properties of bismuth ferrite has been \ndone, few studies have investigated on the photocatalytic activity of Nd-doped BFO and the \ninfluence of synthesis process [42,43]. The goal of this study is to quantify the effect of Nd 3 \ndopant on photocatalytic degradation rates of MO. Furthermore, BFO and BNFO were \ncharacterized for the structure, morphology, energy band-gap. Moreover thermal, ferroelectric, \nleakage current density and magnetic properties of samples are investigated and compared \nbetween SG and HT as-synthesized products. \n2. \nExperimental method\n \n2.1 Sol-gel preparation of pure and Nd-doped BiFeO\n3\n \nIn the sol-gel method, stoichiometric amount of Bi(NO\n3\n)\n3\n.5H\n2\nO, Fe(NO\n3\n)\n3\n.9H\n2\nO and \nNd(NO\n3\n)\n3\n.6H\n2\nO were dissolved in deionized water separately. Meantime, ethylene glycol (EG) \nand 2-methoxyethanol were mixed under stirring. Then, acetic acid was added to the solution \ndropwise (The pH of the mixed solution was adjusted to 1.5). This solution was mixed together \nunder vigorous stirring for 30 min. Then metal nitrate solutions were mixed with fuel solution \nunder constant stirring. The mixture was stirred constantly for 30 min and a dark red mixture \nappears. After stirring at RT for an hour, the temperature was increased to 70 \no\nC. After 3 hours \nheating and stirring, a clear brownish gel was obtained and following a few minutes with a \ntemperature higher than 90 \no\nC, a yellow suspension is formed. The suspension was kept at RT \nfor 10 hours and then it was put at 115 \no\nC for the water evaporation and fuel combustion. \nFinally the obtained powder was calcined at 650 \no\nC for 5 hours before investigating the \ncharacteristics. \n2.2 Hydrothermal synthesis of BFO and BNFO \nIn the next part, BiFeO\n3\n and Bi\n0.8\nNd\n0.2\nFeO\n3\n nanoceramics were synthesized by a \nhydrothermal process. Typically, bismuth (III) nitrate pentahydrate, ferric (III) nitrate \nnonahydrate, and neodymium nitrate hexahydrate (for BNFO) were dissolved in the minimum \namount of deionized water as a specified stoichiometric ratio. The mixture wad dropped into \npotassium hydroxide (4M, 30 ml) under magnetic stirring. After stirring for 30 min, the mixture \nwas placed in a teflon-lined steel autoclave of 50 ml for the hydrothermal reaction with a filling \ncapacity of 80 % and performed at 200\no\nC for 12 hours in an oven and then cooled to RT \nnaturally. The products are collected and washed several times with distilled water and ethanol \nand dried at 110 \no\nC for 2 h before further characterization. \n2.3 Characterization \nThe structural properties of pure and Nd-doped BFO nanoparticles which are synthesized \nvia both sol-gel and hydrothermal methods, were characterized by using XRD analysis (Philips \npowder diffractometer) with Cu-Kα radiation (λ=1.5406 Å) and Furrier transform inferared 4 \n(FTIR) TENSOR27 spectrometer. Crystal sizes also were determined by the Scherrer method. \nTransmission electron microscope (TEM Leo-912-AB) was performed to study the \nmorphology and size of products. The thermal behavior of the as-prepared samples are \nmonitored by thermogravitometric and differential thermal analysis (TG/DTA NETZSCH- PC \nLuxx 409) with the heating rate of 10\no\nC/min up to 1000\no\nC. For the magnetic properties, the \nhysteresis loops are recorded up to 20 KG with the vibrating sample magnetometer (VSM- \nLake Shore model 7410, SAIF) at RT. The optical properties of products were studied by UV-\nvis absorption spectra by using Lambda900 spectrophotometer. The polarization electric field \nP-E hysteresis loops of the prepared pellets were measured by the Sawyer- Tower circuit. \n2.4 Photocatalytic activity measurements \nThe photocatalytic activity of the BFO and BNFO nanoparticles for decomposition of \nmethyl orange (MO) was studied under irradiation of visible light source at the natural pH \nvalue. The reaction temperature was also kept at RT . The initial concentration of MO was 15 \nmgl\n-1\n with dispersing 0.1g BFO or BNFO in 200ml aqueous solution. Before irradiation, in \norder to reach an adsorption equilibrium of MO on products surface, the aqueous suspension \nwas magnetically stirred for 75 min in the dark. Then the lamp was turned on and changes of \nMO concentration were measured by measuring the absorbance of the solution at 554 nm using \na UV-vis spectrophotometer. C/C\n0\n, where C was the concentration of MO at time t, and C\n0\n was \nthe initial concentration of MO, was the photocatalytic degradation ratio of MO which has been \ninvestigated in this study. \n \n3. Results and discussion \n3.1 X-ray diffraction investigation  \nFigs. 1 (a) and (b) show the powder XRD patterns of BiFeO\n3\n (x=0) and Bi\n0.8\nNd\n0.2\nFeO\n3\n \n(x=0.2) nanoparticles prepared by SG method. Analysis of patterns indicated that all products \nhave a single perovskite phase with distorted rhombohedral structure with the space group R\n3\nC. \nAll samples reveal peaks that can be assigned to the standard card of BFO perovskite structure \n(JCPDS card No. 86-1518). By substituting 20% neodymium, a slight shifting of peaks towards \nlower angles occurred and a phase transformation from rhombohedral to tetragonal structure \nwas observed (two major peaks at 30<2θ<33 merged into a one peak) and intensity of peaks \nwere decreased. The width of the BFO peaks also increased with merging the nearby peaks. \nThe size of products was calculated by using the Scherrer formula: D\n=\n\u0000\u0000\n\u0000 \u0000\u0000\u0000\u0000\n , where \nK \nis the 5 \nshape factor that normally measures to be about 0.89, λ stands for the wavelength of X-ray \nsource, β is the width of the observed diffraction peak at its half intensity maximum, and θ is \nthe Bragg angle of each peak. The obtained average nanocrystal sizes were 51 and 46 nm for \nBFO and BNFO respectively.  \n \nFig. 1. (a) XRD patterns of BiFeO\n3\n and Bi\n0.8\nNd\n0.2\nFeO\n3\n nanoparticles prepared by SG method. (b) Same \ngraph \nin the range of 30<2θ<33\n \nXRD patterns of as-prepared samples by HT are depicted in Figs. 2 (a) and (b). Analysis \nof diffraction patterns for the samples confirmed the perovskite single phase. All the major \npeaks of the patterns were possible to index in the rhombohedral phase (R\n3\nC) for BiFeO\n3\n. \nHowever for the case of BNFO, there is a shift in main peaks at 30<2θ<33 and furthermore, it \nhas the tendency to merge in order to form one single widened peak (Fig. 2 (b)). One can relate \nthis behavior to a smaller ionic radius in neodymium compared to bismuth. Moreover, \ncomparison of Figs 1(b) and 2(b) indicates that width of major peacks in HT products are a \nlittle sharper compared to the as-synthesized SG samples.\n \nFinally, the analysis of particle size, pointed out that in the presence of Nd, the average \nsize of nanoparticles decreases (~40 nm for BFO and ~36 nm for BNFO). Neither the \ncharacteristic peaks of Bi\n2\nFe\n4\nO\n9\n nor of Bi\n25\nFeO\n40\n were found in any of BFO and BNFO \npatterns.  \n6 \n \nFig. 2.  (a) XRD patterns of pure and Nd-doped BiFeO\n3\n with x= 0.2 synthesized by HT route and (b) \nXRD patterns of pure BFO and BNFO in the range of 30<2θ<33. \n \n3.2 Furrier transform infrared spectroscopy  \nIn order to further confirm the crystallinity of as-prepared products, Fig. 3 shows the FT-\nIR spectra of BFO and BNFO nanoparticles in the range of 400-4000 cm\n-1\n. In both SG and HT \ncases, analysis of curves and band range 400-600 cm\n-1\n is related to metal- oxygen bond and \nconfirms the existence of perovskite structure for all samples. The vibration of Fe-O at ~450 \ncm\n-1\n and stretching vibration of O-Fe-O bonds at ~550 cm\n-1\n present in the octahedral FeO\n6\n \ngroup and in the  framework can be observed below 600 cm\n-1\n[44]. The broad band at 3200-\n3600 cm \n-1\n is due to antisymmetric and symmetric stretching of H\n2\nO and O-H bond groups \n[45]. A strong band at around 1380 cm\n-1\n was due to the presence of trapped nitrates [46]. \n7 \n \nFig. 3.  (a) FTIR spectra of BFO and BNFO nanoparticles synthesized by SG and (b) products \nsynthesized by HT.  \n3.3 Transmission electron microscope  \n          Transmission electron microscope (TEM) is used for the observation of morphology and \nparticle distribution and size of BFO and BNFO nanoparticles. As shown in Fig. 4, all samples \nconsisted of nano-scale particles, however the morphology of powders depending on the \nsynthesis method is very different. Although nanoparticles which were synthesized by SG are \nsemi-spherical, products obtained via HT are nanorods. Figs. 4 (a) and (b) are BFO and BNFO \nnanoparticles that were prepared by SG method. Particles in both images are semi-spheroid \nand rectangular particles with some irregular shape particles which look denser. In contrast, \nproducts which were obtained from HT method have rod-like shape and look looser (Figs. 4 \n(c) and (d)). After incorporating the BiFeO\n3\n with Nd, it can be seen that there is no significant \nchange in the morphology of both SG and HT products. The size of nanoparticles obtained \nfrom TEM is a little larger than that obtained from the Scherrer equation. According to analysis \nof images, the average size of SG as-prepared products were obtained 55 and 51 nm for BFO \nand BNFO nanoparticles and a thickness length ranging from 45-50 nm and 39-42 nm for HT \nas-prepared BFO and BNFO powders. The length of nanorods was also in a range of 80-300 \nnm. The results demonstrate a clear influence of the preparation method on the structural \nfeatures of the prepared nanocomposites. \n8 \n \nFig. 4.\n  (a) and (b) \nTEM images of BiFeO\n3\n and Bi\n0.8\nNd\n0.2\nFeO\n3 \nsynthesized by SG method. (c) and (d) \nshow BFO and BNFO nanopowders prepared via HT route. \n3.4 Thermal behavior \nThe results of differential thermal analysis (DTA) for pure and 20%Nd-doped bismuth \nferrite nanoparticles synthesized by the sol-gel method are shown in Figure 5 (a). The inset \nalso indicates the thermogravimetric (TG) analysis of samples. In the case of BFO, in DTA \ncurve for a temperature in the range of 180 °C, an exothermic peak can be seen with an instance \nof 0.2% reduction in weight. This peak can be attributed to hydrate thermal and the nitrate \npresent in the course of evaporation as well as evaporating water on nanoparticle’s surface. In \nthe range of 400-600°C, one can observe a small exothermic peak as well as a weight reduction \nof 0.15%. This peak is an indication of  the oxidation reaction between Fe\n3+\n and Bi\n3+\n and it is \nconsidered as an evidence for the crystal phase of BFO [47]. In the range of 820-840 °C, an \nendothermic peak appears which is due to electrical transmission temperature (T\nC\n ~ 830 \no\nC). \nFor Bi\n0.8\nNd\n0.2\nFeO\n3\n in DTA curve, the hydrate and nitrate decomposition peaks were eliminated \nand 0.15% of the weight was declined. Also the phase transition from ferroelectric to Para-\nelectric states occurred with exothermic peak in lower temperature (T\nC\n ~828 \no\nC).  \nInset in Fig. 5 (a) shows TG analysis of the BFO and BNFO nanoparticles. In the curves, \nup to 200\no\nC the loss of physisorbed water is observed. The second part (200-450\no\nC) \n9 \ncorresponds to the loss of surface hydroxyl groups. Finally the weight loss is observed above \n500\no\nC, which is due to the decomposition of the nitrate spices [45].  \n \nFig. 5.  \n(a) DTA and TG (as an inset) curves of BFO and BNFO nanoparticles synthesized by sol-\ngel method. (b) Same curves for as-prepared BFO and BNFO nanoparticles synthesized by \nhydrothermal method.\n \n \n        Same as Fig. 5 (a), in Fig.5 (b) differential thermal analysis curves of BiFeO\n3\n and \nBi\n0.8\nNd\n0.2\nFeO\n3\n nanocrystals are shown. These substances are synthesized by hydrothermal \nmethod. In DTA curves for a temperature at the range up to 200°C, an exothermic peak can be \nobserved. This peak is due to available nitrate and evaporating water on the surface of \nnanoparticles. In the range of 400-600°C a small exothermic peak is seen. This peak indicates \nthe oxidation reaction between Fe\n3+\n and Bi\n3+\n. This can confirm the existence of crystal phase \nfor BFO and 20%Nd-doped BFO. A phase transition at 833 \no\nC for BiFeO\n3\n and 831 \no\nC for \nBi\n0.8\nNd\n0.2\nFeO\n3 \ncan be seen in the measurements, which is related to the Curie temperature.  \n \n3.5 M-H hysteresis loops analysis of BFO and BNFO \nThe RT magnetic hysteresis loops for all samples are shown in Fig. 6. In contrast to the \nantiferromagnetism for bulk BFO, nanoparticle samples showed weak ferromagnetic behavior \nwhich is in agreement with other reports of BFO [10,48,49]. Table 1 shows the information of \nmagnetic characterization tests for the as-prepared BFO and BNFO nanoparticles synthesized \nby SG and HT methods. According to literature, in bismuth ferrite the iron ions (Fe\n3+\n) have a \nstrong relationship with magnetization and these ions are responsible for magnetic properties \n10 \nof BiFeO\n3\n. Surrounding each (Fe\n3+\n) ion with a certain spin there are six other ions with \nnonparallel spins. These spins are not completely nonparallel however they are organized in a \nspiral manner with a period of 62nm, which leads to a magnetization value zero. Breaking the \nspiral organization of the spin is due to nanoparticles size reduction to less than 62 nm and the \nrise of uncompensated spins on the surface of nanocrystals (because of a rise of area relative \nto volume). This can be a justification for the increase in magnetic properties and the decline \nin ferromagnetic properties of bismuth ferrite [5,10]. When neodymium enters in the structure \nof bismuth ferrite, saturated magnetization (M\ns\n) decrease s. However, coercive force (H\nc\n) \nincreases. The remanent magnetization in HT-prepared samples is also increased, but for the \nSG-prepared samples, M\nr\n decreased. \n \nFig. 6. (a) Magnetic hysteresis loops of BNFO nanoparticles with x=0,0.05,0.1,0.15,0.2 at \ntwo  calcination temperatures of 550°C. (b) 650°C  \nBy neodymium doping, the size of nanoparticles has declined. On the other hand, the \nmorphology of nanoparticles are completely different according to synthesis methods. In \ngeneral by decreasing the size of particles and domains, the energy for changing the magnetic \nmoments increases , which is due to changes in mechanism of process. For the HT-prepared \nsamples, loops are not saturated up to 20 KG. The saturated magnetization in SG-prepared \nsamples is much higher compare to the other ones. On the other hand, the coercive force for \nsamples obtained from HT method, is slightly higher than the SG prepared samples.  \n \n \n \n11 Table 1. Information from magnetic characterization for BFO and BNFO synthesized by sol gel and \nhydrothermal\n \nHydrothermal  Sol gel  Sample  \nH\n\u0000\n(\nG\n)\n \nM\n\u0000\n(\nemu\n/\ng\n)\n \nM\n\u0000\n(\nemu\n/\ng\n)\n \nH\n\u0000\n(\nG\n)\n \nM\n\u0000\n(\nemu\n/\ng\n)\n \nM\n\u0000\n(\nemu\n/\ng\n)\n \n208.97   0.0074   0.1093   63.28 0.0027 0.63 BiFeO\n3\n \n332.69   0.0037   0.0096   167.90\n \n0.0068 0.54 Bi\n0.8\nNd\n0.2\nFeO\n3\n \n \n3.6 Ferroelectric properties of BiFeO\n3\n and Bi\n0.8\nNd\n0.2\nFeO\n3\n nanoceramics \nIn order to study the ferroelectric properties of products, here, the as-synthesized BFO and \nBNFO nanoparticles were pressed and were coated with a thin layer (~30 nm) of silver as \nelectrodes by using DC sputtering. Fig. 7 presents the RT polarization-electric field (P-E) \nhysteresis loops of BFO and BNFO nanoparticles synthesis by SG and HT methods under \napplied electric field up to 25 kV/cm. It can be seen that for all products loops are unsaturated. \nBy adding Nd, into the structure of BFO, the saturated polarization is increased. However in \nboth synthesis method results, remanent polarization and coercive field are decreased and \nreduced the leakage current [50,51]. The reduction of leakage current by doping Nd, is due to \na reduction of oxygen vacancies [52,53]. When Nd is doped to BFO, the P-E loop is improved \nshowing an elongated loop compared to the BFO samples.  \n \n \nFig. 7. P-E hysteresis loops of BFO and BNFO nanoparticles synthesized by (a) sol-gel and (b) \nhydrothermal.  \n12 \nIn order to study the influence of Nd on the ferroelectric behavior of all products, leakage \ncurrent density- electric field (J-E) curves of BiFeO\n3\n and Bi\n0.8\nNd\n0.2\nFeO\n3\n nanoparticles \nsynthesized by SG and HT at RT is plotted in Fig. 8. BFO (BNFO) nanoparticles synthesized \nvia SG show high leakage current compared to the BFO nanorods which were synthesized by \nHT. Although the J is decreased with the dopant Nd in the nanoparticles and remained much \nlower than that of BFO, the synthesis method also affects the leakage current and in the case \nof BNFO, the products synthesized by HT route shows the bigger leakage current density. \n \nFig. 8. RT current density- applied electric field (J-E) BFO and BNFO nanoparticles synthesized by \nSG and HT.  \n3.7 Uv-vis spectroscopy analysis \nThe optical band-gap of BiFeO\n3\n and Bi\n0.8\nNd\n0.2\nFeO\n3\n nanoparticles have been calculated \nwith the help of absorbance spectra. Fig. 9 shows the UV-vis absorption spectra of all products. \nBy using Tauc’s equation, the energy band-gap (E\ng\n), absorption coefficient (α), is related by \n(for materials with direct band-gap): (αhν)\n2\n=K(hν-E\ng\n), where K is a constant and hν is the \nphoton energy. An inset in Fig.8 indicates the plot of (αhν)\n2\n vs hν for BiFeO\n3\n and \nBi\n0.8\nNd\n0.2\nFeO\n3\n nanoparticles synthesis by SG and HT methods. The extrapolated straight line \nfitted to the linear part of curves gives the value of E\ng\n. The extracted values of E\ng\n for BFO \n(BNFO) nanoparticles synthesis by SG method is ~ 2.13 eV (2.08 eV) and 2.11 eV (2.04 eV) \n13 \nfor products that obtained from HT reaction route. Slight decrease of E\ng\n value by Nd doping \nwas observed which indicates the narrowing of the optical band-gap and enhanced \nphotocatalytic activity and photovoltaic effects. \n \nFig. 9. UV-vis spectrum of BiFeO\n3\n and \nBi\n0.8\nNd\n0.2\nFeO\n3\n nanoceramics prepared by SG and HT \nmethods. Inset shows \n(αhν)\n2\n versus hν plots of all samples.  \n \n3.8 Photocatalytic activity of BFO and BNFO \nThe photocatalytic activity of products were evaluated by degrading methyl orange (MO) \nin an aqueous solution under visible light irradiation. The reaction rate constant of samples are \nplotted in Fig. 10. All BFO and BNFO nanoparticles have visible light induced photocatalytic \nactivity. The Nd-substitution has a significant effect on the photocatalytic activity of bismuth \nferrite. The first-order rate constant is calculated by the equation ln(C\n0\n/C) = κt, where C\n0\n and \nC are concentrations of MO in solution at the beginning of the tests and at time t. The reaction \nrate constant (κ) is the slope in the apparent of ln(C\n0\n/C) vs. time. The HT-synthesized (SG-\nsynthesized) BFO nanoparticles showed ~ 16% (23%) degradation of MO in 350 min. On the \nother hand, the addition of neodymium inside BFO nanoparticles, enhanced the photocatalytic \n14 \nactivity of BFO nanopowders. The total removal of MO by HT-synthesized (SG-synthesized) \nBNFO is 61% (73%). This result indicates that the morphology of nanoparticles is an important \nfactor that has effects on the photocatalytic activity of BFO or BNFO particles. \nFig. 10. (a) Photocatalytic degradation rates of BFO and BNFO nanoparticles prepared by SG and \nHT methods under visible light irradiation. (b) Represents the pseudo-first-order kinetics curves \nof the MO degradation. \n \n4. Conclusions \n \nIn this paper the influence of synthesis method on the physical properties and \nphotocatalytic activity of BiFeO\n3\n and Bi\n0.8\nNd\n0.2\nFeO\n3\n nanoparticles was investigated by \ndifferent characterization methods. In the BFO samples the main peaks change their shape \nand location after neodymium is involved and they have a tendency to merge and form a \nunit peak. The results from TG/DTA showed that by doping nanoparticles of bismuth ferrite \nthe peak related to phase transition from ferroelectric to para electric took place in a lower \ntemperature. The photocatalytic degradation of methylene orange (MO) under visible light \nirradiation was also implemented. Moreover the effect of neodymium doping into the \nbismuth ferrite, is studied. Results showed that the Nd-substitution, enhanced the \nferroelectric characteristics and reduced the leakage current. 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In the letter, we use blade \nstructure to realize the propagation of SSPPs wave and a \nmatching transition is used to feed energy from coplanar \nwaveguide to the SSPPs.  And the circulator shows good \nnonreciprocal transmission character istics. The simulation result s \nindicate  that in the frequency band from 5 to 6.6 GHz, the \nisolation degree and return loss basically reaches 15dB and the \ninsertion loss is less than 0.5dB. Moreover, the use of confinement \nelectromagnetic waves can decrease  the size of the ferrite and \nshow a broadband characteristic.   \n \nIndex Terms —Circulators, Plasmons , coplanar waveguide  \n \nI. INTRODUCTION  \nhree-port circulator s are  passive nonreciprocal device s, in \nwhich a microwave or radio frequency signal entering any \nport is transmitted only to the next port in rotation [1-5]. \nNowadays , circulators are used more often in microwave area. \nFor instance, circulators are frequently used in \ntelecommunication networks, such as RADARs, cellular \ncommunications and broadcasting s ystems, to connect \ntransmitters and receivers to the same antenna, avoiding \ninterference between them  [6]. In circulator, ferrites are usually \nused to realize the nonreciprocal characteristics. However, to \nget a broadband, junction circulator should have a  long \ntransmission line for impedance matching. And the junction \ncirculator has a bulky ferrite. So making ferrite thin and small is \nstill a challenge.  \nRecently, more efforts have been made towards the \ndevelopment of spoof surface plasmon polaritons  (SSPPs ) for \nmicrowave propagation. SSPPs is the high -confinement SPPs \n(surface plasmon polaritons) at low frequency, in which \nEM(electromagnetic)  waves are bound on the metal/dielectric \ninterface, propagating parallel to the interface and decaying \nexponentially in the direction vertical to the interface, just like \n \nProject is supported by the National Natural Science Foundation of China \n(Grants Nos. 61331005, 11204378, 11274389, 11304393, 61302023), the \nNational Science Foundation for Post -doctoral Scientists of China (Grant Nos. \n2013M532131, 2013M532221) . \nTianshuo Qiu is with College of Science, Air Fo rce Engineering University, \nXi’an, Shaanxi 710051,China (e -mail: 909154790@qq.com).  \nYongfeng Li is with College of Science, Air Force Engineering University, \nXi’an, Shaanxi 710051,China (e -mail: liyf217130@126.com).  \nJiafu Wang is with the College of Scien ce, Air Force Engineering University, \nXi’an, Shaanxi 710051,China (e -mail: wangjiafu1981@126.com).  \nShaobo Qu is with College of Science, Air Force Engineering University, \nXi’an, Shaanxi 710051,China (e -mail: qushaobo@mail.xjtu.edu.cn).  “real SPPs”  [7-10]. J. Pendry et al. reported that structured \nmetal surface can support and propagate SSPPs  [7]. In order to \nproduce high transmission efficiency between single wire and \ncoplanar waveguide (CPW) , many efforts have been made [1 1–\n14]. As we know, SSPPs have good characteristics of high field \nconfinement, planar configuration and deep -subwavelength, so \nwe can take the advantage of the SSPPs to fabrication \ncirculator.  \nIn this paper, we  study a microwave circulator based on \nSSPPs. We propose to utilize metallic blade structure to realize \nSSPPs and replace the stripline and junction with SSPPs. The \naddition of high -confinement electric field reduces the space to \nrealize circulation perfor mance and makes broad bandwidth. \nAnd a matching transition has been proposed, which is \nconstructed by gradient corrugations and flaring ground, to \nmatch both the momentum and impedance of CPW and the \nSSPPs .  \nII. THEORY , AND DESIGN  \nThe circulator is composed of  an inner conductor which has \nthree access. Above and below this inner conductor, there are \ntwo circular discs of ferrite. Dielectric sleeve (NETE C NY 9208 \n𝜀r=2.08 tan δ  =0.001) is around the ferrite for matching then \ntwo ground planes are closing the s tructure, as presented in Fig. \n1.    \n \nFig. 1 .  Geometric model of the proposed SSPPs circulator  \nThe material of the ferri te is yttrium iron garnet (YIG). \nSaturation  magnetization (4πMs ) is 1800Gauss, resonance \nlinewidth is 15Oe and permittivity is 14.3. Ferrite with radius \nr0=4.7 mm and height h0=1.4mm is considered by theory \nanalysis. In the letter, to realize the propagation of SSPPs wave, \nblade structure is used in the inner conductor , as shown in Fig. \n2(a). In order to feed energies into the SSPPs, we use CPW and \npropose a smooth conversion between SSPPs and the CPW, as \nCirculator based on spoof surface plasmon \npolaritons  \nTianshuo Qiu, Yongfeng Li, Jiafu Wang, and Shaobo Qu  \nT  2 \npresented in Fig. 2(b). In the conversion section, the matching \ntransition with gradient grooves and flaring ground are used to \nmake conversion high -efficiency. The groove depth varies from \nh1 =0.9mm to h3= 2.7mm with a step of 0.9mm. The length and \nwidth of the flaring ground are designed as L2 = 2.4 mm, L3= \n0.9 mm, respectively. The parameters for the blade structure are \noptimized to be: the period of the blade p=0.5mm, the width of \nthe grooves a=0.2mm, and the depth of grooves h=0.2mm, the \nouter radius  R2 is 6.35mm, as presented in Fig. 2(c).  \n \nFig. 2.  Geometric model of the proposed inner conductor.  \n(a) Top view of inner conductor.  𝑅2=6.35mm, 𝑅3=3.65mm.  \n(b) The CPW section and the matching transition with gradient grooves \nand flaring ground, in which L=3mm, 𝐿2=2.4mm, 𝐿3=0.9mm, ℎ1 \n=0.09mm, ℎ2=0.18mm and ℎ3=0.27mm.  \n(c) Geometric of the metallic blade structure in detail, in which p= 0.5 mm, \na=0.2 mm, and h = 0.4 mm.  \nFor interconnect CPW to the SSPPs the smooth conversion \nsection is employed. When the groove depth gradually \nincreases and the ground of the matching transit ion is gradually \nflared out, the propagating wave number is changed from k0 to \nkspp, realizing  the perfect momentum matching from CPW (k0)   \nto the blade structure ( kspp). As a result, the transmission \nefficiency of the metallic blade structure has enhanced and the \nimpedance and momentum between SSPPs mode and CPW \nmatch perfectly.  \nAs we all know, ferrite in a magnetic field shows anisotropy \ncharacteristic. To understand th e origin of the anisotropy, the \ninteraction between magnetic moments of the ferrite and a \nmagnetic field needs to be studied. The spectral expression of \nthe equation of motion leads to a complete  permeability tensor  \n00\n=0\n00j\nj\n\n  \n  \n                           (1) \nWhere κ and μ are the tensor elements of the ferrite. The \nelements of the permeability tensor depend on the frequency \nand also depend on the magnitude of the applied field as well as \non the saturation magnetization of the material.  \nWhen t he high -confinement EM wave propagates along the \nblade structure, the constituents of both electric field and \nmagnetic field are in many directions. If the medium around the blade structure is anisotropy such as ferrite, some constituents \nwill interact wit h the medium and then the intensity will be \nchanged. That will make intensity of electromagnetic wave in \nsome section increase or decrease. In this case, we assume the \nenergy is put in port 1. When the metallic blade structure is \nseparated into different d irection. Due to the anisotropy of the \nferrite, the energy propagates from port 1 to port 2 instead of \naverage distribution in the cross point. Obviously, the \ncomponent will realize the circulation performance.  \nHigh -efficiency transmission can be realized as long as the \npropagating wave number has changed from 𝑘0 to 𝑘spp by \nconversion section that interconnect CPW to the SSPPs. And \nthe conversion section can realize wideband transmission by \nitself. So the quarter -wave transmission line for impedance \nmatch ing is unnecessary to realize wideband performance. That \nmakes the circulator smaller undoubtedly, because the \nconversion section that interconnect CPW to the SSPPs is much \nsmaller than quarter -wave transmission line section. Moreover, \nless space are neede d because of the characteristic of the \nhigh-confinement electromagnetic wave. The SSPPs wave are \nconfined close to the blade structure. That can reduce the bulk \nof ferrite.   \n \n \nFig. 3.  The S parameters of the Circulator based on SSPPs.  \nIII. RESULT AND DISCUSSION  \nA high frequency electro -magnetic field simulation software \nbased on a three dimensional finite element method, Ansoft \nHFSS 15.0, is used to analyze the transmission characteristics \nof the circulator. The external magnetic field we choose is \n10000A/M. First of all, the simulated return loss , insertion loss, \nisolation , are studied and the results are shown in Fig. 3. From 5 \nGHz to 6.6 GHz, return loss and isolation degree basically \nreaches 15 dB, the insertion loss is less than 0.5dB. If we \nchange d the design of the circulator, the bandwidth in which \ninsertion loss is less than 1dB can reach 6 GHz (3GHz - 9GHz).  \nTo verify the transmission efficiency of the conversion \nsection, we simulate the conversion structure and blade \nstructure, as shown in Fig .4. Figure 4 illustrates that the \nconventional guided waves in CPW section are turned to \nconfinement electromagnetic wave successfully. A nd the \nSSPPs wave can propagate  along the blade structure.  \n \n 3 \n \nFig.4. Simulated results of near electric field distributions in blade structure section and CPW section.\n \nFig.5. f=6GHz，the distribution of (a) the electric field x-component ( b) the \nelectric field y-component  (c) the magnetic field y-component  \n \nFigure 5 shows  the distribution of the electric field and \nmagnetic field. We can clearly find that as the EM wave \npropagates along the conductor, the electric field z-component  \nis transformed into x -component . And figure 6 shows the \npropagation  of electric field in metallic blade structure of the \ncirculator when energy is input in port 1. As predicted, a \nhigh-confinement electric field exists in SSPPs sections. The \nintensity of the confinement electric field are ne arly the same in \nsection  Ⅰ and Ⅱ. As the confinement EM propagates along the \nX direction, the confinement electric field in section  Ⅰ \nincreases gradually. In the contrary, the confinement electric \nfield in section  Ⅱ decreases. That makes the circulation \nperformanc e. We clearly observe t he high -efficiency \ntransmissions with very small reflections in the figure.  \n \n \nFig.6. Simulated intensity distributions  of the tangential electric field  of the \ncirculator  (f=6GHz) . (a) phase= 50° (b) phase=110° \nIV. CONCLUSION  \nThrough this paper we have applied the metallic blade structure to a [5 -6.6] GHz band circulator, and design a \nhigh-performance circulator. We analyze the characteristics of \nthe circulator. The results in the microwave frequency show the \ngood characteristi cs of the circulator .This study proves that the \ncirculator with SSPPs structure we designed operates with an \nacceptable nonreciprocal transmission characteristics as a \npractical circulator device.  \nIn this letter, the confinement electromagnetic wave is lead \ninto the new kind of circulator based on SSPPs. The \nconfinement electromagnetic wave increases the characteristic \nof the circulator. The new kind of circulator doesn’t need a long \ntransmission line for impedance matching. And the structure \ncan reduce t he bulk of ferrite and then makes a smaller \ncirculator. There are still lots of work to do to improve the \ncharacteristics. For example, we can utilize the center space \nwithout inner conductor to improve the bandwidth. And we can \nmake the guided waves combi ne the SSPPs wave and electric \nand magnetic (TEM) mode to realize broadband characteristic.  \nREFERENCES  \n[1] H. Bosma, “On stripline Y -circulation at UHF,” IEEE Trans. Microw. \nTheory Tech., vol. 12, no. 1, pp. 61 –72, Jan. 1964.  \n[2] E. Schloemann, “Miniature circulat ors,” IEEE Trans. Magn., vol. 25, no. \n5, pp. 3236 –3241, Sep. 1989.  \n[3] S. Ping, H. How, X. Zuo, S. D. Yoon, S. A. Oliver, and C. Vittoria, \n“MMIC circulators using hexaferrites,” IEEE Trans. Magn., vol. 37, no. 5, \npp. 2389 –2391, Sep. 2001.  \n[4] B. K. O’Neil, and J. L. Young, “Experimental investigation of a \nself-biased microstrip circulator,” IEEE Trans. Microw. Theory Tech., \nvol. 57, no. 7, pp. 1669 -1674, Jul. 2009.  \n[5] J. Wang, A. Yang, Y.  Chen, and  Z. Chen et al., “Self Biased Y -Junction \nCirculator at Ku Band,”  IEEE M icrow. Wire. Compon. Lett. , vol. 21, \nno. 6 , pp. 292 -294, Jun. 2011. \n[6] V. G. Harris, A. Geiler, Y. Chen, S. D. Yoon, M. Wu, and A. Yang et al., \n“Recent advances in processing and applications of microwave ferrites,” J. \nMag. Mag. 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Cui, \"Broadba nd and \nhigh-efficiency conversion from guided waves to spoof surface plasmon \npolaritons,\" Laser Photonics Rev., vol. 8, no. 1, pp. 146 –151, Nov. 2014.  \n[12] A. Treizebre, T. Akalin, and B. Bocquet, “ Planar excitation of Goubau \ntransmission lines for THz bioMEMS ,” IEEE Microw. Wire. Compon. \nLett. , vol. 15, no. 12 pp. 886-888, Dec. 200 5. \n[13] T. Akalin, A. Treizebre, and B. Bocquet, “Single -wire transmission lines \nat terahertz frequencies,”  IEEE Trans. Microw. Theory Tech.,  vol. 54, no. \n6, pp. 2762 -2767, Jun. 2006.  \n[14] W. C. Chen, J. J. Mock, D. R. Smith, T. Akalin, and W. J. Padilla, \n“Controlling Gigahertz and Terahertz surface electromagnetic waves with \nmetamaterial resonators,” Phys. Rev. X, vol. 1, no. 021016, pp. 1 -6, Dec. \n2011.  \n"    },    {        "title": "2110.09205v1.Control_of_Electrochemical_Corrosion_Properties_by_Influencing_Mn_Partitioning_through_Intercritically_Annealing_of_Medium_Mn_Steel.pdf",        "content": "   \n \n \n   \nControl of  Electrochemical Corrosion Properties by Influencing \nMn Partitioning through Intercritically Annealing of Medi-\num-Mn Steel  \nRené Daniel Pütz1, Tarek Allam2,3, Junmiao Wang1, Jakub  Nowak1, Christian Haase2, Stefanie Sandlöbes -Haut4, \nUlrich Krupp2 and Daniela Zander1,* \n1 Chair of Corrosion and Corrosion Protection, Foundry Institute, RWTH Aachen University, Intzestr. 5, 52072 \nAachen, Germany; r.puetz@gi.rwth -aachen.de (R.D.P.) ; j.wang@gi.rwth -aachen.de (J.W.); \nj.nowak@gi.rwth -aachen.de (J.N.) ; d.zander@gi.rwth -aachen.de (D.Z.)  \n2 Chair Materials Engineering of Metals , Steel Institute, RWTH Aachen University, Intzestr. 1, 52072 Aachen, \nGermany; tarek.allam@iehk.rwth -aachen.de (T .A.); christian.haase@iehk.rwth -aachen.de  (C.H .); \nkrupp@iehk.rwth -aachen.de (U.K.)  \n3 Department of Metallurgical and Materials Engineering, Suez University, 43528 Suez, Egypt  (T.A.)  \n4 Institute  for Physical Metallurgy and Materials Physics, RWTH Aachen University, Kopernikusstr. 14, 52074 \nAachen, Germany; sandloebes -haut@imm.rwth -aachen.de  (S.S.)  \n* Correspondence: Daniela Zander, d.zander@gi.rwth -aachen.de  (D.Z.)  \nAbstract: Medium -Mn steels exhibit  excellent mechanical properties and lower production costs \ncompared to high -Mn steels, which makes them a potential material for future application in the \nautomotive industry. Intercritical annealing  (ICA)  after cold  rolling allows to control the s tacking \nfault energy (SFE) of austenite, the fraction of ferrite and reverted austenite, and the element parti-\ntioning (especially Mn). Although Mn deteriorates the corrosion behavior of Fe -Mn-Al alloys, the \ninfluence of austenite fraction and element parti tioning of Mn on the electrochemical corrosion \nbehavior has not been investigated yet. Therefore, the electrochemical corrosion behavior in \n0.1 M H2SO 4 of X6MnAl12 -3, which was intercritically annealed for 2 h at 550 °C, 600 °C and 700 °C, \nwas investigated  by potentiodynamic polarization ( PDP ), electrochemical impedance spectroscopy \n(EIS) and mass spectroscopy with inductively coupled plasma (ICP-MS). Additionally, specimens \nafter 1  h and 24  h of immersion were examined via SEM  to visualize the corrosion damage. The \nICA specimens showed a selective dissolution of reverted austenite due to its micro -galvanic \ncoupling with the adjacent ferrite. The severity of the micro -galvanic coupling can be  reduced by \ndecreasing the  interface area as well as the chemical gradient of mainly Mn between ferrite and \nreverted austenite  by ICA.  \nKeywords: medium -Mn steel; intercritical annealing (ICA); reverted austenite; Mn partitioning; \ncorrosion; micro -galvanic coupling;  selective dissolution  \n \n1. Introduction  \nAdvanced high strength steels (AHSS) have been developed since the 1980s to fulfill \nthe low emission and high fuel -efficiency demands in the automotive industry [1]. Sig-\nnificant research has been conducted in the development and application of AHSS \nthroughout the world. Compared to conventional high -strength steels, AHSS strongly \nreduce vehicle weight and show a better combination of strength, ductility , and tough-\nness, which enhances crash performance and vehicle safety . Recently, the third genera-\ntion AHSS is of great interest because of the combination of strength , formability and \nlower production costs in comparison to the first and second generation AHSS [2-4]. \nMedium -Mn steels belong to the third generation of AHSS , which  show excellent me-  2 of 23  \n chanical properties a t reduced Mn content  and production costs compared to high -Mn \nsteels  [5,6] .  \nMedium -Mn steels have an ultrafine grained duplex microstructure containing fer-\nrite (α-Fe) and reverted austenite  (ɣ-Fe), resulting from inter -critical annealing  (ICA) \ntreatment s in the α + γ region.  In addition to Mn, Al and/or Si are important alloying \nelements in alloy design of  medium -Mn steels, since they  play a crucial role in sup-\npressing cementite  formation, expand the ICA temperature -window and control the \nstacking fault energy (SFE) of austenite  [7]. Another important aspect during ICA treat-\nments is  the local element partitioning  that is responsible for the stability of reve rted \naustenite and accordingly tuning the mechanical properties  [8-10]. Many researchers \nhave investigated the influence of ICA on the microstructure and mechanical properties \nof medium -Mn steels. It was  found that the ICA temperature strongly affect s the volume \nfraction, morphology and stability of reverted ɣ-Fe [3,5,11 -14]. Reverted ɣ-Fe transforms \ninto α´ -martensite and/or twins during deformation, influencing the strain -hardening \nbehavior and ductility. It is reported that the ICA time ha s effects on the morphology of \nreverted  ɣ-Fe and the mechanical properties  [5,15] . Besides, the initial microstructure also \nplay s a role in the phase morpholog ies in medium -Mn steels  [14,16 -19]. \nSince corrosion is a  notorious  issue for steels, the corrosion behavior of Mn  steels is \nof high  interest for potential application. Only some  investigation o f the corrosion be-\nhavior of high -Mn steels has been conducted. It is found that high Mn content adversely \naffected the corrosion resistance of high -Mn steels  in various electrolytes like 1 M Na 2SO 4 \n[20] and 30 % NaOH [21] as well as 50 % NaOH [22] due to the instable  Mn-rich oxides \nformed in the corrosion products . Additionally, a high -Mn steel showed also a detri-\nmental corrosion resistance in 0.1 M H 2SO 4 compared to an interstitial -free steel due to \nthe high dissolution tendency of Mn  and Fe in the acidic solution [23]. Addition s of alloy \nelements in high -Mn steels could enhance the corrosion resistance due to the beneficial \neffects on passivation, such as Cr  in 3.5 wt.% NaCl [24,25] , 10–50% HNO 3 [22], and 1 M \nNa 2SO 4 solution [20,22] , Al in 1 M [21] and 5 wt.% Na 2SO 4 [26], 10-50% NaOH [21], 3.5 \nwt.% NaCl [24], 1 N H 2SO 4 [27] and 50% HNO 3 [20] and Si  in 1 N H 2SO 4 [27]. Meanwhile, \nit was observed that ɣ-Fe was more resistant to corrosion compared to α-Fe in 3.5 wt.% \nNaCl solution  [24,28] . The positive influence of grain refinement on the corrosion be-\nhavior of high -Mn steels in 3.5 wt.% was also reported  [29]. With growing interest in \nmedium -Mn steels, the corrosion behavior of medium -Mn draws attention as well.  Sev-\neral authors  [30-32] also reported adverse effects of Mn on the corrosion behavior of me-\ndium -Mn steel containing 5.5 wt.% Mn in simulated seawater and 3.5  wt.% as well as 5 \nwt.%  NaCl solution, which is similar to high -Mn steels. Besides, it was observed that the \naddition of alloy ing elements like Cr, Mo, Cu  and Ni improved the corrosion behavior  in \nsimulated seawater and 5.0 wt.% NaCl  solution  [30,32] . Allam et al. [6] developed an \naustenitic medium -Mn steel with  enha nced corrosion resistance  in 5 wt.% NaCl  solution \ndue to a high er Cr content, ultrafine microstructure and the addition of N. However, \nmore research is needed to comprehensively clarify the corrosion behavior of medi-\num-Mn steels. Especially the influence o f several heat treatment conditions on the corro-\nsion behavior of medium -Mn steels is hardly existent.  \nIn the present study, a medium -Mn steel X6MnAl12 -3 in 0.1 M H 2SO 4 was investi-\ngated in order to evaluat e the influence of intercritical annealing  and the concomitant \nchange in microstructure on the  electrochemical  corrosion behavior  focusing on the in-\nteraction between ferrite and reverted austenite.  \n2. Materials and Methods  \n2.1. Materials processing  \nThe chemical composition of  the medium -Mn steel X6MnAl 12-3 under investigation  \nis listed in Table 1. It was melted in a lab-scale vacuum induction furnace to produce a \ncast-ingot of 140 x 140 x 550 mm³. The cast -ingot underwent  hot forging  at 1150 °C to a  3 of 23  \n thickness of 40 mm  followed by  hot rolling at 1150 °C to manufacture hot -rolled sheets \nwith a thickness of 3 mm . These hot -rolled sheets were subjected to a homogenization \nheat treatment at 1100 °C for two hours and a subsequent water quench . Cold rolling \nprocesses were conducted at room temperature with a to tal thickness reduction of 50 % \nto prepare  cold -rolled  (CR)  sheets of 1.5 mm in thickness. For the subsequent electro-\nchemical investigations, rectangular specimens (11 mm x 10 mm x 1.5 mm) were seg-\nmented from the CR state by means of wire -cut electrical di scharge machining (EDM). \nThe target microstructures of the prepared specimens containing different volume frac-\ntions of austenite phase were adjusted through inter -critical annealing  (ICA) treatment s \nin a salt bath at different temperatures, namely; 550, 60 0 and 700 °C for two hours fol-\nlowed by a water quench.  The I CA treatments were designed based on thermodynamic  \ncalculations performed using Thermo -Calc ® software  (Thermo -Calc Software , Stock-\nholm, SE)  with the database TCFE10 for Steels/Fe -alloys. A schemat ic illustration  of the \nlaboratory processing rout e of the investigated steel X6MnAl12 -3 is shown in Figure 1. \nTable 1. Chemical composition in wt.% of the X6MnAl12 -3 medium -Mn steel  under investigation . \nSteel grade  C Si Mn P S Al \nX6MnAl12 -3 0.064  0.2 11.7 0.006  0.003  2.9 \n \n   \nFigure 1. Schematic illustration of the laboratory processing route of the investigated steel \nX6MnAl12 -3. \n2.2. Characterization  of the microstructures  \nThe microstructures  after cold rolling and annealing  were investigated using a Zeiss \nGemini field -emission  gun scanning electron microscope ( FEG -SEM) (Carl Zeiss Mi-\ncroscopy GmbH, Jena, DE) equipped with  an additional  Oxford X-Max50  energy disper-\nsive X -ray spectrometer (EDS) (Oxford Instruments, Abingdon, UK ). The applied accel-\neration voltages and working distances during SEM investigations were 15 -20 kV  and \n10-20 mm, respectively. The CR and I CA specimens  were prepared following the stand-\nard procedures for preparation of metallographic specimens  starting with mechanical \ngrinding up to  grit 2400 , followed by  mechanical polishing using 6 and 1 µm diamond \npolishing pastes. Afterwards, the polished specimens were subjected to 3% Nital etching \nsolution to reveal the features of the microstructures.  The present phases and their vol-\nume fractions in different states (CR and ICA) were identified and quantified  by means of  \nX-ray diffraction (XRD) technique using a Bruker D8 advanced X -ray diffractometer  \n(Bruker, Billerica, MA, US) . Fe Kα radiation (0.639 nm ) was used to acquire the diffrac-\ntograms.  \n 4 of 23  \n 2.3. Electrochemical testing  and corrosion analysis  \nThe r ectangular specimens were mounted in a non -conductive polymer resin with \nan exposed surface ( RD/TD plane ) area of approximately 1.1 cm2. The specimens were \nstepwise ground and polished till 0.25 µm. Afterwards, the specimens were rinsed with  \ndistilled water , ethanol  and then air-dried . The thin specimens were conductively glued \nwith a Cu -cylinder from the bottom.  A drilled hole acts as a con necting path for an iso-\nlated Cu-cable  to the specimens . A glass tube was glued to the resin above the Cu -cable \npreventing  the contact  of the solution with the Cu -cable and the specimen during the \nelectrochemical measurements. The samples were stored in a d esiccator  after gluing.  A \nsimilar working electrode (WE) setup is described elsewhere [33].   \nThe utilized electrolyte w as 0.1 M H 2SO 4 with a pH of 1. 1 ± 0.1. The electrolyte w as \nprepared with analytic grade chemicals and bi-distilled water. Figure 2 shows the sche-\nmatic illustration of the three -electrode setup for the electrochemical testing. A thermostat \nwas used to maintain a constant electrolyte temperature of 25 °C and a Refe rence 600 po-\ntentiostat  (Gamry Instruments  Inc, Warminster, PA, US)  was utilized to perform  poten-\ntiodynamic polarization (PDP) and  electrochemical impedance spectroscopy (EIS)  meas-\nurements . The specimen was used as the working electrode . A platinum sheet, w hich ex-\nhibits a larger  surface area compared with the working electrode, was used as the counter \nelectrode (CE) . A saturated calomel electrode (SCE), which was connected via a Ha-\nber-Luggin  capillary with the electrolyte in the beaker and placed approximate ly 1 cm in \nfront of  the specimen , was used as reference electrode (RE).  A solution volume of 700 ml \nwas used for PDP  and EIS experiments. The  solutions were purged using Ar for 30 min \nprior to the open circuit potential (OCP) measurement s of 60 min . \n \nFigure 2. Experimental setup (three -electrode cell) for the electrochemical testing adapted from \n[33,34] . \nThe PDP measurements were conducted in 0.1  M H 2SO 4 using a scan rate of 0.167 \nmV/s from  -0.3 VSCE vs. OCP up to 0.25 V SCE vs. OCP . For all polarization tests, iR compen-\nsation was applied. The corrosion current densities (i corr) were determined by graphical \nTafel analysis using the anodic and cathodic branches.  \n 5 of 23  \n Immersion tests with potentiostatic EIS measurements were utilized in 0.1 M H 2SO 4 at \n25 °C  for 24 h. The EIS measurements were performed at the OCP  over a frequency range \nfrom 10 kHz to 0.1 Hz  after 1, 4, 8 and 24 h , using 10 points per decade and applying an AC \namplitude of 5  mV rms.  Separate i mmersion test s without EIS measurements  were per-\nformed  in 0.1 M H 2SO 4 at 25 °C for 24 h . A solution sample of 5 ml was taken at specific \ntime intervals, namely, after 0, 1, 4, 8 and 24 h.  The chemical concentration s (57Fe, 55Mn and \n27Al isotopes ) of the solution samples  were analy zed without dilution or additional acidi-\nfication  by NexION ® 2000 mass spectroscopy with inductively coupled plasma (ICP -MS) \n(Perkin Elmer, Waltham, MA, US)  in kinetic energy discrimination  (KED) mode . The KED \nmode was used to diminish polyatomic ion interferences  [35]. The purity of the used neb-\nulizer and collision gas was Ar  (99.999  vol.%) and He (99.9999 vol .%) with 0  % relative \nhumidity, respectively.  Three solution samples were investigated for each heat treatment \ncondition.  Top view and cross -section images of the rim zone after  the immersion tests \nwere performed using a Supra 55 VP scanning electron microscope (SEM) ( Carl Zeiss Mi-\ncroscopy GmbH , Jena, DE). \n3. Results  \n3.1. Thermodynamic caculations  \nThe equilibrium austenite and ferrite phase fractions as well as  their correspondin g \nchemical compositions  were predicted based on the thermodynamic calculations repre-\nsented in Figure 3. Accordingly, the ICA treatments were designed to adjust the target mi-\ncrostructures with low, intermediate and high austenite fractions within the select ed ICA \nrange starting from 550, 600 to 700 °C as indicated in Figure 3. The evolution of phase frac-\ntions shows an obvious increase in equilibrium austenite fraction  at the expense of ferrite \nphase  within the ICA range  from 0.42 at 550 °C, 0.52 at 600 °C to 0.78 at 700 °C.  Such in-\ncrease in austenite fraction with increasing the temperature within the ICA range is ac-\ncompanied with elemental partitioning phenomena that control the subsequent micro-\nstructural features and t he related electrochemical behaviors. It is clear that Mn and C, \nstrong  austenite stabilizers, generally tend to be enriched in austenite Figure 3 (a) as com-\npared  to ferrite Figure 3 (b). However, Al and Si, ferrite stabilizers, are more enriched in \nferrit e than in austenite. As the ICA temperature increase s, the Mn content in austenite \ndecreases from almost 22 wt.% at 550 °C to around 18 wt.% at 600 °C and 13 wt.% at 700 °C. \nMoreover, a concurrent increase of the  Al content in austenite is observed with in creasing \nthe ICA  temperature  reaching more than 2.2 and 2.6 wt.% at 600 and 700 °C, respectively, \nwhile it maintains  below 2 wt.% at 550 °C. It is worth mentionin g that compared with aus-\ntenite, the Mn content in ferrite  generally  shows relatively low value s below 5 wt.% at 550 \nand 600 °C and slightly higher than 5 wt.% at 700 °C. Meanwhile, the Al content in ferrite \nhardly changes with increasing ICA temperature and shows always higher values (around \n4.5 wt.%) compared with it s values in austenite phase.  Likewise, the Si content  in ferrite  \nstays at a relatively high level of around 1 wt.% compared with its level in austenite that \nshows a slight decrease from around 0.36 to 0.23 wt.% when the ICA temperature increases \nfrom 550 to 700 °C, respectively.   6 of 23  \n  \nFigure 3. Thermodynamic calculations performed using Thermo -Calc ® software package (TCFE10 \nfor Steels/Fe -alloys). The evolution of phase fractions and their corresponding contents of alloying \nelements under thermodynamic equilibrium conditions are represen ted (a) for austenite and (b) \nferrite.  \n3.2 Microstructure evolution during ICA treatment  \nThe SEM micrographs of medium -Mn steel X6MnAl12 -3 in CR as well as in  ICA \nstates are displayed in Figure 4. The  CR state  exhibits a fully martensit ic microstructure \nwith deformed martensite being preferentially elongated parall el to the rolling direction  \n(Figure 4 (a)). However , the ICA states generally show dual -phase  microstructures con-\ntaining reverted austenite and ferrite  grains . The relative amounts of the phases show an \nobvious dependency on the applied ICA temperature s. Figure 4 (b) shows  the developed \ndual -phase microstructure after ICA at 550 °C followed by a water quench. The inset of \nFigure 4 (b) reve als the formation of ultrafine grained  islands of austenite  with a rela-\ntively small amount embedded in the ferrite matrix. Both t he amount and size of austen-\nite phase increase as the applied ICA temperature increases to 600 °C , as represented  in \nFigure 4 (c). Furthermore, it can be observed from the inset of Figure 4 (c) that the  aus-\ntenite phase shows an elongated grain morphology, which is inherited from the de-\nformed martensite grains. Such elongated morphology of austenite  grains  become s less \npronounced and changes to a mixed type of equiaxed and elongated m orphologies  as the \napplied ICA temperature increases further to 700 ° C (inset of Figure 4 (d)). Moreover , the \nICA treatment at 700 °C resulted in a clear increase in the amount of austenite phase . The \n 7 of 23  \n XRD -measurements carried out to quantify the constituti ng phases of the CR and ICA \nstates are shown in Figure 5. The  XRD spectra depicted in Figure 5 (a) demon strate  the \npresence of the characteristic peaks of austenite phase for the ICA states (550, 600 and 700 \n°C), while the characteristic peaks of only ferrite/martensite  appear for the CR state. The \nvolu me fraction s of austenite p hase in the ICA states are displayed as a bar chart analysis \nin Figure  5 (b) indicating the increase of the austenite volume fraction from 21.9 vol.% for \nICA at 550 °C to 40.8 and 58.5 vol.% as the ICA temperature increases to 600 and 700 °C, \nrespectively.  \n \nFigure 4. SEM microgra phs of the investigated medium -Mn steel X6MnAl12 -3 in different ICA \nstates. (a) cold -rolled (CR) state. (b), (c) and (d) are for  the ICA states  annealed  at 550, 600 and \n700 °C, re spectively. M, RA and F denote  martensite, reverted austenite and ferrite, respectively.  \n 8 of 23  \n  \nFigure 5. XRD measurements of the medium -Mn steel X6MnAl12 -3 in different ICA states. (a) the \nrecorded XRD spectra , (b) bar chart of the measured volume fraction of the reverted austenite \nphase.  \n3.3 Electrochemical testing  and corrosion analysis  \nIn Figure 6 (a), the  potentiodynamic polarization curves  of the medium -Mn steel s in \n0.1 M H 2SO 4 are shown. In general,  all investigated medium -Mn steel s revealed an active \ndissolution independently from the processing parameters . In Figure 6 (b), a bar graph  of \nthe determined corrosion current densities (i corr) and the corrosion potentials (E corr) is de-\npicted. Ecorr of CR (-567 ± 2  mV SCE) and ICA 550 °C  (-567 ± 4 mV SCE) are similar, while  Ecorr \nof ICA 600 °C (-570 ± 1 mV SCE) and ICA 700 °C  (-575 ± 2 mV SCE) decrease with increasing \nICA temperature . The (CR) state shows the lowest current density value compared to the \nmedium -Mn steels after intercritical annealing.  In respect to the error bars, t he order of \nthe i corr values is CR < ICA 550 °C and ICA 700 °C < ICA 600 °C.  The i corr values increase \nfrom the CR condition (0.70 ± 0.18 mA/cm2) to ICA 550 °C  (1.19  ± 0.32 mA/c m2) and fur-\nther to  ICA 600 °C (2.69  ± 0.12 mA/cm2), then decrease to ICA 700 °C (1.23 ± 0.24 mA/cm2). \nThe i corr value of ICA 700 °C (1.23 ± 0.24 mA/cm2) is comparable to that of ICA 550 °C (1.19 \n± 0.32 mA/cm2). The determined electrochemical parameters  including the  corrosion cur-\nrent density (i corr), anodic Tafel slope (βa), cathodic  Tafel slop e (β c) and corrosion potential \n(Ecorr) of the potentiodynamic polarization in 0.1 M H 2SO 4 of all material states are listed  in \nTable  2. \n \nFigure 6. (a) Potentiodynamic polarization curves  of the different medium -Mn states  in \n0.1 M H2SO 4 and (b) bar graph of the i corr and E corr values.  \n \nTable 2. Electrochemical parameters in 0.1 M H 2SO 4. \n 9 of 23  \n State  icorr [mA/cm2] βc [V/decade]  βa [V/decade]  Ecorr [mV SCE] \nCR 0.70 ± 0.18 -0.159 ± 0.0 11 0.034 ± 0.0 06 -567 ± 2 \n550 °C 1.19 ± 0.32 -0.167 ± 0.00 0 0.029 ± 0.0 01 -567 ± 4 \n600 °C  2.69 ± 0.12 -0.200 ± 0.00 0 0.038 ± 0.001 -570 ± 1 \n700 °C  1.23 ± 0.24 -0.159 ± 0.0 11 0.034 ± 0.0 02 -575 ± 2 \n \nThe EIS Nyquist plo ts of all states at the open circuit potential in 0.1 M H 2SO 4 after \nan immersion time of 1 h are shown in Figure 7 (a). Each state contains  two various time \nconstants, which are represented by a capacitance semi circle at the mid -section of the \nfrequency range and a semi circle in the low -frequency range. The mid -frequency circle is \nrelated to a n electrochemical  double layer , while the low -frequency circle reveals the ex-\nistence of a pseudo -inductance , which is related to  an adsorption process.  The capacitive \narc at the  mid-frequency region of the different states show various diameters indicating \ndeviating charge transfers and consequently different corrosion resistances. The CR state \nshow s the largest  circle , while t he ICA treatments lead to a decrease of the circle di ame-\nters. ICA 700 °C show s the largest  diameter and ICA 600 °C the lowest, while ICA 550 °C \nis located in between.  \nIn Figure 7 (b), the  Bode plots (phase angle and total impedance magnitude) of all \nstates in 0.1 M H 2SO 4 at 1 h are depicted. The chronology o f the total impedance magni-\ntude of the four states correlat e with the diameters in the Nyquist plots indicating an in-\ncrease of the corrosion resistance from ICA 550 °C, to ICA 600 °C, to ICA 700 °C and fi-\nnally up to the CR state. The phase angle Bode plot indicates the presence of two time \nconstants, which correspond to the capacitive arc in the mid -frequency region and the \npseudo -inductance (adsorption) in the low -frequency region.  \n \nFigure 7. (a) Nyquist  and (b) Bode plot of the specimens in all heat treatment states after 1 h  im-\nmersion in 0.1 M H 2SO 4. \nIn order to fit the obtained EIS data, the equivalent electrochemical circuit model \n(EEC) in Figure 8 is used. The mid -frequency time constant is modelled with a Randles \ncircuit, which is associated to a n electrochemical  double layer. The Randles circuit  (Fig-\nure 8 (a)) contains the ohmic resistance of the solution (R sol), the constant phase element \nof the double layer (CPE DL), the charge transfer resistance (RCT) and a general  element  \nZads, which is placed in series with R CT. Zads represent s the singular adsorbate model of the \nlow-frequency time constant  (pseudo -inductance) using the constant phase element of \nthe adsorbates (CPE ads), the adsorbate resistance (Rads) and the adsorbate conductor (Lads), \nwhich are arranged in parallel  (Figure 8 (b)). The usage of CPE DL and R CT in series with \nRsol were also used in the literature for various Fe -based materials like steels [36,37]  or \niron aluminides [38,39]  in H 2SO 4 containing solutions and are well suitable to fit the data \nin the present study.  \n 10 of 23  \n  \nFigure 8. Electric equivalent circuit (EEC)  containing (a) a Randles circuit  used to evaluate  the EIS \ndata.  (b) The  parallel -connected  general adsorbate element Z ads.  \nIn Figure 9, the  charge transfer resistances over time of all states during immersion \nin 0.1 M H 2SO 4 are shown. The chronology of R CT is in line with the diameter of the \nNyquist plot and the total impedance magnitude in the Bode plot confirming the de-\ncreasing corrosion re sistance from CR, to  ICA 700 °C, to ICA 550 °C , to ICA 600 °C after 1 \nh of immersion. After 4 h of immersion, a partial change in R CT is observed. The charge \ntransfer resistance of CR and ICA 700 °C as well as ICA 550 °C and ICA 600 °C align after \n4 h in t he 0.1 M H 2SO 4 solution , respectively . After 24 h of immersion, ICA 700 °C show s \nthe highest charge transfer resistance, while R CT of CR deceas e further and sort itself in \nbetween ICA 700 °C and ICA 550 °C. R CT of ICA 600 °C is similar with R CT of ICA 550 °C. \n \nFigure 9. EIS resistance of charge transfer  (RCT) of the different heat treatment states at various time \npoints  (1, 4, 8 and 24 h)  during immersion in 0.1 M H 2SO 4. \nThe EIS fitting parameters  RS, R CT, R ads, L ads, the dimensionless fraction exponents of \nthe double layer n DL and adsorbate n ads, the admittance of the double layer Y DL and ad-\nsorbate Y ads, and the fitting variance Χ2 are listed  in Table 3 . \nTable 3. Fitting parameters using EIS data based on the EEC in Figure 8 for various  ICA states in \n0.1 M H 2SO 4. \nTime [h]  Rsol [Ω·cm2] RCT [Ω·cm2] YDL [µS·cm-2·snDL] nDL Rads [Ω·cm2] Yads [µS·cm-2·snads] nads Lads [H·cm2] Χ2 \nCR          \n1 6.3 23.7 87 0.92 4.0 89 1 0.47 1.8·10-4 \n4 6.0 20.9 128 0.92 2.5 162 1 0.14 1.3·10-4 \n8 6.1 16.2 224 0.90 1.9 269 1 0.11 2.5·10-5 \n24 6.1 11.1 525 0.89 1.3 657 1 0.08 6.0·10-5 \n550 °C           \n1 6.2 10.5 129 0.94 0.9 585 1 0.05 8.2·10-5 \n 11 of 23  \n 4 6.1 9.5 250 0.92 1.1 391 1 0.05 6.5·10-5 \n8 6.1 8.5 432 0.91 0.9 747 1 0.04 5.7·10-5 \n24 6.2 7.3 1045  0.86 1.1 1010  1 0.05 6.9·10-5 \n600 °C           \n1 6.6 7.1 264 0.89 0.6 561 1 0.02 6.5·10-5 \n4 6.6 9.1 271 0.92 1.0 531 1 0.07 6.1·10-5 \n8 6.6 8.5 425 0.92 0.8 929 1 0.04 4.3·10-5 \n24 7.0 7.6 1101  0.89 1.0 1271  1 0.05 4.9·10-5 \n700 °C           \n1 6.7 18.9 102 0.92 2.1 146 1 0.13 8.1·10-5 \n4 6.5 17.3 165 0.92 1.9 261 1 0.10 7.9·10-5 \n8 6.5 15.7 276 0.90 1.8 313 1 0.10 7.4·10-5 \n24 6.4 12.6 631 0.90 1.5 638 1 0.08 4.7·10-5 \n \nIn Figure 10, SEM secondary electron (SE) top view images of all states after  1 h and \n24 h of immersion in 0.1  M H2SO 4 are depicted. The typical martensitic lancets are visible \nfor the CR state after 1  h of immersion ( Figure 10 (a)), while after 24  h immersion time  \nsmall dimples are observed  (Figure 10 (b)). Those dimples are also visible for the three \nICA states ICA 550 °C (Figure 10 (d)), ICA 600 °C ( Figure 10 (f)) and ICA 700 °C ( Fig-\nure 10 (h)) after 24 h of immersion. In the top view images of ICA 550 °C ( Figure 10 (c)), \nICA 600 °C (Figure 10 (e)) and ICA 700 °C ( Figure  10 (g)) after 1  h of immersion the re-\nverted austenite and ferrite can be clearly identified due to the apparent selective disso-\nlution of the rever ted austenite  and the consequent appearance of the ir morphologies.   12 of 23  \n  \nFigure 10. SEM secondary electron (SE) top view images after  1 h and 24  h of  immersion in \n0.1 M H2SO 4 of CR (a and b), ICA 550 °C (c and d), ICA 600 °C (e and f), and ICA 700 °C (g and h).  \nIn Figure 11, SEM backscattered electron (BSE) cross -section images of all s tates after \nimmersion in 0.1  M H2SO 4 for 1  h and 24  h are depicted.  The CR state show s a relative  \nuniform material removal after 1 h of immersion ( Figure 11 (a)), while a preferred dis-\nsolution along the martensite lancets may be  indicated after 24 h of immersion in Fig-\nure 11 (b). The cross -section images o f ICA 550 °C after 1 h show s a selective dissolution \nof the rever ted austenite ( Figure  11 (c)), which is more pronounced after an immersion \ntime of 24 h (Figur e 11 (d)). The sel ective dissolution of the rever ted austenite in \nICA 600 °C is enhanced after 1 h ( Figure  11 (e)) and 24 h ( Figure 11 (f)) compared to the \nselective dissolution  in ICA 550 °C. The deep dimples, which are already recognizable in \nthe top view images of ICA  600 °C after an immersion time of 24 h, are also visible in the \ncross -section images due to the increased amount of selectively dissolved reverted aus-\ntenite. The phenomenon of the selective corrosion attack along the reverted austenite is \nstrongly  reduced in  ICA 700 °C after 1 h (Figure  11 (g)) and 24 h (Figure 11 (h)) of im-\nmersion  compared to ICA 550 °C and  ICA 600 °C. It is also notable that the surface of \nICA 700 °C is uniformly dissolved  after 24 h of immersion and only  show s small  signs of \ndimples, which is consistent with the top view images of the same state and immersion \nduration in Figure 10 (h). \n 13 of 23  \n  \nFigure 11. SEM backscatter ed electron (BSE) cross -section  images after immersion of 1 h and 24 h in \n0.1 M H 2SO 4 of CR (a and b), 550 °C (c and d), 600 °C (e and f), and 700 °C (g and h).  \nIn Figure 12, the  cumulative concentration c of Fe, Mn and Al in respect to the sam-\nple surface area A in the solution during immersion in 0.1 M H 2SO 4 is depicted. The ion \nconcentr ation in the solution increase s in all cases over time. After an immersion time of 1 \nh, the total ion concentration in the solution is the lowest for CR. The ion concentration is \nhigher for ICA 550 °C and ICA 600 °C, while the ion concentration in the solu tion for \nICA 700 °C lied in between. The same tendency is observed after 4 h. After an immersion \ntime of 8 h, the ion concentrations in the solution for CR and ICA 700 °C overla p indi-\ncating a change in the dissolution processes. After 24 h of immersion, th e ion concentra-\ntion in the solution for ICA 700 °C is the lowest, while the ion concentration in the solu-\ntion of CR is higher compared to ICA 700 °C and lower compared to ICA 550 °C and ICA \n600 °C. The ion concentration values for the immersion tests of ICA 550 °C and ICA 600 \n°C are similar even after 24 h.  \n 14 of 23  \n  \nFigure 12. ICP-MS analysis of Fe, Mn and Al during immersion in 0.1 M H 2SO 4 (a) at all time steps \nand (b) at 1 and 4 h.  \nIn Figure 13, the factor of dissolution  (Fd) over the immersion time for all states in \n0.1 M H 2SO 4 for the element Fe ( Figure 13 (a)), Mn ( Figure 13 (b)) and Al ( Figure 13 (c)) is \nshown.  The factor of dissolution is calculated using equation  (1), which include s the mass \nfraction (f s) in the solution and the mass  fraction (f a) in the alloy of the respective ele-\nments : \nFd = fs / fa (1) \nA F d value of more than 1 indicates a selective dissolution, while a value of less than \n1 suggests a reduced mass loss  of the respective element. A Fd value of 1 means  that the \nmass loss of an element  is equal to its fraction in the alloy and therefore describes the \nequilibrium state. After 1 h of immersion, the factor s of dissolution of all states show a \nselective dissolution of Fe, while Mn and Al show a lower mass loss compared to the \nrespective fra ction in the alloy. In case of CR, F d (Fe) and Fd (Mn) stay constant after an \nimmersion time of 4 h, while F d (Fe) decrease s, and Fd (Mn)  increase s slightly for \nICA 700 °C. In case of ICA 550 °C and ICA 600 °C, F d (Fe) is decrease d to a value of ap-\nproximately 1 and F d (Mn)  is increased above  1, which indicate s a change in the selective \ndissolution behavior  through the preferential dissolution of Mn. After 8 h of immersion, \nFd (Fe) approaches 1 for CR and ICA 700 °C, while F d (Mn)  is increased above 1. On the \ncontrary, Fd (Mn)  and Fd (Fe) for ICA 550 °C and ICA 600 °C show a contrasting trend \ncompared to CR and ICA 700 °C. After an immersion time of 24 h, the F d values of Fe, Mn \nand Al are equal  for all states. While the Fd (Fe) and Fd (Mn) alternate, F d (Al) is decreased \nover time for all states and stay s constantly below 1. After 1 h of immersion, Fd (Al) for \nCR is slightly higher compared to F d (Al) for the other three states. While after 4 h of \nimmersion Fd (Al) are equal for all states,  Fd (Al) are higher for CR and ICA 700 °C com-\npared to ICA 550 °C and ICA 600 ° C. \n 15 of 23  \n  \nFigure 13. Factor of dissolution ( Fd) of (a) Fe, (b) Mn and (c) Al for all states during immersion in \n0.1 M H 2SO 4. \n4. Discussion  \n4.1. Influence s of ICA treatments on the microstructure evolution  \nThe obtained microstructures of medium -Mn steel X6MnAl12 -3 containing different \namounts of reverted austenite were adjusted by applying ICA treatments at 550, 600 and \n700 °C (between the critical temperatures A1 and A3). During these ICA treatments the \nmartensitic microstructure (of the CR state)  gradually transforms through Austen-\nite-Reverted -Transformation (ART)  [40] into dual -phase microstructures consisting of \nferrite and austenite. The observed ultrafine grained microstructures are attributable to \nthe high density of nucleation sites for ferrite transformation provi ded by the initial de-\nformed martensitic microstructure, as the austenite preferentially nucleates at the packet \nboundaries as well as at the laths interfaces of martensite during ICA  [7,41] . The growth \nof austenite during ICA treatments is believed to be controlled by the partitioning of \naustenite stabilizing elements (here C and Mn) as it was indicated  from the compositional \nchanges of ferrite and austenite in the ICA range based on the thermodynamic calcula-\ntions represented in Figure  3. Such partitioning phenomena that locally occur at the fer-\nrite/austenite interfaces of medium -Mn steel under investigation were previously de-\npicted using atom probe tomography (APT) that enables identifyin g the nano -scale spa-\ntial chemical variations  [42]. The XRD measurements and SEM micrographs demon-\nstrated the incre ase in austenite fractions and the gradual vanishing of the austenite \nelongated grain morphology as the ICA temperature increases. The equiaxed and lamella \n(elongated) grain -types of morphologies were identified for medium -Mn steels of fer-\nrite/austenite mi crostructures processed by ICA treatments  [43]. \n 16 of 23  \n The microstructural observations along with the XRD measurements are in a good \nagreement with phase evolution trend predicted by the equili brium thermodynamic \ncalculations listed in Table 4, although the absolute equilibrium fractions of austenite at \nthe corresponding applied ICA temperatures are higher than the measured counterparts. \nThe difference between the equilibrium austenite fractions  and the measured ones is as-\ncribed to the applied annealing time (2 h in the current study), which is practically too \nshort to allow the equilibrium condition to be reached. In a previous study on the same \nmaterial  [3], energy intensive and longer -time (12 h) ICA treatments were applied, and \nrelatively  close -to-equilibrium austenite fractions were accordingly achieved in the \ntemperature range fr om 555 to 650 °C. However, at 700 °C the final retained austenite \nfraction was higher than the prediction , which was explained by an ultrafine \ngrain -induced austenite stability effect. Likewise, in the current study the equilibrium \nchemical compositions, p articularly the decreasing Mn and C contents at the highest ICA \ntemperatures (700 °C), imply a lower thermal stability of the reverted austenite, which \nwas reflected by the estimated martensite start (M s) temperature of 68 °C as well as with \nthe predicted fraction of possible fresh martensite (f α') of 37 % estimated by Koist-\ninen -Marburger (KM) equation, according to the following equations  [44]:  \nMs = 547.58 − 596.914C − 28.389Mn + 8.827Al − 60.5Vγ −1/3  (2) \nfα' = 1 – exp [−0.011 (M s − T)]  (3) \nwhere C, Mn and Al are in wt.%, V γ is representing the volume of austenite grains (1  µm \nis used as the average grain diameter) and T is the temperature.   \nHowever, the XRD measurements and the microstructure investigations did not  \nreveal the formation of fresh martensite emphasizing the influence of ultrafine grain size \non the austenite stability. In addition, the observed variations in the Mn contents and \naustenite fractions driven by the different applied ICA temperatures have a profound \neffect on the corrosion behavior as will be deciphered in the following section. It was \nreported that the high dissolution rate of Mn leads to an inferior corrosion resistance of \nhigh -Mn steels [45]. \nTable 4. Austenite composition at different inter -critical annealing  (ICA) temperatures  based on \nthe equilibrium thermodynamic calculations , and the predicted martensite start (Ms) temperature \na\ns\n \nw\ne\nl\nl\n \na\ns\n the fresh martensite (fα') and austenite fractions.  \n \n \n \n4.2 Electrochemical testing and corrosion analysis  \n4.2.1 Comparison with high -Mn steel and the role of Al  \nThe corrosion current densities of all investigated states, which were determined by \nTafel slope extrapolation, showed reduced values compared to a high -Mn steel contain-\ning 29.5 wt.% Mn, 3.1 wt.% Al and 1.4 wt.% Si [23] indicating the beneficial corrosion re-\nsistance of medium -Mn steel compared to high -Mn steel in acidic  0.1 M  H2SO 4. Those ICA \ntemp. \n(°C) Thermodynamic equilibrium  \ncalculations  Koistinen -Marburger -based \ncalculations  Measured \naustenite \n(XRD)  Austenite \n(vol.%)  Austenite composition (wt.%)  Ms  \n(°C) fα'  \n(vol.%)  Austenite \n(vol.%)  C Si Mn Al \n550 42 0.151  0.36 22.01  1.98 -225 0 42 21.9 \n600 52 0.123  0.31 18.47  2.22 -106 0 52 40.8 \n700 78 0.082  0.23 13.36  2.65 68 37 41 58.5  17 of 23  \n findings can be generally explained by the reduced Mn content of the X 6MnAl12 -3 ma-\nterial in the current investigation  and the consequent reduction in Mn dissolution , be-\ncause  high Mn content  was reported to highly contribute to the corrosion kinetics [23]. Al \nwas mentioned  to increase the passivity beha vior of Fe -Mn-Al based steels in various  \nsolutions  by several authors  [20-22,27,46] . However, in the acidic solution  0.1 M H 2SO 4 \nwith an approximate  pH of  1.1 ± 0. 1, Al preferentially dissolves as Al3+ according to the \nPourbaix diagram [47] and therefore 2.9 wt.% Al is not considered to contribute  signifi-\ncantly  to the formation of a stable passive film.  That assumption is supported by the ab-\nsence of passive region in the PDP curves, the low resistance in the EIS plots, and the ab-\nsence of measurable corrosion products after immersion . Nevertheless, the formation of  a \nthin layer during immersion at E corr is still expectable. An indication for the formation could  \nbe Fd (Al) that was determined to be constantly below 1 for all states  (Figure 13 (c)) and \nmight  be related to the incorporation  of Al3+ within th e thin  layer.   \n4.2.2 Influence of ICA on corrosion behavior after 1 h in 0.1 M H 2SO 4 \nThe applied ICA treatments resulted in adjusting microstructur es containing  different \namounts of reverted austenite with different content of alloying elements due to  elements \npartitioning , in particular Mn , which consequently led to a divergent corrosion behavior in \nthe investigations including PDP, EIS and ICP -MS. The shift of E corr to more negative  po-\ntentials from CR to elevated ICA temperatures  (Table 2) correlate d with the increase in the \nreverted  austenite fraction , indicating a  more active behavior  due to the increased austenite \ncontent . The chemical composition differences, especially the Mn content between aus-\ntenite and ferrite caused a micro -galvanic coupling , which led to a selective dissolution of \nthe more active austenite  that acted as anode  due to its relatively  high Mn content . The \nselective dissolution of austenite for all ICA states is demonstrated in the SEM images  of \nthe corresponding cross -sections after immersion tests  in Figure  11 (c-h). It is also clearly \nvisible that the severity of the austenite dissolution is di vergent  for the  different  ICA \nstates. Various factors can influence the micro -galvanic coupling of  both phases, which \nneed to be considered.  The influencing factors involve the chemical content (mainly Mn) \nof both phases, the area of interfaces, and the element gradient at the interfaces.   \nCompared with ICA 600 °C, ICA 550 °C revealed a smaller i corr in PDP curves and a \nhigher charge transfer resistance in EIS tests. The thermodynamic ally calculated Mn \ncontent in austenite ( Table 4) is higher for  ICA 550 °C compared to 600 °C , which  con-\nsequently le d to an increased potential difference between austenite  and ferrite, thus an \nincreased driving force for the selective austenite dissolution . However,  with the ICA \ntreatment temperature increasing from 550 °C to 600 °C, the austenite content increased \nand elongated austenite formed as shown in Figure 4 (g), exhibiting a higher area of \naustenite/ferrite interfaces in ICA 600 °C, which provided more corrosion sites. Despite of \nthe higher driving force for micro -galvanic coupling between austenite and ferrite in \nICA 550 °C, a  relatively larger  area of austenite/fe rrite interfaces in ICA 600 °C contrib-\nuted to the increased corrosion susceptibility. In case of ICA 700 °C , the thermodynami-\ncally calculated Mn content in austenite (13.36 wt.%  Mn) approaches to the overall Mn \ncontent of the material  (11.7 wt.% Mn) , which  decrease d the driving force for mi-\ncro-galvanic coupling  and consequently  the selective austenite dissolution  compared to  \nICA 600 °C, which is clearly visible in the SEM cross -section image in Figure 11 (g). The \nreduced galvanic coupling of austenite and ferrite in duplex stainless steel by decreasing \nthe Ni gradient between both phases was also previously reported to reduce the ten-\ndency of the preferential dissolution of the anodic phase [48]. Meanwhile, the increasing \naustenite fraction in ICA 700 °C  resulted in an increasing anode -cathode surface area ra-\ntio, which in turn decreased the  severity of austenite dissolution and  corrosion kinetics . \nBesides, less austenite/ferrite interfaces were expected in ICA 700 °C than in ICA 600  °C, \nconsidering the microstructures shown in Figure 4 (c and d).  Therefore, compared with \nICA 600 °C, ICA 700  °C showed a decreased cumulative concentration of element s (Fe,  18 of 23  \n Mn and Al) in the solution, corrosion current density, and an increased charge transfer \nresistance.  \n \n \n4.2.3 Influence of ICA on corrosion behavior between 1 and  24 h in 0.1 M H 2SO 4 \nThe immersion tests longer than  1 h indicated the ICA 700 °C state t o stay the most \ncorrosion resistant state compared to ICA 550 °C and ICA 600 °C by displaying the low-\nest cumulative concentration of Fe, Mn and Al and the highe st charge transfer resistance. \nMoreover, t he SEM images  (Figure 11 (h)) of ICA 700 °C even showed that the dissolution \nof the surface remained  uniform after 24 h immersion and no intensified selective disso-\nlution of the austenite occurred . ICA 700 °C  showed a  higher corrosion resistance  than \nCR after 8 h immersion with a lower cumulative co ncentration of elements in the solution \n(Figure 12) and a higher charge transfer resistance ( Figure 9). The change  might be re-\nlated to other factors, such as  dislocation defects  in martensite , which were reported to be \ndetrimental on the corrosion behavior  in H 2SO 4 [49,50] . The immersion tests over a time \nrange of 24 h indicated an assimilation of the corrosion behavior of the ICA 55 0 °C and \nICA 600 °C states by the similar concentration of cumulative elements in the solution \n(Figure 12) and similar charge transfer resistances ( Figure 9). Therefore, the mi-\ncro-galvanic coupling seems to be most relevant in the early stage  of the corrosion pro-\ncess during immersion , while other factors  may dominate in the long -term immersion . It \nwas reported that after the dissolution of the anodic phase, the detachment of the ca-\nthodic phase occurred for duplex stainless steel during immersion tests in acidic solution  \n[51]. The dimples  that were found in the SEM images ( Figure 10) could be  related to a \ndetachment of the ferrite when the austenite was selectively dissolved . \n4.2.4 Dissolution  behavior  of Mn and Fe  \nIn order to consider the respective dissolution of Fe  and Mn, the factors of dissolu-\ntion were calculated and plotted over time for the different states  (Figure 13). It needs to \nbe stated that Fd (Mn) and Fd (Fe) were always close to 1  with subtle variations . Never-\ntheless, c onsidering the fact that the Mn -rich rever ted austenite was selectively dissolved , \na preferred dissolution of Mn is expected after 1 h  immersion , which was not the case  in \nthe current results . In contrast, Fd (Fe) was constantly above 1, Fd (Mn) only showed a \nvalue higher than 1 after 4 h for ICA 550 °C and ICA 600 °C as well as after 8 h for CR and \nICA 700 °C. Considering the fact that the selective dissolution of the Mn -rich austenite \nwas more pronounced  for ICA 550 °C and ICA 600 °C , the presence  of a Fd (Mn) value \nabove 1 was found earlier compared to CR and ICA 700 °C. The occurrence of the pre-\nferred dissolution of Mn  after longer times might be explained by a time shift , because  \nthe total  element concentration in the solution was determined after several time steps  \nand small changes consequently might be measurable only  after a certain time. Since the \nfactors of disso lution of Fe and Mn were mostly above 1 and below 1, respectively, the \nselective dissolution of the austenite might not be the only determining factor, especially \nafter longer immersion times. After the austenite was selectively dissolved in the early \nstage due to the micro -galvanic coupling of austenite and ferrite, the distance between \nthe cathode (ferrite) and anode (austenite) increased, which reduce d the driving force for \nthe coupling. Therefore, the dissolution of ferrite , which contained lower amount s of Mn \ncompared to austenite,  will be more pronounced after a certain time.  Due to the larger  \nchemical gradient of Mn between austenite and ferrite and the larger  cathode/anode ratio \nin ICA 550 °C compared to ICA 600 °C, the takeover could  occur earlier f or ICA 550 °C.  \nThe factors of dissolution in Figure 13 were related to the overall composition of the \nmaterial  shown in Table 1. Since the phases differ ed mainly in the Mn content, the factor s \nof dissolution were  recalculated based on the thermodynamically determined  Mn con-\ntent in the austenite from Table 4 and are shown in Figure 14. The CR state was included \nas reference. The lowest F d (Mn) values  were determined for ICA 550 °C and the highest \nfor ICA 700 °C showing a contradict ing trend with respect to the supposed dissolution  19 of 23  \n tendency of the austenite. Con sidering the austenite fraction (Figure 5 (b)), which was the  \nhighest at  ICA 700 °C and the lowest at  ICA 550 °C, the highest  austenite  fraction ap-\npear ed to be responsible for the highest dissolution  of Mn . Furthermore, F d (Mn) re-\nmain ed constant ly below  1, which  indicate d a reduced mass loss of Mn from the austen-\nite.  \n \nFigure 14. Factor of dissolution ( Fd) of Mn based on the thermodynamically calculated Mn  content \nin the reverted austenite  (RA)  for ICA 550 °C, ICA 600 °C and ICA 700 °C. The CR state is included \nas reference.  \n4.2.5 Initiation at the interfaces  \nThe initiation of the selective dissolution occurred  most likely at the austenite/ferrite \ninterface . Bleck et al. [42] found a local Mn segregation at the austenite/ferrite interface of \nintercritically annealed  (700 °C)  medium -Mn steel, which is similar in the chemical \ncomposition compared to the X6MnAl12 -3 material in the current investigation. The Mn \nsegregation leads to a bilateral potential gradient, which is the highest in direction to the \nferrite , since the M n gradient is more pronounced . However, the Mn segregation at the \ninterface makes  it the most active region, which is therefore  more  likely to be dissolved \nfirst. It was stated by several authors [52-54] that the dissolution  in duplex stainless steels  \nin acidic solutions  is also able to be initiated  at the  interfaces of e.g. intermetallic phases  \n[54] or austenite/ferrite  phase  boundar ies [52,53] . Sathirachinda et al. [54] mentioned that \nthe selective dissolution at those interfaces could  correlate to local chemical gradients and \nthe consequent potential difference compared to the adjacent phases.  \n5. Conclusions  \nIn this study, the influence of intercritical annealing (ICA) temperature on the elec-\ntrochemical corrosion behavior  of the medium -Mn steel X6MnAl12 -3 in 0.1 M H 2SO 4 was \ninvestigated by PDP, EIS and ICP -MS. SEM investigations after 1  h and 24  h of immer-\nsion were conducted to visualize d the corrosion damage.  The study  focus ed on the mi-\ncro-galvanic  coupling  between reverted austenite and ferrite in dependence of the ther-\nmodynamically determined Mn partitioning.  The following conclusions are drawn:  \n \n• The ICA treatments at  550, 600 and 700 °C led to a transformation from cold -rolled \nmartensite to a ferrite/austenite (reverted) microstructure with increasing volume \nfraction of austenite and a decrease of Mn content within the austenite.  \n• The Mn partitioning  led to a micro -galvanic coupling between ferrite  (cathode)  and \nreverted  austenite  (anode). The decreasing Mn gradient with increasing ICA tem-\nperature , and accordingly the f raction of reverted austenite,  reduced the driving \nforce for the micro -galvanic coupling. Apart from the Mn gradient, an increase d in-\nterface  area between austenite and ferrite enhanced the dissolution tendency of the \nanodic austenite.  \n• The i mmersion experiments between 1 h and 24 h indicated the decreasing influence \nof selective  austenite dissolution and the transition to  ferrite dissolution when the \n 20 of 23  \n distance between ferrite and austenite is increased due to the selective dissolution of \naustenite in t he early stage.  \n• The increase of the corrosion current density and decrease of the charge transfer re-\nsistance of ICA 600 °C compared to ICA 550 °C were  related to the increase of inter-\nface area between austenite and ferrite due to the higher fraction of aus tenite. Those \nhad a bigger influence on the micro -galvanic coupling than the chemical gradient of \nMn.  \n• In case of ICA 700 °C , the chemical gradient of Mn was the smallest between austen-\nite and ferrite, which changed the corrosion manner from local selectiv e dissolution  \nof reverted austenite  to a uniform corrosion attack. Consequently, the corrosion \ncurrent density and charge transfer resistance was reduced.  \n \nThe control and design of Mn partitioning in reverted austenite  during ICA  of me-\ndium -Mn steels are of a prime importance for tailoring their mechanical properties. The \ncurrent study additionally emphasizes its  crucial role in controlling the  corrosion re-\nsistance.  This opens new opportunities to design new alloying concepts for medi-\num-Mn steels that allow adjusting the characteristics of reverted austenite  to optimize the \ncorrosion properties with maintaining their promising mechanical properties.  To that \nend, r educing the chemical gradient of Mn and the interface area between the reverted  \naustenite and ferrite allow the adjustment of the severity of micro -galvanic coupling and \nconsequent ly the selective dissolution of the austenite. The initiation of dissolution at the \naustenite/ferrite interfaces due to local Mn segregation in dependency of ICA tempera-\ntures and high -resolution surface analysis of potentially formed thin layers  with incor-\nporated Al  are of special interest for future  studies.  \n \nAuthor Contributions:  Conceptualization, R.D.P.  and C.H.; methodology, R.D.P., T.A., J.W. and \nJ.N.; validation, R.D.P., T.A., J.W. and J.N. ; formal analysis, R.D.P., T.A., J.W.  and J.N.; investiga-\ntion, R.D.P., T.A., J.W., J.N. and S.S.; resources, D.Z. and U.K. ; data curation, R.D.P., T.A., J.W.  and \nJ.N.; writing —original draft preparation, R.D.P., T.A. , J.W.  and J.N.; writing —review and editing, \nR.D.P., T.A., J.W., J.N., C.H., S.S., U.K. and D.Z. ; visualization, R.D.P., T.A. , J.W., and J.N.; supervi-\nsion, D.Z. and C.H. ; All authors have read and agreed to the published version of the manuscript.  \nFunding:  This research received no external funding . \nAcknowledgments:  The authors gratefully thank Katharina Utens  (Chair of Corrosion and Corro-\nsion Protection ) for technical support . \nConflicts of Interest:  The authors declare no conflict of interest . \nReferences  \n \n1. 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Study of stainless steels corrosion in a strong acid mixture. Part 2: ano dic \nselective dissolution, weight loss and electrochemical impedance spectro scopy tests. Corros. Sci. 2018 , 130, 12-21, \ndoi:10.1016/j.corsci.2017.10.010.  \n52. Femenia, M.; Pan, J.; Leygraf, C.; Luukkonen, P. In situ study of selective dissolution of duplex stainless steel 2205 by \nelectrochemical scanning tunnelling microscopy. Corr os. Sci. 2001 , 43, 1939 -1951, doi:10.1016/S0010 -938X(00)00180 -3. \n53. Potgieter, J.H.; Olubambi, P.A.; Cornish, L.; Machio, C.N.; Sherif, E. -S.M. Influence of nickel additions on the corrosion \nbehaviour of low nitrogen 22% Cr series duplex stainless steels.  Corros. Sci. 2008 , 50, 2572 -2579, \ndoi:10.1016/j.corsci.2008.05.023.  \n54. Sathirachinda, N.; Pettersson, R.; Pan, J. Depletion effects at phase boundaries in 2205 duplex stainless steel characterized  \nwith SKPFM and TEM/EDS. Corros. Sci. 2009 , 51, 1850 -1860,  doi:10.1016/j.corsci.2009.05.012.  \n "    },    {        "title": "1607.02721v1.Effect_of_isothermal_holding_temperature_on_the_precipitation_hardening_in_Vanadium_microalloyed_steels_with_varying_carbon_and_nitrogen_levels.pdf",        "content": " Effect of isothermal holding temperature on the precipitation \nhardening in Vanadium -microalloyed steels with var ying carbon \nand nitrogen  levels  \nA. Karmakar1*, A. Mandal1, S. Mukherjee2, S. Kundu2, D. Srivastava3, R. Mitra1, \nD. Chakrabarti1 \n1Department of Metallurgical and Materials Engineering, Indian Institute of Technology \nKharagpur, Kharagpur, 721 302, India.  \n \n2Product Research Group, Research Development and Scientific Services, Tata Steel, \nJamshedpur , 831 001, India.  \n \n3Materials Science Division, Bhabha  Atomic Research Centre (BARC), Mumbai 400 085, India.  \n \n*Corresponding author email:  anish.met@gmail.com  \n \nAbstract:  \n \nCombined effect of C and N levels and isothermal holding temperature on the microstructure, \nprecipitation and the tensile  properties of V anadium (V)  microalloyed steels (0.05 wt%. V)  were \nstudied . Two different V steels, one having higher C and lower N content, HCLN steel, and the \nother having lower C and higher N content, LCHN steel, were prepared and subjected to \nisothermal  holding treatment over a temperature range of 500 –750 °C, after hot -deformation. \nMaximum precipitation strengthening from fine V(C,N) precipitates has been found at \nintermediate isothermal holding temperatures (600 –650 °C) in both the steels . In spite of the \nsignificantly smaller fraction of pearlite and bainite , coarser average ferrite grain size and lower \nprecipitation -dislocation interaction  in the ferrite matrix, the yield strength of LCHN steel was \nclose to HCLN steel. This can be attributed to the hi gher preci pitation strengthening (by  ~ 20–\n50 MPa) resulted from the finer V -precipitates in LCHN steel than that of HCLN steel.  \n \nKeywords:  \nVanadium microalloyed steel; Carbon content; Nitrogen content; Isothermal holding; \nPrecipitation strengthening; Hardn ess; Tensile properties.  \n \n \n  1.  Introduction:  \nVanadium microalloy ed high strength low alloy (HSLA) steels are widely being used for \nconstruction, line  pipe, pressure vessel, automobile, naval and defense applications [1,2] . VN or \nV(C,N) precipitates contribute to ferrite grain refinement either by retarding the recrystallizat ion \nof hot -deformed austenite or by acting as the heterogeneous nucleation sites for intragranular \nferrite formation [1,3–10]. Although, the prime o bjective of V addition is to achieve precipitation \nstrengthening from the fine VC or V(C,N) precipitates [1,3–6,11] . \n The nature, size and distribution of V -containing carbide, nitride or carbo -nitride \nprecipitates depend not only on the V content but also on the C and N contents of the steel. \nIncrease in both C and N level s in steel cannot be an effective way of strengthening as that can \nimpart brittleness and hamper the weldability and formability. Therefore, either the s teel can be \nricher in C or richer in N. Effect of C addition and N addition on the precipitation strengthening \nof V-microalloyed steel s has been studied separately and both are found to be beneficial  [1,3,7 –\n9,12–29]. However , the effect of two different alloying strategies , i.e. high-C strategy and high -\nN strategy , on the microstructure and mechanical behaviour of V steels has hardly been \ncompared.  \n V-microalloyed steels are being used both as flat -products like plates and sheets for \nlinepipe and automotive applications and as long products like rods, bars and secti ons for \nconstruction applications [1,3,12–16,19 –41]. Since the fine VC or V(C,N) precipitates primarily \nform during and after the austenite to ferrite transformation, hot -deformed V -microalloyed steels \nneed to be  cooled down at a very slow -rate over the transformation temperature range. The refore, \nhot-rolled plates and sheets of V  microalloyed  steels are coiled over a temperature in the range of \n500-750 °C [24,42 –51]. The effect of coiling temperature on the precipitation in V containing \nsteels has been studied independently for different steel compositions [24,42 –52]. However, the \neffect of coiling temperature on the microstructure, precipitation and mechanical properties of \nsteels  having different alloying strategies (high -C or high -N strategy) has hardly been \ninvestigated and compared earlier. That sets the objective of the present study. Such a study not \nonly fulfills the academic understanding on V precipitation  in steel  but als o provides a guideline \nto the steel industries in selecting the C and N content and coiling temperature for V -\nmicroalloyed steel products.  \n2. Experimental details : \nTwo laboratory cast blocks (20 Kg each) having different C and N contents were \nprovided by Tata Steel Research & Development  and Scientific Services Division , Jamshedpur. \nThe blocks were cast following identical casting process. Chemical compositions of th e steels are \ngiven in Table 1.  High -C, low -N (HCLN) chemistry is typically used in the long-products like \nbars, rods and columns whilst, low -C, high -N (LCHN) chemistry is typically used in the flat \nproducts like plates, sheets and strips.    Cylindrical samp les (1 0 mm diameter and 15 mm height) were prepared from the cast \nmetal blocks and were subjected to thermo -mechanical processing simulation using Gleeble® \n3800 simulator. The processing schedule applied in Gleeble is shown in Fig. 1.  Samples were \nreheated  to 1100  °C, soaked for 5 min, cooled down to 880  °C (cooling rate ~ 10 °C /s) and hot -\ncompression tested at that temperature up to true strain, ε = 1 at a strain rate,  έ = 1/s. After \ncompression testing, the samples were cool ed down ( cooling rate ~ 10 °C/s) to different \ntemperatures in the range of 500 –750 °C at 50  °C interval and isothermally held at that \ntemperature for 1 h, followed by air -cooling. The isothermal holding treatment has been \nperformed  to simulate the industrial coiling treatment, where the steel cools down at a very slow \nrate from the coiling start temperature. Couple of thermocouples attached to the sample \nmonitored the sample temperature during Gleeble® testing at an accuracy of ± 3  °C. \n Cross -section of the compression tested samples near the thermocouple attached location \nwas prepared following standard metallographic techniques . Microstructural study was \nconducted using optical microscope and scanning electron microscope (SEM) and image \nanalysis was used for quantitative metallography. Grain area and equivalent circle diameter \n(ECD) grain size of  400-500 grains were measured from each sample to determine the average \nferrite grain size.  Thin foils were prepared by dimpling and ion -milling process and observed \nunder JEOL -2100 model high resolution transmission electron microscope (HR -TEM) to study \nthe fine V -precipitates.  Solid cylindrical samples (4 mm outer diameter and 10 mm height ) \nprepared from each cast block  were austenitized at  1100  C and continuously cooled down at \n10 C/s to the ambient temperature inside automated quench dilatometer (DIL 805A/D, TA \nInstruments), without applying any deformation.  \n Macro -hardness was measured at 20 Kgf load using a LV -700 model LECO® Vickers \nhardness tester as the average value of 5 readings from each sample. Micro -hardness indents \nwere taken only from the ferrite regions of the investigated samples using UHL VHM -001 model \nmicro -hardness tester (0.01 Kgf load) as the average value of twenty (20 ) readings from  each \nsample. Rectangular samples  (15 mm×20 mm×100 mm) were cut from the cast steel blocks  and \nwere subjected to a processing schedule similar to that shown in Fig. 1.  A forging press was used \nfor applying the compressive deformation and the  isothermal holding treatment was carried out \ninside a muffle furnace.  K–type thermo -couple was attached to the sample s to monitor the \ntemperature. Finally, tensile specimens were prepared from the forged blocks following ASTM E \n8 standard  [53] and tested at ambient  temperature (25 C) and cross -head velocity 0.5  mm /min \nusing Instron®  8862 servo electric test sys tem (10 t).  \n \n3. Results  and discussions : \nThe investigated steels contain a moderate level of V addition (0.05 wt.%) as typically \nused in industrial HSLA steel grades for structural, linepipe  and automotive applications. C and \nN levels are varied such that a comparative assessment can be made between two different alloy \nstrategies, vanadium -carbon strategy and vanadium -nitrogen strategy.  Thermo -Calc® software  was used for the prediction of nat ure, fraction and stability of V -precipitates in the investigated \nsteels, Fig. 2.  As per the prediction, under  equilibrium cooling condition, precipitation starts at \nhigher temperature (~1050  °C) in low -carbon high nitrogen , LCHN , steel than that in high \ncarbon low nitrogen , HCLN , steel (~1010  °C), Fig. 2.  Therefore, at the  soaking temperature  of \n1100  °C all the V precipitates are expected to be dissolved in austenite solution, Fig. 2 . \nAccording to Thermo -Calc®, V -precipitates that form at higher temperatur es in austenite \n(>850  °C) and at lower temperatures during austenite -ferrite transformation in HCLN steel are \npredicted to be richer in N and C, respectively. On the other side, primarily VN precipitates are \nexpected  to form in LCHN steel  irrespective of precipitation temperature . This aspect is reported \nin detail in another study [54]. Precipitate fracti on at the ambient temperature is predicted to be \nhigher in HCLN steel than that in LCHN steel, Fig. 2. \n3.1. Transformation behaviour and microstructures of investigated samples : \nTime -temperature -transformation (TTT) diagram of the investigated steels were predicted \nfrom JMatPro® software version 8.0, Sente software Ltd . The range of isothermal holding \ntemperatures used in the present study are indicated by dotted lines on the TTT diagrams in \nFig. 3(a,b).  The cooling curves obtained  from the dilatometric study showing the sample dilation \nduring cooling from 1100  C (at 10  C/s cooling rate)  without deformation are presented in  \nFig. 3c. Both t he TTT diagrams and the cooling curves indicate  that the ferrite transformation \nstart tempera ture is ~50 -70 °C higher and the incubation time required for the onset of ferrite \ntransformation is ~  8–10s lower in LCHN steel (Ar3720 C) than those in HCLN steel  \n(Ar3650 C), Fig. 3. This is a clear indication that the decrease in C content reduced the \nhardenability and increased  the transformation temperature of LCHN steel as compared to \nHCLN steel. Microstructures of the samples continuously cooled at 10  °C/s inside dilatometer \nare given in Fig. 4.  Microstructure of HCLN steel showed the formation of polygonal ferrite \nalong the  prior -austenite grain boundaries and bainite and martensite at the prior -austenite grain \ninterior locations, Fig. 4a.  Polygonal ferrite formed at a higher quan tity in LCHN steel sample, \nalong with some acicular ferrite and upper bainite, Fig. 4b.  The actual transformation behaviour \nof the investigated steels may not exactly follow Fig. 3  as the samples are hot -deformed (at \n880 °C) before cooling and isothermal h olding. Although Fig. 3  can provide a guideline to \nexplain the difference s in transformation behaviour and the resultant microstructural evolution in \nHCLN and LCHN steels  as identical processing schedules were used for both the steels . \nThe optical micrographs of HCLN steel and LCHN steel samples are presented in Fig. 5  \nand Fig. 6,  respectively, for different isothermal holding  temperatures. Fraction s of the \nmicrostructural constituents determined by the image analysis are  presented in Table. 2.  The \nmicrostructures of LCHN steel were dominated by ferrite, whilst, harder constituents such as, \npearlite and bainite were present at significant proportion in HCLN steel, apart from the ferrite \nmatrix, Fig. 5 and Fig. 6.  The variation in average ferrite grain size and the total fraction of \nharder constituents (pearlite + bainite) with isothermal holding temperature is  presented in Fig. 7 .  Martensite was not present in the isothermally held samples. Average ferrite grain size s were ~ \nhigher in LCHN  steel (7.7-8.9 m) than that in HCLN steel  (5.7-8.4 m), Fig. 7a.  This could be \nattributed to higher transformation temperature and lower incubation time for ferrite \ntransformation in LCHN steel, which allowed greater opportunity for ferrite grain growth,  as \ncompared to HCLN steel. Total fraction of the harder constituents (pearlite + bainite) was ~ 20 -\n30% higher in HCLN steel samples, due to its higher hardenability than LCHN steel, Fig. 7b. \nFerrite fraction in HCLN steel remained within a close range, Table 2.  Besides lamellar pearlite , \ndegenerated pearlite was found in the HCLN samples isothermally held at and below 600  °C. \nUpper bainite was also found in HCLN  steel samples that were held either at high er temperatures \n(700-750 °C) or at low er temperatures (500 -550 °C), Table 2.  On the other side, upper bainite \nformed only at higher holding temperatures (650  °C and higher) in LCHN steel.  \nThe microstructures of the investigated samples can be explained from the transformation \nbehaviour of the st eels. Dilatometry study showed that under 10  °C/s cooling rate, \nallotriomorphic ferrite formation started in HCLN steel and LCHN steel at Ar 3 ~ 650  °C and \nAr3 ~ 720 °C, respectively, Fig. 3 and Fig. 4.  Therefore, some amount of allotriomorphic  ferrite \nformation was expected even during sample cooling when the isothermal holding temperatures \nwere lower than the corresponding Ar 3 temperature. Now ferrite formation along the prior \naustenite grain boundaries during cooling can shift the carbon  towards the austenite grain interior \nlocations and influence the subsequent transformation at those locations during isothermal \nholding. This could have promoted the formation of degenerated pearlite and upper bainite at \nlower holding temperatures in HCLN stee l, Table 2.  The TTT diagrams in Fig. 3(a,b)  indicate \nthat at higher holding temperatures ( 700-750 °C) the decomposition of austenite possibly \nremain ed incomplete even after the prolonged isothermal holding and in such a scenario, pearlite \nand bainite trans formation occurred  from remaining austenite during the final cooling stage (after \nisothermal holding ). This hypothesis  could explain  the formation of bainite at higher holding \ntemperatures in both the steels, Table 2.  Decomposition of austenite in both the  investigated \nsteels  reached completion solely  by the diffusional transformation  at the intermediate holding \ntemperatures (600 –650 °C). \n \n3.2. Precipitate study in the processed samples : \nThe bright field  and dark field TEM images and selected area electron diffraction \n(SAED) analysis of the precipitates present in the ferrite matrix of HCLN steel and LCHN steel \nare given  in Fig. 8 and Fig. 9, respectively . The bright field images of HCLN steel  samples \nisothermally h eld at 500 –650 °C show  the presence of numerous fine precipitates, often \ninteracting with the matrix dislocations, Fig. 8(a, b, d and e) . SAED  analysis  indicated  that the \nprecipitates had FCC crystal structure with a lattice parameter of ~ 4.16  Å. The precipita tes could  \npossibly  be the VC or V(C,N)  precipitates  as 4.16 Å is the lattice parameter of VC . In case of \n600-650 °C holding temperature s, severe dislocation -precipitation interaction can be  noticed,  Fig. 8(d-h). In general, precipitates were found to be distributed in a random fashion up to \n650 °C, Fig. 8(a-h). At higher holding temperatures (700 –750 °C), parallel arrays of interphase \nprecipitation has been found  and the precipitates were also coarsen ed into large r size , Fig. 8i. \nDislocation density and dislocation -precipitation interaction in the ferrite matrix were \ncomparatively lower at the higher holding temperatures  (700–750 °C) than the intermediate \nholding temperatures  (600–650 °C). The V(C,N) precipitates that formed upon holding at 700 -\n750 °C had a lattice parameter of ~ 4.12  Å. Those V(C,N) precipitates were expected to be richer \nin N than C . \n Interphase precipitation of VC or V(C,N) during austenite to ferrite transformation under \nslow -cooling condition or upon isothermal holding at high inter -critical temperature (~700  °C) is \nan established phenomena in V -microalloyed steels [30–33,55]. Lower transformation \ntemperature increases the driving force for austenite to ferrite transformation and accelerates the  \ninterphase migration. As a result,  interphase precipitation becomes difficult and  random \nprecipitation takes place.  The geometrically necessary dislocations (GNDs) that form in ferrite  \nduring austenite to ferrite transformation provide  the heterogeneous nucleation sites for the \nprecipitati on [1]. \nCompared to HCLN steel, the precipitates in LCHN steel were finer, more numerous and \nmore uniformly distributed, Fig. 9. The dislocation density and the extent of precipitation -\ndislocation interaction in the ferrite matrix  of LCHN steel was less se vere than that of HCLN \nsteel, Fig. 9(a-f). This can be attributed to the smaller  fraction of harder microstructural \nconstituents in LCHN steel, which reduced the density of  GNDs  in the adjacent ferrite matrix  of \nLCHN steel as compared to that of HCLN steel . Fine interphase precipitation was found to occur \nat high holding temperature (750  °C) as indicated  in Fig. 9g, while random distribution of \nprecipitates was noticed at lower holding temperature s, Fig. 9(a-f). SAED analysis suggested  that \nthe precipitates could be  richer in N than C , i.e. VN or V(C,N) precipitates,  having a lattice \nparameter of ~ 4.12 Å. \nComparison of average precipitate size, number density and the volume fraction of the \nprecipitates between HCLN and LCHN steels are plotted  in Fig. 10. Decrease in isothermal \nholding temperature from 700 °C to 600  °C not only  decreased the precipitate size  and increased \nthe precipitate density but also  increased the precipitate volume fraction to a small extent , \nFig. 10c. Further decrease in holding temperature to 500 -550 °C refined the precipitate size but \nsignificantly reduced the precipitate volume fraction. T he rate of increase in precipitate size was \nhigher in HCLN steel than that of LCHN steel. Th ese observation s can be explained in view  of \nthe following thermodynamic and kinetic principals : \n The driving force for precipitation increase s with the decrease in holding temperature , \nwhich is expected to increase the precipitate nucleation rate  [56,57] . \n At lower holding temperatures (500 -550 °C) lack of thermal activation for V-diffusion \ncould have affected the precipitation kinetics, which suppressed the precipitation process.    High  holding temperature s (700-750 C) on the other side, provide greater  thermal \nactivation for the diffusion of V (slowest diffusing element), which increases the \nprecipitate size by helping  the precipitate growth and coarsening  [56,57] . \n Diffusional transformation of austenite to ferrite remain ed incomplete after isothermal \nholding at higher temperatures (≥ 700  °C) and it was followed by shear transformation \nduring final cooling. Shear transformation also occu rred during isothermal holding at \nlower temperatures (5 00–550 °C). At intermediate holding temperatures of 600–650 °C \ncomplete diffusional transformation of austenite was expected . Since V(C,N) \nprecipitation is a diffusion controlled process and primarily occurs during diffusional \ntransformation of austenite [1,29,33,36] , the extent of precipitation was higher at \nintermediate holding temperatures.  \n Better coarsening resistance of VN or N -rich V(C,N) precipitates  as compared to VC  or \nC-rich V(C, N) precipitates [3,19,36]  led to finer precipitate size in LCHN steel than that \nin HCLN steel , especially at higher holding temperatures.  \nSimilar to the present study, e arlier studies also reported the formation of numerous fine \nmicroalloy precipitates (VC, NbC and even (Ti, Mo) 2C) at intermediate coiling temperatures \n(500–600 °C) not only in V -microalloyed steels but a lso in presence of other microalloying \nelements [24,42 –51]. \n \n3.3. Hardness and tensile properties of the investigated samples:  \nVariation in Vickers macro -hardness (VHN) and micro -hardness (micro -hardness only \nfrom the ferrite matrix) for  both the steel samples with isothermal holding temperature  are \npresented in Fig. 11. Since macro -hardness and micro -hardness were determined under different \nloads, those readings cannot be compared directly  as indentation size effect as well as the effect \nof local microstructure (i.e. ferrite grain size and presence of harder constituents) can influence \nthe hardness values . One common trend however, can clearly be noticed from Fig. 11 that both \nmacro -hardness and ferrite micro -hardness  were higher  (by ~20-24 VHN ) in the samples \nisothermally hold at intermediate temperatures (600 -650 °C) than the other  holding temperatures, \nFig. 11. \n Tensile stress -strain curves of the samples hot-forged and isothermally h eld at 500  °C, \n600 °C and 700  °C are shown in Fig. 1 2. Yield strength (YS) of HCLN and LCHN steel samples \nvaried i n the range of 5 20-650 MPa and 500-600 MPa, respectively, Fig. 1 2. Therefore, HCLN \nsteel showed ~ 20-50 MPa  higher YS than LCHN steel. LCHN steel on the other hand showed ~ \n3-5 % higher ductility (total elongation) than that of HCLN steel, Fig. 1 2. Higher strength and \nlower ductility of HCLN steel can certainly be attributed to its higher C content and the resu ltant \nhigher pearlite and bainite fraction in that steel (Fig. 7 b), with respect to LCHN steel. The \naverage ferrite grain size was also finer in HCLN steel, Fig. 7 a. The strain hardening ability of  HCLN steel was much stronger than LCHN steel which resulte d in significantly higher ultimate \ntensile strength ( UTS ) (by ~100 –150 MPa) in HCLN steel than that of LCHN steel , Fig. 1 2. \n \n3.4. Discussion on the e ffect of microstructure and precipitation on the tensile  properties : \nThe contribution from different strengthening mechanisms on the overall yield strength \nof ferritic steels can be given by the following equation:  \n                           (1) \nwhere,    is the yield strength of the steel,    is the lattice friction stress (~ 48 MPa),     ,     , \n   and    represent solid solution strengthening, grain boundary strengthening, dislocation \nstrengthening and precipitation strengthening , respectively. Strengthening contribution of each \nmechanism can be determined from a set of equations that are extensively reported in earlier \nstudies [58]. \nChemical composition in the ferrite matrix corresponding to the isothermal holding te mperature \nof the sample has been  predicted from the Thermo -Calc® software and was used to estimate the \nsolid solution strengthening,    . Investigated steels showed similar contributions from       (70-\n80 M Pa). Grain boundary strengthening, as determi ned from  the average ferrite grain sizes \n(given in Fig. 7 a) showed ~ 30 -40 MPa higher contribution in HCLN steel than that in LCHN \nsteel. Higher dislocation density (estimated from the TEM images  following established \nprocedure  [59]) further adds  to higher dislocation strengthening ( by ~20-30 MPa) in HCLN steel \nthan that in LCHN steel. Besides the above mentioned effects, a significant strengthening \ncontribution is expected to arise from the higher fraction of harder microstructural constituents \n(pearlite and bainite) in HCLN steel than that in LCHN steel.  In spite of the greater strengthening \ncontributions coming from the different sources  in HCLN steel, its YS was not significantly \nhigher than the YS of LCHN steel, Fig. 1 2. Study on the precipitation strengthening can help in \nexplaining this difference.  \nStrengthening contribution of incoherent precipitates by dislocation looping mechanism \ncan be determined from the well known Ashby - Orowan equation [60] as stated below:  \n        \n         \n                 (2) \nwhere,    is the average precipitate size (in µm) and f is the volume fraction of precipitates . The \nvalues of    and f can be obtained from Fig. 1 0. The calculated    and    values of the simulated  \nsamples for different holding tem peratures are plotted in Fig. 1 3. According to Fig. 1 3 higher \nprecipitation strengthening (by 30-50 MPa) is expected in LCHN steel which can be attributed \nto its finer precipitate size and higher precipitate density, compared to HCLN steel. The overall \nstrength of ferrite is however, predicted to be higher in HCLN steel due to the greater   contribution s from  grain boundary strengthening and dislocation strengthening, with respect to \nLCHN steel. Fig. 1 3 also justifies the higher hardness and strength in the samples h eld at \nintermediate temperatures (600 –650 °C), which can be attributed to the higher precipitation \nstrengthening contribution, as compared to the other holding temperatures. Higher strain \nhardening ability in HCLN steel can be attributed to the stronger dislocation -precipitation \ninteraction in that steel (mentioned in Section 3.2). The trend shown by the strengthening \ncalculations are therefore, in line with the experimental results except for  the fact that the exact \nstrength levels and the difference in strength of the investigated steels could not be predicted \naccurately. This can be attributed to the following factors:  \n It is difficult to estimate the exact strengthening contribution s of the harder constituents \nlike pearlite and bainite, in presence of ferrite.  \n Presence of harder microstructural constituents can result in preferential strain \npartitioning towards the softer ferrite matrix, which controls the yielding of softer ferrite \nmatrix [61,62] . \n Precipitation strengthening contribution can be different for random precipitation and \ninterphase precipitation [30]. \n Ashby -Orowan equation (eqn. 2) for the prediction of precipitation strengthening is valid \nonly in case of incoherent precipitates, but it is invalid for coherent and shearable \nprecipitates [51], which are very fine (say, < 3 nm [56]) and are difficult to detect and \nquantify.  \n \n4. Conclusions:  \n In order to understand the effect of C and N levels and isothermal holding temperature on \nthe microstructure, precipitation and mechanical properties of V -containing steels (0.05 wt.%), \ntwo different V -steels having different C and N levels , namely high -C low -N (HCLN) and low -C \nhigh-N (LCHN),  were subjected to isothermal holding treatment over a temperature range of \n500–750 °C after hot -deformation and finally air -cooled. Major conclusions derived from the \nstudy are listed below:  \n Maximum precipitation strengthening has been found at intermediate isothermal holding \ntemperatures (600 – 650 °C) in both the investigate d steels  as a result  of the formation of \nfine V(C,N) precipitates at high density.  \n In spite of the significantly smaller fraction of pearlite and bainite (by ~ 20 -30 %), \ncoarser average ferrite grain size (by ~ 2 .0 – 2.5 µm) and lower dislocation density i n the \nferrite matrix, the yield strength of LCHN steel was close to HCLN steel. This can be \nattributed to the higher precipitation strengthening (by ~ 30 –50 MPa) resulted from finer \nV-precipitates in LCHN steel than that of HCLN steel.    Average V (C, N) pre cipitate sizes were  finer in LCHN steel than that of HCLN steel, \nirrespective of the isothermal holding temperature. This can be due to the better \ncoarsening resistance of VN, or N -rich V(C, N), precipitates (present in LCHN steel) as \ncompared to VC, or C -rich V(C, N), precipitates (present in HCLN steel).  \n Presence of harder constituents and the s tronger interaction between incoherent V(C, N) \nprecipitates and the geometrically necessary dislocations contributed to greater strain -\nhardening ability in HCLN steel with respect to LCHN steel.  \nPresent findings indicate that low -C high -N composition in V steel is suitable for flat products \n(thinner plates and sheets)  that are subjected to industrial coiling treatment and require good \ncombination of strength and ductility.  The beneficial effect of intermediate  coiling temperature \n(600 – 650 °C) is evident. In a separate study, high -C low -N composition are found to be suitab le \nfor long products (rod and bars) and thicker plates, which are continuously cooled down after \nrolling and primary require d to achieve the specified strength level  [54]. \n \nAcknowledgements:  \nThe authors duly acknowledge the financial support received from the Research & Development \nand Scientific Services Division of Tata Steel, Jamshedpur, experimental facilities offered by the \nDepartment  of M etallurgical and Materials Engineering , Central Research Facility  and S teel \nTechnology Centre  at IIT Kharagpur and equipment grant provided by S ponsored Research and \nIndustrial Consultancy , IIT Kharagpur  through SGIRG scheme. The authors would also like to \nsincerely thank Mr. S. Neogi  and Dr. G.K. Dey from the Materials Science Division of Bhabha \nAtomic Research Centre, Mumbai for their help and support in performing some TEM studies \nfor this work . \n \nRefer ences:  \n[1] T. 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Abbreviation s: HR : heating rate; CR : cooling rate.  \nFig. 2:  Thermo -Calc® software prediction of equilibrium precipitate stabili ty in the \ninvestigated  steels.   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 Fig. 3 : TTT (Time temperature transformation)  diagram s of (a) HCLN and (b ) LCHN steels as \npredicted from JMatPro® software. (c) D ilation vs. temperature curve s of the investigated  \nsamples upon cooling at a rate of 10°C/s. Austenite to ferrite transformation start temperature s \nduring cooling  (Ar 3) are indicated by arrows.  \nFig. 4: Optical micrographs of dilatometry tested samples of (a) HCLN and (b) LCHN steel s for \nthe cooling rate of 10°C/s. Abbreviations: F: Polygonal ferrit e, AF: Acicular ferrite, P: Pearlite, \nB: Bainite and M: Martensite.  \n \n 3  \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: Optical micrographs of HCLN steel samples for different isothermal holding \ntemperatures as mentioned on the images . Abbreviations: F: Polygonal ferrite ; P: Pearlite, B: \nBainite  and DP: Degenerated pearlite.  High magnification scanning electron  micrographs of \ndegenerated pearlite and bainite are shown in the inset of (c) and (e) respectively.  \n 4  \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: Optical micrographs of LCHN steel samples for different isothermal holding temperatures \nas mentioned on the images. Abbreviations: F: Polygonal ferrite; P: Pearlite, B: Bainite and DP: \nDegenerated pearlite.  High magnification scanning electron  micrographs of degenerated pearlite \nis shown in the inset of (f) respectively.  \n \n 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 Fig. 7: Va riation in (a) the average ferrite grain sizes and (b) the amount of harder constituents \n(pearlite and bainite) as the function of isothermal holding temperatures for the investigated steels.  \n(a) \n (b) \nBF \n DF \n 6  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n600 °C  \n 600 °C  \n 7  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 8: Bright field (BF) and dark field (DF) transmission  electron micrographs (TEM) and the \ncorresponding SAED pattern analysis of the precipitates in the ferrite matrix of HCLN steel for \ndifferent isothermal holding temperatures as mentioned on the micrographs . Energy dispersive \nspectroscopy analysis (EDS) of the fine precipitates is  inserted in (g) . Dislocations , precipitates \nand their mutual interactions are indicated on the micrographs by arrows.  Interphase precipitates \nare shown in (i) and the interphase growth front is marked by an arrow . \n 8  \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. 9: Bright f ield (BF) and dark field (DF) transmission electron micrographs (TEM) and the \ncorresponding SAED pattern analysis of the precipitates in the ferrite matrix of LCHN steel for \ndifferent isothermal holding temperatures as mentioned on the micrographs. Energy dispersive \nspectroscopy analysis (EDS) of the fine precipitates is inserted in (h). Dislocations, precipitates \nand their mutual interactions are indicated on the micrographs by arrows. Interphase \nprecipitates are shown in (g), where one of the  precipitate array s is indicated.  \n 9  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 1 0: Variation in  (a) average precipitate size, (b) precipitate density and the (c) precipitate \nvolume fraction with isothermal holding  temperature  for the investigated steels . \nFig. 1 1: Variation  in (a) average m acro-hardness and (b) average micro -hardness (from ferrite \nregions) of the investigated steels as the function of  isothermal holding temperature . \n(a) \n (b) \n(c) \n(a) \n (b)  10  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 1 2: Tensile stress -strain curves of (a) HCLN and (b) LCHN steel samples isothermally \nheld at 500°C, 600°C and 700°C.  \nFig. 1 3: Estimated  yield strength  levels  (HCLN_YS and LCHN_YS) and precipitation \nstrengthening contributions (HCLN_pptn and LCHN_pptn) of the investigated steels at \ndifferent isothermal holding  temperatures.  \n 11  \n \nTable.1: Chemical composition (wt.%) of the investigated  steels . \n \n \n \n \nTable 2 : Fractions (percenta ge) of different microstructural constituents in the thermo -\nmechanically simulated samples for different isothermal holding temperatures.  \n \n \n  C Mn Si S P V N \nHCLN  0.22 1.7 0.43 0.02 0.02 0.05 0.008  \nLCHN  0.055  1.65 0.42 0.009  0.01 0.05 0.0137  \nIsothermal \nholding  \ntemperature s \n(°C) HCLN  LCHN  \n Ferrite  \n(%) Pearlite  \n(%) Degenerate \npearlite  \n(%) Bainite  \n(%) Ferrite  \n(%) Pearlite  \n(%) Bainite  \n(%) \n500 45±3  25±3  10±3  20±3  81±3  14±3  5±2 \n550 46±2  29±1  25±2  - 79±4  26±4  - \n600 46±1  30±3  24±1  - 77±2  20±3  - \n650 48±3  52±3  - - 73±3  27±3  - \n700 45±1  25±4  - 30±3  70±1  20±3  10±1  \n750 48±3  17±1  - 35±3  72±2  12±4  16±2  "    },    {        "title": "0707.1216v1.Effective_chiral_magnetic_currents__topological_magnetic_charges__and_microwave_vortices_in_a_cavity_with_an_enclosed_ferrite_disk.pdf",        "content": " \n \nEffective chiral magnetic currents, topological magnetic charges, and \nmicrowave vortices in a cavity with an enclosed ferrite disk \n \nMichael Sigalov, E.O. Kame netskii, and Reuven Shavit \n \nDepartment of Electrical a nd Computer Engineering,  \nBen Gurion University of the Negev, Beer Sheva, 84105, Israel \n \n \nAbstract \n \nIn microwaves, a TE-polarized rect angular-waveguide resonator with an inserted thin ferrite disk \ngives an example of a nonintegra ble system. The interplay of re flection and transmission at the \ndisk interfaces together with the material gyrotropy effect gives ri se to whirlpool-like \nelectromagnetic vortices in the proximity of the ferromagnetic resonance. Based on numerical \nsimulation, we show that a charact er of microwave vortices in a cavity can be analyzed by means \nof consideration of equivalent magnetic current s. Maxwell equations allows introduction of a \nmagnetic current as a s ource of the electromagnetic field. Specifically, we f ound that in such \nnonintegrable structures, magnetic  gyrotropy and geometrical factors leads to the effect of \nsymmetry breaking resulting in effective chir al magnetic currents and topological magnetic \ncharges. As an intriguing fact, one can observe  precessing behavior of  the electric-dipole \npolarization inside a ferrite disk.  PACS: 41.20.Jb; 76.50.+g; 78.20.Bh; 84.40.Az   The concept of nonintegrable, i. e. path-dependent, pha se factors is one of the fundamental \nconcepts of electromagnetism. When there is no symmetry with rotational and/or translational \ninvariance and so the wave equatio n cannot be separated in some coordinate system, one has an \nexample of a nonintegrable system. Presently, noni ntegrable systems (such, for example, as Sinai \nbilliards) are the subject for intensive studies in microwave cavity experiments [1]. In view of the \nso-called quantum-classical correspondence, these experiments are useful in studying the quantum \nchaos phenomena. To get microwave billiards with  broken time-reversal symmetry, ferrite samples \nwere introduced into the resonators [2]. In our recent paper [3], we studied the microwave vortices \nin a three dimensional system of a TE-polarized r ectangular-waveguide resona tor with an inserted \nthin ferrite disk based on full Maxwell-equation numerical solutions of the problem. Because of \ninserting a piece of a magnetized ferrite into the resonator domain, a microwave resonator behaves \nunder odd time-reversal symmetry (TRS) and a ferrite  disk acts as a topol ogical defect causing \ninduced vortices. The microwave vortices are define d as lines to which the Poynting vector is \ntangential. The interplay of reflection and transmissi on at the disk interfaces together with material \ngyrotropy effect gives rise to a rich variety of wave phenomena. It was shown that  the power-flow \nlines of the microwave-cavity fi eld interacting with a ferrite disk, in the proximity of its \nferromagnetic resonance, form the whirl pool-like electromagnetic vortices.    \n    In different studies with TE polarized cavities [1  3], the vortex behavior in a vacuum region of \na cavity can be easily understood from an analysis of the field structure. For TE polarized (with an \nelectric field directed along y axis) electromagnetic waves in vac uum, the singular features of the \ncomplex electric field component \n),(zxEy  can be related to those that will subsequently appear in  2the associated two-dimensional time-ave raged real-valued P oynting vector field ),(zxSr\n. For such \nelectromagnetic fields, the Poyn ting vector is represented as ()y yE EcS⊥∇ =r r\n Im8*2\nπω, where yE is \na complex vector of the y-component of the electric field: ()ti\nyc y eE E ω≡ . The fact that for \nelectromagnetic fields invariant with respect to a certain coordinate, a time-average part of the \nPoynting vector can be approximated by a scalar  wave function, allows  analyzing the vortex \nphenomena. For a TE polarized field, we can write ),(  ),( ),( ),(zxi\ny ezx zx zxEχρ ψ = ≡ ,                   \nwhere ρ is an amplitude and χ is a phase of a scalar wave function ψ. This allows rewriting the \nPoynting vector expression as ),( ),(2zx zx S χ ρ⊥∇ =r\n. Such a representation of the Poynting \nvector in a quasi-two-dimensional system gives possi bility to define a phase  singularity as a point \n(x, z) where the amplitude ρ is zero and hence the phase χ is undefined. These singular points of \n),(zxEy correspond to vortices of the power flow Sr\n, around which the power flow circulates. A \ncenter is referred to as a (positive or negative) to pological charge. Since such a center occur in free \nspace without energy absorption, it is evident that 0=⋅∇⊥Sr\n. The singular points of Sr\n (the \nvortex cores) can be dir ectly related to the zero-electric-field  topological features in a vacuum \nregion of the cavity space, but not inside a fe rrite region where one cannot express the Poynting \nvector only by the Er\n-field vector. Moreover, there is a special  interest in an analysis of microwave \nvortices generated by a ferrite sample pla ced in a cavity region of a maximal field ),(zxEy . As it \nwas shown in [3], specifically the cases when a fe rrite disk is placed in a maximum of the cavity \nelectric field show the most pronounced and compact Poynting-vector vortices.  \n    In the standard situation of microwave cavity experiments with  inserted ferrite samples, the \nmechanism behind the TRS breaking effects is believe d to be intimately connected with (and in \nfact generated by) the losses of energy and flux.  Interplay between losses and quantum chaotic \neffects is rather interesting and non-trivial (see e.g. [4]). At the same time, it is known that the \nlosses in the ferrite are always one of the ma in problems to study quantum manifestations of \nclassical chaos in the regime of broken TRS. Th is fact makes a general comparison between the \ntheory and experiment not an ea sy task. For this reason, any e xperimental technique able to \ngenerate TRS-breaking effects with introduci ng minimal losses (or introducing them in a \ncontrollable way) is of considerable interest.      In the ferromagnetic resonance, depending on a quantity of a bias magnetic field and (or) \nfrequency, one has the regions with positive or negative permeability parameters [5]. For a ferrite disk placed in a maximum of the RF electric fiel d in a rectangular-waveguide resonator, there are \nfundamentally different conditions for generati on of microwave vortices in the positive- and \nnegative-parameter regions. For negative permeabili ty parameters, microwave vortices appear only \nwhen the material properties of a disk are charac terized by big losses [6]. Similar situation takes \nplace for a plasmon-resonance nanoparticle illumi nated by the electromagnetic field, where the \nspiral energy flow line trajectories appear for a lossy sample with negative permittivity parameters \n[7]. Contrary, in a case of positive permeability parame ters, the losses may play an indirect role in \nforming microwave vortices. This fact was illustr ated in [3] (see Figs. 15 in [3]): in a case of \npositive permeability parameters one has very slig ht variation of the Poynting-vector vortex \npictures in (and in close vicinity of) a ferrite disk for different losses parameters of a ferrite \nmaterial.       The purpose of this letter is to show that fo r a ferrite disk with positive permeability parameters, \ncreation of microwave vortices in a cavity is due to effective chiral magnetic currents rather than \ndue to additional losses (which can be indeed mi nimal). As a very intriguing fact one can observe  3topological magnetic charges and precessing electric polarization in microwave ferrite samples, \nwhich appear because of the ge ometrical factor and the TRS br eaking. Since the nonintegrable \nnature of the problem precludes exact analytical  results for the eigenvalues and eigenfunctions, \nnumerical approaches are required. We used  the HFSS (the software based on FEM method \nproduced by ANSOFT Company) CAD simulation programs for 3D numerical modeling of \nMaxwell equations [8]. In our num erical experiments, both modulus  and phase of the fields are \ndetermined. We consider a situation when a normally  magnetized ferrite disk (being oriented so \nthat its axis is perpendicular to a wide wall of a waveguide) is placed in  a region of a maximal \ncavity electric field in the middle of the waveguide height [3]. Fi g.1 shows an example of such a \nconfiguration. The ferrite disk parameters are:  diameter D=6 mm, thickne ss t=0.5 mm, saturation \nmagnetization 4 πMo=1880 Gauss, permittivity εr=15. A bias magnetic field of 5030 Oersted and \nthe cavity resonance frequency of 8.7 GHz correspond to the region of positive permeability parameters. To stress the role of a combined eff ect of magnetic gyrotropy and geometrical factors \nwe ignore any material losses in a ferrite disk (we assumed that ∆H=0 and tan δ=0). The only \nlosses taken into account in our numerical simulati ons are the losses in cavity walls: the cavity \nwalls are made from copper.     A typical picture of the P oynting-vector microwave vortex [3] in  air regions closely to a ferrite \ndisk is shown in Fig. 2. In the center point, the Poynting vector is equal to zero. A ferrite disk is \nsurrounded by an air cylinder and th e Poyning vector is pictured on the upper and lower planes of \nthis cylinder. Being aimed to uncover the physics of  such a vortex structure,  we should start with \nconsideration of the electric and ma gnetic fields on the upper and lower planes of this external (air) \ncylinder. Fig. 3 shows the electr ic field vectors corresponding to a certain time phase, which we \nmark as phase ωt=0º. At ωt=180º one has an opposite direction of the electric  field vectors. A thin \nferrite disk slightly perturbs the cavity electric field. With variation of the time phase one has the \no180 -switching of the electric fiel d vector. At the same time, situation with the magnetic field \ndistribution is completely different. Fig. 4 gives the top and side views of magnetic fields on the \nplanes of the external (air) cylinder for different time phases. It becomes evident that for a certain \npoint on the xz plane immediately above a ferrite disk, the magnetic field vector rotates in the \nclockwise direction. In Fig. 4, this can be viewed vi a rotation of the magneti c field vector in a \ngiven point A. The sizes of the in-plane magnetic  field vectors reduce as one moves to a center. \n    The above results show that the main reason leading to creation of th e Poynting vector vortices \nin the external (with respect to a ferrite) regions is strong spiral -like perturbation of the cavity \nmagnetic fields caused by a ferrite disk. This pe rturbation can be described by effective magnetic \ncurrents. The notion of the effective magnetic currents is widely used in different excitation problems of waveguides and cavities [9]. There ar e equivalent surface magnetic currents which are \nformally introduced as a result of  discontinuity of tangential com ponents of an electric field: \n) (\n2 1 t tmE En ir r r r\n− ×≡ , where \n1tEr\nand \n2tEr\n are tangential fields near the interface and nr is a normal to \nthe interface. When we consider the fields insi de a ferrite we find that on the upper and lower \nplanes of a disk there are pronounced tangential compone nts of the electric field vectors. In Fig. 5, \nthe air cylinder has the same height as a ferrite disk and the picture shows the electric field distribution inside a ferrite disk and in the lateral-surf ace air vicinity of a ferrite disk. Comparing \nthe pictures in Figs. 3 and 5 (a bove and below the surfaces of a fe rrite disk), it becomes evident \nthat transitions between the air-cylinder and ferrite -disk planes (the distances between these planes \nare much less than a waveguide height) can be fo rmally considered as discontinuities of the \ntangential components of the electric  field vectors. Fig. 6 shows t op- and side-view time evolutions \nof the electric field in a ferrite disk. On the top-view pictures in  Fig. 6, one sees the upper-plane \nvectors as intense arrows, while the lower-plane v ectors are shown as pale arrows.  The tangential \nelectric field vectors on the lower plane are contra riwise to those on the u pper plane. The in-plane \nelectric field vector rotates in the counter clockw ise direction. For the in-plane rotating clockwise  4magnetic field and counter clockwis e electric field there is zero normal component of the Poynting \nvector in a ferrite disk.      The top-view pictures in Fig. 6 give clear perception of an effective surface magnetic current. In \nevery given point of a ferrite-disk plane, an in-pla ne vector perpendicular to the tangential electric \nfield is the vector of the equivalent magnetic current. Fig. 7 shows the surface magnetic current \ndistributions on the upper plane of  a ferrite disk. One can observe a spiral-like character of the \nsurface magnetic current. Dynamics of the upper-pl ane-vector evolution s hows that during a half \nof a time period a surface magnetic current flow s away from a disk center and so one has a \ndivergent spiral. During the next  half of a period, a surface ma gnetic current flows towards to a \ndisk center. There is a convergent spiral. Since the tangential-electric-field vectors on the upper \nand lower planes are in opposite directions, for the lower-plane-vector evolution (when one looks \nin the direction from the upper disk plane to the lowe r disk plane) there is a convergent spiral at the \nfirst half of the period and a divergent spiral at the s econd half of the period.  \n    Dynamics of surface magnetic currents is co rrelated with the magnetic-field dynamics. The side-\nview pictures of the magnetic fi eld distributions in Figs. 4 clearly show the presence of normal \ncomponents of the magnetic field at certain time phases. More explicit pictures of the magnetic \nfield distributions outside a fe rrite disk (see Fig. 8) clearly  demonstrate the convergent (\no90 =tω ) \nand divergent (o270 =tω ) central locations of magnetic fields on a surface of a ferrite disk. A joint \nanalysis of Figs. 7 and 8 gives an evidence for topological magnetic charges. The topological \nmagnetic charges appear at the phases (o o270 ,90 =tω ) of extreme dynamical symmetry breaking \nwhen circulations of a surface magnetic current are maximal. \n    The main features of the microwave vortex creation appear from  a special charac ter of motion of \nthe electric field vector inside a disk. It is worth noting that in a given point, the electric field \ninside a disk is characterized by a precessional beha vior. An analysis shows that the electric-field \npolarization ellipse depends on the dielectric constant of the disk  material. Fig. 9 (a) gives a picture \nof the in-plane electric-field polarization ellipse for a real ferrite material, having εr=15. In this \ncase an axial ratio of the precession ellipse is 2.35. Fig. 9 (b) shows the in-plane polarization \nellipse (in the same point of a disk) fo r a hypothetical ferri te material with εr=2. In this case an \naxial ratio of the precession ellipse  is equal to 10.72. Because of a processional behavior of an \ninternal electric field, electri c-dipole polarization inside a fe rrite disk should also have a \nprecessional character of motion. This precessional motion of the electric-dipole polarization takes \nplace against a background of the magnetizati on precession causing magnetic gyrotropy of the \nferrite material. There is an evident coupling be tween two precessional moti on processes, electric \nand magnetic.      In conclusion, we have to note that we f ound that in certain nonintegrable structures, magnetic \ngyrotropy and geometrical factors may lead to the effect of symmetry breaking resulting in \neffective chiral magnetic current s and topological magnetic charges.  This is the main reason of \ncreation of microwave vortices in a cavity with an enclosed lossless ferrite disk. We studied a \nnormally magnetized thin-film ferrite sample. In this case of a quasi-2D structure, one has only in-plane components of magnetization, while the fiel ds inside a ferrite ar e three dimensional. \nFollowing our previous results shown in paper [3],  the vortex structure is  strongly dependable on \nthe disk position in a cavity. It is the subject of our future studies to analyze the effective magnetic \ncurrents and topological charges in diffe rent disk positions in a cavity.  \n    One can expect opening a very exciting pros pect in the electromagnetic-vortex applications. \nRecently, a new interesting phenome non of relationships between the near-field phase singularities \n(vortices) in a slit-metal-plate structure and the feat ures of the far-field radiation pattern has been \nobserved [10]. The question of new types of mi crowave devices based on the considered above \nferrite-disk vortex structures may a ppear as a subject of a great future interest. Some examples of \nproper applications of the vortex concept in design of the ferrite- based microwave devices can be  5found in [11]. It is shown, in particular, that for a microwave patch antenna with an enclosed \nferrite disk the \"vortex qu ality\" is strongly correlated with th e far-field antenna characteristics. \n  [1] E. Doron, U. Smilansky, and A. Frenkel, Phys. Rev. Lett. \n65, 3072 (1990); S. Sridhar, Phys. \nRev. Lett. 67, 785 (1991). \n[2] P. So, S.M. Anlage, E. Ott, a nd R. N. Oerter, Phys. Rev. Lett. 74, 2662 (1995); M. Vrani čar, \nM. Barth, G. Veble, M. Robnik, and H.-J . Stöckmann, J. Phys. A: Math. Gen. 35, 4929 (2002); \n     H. Schanze, H.-J. Stöckmann, M. Ma rtinez-Mares, and C.H. Lewenkopf, Phys. Rev. E 71, \n016223 (2005). \n[3] E.O. Kamenetskii, M. Sigalov, and R. Shavit, Phys. Rev. E 74, 036620 (2006). \n[4] Y.V. Fyodorov, D.V. Savin and H.-J .Sommers, J. Phys. A: Math. Gen. 38, 10731 (2005). \n[5] A. Gurevich and G. Melkov, Magnetic Oscillations and Waves  (CRC Press, New York, 1996). \n[6] M. Sigalov, E.O. Kamenetskii, and R. Shavit (unpublished). [7] M. V. Bashevoy, V.A. Fedotov, and N.I. Zheludev, Opt. Express \n13, 8372 (2005). \n[8] High Frequency Structure Simulator, www.hfss. com, Ansoft Corporation, Pittsburgh, PA \n15219. \n[9] L.A. Vainstein, Electromagnetic Waves , 2nd ed. (Radio i Svyas, Moscow, 1988) (in Russian); \nR.E. Collin, Foundations for Microwave Engineering , 2nd ed. (McGraw-Hill, New York, \n1992). \n[10] H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, Phys. Rev. Lett. 93, 173901 \n(2004). \n[11] M. Sigalov, E.O. Kamenetskii, and R. Shavit, eprint, arXiv:0707.0350 [physics.class-ph]. \n Figure captions \nFig. 1. (Color online) A rectangular-waveguide cavity with an enclosed ferrite disk \nFig. 2. (Color online) Top and side views of Poynting vector above and below a ferrite disk \nFig. 3. Electric field above and below a ferrite disk  Fig. 4. Top and side views of magnetic fields above  and below a ferrite disk at different time phases  \nFig. 5. (Color online)  Electric field inside a disk and near a lateral surface of a disk  Fig. 6. (Color online) Top and side views of el ectric fields on the upper and lower planes inside a \nferrite disk at different time phases \nFig. 7. Surface magnetic current distributions Fig. 8. Magnetic field distributions outside a ferrite disk: Evidence for topological magnetic charges \nFig. 9. Tangential electric-field precession in a ferrite disk: (a) \n15=rε , axial ratio of the precession \nellipse is 2.35; (b) 2 =rε , axial ratio of the precession ellipse is 10.72 \n \n \n    \n \n  6\n \n  \n \n \nFig. 1.  (Color online) A rectangular-waveguide cavity with an enclosed ferrite disk \n  \n \n \n \nFig. 2. (Color online)  Top and side views of Poynting vector above and below a ferrite disk \n     7\n \n \nFig. 3. Electric field above and below a ferrite disk at different time phases \n \n  8\n \n \nFig. 4. Top and side views of magnetic fields above and below a ferrite disk at different time phases \n \n   9\n \n \nFig. 5. (Color online)  Electric field inside a disk and near a lateral surface of a disk \n \n \n   10\n \n \nFig. 6. (Color online) Top and side views of elect ric fields on the upper and lower planes inside a \nferrite disk at different time phases \n \n   \n \n \nFig. 7. Surface magnetic current distributions  11\n \n \n                                                                          \n \nFig. 8. Magnetic field distributions outside a ferr ite disk: Evidence for topological magnetic charges \n  \n \n \n                (a)   12\n \n \n                              (b) \n  \nFig. 9. Tangential electric-field precession in a ferrite disk: (a) 15=rε , axial ratio of the precession \nellipse is 2.35; (b) 2=rε , axial ratio of the precession ellipse is 10.72 \n "    },    {        "title": "1907.12076v1.Warren_Averbach_line_broadening_analysis_from_a_time_of_flight_neutron_diffractometer.pdf",        "content": "Warren-Averbach line broadening analysis from a time of flight neutron di \u000bractometer\nD.M. Collinsa,b, A.J. Wilkinsonb, R.I. Toddb\naSchool of Metallurgy &Materials, University of Birmingham, Birmingham, B15 2TT, UK\nbDepartment of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, UK\nAbstract\nThe well known Warren-Averbach theory of di \u000braction line profile broadening is shown to be applicable to time of flight data\nobtained from a neutron spallation source. Without modification, the method is applied to two very di \u000berent examples; a cold\nworked ferritic steel and a thermally stressed alumina-30% SiC composite. Values of root mean square strains averaged over a\nrange of lengths for the ferritic steel were used to estimate dislocation densities; values were found to be in good agreement with\ngeometrically necessary dislocation densities independently measured from similarly orientated grains measured from electron\nbackscatter di \u000braction analysis. An analytical model for the ceramic is described to validate the estimate of root mean square\nstrain.\nKeywords: Neutron di \u000braction, Line profile, Warren-Averbach, Dislocation density, Size and strain broadening\n1. Introduction\nThe seminal theory by Warren and Averbach [1, 2, 3] to\ndescribe the plastic deformation of metals provides a method\nto quantify crystallite size and strain from broadened di \u000brac-\ntion line profiles. Size broadening can arise from both coher-\nent and incoherently di \u000bracting domains, as discovered ear-\nlier by Scherrer [4]. Such domains are microstructural fea-\ntures that may comprise grains, twins, stacking faults, second\nphase particles, subgrain or deformation mosaicity structures.\nStrain broadening results from imperfections in the di \u000bracting\nlattice that, themselves, generate inhomogenous crystalline dis-\ntortions. These typically arise from point or line defects. As\nthe broadening nature is di \u000berent for size and strain, the simul-\ntaneous e \u000bects can be quantified independently. The method\nis hence a powerful characterisation tool that yields a com-\npendium of information from bulk specimens that has the gen-\nerality to be valid for various material classes including ceram-\nics, polymers, composites as well as metals.\nSeparation of size and strain broadening components has\nlong been known to be obtainable from X-ray di \u000braction line\nprofile analysis, but has not been extended to time-of-flight neu-\ntron di \u000braction. This paper adopts the original Warren and\nAverbach line broadening methodology to demonstrate that suit-\nably high resolution time of flight data are also well suited to\nsuch analysis. A derivation of the powder theory for time of\nflight di \u000braction is given, followed by example applications of\nthe Warren-Averbach methodology to time of flight data ob-\ntained from (1) a cold worked steel in which line broadening is\ncaused by dislocation generation, and (2) an alumina-30% SiC\nceramic with spatially varying thermal residual microstresses\nresulting from the thermal expansion mismatch between the\n\u0003d.m.collins@bham.ac.uk\n\u0003\u0003richard.todd@materials.ox.ac.ukphases during cooling after sintering. To demonstrate the va-\nlidity of the results, comparisons are made to geometrically\nnecessary dislocation (GND) densities obtained from electron\nbackscatter di \u000braction (EBSD) analysis for the steel and an an-\nalytical model for the ceramic.\n2. Theory\nWe follow the Warren-Averbach analysis [1, 3] adapting it\nwhere appropriate for use with a neutron time-of-flight di \u000brac-\ntometer. The analysis considers the 00 lreflection of an or-\nthorhombic material [5]. This simplifies the analysis, but it\ncan be shown that the results apply to any set of planes in any\ncrystal structure [6, 7]. A single distorted crystal is considered\nin which the position Rm1m2m3of each unit cell is specified in\nterms of the (undistorted) orthorhombic lattice vectors a1,a2,\na3, plus a displacement \u000em1m2m3from the ideal position in the\nundistorted crystal:\nRm1m2m3=m1a1+m2a2+m3a3+\u000em1m2m3 (1)\nwhere m1,m2andm3are integers specifying a particular cell.\nIf the di \u000braction vector is represented in terms of the recip-\nrocal lattice vectors b1,b2,b3:\ns\u0000s0\n\u0015=h1b1+h2b2+h3b3 (2)\nwhere h1,h2andh3are continuous variables and sands0are\nunit vectors parallel to the incident and di \u000bracted beams respec-\ntively, then the total scattered intensity Irelative to that from the\nPreprint submitted to journal October 20, 2021arXiv:1907.12076v1  [cond-mat.mtrl-sci]  28 Jul 2019reference scattering length is:\nI(h1h2h3)=F2\nSX\nm1X\nm2X\nm3X\nm0\n1X\nm0\n2X\nm0\n3exp\u0014\n2\u0019i\u001a\nh1(m1\u0000m0\n1)\n+h2(m2\u0000m0\n2)+h3(m3\u0000m0\n3)+\u0012s\u0000s0\n\u0015\u0013\n\u0001(\u000em1m2m3\u0000\u000em0\n1m0\n2m0\n3)\u001b\u0015\n(3)\nwhere FSis the structure factor.\nA single small element of a neutron detector array in a time-\nof-flight di \u000bractometer measures the intensity of scattered neu-\ntrons with a fixed 2 \u0012and azimuthal angle, and varying \u0015. The\ncorresponding di \u000braction vector therefore has varying length\nalong a fixed direction. The interference function given in Eq.\n3 represents the spreading of the reciprocal lattice point of an\nindividual crystal in reciprocal space. To obtain the measured\nintensity, it is necessary to sum the contributions from all the\ncrystals in the sample. It is assumed in what follows that the\nextent of the spreading is small compared with the length of the\ndi\u000braction vector and that the e \u000bects of any crystallographic\ntexture or other source of anisotropy are insignificant over the\nsmall range of orientations concerned.\nFigure ??shows a scattering vector of length h3b3whose\ndirection has minimum distance rfrom the centre of a 00 lre-\nciprocal lattice point a crystal. The number of crystals with\nh3b3passing through an annulus centred on the scattering vec-\ntor direction, with thickness d rand radius ris:\nM2\u0019rdr\n4\u0019h2\n3b2\n3=Mrdr\n2h2\n3b2\n3(4)\nwhere Mis the product of the total number of crystals and the\nmultiplicity of the reflection. The average intensity contributed\nby each of the crystals with all possible rotations !around a\nline joining the lattice point to the origin is:\n1\n2\u0019Z2\u0019\n0I(r;!;h3) d! : (5)\nThe total intensity sampled at the tip of the di \u000braction vector is\ntherefore obtained by summing over rto include all crystals:\nI(h3)=ZZI(rwh 3)Mrd!dr\n4\u0019h2\n3b2\n3\n=ZZI(h1h2h3)Mb 1b2dh1dh2\n4\u0019h2\n3b2\n3\n=ZZI(h1h2h3)Mdh1dh2\n4\u0019vah2\n3b3\n3(6)\nwhere vais the volume of the unit cell.\nUsing the de Broglie relation, the length of the reciprocal\nlattice vector for a neutron can be written:\nh3b3=2 sin\u0012\n\u0015=2mLsin\u0012\nhpt(7)where m is the neutron mass, hp, is Planck’s constant, Lis the\nneutron path length and tis the time-of-flight. Substituting for\nI(h1h2h3) and h3in Eq. 6 using Eqs. 3 and 7 gives:\nI(h3)=Mh2\npt2F2\nS\n16\u0019vam(Lsin\u0012)2b3\u00021\nF2\nSZZ\nI(h1h2h3) dh1dh2\n=KZZX\nm1X\nm2X\nm3X\nm0\n1X\nm0\n2X\nm0\n3exp\u0014\n2\u0019i\u001a\nh1(m1\u0000m10)\n+h2(m2\u0000m0\n2)+h3(m3\u0000m0\n3)+\u0012s\u0000s0\n\u0015\u0013\n\u0001(\u000em1m2m3\u0000\u000em0\n1m0\n2m0\n3)\u001b\u0015\ndh1dh2:\n(8)\nwhere K=Mh2\npt2F2\nS\n16\u0019vam(Lsin\u0012)2b3. The important result of this analysis\nis that Kis approximately constant for a time of flight di \u000brac-\ntometer. This is because tandFSvary little close to the re-\nflection and the measured intensities are normalised to account\nfor the variation in incident intensity with wavelength. For a\nsingle elemental neutron detector, Lsin\u0012is obviously constant,\nbut in practice, extended detector banks covering a range of 2 \u0012\nandLare used to increase the total neutron count. These detec-\ntors are “focused”, however, either geometrically or electroni-\ncally, such that Lsin\u0012remains constant, and Eq. 7 shows that\ntis also constant for a given di \u000braction vector (or d-spacing).\nThe Warren-Averbach analysis can therefore be used without\nsignificant modification for extended detector banks providing\nthat anisotropic e \u000bects in the specimen remain small over the\nadditional range of lattice plane orientations sampled.\nEq. 8 gives the intensity measured by a perfect (i.e. free\nfrom instrumental broadening) neutron time-of-flight di \u000brac-\ntometer as a function of h3, which can be calculated easily for a\nparticular di \u000braction peak from the time-of-flight or equivalent\nd-spacing data provided at neutron sources as:\nh3=t0l\nt=d0l\nd(9)\nwhere dis the interplanar spacing which would give a Bragg\npeak at a particular t, and t0andd0are the values of tandd\nrespectively for the centroids of the peak being analysed. In\nreality, instrumental broadening is significant and needs to be\nremoved from the results as described in Section 3.2.\nWith the exception of K, the right hand side of Eq. 8 con-\ncerns only the distorted crystal and is identical to the right hand\nside of Eq. 5 in [1]. Warren [3] goes on to show that, with rea-\nsonable approximations, this can be written as a Fourier series:\nI(h3)=KN1X\nn=\u00001\bAncos 2\u0019nh3+Bnsin 2\u0019nh3\t(10)\nwhere Nis the total number of unit cells per crystal. The cosine\nand sine coe \u000ecients AnandBnare given by:\nAn=Nn\nN3hcos 2\u0019lZni (11)\nBn=Nn\nN3hsin 2\u0019lZni (12)\n2in which N3is the average number of unit cells per crystal in\na column normal to the di \u000bracting planes, Nnis the average\nnumber of cells possessing an nthnearest neighbour cell in the\nsame column, and the displacement along the column length\nbetween nthnearest neighbours due to the distortion is Zna3.\nOnly the cosine coe \u000ecients Anare used in the Warren-Averbach\nanalysis; the information obtainable from the sine coe \u000ecients\nBnis less useful and in any case the broadening is usually close\nto being symmetrical so that analysis around the peak centre\ngives Bnvalues close to zero.\nThe Fourier coe \u000ecients of the physically broadening profile\nare a product of size, AS\nn, and distortion, AD\nncoe\u000ecients, giving\nAn=AS\nnAD\nn (13)\nwhere AS\nn=Nn=N3andAD\nn=hcos 2\u0019lZni. Ifnandlare small,\nZnlmust also be small and the first two non zero terms from a\nMaclaurin series provide the following approximation\nhcos 2\u0019lZni! 1\u00002\u00192l2hZ2\nni: (14)\nIn logarithm form, ln hcos 2\u0019lZni=\u00002\u00192l2hZ2\nni, so that Equa-\ntion 13 becomes\nlnAn(l)=lnAS\nn\u00002\u00192l2hZ2\nni: (15)\nAs the domain size coe \u000ecient is independent of the order, l,\nand strain is dependent on l, the coe \u000ecients can be easily sep-\narated by plotting lnA n(l) against l2with constant nfor 2 or\nmore orders lof the same reflection. The intercept gives the\nstrain coe \u000ecient AS\nnand the gradient gives hZ2\nni.\nTheZnandAS\nnvalues can be related to more conventional\nphysical quantities as follows. The Znvalues relate to pairs of\nunit cells separated by a distance L=na3. The relative dis-\nplacement of the unit cells due to the distortion, \u0001L=a3Zn.\nThe corresponding strain \u000fL=\u0001L\nL=Zn\nn. It is therefore possible\nto use the Znvalues to calculate the rms strain ( \u000f1=2) normal to\nthe di \u000bracting planes and averaged over a distance L. As\nh\"2\nLi1\n2=hZ2\nni1\n2\nn(16)\nLcan be found by recognising that a3is the interplanar spacing\nfor the relevant reflection. For the size coe \u000ecients Warren [3]\nshows that\n\u0000 dAS\nn\ndn!\nn=0=1\nN3: (17)\nHaving obtained N3from the initial gradient of the graph of\nAS\nnversus n, the mean crystallite size normal to the di \u000bracting\nplanes can therefore be calculated as N3a3.\n3. Method\n3.1. Materials Studied\nTwo materials have been studied; (1) a low carbon ferritic\nsteel and (2) an alumina-30vol% SiC composite. Their details\nare as follows:A single phase ferritic steel ‘DX56’ was studied with nomi-\nnal composition Fe-0.12C-0.5Si-0.6Mn-0.1P-0.045S-0.3Ti. The\nmaterial had an initially weak crystallographic texture inherited\nfrom prior rolling during its processing and a mean grain size\nof 15\u0016m. The material was studied in an annealed, strain-free\nreference condition and in a deformed state. The latter was de-\nformed via cold rolling parallel to the initial rolling direction to\nan engineering strain in the rolling direction of 0.5.\nAn alumina-30 vol% SiC composite was made by ball milling\nSumitomo AES-11C \u000b-alumina and F800 \u000b-SiC powders (d 50\n=6.5\u0016m) together in water using Mg-PSZ media with the aid\nof a dispersant (Allied Colloids Dispex A40) and spray dried.\nThe powders were then hot pressed in a graphite die under Ar-\ngon at 1700\u000eC to form a disc 30 mm diameter \u00024 mm thick-\nness. The surface layers on both sides of the disc were ground\no\u000busing diamond abrasives and the resulting disc had a relative\ndensity of>99%. Microstructural examination showed similar\nfeatures to the composites reported previously [8], in which the\napproximately equiaxed and angular SiC particles were essen-\ntially unchanged by incorporation in the composite. An alu-\nmina reference specimen was produced using the same methods\nbut without the SiC addition.\n3.2. Time-of-flight di \u000braction\nAll measurements were made using a 15 mm \u000215 mm neu-\ntron beam. The neutrons sampled by the backscattered detec-\ntors measured direct strains at angles between 2\u000eand 10\u000eto the\nincoming beam. The steel specimens each had a cubic geom-\netry of 8\u00028\u00028 mm3. As the samples were prepared from\nthin sheets, this volume was achieved by layering successive\ncoupons and a \u000exing them with cyanoacrylate adhesive. Sam-\nples were placed with the incoming beam direction parallel to\nthe rolling direction of the sample. This sample orientation rel-\native to the backscatter detectors provided di \u000braction data from\nplanes with normals approximately parallel to the rolling direc-\ntion, providing results that can be related to this deformation\naxis. The alumina-30% SiC sample was positioned with its sur-\nface plane normal at 45\u000eto the incoming beam. The specimen\nwas positioned such that all parts of the beam passed through it.\nThe inclination of the specimen to the beam ensured the mea-\nsurement of strain values between the extremes of the small\namount of anisotropy caused by the alignment of slightly elon-\ngated SiC particles during hot pressing [8].\nFrom the measured di \u000braction data, several preparatory steps\nwere applied to obtain line profile information suitable for the\nWarren-Averbach analysis. Time-of-flight /d-spacings from the\nraw data were firstly converted to h3space (Eq. 9). Next, indi-\nvidual line profiles were extracted from the di \u000braction spectra.\nThese had a h3range of l\u0000\u0001h3tol+\u0001h3where lis the order of\nrefection studied (i.e. 1, 2, ...) and \u0001h3is the interval. The size\nof these intervals corresponded to 4 times the full width at half\nmaximum (FWHM) as recommended by Schwartz and Cohen\n[9].\nPrior to the deconvolution procedure, some practical steps\nwere taken: (1) a uniform h3data point spacing was set; a\nconstant spacing was determined and corresponded to interpo-\nlated intensities, here using a 1-dimensional linear interpolation\n3Figure 1: Raw data collected from HRPD instrument showing a ferritic steel and alumina-30% SiC in the reference and deformed state, with (a) & (d) full spectra,\n(b) & (e) first order line profiles, and (c) & (f) second order line profiles.\nscheme, (2) any background intensity was removed; a linear\nfunction was fitted here to a number of data points at the ex-\ntremes of the range ( \u00180:05\u0002\u0001h3), then deducted from the line\nprofile intensity, and (3) the background corrected intensity data\nwere normalised.\nEach experimentally measured di \u000braction line profile, h(h3)\nis described by the convolution of the specimen line profile,\nf(h3), and instrumental line profile, g(h3). This is written as\nh(h3)=f(h3)\ng(h3) (18)\nwhere\nis the convolution operator. As it is essential to interro-\ngate information only from the sample, f(h3) must be separated.\nEquation 18 may be solved for f(h3) using the convolution the-\norem. Using the Stokes method [10], the Fourier coe \u000ecients of\nf(h3) are given by\nAn=Hc(n)Gc(n)+Hs(n)Gs(n)\nG2c(n)+G2s(n)(19)\nBn=Hs(n)Gc(n)\u0000Hc(n)Gs(n)\nG2c(n)+G2s(n)(20)\nwhere H(n) and G(n) are the Fourier coe \u000ecients of the uncor-\nrected sample peak and the instrumental line profiles, respec-\ntively, for a given harmonic number, n, and subscripts cand\nsrefer to cosine and sine components, respectively. In princi-\nple, the true sample profile f(h3)=I(h3) can be reconstructed\nusing Equation 10, though this is not necessary for the Warren-\nAverbach analysis.\nIn the studies here, h(h3) is a deformed sample, (the cold-\nrolled ferritic steel or the alumina-30 vol% SiC composite) andg(h3) is an undeformed sample (annealed ferritic steel or alu-\nmina reference). The Fourier components were calculated by:\nHc(n)=Zl+\u0001h3\nl\u0000\u0001h3h(h3) cos(2\u0019nh3) dh3 (21)\nHs(n)=Zl+\u0001h3\nl\u0000\u0001h3h(h3) sin(2\u0019nh3) dh3 (22)\nGc(n)=Zl+\u0001h3\nl\u0000\u0001h3g(h3) cos(2\u0019nh3) dh3 (23)\nGs(n)=Zl+\u0001h3\nl\u0000\u0001h3g(h3) sin(2\u0019nh3) dh3: (24)\nThese components were normalised with respect to H0andG0\nwhich were obtained by setting n=0 in Equations 21 and 23,\nrespectively. All integrations were performed using Simpson’s\nrule. These coe \u000ecients were calculated only for values of nfor\nwhich the approximation of ln hcos 2\u0019lZniin Equation 15 led to\nan error of less than 10% in the second order peak.\n3.3. EBSD-based dislocation density validation\nThe ferritic steel in the reference and deformed states was\ncharacterised using EBSD to obtain spatially resolved crystal-\nlographic orientations. Data were obtained using a JEOL6500F\n(FEG) scanning electron microscope operating at an accelerat-\ning voltage of 20 kV with a probe current of \u001815 nA. Di \u000brac-\ntion patterns were acquired with a TSL Digiview II camera at a\n1000\u00021000 pixel2resolution at a\u00181 s acquisition time.\nGeometrically necessary dislocation (GND) density estimates\ncan be obtained via measurement of EBSD pattern shifts [11] to\nobtain lattice curvatures that can be subsequently solved through\n4analysis of the Nye tensor [12]. The process of acquiring an\nEBSD map will typically obtain information from a finite patch\nof microstructure. If multiple grains lie within this finite patch,\nall crystals, irrespective of orientation will be identified. This\nis not equivalent to information obtained from time of flight\ndi\u000braction; only a subset of an EBSD-analysed grain set will\nhave orientations that would obey Bragg’s law. A simple method\nis described here to extract the EBSD estimates of dislocation\ndensities from the orientations equivalent to those measured\nfrom time of flight di \u000braction.\nThe crystal orientation from the EBSD measurements is de-\nscribed through the rotation matrix G=Rx(\u001e1)Rz(\b)Rx(\u001e2)\nwhere\u001e1,\b,\u001e2are the Euler angular operations. The plane\nnormal to each reflection ( hkl) observed from neutron di \u000brac-\ntion, from the crystal reference frame, can be described in the\nglobal reference frame via\nr=G2666666664h\nk\nl3777777775: (25)\nThe HRPD backscatter detector comprises eight octets that\nare positioned radially about the incident neutron beam. Each\noctet is out of plane, accepting a di \u000braction angle of 160\u000e<\n2\u0012<176\u000e. Assuming the incident beam is in the zdirection, the\nradial positions have coordinates x;y. From the vector, r, this\ncorresponds to components rxandry. For any di \u000braction vector,\nQ, detected, the component in the beam direction is given by\nQz=q\nr2x+r2y\nsin 2\u0012: (26)\nThe di \u000braction vector, Q, is given by\nQ=2666666664rx\nry\nQz3777777775: (27)\nIf the vector ris within the range of vectors allowed for Q(for\nthe allowed 2 \u0012range), the crystal will di \u000bract. Practically, this\nenables each data point within an EBSD map to be analysed,\nto deduce whether this location has an orientation that would\npermit time-of-flight di \u000braction, for a given reflection ( hkl).\n4. Results\nThe experimental raw data are shown in Fig. 1a for the fer-\nritic steel and Fig. 1d for the alumina-30% SiC. Traces are shown\nin the reference state (black) and with deformation (red). The\nfirst and second order reflections used for line broadening anal-\nysis are also given, comprising the f211g&f422g(Fig. 1b &\nFig. 1c) for the steel and f102g&f204g(Fig. 1e & Fig. 1f) for\nthe alumina composite. The normalised intensities of these in-\ndividual reflections have been plotted with respect to the dimen-\nsionless parameter, h3.\nThe corrected Fourier coe \u000ecients, Anfor the sample were\ncalculated using Equation 19. This is shown in Fig. 3a & e\nfor the steel and ceramic samples, respectively. For each valueofn, a plot of ln An(l) versus l2yields ln AS\nnat the intercept\nand the gradient is \u00002\u00192hZ2\nni, as shown in Equation 15. These\nare shown in Fig. 3b & f. To deduce the crystallite size from\nthe line broadening, AS\nnis plotted against L, enabling the mean\ncolumn length, N3, to be obtained at the intercept of the ini-\ntial slope on the axis of abscissa. This is shown in Fig. 3c\n& g for the steel and composite samples, respectively. In both\ncases the AS\nnvs.Lplots are essentially horizontal, with all val-\nues close to 1. This shows that the crystallite sizes were so\nlarge that their e \u000bect did not contribute significantly to the peak\nbreadth. Meaningful crystallite sizes cannot therefore be ex-\ntracted. This was confirmed by reconstructing the deconvoluted\nsample peaks, using Equation 10. When plotted as a function\nofh3=l, the peaks from the first and second order peaks were\ncoincident.\nUsing the calculated \u00002\u00192hZ2\nniterms, the strain contribu-\ntion to broadening was calculated. The relationship between\nthis term and distance squared ( L2) is shown in Fig. 2d & (h)\nfor the steel and ceramic samples, respectively. The steel sam-\nple shows a distinct concave \u00002\u00192hZ2\nni-L2trend whereas the\nceramic has a near perfect proportional \u00002\u00192hZ2\nni-L2relation-\nship.\nUsing Equation 16, the rms (root mean squared) strain, h\"2\nLi1\n2,\nwas next calculated with respect to averaging distance, L. The\nresults are shown in Fig. 3. The magnitude of h\"2\nLi1\n2for the fer-\nritic steel is shown to decrease as a function of increasing av-\neraging distance indicating that the strains caused by the dislo-\ncations vary significantly over the range of lengths, Lsampled.\nThis will be used in Section 5.1 to estimate the dislocation den-\nsity.\nThe rms strain in the alumina-SiC composite remains ap-\nproximately constant at \u00181:3\u000210\u00003over the full range of av-\neraging distance, L. This is because the thermal residual strains\nvary on the scale of the microstructure i.e. several micrometres,\nwhich is much greater than the length scales of \u001810 nm acces-\nsible to the Warren-Averbach analysis. The large grain size is\nalso the reason that the particle size broadening is negligible for\nthis specimen.\n5. Discussion\n5.1. Analysis and validation of results: steel\nA measurement of rms strain with respect to increasing av-\neraging distance can be used to estimate the dislocation density\n[13, 14]; this has been performed for the steel sample exam-\nined in this study. A dislocation density estimate that assumes\nstraight, parallel, randomly distributed screw dislocations was\nproposed by Krivoglaz and Ryaboshapk [15]:\nh\"2\nLi\u001b\u001aCb2\n4\u0019ln\u0012Re\nL\u0013\n(28)\nwhere\u001ais the dislocation density, Cis a contrast factor, bis\nthe Burgers vector and Reis the dislocation cuto \u000bradius. The\nexpression assumes Re>L. The contrast factor is dependent on\nthe di \u000braction vector, dislocation line vector and Burgers vec-\ntor and may be calculated numerically [13]. For a cubic system,\n5(a) (b) (c)(d)\n(e) (f) (g) (h)00.20.40.60.81\n00.20.40.60.81-0.3-0.2-0.10\n0 1 400.20.40.60.81\n0 20 40 0 20 40\n0 50 100 15000.20.40.60.81\n-0.3-0.2-0.10-0.04-0.020\n-0.06\n-0.08\n-0.04-0.020\n-0.06\n-0.08\n0 1 4 0 20 401030 1 2 3\n0 1 2104Figure 2: Calculated Fourier coe \u000ecients for the deconvoluted f(h3) line profiles for the first order ( l=1) and second order ( l=2) reflections, for ferritic steel (a)\nand alumina-30% SiC (b). Particle size and strain e \u000bects can be separated with the logarithmic plots shown in (b) and (f) for multiple orders. Data used for particle\nsize calculation are shown in (c) & (g) and strain calculations can be calculated from the relationship between \u00002\u00192hZ2\nniandL2, (d) & (h).\n1.6\n1.5\n1.4\n1.3\n1.2\n1.1\n1.0\n0 10 20 30 4010-3\n1.7\n50\nFigure 3: Root mean squared strain for ferritic steel and alumina composite\nsamples.\nan average contrast factor, ¯Ccan be used in a simpler formula-\ntion [16]; this method accounts for the elastic anisotropy of the\nmaterial and the di \u000braction vector relative to the slip systems,\nassumed to bef110gh¯111islip alone for the ferritic steel. For\nsteel, the average contrast factor calculated for the f211gre-\nflection was 0.1040 assuming screw dislocations only. As the\nferritic steel studied has a BCC crystal structure, deformation\nis typically mediated by screw dislocations due to the availabil-\nity of multiple slips planes ( f110g,f112gorf123g) onto which\nthe characteristic non-planar cores dislocations cores can exist\nwhilst preserving the slip direction, h111i[17]. Therefore, as-\nsuming all dislocations are of screw type is deemed appropriate\nas an estimate. This may be extended to account for edge /screw\ndislocation ratio [18]. Further details of the Equation 28 ap-proximation including a comprehensive description of the rele-\nvant supporting literature is available elsewhere (e.g. [19]).\nIf one plotsh\"2\nLiversus ln Lthe gradient is\u0000\u001aCb2\n4\u0019andRecan\nbe calculated from the intercept. An example fit for the ferritic\nsteel is shown in Fig. 4. The measured dislocation density is\n1:78\u00021015\u00060:05\u00021015m\u00002andReis 27\u000618 nm (errors\ncalculated from 95% confidence of fit).\n-19.6 -19.4 -19.2 -19.010-6\n1.6\n1.5\n1.41.71.81.9\nFigure 4: Fitted ferritic steel data for dislocation density calculation.\nWilliamson and Smallman [20] presented an alternative dis-\nlocation density estimation with no Ldependency;\n\u001a=k\nF\u00182\nb2(29)\nwhere kis a crystal structure dependent constant; this is 14.4 for\nthe BCC crystal structure [20] and Fis an interaction factor of\n6order 1 for widely distributed dislocations. This value assumes\nan idealised organisation of dislocations; the metal comprises\na set of blocks with a single dislocation located at each block\nboundary.\u0018is the strain distribution integral breadth. Assum-\ning a Gaussian strain distribution this corresponds top\n2\u0019h\"2\nLi1\n2\nwhen n=1. This method gives a dislocation density estimate\nin the ferritic steel of 3 :3\u00021015m\u00002.\nA validation test is given here for the dislocation density\nmeasurement of the steel sample. The undeformed and de-\nformed states of the ferritic steel examined with time of flight\nneutrons were characterised with EBSD. Inverse pole figure\nplots with respect to the Z-direction (normal direction, ND) are\nshown in Fig. 5a & Fig. 5b. Grain boundaries of misorienta-\ntion>10\u000eare shown with continuous black lines. The corre-\nsponding GND density maps are shown in Fig. 5c & Fig. 5d.\nThe undeformed sample possesses GND densities in the 1012-\n1013m\u00002region, which is several orders of magnitude smaller\nthan the deformed specimen (1014- 1016m\u00002). The regions\nof these maps that have crystal orientations that would di \u000bract\nfrom thef211glattice planes using the method proposed earlier\nare shown in Fig. 5e & Fig. 5f. To aid a comparison of this data\nto dislocation density estimates made via line profile broaden-\ning, histograms of the GND densities (for the subset of grains\nthat satisfy the HRPD f211greflection di \u000braction condition),\nfor each of the deformation states, are shown in Fig. 5g. The\ndislocation density estimates obtained from time of flight line\nbroadening are superimposed. These values sits within the dis-\ntribution of the deformed specimen, and are both slightly higher\nthan the mean GND density of 1 :2\u00021015m\u00002. A higher value\nis expected from Warren-Averbach analysis because this will\nsample statistically stored as well as geometrically necessary\ndislocations. The neutron results are therefore good estimators\nfor the mean dislocation density of plastically deformed metals.\nData presented in this manner emphasise that a single value for\nthe dislocation density may be a good estimator of the mean\ndislocation density, but neglects the dislocation density range,\nor indeed dislocation density hotspots which may be of greater\ninterest in a structural metallic material.\nAlthough the dislocation density estimates from time of flight\ndi\u000braction show good agreement to the GND results, the GND\nanalysis from EBSD data shows that the ferritic steel can ex-\nhibit highly inhomogeneous deformation (Fig 6d), with cer-\ntain grains accumulating much higher dislocation densities than\nneighbours. This was evident from the the bimodal distribution\nof the estimated GND density obtained (shown in Fig. 5g). Both\nmodels relating rms strain to dislocation density assume a uni-\nform dislocation distribution and such results should therefore\nbe treated as approximate.\n5.2. Analysis and validation of results: alumina-SiC composite\nIn order to test the validity of the strain analysis for the\nalumina-SiC composite, we estimate an approximate relation-\nship between the mean strain in the alumina matrix of the com-\nposite, which can be measured straightforwardly from di \u000brac-\ntion peak shifts relative to the pure alumina reference specimen,\nand the rms strain measured using the Warren-Averbach anal-\nysis. Using the mean field approximation of Tanaka and Mori[21], the thermal residual stress in the vicinity of a particle in a\ncomposite consisting of a finite volume fraction fof particles\ndispersed in a matrix can be approximated by the stress which\nwould result from the presence of a single particle in an infinite\nmatrix, plus a background stress which arises due to the actual,\nfinite volume fraction of particles, and which acts as though ap-\nplied externally to maintain the force balance within the com-\nposite. Assuming the particles to be spherical, the stress \u001bp1\nwithin a single particle in an infinite matrix is hydrostatic. A\nsimplification can be introduced by noticing that the bulk mod-\nuli of alumina and SiC are the same to within a few % (248 GPa\nand 244 GPa using data in ref [22]). The background stress is\ntherefore uniform and hydrostatic with magnitude \u0000f\u001bp1. This\ngives principal stresses in the matrix at its interface with the par-\nticle [23]:\n\u001bmr(int)=\u001bp1(1\u0000f) (30)\n\u001bm\u0012(int)=\u001bm\u001e(int)=\u001bp1 \n\u00001\n2\u0000f!\n(31)\nwhere r,\u0012,\u001eare spherical polar coordinates. The di \u000berence\nbetween the corresponding principal strains gives an estimate of\nthe maximum range of normal strains sampled during neutron\ndi\u000braction:\n\u0001\"max=\u00003\u001bp1(1\u0000\u0017m)\n2Em(32)\nwhere Emand\u0017mare the Young’s modulus and Poisson’s ratio\nof the matrix. The mean strain in the matrix is:\nh¯\"mi=\u0000f\u001bp1(1\u00002\u0017m)\nEm(33)\nand combining Equations 32 and 33 gives an estimate of the\nmaximum half width of the strain distribution around a central\nposition:\n\u0006\u0001\"max\n2=3h¯\"mi(1+\u0017m)\n4f(1\u00002\u0017m): (34)\nh¯\"miwas measured from the di \u000braction peak shift between the\ncomposite specimen and the alumina reference, and was equal\nto 3:4\u00060:5\u000210\u00004. With f=0:3 and\u0017m=0:23, equation\n5 gives \u0001\"max=2=1:9\u000210\u00003. This maximum half width is\nexpected to be of the same order, but a little larger than the\nrms strainh\"2\nLi1\n2. The value obtained is about 50% larger than\nthe rms strain,h\"2\nLi1\n2of 1:3\u000210\u00003measured using the Warren-\nAverbach analysis, giving strong support to the validity of the\nmethod.\n5.3. Experimental considerations\nThe Warren theory applies in principle to any set of planes.\nHowever, not all reflections were found to be suitable for anal-\nysis in this study. For the ferritic steel, selecting the f110g\nreflection was unsuitable as the background was highly non-\nuniform. Accounting for this with confidence was deemed to\nbe too di \u000ecult. Evidence of thermal di \u000buse scattering (TDS) in\nthe vicinity of low index reflections was apparent. In particu-\nlar these scattering artefacts were seen to flank the shoulders of\nthef200greflection in the ferritic steel, labelled ( ?) in Fig.1a.\n7Figure 5: EBSD IPF-Z plots for ferritic steel in the (a) annealed and (b) cold-rolled states. The corresponding GND density maps are shown in (c) & (d). In (e) &\n(f), the GND density maps are plotted only for regions that satisfy the HRPD geometry for di \u000braction from thef211glattice planes. GND density histograms of the\nannealed and cold-rolled steels are also shown (g) together with peak-broadening estimates of dislocation density.\nWhilst the intensity of such scattering increases with smaller d-\nspacing [24], for a polycrystalline sample this incoherent scat-\ntering becomes less structured, becoming more like a uniform\nbackground, as the volume of reciprocal space over which in-\ntegration occurs increases. For the f211g&f422greflections\nused, there was no clear evidence of TDS in the vicinity of the\nline profiles. Considering higher index reflections is valid, but\ntheir use becomes increasingly di \u000ecult as (1) the line profiles\ndo not decay to the background level before encroaching on a\nneighbouring reflection in close proximity, and (2) noisy (low\nsignal to noise ratio) line profiles provide unreliable values of\nthe Fourier coe \u000ecients, An, making subsequent analysis less ac-\ncurate. The methodology adopted in this study, does, however\ninclude strategies to judiciously select appropriate reflections to\nobtain valid results.\nFor thef211gtype reflections chosen in this study the Schwartz\nand Cohen [9] recommendation of \u00184 FWHM for the tails\ncould be achieved without encroaching on the neighbouring\npeaks and the above validation indicates that meaningful results\nwere obtained. However, the h\"2\nLi1\n2estimates for low nshowed\nsensitivity to variations in the hrange used in the analysis, indi-\ncating the validity can only just be achieved within the conflict-\ning requirements described above. For a range of more than\nFWHMs it was evident that tails of neighbouring peaks were\nbeing included in the analysis. Variations in the low nstrain val-\nues were also found for tail widths of less than 4 FWHM, this\nindicates that the tails were being truncated. This indicates that\nlowLvalues of rms strain are less reliable and the dislocation\nanalysis, using Equation 28, cannot be reliably obtained if the\ncorresponding rms values are included. Rothman and Cohen\n[25] found that strains around dislocations must yield a linear\ndecrease in mean-square strains, h\"2\nLi, with averaging distance,L. In practice, this was not observed for the steel sample for\nlowLvalues, owing to the longs tails, and thus the dislocation\ndensity approximation from Equation 28, with results shown in\nFig. 5, used values only where a linear h\"2\nLi-Lrelationship\nexists.\n6. Conclusions\nThe line profile broadening measured from high resolution\ntime-of-flight neutron di \u000braction data has been quantified from\ndeformed samples of a single phase ferritic steel and an alumina-\n30% SiC ceramic composite. The following conclusions can be\ndrawn:\n1. The Warren-Averbach theory for analysing peak broad-\nening from X-ray di \u000braction data has been shown to be\nsuitable for time of flight data.\n2. The data analysis method was capable of describing the\nroot mean square strain, h\"2\nLi1\n2, of two materials deform-\ning in di \u000berent ways. The plastically deformed steel ex-\nhibits a rapid decay of strain with averaging distance,\ncharacteristic of plastic deformation. A composite ce-\nramic exhibited an Al 2O3phase subjected to thermal resid-\nual strains varying on the scale of the microstructure,\ngave a near uniform h\"2\nLi1\n2with distance over the mea-\nsured range.\n3. Obtaining an appropriate estimate of the dislocation den-\nsity and rms strain requires a best practice methodology\nof selecting an appropriate h3interval that is both su \u000e-\nciently wide to capture the line profile shape, but not too\nwide that neighbouring reflections or other background\nartefacts are included. The analysis should also consider\n8the sensitivity of low Lrms strain values due to the tails\nof the broadened line profile.\n4. A method to compare the dislocation density estimated\nfrom the Warren-Averbach method and lattice curvatures\nmeasured from EBSD characterisation has been devised\nthat accounts for grains in each case that are similarly\norientated. Good agreement was seen between the dis-\nlocation density estimated between the two methods for\ndeformed ferritic steel.\n5. A numerical estimate of the uniform elastic strain in an\nAl2O3- SiC composite provides good agreement with\nthe rms strain measured by the time of flight Warren-\nAverbach analysis method.\n7. Data Statement\nAn open-source Matlab version of the time of flight Warren-\nAverbach analysis code is freely available for download via\nGitHub (https: //github.com /d-m-collins /ToF-WarrenAverbach).\nThe ferritic steel raw data presented in this paper is made avail-\nable as an example.\n8. Acknowledgments\nFinancial support of this work was provided by the EP-\nSRC (grant EP /I021043 /1) and steel provided by BMW-MINI.\nBeamline time at ISIS (RB1220050) and experimental support\nfrom Dr S. Kabra & Dr A. Daoud-Aladine is gratefully ac-\nknowledged. With thanks also to Prof. Andrew Goodwin for\nhis guidance in TDS observations.\nReferences\nReferences\n[1] Warren, B.E., Averbach, B.L.. J Appl Phys 1950;21:595–599.\n[2] Warren, B.E., Averbach, B.L.. J Appl Phys 1952;23:497.\n[3] Warren, B.. Progress in Metal Physics 1959;8:147–202.\n[4] Scherrer, P.. Nachr G ¨ott 1918;2:98–100.\n[5] Stokes, A., Wilson, A.. Proc Phys Soc London 1944;56:174.\n[6] Stokes, A., Wilson, A.. Proc Camb Phil Soc 1944;40:197–198.\n[7] Warren, B.. Acta Cryst 1955;8:483–486.\n[8] Todd, R., Derby, B.. Acta Mater 2004;53:1621–1629.\n[9] Schwartz, L., Cohen, J.. Di \u000braction from Materials. Academic Press;\n1977.\n[10] Stokes, A.. Proc Phys Soc London 1948;61:382.\n[11] Wilkinson, A., Randman, D.. Philos Mag 2010;90:1159–1177.\n[12] Nye, J.. Acta Metall 1953;1:153–162.\n[13] Wilkens, M.. Physica Status Solidi A 1970;2:359–370.\n[14] Groma, I., Ung ´ar, T., Wilkens, M.. J Appl Cryst 1988;21:47–54.\n[15] Krivoglaz, M., Ryaboshapka, K.. Fiz Met Metalloved 1963;15:18–31.\n[16] Ung ´ar, T., Dragomir, I., R ´ev´esz, ´A., Borb ´ely, A.. J Appl Crystall\n1999;32:992–1002.\n[17] Hull, D., Bacon, D.. Introduction to Dislocations. Butterworth-\nHeinemann; 2001.\n[18] Ung ´ar, T., Borb ´ely, A.. Appl Phys Let 1996;69:3173–3175.\n[19] Ung ´ar, T.. Mater Sci Eng A 2001;309-310:14–22.\n[20] Williamson, G.K., Smallman, R.E.. Philos Mag 1956;1:34–46.\n[21] Tanaka, K., Mori, T.. Acta Metall 1970;18:931 – 941.\n[22] Todd, R., Bourke, M., Borsa, C., Brook, R.. Acta Mater 1997;45(1791-\n1800).[23] Timoshenko, S., Goodier, J.. Theory of Elasticity. McGraw-Hill; 3rd\ned.; 1982.\n[24] Hutchings, M., Withers, P., Holden, T., Lorentzen, T., editors. Intro-\nduction to the Characterization of Residual Stress by Neutron Di \u000braction.\nTaylor and Francis; 2005.\n[25] Rothman, R., Cohen, J.. J Appl Phys 1971;42(971-979).\n9"    },    {        "title": "1811.05762v1.Enhancing_nanoparticle_diffusion_on_a_unidirectional_domain_wall_magnetic_ratchet.pdf",        "content": "Enhancing nanoparticle diffusion on a\nunidirectional domain wall magnetic ratchet\nRalph L. Stoop,yArthur V. Straube,z,yand Pietro Tierno\u0003,y,{,x\nyDepartament de Física de la Matèria Condensada, Universitat de Barcelona, Av.\nDiagonal 647, 08028 Barcelona, Spain\nzFreie Universität Berlin, Department of Mathematics and Computer Science, Germany\n{Institut de Nanociència i Nanotecnologia, Universitat de Barcelona, Barcelona, Spain\nxUniversitat de Barcelona Institute of Complex Systems (UBICS), Barcelona, Spain\nE-mail: ptierno@ub.edu\nKeywords: Diffusion, Magnetic fields, Domain walls, Microfluidics, Ratchet effect.\nAbstract\nThe performance of nanoscale magnetic devices is often limited by the presence\nof thermal fluctuations, while in micro-nanofluidic applications the same fluctuations\nmay be used to spread reactants or drugs. Here we demonstrate the controlled motion\nand the enhancement of diffusion of magnetic nanoparticles that are manipulated and\ndriven across a series of Bloch walls within an epitaxially grown ferrite garnet film.\nWe use a rotating magnetic field to generate a traveling wave potential that unidirec-\ntionally transports the nanoparticles at a frequency tunable speed. Strikingly, we find\nan enhancement of diffusion along the propulsion direction and a frequency dependent\ndiffusion coefficient that can be precisely controlled by varying the system parameters.\nTo explain the reported phenomena, we develop a theoretical approach that shows a\nfair agreement with the experimental data enabling an exact analytical expression for\n1arXiv:1811.05762v1  [cond-mat.soft]  14 Nov 2018the enhanced diffusivity above the magnetically modulated periodic landscape. Our\ntechnique to control thermal fluctuations of driven magnetic nanoparticles represents a\nversatile and powerful way to programmably transport magnetic colloidal matter in a\nfluid, opening the doors to different fluidic applications based on exploiting magnetic\ndomain wall ratchets.\nThe ratchet effect emerged in the past as a powerful way to transport matter at the micro\nand nanoscale, taking advantage of Brownian motion.1,2The success of such concept comes\nfrom the possibility of using thermal fluctuations to obtain useful work out of a thermody-\nnamic system, although such fluctuations produce noise, heat and randomize the motion of\nnanoparticles which limits the efficiency of any device operating at such scale.3–5Reducing\nor controlling thermal fluctuations in nanoparticle systems may present different technolog-\nical advantages apart from providing important fundamental insight into the dynamics and\ninteractions of matter at such scale. In the first case, diffusion can be used as a mean for\nmixing streams of fluids or for spreading reactants, drugs and biological species6,7in micro-\nand nano-fluidic applications.8–10On a more fundamental level, the search for strategies\nthat enable controlling diffusion and noise has fascinated scientists for long time, since the\npioneering seminar of Richard Feynman on Brownian ratchet.11\nIn a typical ratchet system the random fluctuations of nanoparticles can be rectified\ninto a directed motion by an external potential. In overdamped systems, different ratchet\nmechanisms can be sorted in two general classes, depending on the nature of the external\npotential.2One class refers to the tilting ratchets that are typically specified by a station-\nary, spatially asymmetric potential landscape often accompanied by an external force. The\nasymmetry of the landscape is able to rectify thermal fluctuations into net motion, and the\nforce, which determines the tilt of the total potential, can be used to further enhance the\nparticle current. Such a ratchet scheme can be realized by using, for example, the walls or\nbarriers in a microfluidic device.12Another broad class are the pulsating ratchets that are\ncharacterized by a time-dependent landscape, in which the net particle flux arises from the\n2periodic or random evolution of the landscape. Their particular subclasses are the flashing\nratchet with the landscape switching between two states, and the traveling wave ratchet,\nwhere the landscape translates at a given speed. The latter allows dragging the particles\nthat are trapped in the energy minima of the moving landscape.\nThe generation of translating potential landscapes usually requires an external field,\nwhich makes it appealing due to its programmable nature, that enables to remotely control,\ndirect and reverse the flux of particles at will.13Recent realizations in this direction include\nthe use of electric,14–18optical19–22or magnetic23–25fields to transport microscopic particles.\nHowever, most of the proposed approaches remain difficult to apply at the nanoscale, due to\nthelargefieldgradientsrequiredtoovercomethermalforces. Plasmoniclandscapeshavebeen\nproposed as a successful strategy,21,26,27however the created patterns are often composed of\nfixed lithographic gold structures that cannot be easily changed by an external control. An\nalternative solution is the use of patterned magnetic substrates that present high contrast in\nthe magnetic susceptibility on the particle scale and thus generate strong and localized field\ngradients.28–31While the trapping of magnetic nanoscale matter was recently demonstrated\nwith lithographic32and epithaxial33films, the complete control of particle transport, speed\nand diffusion within the same functional platform still remains a challenging issue.\nIn this article we demonstrate all these features by using a periodic patterned substrate,\ni.e. a uniaxial ferrite garnet film (FGF), composed of a series of mobile magnetic Bloch\nwalls. Controlling the dynamics of magnetic domain walls in thin films has recently led to\nnovelapplicationsindisparatefields, includingspintronics,34logicdevices,35,36nanowires37,38\nand ultracold atoms.39Here we use such magnetic walls to achieve the controlled transport\nand, in particular, the diffusion enhancement of magnetic nanoparticles. The application of\nan external rotating magnetic field induces the unidirectional translation of the otherwise\nstationary magnetic landscape generated by the FGF. Due to the magnetic interaction with\nthe nanoparticles, the translation of the landscape is converted into directed particle motion.\nThis technique allows to rapidly traps and steers magnetic nanoparticles deposited on top of\n3it thus, with in situ real time control. Moreover, our magnetic ratchet allows us to precisely\ncontrolalsothediffusivityofthenanoparticles, toenhance, tosuppressit, bysimplychanging\nthe driving frequency of the applied field.\nOur magnetic ratchet scheme is shown in Figure 1(a) and it is based on a series of\nBloch walls (BWs), narrow ( \u001820 nmwide) transition regions where the film magnetization\nchanges orientation by performing a 180 degrees rotation in the particle plane (x;y). Such\nBWs emerge spontaneously in a single crystal FGF film after an epitaxial growth process\nand they generate strong local stray fields on the surface. In this study we use an FGF of\ncomposition Y 2:5Bi0:5Fe5\u0000qGaqO12(q= 0:5\u00001), that was grown by dipping liquid phase\nepitaxy on ah111ioriented single crystal gadolinium gallium garnet (Gd 3Ga 5O12) substrate.\nIn absence of external field, the BWs in the FGF are equally spaced at a distance \u0015=2being\n\u0015= 2:6µmthe spatial periodicity, and separates domains of opposite magnetization, with\nsaturation magnetization Ms= 1:3\u0002104A m\u00001.\nAbove this film we demonstrate the controlled transport of three three types of param-\nagnetic polystyrene nanoparticles with diameters d= 540 nm ,360 nmand270 nm, charac-\nterized by a\u001840% wt:iron oxide content (Microparticles GmbH). When placed above the\nFGF, the nanoparticles show simple diffusive dynamics and are able to easily pass the BWs\ndue to the presence of a 1µmthick polymer coating that prevents sticking to the FGF sur-\nface, see the Materials and Methods section. We apply an external magnetic field rotating in\nthe(x;z)plane with frequency fand amplitudes (Hx;Hz) =H0(p1 +\f;p1\u0000\f), such that\nHac(t) = (Hxcos (2\u0019ft);0;\u0000Hzsin (2\u0019ft)), Figure 1(a). The polarization of the applied\nfield is slightly elliptical, Hx6=Hz, as characterized by the amplitude H0=p\n(H2\nx+H2\nz)=2\nand an ellipticity parameter \f= (H2\nx\u0000H2\nz)=(H2\nx+H2\nz). The value of \f=\u00001=3is chosen to\nminimize dipolar interactions between nearest particles,40as they can promote the undesired\nchaining at high density.41Although for self-consistency we account for this small ellipticity\nin a full numerical model, a more tractable semi-analytical model for the special case of\n\f= 0is shown to work quantitatively well.\n4The applied field modulates the stray magnetic field generated by the FGF, leading to a\nspatially periodic magnetic energy landscape Um(x;t) =\u0000U0cos(k(x\u0000v0t))that translates\nat a constant speed, v0=\u0015f, Figure 1(a), see Materials and Methods section for further\ndetails and definition of U0andk. Its evolution at the elevation corresponding to the 270 nm\nparticles is shown in Figure 1(b). One may intuitively expect that the particles would follow\nthe energy minima whose locations are given by the blue regions, and travel the distance of\none wavelength during one time period. However, the nature of motion and the velocity of\nmean drift across the BWs,\nhvi= lim\nt!1hX(t)i\nt; X (t) =x(t)\u0000x(0); (1)\ndepend on the external frequency, as becomes evident for the simple case of no thermal\nfluctuations, see Equation 4. The variation of hviwith the frequency fis also shown in Fig-\nure 1(c), where the experimental data (open symbols) are plotted along with the theoretical\npredictions (dashed and solid lines) for a 270 nmparticle.\nAsitfollowsfromourmodel(seeEquation4inMaterialsandMethods), andasconfirmed\nby the experimental data, there exist two different dynamic states separated by the critical\nfrequencyfc. At low frequencies, here f < fc\u001913 Hz, the nanoparticles indeed translate\nconsistently with the landscape with the maximum possible speed, _x(t) =hvi=v0=\u0015f; see\nthe range of perfectly linear increase of hviwithfin Figure 1(c). We also note that at any\n0< f < f cin this “locked” regime, the particle lags behind the minimum of the potential\nenergylandscape Um, whoserelativepositionisdeterminedbyminimizingthepotentialgiven\nby Equation 5. For f > fc, the particle is unable to move together with the landscape. It\nstarts to slip and cannot remain localized within the same minimum. In this “sliding” regime,\nevery time period the particle covers a distance smaller than \u0015, resulting in a reduction of the\nmean speed. At high frequencies f, the speed of mean drift decays as hvi'\u0015f2\nc=(2f). Note\nthat at low and high frequencies, thus far away from fc, the predictions of the model with\n5thermal fluctuations and the experimental data are highly consistent with the deterministic\n(zero temperature) results given by Equation (4). However, the deterministic model overshot\nthe experimental data close to fc. Therefore, we conclude that thermal fluctuations start\nto significantly affect the particle speed close to the critical frequency, blurring the sharp\ntransition from the locked to the sliding dynamics and effectively decreasing the value of fc\nas compared to the relative to the deterministic model.\nWhile our domain wall magnetic ratchet enables full control over the mean speed of\nparticle moving across BWs, the instant position of the particles is affected by unavoidable\nthermal fluctuations, as evidenced by the particle trajectory in Figure 1(d). The role of such\nfluctuations is typically quantified by measuring the mean squared displacement (MSD),\nwhich can be calculated from the positions of the nanoparticles. Since we are interested in\nthegeneraleffectofhowthediffusivepropertiesofananoparticleareinfluencedbytheratchet\nmechanism, we focus on investigating these quantities along the propulsion direction, namely\nthexaxis. Because of the non vanishing mean drift, we define the MSD as the variance\nof the corresponding particle position, \u001b2\nx(t) =h\u000ex2(t)i\u0018t\u000b, where\u000ex(t) =X(t)\u0000hX(t)i.\nFor our statistical analysis we averaged over more than \u001850independent experimentally\nmeasured trajectories, for each of which we subtracted its mean drift, hX(t)i=hvit. The\nexponent of the power law \u000bcan be used to distinguish the diffusive \u000b= 1from anomalous\n(sub-diffusive \u000b < 1or super-diffusive \u000b > 1) dynamics. Our case corresponds to normal\ndiffusion, with an effective diffusion coefficient across the BWs evaluated as\nDeff= lim\nt!1\u001b2\nx(t)\n2t: (2)\nIn Figure 2 we show that effectively, the magnetic ratchet provides a strong enhancement\nof the diffusion coefficient at an optimal frequency. At low frequencies, the diffusive motion\nof the nanoparticles with d= 270 nm corresponds to a Deffmuch lower than the diffusivity\nin absence of the external field, D0= 1:0µm2s\u00001. At high frequencies, the diffusivity\n6becomes close to D0. At intermediate frequencies, the effective diffusivity increases (note\nthatDeff\u00190:6D0forf= 11:5 Hzandf= 70 Hz) with an enhancement of almost one order\nof magnitude relative to free diffusivity, Deff\u00198:5D0, at a frequency of f= 15:5 Hz. In\nparticular, as shown in the inset of Figure 2, after subtracting the mean drift we calculate\nthe effective diffusion coefficient using Eq. 2 and performing a linear fit only for the region\nof data where the mean square displacement displays a power law behavior with \u000b= 1. This\nregime is always found in the long-time limit. Overall, the dependence of Deffonffollows a\nwell defined trend, characterized by an initial sharp raise above a value of f\u00198 Hz, and an\nexponential-like decay above a peak at 15:5 Hz, which is close to the critical frequency value\nmeasured for this size of nanoparticle, fc= 13:4 Hz.\nThe observed enhancement of diffusion can be explained by formulating a reduced theo-\nretical model that explicitly takes into account the interaction of the nanoparticle with the\nmagnetic landscape generated by the FGF surface and the thermal noise. Details of the\nderivation are given in the Materials and Methods section. In the reference frame moving\nwith the magnetic potential, the behavior of the nanoparticle is equivalent to the motion\nin an effective tilted periodic potential V(u)as given by Equation 5. For such a potential,\nthe effective diffusion coefficient, defined via Equation 2, can be expressed as Equation 7,\nwhich admits an accurate numeric evaluation. We used Equation 7 in Figures 3(a)-3(c) to\nfit the experimental data for the three types of nanoparticles, and find that our theoretical\napproach captures very well the observed diffusion enhancement for all cases.\nThe explicit form of the effective potential, Equation 5, with the tilt /fand the am-\nplitude of the landscape /fcadmits an intuitively clear interpretation of the observed\nfrequency-dependent diffusive regimes, as also illustrated in Figures 3(d)-3(f). Indeed, at\nsubcritical frequencies, f\u0014fc, the potential barrier the particle needs to overcome to es-\ncape from a minimum drops with the frequency as \u0001V=kBT\u0019\u0001\u001dm(1\u0000f=fc)3=2, where\n\u0001\u001dm=\u00152fc=(\u0019D 0)is double the amplitude of the landscape relative to thermal energy, cf.\nEquation 5. At low frequencies ( f\u001cfc), the tilt is negligible and the particle is strongly\n7trapped in a minimum of the magnetic landscape, as shown in Figure 3(d). Since in this\ncase the potential barrier is maximum, \u0001V=kBT\u0019\u0001\u001dm, and is too high for the particle to\nescape, the effective diffusivity is nearly vanishing, Deff\u001cD0. At intermediate frequencies,\nasftends tofcbut remains smaller than fc, the potential barrier decreases with fand dis-\nappears atf=fc. In this frequency range, the barriers become progressively more accessible\nfor the Brownian particle, explaining the observed significant enhancement of the effective\ndiffusivity at frequencies f\u0019fc, see Figure 3(e). Beyond f=fc, there exits no minima in\nthe potential, and at high frequencies f\u001dfc, the particle does not feel the landscape, as\nshown in Figure 3(f). As a result, the diffusive motion occurs effectively at a constant force\n/fand therefore corresponds to free diffusion, when ideally Deff=D0.\nFurther, since the reduced model was developed for the zero field ellipticity, \f= 0, we\nhave performed control Brownian dynamic simulations of the full system with \f=\u00001=3to\nconfirm the negligible effect of \fon the basic physics. The simulation (blue lines) agree very\nwell with the model and the experimental data. The only difference is that the system with\n\f=\u00001=3is characterized by slightly higher values of the critical frequency compared to the\ncase\f= 0, which is in agreement with our earlier observation that the value of fcat any\n\f6= 0is generally smaller than that for \f= 0at otherwise identical conditions.40Taken\ntogether, our findings show that each type of nanoparticle investigated exhibit an enhanced\ndiffusive behavior, with the highest diffusion coefficient measured for the intermediate size,\nd= 360 nm . This outcome can be understood by considering the balance between two\nopposite effects. First, reducing the nanoparticle diameter dincreases thermal fluctuations\nand its free diffusion coefficient D0=kBT=\u0010, where\u0010is the friction coefficient of a spherical\nparticle immersed in water. Note that if in the bulk \u0010= 3\u0019\u0011dwith the dynamic viscosity\n\u0011= 10\u00003Pa s, the presence of the FGF surface leads to effectively larger values of \u0010. Second,\nfor our system, smaller particles come closer to the FGF surface, and thus they are strongly\nattracted by its stray field, which results in a reduction of their thermal fluctuations. For\nexample, for a 270 nmparticles an effective enhancement of the friction coefficient due to\n8the presence of the FGF surface is estimated to be 25\u000030%, as follows from the reduction\nof the diffusion coefficient (cf. the inset of Figure 2) relative to the bulk diffusivity.\nWhile we have analyzed the transport of single particles, our magnetic ratchet enables\nalso the collective motion for an ensemble of nanoparticles. We demonstrate this feature\nin Figure 4(a), where a dense suspension of 270 nmparticles are trapped and transported\nacrosstheFGFsurface,seealsocorrespondingVideoS2intheSupportingInformation. Inthe\nabsence of applied field (ratchet off) the colloidal suspension shows simple diffusive dynamics\nas illustrated by the Gaussian distribution of the displacements perpendicular (along the x\naxis) and parallel (along the yaxis) to the magnetic stripes, Figures 4(b,c). Here we used\nP(\u000ex) = (2\u0019\u001b2\nx)\u00001exp(\u0000\u000ex2=2\u001b2\nx)with the variance \u001b2\nx(t) =h\u000ex2(t)i,\u000ex(t) =X(t)\u0000hX(t)i\nandX(t) =x(t)\u0000x(0)defined as earlier and considered at a sufficiently large time. The\ndistribution P(\u000ey)is defined in a similar way, for which hYi\u00110, implying no mean drift and\nfree diffusion in the ydirection. We note that when the ratchet is off, H0= 0, the magnetic\nlandscapeUm/H0becomes effectively flat and does not affect the dynamics of the particle.\nIn this situation, the mean drift disappears, hvi= 0, and thereforehXi=hvit= 0. As a\nresult, when the ratchet is off, both distributions (greens lines in Figures 4(b,c)) are identical,\nwith\u001b2\nx=\u001b2\nyand thus featuring an isotropic free diffusion dynamics.\nUpon application of the rotating field, H0>0, the particles are immediately localized\nalong the BWs, forming parallel chains and being consecutively transported across the mag-\nnetic platform. The ratchet transport features a similar dispersion along the perpendicular\ndirection as in the free case, as shown by the blue line in Figure 4(b). Along the transport\ndirections we observe a stronger confinement with a narrower distribution of displacement\nP(\u000ex), where the mean drift is subtracted by putting hX(t)i=hvit. VideoS2 in the sup-\nporting information shows the versatility of our magnetic ratchet approach, as now the\nnanoparticles can be easily trapped or released and transported to the right or left by just\ninverting the chirality of the rotating field, Hx!\u0000Hx, which keeps H0and\funchanged.\nWhile these features have been previously reported for micro-scale systems,23,24,29,30our\n9experimental realization proves its potential for further miniaturization of the transported\nelements and opens the doors towards applications in magnetic drug delivery systems using\nthe higher surface to volume ratio of nanoparticles.\nTo conclude, we have reported the controlled transport and diffusion enhancement of\nnanoparticles in a magnetic ratchet generated at the surface of a ferrite garnet film. In con-\ntrast to previous experimental works on microscopic particles confined in a rotating optical\nring,42,43above patterned plasmonic21or lithographic44landscapes, our nanoparticles are\ntrapped and controlled on an extended surface in absence of any topographic relief that can\nperturb the transport via steric interactions. The advantage of using the garnet film as a\nsource of magnetic background potential is that the domain wall motion is essentially free\nfrom intermittent behavior and hysteresis. This makes the cyclic displacements and the de-\nvice performance fully reversible. We have also reported in the past the giant diffusion in a\nmagnetic ratchet of microscopic colloids.45However, the significant advantage of the present\nimplementation is that the particle fluctuations can also be controlled along the direction\nof motion and remain completely independent of the deformation of the BWs in the lateral\ndirections. This crucial difference has important implications for the design of channel-free\nnanofluidic devices where the colloidal motion can be confined to a straight line. Further,\nour work goes beyond a previous one,33centered on trapping fluctuating nanoparticles along\nthe BWs in an FGF. Here, we not only demonstrate the possibility to transport nanoscale\nobjects at a well defined speed, but also that our magnetic ratchet system can be used to\ntune the diffusive properties of the particles, increasing their effective diffusion constant by\nalmost an order of magnitude with respect to the previous case. Another future avenue of\nthis work is to investigate the dynamics of dense suspensions of interacting nanoparticles,\nand how collective effects alters the average particle flow. Increasing their density, however,\nrequires a visualization procedure different from fluorescent labeling, to avoid artifacts dur-\ning the tracking mechanism. While with relatively larger particles, the average speed of a\ncolloidal monolayer was found to decrease with respect to the individual case,46the presence\n10of stronger thermal fluctuations for smaller particles may cause different non-trivial effects\non the overall system dynamics. Finally, our controlled enhanced diffusion in a transported\nratchet may be used as a pilot system for fundamental studies related to transport, diffusion\nand their complex relationship at the nanoscale.\nMaterials and Methods\nExperimental System and Methods\nWe give further details on the sample preparation and experimental setup. In order to\ndecrease the strong magnetic attraction, we coat the FGF film with a h= 1µmthick layer\nof a photoresist (AZ-1512 Microchem, Newton, MA), i.e. a light curable polymer matrix,\nusing spin coating at 3000 rpm for30 s(Spinner Ws-650Sz, Laurell). Before the experiments,\neach type of particle is diluted in highly deionized water and deposited above the FGF, where\nthey sediment due to the magnetic attraction to the BWs.\nExternal magnetic fields were applied via custom-made Helmholtz coils connected to\ntwo independent power amplifiers (AMP-1800, Akiyama), which are controlled by a wave\ngenerator (TGA1244, TTi). Particle positions and dynamics are recorded using an upright\noptical microscope (Eclipse Ni, Nikon) equipped with a 100 \u00021.3 NA oil immersion objective\nand a CCD camera (Basler Scout scA640-74fc, Basler) working at 75frames per second. The\nresulting field of view is 65\u000248µm2.\nVideomicroscopyandparticletrackingroutines47areusedtoextracttheparticlepositions\nfxi(t);yi(t)g, withi= 1;:::;N, from which the mean speed hviis obtained performing\nboth time and ensemble averages. To calculate the mean square displacement and diffusion\ncoefficient, we use Ntrajectories with length lthreshold = 200frames, corresponding to a\nmeasurement time of \u0001t= 200=75 = 2:6s.\n11Theoretical model\nThe motion of paramagnetic colloidal particles above the FGF can be well interpreted within\na simple model. In an external magnetic field H, a spherical magnetically polarizable particle\nof volume Vbehaves as an induced magnetic dipole with the moment m=V\u001fHand\nthe energy of interaction Um=\u0000\u00160m\u0001H=2,48where\u001fis the effective susceptibility and\n\u00160= 4\u0019\u000210\u00007H m\u00001is the magnetic permeability of free space. The total magnetic field\nabove the FGF His given by the superposition Hac+Hsubof the external modulation\nof elliptic polarization, Hac=H0(p1 +\fcos!t;0;\u0000p1\u0000\fsin!t), and the stray field of\nthe garnet film Hsub\u0019(4Ms=\u0019) exp(\u0000kz)(coskx;0;\u0000sinkx),49whereMsis the saturation\nmagnetization, != 2\u0019fis the angular frequency, k= 2\u0019=\u0015is the wave number, H0is the\namplitude and \fis the ellipticity of the modulation.\nBeing interested in the transport properties across the magnetic stripes and evaluating\nthe magnetic force exerted on the particle, F(x;t) =\u0000@xUm(x;t)withUm=\u0000\u00160V\u001f(Hac+\nHsub)2=2, in the overdamped limit we obtain an equation of motion,\n_x(t) =\u0010\u00001F(x;t) +\u0018(t); (3)\nwithF(x;t) =\u0000\u0010\u0015fc(p1 +\fcos!tcoskx\u0000p1\u0000\fsin!tsinkx)and a random variable \u0018(t)\ntaking account of thermal fluctuations. Here, fc= 8\u00160\u001fVMsH0exp(\u0000kz)=(\u0010\u00152), and\u0018(t)\nobeys the properties h\u0018(t)i= 0,h\u0018(t)\u0018(t0)i= 2D0\u000e(t\u0000t0), whereD0is the free diffusion\ncoefficient, kBTis the thermal energy and \u0010is the friction coefficient. By using fcandD0as\nfitting parameters, we numerically integrate Equation 3 and evaluate the velocity of mean\ndrifthviand effective diffusion Deff, which capture well the experimental data.\nTo gain further insights, we approximate the general model for an arbitrary \f, Equation\n3, by a more tractable special case \f= 0, which corresponds to a traveling wave potential,\nUm(x;t) =Um(x\u0000v0t)/cos(k(x\u0000v0t)). In the reference frame co-moving with the speed\nv0=\u0015f, in terms of a new variable u(t) =\u0000x(t)+v0twe obtain, _u(t) =\u0015f\u0000\u0015fcsinku+\u0018(t).\n12The deterministic velocity of mean drift is known to be:49\nhviT=0=8\n><\n>:\u0015f; f \u0014fc;\n\u0015f\u0000\u0015p\nf2\u0000f2\nc; f >f c:(4)\nHere,fcplays the role of the critical frequency that separates the low frequency domain\nwith the maximum possible speed of mean drift, hvi=v0/f(f < f c), from the high\nfrequencydomainwhereitsefficiencyprogressivelydropsdown, hvi/f\u00001<v 0(f >fc). For\nnanoparticles, thermal fluctuations are, however, inevitable, and we consider their thermal\nmotion in the associated potential,\nV(u)\nkBT=\u0000\u0015f\nD0u\u0000\u0015fc\nD0kcosku; (5)\nwhich admits evaluation of the velocity of mean drift and effective diffusion50,51\nhvi=\u0015f\u0000D0\n\u00151\u0000exp(\u0000\u00152f=D 0)\nhI\u0006(u)iu; (6)\nDeff=D0hI2\n\u0006(u)I\u0007(u)iu\nhI\u0006(u)i3\nu: (7)\nHere,I\u0006(u) =hexp[\u0006(V(u)\u0000V(u\u0007u0))=kBT]iu0are evaluated for potential (Equation 5)\nandh\u0001iu=\u0015\u00001R\u0015\n0\u0001dudenotes the average over one wavelength of the landscape.\nAcknowledgment\nR. L. S. acknowledges support from the Swiss National Science Foundation grant No.\n172065. A. V. S. acknowledges partial support by Deutsche Forschungsgemeinschaft (DFG)\nthrough grant SFB 1114, Project C01. P. T. acknowledges support from the ERC starting\ngrant \"DynaMO\" (335040) and from MINECO (FIS2016-78507-C2) and DURSI (FIS2016-\n78507-C2).\n13Supporting Information Available\nThe following files are available free of charge. Two experimental videos (.WMF) showing\nthe dynamics of magnetic colloidal particles above the FGF film.\nReferences\n(1) Hänggi, P.; Marchesoni, F. Rev. Mod. Phys. 2009,81, 387–442.\n(2) Reimann, P. Phys. Rep. 2002,361, 57–265.\n(3) by Eric R. Hedin, E.; Joe, Y. S. Nanoscale Device Physics: Science and Engineering\nFundamentals , 1st ed.; Pan Stanford Publishing: Oxford, United Kingdom, 2013.\n(4) Crider, P. S.; Israeloff, N. E. Nano Letters 2006,6, 887–889.\n(5) Tiwari, S. Nanoscale Device Physics: Science and Engineering Fundamentals , 1st ed.;\nOxford University Press: Oxford, United Kingdom, 2017.\n(6) Walder, R.; Nelson, N.; Schwartz, D. K. Phys. Rev. Lett. 2011,107, 156102.\n(7) Skaug, M. J.; Wang, L.; Ding, Y.; Schwartz, D. K. 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J. Phys. 2002,70, 1194–1204.\n(49) Straube, A. V.; Tierno, P. Europhys. Lett. 2013,103, 28001.\n(50) Reimann, P.; den Broeck, C. V.; Linke, H.; Hänggi, P.; Rubi, M.; Pérez-Madrid, A.\nPhys. Rev. Lett. 2001,87, 010602.\n(51) Burada, P. S.; Hänggi, P.; Marchesoni, F.; Schmid, G.; Talkner., P. Chem. Phys. Chem.\n2009,10, 45–54.\n17Figure 1: (a) Schematic illustrating a single magnetic nanoparticle above an FGF subjected\nto a rotating magnetic field polarized in the ( x;z) plane. The applied field helps to generate\na periodic potential (blue line) that translates at constant speed v0=\u0015fand transports the\nnanoparticle. Bottom illustrates the rotation of magnetization in a Bloch wall. (b) Color\ncoded energy landscape ( Um=kBT) versus particle position and time calculated for a 270 nm\nnanoparticle (effective magnetic volume susceptibility \u001f= 1, elevation from surface z=\n1:34µm,f= 2 Hz), low energies in blue and high energies in yellow. The particle transport\noccurs consistently with the blue regions. (c) Particle mean speed hviacross the BWs versus\nfrequencyf(top) and normalized with respect to \u0015f(bottom), for a 270 nmnanoparticle\nsubjected to a rotating field with amplitude H0= 1200 A m\u00001. The experimental data\n(empty circles \f=\u00001=3) are plotted together with the theoretical lines: The dashed line\nshows the deterministic limit, Equation 4, while the red solid line also accounts for the effect\nof thermal fluctuations, Equation 6, see the Materials and Methods section. (d) Polarization\nmicroscope image showing a 270 nmparticle trajectory transported by the rotating field at\nan average speed hvi= 5:2µm s\u00001. The field parameters are f= 2 Hz,H0= 1200 A m\u00001.\nThe trajectory (red line) is superimposed to the image, \u0015= 2:6µm, see VideoS1 in the\nSupporting Information.\n18Figure 2: Effective diffusion coefficient Deffmeasured along the propulsion direction versus\ndriving frequency ffor particles with d= 270 nm . For all data, the particles were driven\nabove the ratchet by a rotating field with amplitude H0= 1200 A m\u00001and\f=\u00001=3. The\ngreen dashed line corresponds to the free diffusion coefficient measured above the FGF and\nin absence of applied field. The small inset displays the mean square displacements (MSDs)\ndivided by the time, h\u000ex2i=(2t)for three different frequencies with locations indicated by\nopen symbols in the main plot. The continuous red line through the f= 14:5Hzdata shows\na linear fit to calculate Deff.\n19Figure 3: (a-c) Effective diffusion coefficient Deffversus frequency ffor three types of parti-\ncles with diameters 270 nm(a),3600 nm (b) and 540 nm(c). All panels show the experimen-\ntal data (open circles), predictions of the reduced theoretical model with \f= 0according\nto Equation 7 (red line) and estimates of the effective diffusion coefficient from numerical\nsimulations of the full model with \f=\u00001=3, Equation 3 (blue lines). The experimental\nparameters used are H0= 1200 A m\u00001and\f=\u00001=3that reflect the values used for the\nsimulations. For the three types of particles we also indicate the critical frequencies fcused\nfor the model (red) and simulation (blue). (d-f) Magnetic potential V(u)in units of thermal\nenergykBTin the moving reference frame (Equation 5) evaluated for a 270nmparticle and\nat a frequency of f= 1Hz(d),f= 11Hz(e) andf= 15Hz(f). Here the critical frequency\nisfc= 13Hz.\n20Figure 4: (a) Sequence of snapshots showing the collective transport of 270 nmparticles.\nAt timet= 7:2 sthe magnetic ratchet is switched on and all the nanoparticles are trans-\nported to the right (arrow at bottom) at an average speed hvi= 2:6µm s\u00001(field parameters\nf= 1 Hz,H0= 1200 A m\u00001,\f=\u00001=3). The green (middle figure) and blue (bottom figure)\nlines are trajectories of a single particle. Scale bar is 5µmfor all images, see also VideoS2\nin the Supporting Information. (b) and (c) Probability distributions of the particle position\nperpendicular P(\u000ex)(b) and parallel P(\u000ey)(c) to the BWs. The scattered points are exper-\nimental data while the solid lines are Gaussian fits with \u001b2\nxand\u001b2\nyfor the variance in the\ncorresponding directions. In (b) the distribution was calculated by subtracting the drift as\ndescribed in the text.\n21"    },    {        "title": "1307.7561v1.Simple_concentration_dependent_pair_interaction_model_for_large_scale_simulations_of_Fe_Cr_alloys.pdf",        "content": "arXiv:1307.7561v1  [cond-mat.mtrl-sci]  29 Jul 2013A simple concentration-dependent pair interaction model f or large-scale simulations of\nFe-Cr alloys\nM. Levesque,1E. Martínez,1,∗C-C. Fu,1M. Nastar,1and F. Soisson1\n1CEA, DEN, Service de Recherches de Métallurgie Physique, F- 91191 Gif-sur-Yvette, France\nThis work is motivated by the need for large scale simulation s to extract physical information on\nthe iron-chromium system which is a binary model alloy for fe rritic steels used or proposed in many\nnuclear applications. From first principle calculations an d the experimental critical temperature\nwe build a new energetic rigid lattice model based on pair int eractions with concentration and\ntemperature dependence. Density Functional Theory calcul ations in both norm-conserving and\nprojector augmented wave approaches have been performed. A thorough comparison of these two\ndifferent ab initio techniques leads to a robust parameterization of the Fe-Cr H amiltonian. Mean field\napproximations and Monte Carlo calculations are then used t o account for temperature effects. The\npredictions of the model are in agreement with the most recen t phase diagram at all temperatures\nand compositions. The solubility of Cr in Fe below 700 K remai ns in the range of about 6% to\n12%. It reproduces the transition between the ordering and d emixing tendency and the spinodal\ndecomposition limits are also in agreement with the values g iven in the literature.\nPACS numbers: 61.80.Az,61.82.Bg,61.66.Dk, 61.50.Lt\nI. INTRODUCTION\nAs it has been extensively reported in the literature1–4,\nferritic steels with a content in Cr ranging from 5 to 13\nat. % present a set of features concerning their radiation\ndamage resistance that makes them the strongest candi-\ndates for future nuclear energy applications as structural\nmaterials. To develop new materials capable of working\nat high irradiation doses we need to understand both\ntheir thermodynamic and kinetic properties. At high\ntemperature, the kinetic evolution is rapid enough to ob-\nserve the formation of the γphase and the decomposition\nof Fe-Cr alloys into two body centered cubic (BCC) solid\nsolutions, αandα′and therefore the phase diagram is\nwell-known. Below 700 K, the equilibrium state is still in\ndebate. The phase diagrams available in usual compila-\ntions and database like CALPHAD5–7are derived from\nhigh temperature experiments8–12. They display an al-\nmost symmetrical α-α′miscibility gap and yield a zero\nsolubility limit of Cr in Fe at low temperature. How-\never, first principles calculations by Hennion13in 1983,\nconfirmed a few years later by a neutron study of short\nrange order (SRO)14,15have shown that Fe-Cr alloys dis-\nplay an ordering tendency for low chromium contents.\nThis anomaly has been extensively studied using vari-\nousab initio methods and is now well understood, al-\nthough the Cr concentration at which the sign of the\nmixing energy changes depends on the approach (from\n5% up to approximately 10%)16–19. This behavior has\nbeen rationalized in terms of an anti-alignment of the\nmagnetic moment of Cr in the Fe matrix, the repulsion\nbetween first nearest neighbor (1nn) Cr, and the order-\ning tendency observed at low concentrations18. More-\nover, a few experimental observations in alloys submitted\nto irradiation have been recently reviewed by Bonny et\nal.20. They suggest that the chromium solubility remains\nabove 8%, even at low temperatures. This interpretationis based on the assumption that irradiation only results\nin an enhancement of diffusion and that more complex\neffects that could modify the solubility limit, such as bal-\nlistic disordering or radiation induced segregation, can b e\nneglected. Recent critical reviews have therefore high-\nlighted the need to modify the Fe-Cr phase diagram at\nlow temperature20–22.\nSeveral atomistic models have been proposed in this\ncontext to reproduce the complex thermodynamic behav-\nior of Fe-Cr alloys. Semi-empirical potentials have been\ndeveloped that take into account the change of sign of\nthe mixing energy, such as the concentration dependent\nmodel (CDM) of Caro et al.1or the Two-Band Model\n(2BM) of Olsson et al.23recently updated by Bonny et\nal.24. A lot of work has been done in order to assess\ntheir thermodynamic properties as well as their dynam-\nical behavior25–28. However, it remains difficult to de-\nvelop a potential fitting simultaneously all the key prop-\nerties that control the thermodynamics and kinetics of\nthe Fe-Cr decomposition (such as the mixing energies, the\npoint defects formation energies and migration barriers,\nwith their dependence on the local atomic distribution\nand with the corresponding vibrational entropy contri-\nbutions). Furthermore, magnetic contributions have not\nbeen introduced in these potentials. Because the kinetic\nmodeling of phase transformations including atomic re-\nlaxations and vibrational contributions is a challenging\ntask29, these potentials are usually mapped on a rigid\nlattice model which in turn affects their thermodynamic\nand kinetic properties.\nCluster expansion (CE) techniques, based on a\nrigid lattice approximation with N-body concentration-\nindependent interactions, have been proposed to model\nthe thermodynamics of Fe-Cr alloys30–32. However, to\nbe able to reproduce the ab initio mixing energies of Fe-\nCr, a purely chemical CE ( i.e.that does not take ex-\nplicitly into account the magnetic moments) requires a\nlarge set of many-body interactions31,32. The difficulty2\nof obtaining a small set of effective interactions was also\nreported by authors using a screened generalized pertur-\nbation method33. Therefore, a cluster interaction model,\nalthough restricted to a rigid lattice description, remain s\nquite heavy numerically. The whole corresponding phase\ndiagram was not published, but following the trend of the\nempirical concentration-dependent energy models, it is\nexpected that the critical temperature for the miscibility\ngap is way too high compared to experiments and classi-\ncal CALPHAD database5,10. Two missing ingredients in\nthose CE models are the vibrational entropy (which is sig-\nnificant in the Fe-Cr system34,35) and the magnetic con-\ntributions. The screen generalized perturbation model\nhas been used to show how the effective cluster inter-\nactions depend on the magnetic state of the alloy and\ntherefore on the temperature33and the composition36.\nMixed models including chemical and magnetic interac-\ntions have been proposed30,37–39. The Ising model by\nAckland, with magnetic moments of constant amplitude,\nreproduces some key features of the Fe-Cr alloys. The\nmagnetic cluster expansion of Lavrentiev et al.38,39is\nable to reproduce the ab initio mixing energies with much\nfewer interactions than a purely chemical CE and it can\ntake into account the variation of the magnetic moments\nwith the concentration, but its phase diagram has never\nbeen calculated in the α-α′region. The last model we\nwanted to mention is the Stoner Hamiltonian developed\nby Nguyen-Manh and Dudarev40. It is shown that all\nthe significant features of the Fe-Cr alloys can be ex-\nplained in terms of bonding effects involving 3 delec-\ntron orbitals and magnetic symmetry-breaking effects re-\nsulting from intra-atomic on-site Stoner exchange. The\ncomplete phase diagram has not been reported for this\nHamiltonian and, as it is said in the manuscript, further\napproximate computational algorithms will have to be\ndeveloped suitable for large scale simulations.\nFinally, it is worth noting that using a magnetic model\nin a kinetic simulation of the α-α′decomposition (such\nas a kinetic Monte Carlo simulation) would require the\nrelaxation of the atom-vacancy exchange events and mag-\nnetic moments transitions which probably occur at very\ndifferent time scale. Even with simplifying assumption\n(e.g. if the relaxation time of the magnetic moment is\nnegligible), it would make the simulation much more time\nconsuming than for a non-magnetic model. We propose\nhere an alternative model: a concentration and tempera-\nture dependent pair interaction model fitted on ab initio\ncalculations and the experimental critical temperature of\nFe-Cr alloys. The goal is to keep the model simple enough\nto be used in kinetic Monte Carlo simulations such as the\none of Ref.41.\nThe manuscript is organized as follows. In Sec.II the\nresults from density functional theory (DFT) calcula-\ntions on the energetics of the Fe-Cr system, that have\nbeen used to parameterize the interaction model, are re-\nported. Because energy values, in the magnetic Fe-Cr\nsystem, depend on the method, we have performed our\nownab initio calculations using two different methods.In Sec.III we present the concentration and temperature\ndependent pair interaction model and its phase diagram,\ncomputed in a mean-field approximation and by Monte\nCarlo simulations. The phase diagram, including the\nspinodal decomposition region, and the short range order\nare compared with available experimental data. Compar-\nison with other models are discussed in Sec. IV. Finally,\nsome conclusions and perspectives are highlighted.\nII. DFT CALCULATIONS\nSeveral studies have already been devoted to the calcu-\nlations of energetic properties of FeCr alloys16–19,26. We\nhave nevertheless performed a new systematic first princi-\nples study in order to parameterize our interaction model\nin a self-consistent way. In particular, we have calculated\nthe enthalpy of mixing of the FeCr alloy to account for its\nbehavior in the whole concentration range. We have also\nestimated the interactions between two Cr(Fe) impuri-\nties in a bcc Fe(Cr) matrix, which allows to determine\nthe cutoff of interactions distance of the pair-interaction\nmodel. Ferromagnetic (FM) Fe and (100)-layered antifer-\nromagnetic (AF) Cr have been taken as reference states\nto obtain the values mentioned above. Note that even\nthough the experimental magnetic ground state of pure\nbcc Cr is an incommensurate spin-density wave (SDW),\nthe presence of Fe atoms seems to reduce the stability\nof such a long-ranged state. It, indeed, becomes unsta-\nble against the formation of AF structures with 1.6 %\nof Fe42. Because our interest is mainly focused on the\nFe-rich side of the alloy, we assume the AF state for Cr\nin the present study.\nCalculations are performed in the framework of Den-\nsity Functional Theory as implemented in the PWSCF\ncode43. They are spin polarized within the Gener-\nalized Gradient approximation (GGA) with the PBE\nparametrization44. We have used the Projector Aug-\nmented Wave (PAW) potential instead of pseudopoten-\ntials. The kinetic energy cutoff chosen for the plane-\nwave basis set was 544 eV. All the calculations are fully\nrelaxed, i.e., both atomic positions and simulation-cell\nvolumes are optimized. The corresponding residual force\nand stress tolerances are respectively 0.04 eV/Å and 5\nkbar. We have also calculated the mixing enthalpies us-\ning norm-conserving (NC) pseudopotentials and localized\nbasis sets, as implemented in the SIESTA code45. This\napproach has been shown to give results of equivalent ac-\ncuracy as plane-wave DFT methods. In particular, prop-\nerties of defects in various Fe based systems have been\nsatisfactorly predicted41,46–48. It is however less compu-\ntationally demanding thanks to significant reduction of\nthe basis size. The aim is to check the ability of this\nless standard DFT approach for quantitative prediction\nof properties in FeCr alloys, where the energetics may be\nextremely sensitive to magnetic couplings18,49.3\n0 0.2 0.4 0.6 0.8 1\nCr atomic fraction-50050100150200Mixing Enthalpy (meV)\nFIG. 1. Enthalpy of mixing for the Fe-Cr system as a function\nof Cr atomic fraction. PAW-GGA (NC-GGA) calculations\nare in black (red). Full symbols are for SQS structures and\ncrosses for ordered structures. The lines give the fit of the\nenthalpy of mixing of the SQS by the Redlich-Kister formula\n(see section III).\nA. Mixing Enthalpy\nThe enthalpy of mixing is defined as:\n∆Hmix=E[nFe+mCr]−{nE[Fe]+mE[Cr]}\nn+m(1)\nwhereE[nFe+mCr]is the total energy of a mixed sys-\ntem containing nFe atoms and mCr atoms. E(Fe)and\nE(Cr)are energies per atom of the Fe and Cr reference\nsystems. Because the calculations are performed at zero\npressure this value is also equivalent to the mixing energy.\nThe supercells used for different concentrations have\nbeen generated using two methods. The first generation\nmethod is \"user\"-chosen. It consists of a large set of\nordered structures devised to explore various energetic\nlandscapes ( DO3, B2, Fe n−1Cr, FeCr m−1etc.), the same\nas considered in a previous work on Fe-Cu alloys41. The\nother generation method is based on the special quasi-\nrandom structure methodology (SQS)50) which allows to\ngenerate a supercell with as small short range order as\npossible. These supercells are thus the best representa-\ntive configurations of a random solid solution for each\nconcentration. The number of atoms in each supercell is\neither 54 or 128.\nThe resulting mixing enthalpies for both ordered and\nSQS structures are shown in Fig. 1.\nIn good agreement with previous DFT results18,51, we\nnote a change of sign of the mixing enthalpy, showing\nnegative values for low Cr concentrations according to\nboth PWSCF and SIESTA approaches. However, the\nrange of this negative part of the enthalpy of mixing as\nwell as its depth strongly depend on the approach. From\nthe SIESTA-NC calculations the change in sign is aroundxCr= 0.15while with the PWSCF-PAW approach, the\nvalue is about xCr= 0.07.\nFirst of all, we focus on two extreme cases, i.e., infinite\ndilution in Fe and Cr respectively, where we may also\ndefine the solution energy as:\nEXinY\nsol=E[Yn−1X1]−{(n−1)E[Y]+E[X]}(2)\nwhere X and Y are either Fe or Cr, E[Yn−1X1]is the total\nenergy of a supercell containing n-1atoms of Y and one\natom of X and E[Y]andE[X]are the energy per atom\nof the pure systems: bcc FM Fe and AF Cr. The solu-\ntion energies are well converged within 1meV for n= 128\natoms. For the case of Cr dissolution in Fe we find a\nvalue of Esol=−0.20eV using PWSCF while SIESTA\npredicts Esol=−0.47eV. They are consistent with pre-\nvious DFT values ranging between Esol=−0.12eV and\nEsol=−0.46eV18,51. We see that all DFT results pre-\ndict Cr dissolution to be exothermic, i.e., energeticaly\nfavorable to insert one substitutional Cr in the Fe ma-\ntrix. However, the precise value reveals to be method\ndependent. In particular, the SIESTA-NC result overes-\ntimates the solution tendency of Cr in Fe with respect\nto the PWSCF-PAW data. In order to gain more insight\ninto the origin of this overstimation, we have performed\ncomplementary PWSCF calculations using a NC pseu-\ndopotential (PWSCF-NC) as close as possible to that of\nSIESTA. The obtained Cr solution energy in Fe is -0.49\neV, very close to the SIESTA value. This comparison\nsuggests that the overestimation of Cr solution energy\nis essentially due to the NC-pseudopotential approxima-\ntion rather than the use of localized basis functions in\nthe SIESTA approach. We have also checked that the\nmagnitude of Cr solution energy is indeed closely cor-\nrelated with the local magnetic moments of the Cr in\nFe. The corresponding values from the PWSCF-PAW,\nPWSCF-NC and SIESTA-NC studies are 2.1 µB, 2.6µB\nand 2.5 µBrespectively. It is interesting to point out\nthat the overstimation of the Cr solution energy from a\nNC pseudopotential prediction is closely correlated to the\nobtained higher value of Cr local moments with respect\nto the PAW value.\nWhen a Cr atom is substituted by one Fe atom in the\nCr matrix, the solution energy obtained was Esol= 0.45\neV using PWSCF and Esol= 0.29eV with SIESTA, indi-\ncating an endothermic reaction. In this case, the Fe local\nmagnetic moment found for the Fe impurity is within the\nprecision limits, 0.02 µBwhile SIESTA gives 0.14 µB. In\nboth cases the magnetic moment of the Fe solute is anti-\nalligned to the local moment of its first nearest neighbors.\nThe local magnetic moment of all the Cr atoms remains,\nas expected, practically the same as in pure AF Cr. The\nsmall moment of Fe may be explained as a consequence of\nmagnetic frustration resulting from the competition be-\ntween the Fe and its first- and second-nearest Cr neigh-\nbors. As also suggested by a previous study18, Fe and\nCr first and second nearest neighbors prefer an antiferro-\nmagnetic coupling, which can clearly not be satisfied for\nan isolated Fe in a bcc AF Cr lattice.4\nBeyond the infinite diluted cases, ordered structures\nwith a mixing energy lower than the SQS-random con-\nfigurations ( ∆Hmix=−xECrinFe\nsol) are observed at low Cr\nconcentrations with both PWSCF and SIESTA. In par-\nticular, the Fe52Cr2system with the 2 Cr atoms sep-\narated by (1.5,1.5,1.5) times the bcc lattice parameter\n(a0) has an energy lower than the solid solution of the\nsame composition, suggesting the possible formation of\nan intermetallic phase for that concentration at low tem-\nperatures. Indeed, the same Fe52Cr2structure has also\nbeen pointed out by Erhart et al. as a possible inter-\nmetallic system26. Other DFT calculations predict that\nthe Fe 15Cr31or the Fe 14Cr52ordered structures could be\nthe ones forming the intermetallic compounds. However,\nit should be noticed that the relative stability of such\nphases is difficult to assess because their difference in\nformation energies is very close to the DFT uncertainties\nand because it remains to be verified whether they may\nexist at finite temperatures when the entropy becomes\nrelevant.\nOn the other hand, at higher Cr concentrations, SQS\nsystems show overall lower energy than the ordered con-\nfigurations. This is indeed consistent with the positive\nmixing enthalpies suggesting a tendency to phase sepa-\nration rather than ordering.\nB. Impurity Interactions\nIn order to determine the cutoff distance of the pair\ninteraction model for FeCr, we have also evaluated the\ninteraction between two Cr(Fe) impurities in a bcc Fe(Cr)\nmatrix. Binding energy between two X atoms ithnearest\nneighbors in a bcc lattice of Y atoms is defined as follows,\nwhere positive values mean attraction:\nEb(X−X) =−E[YN−2+X2]−E[YN]+2E[YN−1X1](3)\nwhereE[YN−2+ X2]is the total energy of the system\nwith the two X atoms at a ithnearest neighbor distance,\nE[YN]is the total energy of NY-atoms in the corre-\nsponding reference system (either ferromagnetic bcc Fe\nor antiferromagnetic bcc Cr), and E[YN−1X1]is the total\nenergy of the system of Natoms with just one impurity\natom. The values for the binding energies of Cr-Cr in Fe\nand Fe-Fe in Cr are shown in Fig. 2. The calculations\nhave been done within the more accurate PAW approach\nusing 128-atom supercells.\nConsistent with previous DFT calculations51and with\nthe experimentally observed ordering tendency at low Cr\ncontent, we find that two Cr atoms repel each other in a\ndilute FeCr alloy. Such repulsion is particularly strong fo r\n1nnand2nninteractions. The binding energies are -0.32\neV and -0.15 eV respectively from the PWSCF calcula-\ntions. For 3nnto5nnthe Cr repulsion becomes signifi-\ncantly weaker (around -0.04 eV) according to our results\n(Fig. 2). It vanishes for farther Cr-Cr distances within\nthe estimated error bar of ±0.025 eV. As explained inprevious studies18, this Cr-Cr repulsion is directly corre-\nlated to the corresponding local magnetic structure. Lo-\ncal magnetic moments of both Cr atoms are found to\nbe parallel to each other when they are close neighbors.\nAlso, their moment amplitudes are reduced as compared\nwith that of an isolated Cr (2.2 µB). For instance, we\nfind local moment reductions of around 0.1 µBfor two\n1nnand2nnCr atoms with respect to an isolated Cr.\nThis can be understood as a magnetic frustration result-\ning from competition between various magnetic coupling\ntendencies, i.e., antiferromagnetic for Fe-Cr and Cr-Cr\nand ferromagnetic for Fe-Fe pairs. Indeed, when per-\nforming complementary calculations constraining all the\nCr local moments to zero, the resulting Cr-Cr binding\nenergies become negligible.\nOn the other hand, in the case of two Fe impurities\nin a Cr matrix, their binding energy is slightly positive\nfor a1nnseparation (0.06 eV), whereas it is negative for\nthe2nnFe-Fe pair (-0.05 eV). Beyond, all the values are\nrepulsive, but their magnitudes are smaller than 0.03 eV,\nclose to our estimated error bar (Fig. 2). It is interesting\nto mention that the change of interaction between the\n1nnto the2nnseparations , i.e., from an attraction to\na repulsion, may be linked to a change of local magnetic\nmoments of the respective Fe atoms. Indeed, as discussed\nin Sec. II A, an isolated Fe in the AF-Cr shows a small\nmoment due to the magnetic frustration. It is also the\ncase for all the Fe atoms separated by a 2nndistance or\nfarther. However, the magnetic state can be expected to\nchange when Fe atoms get close to each other. For in-\nstance, when they are first nearest neighbors, one of the\ntwo Fe atoms adopts a high moment of 2.11 µBwhereas\nthe other remains at a low-moment state (0.53 µB). Both\nFe moments align parallel to each other, but only the\nhigh-moment Fe is antiferromagnetically coupled to its\nCr first-nearest neighbors. This assymetric configuration\nsuggests that at least the magnetic frustration of one iron\natom, i.e., the high-moment Fe, is partly relaxed, induc-\ning a decrease of the system energy. It is worth men-\ntioning that other metastable states may also exist for\nthe1nnFe-Fe case. For instance, we have found another\nmagnetic configuration where both Fe atoms have low\nlocal moments. The corresponding binding energies is\npractically zero.\nEven though the absolute values of Cr-Cr interaction\nenergies in Fe are overall larger than the corresponding\nFe-Fe values, in both cases, the range of the significant\ninteractions is up to a second-nearest neighbor distance,\nwhich may therefore be reasonably considered as the cut-\noff distance for our pairwise energetic model as described\nbelow (Sec. III).5\n1 1.2 1.4 1.6 1.8 \ndistance (lattice parameter) -0.3 -0.2 -0.1 0Binding energy (eV) 2nn 3nn \n1nn 4nn 5nn \nCr-Cr \nFe-Fe \nFIG. 2. Binding energy of two Cr (Fe) impurities in an Fe (Cr)\nmatrix where inn stands for the ithnearest neighbor between\nthe impurities in a bcc lattice.\nIII. THERMODYNAMIC MODEL\nA. Constant pair interactions (Ising model)\nOur objective is now to build an interaction model able\nto take into account the key energetic properties revealed\nby DFT calculations and to predict a phase diagram in\nagreement with the experimental one. The most simple\nmodel of phase separation in a binary A-B alloy is proba-\nbly the Ising model, with constant pair interactions ǫ(i)\nAA,\nǫ(i)\nABandǫ(i)\nBBbetween A and B atoms on ithneighbor\nsites. The mixing free enthalpy of a solid solution can\nbe computed using mean-field (MF) approximations (see\ne.g.53). With the simplest Bragg-Williams (BW) approx-\nimation and when ǫ(i)\nAA+ǫ(i)\nBB−2ǫ(i)\nAB<0, one gets for\nthe mixing enthalpy:\n∆Hmix=−Ωx(1−x) (4)\nWhile the configurational entropy of mixing is given by:\n∆Smix=−kB[(1−x)ln(1−x)+xlnx](5)\nwherexis the B atomic fraction and kBthe Boltzmann\nconstant,\nΩ =/summationdisplay\ni/bracketleftbiggz(i)\n2/parenleftBig\nǫ(i)\nAA+ǫ(i)\nBB−2ǫ(i)\nAB/parenrightBig/bracketrightbigg\n(6)\nis the ordering energy and z(i)the coordination num-\nber of shell i. The minimization of the free enthalpy\n∆Gmix= ∆Hmix−T∆Smixgives a symmetrical misci-\nbility gap, with a critical temperature Tc=−Ω/2kB.In the BW approximation, when all the combination\nǫ(i)\nAA+ǫ(i)\nBB−2ǫ(i)\nABare negative, the phase diagram de-\npends exclusively on Ωand not on the distribution of\nthe interactions among the different coordination shells\n(i). This approach neglects the short range order in the\nsolid solution. In the specific case of alloys with nearest\nneighbors interactions, the BW critical temperature is\n20% larger than the Monte Carlo reference value54. The\ndiscrepancy decreases with the range of interactions (for\ninfinitely long-range interactions, mean-field approxima-\ntions become exact55).\nB. Composition-dependent pair interactions\nA constant pair interaction model always gives sym-\nmetrical mixing energies and phase diagrams and there-\nfore cannot reproduce the DFT mixing energies of Fe-Cr\nalloys (Fig. 1), with negative mixing energies in the Fe-\nrich configurations only. To be able to reproduce the\nmixing enthalpy in the whole concentration range we in-\ntroduce pair interactions that depend on the local com-\nposition, using a polynomial expression. In the BW ap-\nproximation, the mixing enthalpy is given by:\n∆Hmix=−Ω(x)x(1−x) =−x(1−x)n/summationdisplay\np=0L(p)(1−2x)p\n(7)\nalso known as the Redlich-Kister formalism56.nis the\nmaximum order of the parametrization and L(p)is called\ninteraction parameter of order pand it has the form:\nL(p)=a(p)+b(p)T (8)\nTheL(p)parameters at 0 K (i.e. the a(p)parameters)\nare fitted on the mixing energies of the SQS structures\npresented in sec. II. Ordered structures are not taken into\naccount because the SQS configurations are more repre-\nsentative of a random solid solution described by the BW\napproximation. The best fit we have found (see Fig. 1)\nis of the form:\nΩ(x) = (x−α)(βx2+γx+δ) (9)\nwhere the values of α,β,γandδfor the PWSCF-PAW\nand SIESTA-NC results are given in table I. The maxi-\nmum of the ∆Hmixin the PWSCF-PAW fit is at x= 0.48\nwith a value of 0.089 eV, whereas for the SIESTA-NC\nx= 0.52and the value is 0.071 eV.\nThe corresponding phase diagram has been first com-\nputed in the BW approximation, with b(p)= 0, i.e. with\ntemperature independent pair interactions (see Fig.3).\nWe observe that the solubility limits are non-symmetric.\nThe solubility of Cr in Fe does not vanish at 0K. On the\nother hand, the Fe solubility in Cr is negligible at that\ntemperature. The non-zero solubility limit on the Fe rich\nside is in contradiction with the reference phase diagram\ngiven by CALPHAD10but in accord with more recent6\nPWSCF-PAW SIESTA-NC\nα 0.070 0.160\nβ(eV)−2.288 −2.348\nγ(eV)4.439 4.381\nδ(eV)−2.480 −2.480\nTABLE I. Fitting parameters for the ordering energy Ω(x)\nobtained from PWSCF-PAW and SIESTA-NC calculations\n0 0.2 0.4 0.6 0.8 1\nCr atomic fraction010002000300040005000Temperature (K)\nFIG. 3. Mean field Fe-Cr phase diagram with the\nconcentration-dependent interaction model (no temperatu re\ndependence) fitted on the ab initio PAW (solid black lines)\nand NC (solid red lines) calculations. Spinodal decomposi-\ntion limits are shown in dashed lines.\nstudies20–22,24. The critical temperature (about 4200 K\nfor the PWSCF-PAW parametrization and 3800 K for the\nSIESTA-NC values) is much higher than the experimen-\ntal one (approximately 1000 K21). The spinodal limits\nfor both parametrizations, defined as:\n∂2∆Gmix\n∂2x= 0 (10)\nare shown in the same Fig. 3. We observe an unusual\nlocal minimum (for x≈0.8) on the Cr rich side. This\nphenomenon occurs at temperatures lower than 500 K,\nregime where data is difficult to obtain experimentally\ndue to slow kinetics. The phase diagram obtained with\nthe PWSCF-PAW and SIESTA-NC parameters are qual-\nitatively similar. Since the PWSCF-PAW is more reliable\nand for the sake of clarity, we will only discuss the corre-\nsponding results in the following.\nThe phase diagram has also been computed by\nMonte Carlo simulations in the semi-grand canonical\nensemble57. In the BW approximation, as for the con-\nstant interaction model, the phase diagram only depends\non the ordering enthalpy (Eq. 9). For the Monte Carlo\nsimulations for the same ordering energy, one must con-\nsider the pair interactions ǫ(i)\nFeFe,ǫ(i)\nCrCr andǫ(i)\nFeCr, the\nrange of interactions and the way they decrease with01000 2000 3000 4000 5000 \n0.0 0.2 0.4 0.6 0.8 1.0  BW approximation \n Monte Carlo (1 st  - 2 nd  nn interactions) \n Monte Carlo (1 st  - 5 th  nn interactions)  \n \nCr atomic fraction Temperature (K) \nFIG. 4. The Fe-Cr phase diagram with a composition depen-\ndent pair interaction model fitted on the PWSCF-PAW mix-\ning energies (no temperature dependence). The continuous\nline gives the solubility limit computed in the BW approxima -\ntion. The open circles give the Monte Carlo results with first\nand second neighbor interactions, the full circles the Mont e\nCarlo results with interactions up to the fifth neighbors.\nthe distance. For the sake of simplicity we have chosen\nthat cross interactions ǫ(i)\nFeCrcarry the dependency on\nthe local concentration. The self-interactions ǫ(i)\nFeFeand\nǫ(i)\nCrCrare considered as constants given by the cohesive\nenergies of the pure elements, according to Ecoh(A) =\n−/summationtext\niz(i)ǫ(i)\nAA(Caro and coworkers followed the same\nstrategy in the development of their CDM model1). The\nlocal Cr concentration around a Fe-Cr pair is defined as\nthe fraction of Cr atoms among their neighbors. If the\ninteractions are limited to the rthnearest-neighbors, the\nlocal chromium concentration around a Fe atom on site\niand a Cr atom on site jis defined as:\nc(FeiCrj) =/summationtextr\nn=0/summationtextz(n)\nk=1p(n)\nik+/summationtextr\nn=0/summationtextz(n)\nk=1p(n)\njk\n2/summationtextr\nn=0z(n)\nwherep(n)\nik= 1when the kthneighbor of the site iat a\nnthneighbor position is occupied by a Cr atom. We also\ninclude in the calculation the type of the atoms in sites i\nandj.\nTo assess the effect of the interaction range, we have\nused two sets of pair interactions. One has to consider\nenough neighbors to get a sufficient discretization of the\nmixing energy with its change of sign at 7%, so that first\nnearest-neighbor interactions are not enough. Therefore,\nthe first set of parameters is limited to first and second\nneighbor interactions, with the second neighbor interac-\ntions two times smaller than the first ones ( ǫ(2)\nXY=ǫ(1)\nXY/2.\nThe second set includes up to the fifth nearest-neighbors\ninteractions and they decrease more slowly, as the inverse\nof interatomic distance.\nThe resulting phase diagrams are compared with the7\nBW approximation on Fig. 4: at low temperature, the\nBW approximation is close to the Monte Carlo results.\nAt high temperature it underestimates the mutual sol-\nubility of Fe and Cr and overestimates the critical tem-\nperature, by approximately 40% when the interactions\nare limited to the first and second neighbor shells. The\ndiscrepancy is then two times larger than for the usual\nIsing model. The critical temperature of the Monte Carlo\nsimulations is significantly higher with interactions up to\nthe fifth neighbors. This is in agreement with the usual\ntendency, where the mean field and Monte Carlo results\nconverge for infinite interaction range55.\nC. Temperature dependence\nThe critical temperatures calculated by the\ncomposition-dependent approach, shown in Figs. 3\nand 4, lay well above the critical temperature observed\nexperimentally for this system, of about 1000 K21. The\nCDM potential shows the same deviation, as does the\nchemical CE (see section IV).\nWe rationalize this difference in terms of the mayor\neffects that are not taken into account:\n- The competition between magnetic and chemical in-\nteractions.\n- The intrinsic nature and the amplitude of the atomic\nmagnetic moments change. The magnetic moments de-\ncrease with the temperature what in turn decreases the\npair interaction strength58.\n- The vibrational entropy.\n- The magnetic entropy.\nOne could in principle evaluate the vibrational entropy\nfrom DFT calculations, for instance in the harmonic ap-\nproximation, but would be obliged to take the rest of\nthe temperature effects empirically. We consequently de-\ncided to introduce an empirical temperature dependency\non the ordering energy to compensate all the effects of\nthe non-configurational entropies and magnetic contri-\nbutions. In order to keep the simplicity of the model\nand to get a phase diagram closer to the experimental\none we assign to the ordering energy Ωa simple linear\ndependency on temperature:\nΩ(x,T) = Ω(x)/parenleftbigg\n1−T\nΘ/parenrightbigg\n(11)\nwhereΘhas units of temperature and it is adjusted such\nthat the Monte Carlo simulations yield the experimen-\ntal critical temperature ( ≈1000 K). This effect is again\ntaken into account solely to fit the pair interaction cross\ntermǫ(i)\nFeCr. We find Θ = 1480 K using the PWSCF-\nPAW parameters. The phase diagram as given by this\nmodel is shown in Fig. 5. It is worth noting that it is not\njust a temperature rescaling of the one in Fig. 4, since\nthe configurational entropy is not changed. As a conse-\nquence the BW and the Monte Carlo results are closer\nthan before. These results match quite well the collec-\ntion of experimental results reported by Xiong et al.210.0 0.2 0.4 0.6 0.8 1.0 200 400 600 800 1000 1200 \n BW approximation \n Monte Carlo (1 st  - 2 nd  nn interactions) \n Monte Carlo (1 st  - 5 th  nn interactions)  \n Temperature (K) \nCr atomic fraction \nFIG. 5. The Fe-Cr phase diagram with a composition and\ntemperature dependent pair interaction model fitted on the\nPWSCF-PAW mixing energies and the experimental critical\ntemperature. The continuous line gives the solubility limi t\ncomputed in the BW approximation. The full circles give the\nMonte Carlo results with first and second neighbor interac-\ntions, the open circles the Monte Carlo results with interac -\ntions up to the fifth nearest neighbors.\nand Bonny et al.20(Fig. 6). The Cr solubility in Fe is\nlarger than 0 at low temperatures. Following the results\nby Xiong et al. the solubility limit at 250 K should be\nbetween 1 and 7%. We have obtained a value of around\n7% with our model. The solubility limit in the Cr rich\nside is lower than 1% at temperatures below 600 K. At\nvery low temperatures the Monte Carlo results show a\nsolubility of Cr in Fe of about 6%.\nLong runs of Monte Carlo simulations at lower temper-\natures seem to confirm that the Cr solubility in Fe at zero\ntemperature is different from zero. This unusual aspect\nof a demixing alloy can be explained by the change of sign\nof the mixing energy, as already mentioned. No evidence\nof long-range ordered structure has been observed above\n200 K. At lower temperature, the stability of such struc-\ntures is difficult to study because the efficiency of the\nMonte Carlo algorithm decreases. A negative ordering\nenergy invariant with respect to the local concentration\nwould imply the stabilization of an ordered phase. At 0\nK, the solid solution would be less energetically favorable\nthan the two-phase system formed by the ordered phase\nand a pure phase. However, in the case of a concen-\ntration dependent model, it is possible that in the small\nconcentration range associated with a negative value of\nthe ordering energy there is no formation of an ordered\nphase and then stabilization of the solid solution.\nThe spinodal decomposition limits as given by our con-\ncentration and temperature dependent model are shown\nin Fig. 7 (computed in the BW approximation) where\nthey are compared to the experimental data compiled by\nXiong et al.21obtained in the temperature range of 650\nto 800 K. The existence of a strictly defined limit between\ntwo kinetic regimes (nucleation and growth and spinodal8\n0.0 0.2 0.4 0.6 0.8 1.0 200 400 600 800 1000 1200 \n  \n Bonny et al. (2008) \n Xiong et al. (2010) \n BW approximation \n Monte Carlo (1 st  and 2 nd  nn interactions) Temperature (K) \nCr atomic fraction \nFIG. 6. The Fe-Cr phase diagram. Comparison between\nthe composition and temperature dependent pair interactio n\nmodel (Monte Carlo simulations with first and second neigh-\nbor interactions and BW approximations fitted on PWSCF-\nPAW mixing energies) and the critical reviews of Bonny et\nal.20(dotted line) and Xiong et al.21(shaded region).\n0 0.2 0.4 0.6 0.8 1\nCr atomic fraction60070080090010001100Temperature (K)\nFIG. 7. Miscibility gap and spinodal limit of the compositio n\nand temperature dependent pair interaction model (PWSCF-\nPAW parameters) computed in the BW approximation. The\nexperimental data for the nucleation and growth regime are\ngiven by square red dots and the spinodal decomposition\nregime is given by the blue triangles. The experimental valu es\nhave been collected by Xiong et al.21.\ndecomposition) is debatable59. Nevertheless, we observe\nin Fig. 7 that the spinodal limits we are proposing are\nin good agreement with the existing experimental data.\nD. Short Range order\nThe negative part of the enthalpy of mixing at low\nCr concentrations induces the formation of short range\norder (SRO) structures in the Fe-Cr alloy, as shown ex-perimentally by Mirebeau et al.14,15who measured the\nCowley-Warren SRO parameter for different Cr contents\nvia neutron diffraction at 703 K. They observed a change\nin sign in the parameter at around 10% Cr, showing a\nminimum close to 5%. This inversion of sign was earlier\npredicted by Hennion13carrying out ab initio calcula-\ntions on ferromagnetic systems.\nThe analysis of the SRO parameter is of technological\nimportance because of its implications on the mechanical\nproperties of the alloy. It is usually defined following the\nCowley’s notation60,61where the expression for the ith\natomic shell of a B atom in an A-B binary allow is given\nby:\nα(i)\nB= 1−z(i)\nA\nz(i)(1−xB)(12)\nwherez(i)\nAdenotes the number of A atoms in the ithshell\nfrom a B atom, z(i)is the total number of atoms in the\nithshell and xBis the global concentration of B atoms.\nThe value of this parameter will tend to 1 in a segregated\nalloy and it will be close to 0 for a random solution. For a\nsystem with ordering tendency the value will be negative,\nwith a minimum given by:\nα(i)\nB=−xB\n1−xB(13)\nThis latter value indicates the maximum degree of short\nrange order that an alloy can possibly attain. In the\nstudies by Mirebeau et al. the parameter that is actually\nmeasured is specific for BCC structures, defined for the\nFe-Cr system as:\nβ=8α(1)\nCr+6α(2)\nCr\n14(14)\nTo be able to compare to the experimental measurements\nand to the recently published data based on the empiri-\ncal energetic models described above, we have performed\nequilibrium Monte Carlo calculations in the semigrand-\ncanonical ensemble57and measured the parameter de-\nscribed in eq. 14 for different Cr concentrations. Results\nare shown in Fig. 8 where the concentration and temper-\nature dependent model was used with interactions up to\nthe2nn. We observe that the model slightly overstimates\nthe ordering tendency but captures the ordering trend\nof the alloy for small Cr concentrations. The βvalues\ntend to0with temperature due to entropic effects. The\nsolubility limits are also shown in the figure. Beyond\nthe solubility limit, the SRO parameters are measured\nin metastable solid solutions which remain homogeneous\nduring the simulation.\nWe have shown how the model captures the ordering\ntendency of Cr in the Fe matrix. The βparameter be-\ncomes negative for low Cr concentrations. Experiments\nshow the same trend, with negative values for low Cr\nconcentration and a change in sign at around 11%. This\ninversion of sign observed experimentally is probably due9\n-0.15-0.10-0.0500.050.100.15\n246810121416SRO parameter β\nCr concentration (%)800 K\n700 K\n600 K\n500 K\n400 K\n300 K CTD model\n Expa\nExpb\n Solubility limit\n maximum SRO\nFIG. 8. Short range order of the FeCr system as given by\nthe concentration and temperature dependent model (CTD\nmodel) with interactions up to 2nn. The results of the model\nare compared to available experimental data (blue squares\nand black triangles)14,15performed at 703 K.\nto the presence of a secondary α′phase, as explained by\nErhart et al.27. Our results are in very good agreement\nwith those presented in27using the CDM semiempirical\npotential1.\nIV. DISCUSSION\nThe results given above show that the simple pair inter-\naction model described along the manuscript is able to\nreproduce the main features of the experimental phase\ndiagram. In the following we are going to analyze the\nmatches and the disagreements between our model and\nexisting models in the literature with special attention to\nthe Cr solubility in Fe, since it controls the precipitation\ndriving force.\nOur model has been fitted to the experimental criti-\ncal temperature following the CALPHAD approach by\nAndersson and Sundman10(dotted line in Fig. 9) and\ntherefore it reproduces the value given by the regular\nsolution results but with a solubility limit of Cr in Fe\ndifferent from zero. The magnetic model proposed by\nInden and Sch ¨on30following a cluster variation method\nwas fitted to high temperature values of the experimental\nphase diagram. It reproduces the magnetic phase transi-\ntions with a critical temperature for the miscibility gap\nof around 880 K. The solubility limits at low tempera-\nture tend to 0in both sides of the phase diagram. It\nwould be interesting to see the results of this model with\nthe parameters fitted to ab initio results. In the mag-\nnetic Ising model by Ackland37the temperature is not in\nreal units and it is hard to compare. However, it repro-\nduces the magnetic transitions in spite of its simplicity\nand results in an asymmetric phase diagram. Concern-\ning the non-magnetic CE of Lavrentiev et al.25,31, the\nsolubility of Cr in Fe at low temperatures reported inthat study matches the values obtained with our model\n(see Fig. 9), even though the curves deviate for tem-\nperatures above 400 K, with the solubility predicted by\nour approach larger than the one given by the CE. This\nCE development does not take into account the vibra-\ntional or magnetic entropy which results in low solubility\nat high temperatures.\nThe semi-empirical interatomic potentials existing in\nthe literature and described in Sec. I have not been fit-\nted to the phase diagram itself, but only to the enthalpy\nof mixing at 0 K. The original 2BM predicts a symmet-\nric mixing enthalpy, with two changes of sign at low and\nhigh Cr concentration implying a non-zero Fe concentra-\ntion in Cr at low temperatures. The vibrational entropy\nis found to be very high what implies a decrease in the\ncritical temperature to around 750 K (EAM Olsson in\nFig. 9). For the new version of the potential, the mixing\nenthalpy is non-symmetric, following the DFT results by\nOlsson et al51using SQS structures. Therefore, the sol-\nubility of Fe in Cr is closer to the experimental values.\nThe vibrational entropy is lower in this case which in-\ncreases the critical temperature to a value close to 1100\nK (EAM Bonny in Fig. 9). The CDM is fitted to the en-\nthalpy of mixing of the alloy as given by exact muffin-tin\norbitals theory within the coherent potential approxima-\ntion (EMTO-CPA) calculations16. The maximum value\nis, in this case, higher than using SQS-PAW structures.\nThis effect, added to the fact that the vibrational entropy\nis lower, results in a critical temperature above the ex-\nperimental melting temperature (EAM Caro in Fig. 9).\nThese models have been fitted to 0 K enthalpy of mixing\ncurves. Both Fe and Cr undergo a magnetic transition\nat high temperatures (around 1043 K for Fe and 312 K\nfor Cr). This means that calculations beyond the mag-\nnetic critical temperature of the alloy are out of their\nscope. In a kinetic calculation in a rigid lattice62,63us-\ning these potentials directly, the vibrational contributi on\nto the entropy is not taken into account, which modifies\nthe solubility limits and therefore the chemical composi-\ntion in each phase. More specifically, the experimental\nsolubility limit at 773 K in the Fe rich side of the phase\ndiagram is about 14-15%21,25,64. On the other hand, the\noriginal 2BM potential (according to the latest reported\nvalues24) gives no miscibility gap at 773 K, while, if the\ndata shown in Ref. 25 is still valid, the solubility limit\nat the same temperature without the vibrational contri-\nbution is around 8%. The new version of the potential\ngives a solubility about 20% and it seems more suitable\nfor these kind of kinetic calculations because its vibra-\ntional entropy is lower. However, there is no information\nabout the values for the solubility without taking into\nconsideration the vibrational entropy. For the CDM po-\ntential the solubility at this temperature is already too\nlow. Its vibrational entropy contribution is small and,\ntherefore, the values for the solubility limits in a rigid\nlattice model will not be strongly modified. Although,\nthe variation will persist and care should be taken if ki-\nnetic simulations on a rigid lattice are to be performed.10\nThis discrepancy between the relaxed models versus the\nrigid lattice approximations will result in the wrong ther-\nmodynamic forces as taken into account in the kinetic\ncalculation. In the studies on precipitation kinetics pub-\nlished in Refs. 28, 52, 62, and 63 nucleation starts around\n10%, in disagreement with experiments. It is worth not-\ning that describing this concentration region accurately\nis important for industrial applications.\nOur model avoids such a drawback which makes it\nmore useful for kinetic calculations. In our model, the\npair interactions depend on the temperature taking in\nthis way into account the magnetic and vibrational con-\ntributions to the entropy. This approximation will not\nbe able to reproduce the magnetic transitions either\nfor the pure elements or the alloy. Even though the\nferromagnetic-paramagnetic transition in Fe is not lin-\near with the temperature and neither it is in Cr or the\nalloy, the simple model described is able to reproduce\nthe experimental phase diagram by adding just one ex-\ntra degree of freedom. This extra degree of freedom does\nnot affect the computational performance and makes it\nsuitable for large-scale calculations.\n 300 400 500 600 700 800 900 1000\n 0 0.05  0.1  0.15  0.2Temperature (K)\nCr atomic fractionThis study (MC 2nn)\ncalphad\ncluster expansion\nEAM Olsson (2BM)\nEAM Caro (CDM)\nEAM Bonny (2BM)\nFIG. 9. Solubility limits of Cr in iron as given by different\nmodels. The Calphad values are taken from Ref. 10. The\nsolubility limit for the CE and for the CDM were presented\nin Ref. 25. The values for both 2BM potentials have been\nobtained from Ref. 24.\nV. CONCLUSION\nWe propose in this article a rigid lattice model based\non concentration and temperature dependent pair inter-\nactions to describe the thermodynamics of Fe-Cr systemin the whole concentration range. It is fitted to both ab\ninitio calculations of the enthalpy of mixing at 0 K and\nto the experimental critical temperature. Only the cross\nterms of the atomic pair interactions depend on both the\nlocal concentration and the temperature, while the self\ninteraction terms are fitted to the respective cohesive en-\nergies. In order to check the sensitivity of energetic value s\nin FeCr alloys against DFT implementations with differ-\nent approximations, and to choose the most accurate val-\nues of the enthalpy of mixing, we have performed a set\nof first principle calculations. We carried out the calcu-\nlations from two different kind of approaches. The first\none was the norm-conserving pseudopotential approach\nas implemented in the efficient SIESTA-NC code and\nthe second the more robust projector-augmented wave\nas implemented in the PWSCF code. Both approaches\ngive the same qualitative trend, but different quantitative\nmixing energies at 0 K. The resulting models are similar,\nalthough the SIESTA values overestimate slightly the Cr\nsolubility in Fe.\nAlthough its simplicity and even though it does not\nexplicitly consider the magnetic degrees of freedom, this\napproach captures the main features of the Fe-Cr ther-\nmodynamics: thanks to the concentration dependence of\nthe pair interactions, it reproduces the transition betwee n\nthe ordering and demixing tendency, and the trend in\nthe short-range order parameter when the Cr content in-\ncreases. The magnetic and non-configurational entropic\ncontributions are taken into account by a linear temper-\nature dependence of the pair interactions. The resulting\nphase diagram is in very good qualitative and quantita-\ntive agreement with the experimental results. Finally,\nthe model remains simple enough to be used in Monte\nCarlo simulations of the solid solution decomposition ki-\nnetics (preliminary results can be found in Ref. 65).\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge E. Clouet for use-\nful discussions. This research has received partial fund-\ning from the European Atomic Energy Community’s 7th\nFramework Programme (FP7/2007-2011), under grant\nagreement number 212175 (GetMat project). Part of this\nwork was performed using HPC ressources from GENCI-\nCINES (Grant 2011-x2011096020). E. Martinez wants to\nthank the Spanish Ministry of Science and Innovation,\nsubprogram Juan de la Cierva and the Energy Frontier\nResearch Center, Center for Materials at Irradiation and\nMechanical Extremes at Los Alamos National Labora-\ntory (DOE-BES) for partial funding.\n∗Correspondent author. 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Soisson, Solid State Phenomena 172-174 , 1146 (2011)."    },    {        "title": "1703.06129v2.Rotation_and_Neoclassical_Ripple_Transport_in_ITER.pdf",        "content": "arXiv:1703.06129v2  [physics.plasm-ph]  24 Jul 2017Rotation and Neoclassical Ripple Transport in ITER\nE. J. Paul\nDepartment of Physics, University of Maryland, College Par k, MD 20742, USA∗\nM. Landreman\nInstitute for Research in Electronics and Applied Physics,\nUniversity of Maryland, College Park, MD 20742, USA\nF. M. Poli\nPrinceton Plasma Physics Laboratory, Princeton, NJ 08543, USA\nD. A. Spong\nOak Ridge National Laboratory, Oak Ridge, TN 37831, USA\nH. M. Smith\nMax-Planck-Institut f¨ ur Plasmaphysik, 17491 Greifswald , Germany\nW. Dorland\nDepartment of Physics, University of Maryland, College Par k, MD 20742, USA\n1Abstract\nNeoclassical transport in the presence of non-axisymmetri c magnetic fields causes a toroidal\ntorque known as neoclassical toroidal viscosity (NTV). The toroidal symmetry of ITER will be\nbroken by the finite number of toroidal field coils and by test b lanket modules (TBMs). The ad-\ndition of ferritic inserts (FIs) will decrease the magnitud e of the toroidal field ripple. 3D magnetic\nequilibria with toroidal field ripple and ferromagnetic str uctures are calculated for an ITER steady-\nstate scenario using the Variational Moments Equilibrium C ode (VMEC). Neoclassical transport\nquantities in the presence of these error fields are calculat ed using the Stellarator Fokker-Planck\nIterative Neoclassical Conservative Solver (SFINCS). The se calculations fully account for Er, flux\nsurface shaping, multiple species, magnitude of ripple, an d collisionality rather than applying ap-\nproximate analytic NTV formulae. As NTV is a complicated non linear function of Er, we study\nits behavior over a plausible range of Er. We estimate the toroidal flow, and hence Er, using a\nsemi-analytic turbulentintrinsicrotation modelandNUBE AM calculations of neutralbeamtorque.\nThe NTV from the |n|= 18 ripple dominates that from lower nperturbations of the TBMs. With\nthe inclusion of FIs, the magnitude of NTV torque is reduced b y about 75% near the edge. We\npresent comparisons of several models of tangential magnet ic drifts, finding appreciable differences\nonly for superbanana-plateau transport at small Er. We find the scaling of calculated NTV torque\nwith ripple magnitude to indicate that ripple-trapping may be a significant mechanism for NTV\nin ITER. The computed NTV torque without ferritic component s is comparable in magnitude to\nthe NBI and intrinsic turbulent torques and will likely damp rotation, but the NTV torque is\nsignificantly reduced by the planned ferritic inserts.\n∗ejpaul@umd.edu\n2I. INTRODUCTION\nToroidal rotation is critical to the experimental control of tokam aks: the magnitude of\nrotation is known to affect resistive wall modes [1, 2], while rotation sh ear can decrease\nmicroinstabilities and promote the formation of transport barriers [3, 4]. As some ITER\nscenarios will be above the no-wall stability limit [5], it is important to und erstand the\nsources and sinks of angular momentum for stabilization of externa l kink modes. One such\nsink (or possible source) is the toroidal torque caused by 3D non-r esonant error fields, known\nasneoclassicaltoroidalviscosity(NTV).DedicatedNTVexperimen tshavebeenconductedin\nthe Mega Amp Spherical Tokamak (MAST) [6], the Joint European Tok amak (JET) [7, 8],\nAlcator C-MOD [9], DIII-D [10, 11], JT-60U [12], and the National Sphe rical Tokamak\nExperiment (NSTX) [13].\nIn addition to the ripple due to the finite number (18) of toroidal field (TF) coils, the\nmagnetic field in ITER will be perturbed by ferromagnetic component s including ferritic\ninserts (FIs) and test blanket modules (TBMs). TBMs will be installed in three equatorial\nports to test tritium breeding and extraction of heat from the blan ket. The structural\nmaterial for these modules is ferritic steel and will produce addition al error fields in response\nto the background field. The TBMs will be installed during the H/He pha se in order to\ntest their performance in addition to their possible effects on confin ement and transport\n[14]. It is important to understand their effect on rotation during th e early phases of ITER.\nExperiments at DIII-D using mock-ups of TBMs found a reduction in toroidal rotation by\nas much as 60% due to an n= 1 locked mode [15]. Here nis the toroidal mode number.\nFurther experiments showed compensation by n= 1 control coils may enable access to low\nNBI torque (1.1 Nm) regimes relevant to ITER without rotation collap se [16]. In addition to\nTBMs, ferritic steel plates (FIs) will be installed in each of the TF coil sections in order to\nmitigate energetic particle loss due to TF ripple [17]. Experiments includ ing FIs on JT-60U\n[18] and JFT-2M [19] have found a reduction in counter-current ro tation with the addition\nof FIs. As FIs will decrease TF ripple, they may decrease the NTV in I TER.\nWhile the bounce-averaged radial drift vanishes in a tokamak, trap ped particles may\nwander offtheflux surfaceinthepresence ofnon-axisymmetric er ror fields. Particles trapped\npoloidally (bananas) can drift radially as the parallel adiabatic invarian t,J||=/contintegraltext\ndlv||,\n3becomes a function of toroidal angle in broken symmetry. Here v||is the velocity coordinate\nparallel to b=B/Band integration is taken along the field between bounce points. If\nlocal ripple wells exist along a field line and the collisionality is small enough t hat helically\ntrapped particles can complete their collisionless orbits, these trap ped particles may grad- B\ndrift away from the flux surface [20]. The TF ripple in ITER causes loca l wells along the\nfield line, corresponding to α=ǫ/(qnδB)<1 [20]. Here ǫ=r/Ris the inverse aspect ratio,\nris the minor radius, Ris the major radius, qis the safety factor, and δBis a measure of the\namplitude of the ripple. Because of ITER’s low collisionality, ν∗≪(δB/ǫ)3/2, ripple-trapped\nparticles can complete their collisionless orbits [21]. Here the normalize d collision frequency\nisν∗=qRvti/(νiiǫ3/2) where the ion-ion collision frequency is νii. The ion thermal velocity\nisvti=/radicalbig\n2Ti/miwhereTiis the ion temperature and miis the ion mass. Therefore,\nboth ripple trapping and banana diffusion should be considered for NT V in ITER. For a\ngeneral electric field, the neoclassical electron and ion fluxes are n ot necessarily identical\nin broken symmetry. The resulting radial current induces a J×Btorque which is often\ncounter-current.\nAnalytic expressions for neoclassical fluxes in several rippled toka mak regimes have been\nderived by various authors, making assumptions about the magnitu de of the perturbing\nfield, electric field, magnetic geometry, collisionality, and the collision o perator. Multiple\nregimes are typically needed to describe all radial positions, classes of particles, and helicities\nof the magnetic field for a single discharge. When collisions set the rad ial step size of\ntrapped particles, the transport scales as 1 /νwhereνis the collision frequency. The 1 /ν\nregime can be relevant for both ripple trapped and banana particles with small radial electric\nfield. With a non-zero radial electric field, transport from the collisio nal trapped-passing\nboundary layer leads to fluxes that scale as√ν. When the collisionality is sufficiently\nlow, the collisionless detrapping/trapping layer becomes significant, where fluxes scale as\nν. Here bananas can become passing particles due to the variation of Bmaxalong their\ndrift trajectories [22], and ripple trapped particles can experience collisionless detrapping\nfrom ripple wells to become bananas [23, 24]. If the collisionality is small c ompared with\nthe typical toroidal precession frequency of trapped particles, the resonant velocity space\nlayer where the bounce-averaged toroidal drift vanishes can dom inate the neoclassical fluxes,\nleadingtosuperbanana-plateautransport[25]. Inthepresenceo fastrongradialelectricfield,\n4the resonance between the parallel bounce motion and drift motion of trapped particles can\nalsoresult inenhancedtransport, knownasthebounce-harmonic resonance [26, 27]. The1 /ν\nand√νstellarator regimes for helically-trapped particles have been formu lated by Galeev\nand Sagdeev [28], Ho and Kulsrud [29], and Frieman [30]. These results w ere generalized to\nrippled tokamaks in the1 /νregime by Stringer [20], Connor andHastie [31], andYushmanov\n[32]. Kadomtsev and Pogutse [33] and Stringer [20] presented the s caling of ripple diffusion\nincluding trapping/detrapping by poloidal rotation, where fluxes sc ale asν. This regime is\nlikely to be applicable for ITER’s low collisionality and strong radial electr ic field. Banana\ndiffusion in the 1 /νregime has been evaluated by Davidson [34], Linkser and Boozer [26],\nand Tsang [35]. The corresponding νtransport was studied by Tsang [35] and Linsker\nand Boozer [26]. Shaing emphasized the relationship between nonaxis ymmetric neoclassical\ntransport andtoroidal viscosity [36]. The theory for NTV torque d ue to banana diffusion has\nbeen formulated in the 1 /ν[21],ν−√ν[37],ν[22], and superbanana-plateau [25] regimes\nin addition to an approximate analytic formula which connects these r egimes [38].\nThe calculation of NTV torque requires two steps: (i) determine the equilibrium magnetic\nfield in the presence of ripple and (ii) solve a drift kinetic equation (DKE ) with the magne-\ntohydrodynamic (MHD) equilibrium or apply reduced analytic formulae . The first step can\nbe performed using various levels of approximation. The simplest met hod is to superimpose\n3D ripple vacuum fields on an axisymmetric equilibrium, ignoring the plasm a response. A\nsecond level of approximation is to use a linearized 3D equilibrium code s uch as the Ideal\nPerturbed Equilibrium Code (IPEC) [27] or linear M3D-C1 [39]. A third lev el of approxima-\ntion is to solve nonlinear MHD force balance using a code such as the Va riational Moments\nEquilibrium Code (VMEC) [40] or M3D-C1 [41] run in nonlinear mode. In th is paper we\nuse free-boundary VMEC to find the MHD equilibrium in the presence o f TF ripple, FIs,\nand TBMs.\nMany previous NTV calculations [6, 13, 27, 42] have been performed using reduced ana-\nlytic models with severe approximations. Solutions of the bounce-av eraged kinetic equation\nhave been found to agree with Shaing’s analytic theory except in the transition between\nregimes [43]. However, the standard bounce-averaged kinetic equ ation does not include\ncontributions from bounce and transit resonances. Discrepancie s have been found between\nnumerical evaluation of NTV using the Monte Carlo neoclassical solve r FORTEC-3D and\n5analytic formulae for the 1 /νand superbanana-plateau regimes [44, 45]. NTV calculations\nwith quasilinear NEO-2 differ from Shaing’s connected formulae [38], es pecially in the edge\nwhere the large aspect ratio assumption breaks down [46]. Rather t han applying such re-\nduced models, in this paper a DKE is solved using the Stellarator Fokke r-Planck Iterative\nNeoclassical Conservative Solver (SFINCS) [47] to calculate neocla ssical particle and heat\nfluxes for an ITER steady-state scenario. The SFINCS code does not exploit any expansions\nin collisionality, size of perturbing field, or magnitude of the radial elec tric field (beyond\nthe assumption of small Mach number). It also allows for realistic exp erimental magnetic\ngeometry rather than using simplified flux surface shapes. All trap ped particle effects in-\ncluding ripple-trapping [20], banana diffusion [26], and bounce-resona nce [26] are accounted\nfor in these calculations. The DKE solved by SFINCS ensures intrinsic ambipolarity for\naxisymmetric or quasisymmetric flux surfaces in the presence of a r adial electric field while\nthis property is not satisfied by other codes such as DKES [48, 49]. T his prevents spurious\nNTV torque density, which is proportional to the radial current. A s SFINCS makes no as-\nsumption about the size of ripple, it can account for non-quasilinear transport, such as ripple\ntrapping, rather than assuming that the Fourier modes of the ripp le can be decoupled. For\nTF ripple, the deviation from the quasilinear assumption has been fou nd to be significant in\nbenchmarks between SFINCS and NEO-2 [46].\nIn addition to NTV, neutral beams will provide an angular momentum s ource for ITER.\nAs NBI torque scales as P/E1/2for input power Pand particle energy E, ITER’s neutral\nbeams, with E= 1 MeV and P= 33 MW, will provide less momentum than in other\ntokamaks such as JET, with E= 125 keV for P= 34 MW [50]. NBI-driven rotation\nwill also be smaller in ITER because of its relatively large moment of inert ia, withR= 6\nm compared to 3 m for JET. However, spontaneous rotation may be significant in ITER.\nTurbulence can drive significant flows in the absence of external mo mentum injection, known\nas intrinsic or spontaneous rotation. This can be understood as a t urbulent redistribution\nof toroidal angular momentum to produce large directed flows. For perturbed tokamaks this\nmust be in the approximate symmetry direction. According to gyrok inetic orderings and\ninter-machine comparisons by Parra et al[51], intrinsic toroidal rotation is expected to scale\nasVζ∼Ti/IpwhereIpis the plasma current, and core rotations may be on the order of\n100 km/s (ion sonic Mach number Mi≈8%) in ITER. Scalings with βN=βTaBT/IPby\n6Riceet al[52] predict rotations of a slightly larger scale, Vζ≈400 km/s (Mi≈30%). Here\nβT= 2µ0p/B2\nT,BTis the toroidal magnetic field in tesla, ais the minor radius at the edge in\nmeters, and pis the plasma pressure. Co-current toroidal rotation appears to be a common\nfeature of H-mode plasmas and has been observed in electron cyclo tron (EC) [53], ohmic\n[53], and ion cyclotron range of frequencies (ICRF) [54] heated plas mas. Gyrokinetic GS2\nsimulations with H-mode parameters find an inward intrinsic momentum flux, corresponding\nto a rotation profile peaked in the core toward the co-current dire ction [55]. In an up-down\nsymmetric tokamak, the radial intrinsic angular momentum flux can b e shown to vanish to\nlowest order in ρ∗=ρi/a, but neoclassical departures from an equilibrium Maxwellian can\nbreak this symmetry and cause non-zero rotation in the absence o f input momentum [56].\nHereρi=vtimi/ZieBis the gyroradius where Ziis the ion species charge.\nIn section II we present the ITER steady state scenario and free boundary MHD equi-\nlibrium in the presence of field ripple. In section III we estimate rotat ion driven by NBI\nand turbulence. This flow velocity is related to Erin section IV. The NTV torque due to\nTF ripple, TBMs, and FIs is evaluated in section V. In section VI the sc aling of transport\ncalculated with SFINCS with ripple magnitude is compared with that pre dicted by NTV\ntheory, and in section VII neoclassical heat fluxes in the presence of ripple are presented.\nIn section VIII, we assess several tangential magnetic drift mod els on the transport for this\nITER scenario and a radial torque profile is presented. In section I X we summarize the\nresults and conclude.\nII. ITER STEADY STATE SCENARIO AND FREE BOUNDARY EQUILIBRIUM\nCALCULATIONS\nWe consider an advanced ITER steady state scenario with significan t bootstrap current\nand reversed magnetic shear [57]. The input power includes 33 MW NBI , 20 MW EC, and\n20 MW lower hybrid (LH) heating for a fusion gain of Q= 5. This 9 MA non-inductive\nscenario is achieved with operation close to the Greenwald density limit . The discharge\nwas simulated using the Tokamak Simulation Code (TSC) in the IPS [58] f ramework for\nthe calculation of the free-boundary equilibrium and the RF calculatio ns, and TRANSP\nfor calculations of the NBI heating and current and torque. The dis charge was simulated\n7FIG. 1: Radial profiles of temperature, density, safety factor, total plasma current, and\nbootstrap current for the ITER steady state scenario [57]. Black dashed lines indicate the\nradial locations that will be considered for neoclassical calculations .\nusing the Tokamak Simulation Code (TSC) [59] and TRANSP [60] using a c urrent diffusive\nballooning mode (CDBM) [61, 62] transport model and EPED1 [63] ped estal modeling. The\nNBI source is modeled using NUBEAM [64, 65] with 1 MeV particles. The b eams aresteered\nwith one on-axis and one off-axis, which avoids heating on the midplane wall gap and excess\nheat deposition above or below the midplane. Further details of the s teady state scenario\nmodeling can be found in table 1 of [57].\nThe density ( n), temperature ( T), safety factor ( q), total plasma current, and bootstrap\ncurrent profiles are shown in figure 1. Neoclassical transport will b e analyzed in detail at\n8the radial locations indicated by dashed horizontal lines ( r/a= 0.5,0.7,0.9). Throughout\nwe will use the radial coordinate r/a∝√ΨTwhere Ψ Tis the toroidal flux.\nThe VMEC free boundary [40] magnetic equilibrium was computed using the TRANSP\nprofiles along with filamentary models of the toroidal field (TF), poloid al field (PF), and\ncentral solenoid (CS) coils and their corresponding currents. The vacuum fields produced\nby the three TBMs and the FIs have been modeled using FEMAG [66]. Th e equilibrium\nis computed for four geometries: (i) including only the TF ripple, (ii) inc luding TF ripple,\nTBMs, and FIs, (iii) TF ripple and FIs, and (iv) axisymmetric geometry . We define the\nmagnitude of the magnetic field ripple to be\nδB= (Bmax−Bmin)/(Bmax+Bmin), (1)\nwhere the maximum and minimum are evaluated at fixed radius and VMEC poloidal angle\nθ. In figure 2, δBis plotted on the poloidal plane for the three rippled VMEC equilibria. A\nfourth case is also shown in which the component of Bwith|n|= 18 was removed from the\ngeometrywithTBMsandFIsinordertoconsider the |n|<18ripplefromtheTBMs(bottom\nright). When only TF ripple is present, significant ripple persists over the entire outboard\nside, while in the configurations with FIsthe ripple is much more localized inθ. When TBMs\narepresent, theripple is higher in magnitude near the outboardmidp lane (δB≈1.4%), while\nin the other magnetic configurations δB≈1% near the outboard midplane. For comparison,\nthe TF ripple during standard operations is 0 .08% in JET [8] and 0 .6% in ASDEX Upgrade\n[46]. In JT-60U the amplitude of TF ripple is reduced from δB≈1.7% toδB≈1% by FIs\n[18].\nIn figure 3, the magnitude of Bis plotted as a function of toroidal angle ζatθ= 0 and\nθ=π/4. Away from the midplane ( θ=π/4) the FIs greatly decrease the magnitude of the\nTF ripple. Near the midplane the FIs do not decrease the magnitude o f the toroidal ripple as\nstrongly, as the number of steel plates is reduced near the midplan e [66]. The ferromagnetic\nsteel of the TBMs concentrates magnetic flux and locally decrease sBin the plasma near\ntheir location. This causes enhancement of δBnearθ= 0.\n9FIG. 2: Magnetic field ripple, δB= (Bmax−Bmin)/(Bmax+Bmin), is plotted on the\npoloidal plane for VMEC free boundary equilibria including (i) only TF ripp le (top left),\n(ii) TF ripple, TBMs, and FIs (top right), (iii) TF ripple and FIs (bottom left), and (iv)\nwith TBMs only (bottom right). FIs decrease the poloidal extent of the ripple, while\nTBMs add an additional ripple near the outboard midplane.\n10FIG. 3: The magnitude of Bas a function of toroidal angle ( ζ) atr/a= 1,θ= 0 andπ/4.\nVertical dashed lines indicate the toroidal locations of the TBM port s. The mitigating\neffect of the FIs is stronger away from the midplane, where an incre ased number of steel\nplates are inserted. The TBMs add an additional ripple near their loca tions atθ= 0.\n11III. ESTIMATING TOROIDAL ROTATION\nIn order to predict the ripple transport in ITER, the radial electric field,Er=−Φ′(r),\nmust be estimated, as particle and heat fluxes are nonlinear functio ns ofEr. This is equiv-\nalent to predicting the parallel flow velocity, V||, which scales monotonically with Er. As\nwe simply wish to determine a plausible value of Er, the difference between V||andVζ, the\ntoroidal flow, will be unimportant for our estimates. We define Vζin terms of the toroidal\nrotation frequency, Vζ= ΩζR, where Ω ζ≈Ωζ(r). AsIPand the toroidal magnetic field are\nboth directed clockwise when viewed from above, VζandV||will point in the same direction.\nHere we use the convention that positive Vζcorresponds to co-current rotation. For this\nrotation calculation, angular momentum transport due to neutral beams and turbulence will\nbe considered. There is an additional torque caused by the radial c urrent of orbit-lost alphas\n[67], but it will be negligible ( ≈0.006 Nm/m3). The following time-independent momentum\nbalance equation is considered in determining Ω ζ(r),\n∇·Πturb\nζ(Ωζ)+∇·ΠNC\nζ(Ωζ) =τNBI, (2)\nwhere Πturb\nζand ΠNC\nζare the toroidal angular momentum flux densities due to turbulent a nd\nneoclassical transport and τNBIis the NBI torque density. For this paper the feedback of\nΠNC\nζon Ωζwill not be calculated. Determining the change in rotation due to NTV w ould\nrequire iteratively solving this equation for Ω ζ, as ΠNC\nζis a nonlinear function of Ω ζ.\nThe quantity Πturb\nζconsists of a diffusive term as well as a term independent of Ω ζwhich\naccounts for turbulent intrinsic rotation,\nΠturb\nζ=−miniχζ∝angb∇acketleftR2∝angb∇acket∇ight∂Ωζ\n∂r+Πint. (3)\nFor simplicity, an angular momentum pinch, Pζ, will not be considered for this analysis. As\nRPζ/χζ≈2, there would be a factor of 2 difference in rotation peaking at the c ore due to the\nturbulent momentum source at the edge [68]. Here χζis the toroidal ion angular momentum\ndiffusivity. The flux surface average is denoted by ∝angb∇acketleft...∝angb∇acket∇ight,\n∝angb∇acketleft...∝angb∇acket∇ight=1\nV′/integraldisplay2π\n0dθ/integraldisplay2π\n0dζ√g(...) (4)\nV′=/integraldisplay2π\n0dθ/integraldisplay2π\n0dζ√g, (5)\n12where√gis the Jacobian. Ignoring NTV torque, we will solve the following angula r mo-\nmentum balance equation,\n−mi1\nV′∂\n∂r/parenleftbigg\nV′niχζ∝angb∇acketleftR2∝angb∇acket∇ight∂Ωζ\n∂r/parenrightbigg\n=−1\nV′∂\n∂r/parenleftbig\nV′Πint/parenrightbig\n+τNBI. (6)\nEquation 6 is a linear inhomogeneous equation for Ω ζ, as the right hand side is independent\nof Ωζ. We can therefore solve for the rotation due to each of the sourc e terms individually\nand add the results to obtain the rotation due to both NBI torque a nd turbulent intrinsic\ntorque.\nThe NBI-driven rotation profile is evolved by TRANSP assuming χζ=χi, the ion heat\ndiffusivity. The total beam torque density, τNBI, is calculated by NUBEAM including colli-\nsional,J×B, thermalization, and recombination torques. The following momentu m balance\nequation is solved to compute Ω ζdriven by NBI,\nτNBI=−1\nV′∂\n∂r/parenleftbigg\nV′miniχi∝angb∇acketleftR2∝angb∇acket∇ight∂Ωζ\n∂r/parenrightbigg\n. (7)\nWe consider a semi-analytic intrinsic rotation model to determine the turbulent-driven\nrotation [69],\nΩζ(r) =−/integraldisplaya\nrvtiρ∗,θ\n2PrL2\nT/tildewideΠ(ν∗)dr′, (8)\nwhereρ∗,θ=vtimi/(ZieBθ∝angb∇acketleftR∝angb∇acket∇ight) is the poloidal normalized gyroradius, Bθ=B·∂r/∂θ,\nandLT=−/parenleftbig\n∂lnTi/∂r/parenrightbig−1is the temperature gradient scale length. The Prandtl number\nPr=χζ/χiis again taken to be 1. Equation 8 is obtained assuming that Π intbalances\nturbulent momentum diffusion in steady state, Π int=miniχζ∝angb∇acketleftR2∝angb∇acket∇ight∂Ωζ/∂r. This model\nconsiders the intrinsic torque driven by the neoclassical diamagnet ic flows, such that Ω ζ∼\nρ∗,θvti/∝angb∇acketleftR∝angb∇acket∇ightand ΩζΠint/Qi∼ρ∗,θwhereQi=niTiχiLTis the turbulent energy flux. We\nalso take Ω ζ(a) = 0. The quantity /tildewideΠ(ν∗) is an order unity function which characterizes\nthe collisionality dependence of rotation reversals, determined fro m gyrokinetic turbulence\nsimulations [56],\n/tildewideΠ(ν∗) =(ν∗/νc−1)\n1+(ν∗/νc), (9)\nwhereνc= 1.7. Because of ITER’s low collisionality, we do not expect a rotation rev ersal,\nwhich is correlated with transitioning between the banana and platea u regimes. Equation 8\nwas integrated using profiles for the ITER steady state scenario.\n13The flux-surface averaged toroidal rotation, ∝angb∇acketleftVζ∝angb∇acket∇ight= Ωζ(r)∝angb∇acketleftR∝angb∇acket∇ight, predicted by these models\nis shown in figure 4. NBI torque contributes to significant rotation a tr/a/lessorsimilar0.4 where the\ntorque density also peaks (see figure 13), while turbulent torque p roduces rotation in the\npedestal due to the L−2\nTscaling of our model. The intrinsic rotation calculated is comparable\nto that predicted from theoretical scaling arguments by Parra et al[51],Vζ≈100 km/s. At\nthe radii that will be considered for neoclassical calculations (indica ted by dashed vertical\nlines), intrinsic turbulent rotation may dominate over that due to NB I. However, we em-\nphasize that it is an estimate based on scaling arguments, as much un certainty is inherent\nin predicting turbulent rotation. The volume-averaged toroidal ro tation due to both NBI\nand turbulent torques, 113 km/s, is slightly larger than that predic ted from dimensionless\nparameter scans on DIII-D, 87 km/s [70].\nFIG. 4: Flux-surface averaged toroidal rotation, ∝angb∇acketleftVζ∝angb∇acket∇ight, due to turbulence and NBI (top) is\nshown along with corresponding Alfv` en Mach number (bottom, solid ), and ion sonic Mach\nnumber (bottom, bulleted). The intrinsic rotation calculation uses a semi-analytic model of\nturbulent momentum redistribution [69]. The NBI rotation is calculate d from turbulent\ndiffusion of NBI torque using NUBEAM and TRANSP [57]. Dashed vertica l lines indicate\nthe radial positions where SFINCS calculations are performed.\n14Forstabilization of theresistive wall mode(RWM) in ITER, it hasbeen e stimated [5] that\na critical central Mach number, MA= Ωζ(0)/ωA/greaterorsimilar5%, must be achieved given a peaked\nrotation profile. Here ωA=B/(∝angb∇acketleftR∝angb∇acket∇ight√µ0mini) is the Alfv` en frequency. With a central\nrotation frequency Ω ζ(0)≈2%ωAas shown in figure 4, it may be difficult to suppress the\nRWM in ITER with rotation alone. As this calculation does not take into a ccount NTV\ntorque,MAis likely to be smaller than what is shown. Additionally, the TBM are known to\nincrease the critical rotation frequency as they have a much shor ter resistive time scale than\nthe wall [5]. More recent analysis has shown that even above such a c ritical rotation value,\nthe plasma can become unstable due to resonances between the dr ift frequency and bounce\nfrequency [71, 72].\nIV. RELATIONSHIP BETWEEN ErANDV||\nNeoclassical theory predicts a specific linear-plus-offset relations hip between V||andEr,\nbut it does not predict a particular value for either V||orErin a tokamak. Neoclassical\ncalculationsof V||aremadeinordertodetermine an Erprofileconsistent withourestimateof\nVζ≈ ∝angb∇acketleftV||B∝angb∇acket∇ight/∝angb∇acketleftB2∝angb∇acket∇ight1/2made in section III. The parallel flow velocity for species ais computed\nfrom the neoclassical distribution function,\nVa\n||=/parenleftbigg1\nna/parenrightbigg/integraldisplay\nd3vv||fa, (10)\nwhich we calculate with the SFINCS [47] code. SFINCS is used to solve a radially-local DKE\nfor the gyro-averaged distribution function, fa1, on a single flux surface including coupling\nbetween species.\n(v||b+vE+vma)·(∇fa1)−C(fa1) =−vma·∇ψ/parenleftbigg∂fa0\n∂ψ/parenrightbigg\n+Zaev||B∝angb∇acketleftE||B∝angb∇acket∇ight\nTa∝angb∇acketleftB2∝angb∇acket∇ightfa0(11)\nHereaindicates species, fa0is an equilibrium Maxwellian, ψ= ΨT/2π,Zindicates charge,\nandCisthelinearizedFokker-Planckcollisionoperator. Gradientsareper formedatconstant\nW=mav2/2+ZaeΦ andµ=v2\n⊥/(2B). TheE×Bdrift is\nvE=1\nB2B×∇Φ (12)\nand the radial magnetic drift is\nvma·∇ψ=1\nΩaB/parenleftBigg\nv2\n||+v2\n⊥\n2/parenrightBigg\nb×∇B·∇ψ, (13)\n15wherev⊥is the velocity coordinate perpendicular to b. The quantity Ω a=ZaeB/mais the\ngyrofrequency. Transport quantities have been calculated using the steady state scenario ion\nand electron profiles and VMEC geometry. We consider a three spec ies plasma (D, T, and\nelectrons), and we assume that nD=nT=ne/2. The second term on the right hand side of\nequation 11 proportional to E||is negligible for this non-inductive scenario with loop voltage\n≈10−4V. For the calculations presented in sections IV, V, VI, and VII, vma·∇fa1is not\nincluded. The effect of keeping this term is shown to be small in section VIII.\nThe relationship between Erand∝angb∇acketleftV||B∝angb∇acket∇ight/∝angb∇acketleftB2∝angb∇acket∇ight1/2for electrons and ions at r/a= 0.9 is\nshown in figure 5. Only one curve is shown for each species as the add ition of ripple fields\ndoes not change the dependence of V||onErsignificantly ( /lessorsimilar5%). While radial transport of\nheat and particles changes substantially in the presence of small rip ple fields (see sections V,\nVI, and VII), the parallel flow is much less sensitive to the perturbin g field. Note that the\nparallel flow is non-zero in axisymmetry while the radial current vanis hes without symmetry-\nbreaking.\nFIG. 5: SFINCS calculation of the flux surface averaged parallel flo w,∝angb∇acketleftBV||∝angb∇acket∇ight/∝angb∇acketleftB2∝angb∇acket∇ight1/2, at\nr/a= 0.9 for ions and electrons. The addition of ripple does not change the t okamak\nneoclassical relationship between ErandV||by a discernible amount on this scale although\nthe radial particle fluxes, Γ ψ, are sensitive to the perturbing field.\n16In a tokamak we can write ∝angb∇acketleftVa\n||B∝angb∇acket∇ightin terms of a dimensionless parallel flow coefficient, k||,\nand thermodynamic drives,\n∝angb∇acketleftVa\n||B∝angb∇acket∇ight=−G\nZaena/bracketleftbigg1\nnd(nT)\ndψP+ZaedΦ\ndψP−k||dT\ndψP/bracketrightbigg\n, (14)\nwhere 2πψPis the poloidal flux, G(ψ) =RBζ, andBζ=B·∂r/∂ζ. The low collisionality,\nlarge aspect ratio limit [73, 74] k||≈1.17 is often assumed in NTV theory [75, 76] in relating\nanalytic expressions of torque density to toroidal rotation frequ ency. The value of the ion k||\ncalculated by SFINCS for ITER parameters varies between 0.5 near the edge and 0.9 near\nthe core. The bootstrap current computed with SFINCS,\nJBS=/summationdisplay\nanaZae∝angb∇acketleftVa\n||B∝angb∇acket∇ight, (15)\nis consistent with that computed by TRANSP within 10% for r/a≥0.5. Though there is\nsome discrepancy in the core, they have the same qualitative behav ior and similar maxima.\nThe bootstrap current in TRANSP is computed using a Sauter model [77], an analytic fit\nto numerical solutions of the Fokker-Planck equation.\nV. TORQUE CALCULATION\nThe NTV torque density, τNTV, is calculated from radial particle fluxes, Γ ψ,\nΓψ,a=/angbracketleftbigg/integraldisplay\nd3v(vma·∇ψ)fa/angbracketrightbigg\n, (16)\nusing the flux-force relation,\nτNTV=−Bθ√g/summationdisplay\naZaeΓψ,a, (17)\nwhereBθ=B· ∇θand the summation is performed over species. This expression re-\nlates radial particle transport to a toroidal angular momentum sou rce caused by the non-\naxisymmetric field. This relationship can be derived from action-angle coordinates [78],\nneoclassical moment equations [79], or from the definition of the drif t-driven flux [80].\nThe calculation of τNTVfor three geometries at r/a= 0.9 is shown in figure 6. Here\npositive corresponds to the co-current direction. The numerically computed NTV torque\nis found to vanish in axisymmetric geometry, as expected. Overall, t he magnitude of τNTV\n17with only TF ripple is larger than that with the addition of both the FIs a nd the TBMs. In\nfigure 8 we show that the |n|<18 TBM ripple produces much less torque than the |n|= 18\nripple, so the decrease in τNTVmagnitude with both FIs and TBMs can be attributed to\nthe decrease in ripple in the presence of FIs. As will be discussed in se ction VI, neoclassical\nripple transport in most regimes scales positively with δB. The addition of FIs significantly\ndecreasesthemagnitudeof δBacrossmostoftheoutboardside, andasaresultthemagnitude\nofτNTVis reduced. The dashed vertical line indicates the value of ∝angb∇acketleftV||B∝angb∇acket∇ight/∝angb∇acketleftB2∝angb∇acket∇ight1/2andEr\npredicted from the intrinsic and NBI rotation model. At this value of Erthe presence of\nferritic components decreases the magnitude of the torque dens ity by about 75%.\nThe circle indicates the offset rotation at the ambipolar Er. If no other angular mo-\nmentum source were present in the system, τNTVwould drive the plasma to rotate at this\nvelocity. Although τNTVdiffers significantly between the two geometries they have similar\noffset rotation velocities, Vζ= -10 km/s with TF ripple only and -6 km/s with TBMs and\nFIs. Note that for Ergreater than this ambipolar value, τNTVis counter-current while neu-\ntral beams and turbulence drive rotation in the co-current direct ion, soτNTVis a damping\ntorque. The NTV torque due to TF ripple only is larger in magnitude tha nτNBIandτturb\nwhile that with TBMs and FIs is of similar magnitude (see figure 13). The refore, NTV\ntorque may be key in determining the edge rotation in ITER.\nThe magnitude of τNTVpeaks atEr= 0 where 1 /νtransport becomes dominant. Al-\nthoughν∗is sufficiently small such that the superbanana-plateau regime beco mes relevant,\nthe physics of superbanana formation is not accounted for in thes e SFINCS calculations\nwhich do not include vm·∇f1. Superbanana-plateau transport will be considered in section\nVIII. Atr/a= 0.9, the 1/νregime applies for |Er|/lessorsimilar0.2 kV/m where the effective collision\nfrequency of trapped particles is larger than the E×Bprecession frequency. The peak at\nsmall|Er|also corresponds to the region of 1 /νtransport of particles trapped in local ripple\nwells. Much NTV literature is based on banana diffusion and ripple trapp ing in the 1 /ν\nregime [20, 81], which is not applicable for the range of Erpredicted for ITER. For the\nrange of applicable Er, bounce-harmonic resonance may occur. The l= 1,n= 18 resonance\ncondition,ωb−n(ωE+ωB) = 0, will be satisfied for v||≈vtiatEr≈7 kV/m. Here ωbis\nthe bounce frequency, ωEis theE×Bprecession frequency, and ωBis the toroidal magnetic\ndrift precession [27]. Note that here vm·∇f1is not included in the kinetic equation ( ωB= 0),\n18but the physics of the bounce harmonic resonance between ωEandωbis still accounted for\nin our calculation. However, we see no evidence of enhanced τNTVnear thisErthat would\nbe indicative of a bounce-harmonic resonance.\nNTV torque is often expressed in terms of a toroidal damping frequ ency,νζ,\nτNTV=−νζ∝angb∇acketleftR2∝angb∇acket∇ightmn(Ωζ−Ωζ,offset), (18)\nwhere Ω ζ,offsetis the offset rotation frequency. We note that τNTVdoes appear to scale\nlinearlywith Er(andthus Ω ζ)for|Er|/greaterorsimilar30kV/m. However, τNTVisacomplicatednonlinear\nfunction of Ω ζfor|Er|/lessorsimilar30 kV/m at the transition between collision-limited 1 /νtransport\nandν−√νtransport, so equation 18 is not a very useful representation in t his context.\nFIG. 6: SFINCS calculation of NTV torque density as a function of Erand ion\n∝angb∇acketleftV||B∝angb∇acket∇ight/∝angb∇acketleftB2∝angb∇acket∇ight1/2atr/a= 0.9 is shown for 3 VMEC geometries: (i) axisymmetric (blue\ndashed), (ii) with TF ripple only (orange dash-dot), and (iii) TF ripple w ith FIs and TBMs\n(green solid). The vertical dashed line indicates the estimate of ErandVζ≈ ∝angb∇acketleftV||B∝angb∇acket∇ight/∝angb∇acketleftB2∝angb∇acket∇ight1/2\nbased on the intrinsic and NBI rotation model. The circle denotes the offset rotation at\nV||≈ −10 km/s. The magnitude of τNTVat this radius is of similar magnitude to the NBI\nand turbulent torques but is opposite in direction (see figure 13).\nIn figure 7 we present τNTVatr/a= 0.9 due to the electron and ion radial current in the\n19presence of TF ripple only (left) and TF ripple with ferromagnetic com ponents (right). The\nErcorresponding to the offset rotation frequency for the electron s is positive while that of\nthe ions is negative. At the predicted Er,τNTVdue to the electron particle flux is positive\nwhilethatduetoionparticlefluxisnegative. At allradiallocationsthe electroncontribution\ntoτNTVis less than 10% of the total torque density.\nFIG. 7: Total (blue dashed), electron (yellow dash-dot), and ion ( green solid)\ncontributions to NTV torque density at r/a= 0.9 for TF ripple only geometry (left) and\nTF ripple with ferromagnetic components (right). The dashed vert ical line indicates the Er\npredicted by the intrinsic and NBI rotation model. The electrons hav e a co-current\nneoclassical offset rotation and contribute a small co-current NT V torque density at the Er\npredicted by the rotation model.\nIn order to decouple the influence of the FI ripple and the TBM ripple, τNTVatr/a= 0.9\nis calculated for toroidal modes (i) |n| ≤18, (ii)|n|= 18, and (iii) |n|<18, shown in figure\n8. For|n| ≤18 and|n|= 18, VMEC free boundary equilibria were computed including these\ntoroidal modes. For |n|<18, the SFINCS calculation was performed including the desired\nnfrom the VMEC fields. Here Bis decomposed as,\nB=/summationdisplay\nm,nbc\nmncos(mθ−nζ)+bs\nmnsin(mθ−nζ), (19)\n20whereθandζare VMEC angles. The covariant and contravariant components of Balong\nwith their partial derivatives and√gare similarly decomposed such that the DKE can be\nsolved for the desired toroidal modes.\nThe TBM produces a wide spectrum of toroidal perturbations, inclu ding|n|= 1 and\n|n|= 18. While the FIs decrease the magnitude of the |n|= 18 ripple, the TBM contributes\nmost strongly to low mode numbers. As SFINCS is not linearized in the p erturbing field, the\ntorque due to |n| ≤18 is the not the sum of the torques due to |n|= 18 and |n|<18. We find\nthat the|n|= 18 ripple drives about 100 times more torque than the lower nripple. This\nresult is in agreement with most relevant rippled tokamak transport regimes, which feature\npositive scaling with n[21, 37]. For tokamak banana diffusion, in the√νboundary layer [37]\nion transport scales as Γ ψ∼√nand in the 1 /νregime [21] Γ ψ∼n2. Moreover, it is more\ndifficult to form ripple wells along a field line from low- nripple, so ripple trapping cannot\ncontribute as strongly to transport. This matches our findings th at the higher harmonic\n|n|= 18 ripple contributes more strongly to τNTVthan the |n|<18 ripple.\nIn figure 9, the SFINCS calculation of τNTVwith TF ripple only is shown at r/a=\n0.5, 0.7, and 0.9. For these three radii the maximum δB= 0.26%, 0.51%, and 0.82%\nrespectively. As τNTVscales with a positive power of δBin most rippled tokamak regimes, it\nis reasonable to expect that the magnitude of τNTVwould decrease with decreasing radius.\nOn the other hand, transport scales strongly with Ti. In the√νbanana diffusion regime [37]\nΓψ∼v4\nti√νii∼T5/4\ni. The combined effect of decreased ripple and increased temperatu re\nwith decreasing radius leads to comparable torques with decreasing radius in the presence\nof significant Er. The scaling with Tiis even stronger in the 1 /νregime [20, 21], where\nΓψ∼v4\nti/νii∼T7/2\ni. Indeed, we find that the magnitude of τNTVatEr= 0 increases with\ndecreasing radius.\n21FIG. 8: The NTV torque density at r/a= 0.9 for toroidal mode numbers (i) |n|= 18\n(purple solid), (ii) |n|<18 (brown dash dot), and (iii) |n| ≤18 (green dashed). The TBM\nripple contributes most strongly to low |n|, while the FIs and TF ripple only contribute to\n|n|= 18. The low nTBM ripple does not contribute as strongly to the NTV torque densit y\nas the|n|= 18 ripple does.\n22FIG. 9: SFINCS calculation of NTV torque density ( τNTV) as a function of ion\n∝angb∇acketleftV||B∝angb∇acket∇ight/∝angb∇acketleftB2∝angb∇acket∇ight1/2for VMEC geometry with TF ripple only at r/a= 0.5 (blue dashed), 0.7\n(red solid), and 0.9 (green dash-dot). Although the field ripple decr eases with radius\n(maximum δB= 0.82% atr/a= 0.9,δB= 0.51% atr/a= 0.7,δB= 0.26% atr/a= 0.5),\ntransport near Er= 0 increases with decreasing radius because of strong scaling of τNTV\nwithTi[20, 21].\n23VI. SCALING WITH RIPPLE MAGNITUDE\nIn figure 10, the NTV torque density calculated by SFINCS is shown a s a function of\nthe magnitude of the ripple, δB, for TF only geometry. The additional ferromagnetic ripple\nis not included, while the |n|= 18 components of B, its derivatives, and√gare rescaled\nas described above. The quantity τNTVis calculated at r/a= 0.9 withEr= 30 kV/m,\ncorresponding to the intrinsic rotation estimate. The color-shade d background indicates\nthe approximate regions of applicability of the collisional boundary lay er (ν−√ν) and\nthe collisionless detrapping/trapping ( ν) regimes. The boundary between these regimes\ncorresponds to the δBfor which the width in pitch angle of the detrapping/retrapping layer\nis similar to the width of the collisional boundary layer, ( δB/ǫ)∼(ν/(ǫωE))1/2. The 1/ν\nregime [21] does not apply at this Er, asωE≫ν/ǫwhereωE=Er/Bθis theE×B\nprecession frequency. The radial electric field isalso largeenought hat the resonance between\nvEandvmcannot occur, so the superbanana-plateau [25] and superbanan a [82] regimes are\navoided. This significant Ermay also allow the bounce-harmonic resonance to occur [27].\nTransport from ripple-trapped particles in the ν−√νregime may also be significant for\nthese parameters.\nThe observed scaling appears somewhat consistent with ripple trap ping in the stellarator\n√νregime [29] which predicts Γ ψ∼δ3/2\nB. However, this result is inconsistent with predic-\ntions for tokamak ripple transport in the νregime, Γ ψ∼δ0\nB[26, 35]. Contributions from\nother transport regimes may also influence the observed scaling. I n the banana diffusion√ν\nregimeτNTV∼δ2\nBand in theνregimeτNTV∼δB. Bounce-harmonic resonant fluxes scale as\nδ2\nB[27]. A scaling between δ0\nBandδ2\nBhas been predicted for plasmas close to symmetry with\nlarge gradient ripple in the absence of Er[83]. ForδBsmaller than 0 .82%, the actual value of\nripple atr/a= 0.9 for ITER geometry, the scaling of τNTVwithδBappears similar to δ3/2\nB.\nThe disagreement between the SFINCS calculations and the quasiline ar prediction, Γ ψ∼δ2\nB,\nindicates the presence of nonlinear effects such as local ripple trap ping and collisionless de-\ntrapping. The departure from quasilinear scaling increases with δB, which is consistent with\ncomparisons of SFINCS with quasilinear NEO-2 [46]. We see that τNTVshows very shallow\nscaling between δB= 0.05 andδB= 0.2. This could be in agreement with a scaling of δ0\nB\npredicted for νregime ripple-trapping in tokamaks [20, 33]. In this region the collisionle ss\n24detrapping boundary layer and collisional boundary layer are of com parable widths, so it is\npossible that the transport here is not described well by any of the displayed scalings. Fur-\nthermore, near the collisionless detrapping-trapping regime, δBbecomes comparable to the\ninverse aspect ratio and the assumptions made for rippled tokamak theory are not satisfied.\nFIG. 10: SFINCS calculations of NTV torque density as a function of δBatr/a= 0.9. A\nsingle value of Er= 30 kV/m is used corresponding to the intrinsic rotation estimate. T he\ncolor-shading indicates the approximate regions of applicability for r ippled tokamak\nbanana diffusion ν−√νregime [37] where τNTV∼δ2\nBand the collisionless\ndetrapping/trapping νregime [22] where τNTV∼δB.\nVII. HEAT FLUX CALCULATION\nAs well as driving non-ambipolar particle fluxes, the breaking of toro idal symmetry drives\nan additional neoclassical heat flux. In figure 11, the SFINCS calcu lation of the heat flux,\nQNC, is shown for three magnetic geometries: (i) axisymmetric (blue solid ), (ii) with TF\nripple only (red dash-dot), and (iii) TF ripple with TBMs and FIs (green dashed). In the\npresence of TF ripple, the ripple drives an additional heat flux that is comparable to the\naxisymmetric heat flux. However, with the addition of the FIs the he at flux is reduced to the\n25magnitude of the axisymmetric value, except near Er= 0 where 1 /νtransport dominates.\nWhile the radial ripple-driven particle fluxes will significantly alter the I TER angular\nmomentum transport, the neoclassical heat fluxes are insignifican t in comparison to the\nturbulent heat flux. Note that the neoclassical heat flux is /lessorsimilar5% of the heat flux calculated\nfrom heating and fusion rate profiles (see appendix A), Q≈0.2 MW/m2. Thus we can\nattribute /greaterorsimilar95% of the heat transport to turbulence. If ITER ripple were scale d up to\nδB/greaterorsimilar30%, the neoclassical ripple heat transport would be comparable to the anomalous\ntransport at this radius.\nFIG. 11: SFINCS calculation of neoclassical heat flux, QNCatr/a= 0.9 for three\nmagnetic geometries: (i) axisymmetric (blue solid), (ii) with TF ripple on ly (red dash-dot),\nand (iii) TF ripple with TBMs and FIs (green dashed). The vertical das hed line\ncorresponds to the intrinsic and NBI rotation estimate for Er. Note that QNCis much\nsmaller than the estimated anomalous heat transport, Q≈0.2 MW/m2,\nVIII. TANGENTIAL MAGNETIC DRIFTS\nAlthough ( vE+vm)· ∇f1is formally of higher order than the other terms in equation\n11, it has been found to be important when ν∗/lessorsimilarρ∗[84, 85] and has been included in other\n26calculations of 3D neoclassical transport. In the SFINCS calculatio ns shown in sections IV,\nV, VI, and VII, vm·∇f1has not been included, but now we examine the effect of including\nparts of this term. As SFINCS does not maintain radial coupling of f1, only the poloidal and\ntoroidal components of this magnetic drift term can be retained wh ile the radial component\ncannot. Note that the radial magnetic drift is retained in vm· ∇f0. We first implement\nvm·∇θandvm·∇ζusing the following form of the magnetic drifts,\nvma=v2\n2ΩaB2(1+ξ2)B×∇B+v2\nΩaBξ2∇×B, (20)\nwhereξ=v||/v. However, a coordinate-dependence can be introduced as we simp ly drop\none component of vm. For a coordinate-independent form, one must project vmonto the flux\nsurface. Additionally, when poloidal and toroidal drifts are retaine d, the effective particle\ntrajectories do not necessarily conserve µwhenµ= 0. The drifts can be regularized in\norder to satisfy ˙ξ(ξ=±1) = 0. Regularization also eliminates the need for additional\nparticle and heat sources due to the radially local assumption and pr eserves ambipolarity\nof axisymmetric systems [86]. To this end, we also implement a coordina te-independent\nmagnetic drift perpendicular to ∇ψ,\nv⊥\nma=∇ψ×(vma×∇ψ)\n|∇ψ|2=v2\n2ΩaB2(B×∇ψ)\n|∇ψ|2∇ψ·/bracketleftbig\n(1−ξ2)∇B+2Bξ2(b·∇b)/bracketrightbig\n.(21)\nNotethatthe ∇Bdrifttermisregularizedwhilethecurvaturedrifttermisnot. Astan gential\ndrifts are important for the trapped portion of velocity space, we can consider ξ2≪1. For\nthis reason we drop the curvature drift for regularization,\nv⊥\nma=v2\n2ΩaB2(B×∇ψ)(1−ξ2)(∇ψ·∇B)\n|∇ψ|2. (22)\nThis is similar to the form presented by Sugama [86], but we have chose n a different form\nof regularization. This choice for v⊥\nmadoes not alter the conservation properties shown by\nSugama, as it remains in the B×∇ψdirection and vanishes at ξ=±1. We note that the\nphase space conservation properties rely on the choice of a modifie d Jacobian in the presence\nof tangential magnetic drifts. In SFINCS we have not implemented s uch a modification.\nHowever, as Sugama shows, the correction to the Jacobian is an or derρ∗correction. As\nparticle and heat sources have been implemented in SFINCS, we have confirmed that the\naddition of tangential magnetic drifts does not necessitate the us e of appreciable source\n27terms. We note that this form of the tangential magnetic drifts we have chosen does not\ninclude a magnetic shear term which is present in the bounce-averag ed radial drift. This\nnon-local modification has been found to significantly alter superba nana transport [78, 87]\nand the drift-orbit resonance [46].\nAnErscan atr/a= 0.7, whereρ∗becomes comparable to ν∗, is shown in figure 12. When\nvm·∇f1is added to the kinetic equation, the 1 /νpeak atEr= 0 is shifted toward a slightly\nnegativeEr, corresponding to the region where ( vE+vM)· ∇ζ≈0, where superbanana-\nplateau transport takes place. For ITER parameters at this radiu s, the collisionality is large\nenough that superbananas cannot complete their collisionless traj ectories but small enough\nthat non-resonant trapped particles precess, νSB\n∗≪ν∗≪νSBP\n∗, whereνSBP\n∗=ρ∗q2/ǫ1/2and\nνSB\n∗=ρ∗δ3/2\nBq2/ǫ2[25, 82], thus superbanana-plateau transport is relevant.\nWhen the in-surface magnetic drifts are present, the depth of th e resonant peak is di-\nminished. In the absence of tangential drifts, the bounce-avera ged toroidal drift vanishes\natEr= 0 for all particles regardless of pitch angle and energy. When tang ential drifts\nare added to the DKE, the resonant peak will occur at the Erfor which thermal trapped\nparticles satisfy the resonance condition. However, only particles above a certain energy and\nat the resonant pitch angle will participate in the superbanana-plat eau transport, thus the\ndepth of the peak is diminished. Note that local ripple trapping might a lso contribute to the\n1/νtransport at small |Er|. For|Er|>20 kV/m, the range relevant for ITER, the addition\nofvm·∇f1has a negligible effect on τNTV. The addition of tangential magnetic drifts would\nnot dramatically change the results in previous sections.\n28FIG. 12: Calculation of NTV torque density, τNTV, as a function of Eratr/a= 0.7. The\nblue dashed curve corresponds to a SFINCS calculation without vm·∇f1in the DKE. The\norange dash-dot curve corresponds to the addition of vm·∇f1as given in equation 20. The\ngreen solid curve corresponds to the addition of the projected an d regularized drift,\nv⊥\nm·∇f1, as given in equation 22.\nWe compute a radial profile of τNTVdue to TF ripple including v⊥\nm·∇f1. The intrinsic\nrotation model and NBI rotation model are used to estimate Erat each radius, as shown in\ntheErprofile in figure 13. As Ercrosses through 0 at r/a= 0.56 for the NBI rotationmodel,\ntangential drifts will affect the transport. In figure 13, we compa re the magnitude of τNTV\ndue to TF ripple with τNBIandτturb=−∇·Πint, the turbulent momentum source causing\nintrinsic rotation. The τNBIprofile was computed by NUBEAM as used in section III, and\nτturbis estimated using Π int∼(ρθ/LT)/tildewideΠ(ν∗)Q(∝angb∇acketleftR∝angb∇acket∇ight/vti) (see appendix A). At r/a= 0.62\nsuperbanana-plateau transport dominates when NBI rotation is c onsidered, and τNTVis\nabout 6 times larger than when the higher-rotation turbulent torq ueEris considered. For\nbothErestimates/vextendsingle/vextendsingleτNTV/vextendsingle/vextendsingleincreases with decreasing radius due to the scaling with Tias\ndiscussedinsectionV. Notethattheturbulenttorqueproducesm uchrotationinthepedestal\naccording to this model as τturb∝1/LT. The integrated NTV torque, -45.6 Nm with the\nturbulent rotation model and -71 Nm with the NBI rotation model, is la rger in magnitude\n29than the NBI torque, 35 Nm, but smaller than the turbulent torque , 93 Nm. Here the\nintegratedτturbis significantly larger than that obtained from dimensionless paramet er scans\non DIII-D, 33 Nm [70]. This is possibly due to the assumed scaling in our t urbulent rotation\nmodel, which may not be physical near the edge.\nIn the region 0 .5/lessorsimilarr/a/lessorsimilar0.9, the magnitude of τNTVis comparable to τturband greater\nthanτNBI. The NTV torque will likely significantly damp rotation in the absence of inserts,\ndecreasing MHD stability. However, the resulting rotation profile ma y be sheared because\nof the significant counter-current NTV source at the edge and co -current NBI source in the\ncore. We estimate the rotation shear, γ= ∆Vζ/∆r≈0.4(vti/R), using the neoclassical\noffset atr/a= 0.7 and the NBI-driven rotation at r/a= 0.4. Assuming the maximum linear\ngrowth rate for drift wave instabilities, γLin≈vti/R[88], this rotation shear may be large\nenough to suppress microturbulence. In concert with reversed m agnetic shear sustained by\nheating and current drive sources [57], rotation shear may suppor t the formation of an ITB\n[89] for this steady state scenario.\n30(a)\n(b)\nFIG. 13: (a) Profiles of NTV torque density ( τNTV) due to TF ripple without ferritic\ncomponents calculated with SFINCS, NBI torque density calculated from NUBEAM\n(τNBI), and estimate of turbulent intrinsic rotation momentum source ( τturb). The quantity\nτNTVis calculated using the Erdetermined by the intrinsic rotation model and NBI\nrotation model described in section III. Turbulent torque is estima ted using\nτturb∼ −Πint/awhere Π int∼ρ∗,θ/tildewideΠ(ν∗)Q∝angb∇acketleftR∝angb∇acket∇ight/vti(see appendix A for details). (b) Profiles of\nradial electric field ( Er) due to NBI torque and turbulent intrinsic rotation. Here toroidal\nrotation is computed with the model described in section III and Eris computed with\nSFINCS as described in section IV.\n31IX. SUMMARY\nWe calculate neoclassical transport in the presence of 3D magnetic fields, including\ntoroidal field ripple and ferromagnetic components, for an ITER st eady state scenario. We\nuseanNBIandintrinsicturbulentrotationmodeltoestimate Erforneoclassicalcalculations.\nWe find that without considering τNTV, toroidal rotation with MA/lessorsimilar2% is to be expected,\nwhich is likely not large enough to suppress resistive wall modes [5]. We u se VMEC free\nboundary equilibria in the presence of ripple fields to calculate neoclas sical particle and heat\nfluxes using the drift-kinetic solver, SFINCS. At large radii r/a/greaterorsimilar0.5,τNTVdue to TF ripple\nwithout ferritic components is comparable to τNBIandτturbin magnitude but opposite in\nsign, which may result in flow damping at the edge and a decrease in MHD stability. As\nthe integral NTV torque is similar in magnitude to the NBI torque, non -resonant magnetic\nbraking cannot be ignored in analysis of ITER rotation. The torque p rofile may also result\nin a significant rotation shear which could suppress turbulent trans port. While the addition\nof FIs significantly reduces the transport ( ≈75% reduction at r/a= 0.9), the low nper-\nturbation of the TBM produces very little NTV torque. The neoclass ical heat flux caused\nby ripple is insignificant in comparison to the turbulent heat flux. Thou gh NTV torque has\nbeen shown to be important for ITER angular momentum balance, ite ratively solving for\nthe rotation profile with τNTVwill be left for future consideration.\nSeveral transportregimesmust beconsidered forITERNTV: the ν−√νbananadiffusion,\nbounce-resonance, and ν−√νripple trapping regimes. The calculated scaling of τNTVwith\nδBis between the δ3/2\nBscaling of the ripple trapping√νregime and the δ2\nBscaling predicted\nin theν−√νregime at small δB. There is room for further comparison between SFINCS\ncalculations of τNTVand analytic fomulae. However, we note that the analytic theory fo r\ntransport of ripple-trapped particles in a tokamak close to axisymm etry in the presence of\nEris not fully developed.\nAppendix A: Approximate Turbulent Heat Flux and Torque\nAs Πintis proportional to Qin our model, we must estimate Qusing the input heating\npower and D-T fusion rates calculated with TRANSP. The LH, NBI, an d ECH power densi-\nties (PLH,PNBI, andPECH) are integrated along with the fusion reaction rate density ( RDT)\n32to calculate the total integrated heating source, H(r),\n/integraldisplayr\n0dV(r′)H(r′) =/integraldisplayr\n0dV(r′)/parenleftbig\nPLH+PNBI+PECH+RDT(3.5MeV)/parenrightbig\n.(A1)\nAs/integraltext\nQdS=/integraltext\nHdV,\nQ(r) =/integraltextr\n0dV(r′)/parenleftbig\nPLH+PNBI+PECH+RDT(3.5MeV)/parenrightbig\nA(r), (A2)\nwhereA(r) is the flux surface area and V(r) is the volume enclosed by flux surface r. We\nhave shown in section VII that the neoclassical heat flux is insignifica nt in comparison to\nQ(r), so we can attribute Q(r) to turbulent heat transport. The calculated Qis shown in\nfigure 14.\nFIG. 14: Heat flux Qcalculated with input heating and fusion rate profiles from TRANSP\nand TSC.\nWe estimate τturb=−∇·Πint∼ −Πint/ausing\nΠint∼ρθ/tildewideΠ(ν∗)Q∝angb∇acketleftR∝angb∇acket∇ight\nvtiLT. 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In particular, we show that\nour system can be implemented by using a magnetic photonic crystal structure that\nemploys a square ray of ferrite rods, and as the \feld response of ferrites can be tuned by\nexternal magnetic \felds, we end up with an electromagnetic gateway that can be open\nor shut using external \felds. The functionality is also robust against the positional\ndisorder of the rods that made up the photonic crystal.\nPACS numbers: 41.20.Jb, 42.25.Fx, 42.25.GyarXiv:0905.1273v1  [physics.optics]  8 May 2009A simple route to a tunable electromagnetic gateway 2\n1. Introduction\nTransformation optics [1, 2, 3] has paved the way for the development of optical devices\nthat can realize functionalities that were thought to be possible only in science \fctions\n[4]-[14]. One such conceptual device that has attracted great public interest is a gateway\nthat can block electromagnetic waves but that allows the passage of other entities. This\ndevice can be viewed as an implementation of a \\hidden portal\" mentioned in \fctions\n[15, 16]. However, the feasibility of such devices is limited by the very complex material\nparameters and the narrow bandwidth. Here, we show that gateway-type devices can\nactually be realized with simple parameters and they can have wider band widths such\nthat the concept is closer to reality [17]-[22] than previously thought. The structure\ncan be implemented by using the magnetic photonic crystal structures that are \feld\ntunable, resulting in an invisible electromagnetic gateway that can be open or shut\nusing magnetic \felds.\n2. A new \\superscatterer\"\nWe start from a simple implementation of transformation optics. In Fig. 1(a), an\nobject (colored in blue) is placed to the left side of a double negative medium (DNM)\nwith\"=\u0016=\u00001. Let the object be of permittivity \"0and permeability \u00160. The\ndetailed shape and the length scale of the structure are shown in Fig. 1(a). From the\nviewpoint of transformation optics, the whole structure is optically equivalent to another\nobject described in Fig. 1(b) for far \feld observers. The equivalent permittivity and\npermeability tensors are$\"=\"0$cand$\u0016=\u00160$cwith a constant tensor$c. We introduce\nsome parameters for ease of reference following the coordinates x1,x2,y1,y2and the\nangle\u000bin Fig. 1. Let \u0001 = x2\u0000x1be the waist of the object in the x-direction,\nr=\u0001\n2x1+\u0001be a coordinate compression ratio and\np=8\n>><\n>>:1\ntan\u000b2(x1+\u0001)\n2x1+\u0001; y1<y<y 2;\u0000x2y2\u0000y\ny2\u0000y1<x<x 1y2\u0000y\ny2\u0000y1;\n0;jyj<y 1;\u0000x2<x<x 1;\n\u00001\ntan\u000b2(x1+\u0001)\n2x1+\u0001;\u0000y2<y<\u0000y1;\u0000x2y2+y\ny2\u0000y1<x<x 1y2+y\ny2\u0000y1;(1)\nwhere (x; y) are the position coordinates of the equivalent transformation medium in\nFig. 1(b). The inequalities for the de\fnition of the parameter, p, come from three\ndi\u000berent coordinate transformations and the three regions are distinguished by the two\ndashed lines in Fig. 1(b).\nIt can be proved that (see Appendix A):\n$c=2\n6641+p2\nr\u0000p0\n\u0000p r 0\n0 0r3\n775: (2)\nThe coordinate transformation is based on the recent work on transformation optics\nwith complementary medium (or folded geometries) [23, 24]. The original object isA simple route to a tunable electromagnetic gateway 3\nFigure 1. A simple electromagnetic structure comprising of two\nhomogenous regions (left panel) and an equivalent structure that gives the\nsame scattering (right panel). (a) The schematic plot for the scatterer that appears\nelectromagnetically larger than its physical size. The red region is the DNM, the blue\nregion is the original isotropic material that is to the left of the DNM, and the green\nregion represents the air background. x1,x2,y1,y2and\u000bare the detailed coordinates\nof the structure. The dashed lines in the region ( x>0) give the virtual boundary for\nthe equivalent transformation medium. (b) The equivalent transformation medium is\nindicated by the yellow region, whose parameters are implemented by equation (2).\nThe dashed lines distinguish the three di\u000berent regions for the de\fnition of p.\nmapped into the equivalent transformation medium. The DNM together with its mirror\nimage (the boundary is marked by the dashed line in Fig. 1(a)) of air form a pair of\ncomplementary media, in the sense the phase accumulated in one segment is exactly\ncancelled by another. A more detailed discussion of the optical property of this speci\fc\ngeometry and the associated coordinate transformation can be found in Appendix A.\nThe structure in Fig. 1(a) can be treated as a form of a scattering ampli\fer [25] because\nits scattering cross section can be much larger than its geometric cross section. The\nampli\fcation e\u000bect originates from the excitation of surface plasmons in the DNM and\nair interface.\nAs a concrete example, we suppose that the original isotropic material (the blue\nregion in Fig. 1(a)) has a large value of permittivity, \"0=\u000010000 and\u00160= 1, which\nmay be treated approximately as a perfect electric conductor (PEC). Figure 2(a) shows\nthe scattering \feld pattern of the structure in Fig. 1(a). The plane wave is incident from\nthe top to the bottom and has transverse electric (TE) polarization, for which the E \feld\nis alongz-direction. In this paper, we only consider the TE modes for simplicity. Note\nthat the same idea works for transverse magnetic (TM) modes as well. The frequency\nis 2GHz andx1=y1= 0:1m,x2=y2= 0:2mand\u000b=\u0019=4. The structure behaves\nlike an equivalent material with parameters described by equation (2) to the far-\feld\nobservers. Due to the large mismatch of the impedance of the equivalent material with\nthe air background, we expect that the equivalent object (the yellow domain in Fig.A simple route to a tunable electromagnetic gateway 4\nFigure 2. The scattering pattern for the scattering object and the equivalent\nscatterer. (a) The scattering pattern of the structure depicted in Fig. 1(a). (b) The\nscattering pattern of the equivalent material but with the PEC replacing the material\nimplemented by equation (2).\n1(b)) should scatter like a PEC, and for that reason we choose for comparison in Fig.\n2(b) the scattering \feld pattern of a perfect conductor \flling up the entire domain of the\nvirtual object in Fig. 1(b). The similar far-\feld scattered patterns between Fig. 2(a)\nand Fig. 2(b) con\frm the strong scattering e\u000bect. The permittivity and permeability\nof the DNM are actually chosen as \u00001+0:0001\u0002iin the simulations to avoid numerical\ndivergence problems [24]. The PEC-like scatters are always chosen as the materials of\npermittivity at\u000010000 and the permeability is taken to be 1 in this paper for simplicity.\n3. An invisible electromagnetic gateway\nThe ampli\fed scattering e\u000bect can be utilized to make an invisible gateway [16]. Suppose\nthat a PEC wall separates the whole space into two regions, the upper domain and the\nlower domain. If there are channels (or gateways) opened in the PEC wall, people in\nthe two di\u000berent spaces can communicate with each other, both physically and through\nEM waves. However, if we replace the doors with the above-described con\fguration at\na speci\fc frequency, the communication for that frequency will be blocked because the\nsystems behave like PECs. The most amazing fact is that the channel is in fact physically\nempty. There is nothing but air in the channel so objects can \\walk through\" but the\nchannel is blocked as perceived by the eye because light at the designated frequency\ncannot penetrate. Figure 3 is a schematic plot of such a gateway based on the above\nidea. With the same scale as the one in Fig. 2(a), we demonstrate the properties of\nsuch a gateway. We suppose that there is a line source located at (0 :05m;0:4m) with a\nfrequency of 2 GHz in the upper domain. Figure 4(a) shows that the waves cannot pass\nthrough the gateway and are excluded from the lower domain. However, without the\nDNM, the waves can propagate into the lower region as shown in Fig. 4(b).\nWe now consider the band-width of such a device. To be more concrete, we considerA simple route to a tunable electromagnetic gateway 5\nFigure 3. The computation domains for the electromagnetic gateway. The\nyellow regions are the perfectly matched layers (PMLs), the red region is the DNM,\nthe green region is air, the blue regions are PECs. The point source is located at above\nthe gateway and it is denoted by a brown circle. The region between the DNM and\nthe right PEC is the so called \\invisible gateway\".\nthe following dispersions for the DNM:\n\"= 1\u0000f2\np\nf(f+i\u0000);\n\u0016= 1\u0000Ff2\nf2\u0000f2\n0+i\rf;(3)\nwherefp= 2:828GHz , \u0000 = 0:1MHz ,F= 1:5,f0= 1GHz and\r= 0:075MHz .\nWhenf= 2GHz , the relative permittivity and permeability return to \u00001 + 0:0001\u0002i\nused in the above simulations. When f= 1:6GHz , the impedances do not match at the\ninterface of the air and the DNM while the refractive index of the DNM is about \u00001:76.\nFigure 4(c) shows the electric \feld pattern while Fig. 4(d) shows the \feld without the\nDNM. The DNM can reduce the penetration of the waves from the upper region when\ncompared with Fig. 4(c) and Fig. 4(d). However, as n <\u00001, there is still a \\slit\"\nbetween the virtual boundary and the right PEC, which allows the waves to propagate\nsomewhat into the lower region. If n=\u00001, the virtual boundary is simply the mirror of\nthe interface of the DNM and PEC on the left with x= 0 as the mirror. The position\nof the virtual boundary can be obtained heuristically from the image-forming principle.\nWhenf= 2:4GHz , the impedances are still mismatched while the refractive index of\nthe DNM is about \u00000:56. As the absolute value of ndecreases, the virtual boundary of\nthe image expands and there will be no passage for the waves to penetrate because theA simple route to a tunable electromagnetic gateway 6\nvirtual boundary overlaps with the PEC on the right-band side. For example, we plot\nthe electric \feld pattern in Fig. 4(e) and the case without the DNM in Fig. 4(f). We\n\fnd that the DNM can eliminate the penetration of the waves from the upper region.\nThat means that the negative band of the DNM beyond f= 2GHz (or\u00001< n < 0)\nis the working frequency of the designed gateway. However, we also \fnd that the wave\nblocking e\u000bect will become weaker with higher frequencies (in this case, about 2 :5GHz )\nin our simulations. As negative index media are intrinsically dispersive, the bandwidth\nhas to be \fnite as the parameters will deviate progressively from those required by\ntransformation optics. But the present gateway is shown to be relatively robust and\nhas a broad operation bandwidth of about 20%, which can be regarded as a broadband\ndevice. To have a broader bandwidth, we can simply reduce the distance of the DNM\nand the right PEC to enhance the overlapping while the functionality of the gateway\nis sacri\fced. As there are extensive designs of the DNMs at various wavelengths (both\ntheoretically and experimentally) [17]-[22], it would be reasonably feasible for the present\ngateway to be realized in the future.\nIf the losses of the metamaterial are large (i.e., the imaginary parts of the parameters\nof the DNM are larger than 0 :1), the bandwidth of the gateway will be small and the\nfunctionality will be compromised. The absorption of the DNM is the key di\u000eculty for\nboth the gateway described here as for other devices such as the perfect lens. All our\nsimulation results are calculated using the COMSOL Multiphysics \fnite element-based\nelectromagnetics solver.\n4. The implementation of a tunable electromagnetic gateway\nVery recently, it has been demonstrated that the DNM can be realized with a simple\narray of ferrite rods without any metallic components [26], which can be used to\nimplement the present gateway. One of the unique merits of this magnetic metamaterial\nis that its optical properties are magnetically tunable. As such, the gateway can be\nmanipulated using external magnetic \felds. The magnetic metamaterial is a periodic\nsquare array of subwavelength ferrite rods with radii of r= 3:5mmand a lattice constant\nofa= 10mm. The permittivity is taken to be \"= 25, and the permeability of the\nferrite rods has the form ^ \u0016=0\nBB@\u0016r\u0000i\u0016\u00140\ni\u0016\u0014\u0016r 0\n0 0 11\nCCAwith\u0016r= 1 +!m(!0\u00002\u0019i\u000b1f)\n(!0\u00002\u0019i\u000b1f)2\u00004\u00192f2and\n\u0016\u0014=2\u0019!mf\n(!0\u00002\u0019i\u000b1f)2\u00004\u00192f2, where\u000b1is the damping coe\u000ecient, !0=\r1H0is the resonance\nfrequency with \r1being the gyromagnetic ratio; H0is the sum of the external magnetic\n\feld and shape anisotropy \feld along the z-direction,!m=\r1Msis the characteristic\nfrequency with Msthe saturation magnetization and it is taken as Ms= 1750G, typical\nfor single-crystal yttrium-iron-garnet (YIG). As the absorption of single-crystal YIG is\nextremely low, we can set \u000b1= 0.\nThe numerical simulations on the gateway under di\u000berent conditions are performed\nby using the multiple scattering method [26]. Figure 5 shows the electric \feld intensity inA simple route to a tunable electromagnetic gateway 7\nFigure 4. The functionalities of the gateway (with or without the DNM) at\ndi\u000berent frequencies. (a) The electric \feld pattern for the gateway at 2 GHz . (b)\nThe electric \feld pattern for the gateway without the DNM at 2 GHz . (c) Same as (a)\nbut for 1:6GHz . (d) Same as (b) but for 1 :6GHz . (e) Same as (a) but for 2 :4GHz .\n(f) Same as (b) but for 2 :4GHz .A simple route to a tunable electromagnetic gateway 8\nFigure 5. The implementation of the invisible gateway using magnetic\nphotonic crystals. Here, we show the electric \feld intensity pattern for\ndi\u000berent geometries working at di\u000berent values of the external magnetic\n\feld. (a)H0= 500Oe, (b)H0= 475Oe, (c) without magnetic metamaterials, (d)\nwith magnetic metamaterial replaced by PEC.\nthe logarithmic scale. The bulk PEC is replaced by a discrete system of the periodically\narranged PEC rods (marked with dark green solid circles in the \fgure), and the ferrite\nrods are denoted as hollow circles. Under H0= 500Oeand the frequency f= 2:55GHz ,\nthe e\u000bective refractive index of the magnetic metamaterial is n=\u00001(\"eff=\u0016eff=\u00001).\nIn Fig. 5(a), it can be seen that the electric \feld is excluded from the air channel, so that\nthe open channel appears to be blocked to the eye at this frequency. This demonstrates\nthat the gateway can be implemented by a photonic crystal type structure in which each\nelement (each ferrite rod in the periodic array) is identical. By changing the external\n\feld toH0= 475Oe, the refractive index of the magnetic metamaterial becomes n= 1(\n\"eff=\u0016eff= 1). The electric \feld intensity under such conditions is shown in Fig.\n5(b), which shows that the channel is open for EM wave passage, or the passage appearsA simple route to a tunable electromagnetic gateway 9\nto be open to the eye. Our results demonstrate that the gateway can be implemented by\na very simple con\fguration and the e\u000bect is tunable by an external \feld. As shown in\nAppendix B, the e\u000bect is fairly robust to the disorder of the position of ferrite cylinders.\nFigures 5(c) and 5(d) illustrate the fact that the channel is electromagnetically open if\nthe ferrite rods are replaced by air and by PEC rods. In addition to the simple geometry\nand weak absorption, the magnetic metamaterial has a reasonably broad bandwidth as\na DNM.\n5. Conclusion\nIn conclusion, we showed that a robust and tunable electromagnetic gateway can\nbe constructed using simple material parameters. The idea is conceived through\nconsideration of transformation optics and can be realized using a photonic crystal\ntype structure.\nAcknowledgments\nWe thank Dr. Yun Lai and Dr. Jack Ng for their helpful discussions. This work was\nsupported by Hong Kong Central Allocation Grant No. HKUST3/06C. Computation\nresources were supported by the Shun Hing Education and Charity Fund. S.Y.L and\nZ.F.L were supported by CNKBRSF, NNSFC, PCSIRT, MOE of China (B06011), and\nShanghai Science and Technology Commission.\nAppendix A\nIn this appendix, we give the detailed coordinate transformation to produce the folded\ngeometry used in the text. Let us \frst consider the following coordinate transformation,\nx=8\n>><\n>>:\u0000xy\n2+x2+x1\nx2\u0000x1\u0002(x0+xy\n2);\u0000xy\n2<x0<\u0000xy\n1;\n\u0000x0;\u0000xy\n1<x0<0;\nx0; else;\ny0=y; z0=z;(A.1)\nwhere\n(xy\n1; xy\n2) =8\n>><\n>>:(x1y2\u0000y\ny2\u0000y1; x2y2\u0000y\ny2\u0000y1); y1<y<y 2\n(x1; x2);\u0000y1<y<y 1\n(x1y2+y\ny2\u0000y1; x2y2+y\ny2\u0000y1);\u0000y2<y<\u0000y1; (A.2)\nThis coordinate transformation maps the virtual space ( x-space) described in Fig.\n1(b) into the physical space ( x0-space) described in Fig. 1(a). If the virtual space is\na vacuum, we can obtain the corresponding material parameters in the physical space.\nIt contains two kinds of materials in the vacuum background, one a double negative\nmedium (DNM) with \"=\u0016=\u00001 (shown in red in Fig. 1(a)), the other an anisotropicA simple route to a tunable electromagnetic gateway 10\nFigure A1. The folded geometry used in the text. (a)A obliquely incident TE\nplane wave in free space. (b)The structure with the material described by equation\n(A.4) and the DNM are invisible.\nmedium (shown in blue in Fig. 1(a)) whose parameters are derived in the following\nsection.\nWe can easily obtain the Jacobian transformation matrix,\n\u0003 =2\n664@x0\n@x@x0\n@y0\n0 1 0\n0 0 13\n775 (A.3)\nfrom equations (A.1) and (A.2) for the mapping between the blue region in Fig. 1(a)\nand the yellow region in Fig. 1(b), and we can also prove that@x0\n@x=rand@x0\n@y=pin\nthe text. From transformation optics, we can obtain the material parameters for the\nblue region,\n$\"p=$\u0016p=2\n664r2+p2\nrp\nr0\np\nr1\nr0\n0 01\nr3\n775: (A.4)\nFigure A1(a) shows the propagation of the incident transverse electric (TE)\npolarization plane wave from the left to the right but at an angle of 45 degrees from\nthex-direction in free space. The frequency is 2 GHz . We consider the object described\nin equation (A.4) together with an DNM domian. The scales of the structure in our\nsimulations are the same as the ones used in the text. Figure A1(b) shows the scattering\npattern of such a structure and demonstrates that the structure itself is invisible to the\nfar-\feld observers.\nA modi\fcation of the above folded geometry can lead to a variety of conceptual\ndevices. For example, if the virtual space is not a vacuum but an object of permittivity\n\"sand permeability \u0016s, then the parameters in the blue region should be a permittivity\ntensor,\"s$\"p, and a permeability tensor, \u0016s$\u0016p. The object in the blue region together\nwith the DNM domain will have a scattering cross section that can be much largerA simple route to a tunable electromagnetic gateway 11\nFigure A2. The scattering pattern of an equivalent scatterer and the real\nscatterer. (a) The scattering pattern of an object of permittivity \"s= 2:25. (b) The\ncorresponding scattering pattern of the scattering ampli\fer.\nthan its geometric cross section. Figure A2(a) shows the scattering pattern of an\nobject with permittivity \"s= 2:25 and permeability \u0016s= 1. Figure A2(b) shows\nthe scattering pattern of the structure including the object with permittivity tensor\n\"s$\"pand permeability tensor \u0016s$\u0016pand the DNM. The two similar far-\feld patterns in\nFigs. A2(a) and A2(b) con\frm the enhanced scattering e\u000bect. In the text, we apply\nthis structure in an inverse manner. We suppose that the object in the blue region is of\npermittivity \"0and permeability \u00160. Then, the whole structure is equivalent to another\nobject with its permittivity and permeability tensors,$\"=\"0$cand$\u0016=\u00160$c(described\nin the text). We can easily obtain equation (2) by performing an inverse transformation.\nAppendix B\nIn this appendix, we demonstrate the robustness for the implementation of the gateway\nusing the magnetic photonic crystal structure. We introduce disorder by displacing the\nferrite rods from their original positions with\n\u0001r= (a\u00002r)\u0002s\u0002rn; (B.1)\nwhereais the lattice constant, ris the radii of the ferrite rods, rnis a uniform random\nvariable on the interval ( \u00000:5;0:5) (assures no overlap of cylinders), sis the strength\nof disorder. Figure B1(a) corresponds to Fig. 5(a) but with a disorder of s= 0:5\nintroduced. The gateway remains blocked to the eye for this strength of disorder.\nWhenH0changes to 475 Oe, the gateway is open again as shown in Fig. B1(b). Figure\nB1(c) corresponds to Fig. 5(a) but with a strong disorder of s= 1:0 introduced. The\nfunctionality of the gateway is now slightly compromised but the blocking is at least\npartially e\u000bective at such a strength of disorder. With H0= 475Oe, the gateway is\nopen again as shown in Fig. B1(d). The results indicate that the functionality is robust\nto structural perturbations.A simple route to a tunable electromagnetic gateway 12\nFigure B1. The robust e\u000bect from the disorder. The electric \feld intensity\npattern for the gateway with di\u000berent levels of disorder working at di\u000berent\nexternal magnetic \felds. (a)s= 0:5,H0= 500Oe; (b)s= 0:5,H0= 475Oe; (c)\ns= 1:0,H0= 500Oe; (d)s= 1:0,H0= 475Oe.\nReferences\n[1] Leonhardt U 2006 Science 3121777\n[2] Pendry J B, Schurig D and Smith D R 2006 Science 3121780\n[3] Greenleaf A, Lassas M and Uhlmann G 2003 Math. Res. Lett. 10685\n[4] Schurig D, Mock J J, Justice B J, Cummer S A, Pendry J B, Starr A F and Smith D R 2006\nScience 314977\n[5] Cai W, Chettiar U K, Kildishev A V and Shalaev V M 2007 Nat. Photonics 1224\n[6] Rahm M, Schurig D, Roberts D A, Cummer S A, Smith D R and Pendry J B 2008 Photon.\nNanostruct.: Fundam. Applic. 687\n[7] Chen H Y and Chan C T 2007 Appl. Phys. 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Lett. 102093901\n[15] Rowling J R 1998 Harry Potter and the Chamber of Secrets (London: Bloomsbury)\n[16] Luo X D, Yang T, Gu Y, Chen H Y and Ma H R 2008 Conceal an entrance by means of\nsuperscatterer Preprint physics.optics/0809.1823\n[17] Shelby R A, Smith D R and Schultz S 2001 Science 29277\n[18] Smith D R, Pendry J B and Wiltshire M C K 2004 Science 305788\n[19] Soukoulis C M, Linden S and Wegener M 2007 Science 31547\n[20] Lezec H J, Dionne J A and Atwater H A 2007 Science 316430\n[21] Yao J, Liu Z, Liu Y, Wang Y, Sun C, Bartal G, Stacy A and Zhang X 2008 Science 321930\n[22] Valentine J, Zhang S, Zentgraf T, Ulin-Avila E, Genov D A, Bartal G and Zhang X 2008 Nature\n455376\n[23] Leonhardt U and Philbin T G 2006 New J. Phys. 8247\n[24] Pendry J B and Ramakrishna S A 2003 J. Phys. Condens. Matter 156345\n[25] Yang T, Chen H Y, Luo X D and Ma H R 2008 Opt. Express 1618545\n[26] Liu S, Chen W, Du J, Lin Z F, Chui S T and Chan C T 2008 Phys. Rev. Lett. 101157407"    },    {        "title": "1403.5157v1.Magnetization_of_densely_packed_interacting_magnetic_nanoparticles_with_cubic_and_uniaxial_anisotropies__A_Monte_Carlo_study.pdf",        "content": "arXiv:1403.5157v1  [cond-mat.mtrl-sci]  20 Mar 2014Magnetization of densely packed interacting magnetic nano particles\nwith cubic and uniaxial anisotropies: a Monte Carlo study.\nV. Russiera), C. de-Montferrandb), Y. Lalatonneb)and L. Motteb).\na)ICMPE, UMR 7182 CNRS and UPEC,\n2-8 rue Henri Dunant 94320 Thiais, France. and\nb)CSPBAT UMR 7244 CNRS and University Paris 13, 93017 Bobigny, France.\n(Dated: August 10, 2021)\nThe magnetization curves of densely packed single domain magnetic n anoparticles (MNP)\nare investigated by Monte Carlo simulations in the framework of an eff ective one spin model.\nTheparticleswhosesizepolydispersityistakenintoaccountarearr angedinsphericalclusters\nand both dipole dipole interactions (DDI) and magnetic anisotropy en ergy (MAE) are in-\ncluded in the total energy. Having in mind the special case of spinel f errites of intrinsic cubic\nsymmetry, combined cubic and uniaxial magnetocrystalline anisotro pies are considered with\ndifferent configurations for the orientations of the cubic and uniax ial axes. It is found that\nthe DDI, together with a marked reduction of the linear susceptibilit y are responsible for a\ndamping of the peculiarities due to the MAE cubic component on the ma gnetization. As an\napplication, we show that the simulated magnetization curves compa re well to experimental\nresults for γ–Fe2O3MNP for small to moderate values of the field.\nI. INTRODUCTION\nMagnetic nanoparticles (MNP) assemblies present a fundame ntal interest in the devel-\nopment of nanoscale magnetism research and are promising ca ndidates in a wide range of\npotential applications going from high density recording t o bio-medicine1–5. Experimen-\ntally MNP can be obtained either as colloidal suspensions wh ere the concentration can be\nvaried at will, embedded in non magnetic material where one c an tune the interparticle\ninteractions or as powder samples where they form densely pa cked systems. The case of\niron oxide particles, which aretypical cubicspinel ferrit es take acentral place especially for\nbio-medical applications because of their biocompatibili ty and suitable superparamagnetic\nproperties6,7.\nThe magnetic behavior of nanostructured materials or syste ms including nanoscale\nmagnetic particles is a multiscale problem since the local m agnetic structure within NP\nat the atomic site scale presents non trivial features8–11and the interactions between\nparticles play an important role. A simplification occurs fo r NP of diameter below some\ncritical value of typically few tens of nanometers since the y then reach the single domain\nregime and can be described through an effective one spin model (EOS) where each NP\nis characterized by its moment and anisotropy energy. Howev er both the moment value2\nand the anisotropy energy function are to be understood as effe ctive quantities which take\ninto account some of the atomic scale characteristics12–15. The EOS type of approach is a\nsimplifying but necessary step for the description of inter acting MNP assemblies. In the\nframeworkoftheEOSmodels, thetotal energyincludesonthe onehandtheNPanisotropy\nenergy through a one-body term and on the other hand the inter particle interactions. It\nis generally assumed for frozen systems of well separated NP that the leading term in the\ninterparticle interactions is the dipolar interaction (DD I) between the macrospins which\nis totally determined once the NP saturation magnetization and the size distribution are\nknown. Conversely modeling the anisotropy energy is not str aightforward since in finite\nsized particles it comes from different origins. The intrinsi c contribution which stems form\nthe bulk material is a prioriknown experimentally without ambiguity. It can be of either\nuniaxialorcubicsymmetryaccordingtothecrystallinestr ucturewithanisotropyconstants\nwhosemagnitudeandevensigndependontemperature. Inthew idelystudiedcaseofoxide\nspinel ferrites at room temperature the intrinsic anisotro py is of cubic symmetry16–20with\nin general a negative constant Kc, leading to the moment preferentially oriented along\nthe{111}directions of the crystallites. Then for NP not strictly sph erical one has to\nadd the shape anisotropy term resulting from demagnetizing effect at the particle scale\nwhich for ellipsoidal NP is uniaxial with a shape anisotropy constant proportional to\nthe NP volume1. Finally the finite size of the NP is the source of surface anis otropy\nresulting from symmetry breaking, surface defects or chemi cal bonding of the coating\nlayer. When modeled by a transverse anisotropy or the N´ eel s urface anisotropy model21,\nthe resulting non collinearities of the surface spins can be represented through a cubic\nterm in the framework of the EOS14,15. Concerning spherical iron oxide nanoparticles, the\ngeneral experimental observation is that the uniaxial anis otropy dominates with however a\nrather large dispersion in the effective annisotropy constan t valueKeff13,22–27. Moreover\na small value of Keffis interpreted as a small amount of crystalline defects with in or\nat the nanoparticle surface24,27–29. In any case the effective uniaxial anisotropy constant\ncannot be compared to the intrinsic, or bulk one, since the la tter corresponds to the\ncubic symmetry and is negative at room temperature. Further more from the particle\nsize dependence it is generally concluded that the uniaxial anisotropy is predominantly a\nsurface anisotropy with a related constant Ks= (d/6)Keff13,22,26.\nAt the atomic scale the well known N´ eel model of pair anisotr opy21is often invoked to\ndeduce surface anisotropy either in thin film geometry or in 3 D NP. In the framework of\nEOSapproach, thedeviationfromthesphericalshapetransl atesintheN´ eel pairanisotropy\nmodel, in addition to the magnetic dipolar term responsible for the shape anisotropy, in a\ncontribution with the same symmetry and proportional to the NP surface because of its\nshort range character. This leads for ellipsoidal NP to a sur face contribution of uniaxial3\nsymmetry. One has to keep in mind however that the N´ eel model although useful in the\nsensethatitreproducesthecorrectdescriptionofthesymm etryofthemagneticanisotropy,\ndoes not provide the physical understanding of the single io n anisotropy30. Thus when\ndealing with spinel ferrites oxides as well as with Fe or Ni si ngle domain nearly spherical\nnanoparticles in the framework of a EOS model, combined unia xial and cubic anisotropies\nshould be taken into account because of the intrinsic cubic a nisotropy on the one hand,\nthe shape (uniaxial) and surface contributions (uniaxial a nd/or cubic) on the other hand.\nIn Ref. [31] the expansion of the linear and non linear suscep tibility for non interacting\nassembly with either uniaxial or cubic anisotropy has been p erformed with the result that\nwhen the 3 axes of the cubic contribution are randomly distri buted both the linear and the\nfirst non linear susceptibilities are anisotropy independe nt. In Ref. [32] the LLG equation\nis considered to calculate the hysteresis curve at vanishin g temperature of non interacting\nNP, with randomly oriented cubic axes. The easy axis of the un iaxial term is either fixed\nat conveniently chosen direction or randomly distributed. In Ref. [33] an assembly of\nweakly interacting NP is considered both from perturbation theory and MC simulations\nwith cubic anisotropy relative to the same cubic axes for all the NP combined with an\nuniaxial anisotropy with a random distribution of easy axes .\nIn the present work, we perform MC simulation of NP assemblie s interacting through\nDDI with cubic and uniaxial contributions to the anisotropy energy. Having in mind the\ncase of strongly interacting powder samples of NP dispersed at zero field, we consider the\ncase of NP with cubic axes randomly distributed. The uniaxia l easy axis on the other\nhand is either randomly distributed independently of the cu bic axes or oriented along a\nparticular crystallographic orientation of the particles for which two cases are considered,\nnamely{100}or{111}. Themainpurposeofthepresent workis toinvestigate wheth erthe\ncubic contribution to the anisotropy leads to an observable deviation to the magnetization\ncurve in the superparamagnetic regime. We also revisit the c onsequences of the DDI in\nthe strong coupling regime, in particular on the linear susc eptibility at low field, through\nthe comparison of simulations performed either with free bo undary conditions on spherical\ncluster or with periodic boundary conditions to simulate an infinite system.\nIn section II, we give the details of the model and explicit th e different energy contribu-\ntions. Section III is devoted to the results and the comparis on with experimental results\nand we briefly conclude in section IV.4\nII. MODEL FOR DENSELY PACKED ASSEMBLIES\nWe consider a EOS model with nanoparticles described as non o verlapping spheres\nbearing at their center a permanent point dipole representi ng the uniform magnetization\nof the particle (macro spin). The moment of each particle is e qual to its volume times\nthe bulk magnetization, Ms, which means that no spin canting effect is explicitly taken\ninto account. The particles are surrounded by a non magnetic layer of thickness ∆/2,\nrepresenting the usual coating by organic surfactant molec ules. The particle diameters,\n{di}are distributed according to a log-normal law defined by the m edian diameter dmand\nthe standard deviation σofln(d),\nf(d) =1\nd√\n2πσexp/parenleftbigg\n−(ln(d/dm))2\n2σ2/parenrightbigg\n(1)\nIn the following, we use dmas the unit of length, and the distribution function in reduc ed\nunit is totally determined by the single parameter σwhich characterizes the system poly-\ndispersity. When dealing with interacting particles, we ma inly have in mind the case of\nlyophilized powders samples or high concentration nanopar ticles assemblies embedded in\nnon magnetic matrix. Accordingly the coated particles are d istributed in densely packed\nclusters whose external shape is spherical in order to avoid the demagnetizing effects due\nto the system shape with the free boundary conditions. We emp hasize that this NP\nconfiguration has an experimental justification since upon d rying the NP are likely to ag-\ngregate in spherical shaped large clusters which has been co nfirmed from simulations34.\nMoreover, we consider mainly the superparamagneticregime , wherewe simulate only equi-\nlibrium magnetization curves corresponding to the static o r infinite time measurements\n(τm→ ∞).\nWe include only the leading terms of the anisotropy energy; t he cubic symmetry con-\ntribution for particle say iof moment v(di)Msˆmican be written as\nE(i)\nc\nv(di)=Kc/parenleftbig\nm2\nxim2\nyi+m2\nyim2\nzi+m2\nxim2\nzi/parenrightbig\n=Kc\n2/parenleftBigg\n1−/summationdisplay\nα=x,y,zm4\nαi/parenrightBigg\n(2)\nwhere we have used the unitarity of ˆ miin the second equality. Here and in the following\nhated letters denote unit vectors. In equation (2) mαirefer to the ˆ micomponents in the\nlocal cubic frame of the particle considered. Let us denote b y{ˆxαi},α= 1,3 this local\ncubic frame; dropping an irrelevant constant, the total cub ic anisotropy of the system can\nbe written\nEc=−Kc\n2/summationdisplay\niv(di)/summationdisplay\nα(ˆxαiˆmi)4(3)5\nThe local axes {ˆxαi}can be oriented in different ways according to the physical sys tem\nunder study; for non textured distributions of particles we have to consider a random\ndistribution of the {ˆxαi}. Most of our simulations are performed in the case. The effect o f\nthe texturation is nevertheless examined by considering th at the [111] directions {ˆx1+ˆx2+\nˆx3}ior the{ˆx3}iaxes are confined in a cone along the ˆ z-axis according to the following\nprobability distribution for polar angles\nP(θ) =Csin(θ)exp(−(θ/σθ)2/2) ; (4)\nThe configuration with the {ˆxαi}fixed parallel to the system frame (ˆ x,ˆy,ˆz) is also consid-\nered. This later case is the same as that Margaris et al.33considered for which a strong\neffect of the cubic term is obtained while Usov and Barandiar´ a n32consider cubic axes\nrandomly distributed. Although we have in mind particles of intrinsic cubic anisotropy,\nwe are aware of a possible surface contribution to Ecas shown in Ref. [14] resulting from\nthe non collinearity of the surface spins; as a result the val ue ofKcmay differ form the\nbulk one. The uniaxial term is proportional either to the vol umev(d) or to the surface\ns(d) of the particle. The volume part stems a priorifrom the shape anisotropy where for\nellipsoidal particles K(u)\nsh, given by J2\ns(1−3Nu)/(4µ0) withJs=µ0MsandNu, the demag-\nnetizing factor along the revolution axis, can be deduced fr om the knowledge of the aspect\nratioξ. Notice that one can imagine easily a situation where the dev iation from sphericity\nis not characterized by the same aspect ratio for all particl es leading to a size dependence\nofKsh. For instance one cannot rule out the situation where the dev iation from sphericity\nfollows from a major axis of the form c= (d/2 +δ) and minor axes a=b=d/2, with\na size independent corrugation δ. Then from the demagnetizing factor in the major axis\ndirection\nNu=1−ǫ2\n2ǫ3/bracketleftbigg\nln/parenleftbigg1+ǫ\n1−ǫ/parenrightbigg\n−2ǫ/bracketrightbigg\n; withǫ= (1−1/ξ2)1/2andξ=c/a= 1+2δ/d ,\nthe shape anisotropy may transform in a surface uniaxial ani sotropy with an anisotropy\nconstant given by K(u)\ns≃J2\nsδ/(30µ0) from an expansion of K(u)\nshat order δ/d.\nTheeasy axes {ˆn}iare either randomlydistributedindependently of theparti cles frame\nor aligned along one specified crystallographic axis of the c rystallite. Different origin for\nsuch a easy axes distribution can be invoked. In the framewor k of the uniaxial anisotropy\noriginatingfromthedeviation tosphericityit correspond stoapreferential crystallographic\norientation for crystallite growth, whilein the framework of the uniaxial surfaceanisotropy\nthis may result from a preferential crystallographic orien tation for chemical bondingat the\nparticle surface. We have considered two possibilities, na mely ˆni={001}ior ˆni={111}i.\nIn the total energy, we include formally both surface and vol ume terms in the uniaxial6\ncontribution, with anisotropy constants K(u)\nvandK(u)\nsrespectively and at most one of\nthese is non zero in the simulations. The total energy thus in cludes the DDI, the one-body\nanisotropy term and the Zeeman term corresponding to the int eraction with the external\napplied field /vectorHa=Haˆh. Let{/vector ri},{v(i)},{/vector mi}and{/vector ni}denote the particles locations,\nvolumes, moments and easy axes respectively. The total ener gy of the cluster reads\nE=µ0\n4π/summationdisplay\ni<jmimjˆmiˆmj−3(ˆmiˆrij)(ˆmjˆrij)\nr3\nij−/summationdisplay\ni(K(u)\nvv(i)+K(u)\nss(i))(ˆniˆmi)2\n−Kc\n2/summationdisplay\niv(i)/summationdisplay\nα(ˆmiˆxαi)4−µ0Ha/summationdisplay\nimiˆmiˆh(5)\nmiare the moment magnitudes, rij=|/vector ri−/vector rj|. It is worth mentioning that the considera-\ntion of the anisotropy term with a fixed easy axes distributio n means that the magnetiza-\ntion relax according to a N´ eel process16,35, namely the particles are considered fixed while\ntheir moment relaxes relative to their easy axis. In the foll owing we usereduced quantities;\nfirst the energy is written in kBT0units,T0being a suitable temperature ( T0= 300Kin\nthe present work) and we introduce a reference diameter, dref. The reference diameter,\ndrefis a length unit independent of the size distribution, usefu l for the energy couplings,\nand can be chosen from a convenient criterion independently of the actual structure of the\nMNP assembly. The reduced total energy is given by\nβ0E=−ǫ(0)\nuv/parenleftbiggdm\ndref/parenrightbigg3/summationdisplay\nid∗3\ni(ˆniˆmi)2−ǫ(0)\nus/parenleftbiggdm\ndref/parenrightbigg2/summationdisplay\nid∗2\ni(ˆniˆmi)2\n−ǫ(0)\nc\n2/parenleftbiggdm\ndref/parenrightbigg3/summationdisplay\nid∗3\ni/summationdisplay\nα(ˆmiˆxαi)4−ǫ(0)\nd/parenleftbiggdm\ndref/parenrightbigg3/summationdisplay\ni<jd∗3\nid∗3\njˆmiˆmj−3(ˆmiˆrij)(ˆmjˆrij)\nr∗3\nij\n−h/summationdisplay\nid∗3\niˆmiˆh , (6)\nwith\nǫ(0)\nuv=β0K(u)\nvv(dref) ;ǫ(0)\nus=β0K(u)\nss(dref) ;ǫ(0)\nc=β0Kcv(dref)\nǫ(0)\nd=β0µ0\n4π(π/6)2M2\nsd3\nref;h=β0µ0Ms(π/6)d3\nmHa≡/parenleftbiggdm\ndref/parenrightbigg3Ha\nHref(7)\nwhereβ0= (kBT0)−1and the stared lengths are in dmunit. The dimensionless dipolar\ncoupling constant is then ǫd= (dm/dref)3ǫ(0)\ndand the dimensionless anisotropy constants\nareǫuv= (dm/dref)3ǫ(0)\nuv,ǫus= (dm/dref)2ǫ(0)\nus, andǫc= (dm/dref)3ǫ(0)\ncfor the volume and\nsurface uniaxial and cubic contributions respectively. Th e reference diameter, drefcan be\nchosen such that ǫd(dm=dref)≡ǫ(0)\nd= 1 ; the reduced external field hcoincides with the\nusual Langevin variable at temperature T0for a monodisperse distribution with d=dm7\nIn equation (7), we also introduce the reference external fie ld,Hreffor convenience.\nConcerning the structure in position, the nanoparticles su rrounded by their coating\nlayer of thickness ∆/2 form an assembly of hard spheres of effec tive diameters {di+∆}\nwhich are arranged in large densely packed clusters with eit her a random or a well ordered\nsimple structure (simple cubic lattice) The clusters are bu ilt as in Ref. [36]. First a\nlarge stacking of the coated spheres is made in a parallepipe dic box with the desired\nstructure, random or well ordered. Once this first step is per formed, we cut within the\nglobal stacking the cluster we want to study by imposing both the external shape and the\nnumberof particles Np, with typically Np≃1000. Because of thecoating layer of thickness\n∆/2 the closest distance of approach between particles i,jis shifted from ( di+dj)/2 to\n(di+dj)/2 + ∆ and we therefore define an effective dipolar constant corre sponding the\nparticles uncoated at contact\nǫeff\nd=ǫd1\n(1+∆/dm)3=ǫdφ\nφmϕ(σ,∆/dm) (8)\nwhereφis the volume fraction and φm=φ(∆ = 0) is the maximum value of φfor a\ngiven structure. The function ϕ(see appendix) in equation (8) is equal to 1 for σ= 0 and\nremains very close to 1 for σ <0.1. Two systems differing by ∆ or σand characterized\nby the same value of ǫeff\ndcorrespond to the same intensity of DDI. Notice that for weak\npolydispersity, ǫeff\ndis related to the parameter ywidely used in the works dealing with\nthe dipolar hard sphere fluid (DHS) which in our notations rea dsy= 8β∗φǫd/3.\nAlthough we do not limit our simulations to a specific experim ental system, we have\nin mind iron oxide NP to guide our choice of the physical param eters entering the model.\nAt room temperature the bulk anisotropy constant is Kc=-11.0 to -13.0 kJm−3and -\n4.70 kJm−3for magnetite and maghemite respectively. The saturation m agnetization\nfor these two materials are quite close and lead to Js∼0.50T. Therefore in this work\nwe useJs= 0.50 T, which corresponds to dref= 10nmwhen this later is fixed from\nǫd(dref) = 1.0. The shape anisotropy constant can be estimated for el lipsoidal NP once\nthe aspect ratio is known; using Js= 0.5Twe getK(u)\nsh≃50(1−3Nl) kJm−3which leads\ntoK(u)\nsh<7.0 kJm−3for NP characterized by an aspect ratio ξ <1.20, and accordingly\nǫ(0)\nuv= 0.95. Concerning the surface anisotropy constant, we conside r the experimental\nvalues ranging from Ks= 5.5 10−6Jm−2toKs= 2.7 10−5Jm−2for maghemite13,22,27;\nthusKs= 2.7 10−5Jm−2is considered somewhat as an upper bond for iron oxide NP.\nNotice that if we consider the deviation from sphericity res ulting from a size independent\ncorrugation δas outlined above, we get K(s)\nsh≃2.70 10−5Jm−2withδ=2nm, which\ncorresponds to an aspect ratio ranging from 1.4 to 1.2 for NP o f diameter ranging from8\n10 to 20 nm. The corresponding values of the reduced parameters for Iro n oxide NP of\nc.a.10 to 20 nmin diameter are summarized in table I. In the following we can consider\nthat a characteristic value for the uniaxial anisotropy is a boutǫuv∼5 while a maximum\nvalue for the cubic anisotropy constant is |kc|= 15.\ndm/dref11.201.331.712.0\nǫca)-1.65-2.85-3.88-8.25-13.20\nǫcb)-0.60-1.05-1.41-3.0-4.8\nǫuvc)0.6251.081.473.135.00\nǫuvd)1.001.732.375.08.00\nǫuse)2.052.953.626.08.20\nǫd1.01.732.375.08.00\nǫeff\ndf)0.601.091.563.66.00\nǫeff\ndg)1.56\nTable I: Reduced values for the parameters of the model corresp onding to iron oxide NP. with Kc\n= - 13 kJm−3a); -4.7 13 kJm−3b); an aspect ratio ξ= 1.135c); or 1.20d); or surface anisotropy\nconstant Ks= 2.70 10−5Jm−2e).ǫeff\ndfrom equ. (8) with ∆ =2 nmf)or 14.5 nmg).\nThe main effect of the DDI on the magnetization curve at low field is a strong reduc-\ntion of the initial slope of M(H) versus H, namely of the linear susceptibility36,37. This\nis directly related to the well known plateau in the FC magnet ization in terms of the\ntemperature for T < T Boccurring in strongly interacting NP and also the the platea u\nin theχ(dm) curve obtained in Ref. [36] for densely packed clusters of N P. Indeed, in the\nabsence of anisotropy, the hamiltonian (6) can be rewritten , withhr=Ha/Href\nβE=β∗β0E=−β∗ǫ(0)\nd/parenleftbiggdm\ndref/parenrightbigg31\n(1+∆/dm)3/summationdisplay\ni<jd∗3\nid∗3\njˆmiˆmj−3(ˆmiˆrij)(ˆmjˆrij)\n(r∗\nij/(1+∆/dm))3\n−hrβ∗/parenleftbiggdm\ndref/parenrightbigg3/summationdisplay\nid∗3\niˆmiˆh (9)\nwhere we have introduced the geometrical sum of the reduced D DI of the most con-\ncentrated cluster (∆ =0) of the structure considered, namel y where particles can get at\ncontact which thus depends neither on dmand ∆. Introducing the dimensionless variable9\nλ=β∗(dm/dref)3we get\nβE=−λǫeff(0)\nd/summationdisplay\ni<jd∗3\nid∗3\njˆmiˆmj−3(ˆmiˆrij)(ˆmjˆrij)\n(r∗\nij/(1+∆/dm))3−hrλ/summationdisplay\nid∗3\niˆmiˆh (10)\nFrom equation (10), we can conclude that, when the field is van ishingly small, the leading\ncontribution to the magnetization linear in hrdepends on hrandβ∗only through λhrand\nλǫeff(0)\nd. More precisely, M/Ms≃λhrf(λǫeff(0)\nd) withfa scaling function. Thus the\nlinear susceptibility, χ=∂M/∂H a≃M/Haat vanishing Hamust be in the form\nχ=Ms\nHrefλf(λǫeff(0)\nd) (11)\nIn the limit of zero coupling f(x= 0) is a finite constant and we recover the Langevin\nresult,χ∝λ= (6/π)(T0/T)v(dm). In the interacting system, the strong coupling limit\nλǫeff(0)\nd>>1 or equivalently y >>1 is obtained through the increase of either the DDI\ncoupling, ǫ(0)\ndorβ∗(decrease of T). In this case, the limiting value of the susceptibility\ncan be obtained. We note that the linear susceptibility we de al with is the external one,\nrelating the magnetization to the external, or applied field Haand since we consider the\nmagnetization per unit magnetic volume, the magnetization per unit volume is Mv=Mφ.\nThus the internal field is related to the external one through Hi=Ha−DhφMwhereDh\nis the demagnetizing factor of the sample in the direction of the field. Hence we can relate\nχto the internal susceptibility, χithrough the usual way16\nχ=χi\n1+Dhφχi(12)\nWe can also introduce the relative permeability, µ= (1+φχi) to get\nφχ=µ−1\n1+Dh(µ−1)(13)\nIn the case of a spherical system as those considered here, Dh= 1/3 and equation (13)\nreads\nφχ=3(µ−1)\nµ+2(14)\nItisworthmentioningthat χisrelated tothemomentfluctuationsthroughthefluctuation -\ndissipation theorem40as already used in [36]. We have in an isotropic system\n∂(M/Ms)\n∂h=χr=β∗N¯v\n3v(dm)\n/angbracketleftBig\n(|Σ/vector mi|)2/angbracketrightBig\n(Σmi)2−|∝an}bracketle{tΣ/vector mi∝an}bracketri}ht|2\n(Σmi)2\n≡β∗¯v g\n3v(dm)(15)10\nwhich introduces the factor gand where ¯ vis the average value of the particle volume over\nthe distribution function. From equation (15) we rewrite (1 4) in the equivalent form\n3(µ−1)\nµ+2=φχ= 8φǫdβ∗¯v\nv(dm)g (16)\nNow in the strong coupling limit we expect the system to reach a ferromagnetic transition\nas is the case for the DHS fluid38,39. In this limit the permeability µ→ ∞and a limiting\nvalue for χand thus a plateau in the FC magnetization when the temperatu re is decreased\nis obtained with, from equation (14)\nχ→3\nφor ˜χ→3\n8ǫ(0)\ndφwith ˜χ=Href\nMsχ (17)\nThis is quite well reproduced by the present simulations (se e section (III)) and in total\nagreement with the behavior of ˜ χin terms of the particle size dmwe obtained in Ref. [36]\nin the quasi monodisperse case where ϕ≃1 which is easily deduced from (17) by writing\nφin terms of ∆ /dm\n˜χ→ϕ(1+∆/dm)3\n8ǫ(0)\ndφm(18)\nIt is important to note that equation (14) is the well known re lation between the dielectric\nconstant and the polarization susceptibility in the DHS flui d in the case of an infinite\nspherical system embedded in vacuum, i.e.surrounded by a medium of dielectric constant\nǫs= 1. Indeed the magnetic permeability plays the role of the di electric constant of the\nDHS and the polarization susceptibility is related to the flu ctuations or the Kirkwood\nfactorgK(ǫs), equivalent to the factor gintroduced above; in the monodisperse case, with\nthe dielectric constant, ǫ, in place of µthe DHS satisfies40,41\nµ−1\nµ+2=ygK(ǫs= 1) ; or µ−1 = 3ygK(ǫs=∞) (19)\nNotice that the second equation (19) is the equivalent of (13 ) written for Dh= 0 and cor-\nrespondsto the case whereeither through the boundarycondi tions (ǫs=∞) or the system\nshape(Dh= 0) the system can be uniformly polarized. Equation (19) is s trictly equivalent\nto (16) since in the present model we have, in the monodispers e case,χr=β∗g/3. The\nDHS undergoes a ferromagnetic transition at which the diele ctric constant diverges and as\na result38,42,43, one expects a limiting value for the Kirkwood factor gK(ǫs= 1)→1/y\nand accordingly χr→β∗/(3y) orχ→3/φin agreement with equation (17).\nThe plateau in the FC magnetization at low temperature and lo w field is a behavior\nobserved in the framework of the FC/ZFC procedure25,29,44–46generally related to a col-11\nlective behavior of the dipoles leading to a frozen state. He re, by analogy with the known\nbehavior of the DHS fluid, we relate this plateau to the approa ch of the onset of the fer-\nromagnetic transition at least for σ <<1 and in the absence of MAE. We emphasize that\nas can be deduced from equation (14), in the case of a spherica l system surrounded by\nvacuum, χbecomes nearly independent of µwhenµincreases beyond a sufficiently high\n(µ∼35) but still finite value. As a result χgets close to its limiting value before the\nferromagnetic transition.\nThe Monte Carlo simulations are performed according to the u sual Metropolis\nscheme40,41,47. The trial move of each moment is performed within a solid ang le cen-\ntered on its old position. Since we seek equilibrium configur ations, the maximum solid\nangle of the move is only restricted by the acceptance ratio, R∼0.35–0.50. Moreover we\nuse a annealing scheme at all values of the field in the range wh ere we expect an hysteresis.\nThe averages are performed on 10 to 30 independent runs (up to 70 runs for low tempera-\nture and/or large DDi couplings) with 3104to 4104thermalisation MC steps followed by\nanother set of 3104to 4104MC steps to compute the averages.\nIII. RESULTS\nNon interacting system\nIn this section we deal with the case free of DDI. We first have c hecked that as h→0\nwith volume uniaxial MAE and a random easy axes distribution the linear susceptibility is\nǫuvindependentwhile with cubicMAE and randomly distributed a xes, both the linear and\nthe first non linear susceptibilities are kcindependent and accordingly we get a nearly kc\nindependent M(h) beyond the very vicinity of h=0. This is shown in figure 1 in terms of\nthe inverse reduced temperature β∗. Moreover we also check in figure 1 that the deviation\nofM(h) relative to the isotropic case is negative whatever the sig n ofkcwith the random\ndistribution of cubic axes. This is no more the case when the c ubic axes of the particles\nare fixed where on the one hand only the linear susceptibility iskcindependent and on\nthe other hand the sign of ( M(h,kc)−M(h,kc= 0)) depends on the sign of kc. The same\nresult holds when ǫd∝ne}ationslash= 0.\nFor randomly distributed cubic axes, the cubic MAE has only a negligible effect on\ntheM(h) curve. On the opposite, as shown in figure 2, when the cubic ax es are fixed\nalong the system frame, the cubic MAE has a strong effect on the M(h) curve. Moreover,\nas noted above in the low field region, the sign of the anisotro py induced deviation of\nM(h) depends on the sign of kc. This is expected since a positive value of kcwill favor\nthe principal frame directions for the moments; for an appli ed field along one of these12\ndirections, say ˆh=ˆz,kc>0 leads to a positive deviation of M(h) andvice versa . The\nresults displayed in figure 2 are in agreement with those of Re f. [33] (notice that our kc\ncorresponds to w/2 of Ref. [33]).\nThe effect of the texturation through the preferential orient ation along the ˆ z-axis of\nthe crystallites [111] direction according to the probabil ity density (4) is shown in figure 3\nfor the polydisperse and monodisperse cases.\nConcerning the uniaxial anisotropy, we note that the surfac e contribution can be very\nwell approached by the volume term with the introduction of a n effective volume uniaxial\nconstant, ǫeff\nuvtaking into account the polydispersity. In equation (6), we rewrite the\nuniaxial energy terms by introducing the reduced n−thorder moments d∗\nnof the diameter\ndistribution function and under the hypothesis that (/summationtextd∗n\ni(ˆniˆmi)2)/d∗\nnis independent of\nnat least for n≤3 we get\nǫeff\nuv=d∗\n2\nd∗\n3ǫus= exp(−5σ2/2)ǫus (20)\nwhere we have used the analytical result for the d∗\nnof the lognormal law. The same\nconclusion holds in presence of DDI; in figure 4 we compare the deviation of M(h) due to\nthe surface uniaxial MAE with that due to the volume uniaxial MAE with ǫuv=ǫeff\nuv\ntaken from (20) in the case of a polydisperse interacting sys tem.\nWe now consider the case of combined uniaxial and cubic aniso tropies. The result is\nshown for a typical set of parameters, ǫuv= 5 and |kc|= 15 in figure 5. As is the case\nwhen only the cubic anisotropy is taken into account, we find t hat the effect on M(h) of\nthe cubic anisotropy with random distributed cubic axes is v ery small when the uniaxial\neasy axes are also randomly distributed and uncorrelated fr om the cubic ones. This is no\nmore the case when, still for a random distribution of cubic a xes, the easy axes {ˆn}iare\nalong a specified crystallographic orientation of the cryst allites. The cubic MAE enhances\nthe uniaxial one when ǫc>0 and{ˆn}i= [001], or when ǫc<0 and{ˆn}i=[111]. This is\nqualitatively expected since then the two components of the MAE tend to favor the same\nlocal orientation for the moment.\nA shoulder in M(h) is clearly observed when {ˆn}i= [001] and ǫc>0 or{ˆn}i=[111]\nandǫc<0. This can be compared to the behavior of the hysteresis curv es determined\nby Usov and Barandiar´ an [32] when the easy axis of the uniaxi al MAE component is fixed\nrelative to the NP frame. This shoulder is enhanced when eith er the inverse temperature β\nincreases or when the polydispersity σincreases (see figure 6). This latter point is simply\nduetothepresenceoflargerparticlesinthedistributionw henσincreases, withaccordingly\nlarger anisotropy energies. We can be interpret this featur e as the coherent contributions\nof uniaxial and cubic terms. In the case ǫc<0 where the favorable orientations are the13\n{111}axes, we find that the cubic contribution remains to enhance t he uniaxial anisotropy\nconstant by a factor of roughly |ǫc|/5 as shown in the inset of figure (5).\nInteracting systems\nMost of our simulations with DDI are performed with free boun dary conditions (FBC)\non large spherical NP clusters of Np∼1000 particles. In order to check the validity of the\nmethod, we have performed simulations with periodic bounda ry conditions (PBC) with\nEwald sums for the DDI in both the conducting or the vacuum ext ernal boundary condi-\ntions40,41. This is done by using either ǫs=1 orǫs=∞for the surrounding permeability\n(or dielectric constant in the electric dipolar case). Here we are interested in the determi-\nnation of the linear susceptibility for the infinite system e mbedded in vacuum, as we seek\nthe magnetic response in terms of the external field. Therefo re, we check that one can get\nχr(ǫs= 1) from simulations on a large spherical NP cluster with FBC , or by using PBC\nwith Ewald sums in either the conducting or the vacuum bounda ry conditions. The value\nofχr(ǫs= 1) can be obtained from a simulation with external conducti ng conditions by\nexploiting in equation (19) the independence of µwith respect of ǫsas it is an intrinsic\nproperty ,\nχr(ǫs= 1) =χr(ǫs=∞)/(1+8φǫdχr(ǫs=∞)). (21)\nThe comparison of χr(ǫs= 1) from the three routes is shown in figure 7 in the absence of\nanisotropy and in the quasi monodisperse case ( σ=0.05). We have used the same initial\ncluster and extracted either a spherical cluster for FBC or a cubic simulation box for PBC\nwith a value of ∆ fitted on the volume fraction φ. Moreover we have checked that for\nmoderate values of the DDI coupling the permeability obtain ed from these three routes\nleads to similar values. These two points show the coherence of our simulations with DDI.\nWhen compared to the results of Klapp and Patey48the curve µ(y) we get at φ=0.385\nlies in between the ones of the frozen model with correlation and of the frozen model with\nquenched disorder, much closer to the former and in fact very close to that of the DHS\nfluid.\nBeside the strong reduction of the initial susceptibility, the DDI reduce also the devi-\nation of the M(h) curves due to MAE, as can be seen in figure 8. As expected the cu bic\nanisotropy has nearly no influence on the M(h) when the easy axes and the cubic axes are\nindependently randomly distributed; on the other hand the c hange in the M(h) curve due\nto the cubic contribution when {ˆn}iare along the crystallites [111] with kc<0 or along\nthe[001] with kc>0is smaller than in the absence of DDI. Nevertheless, the con tribution14\nof the cubic anisotropy may be not negligible under the condi tion of a coherence with the\nuniaxial term. Moreover, we do find that in order for the cubic term to give a noticeable\neffect a rather large value of the cubic anisotropy constant, kcis necessary.\nIn opposite to what we get in the absence of DDI, we do not find an y distinctive\nfeature of either the cubic or the uniaxial symmetry on the M(h) curve if the cubic axes\narerandomlydistributedinthecaseofcombinedoronlyunia xialanisotropy. Thisisshown\nin figure9 wheredifferent combinations of anisotropies leadi ng to comparable M(h) curves\nare considered for ǫeff\nd= 1.\nFinally we consider the comparison with the experimental ma gnetization curves of\nRef. [49] on powder samples of maghemite NP differing by their s ize. These samples are\ncharacterized by a polydispersity σ∼0.27 and the estimated coating layer thickness is\nc.a.2nm. The behavior of the M(Ha) curve being controlled by the DDI and the MAE\nat low and intermediate values of the applied field respectiv ely, we fit the value of ∆ by\nthe slope at Ha∼0 and the anisotropy constants on the behavior of M(Ha) at higher\nvalues of Ha. We find that the region Ha∼0 is well reproduced with ∆ = 2 nmfor\ndm= 10nmand 21nm, and ∆ = 2.4 nmfor 12nm, which does not differ much from\nthe estimated experimental value. Concerning the cubic ani sotropy since the experimental\nsamplesarenottexturedweconsideronlyarandomdistribut ionofcubicaxes. Thevalueof\nthe corresponding anisotropy constant may differ from its kno wn bulk value due to surface\neffects; however, we consider the bulk value as a starting poin t. Inany case, since thecubic\nanisotropy constant for iron oxide is rather small, we expec t only a small effect of the cubic\ncontribution to the MAE and accordingly we consider only the case where the cubic and\nthe uniaxial components of the MAE reinforce each other. Wit hǫc<0, this means that\nwe limit ourselves to a easy axes distribution {ˆn}i= [111] i. For the uniaxial MAE we\nhave to choose either a surface or volume dependent MAE (see e quation (6)); however, we\nhave shown that the surface dependent MAE can be reproduced b y the volume dependent\none through the effective constant of (20). Hence, starting fr om the bulk value for ǫc\nwe are left with ǫuvas the only fitting parameter. We find ǫuv= 4.00 for dm= 10nm\nby fitting M(Ha) in the intermediate field range; then, the same quality of ag reement\nbetween the model and the experimental curves is obtained fo rdm= 12nmand 21nm\nby using a value of ǫuvscaling as d3\nm, namely ǫuv= 6.912 and 32.0 for dm/dref= 1.2 and\n2.0 respectively, i.e.Kv= 31.6 kJm−3(we use the simulated curve for dm/dref= 2 for\ncomparison of the experimental curves of the samples with dm18 and 21 nm; only the\nsecond is presented here). Notice the weak hysteresis cycle for the experimental sample\ncharacterized by dm= 21nm; this is due to the largest particles in the distributi on and\nis not reproduced by the M.C. simulations, since we have chos en to perform equilibrium\n(τm=∞) simulations only. The cubic MAE gives only a small contribu tion toM(Ha)15\nas illustrated by the difference obtained using ǫcdeduced from either the magnetite or\nthe maghemite bulk values given in Table I (see figures 11 and 1 2). Therefore, we find\nthat using the iron oxide bulk value for the cubic MAE constan t the experimental NP of\nRef. [49] can be modeled excepted in the high field region, by N P presenting a volume\ndependent uniaxial anisotropy with Kv= 31.6kJm−3. However, as we have shown, we can\nget similar M(Ha)curveswithdifferent combinations ofcubicanduniaxialMAE especially\nwith the DDI which weaken the peculiar features of the cubic c ontribution. Hence, we\ncan get the same agreement with experiment by using on the one hand a uniaxial MAE\nscaling as d2\nmcorresponding to a surface anisotropy and on the other hand a fitted cubic\ncontribution. Starting from ǫuv= 4 fordm/dref= 1.0 this gives ǫuv= 5.76 for dm/dref1.2\n(which translates to ǫus= 7.03 for σ= 0.28 and Ks= 6.4510−5Jm−2). Thecorresponding\ncubic component is obtained from our finding that an increase of|ǫc|corresponds to an\nincrease of ǫuvof roughly |ǫc|/5, leading to ǫc= -9 and Kc= -41kJm−3. We have also\nconsidered a fitted cubic MAE with a positive ǫc, and ˆni= [001] ifor which we find\nǫc= 5.0(Kc= 25.15 kJm−3). The results is shown in figure 11. Doing this means that the\ncubic anisotropy energy present an anomalous component, na mely/vextendsingle/vextendsingleKc−Kbulk\nc/vextendsingle/vextendsingle, scaling as\nthe NP volume while it should be understood as a surface effect. Hence, although it seems\ndifficult to conclude on the best fit of the experimental set con sidered, it may be better\nto avoid the latter contradiction and consider these NP as pr esenting a volume dependent\nuniaxial MAE; however, we then get a value for the effective ani sotropy constant too large\nto be explained only as a shape anisotropy. It is nevertheles s still in the range of what\nis obtained experimentally from TBfor iron oxide NP. In any case, we have to take such\nconclusions with care given the simplicity of the model. Sim ilarly, the high field range\ncannot be reproduced with the simple OSP model and necessita tes a the inclusion of a\nfield dependent description of the individual NP.\nIV. CONCLUSION\nIn this work, we have performed Monte Carlo simulations of ro om temperature magne-\ntization curves in the superparamagnetic regime, with a par ticular attention paid to the\niron oxide based NP. We focused on the search for a peculiar fe ature of the cubic MAE\ncomponent on the M(Ha) curve since iron oxide and spinel ferrites in general prese nts an\nintrinsic MAE with cubic symmetry while from experiments a u niaxial MAE is generally\nfound. Our result is that a peculiar feature of the cubic comp onent can be obtained only\ni)if the the cubic and the uniaxial components are correlated t hrough the alignment of\nthe NP easy axes on a specified crystallographic orientation of the crystallites; ii)if the\nDDI are negligible viaa small NP volume fraction. Nevertheless a large value of the cubic16\nMAE constant compared the uniaxial one is necessary for the f ormer to give a noticeable\neffect on the room temperature M(Ha).\nV. APPENDIX A\nInthisappendixweexplicitthefunction ϕintroducedinequation (8). Thevolumefraction\nis defined as\nφ=Np1\nV/integraldisplay∞\n0f(d)π\n6d3d(d) =πd3\nm\n6Vd∗\n3 (A.1)\nwhereVis the total volume and d∗\nnis the reduced n−thmoment of f(d). Each particle\nof diameter dis surrounded by a coating layer of thickness ∆ /2; the maximum value of\nthe volume fraction, φmis obtained as the volume fraction of the spheres including b oth\nthe particles and the coating layer, namely by replacing din (A.1) by ( d+ ∆) with the\nsame distribution function. Defining ∆∗= ∆/dmwe get\nφm=Np1\nV/integraldisplay∞\n0f(d)π\n6(d+∆)3d(d)\n=πd3\nm\n6Vd∗\n3(1+∆∗)3/bracketleftbigg1+3∆∗(d∗\n2/d∗\n3)+3∆∗2(d∗\n1/d∗\n3)+∆∗3(1/d∗\n3)\n(1+∆∗)3/bracketrightbigg\n(A.2)\nwhich defines the function ϕas the expression in square brackets. 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Patey, The Journal of Chemical Physics 115, 4718 (2001).\n[49] C. de Montferrand, Y. Lalatonne, D. Bonnin, N. Li` evre, M. L ecouvey, P. Monod, V. Russier,\nand L. Motte, Small 8, 1945 (2012).19\n-0.03-0.02-0.01 0 0.01 0.02 0.03\n 0  0.5  1  1.5  2M/Ms - M/Ms(εc = 0)\nβ*\nFigure 1: Deviation of the reduced magnetization M/Msdue to MAE at h= 0.20 for a non inter-\nacting system with cubic anisotropy. Polydispersity: σ= 0.28. Cubic axes randomly distributed\nandǫc=15, solid circles; ǫc= -15, open circles. Cubic axes fixed and parallel to the system fram e\nwithǫc= 15, solid squares; ǫc= -15, open squares.\n 0 0.2 0.4 0.6 0.8 1\n 0 2 4 6 8 10 12 14M/Ms\nhεc < 0εc > 0\nFigure 2: Magnetization curve for a monodisperse non interacting s ystem with cubic anisotropy.\nThe cubic anisotropy axes are fixed along the system frame with ǫc=±15 long dashed; ±12\ndashed; and ±8 short dasched. The sign of ǫcis as indicated. The case with random distribution\nof the cubic axes is shown for comparison with ǫc= 15, thin solid line; and ǫc=-15, thin dotted\nline. The thick solid line is the reference ǫc= 0 case. β∗= 1.20\n 0 0.2 0.4 0.6 0.8 1\n 0 2 4 6 8 10 12 14 16 0.2 0.4 0.6 0.8 1M/Ms\n  h σ = 0.28\nσ = 0\nFigure 3: Magnetization curve for non interacting system with cubic anisotropy, |ǫc|= 15 and\nβ∗= 1. The [111] direction of the cristallites are prefentially oriented alo ng thezaxis (which is\nalso the direction of the field) with the probability distribution of equa tion (4). Polydisperse case\n(σ= 0.28) with ǫc= -15 and σθ= 0.015, long dash; π/10, short dash; π/2, solid line. Same with\nǫc= 15 and σθ=π/2, dotted line; π/10, short dash dot; 0.015, long dash dot. Monodisperse case\n(σ= 0) with ǫc= 15 and σθ= 0.015, open triangles; π/10, open squares; π/2, open circles.\n 0 0.2 0.4 0.6 0.8 1\n 0  2  4  6  8  10  12M/Ms\nh\nFigure4: M(h)foraninteractingsystemcharacterizedby ǫd=2.37,∆ /dref=0.20,dm/dref=1.33\nandβ∗= 1. Without anisotropy: solid line. In the presence of uniaxial anisot ropy with ǫuv= 5.64\nandǫus= 0.0, solid squares; ǫuv= 0.0 and ǫus= 6.88, open circles. (The value ǫus= 6.88\ncorrespondsto ǫuv(d∗\n3(σ)/d∗\n2(σ)) withǫuv=5.64,d∗\nnisthen-thmomentofthediameterdistribution\nfunction.)21\n 0 0.2 0.4 0.6 0.8 1\n 0  2  4  6  8 10  12  14M/Ms\nhεc = 0\nεc =  15; rand\nεc = -15; rand\nεc =  15; [111]\nεc = -15; [111]\nεc =  15; [001] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0 1 2 3 4 5\nFigure 5: Magnetization curve for non interacting system with uniax ial and cubic anisotropieswith\nβ∗= 1,ǫuv= 5 ,ǫus= 0 and |ǫc|= 15. Polydispersity : σ= 0.28. Open circles: case free of\nanisotropy for comparison. ǫcand easy axes distributions as indicated. Inset : comparison of the\nM(h) curves for ǫuv= 5 and ǫc=−15, long dash dotted line and for ǫuv= 8 and ǫc= 0, solid line.\n 0 0.2 0.4 0.6 0.8\n 0 2 4 6 8 10 12 14 16 18M/Ms\nha)\n 0.2 0.4 0.6 0.8\n 0  2  4  6  8  10b)\nFigure 6: Reduced magnetization for a non interacting system with ǫuv= 5.0,ǫc= -15, cubic\naxes randomly distributed, easy axes along the [111] NP cristallogra phic orientations and different\nvalues ofthe reduced inversetemperature β∗.β∗= 0.5, dash dotted line; 0.75, dotted line; 1.0, long\ndashed line; 2.0, solid line; 4.0short dashed line. a)monodisperse system ( σ= 0);b)polydispersity\nσ= 0.28.22\n 0 0.05 0.1 0.15 0.2 0.25\n 0  0.5  1  1.5  2  2.5χr \nβ∗εd = 1.33\nεd = 2.66\nFBC\nPBC (εs = 1)\nPBC (εs = ∞)\nFigure 7: Reduced linear susceptibility, χrversus the inverse reduced temperature β∗in the quasi\nmodisperse case, σ= 0.05 for a volumic fraction φ= 0.385,ǫd= 1.33 and 2.66 ( ǫeff\nd= 1.0 and\n2.0 respectively). Different boundary conditions are considered. I n the PBC with Ewald sums, the\nnumber of particles is Np= 600 while the clusters for the FBC include Np= 1000 particles. Solid\nline :M/Msforh= 1. Solid horizontal lines indicate the limit for y→ ∞, (equation (17). The\nsolid triangle at β∗= 1 indicates the value of χrforǫeff\nd= 1.0 in the polydisperse case σ= 0.28\n(ǫd= 1.73; ∆ /rm= 0.40).\n 0 0.2 0.4 0.6 0.8 1\n 0 2 4 6 8 10 12 14 16 18M/Ms\nh\nFigure 8: Reduced magnetization for a polydisperse interacting sys tem with β∗= 1,ǫd= 2.37,\n∆/dm= 0.15, polydispersity σ= 0.28 and different sets of MAE constants. ǫuv= 0.0 and ǫc= 0,\ndotted line; ǫuv= 5.0 and ǫc= 0, solid line; ǫuv= 5.0,ǫc= 15 and ˆ n= random, open squares;\nǫuv= 5.0,ǫc= -15 and ˆ n= random, solid squares; ǫuv= 5.0,ǫc= 15 and ˆ n= [111], short dashed\nline;ǫuv= 5.0,ǫc= -15 and ˆ n= [111], long dashed line; ǫuv= 5.0,ǫc= -15 and ˆ n= [001], solid\ncircles;ǫuv= 5.0,ǫc= 15 and ˆ n= [001], open circles.23\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 0  2  4  6  8  10  12M/Ms\nh \nFigure 9: Reduced magnetization for the effective DDI coupling cons tantǫeff\nd= 1.0,β∗= 1,\npolydispersity σ= 0.28, open symbols or σ= 0.05, solid symbols. ǫuv= 6.30 and ǫc= 0, circles;\nǫuv= 4.0,ǫc= -12.0 and ˆ n= [111], squares; ǫuv= 4.0,ǫc= 50, and ˆ n= [001], triangles. ǫuv= 0.0,\nǫc= 0 and σ= 0.28, solid line. The dotted lines are guides to the eyes.\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 0  40  80  120  160  200  240M/Ms\nH(kA/m)\nFigure 10: Comparison of the experimental reduced magnetization curve of a maghemite powder\nsample49withdm= 10nm, open circles with the M.C. simulation, solid line. The parameters\nused in the MC simulation are σ= 0.28,ǫd= 1.0, ∆ /dm= 0.20,ǫuv= 4.0 andǫc=−1.5 with\nˆni= [111]. β∗= 1.24\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 0  40  80  120  160  200M/Ms\nH (kA/m) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 50  100  150\nFigure 11: Same as figure 10 for dm= 12nm. Experiments49, open circles. The M.C. simula-\ntions are performed with σ= 0.28,ǫd= 1.733, ∆ /dm= 0.20 and different sets of MAE param-\neters.ǫuv= 6.912,ǫc=−2.85 and ˆni= [111], solid line. ǫuv= 6.912,ǫc=−1.1\nand ˆni= [111], open triangles. Inset: Comparison of the simulated M(Ha)/Mscurves with\nǫuv= 6.912, ˆni= [111] and ǫc=−2.85, solid line; ǫuv= 5.76, ˆni= [111] and ǫc=−9.00,\nopen triangles; ǫuv= 5.76, ˆni= [001] and ǫc= 5.50, open squares.\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 0  40  80  120  160  200M/Ms\nH (kA/m) 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0  10  20  30  40  50\nFigure 12: Same as figure 10 for dm= 21nm. Experiments49, open circles. The M.C. simulations\nare performed with dm/dref= 2,σ= 0.28,ǫd= 8.0, ∆ /dm= 0.10,ǫuv= 32.00, ˆni= [111] and\nǫc= -13.2, solid squares or ǫc= -5.0, open triangles. The thin solid line is a guide to the eyes."    },    {        "title": "1207.4799v1.Raman_study_of_the_phonon_symmetries_in_BiFeO__3__single_crystals.pdf",        "content": "arXiv:1207.4799v1  [cond-mat.mtrl-sci]  19 Jul 2012Ramanstudy ofthephonon symmetries inBiFeO 3singlecrystals\nC. Beekman1, A.A. Reijnders1, Y.S. Oh2, S.W. Cheong2and K.S. Burch1\n1Department of Physics & Institute of Optical Sciences,\nUniversity of Toronto, 60 St. George Street, Toronto, ON M5S 1A7\n2Rutgers Center for Emergent Materials and Department of Phy sics and Astronomy,\nRutgers University, 136 Frelinghuysen Road,\nPiscataway, NJ 08854, USA.\n∗\nIn Bismuth ferrite (BiFeO 3), antiferromagnetic and ferroelectric order coexist at ro om temperature, making\nit of particular interest for studying magneto-electric co upling. The mutual control of magnetic and electric\nproperties is very useful for a wide variety of applications . This has led to an enormous amount of research\ninto the properties of BiFeO 3. Nonetheless, one of the most fundamental aspects of this ma terial, namely the\nsymmetriesofthelatticevibrations,remainscontroversi al. WepresentacomprehensiveRamanstudyofBiFeO 3\nsinglecrystalswiththenovelapproachofmonitoringtheRa manspectrawhilerotatingthepolarizationdirection\nof the excitation laser. Our method results inunambiguous a ssignment of the phonon symmetries, and explains\nthe origin of the controversy in the literature. Furthermor e, it provides access to the Raman tensor elements\nenabling direct comparison with theoretical calculations . Hence, this allows the study of symmetry breaking\nand coupling mechanisms in a wide range of complex materials and may lead to a non-invasive, all-optical\nmethodtodetermine the orientationand magnitude of the fer roelectric polarization.\nMultiferroicBiFeO 3(BFO)isoneofthefewmaterialsthat\nsimultaneously exhibits a robust magnetic orderingand lar ge\nspontaneousferroelectricpolarizationatroomtemperatu re[1],\nmaking it of particular interest for studying magneto-elec tric\ncoupling [2–4]. The mutual control of magnetic and elec-\ntric properties is of great interest for applications in spi n-\ntronics and magnetic storage media [5]. This has triggered\nsignificant interest in BFO, resulting in numerous studies i n-\ncluding optical [6, 7], and Raman spectroscopy[8–10], the-\noretical calculations[11, 12], thin film devices [1, 13] and\nelectrical control of magnetic excitations[14–18]. Among st\nthese various techniques, the Raman spectrum of BFO is\none of the most widely studied as it is a powerful tool\nto investigate phonons, magnons and their interaction (i.e .\nelectromagnons).[16–19] Moreover,proper phononmode as-\nsignment is necessary to describe the phononscritical for t he\nmultiferroicbehavior. However,evenformeasurementstak en\nalong the high symmetry directions of single crystals, con-\ntroversy in the symmetry assignments of the phonon modes\nremains. The discrepancies have previously been ascribed t o\nviolation of Raman selection rules due to variations in stra in\nfields[9] (i.e multidomain states) caused by polishing of th e\ncrystal surface. Once the symmetries are unambiguously as-\nsigned,deviationsinphononmodebehaviorscouldbeusedto\ndetectthepresenceofsymmetrybreaking,multidomainstat es\nand phonon-magnoninteractions. Furthermore,simplydete r-\nminingthemodesymmetryonlyallowsforaqualitativecom-\nparison with theoretical calculations. Whereas a quantita tive\ncomparison is enabled by measuring the Raman tensor ele-\nments.\nTo this end we have performed a comprehensive set of\npolarized micro-Raman spectroscopic studies of BFO single\ncrystals with uniform ferroelectric polarization. Carefu l ex-\namination and proper modeling of the rotational dependence\nof the Raman intensity enables us to unambiguously assignthe (A 1, Exand Ey) modes. Furthermore, we use the pre-\nsented model to show that slight misalignment of the crys-\ntal leads to ambiguity in the symmetry assignments. Indeed,\nour data reveal that comparison of spectra obtained for dif-\nferent scattering geometries at a single polarization vect or of\ntheincominglightisnotsufficienttohavetrulyunambiguou s\nmodeassignment. Nonetheless,unambiguousassignmentcan\nbe reached on the as grown single crystal when the Raman\nmode intensities as function of crystal rotation are measur ed\n(consistentwith previousworkonsapphire[20]). Hence,wi th\nthe presented method polishing is omitted and the resulting\nambiguityfrommisalignmentcanbeavoided.\nThe as-grown BFO single crystals used in this work have\npseudocubic[100] pcfacetswith a ferroelectricsingle domain\nstate[21] (see Supplemental Material[22]). The crystal st ruc-\nture of BFO (rhombohedraldistorted perovskite, R3c) shows\na transition from high to low symmetry accompanied by the\nformationofspontaneouselectricpolarizationbelowthet ran-\nsition temperature T C∼1100 K[23]. The ferroelectricity is\nascribedtolatticedistortions(i.e. off-centeringofthe Bi-ions)\nandresultsfromsofteningandsubsequentfreezingofthelo w-\nest frequencypolar-phononmode. The antiferromagneticor -\ndering sets in below T N∼640 K with a large magnetic mo-\nment of 4 µBon the Fe-ions. Canting of the spins leads to\na cycloidal spin structure with large period (62 nm)[23, 24]\nrotating in the plane containing the electric polarization vec-\ntorPand cycloid wavevector q. At room temperature BFO\nhasaperovskitepseudocubicunitcell(a ∼3.96˙A)elongated\nalongthe(111) pcdirectioncoincidingwith P.Thepointgroup\nis C3v, with 13 Raman active modes, of which four have A 1\nsymmetry(i.epropagatealongthec-axis)andninehaveeith er\nExor Eysymmetry (i.e. propagate in the x-y plane), which\naredoublydegenerate. Whenthelaserisnotalongthec-axis ,\nphonons can propagate in the x-z plane, which could lead to\nLO-TOsplitting(i.e. liftsthedegeneracy)andhence,thep res-2\nence of A(TO) modes in the XXand E(LO) modes in the XY\ngeometry[25���27],whichfurthercomplicatestheanalysis.\nThe Raman spectra were taken in a backscattering config-\nuration with a Horiba Jobin Yvon LabRam microscope with\na 532 nm excitation source and a 100x objective (0.8 NA),\nresulting in a collection area of ∼1µm (see Supplemental\nMaterial [22]). All data presented in this work are taken at\nroom temperature. Furthermore, we investigate the polariz a-\ntion dependence of the Raman spectra by linearly polarizing\ntheexcitationlaserintheplaneofthesampleandrotatingt he\npolarization direction with steps of 10 degrees over a total of\n180 degrees. The rotation is accomplished via a λ/2 Fresnel\nRhomb and is fully equivalent to an in-plane rotation of the\nsample (see Fig. 1a and Supplemental Material[22]). A sec-\nond polarizer is used to analyze the scattered light, which i s\neitherparallel( XX)orperpendicular( XY)totheincomingpo-\nlarizationdirection.\nFIG. 1: a) Experimental setup with the Fresnel Rhomb (FR) used to rot ate\nthe polarization of the incoming laser. Thegreen beam is the excitation laser\nand the red the Raman scattered light, with polarizers (P), N otch filter (BS),\nobjective (O) to focus down the laser and the sample (BFO). b) Typical sin-\ngle phonon spectra in XXgeometry for two different [100] pccrystals (black:\ncrystalIandred: crystalII)forRamanshiftsbetween0and6 50cm−1. Inset:\nfull range up to 1500 cm−1. Thecurves are vertically translated for clarity.\nFig. 1b showstypical Ramanspectra( XXscatteringgeom-\netry) taken on two different crystals (both with a [100] pcsur-\nface). The modes below 600 cm−1are single phonon modes\nand the broad featuresabove600 cm−1(see inset Fig.1b) are\nascribed to 2 phonon excitations, which is in agreement with\nprevious reports [9]. The spectrum taken on crystal I shows\na total of 11 single phononmodes(see Table I), while crystal\nII shows a total of 13 single phonon modes (i.e. all modes\nobserved in crystal I and two additional modes at 53 and 77\ncm−1, which can be seen due to the use of a better filter with\na lower cutoff frequency). Raman intensities taken on diffe r-\nentlocationsononecrystalandondifferentcrystals(Fig. 1b)\nshow similar polarizationdependencies(i.e the symmetry a s-\nsignments are consistent), which confirms the single domain\ncharacter of the crystals. By comparing the polarization de -\npendenceoftheRamanintensitiesofcrystalIandIIweshowhow a different (but homogeneous) ferroelectric polarizat ion\ndirectioninfluencesthephononmodebehaviors(discussedi n\ndetail below). In Fig. 2 we show the evolution of the Raman\nFIG.2:Theevolution of Raman spectra as the Fresnel Rhomb is rotate d. a)\nXXscattering geometry. b) XYscattering geometry. The spectra were taken\non the as-grown [100] pcsurface of crystal I.\nspectra as a function of in-plane crystal rotation (i.e. rot a-\ntion of the polarization direction) taken on the [100] pcsur-\nface of crystal I for the XX(Fig.2a) and XY(Fig.2b) scatter-\ning geometries. Furthermore, we normalize the Raman spec-\ntra at 1500 cm−1to correct for any power fluctuations of the\nlaserandforpolarizationdependenceofthereflectivityof the\ncrystal. We have also confirmed that the anisotropy of the\noptical constants [28] does not significantly influence pola r-\nization dependence of the Raman spectra (see Supplemental\nMaterial [22]). To quantitatively analyze the data the spec -\ntra are fit with multiple Lorentzian oscillators of the form:\nI(ω) =I0+/summationtext\ni(AiΓi\n(4(ω−Ei)2+Γ2\ni))whereiisthepeaknumber,\nI0accounts for the background, Eiis the center frequency,\nΓiis the width, and Aiis the area of peak i. The fitting is\ndone with fixed mode positions, extracting mode peak inten-\nsity(Ii(Ei))fromtheratiobetweenareaandwidthofthefitted\noscillators (i.e. I i(Ei)=Ai/Γi). In Fig. 3 we show the mode\nintensities as determined from fitting the Raman spectra as\nfunctionof polarizationangle for three representativemo des.\nThe polar plots indicate the presence of exactly three diffe r-\nentmodesymmetries. Notsurprisinglywehavefoundthatall\nmodescanbesortedintooneofthesethreetypes. Indeed,the\nfitsinFig. 3showthatthesethreetypesmatchwellwithwhat\nweexpectfortheA,E xandEysymmetries. Herewenotethat\nthedifferencesbetweenthemodebehaviorscanbesubtle,fo r\nexample the XYcurves for the A and the E ymodes (see Fig.\n3bandd)looksimilar. Hence,simultaneousmodellingofthe\nfull polarization curves for both XXandXYis necessary and3\nFIG.3:Polarplotsofthemodeintensities determined fromtheRama nspec-\ntra (left:XXand right: XY) as function of polarization rotation for three rep-\nresentative modes (a) and b): mode @ 350 cm−1, c) and d): mode @ 140\ncm−1and e) and f): mode @ 471 cm−1) measured on crystal I. The solid\nlines arefits (350cm−1: A,140 cm−1:Eyand 471 cm−1: Ex)ofwhich the\ntensor elements areindicated in TableI.\nonlythenresultsinunambiguousassignmentofthe modes.\nMoreover,wefindnoevidencethatweareprobingphonons\nthat propagate in the x-z plane (i.e. oblique phonons[26,\n27]). Indeed, such modes would exhibit LO-TO splitting\nas seen previously, with the presence of A(TO) and E(LO)\nmodesleadingto reducedintensitiesoftheA(LO)andE(TO)\nmodes[27]. Hence, we observe the 13 modes expected from\ngrouptheoryand the modelingshowsthat the phononstrans-\nformaccordingtothezonecentermodesirreduciblereprese n-\ntations. Furthermore,ona polishedc-axissurfacewe find th e\nsame number of modes as for the as grown surface with the\nmodesatthesame frequencies(withinourresolution).\nWithacloserlookatthemodelweusedforthefitsinFig. 3\nwe can explain why there is controversyin the phonon mode\nassignmentintheliterature. Theextractedmodeintensiti esas\nfunction of polarization angle are modelled using the Raman\ntensorsfor the C 3vpoint group(i.e. the trigonalsymmetryof\nthe lattice). The Raman intensity as function of polarizati on\nanglecanbecalculatedusingtheequation[20, 29],\nI=|e†\nsR†αRei|2(1)\nFIG.4:a)-c) Polarplots ofcalculated modeintensity variations a sfunction\nof polarization angle for A , E yand Exsymmetry, respectively, in the case\nof the perfect alignment of the poynting vector with c-axis ( black:XX, red:\nXY). The dotted lines in b and c indicate angles for which E-mode s will be\nmistakingly assigned as A-modes. (d) - f) Linear plots of the calculated in-\ntensity variations in the A, E xand Eymodes, respectively (for XYscattering\ngeometry) as function of the deviation of the c-axis from the surface normal.\nThe dotted lines indicate misalignment angles (0.9 and 1.5◦) for which the\nsymmetry of 2 outof 3 modes will bemisassigned.\ninwhichαaretheC 3vRamantensorsina trigonalbasis[30],\nA=\na0 0\n0a0\n0 0b\nEx=\n0d0\nd0e\n0f0\nEy=\nd0−e\n0−d0\n−f0 0\n\n(2)\nwithRthematrixthatrotatesfromcubictothetrigonalorien-\ntation,andwith esandeithepolarizationvectorsthatdescribe\nthescatteredandincominglight,respectively. Thepolarp lots\nfor the calculated Raman mode intensity as a function of po-\nlarization rotation are shown in Figs. 4a-c. Here we have as-\nsumedthepoyntingvectorisperfectlyparrallelwiththec- axis\n[111] of the crystal resulting in easy to distinguish behavi ors\nof the modes. Indeed, in this case determination of A modes\nshould be easy (Fig. 4a); their intensity is independentof t he\npolarizationangleandshouldhavenomeasurableintensity in\ntheXYgeometry. However,at certain anglesof crystal orien-\ntation(indicatedbydottedlinesinFigs. 4bandconecansti ll\nmistake an E mode for an A mode (i.e. the mode has inten-\nsity in the XXgeometry but disappears in the XYgeometry).\nAdditionalerrorcan come from slight misalignmentof the c-\naxis with respect to the propagation of the Raman laser light\n(i.e. surface normal). Figs. 4d-f demonstrate that, for the A,\nExand Eymodes respectively, introduction of a few degrees\nmisalignment can lead to large variations in the intensity o f\nthephononmodes(inthe XYgeometry),andhencecrossover4\ninmodesymmetryassignments. ThetwodottedlinesinFigs.\n4d-f are examples of misalignment angles (0.9 and 1.5◦re-\nspectively) for which the symmetry of 2 out of 3 modes will\nbe misassigned. Also, the Raman beam is typically focused\ndown, resulting in an average of incident angles, which al-\nready introduces some misalignment. Here we note that our\nmethodresultsinthesamemodesymmetryassignmentscom-\nparedtoPalaietal.[9],measuredonapolishedc-axissurfa ce.\nHowever,wealsoshowthatthestandardmethodofjustmon-\nitoring the disappearance of modes when switching from XX\ntoXYscatteringgeometrydoesnotprovideadequateinforma-\ntion to unambiguously assign the phonon mode symmetries.\nOurresultsnotonlyunambiguouslydeterminethemodesym-\nmetries but also explainsthe controversyin the literature and\nleadsto directdeterminationofthe Ramantensorelements.\nInTableIweshowtheobservedphononmodefrequencies,\nRaman tensor elements ( a,b,d,eandf) as obtained from\nthe fits and and their symmetry assignments for the [100] pc\nsurface of crystal I. We provide the corresponding polariza -\ntion curves in the Supplemental Material[22]. Moreover, on\ncrystal II, 13 modes were observed (i.e. the correct amount\naccording to group theory) at the same frequencies and with\nthe same assignments as presented in Table I; two additional\nmodeswereobservedat 53and77cm−1, thetensorelements\nand symmetry assignments for all modes observed on crys-\ntalIIareshowninTableTIintheSupplementalMaterial[22] .\nHere we note that the symmetry assignment of the mode at\n279 cm−1remains challenging, because it is very weak and\nshouldering the very strong E ymode at 288 cm−1.We have\nalso checkedthe Raman spectra ona polished[111] pc(i.e. c-\naxis) surface (data not shown) on which we observed a total\nof 13 modes (the mode at 53 cm−1disappeared while an ad-\nditional mode appeared at 70 cm−1). The mode at 70 cm−1\nremainsunassigned,itprobablyalsoexistsonthe[100] pcsur-\nfacebutistooweakandclosetoastrongE-modetobeclearly\nvisible. Furthermore,itispossiblethatthemodeat53cm−1is\nindicativeofa violationofRamanselectionrulesduetosym -\nmetry breaking. Modes at this Raman shift have been previ-\nously assigned as A(TO) modes[9], however they should not\nbe visible in our scattering geometry and we do not see evi-\ndenceof the otherA(TO)modesin our spectra. Alternatively\nthis mode may be an electromagnon[17]. Future low temper-\nature studies, where the linewidths are narrow, would help t o\nbetter assign these modes. Nonetheless, using the presente d\nmethod we have unambigously assigned the phonon modes\nand extracted the Raman tensor elements providingquantita -\ntive information for direct comparison with theoretical pr ed-\nications. Furthermore, the ratio between the Raman tensor\nelements aandbare identical for the A-modes observed on\nboth crystals. However, we do observe some differences be-\ntween the ratios of the E-mode tensor elements between the\nmeasurements taken on crystal I and II, which in no way in-\nfluencesthe consistencyofthe symmetryassignments. These\ndifferences may indicate that the two crystals (both are sin -\ngle domain) have a different direction and/or magnitude of\nthe ferroelectric polarization. This would indeed affect t heE-modes but not the A-modes, since the A-modes are fully\nsymmetrical and constitute vibrations along the c-axis (i. e.\nparallel to the ferroelectric polarization direction). Ho wever,\nthis could mean that changes in the ferroelectric polarizat ion\ndirection leave the mode symmetries unaltered. Hence, one\nneeds the method presented here to observe this subtle effec t\n(i.e. changes in the tensor element ratios of the E-modes) of\ndifferentferroelectricpolarizationontheRamanintensi ties.\nTABLE I: Phonon mode frequencies, the Raman tensor elements ( a,b,d,e\nandf) for the modes as obtained from the fits and the symmetry assig nments\nfor crystal I. The data for all the modes are presented in the S upplemental\nMaterial [22].\nWe have measuredthe evolution of polarizedRaman spec-\ntra of BFO single crystals and extracted the polarization\ncurves for every single phonon mode for both the XXand\ntheXYscattering geometry. We fit the XXandXYcurves\nsimultaneously for each mode using a model based on the\nRaman tensors of the C 3vpoint group (eq. 1). As a result\nunambiguoussymmetryassignmentanddeterminationofRa-\nman tensor elements of the phonon modes is accomplished\neven on the as grown [1 0 0] pcsurface. In Fig. 3 the ex-\ncellent and unambiguous agreement between the experimen-\ntal results and our calculations for a [100] pcsurface demon-\nstratetheimportanceofperformingtheRamanmeasurements\nover a full rotation of the crystal. Whereas the calculation s\nin Fig. 4 demonstrate that only measuring the XXandXY\nspectraona[111] pcsurfaceforasinglepolarizationdirection\n(as is typically done), can easily lead to wrong assignments\nof the phononsdue to misalignment of the crystal. It is clear\nthat unambiguous mode assignment can only be reached if\none monitors the Raman signal as function of rotation of the\ncrystal. Simply comparing XXandXYscattering geometries\nforonepolarizationangleisnotenoughevenforac-axissur -\nface. Besides obtaining unambiguous mode assignment for\nBFO, this work has wider implications as well. The method\ncan be used on any material to check crystal symmetry and\nassign the phonon modes. Furthermore, once unambiguous\nassignment has been accomplishedone can use the presented\nmethodtoinvestigatesymmetrybreakingaswell,forexampl e5\nby studying deviations in the tensor element ratios, the sym -\nmetry assignments and through observation of more than the\npredicted number of modes. This gives us a powerful tool\ntoinvestigateoccurrenceof(electro)magnonsandcompare to\nexisting reports[17, 18]. Moreover it would allow study of\ncoupling mechanisms in complex materials such as multifer-\nroics, as well as provides quantitative information for dir ect\ncomparisonwiththeoreticalpredications.\nWearegratefulfornumerousdiscussionswithR.deSousa,\nN.B Perkins, and H.Y. Kee and we thank Harim Kim for de-\nsigning Fig. 1a. Work at the University of Toronto was sup-\nportedbyNSERC,CFI,andORF;workatRutgersUniversity\nwassupportedbytheNSF undergrantNSF-DMR-1104484.\n∗Electronicaddress: beekmanc@ornl.gov\n[1] R.Ramesh and N.A. Spaldin, Nat. 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Nine stellarator con\fgurations, three quasiaxisymmetric, three quasiheli-\ncally symmetric and three non-quasisymmetric are scaled to ARIES-CS size and an-\nalyzed for energetic particle content. The best performing con\fgurations with regard\nto energetic particle con\fnement also perform the best on the neoclassical \u0000 cmetric,\nwhich attempts to align contours of the second adiabatic invariant with \rux surfaces.\nQuasisymmetric con\fgurations that simultaneously perform well on \u0000 cand quasisym-\nmetry have the best overall con\fnement, with collisional losses under 3%, approaching\nthe performance of ITER with ferritic inserts.\n1. Introduction\nCon\fning energetic particles, especially alpha particles born in nuclear fusion reactions,\nis of key importance for magnetic con\fnement fusion reactors. In con\fgurations where\naxisymmetry is not present, either tokamaks with non-axisymmetric perturbations,\nor stellarators, some particle orbits will have a non-zero bounce-averaged radial drift\ncausing them to leave the con\fned region, sometimes very quickly. These promptly lost\nparticles can cause signi\fcant damage to plasma facing surfaces and reduce the lifetime\nof the plasma wall [1]. It will never be possible to con\fne all alpha particles in a fusion\ndevice, but reducing the losses, especially prompt losses, is crucial for the longevity of\nthe device. This paper will show collisional energetic particle transport results from\nvarious stellarator con\fgurations at the reactor scale with the goal of identifying the\nproperties of con\fgurations with good con\fnement.\nSeveral metrics have been developed for neo-classical con\fnement in 3D systems\n[2]. Some con\fgurations possess a symmetry in the magnetic \feld strength, jBj, andarXiv:2106.00716v1  [physics.plasm-ph]  1 Jun 2021Energetic particle transport in optimized stellarators 2\ntherefore are isomorphic to axisymmetric systems. These con\fgurations are called\nquasi-symmetric, because they possess a symmetry in jBjsimilar to an axisymmetric\nsystem [3]. This paper includes con\fgurations of two quasi-symmetric types, quasi-\naxisymmetric (QA) where jBjcontours connect toroidally, and quasi-helical (QH) where\njBjcontours connect helically [4]. In all the con\fgurations exact quasisymmetry is\nnot present, but rather an approximate symmetry exists. The deviation from a strict\nsymmetry is referred to in this paper as quasisymmetric deviation. Mathematically this\nis obtained by transforming coordinates into the Boozer coordinate system, and then\ncalculating the energy in the non-symmetric modes.\nIt has been shown experimentally that quasi-helical con\fgurations improve\nneoclassical con\fnement [5]. Additionally, many numerical explorations of\nquasisymmetric con\fgurations of all types exist as well. Speci\fcally, recent results\nindicate that low quasi-symmetric deviation can help alpha con\fnement in both\nquasiaxisymmetric [6] and quasi-helically symmetric con\fgurations [7].\nThere are alternative methods for improving neoclassical con\fnement in the absence\nof quasisymmetry exist. A broader class of con\fgurations are omnigenous, in which the\nsecond adiabatic invariant, Jk=Hvkdl, is constant on a \rux surface. A consequence\nof this optimization is that all the maxima and minima of a \feld line on a \rux surface\nare the same. Con\fgurations that approximate omnigeneity are called quasi-omnigenous\n(QO). If, in addition, \rux surfaces close poloidally, the con\fguration is quasi-isodynamic.\nA consequence of this optimization is that drifts are purely poloidal, and bootstrap and\nP\frsch-Schl uter currents vanish. This optimization was used to produce W7-X [8].\nCon\fnement of energetic particles in quasi-isodynamic con\fgurations is expected to\nimprove at high pressure when the alignment between Jkand \rux surfaces improves [9].\nAn even less restrictive optimization for improved con\fnement is described as \u001b-\noptimization [10]. In LHD it is possible to achieve an equilibria where \u001b= 1, where\nfor a given \rux surface the minima of each \feld line are equivalent, but the maxima do\nnot. This is achieved in LHD by shifting the axis inward, creating the \"inward shifted\"\ncon\fguration [11]. In this optimization, collisionless drift orbits are not fully con\fned,\nbut the coe\u000ecient of neoclassical transport drops signi\fcantly.\nVarious metrics have been used to quantify the degree of neoclassical transport.\nOne metric, \u000fe\u000b, is the coe\u000ecient of the neoclassical di\u000busion in the low-collisionality\n(\u00181=\u0017) regime has been regularly used for stellarator optimization [12]. However,\nrecent results indicate that there is little correlation between \u000fe\u000band good energetic\nparticle con\fnement [7]. However, a di\u000berent metric \u0000 c[13, 14], that seeks explicitly to\nalign contours of Jkwith \rux surfaces similar to the omnigeneity constraint, has been\nshown to correlate better with energetic particle con\fnement. This metric has been\nused to optimize quasi-helically symmetric con\fgurations with good energetic particle\ncon\fnement [7, 15].\nIn previous publications stellarators were compared only between variations of\nsimilar classes (QA, QH, QO). Comparisons between stellarators of di\u000berent classes,\nhave often been hampered by di\u000berent choices of magnetic \feld strength, size, andEnergetic particle transport in optimized stellarators 3\nTable 1: A list of con\fgurations along with relevant properties\nName Type Periods Aspect ratio \f\nWistell-A QH 4 6.7 Vacuum\nWistell-B QH 5 6.6 Vacuum\nKu5 QH 4 10.0 10.0%\nARIES-CS QA 3 4.5 4.0%\nNCSX QA 3 4.4 4.3%\nSimsopt QA 2 6.0 Vacuum\nW7-X QI 5 10.5 4.4%\nLHD st Torsotron 10 6.5 Vacuum\nLHD in. Torsotron 10 6.2 Vacuum\nITER Tokamak N/A 2.5 2.2%\npro\fles of density and temperature. Some calculations include collisions where others\ndo not. Comparisons between published results is therefore very di\u000ecult. This\npaper attempts to rectify the situation by providing consistent scalings across a broad\nclass of con\fgurations and then comparing the energetic particle con\fnements both\ncollisionlessly and with collisions.\nThe layout of this paper is as follows. In Section 2, we will brie\ry describe\nthe con\fgurations used in this paper. Section 3 will explain how the reactor scale\ncon\fgurations were constructed. Section 4 will show results from both collisional and\ncollisionless calculations of alpha particles. Section 5 compares the alpha particle losses\nfor the metrics of interest in each con\fguration. Section 6 will discuss the results and\ndescribe the limitations of the current work. Section 7 will conclude the paper and\nprovide areas for future research.\n2. Con\fgurations\nThis paper considers three quasi-helically symmetric con\fgurations, three quasi-\naxisymmetric con\fgurations, a W7-X like con\fguration, and two LHD-like con\fgura-\ntions. An ITER con\fguration is included for comparison. A table of all the con\fgura-\ntions and their relevant properties has been included in table 1.\nThe quasi-helically symmetric con\fgurations are: the \"Wistell-A\" con\fguration\nwhich has been described in a previous publication [15]; the \"Wistell-B\" con\fguration,\na \fve-\feld period vacuum con\fguration optimized with the ROSE code [16] explicitly\nfor quasisymmetry and \u0000 c; \"Ku4\" a four \feld period con\fguration from [17] that\nwas optimized for quasisymmetry at high normalized pressure, \f. The three quasi-\naxisymmetric con\fgurations are comprised of the NCSX (speci\fcally \"li383\") [18, 19]\nand ARIES-CS (speci\fcally: \"n3are\") [20, 21] con\fgurations. Also included amongst\nquasiaxisymmetric con\fgurations is a more recent vacuum con\fguration. called\nSimsopt, which was optimized solely for quasisymmetry at the s= 0:5 surface using theEnergetic particle transport in optimized stellarators 4\nFigure 1: Boundary \rux surfaces for the scaled con\fgurations are shown. Left:\nQuasihelically-symmetric con\fgurations. Center: Quasiaxisymmetric con\fgurations\n(including ITER) Right: non-quasisymmetric con\fgurations\nSIMSOPT optimizer [22]. Here and throughout the paper, srepresents the normalized\ntoroidal \rux,  = edge. The W7-X like con\fguration is a high-mirror con\fguration\ndesigned for improved energetic particle con\fnement [23], with coe\u000ecients given by\nTable IV in [24]. The two LHD [11] con\fgurations are vacuum con\fgurations, with one\nin the standard (outward) con\fguration and the other in an inward shifted con\fguration\nwhich is known to have improved con\fnement properties [10]. Finally the ITER\ncon\fguration is a near-axisymmetric con\fguration, although this equilibrium includes\ncoil ripple, blanket modules and ferritic inserts [25].\n3. Scaling to Reactor size\nIn order to properly scale con\fgurations to each other it is necessary to adjust to a\nbenchmark size. The ARIES-CS parameters are used for the scaling, representing a\nfairly compact reactor size. There are two possible ways to scale the con\fgurations.\nOne option is to scale the con\fgurations to have the same volume (444 m3) the other\nis to scale the minor radii to the same value (1.7m). For this paper we only show\nresults using the volume scaling, although the main conclusions do not change when the\ncon\fgurations are scaled to have equivalent minor radii.\nAll con\fgurations are represented by VMEC equilibria [26], and the size scaling is\naccomplished by adjusting the boundary coe\u000ecients such that all con\fgurations have\nthe same volume. The magnetic \feld strengths are made equivalent by ensuring the\nvolume averaged magnetic \feld is equivalent across all con\fgurations (5.86 T). For non-\nvacuum con\fgurations, the rotational transform are adjusted for each con\fguration by\nusingI/RB, whereIis the total current in the plasma. The normalized pressure, \f\nis similarly kept constant through the scaling procedure. The boundary \rux surfaces\nfor the con\fgurations at the \u001e=0 plane, often referred to as the \"bean\" or \"crescent\",\nare plotted in \fgure 1\nFor all con\fgurations, with three exceptions, the scaling is accomplished by startingEnergetic particle transport in optimized stellarators 5\nfrom an idealized \fxed boundary equilibrium and scaling the coe\u000ecients. These \fxed\nboundary equilibria are usually generated through optimization and do not include\ne\u000bects from \fnite coils. One exception is the ITER equilibrium which includes coil ripple\nand the e\u000bect from ferritic inserts and blanket modules. The scaled ITER equilibrium\nis a direct replica of the unscaled ITER equilibrium, but with slightly larger volume\nand higher \feld. There was no attempt to recalculate the e\u000bects of blanket modules\nand ferritic inserts on the larger size. The other exceptions are the two LHD equilibria\nwhich represent two con\fgurations very similar to those generated in the actual LHD\ndevice. In these cases, the coils used to generate the equilibria were adjusted and new\nfree-boundary equilibria were generated from the enlarged coils. Results from these\nfree-boundary equilibria are presented in this paper. A second set of LHD calculations\nwere undertaken with a scaled \fxed-boundary equilibrium and the di\u000berences were not\nnoticeable, and are not included here.\nIn order to perform the collisional calculations, it is necessary to de\fne the\ntemperature and density pro\fles. These pro\fles determine the initial launch points\nfor collisional calculations as well as the slowing down behavior of the alpha particles.\nThe density pro\fle chosen for these simulations is mostly \rat with n=n0(1\u0000s5), the\ntemperature pro\fle is more peaked with T=T0(1\u0000s). These pro\fles are roughly\nconsistent with those chosen for the ARIES-CS studies [20]. In the previous equations,\nsrepresents the normalized toroidal \rux,  = edge. The values of the core temperatures,\nT0andn0are approximately equivalent to those of ARIES-CS: ne;0= 4:8\u00021020\nandnD;0=nT;0= 2:25\u00021020, withTe;0=Ti;0= 11:5 keV. The di\u000berence between\nnD+nTandnearises from a \rat pro\fle of Ze\u000b= 1:13 as in the ARIES-CS equilibrium.\nHowever, collisions with impurity ions are not included in these calculations. Once the\ntemperature pro\fles are chosen, the reaction pro\fle is determined. The temperature,\ndensity and reaction pro\fles are shown in \fgure 2. The same reaction pro\fle is used for\neach equilibrium, and is estimated as\nRdV\nds=nDnTh\u001bvidV\nds;h\u001bvi= 3:6\u000210\u000018T\u00002=3exp\u0010\n\u000019:94\u0003T\u00001=3\u0011\nm3=sec (1)\nwherenDandnTare the deuterium and tritium concentrations, Tis the\ntemperature in keV and dV=ds is the derivative of the volume with respect to normalized\ntoroidal \rux sfor the ARIES-CS equilibrium. Even though the reaction pro\fle varies\nslightly from equilibrium to equilibrium due to variations in dV=ds , the same fusion\nreaction pro\fle is maintained in order to keep consistent particle launch pro\fles across\nequilibria.\nSince the con\fgurations vary in pressure from vacuum con\fgurations to normalized\npressures of 10%, it is impossible to choose pro\fles consistent across con\fgurations\nthat also match the pressure pro\fles from each con\fguration. The main goal is to\ndetermine what magnetic con\fguration properties a\u000bect alpha particle con\fnement\nrather than to do self-consistent studies of each of the con\fgurations. Therefore, the\nsame temperature and density pro\fles are used in all con\fgurations for alpha particleEnergetic particle transport in optimized stellarators 6\nFigure 2: Left: Temperature (red) and density (blue) pro\fles as a function of normalized\ntoroidal \rux s. Right: The derived reaction rate given as a function of temperature and\ndensity for the ARIES-CS equilibrium.\ncon\fnement calculations even though there is no self-consistency with the plasma\npressure used in the equilibrium.\n4. Alpha Particle Losses\nParticles are sourced by \frst choosing a radial location such that the distribution\nmatches the reaction pro\fle given in \fgure 2. Next a random location on the surface and\nthe velocity pitch angle is chosen in the same manner as described in [7]. The guiding\ncenters of the particles are followed using an Adams-Bashford integration scheme and\ncan under go both slowing down and pitch angle scattering. The ANTS code is used for\nall particle following calculations [27]. If a particle passes beyond the penultimate \rux\nsurface at any point in time it is considered lost. If the particle's energy is the same as\nthe background thermal particles it is considered con\fned and is no longer followed.\nThe results from the collisional calculation are shown in Figure 3. The line style\nindicates the con\fguration type, with solid lines indicating QH, dotted lines QA, dashed\nlines for the LHD-like con\fgurations and both ITER and W7-X use dashed-dotted\nlines. To help the reader, throughout the paper consistent colors and linestyles (where\npossible) for each con\fguration are used. Among the quasisymmetric con\fguration,\nthe QHs strongly outperform the QAs with the exception of the Simsopt con\fguration\nwhich performs as well as the best QHs. The three best performing con\fgurations shown\n(outside of ITER) are the Ku5 con\fguration (QH), the Wistell-B con\fguration (QH)\nand the Simsopt con\fguration (QA). W7-X performs about equivalently to both the\nWISTELL-A con\fguration and the inward shifted LHD con\fguration. This behavior\nwill be examined in depth later. Note that due to di\u000berences in machine size, magnetic\n\feld, and particle sourcing, the results shown here may di\u000ber from previously published\nresults on energetic particle con\fnement.Energetic particle transport in optimized stellarators 7\nFigure 3: Energy loss from alpha particles as a function of time for all con\fgurations.\nIn addition to the collisional calculation, calculations without collisions are also\npresented in Figure 4. For these calculations particles were started on a speci\fc \rux\nsurface, in this case s=0.3, representing a surface just outside the midradius is chosen.\nParticles are launched on this surface and followed until they are lost or 200 ms have\nelapsed, corresponding to several ( \u00183) slowing down times. Figure 4 shows the particle\nloss versus time rather than the energy loss, but because no collisions are included,\nall lost particles have the full energy. Collisionless calculations were previously used\nto distinguish between con\fgurations [7], and they are very useful to highlight the\nspeci\fc loss behaviors of the con\fgurations, which will be examined below. Note that\nthe addition of collisions tend to enhance energetic particle losses due to pitch angle\nscattering onto lost orbits.\nBefore a more detailed look at the con\fgurations, it will be useful to distinguish\nbetween the di\u000berent types of losses seen. Some particles are born on lost orbits and leave\nthe con\fned region at almost the full energy values even in the collisional calculation.\nParticles born on the outer regions of the plasma are likely to be lost in this manner.\nIn fact all the losses from ITER are particles born near the edge that are promptly lost.\nWe will refer to these particles as \\prompt\" losses.\nThere is a second class of lost particles that undergo many orbits before being lost.\nSometimes this is the result from di\u000busive properties, especially pitch-angle scattering\nwhich becomes increasingly important at low energies. However, these slow losses can\nalso occur in collisionless calculations discussed below. As such we will refer to all losses\non extended time scales as \"stochastic\" losses, following the convention in [28].\nThe exact boundary between prompt and stochastic losses is not clear in all\ncon\fgurations, but it is often easy to see the distinction in some of the con\fgurations.\nThe W7-X collisionless losses at, say, s= 0:3 are particularly clear. The W7-XEnergetic particle transport in optimized stellarators 8\nFigure 4: Particle loss as a function of time for all con\fgurations for particles born on\nthes=0.3 surface.\ncon\fguration loses about 2% of launched particles born at s= 0:3 before 0.2 ms. There\nare almost no additional losses until about 1 ms when additional stochastic losses begin\naccumulating again. Many of the particles are lost stochastically, however the precise\nbehavior is important. Slow stochastic losses are less problematic because particles\nwill be able to deposit most of their energy. The same distinction between prompt\nand stochastic losses in W7-X exists with collisionless losses on other \rux surfaces (not\nshown). It also is visible in the collisional losses, however, with collisional losses, di\u000busive\nbehavior causes there to be some particles losses between 0.2 and 1 ms. A more detailed\ndiscussion about prompt versus stochastic losses is in sections 6.2 and 6.3.\n5. Alpha Particle Loss Metrics\nWe consider two metrics for alpha particle losses, quasisymmetry and \u0000 c. A given\ncon\fguration is quasisymmetric if the variation of jBjalong a \feld line is the same\nfor all \feld lines on a \rux surface [29]. Quasisymmetry can be determined by Fourier\ndecomposingjBjon a \rux surface in the straight \feld line coordinate system known as\nBoozer coordinates [30]. If the only modes present are ones where the ratio of the toroidal\nmodento the poloidal mode mis constant, the con\fguration is quasisymmetric. For\nquasiaxisymmetric equilibria, n=m = 0 and a perfectly quasiaxisymmetric equilibrium\nwill only have modes with n= 0. For quasihelically-symmetric equilibria, the ratio\nn=m is usually equal to the number of \feld periods (modulo a sign). So for a perfectly\nquasihelically-symmetric equilibrium with four periods, the only modes present are ones\nwheren=m = 4. A third symmetry, quasipoloidal symmetry, where only modes withEnergetic particle transport in optimized stellarators 9\nFigure 5: The deviation from quasisymmetry as a function of normalized toroidal \rux, s\nfor quasiaxisymmetric con\fgurations (left) and quasihelically symmetric con\fgurations\n(right)\nm= 0 are present, is not considered in this paper.\nExcepting precisely axisymmetric con\fgurations, it is conjectured that perfect\nquasisymmetry can only be achieved on a single \rux surface [31], and a metric is\nneeded to describe the deviation from perfect quasisymmetry. The metric used in this\npaper is calculated by \frst Fourier decomposing the two dimensional \rux surface in\nBoozer coordinates, and then summing the magnetic energy in all non-symmetric modes\nnormalized to the m= 0;n= 0 mode, which is representative of the background \feld\nstrength. That is,\nQqs(s) =1\nB0;0(s)0\n@X\nm=n6=CqsB2\nm;n(s)1\nA1=2\n(2)\nwhereCqsrepresents the target for quasisymmetry, 0 for quasiaxisymmetry and the\nnumber of \feld periods for quasihelical symmetry. Lower values of Qqsindicate better\nquasisymmetry.\nFigure 5 shows the results of Qqsfor quasiaxisymmetric con\fgurations (left) and\nquasihelically symmetric con\fgurations (right) as a function of \rux surface. There\nis clear separation among the quasiaxisymmetric con\fgurations. At all svalues, the\nARIES-CS con\fguration is the least quasisymmetric and the Simsopt con\fguration is\nthe most quasisymmetric. The story is less clear for the quasihelical con\fgurations. The\nKu5 con\fguration is the most quasisymmetric in the core and the least quasisymmetric\nin the edge. Overall the Wistell-B con\fguration has the best average quasisymmetry.\nThe second metric, \u0000 cwas introduced by Nemov [14] (eqs 61, 50, and 36) as a\nmeasure of the energetic ion con\fnement properties and is given by,\n\u0000c=\u0019p\n8lim\nLs!1 ZLs\n0ds\nB!\u00001ZBmax=Bmin\n1db0X\nwell j\r2\ncv\u001cb;j\n4Bminb02;\rc=2\n\u0019arctanvr\nv\u0012(3)Energetic particle transport in optimized stellarators 10\nFigure 6: \u0000 cas a function of normalized toroidal \rux, sfor quasisymmetric\ncon\fgurations (left) and non-quasisymmetric con\fgurations (right)\nHere,vrandv\u0012are the bounce average radial and poloidal drifts respectively; v\nis the particle velocity; \u001cbis the bounce time, BmaxandBminare the maximum and\nminimum \feld strength on a \rux surface or suitably long \feld line; b0represents a\nnormalized \feld strength, here equivalent to jBj=BminandLsis the length along a \feld\nline. The summation is over every well along a \feld line, where the boundaries of the\nwells are themselves a function of the integrating variable, b0. When \u0000 cis small, contours\nofJkalign with \rux surfaces, and the bounce average radial drift goes to zero. More\ninformation about \u0000 cand its use for stellarator optimization can be found in [7]. Unlike\nquasisymmetry, the \u0000 cmetric can be calculated for all stellarator con\fgurations. All\ncalculations for both \u0000 cand quasisymmetry were carried out using the ROSE code. Due\nto an unresolved di\u000eculty with handling single \feld period equilibria, \u0000 cfor the ITER\ncalculation is unavailable for this paper.\nNine con\fgurations are represented in \fgure 6 in the two plots showing \u0000 cas\na function of normalized toroidal \rux s. The six quasisymmetric con\fgurations are\nplotted on the left, and the three non-quasisymmetric con\fgurations are plotted on the\nright. Looking at the non-quasisymmetric con\fgurations \frst, there is a clear distinction\nbetween the optimized con\fguration W7-X and the two LHD con\fgurations. There is\nalso a clear improvement between the LHD inward shifted con\fguration compared to\nthe outward shifted con\fguration. However, the LHD inward shifted con\fguration has\nroughly the same magnitude of \u0000 cas the worst of the quasisymmetric con\fgurations\nNCSX (note the di\u000berence in y-axis scale).\nThe quasisymmetric con\fgurations also show considerable spread in the \u0000 cmetric.\nOnce again there is clear separation among the three quasiaxisymmetric con\fgurations.\nThe best performing case is the Simsopt equilibrium. Contrary to the quasisymmetry\nresult, ARIES-CS outperforms NCSX with regard to the \u0000 cmetric. This behavior is\nnot surprising since the ARIES-CS optimization explicitly degraded quasisymmetry in\norder to improve energetic particle con\fnement [21]. The particle loss results shown inEnergetic particle transport in optimized stellarators 11\nFigure 7: Values of total collisional energy lost as a function of quasisymmetry on the\ns=0.6 surface (left) and \u0000 con thes=0.6 surface (right)\n\fgures 3 and 4 indicate that this optimization was successful.\nFor the three quasihelically symmetric con\fgurations the Wistell-A and Wistell-B\ncon\fgurations have almost identical values of \u0000 c(these are similar in scale to the W7-\nX value). The Ku5 con\fguration has a larger value in the core, but a lower value in\nthe outer half of the plasma. Since the Ku5 and Simsopt con\fgurations represent the\nbest performing con\fgurations, it appears that the edge values in the outer half may\nbe more important. The importance of the quasisymmetric values on the outer half of\nthe plasma has already been discussed with respect to optimizations of quasisymmetry\n[6] and the results presented here indicate that the values of \u0000 cin the outer half of the\nplasma may be more closely related to energetic particle con\fnement as well.\nFigure 7 shows the total energy loss for each con\fguration plotted against the value\nof a parameter of interest evaluated at s=0.6. The deviation from quasisymmetry (for\nthe QS con\fgurations) is plotted on the left hand plot and \u0000 cis in the right hand\nplot. A correlation between alpha energy con\fnement and \u0000 cis clear from the right\nhand plot. Although a perfect correlation does not exist, it is clear that the best/worst\nperforming con\fgurations also have the best/worst performance with regard to this\nmetric. For quasisymmetry the correlation is weaker. While the Simsopt and Wistell-B\ncon\fguration both perform very well on this metric, the Ku5 con\fguration performs\nworse despite having the best overall energetic particle con\fnement.\nAmong the QA con\fgurations the best performing con\fguration is the Simsopt QA,\nwhich also performs the best on both metrics, despite only optimizing for quasisymmetry.\nAs noted above, ARIES-CS performs worse in quasisymmetry but better in both \u0000 cand\nenergetic particle con\fnement.\nThe QH con\fgurations include two stellarator con\fgurations with excellent\ncon\fnement, Ku5 and Wistell-B. For the QH con\fgurations, Wistell-A performs worse in\nthe quasisymmetry metric but approximately as well in \u0000 c. Both Wistell-A and Wistell-\nB were optimized including both quasisymmetry and \u0000 cin the target function. TheEnergetic particle transport in optimized stellarators 12\nFigure 8: Histogram of number of particles lost (per 10k) as a function of the loss energy\nfor Wistell-A (black), W7-X (magenta dashed) and LHD-inward shifted (green dashed).\nPrompt losses at 3.5 MeV are at the far right side of the graph.\nKu5 con\fguration only optimized for quasisymmetry, but despite this has the lowest \u0000 c.\nThere is a caveat to the performances of the two high performing QH con\fgurations. In\nboth the Wistell-B and Ku4 con\fgurations, strong indentations in the plasma boundary\nmake designing coils extremely challenging (see for example \fgure 13 in [17]). However,\ncoils that reproduce the energetic particle properties have already been designed for the\nWistell-A con\fguration [15].\nThe W7-X con\fguration was speci\fcally designed for good energetic particle\ntransport [23] and indeed it outperforms ARIES-CS and is on par with the Wistell-\nA con\fguration. Interestingly, the W7-X and Wistell-A con\fguration also have almost\nthe same minimum value for \u0000 cso the points are very close in Figure 7b.\nThe inward shifted LHD-like con\fguration has properties that deserve some\nattention. The overall losses for this con\fguration are on par with both W7-X and\nWistell-A. Even more interesting is that this con\fguration does exceedingly well at\ncon\fning prompt losses. This good performance in the collisional results appears despite\nnot performing particularly well on the \u0000 cmetric. Figure 8 illustrates this by plotting a\nhistogram of the number of lost particles in the collisional calculation against the energy\nat which they are lost. Prompt losses are on the far right of the graph. These prompt\nlosses are lowest for the LHD inward shifted con\fguration compared to both Wistell-A\nand W7-X. The LHD losses reach their maximum between 2.5 and 3.0 MeV after which\nthey fall to a very low level. The Wistell-A and W7-X, in contrast have fewer particles\nlost between 2.5 and 3.0 MeV but considerably more particles lost at energies under 1\nMeV. These con\fgurations will be examined closer in sections 6.2 and 6.3.\n6. Discussion\n6.1. Optimization\nOne salient feature of the con\fguration scan is that two of the con\fgurations with lowest\nachieved values of \u0000 cwere not actually optimized for \u0000 cbut rather for quasisymmetry\nonly. These are the Ku5 and Simsopt con\fgurations. Since perfect quasisymmetry willEnergetic particle transport in optimized stellarators 13\ncon\fne all particles, and a perfectly quasisymmetric con\fguration will have \u0000 c= 0,\nan optimization for quasisymmetry will almost always improve \u0000 cas well. The cases\nwhere this does not occur, such as one of the con\fgurations presented in [7] are fairly\nuncommon. In fact, in several steps of the Wistell-B optimization it was found that the\nbest improvement on both metrics, quasisymmetry and \u0000 cwas obtained when the \u0000 c\noptimization was turned o\u000b in the optimizer.\nThe reverse is not true. Optimization for \u0000 calone will almost never improve\nquasisymmetry. There are many pathways to improving \u0000 cthat do not include\nquasisymmetry, for example, the optimizations that lead to W7-X.\nAnother important point of consideration is that many of the con\fgurations were\ndesigned with additional metrics included. Both the NCSX and ARIES-CS equilibria\nplaced strong emphasis on stability properties at \fnite \f. Among other things, this\ngenerates a strong crescent shape at the \u001e= 0 plane. In contrast the Simsopt\nQA is a vacuum con\fguration without a vacuum magnetic well and no attempt to\nprovide stability at high pressure. The Simsopt con\fguration should be viewed as\nwhat is possible if you attempt to make the most quasiaxisymmetric con\fguration\npossible perhaps in opposition to other desired or even necessary properties. As always,\nsigni\fcant e\u000bort is needed to weigh di\u000berent optimizations considerations together to\nproduce the ideal con\fguration for an experiment or reactor.\nA similar story exists in the QH con\fgurations. While Wistell-A and Wistell-B are\nboth vacuum con\fgurations, Wistell-A has a vacuum magnetic well and Wistell-B does\nnot. Furthermore, attempts to generate coils to reproduce the Wistell-A con\fguration\nwere successful, while attempts to produce coils for Wistell-B or Ku5 have not been\nsuccessful to date. Of course this does not mean that it is impossible to \fnd coils for\nthese con\fgurations, just that it is comparatively easier to design coils for Wistell-A.\nDespite having entirely di\u000berent optimization schemes, Wistell-B and Ku5 have similar\nfeatures. Speci\fcally, both have a strong indentation in the teardrop shape that is very\ndi\u000ecult to reproduce with coils. Future work that incorporates coil buildability should\nexamine whether this feature is necessary for good con\fnement or not.\nFinally we note that several of the con\fgurations, namely the LHD con\fgurations\nare actually built machines. It is much easier to design a con\fguration with good\nparameters than to actually build one. It is possible that the performance of the other\ncon\fgurations would degrade due to accumulated errors in the construction process.\nE\u000borts to optimize taking into account manufacturing errors are being undertaken by\nothers [32] and will not be discussed further here.\n6.2. LHD Inward Shifted\nA surprising result from the con\fguration scan was the performance achieved by\nthe LHD con\fguration. While it has been theorized and experimentally veri\fed\nthat the con\fnement improves in LHD with inward-shifted con\fgurations, the actual\nperformance deserves some additional discussion here.Energetic particle transport in optimized stellarators 14\nFigure 9: Magnitude of the magnetic \feld, jBjalong a \feld line for the LHD inward\nshifted (green) and outward shifted (yellow) con\fgurations\nThe speci\fc optimization in question aligns the minimal values of the magnetic\n\feld on the surface, referred to as \u001b= 1 optimization. This can be seen in \fgure\n9 where a \feld line from both the outward and inward shifted cases are plotted as a\nfunction of toroidal angle. The minima align for the inward shifted case (green) but\ndo not align for the outward shifted case (yellow). Another feature of these LHD-like\nequilibria is because the \feld is generated with helical coils, the \feld strength is smoothly\nvarying with no local minima above the global minimum value. These local minima are\nproblematic for particle con\fnement and can lead to promptly lost particles, similar to\nripple trapped particles in a tokamak.\nSince the maxima along the \feld line do not align, particles in the the LHD inward\nshifted con\fguration have a \fnite radial drift. In fact, as visible in \fgure 4 even in the\nabsence in collisions, all trapped particles in both LHD con\fgurations are eventually\nstochastically lost. The end result is a con\fguration which has no prompt losses due to\nthe\u001b= 1 optimization, but eventually loses all the particles. The parameters used for\nthis calculation use the ARIES-CS parameters which have high plasma density and low\ntemperature giving a slowing down time of \u001950 ms in the core, and considerably lower\nin the edge. When examining the collisional results under these conditions, the LHD\ninward shifted con\fguration compares favorably to con\fgurations such as Wistell-A even\nthough stochastic losses are very low in Wistell-A.\n6.3. QH: Collisionless vs Collisional losses\nThe performance of Wistell-A is worth looking at closer. In the collisionless losses\n(\fgure 4) the total losses are low, below all other con\fgurations except for Wistell-B\nand ITER. Furthermore, almost all the losses that do exist are prompt, occurring well\nbefore 1 ms. Yet, when collisions are added, Wistell-A performs signi\fcantly poorer to\nWistell-B and Ku5 and instead performs equally well to W7-X which has more prompt\nlosses and signi\fcantly more collisionless stochastic losses. The performance of Wistell-\nA is actually slightly worse than LHD, which has fewer prompt losses, but very large\nvalues of collisionless stochastic losses.Energetic particle transport in optimized stellarators 15\nFigure 10: Histogram of collisionless prompt losses (less than 0.1 s) as a function of\nstarting pitch for Wistell-A (black) and W7-X (magenta). The vertical dotted lines\nrepresent the minimum possible value of E=\u0016 and the vertical dashed lines represent\nthe trapped-passing boundaries for each con\fguration. The large population for W7-X\ncorresponds to deeply trapped particles, while the other peak, larger on Wistell-A is\nnear the trapped-passing boundary.\nTo understand this behavior it is necessary to not only distinguish between prompt\nand stochastic losses, but between the pitch angle of promptly lost particles. Particles\ncan di\u000buse through phase space by pitch-angle scattering. Although pitch-angle\nscattering is small for 3.5 MeV alpha particles compared to momentum loss (by roughly\na factor of 20), it still exists. If a particle di\u000buses into a region of phase space which is\npromptly lost, it will likely be lost before it can di\u000buse out. The distribution of these\nloss regions in phase space is important. If there is one major region of losses, such as\nall deeply trapped particles, the only particles that will be lost are those born in the\nregion or close to it. However, if the prompt-loss regions are scattered around phase\nspace, even if the total volume is lower, the amount of particles that may drift through\na prompt-loss may be higher. Although veri\fcation will require statistical analysis tools\nbeyond the scope of this paper, some basic analysis can be done by examining the\npitches of promptly lost particles. Figure 10 shows a histogram of prompt (within 1 ms)\ncollisionless particle losses for W7-X and Wistell-A as a function of pitch. The pitch\nparameter is given as E=\u0016 where E is the particle energy and \u0016is the \frst adiabatic\ninvariant. This ratio is the maximum \feld a particle can reach before re\recting. The\ntrapped-passing boundary is slightly di\u000berent for the con\fgurations and is shown with\nvertical dashed lines. All the lost particles are trapped. Most of the losses from W7-X\nare from deeply trapped particles. All the losses for Wistell-A are near the trapped-\npassing boundary. While there are some losses near the trapped passing boundary for\nW7-X, these losses are signi\fcantly less in number than for Wistell-A.\nOne explanation for the relatively poor performance of Wistell-A compared to\nthe expectations from collisionless losses is as follows. The LHD-inward shifted has\nno prompt losses and the collisional results appear similar to the collisionless results.\nMost particles are lost, but they are lost slowly. W7-X does have prompt losses, butEnergetic particle transport in optimized stellarators 16\nthese occur mostly in the deeply trapped particles. Only particles that are close to\nthe deeply trapped region can di\u000buse onto lost orbits and be lost. In contrast, the\nlosses from Wistell-A occur mostly near the trapped-passing boundary. The phase\nspace volume near the trapped-passing boundary is signi\fcantly larger than the deeply\ntrapped volume. For this reason, it is easier to di\u000buse into loss regions near the trapped-\npassing boundary and the losses are enhanced for Wistell-A when collisions are included.\nOne result from this analysis is that if a con\fguration is to have prompt losses, it is far\nbetter to have them in deeply trapped regions.\n6.4. Limitations and Caveats\nThe analysis presented in this paper is useful particularly for comparing di\u000berent\ncon\fgurations, but to actually calculate losses in reactor con\fgurations additional steps\nneed to be taken. This section outlines some of the limitations of the calculations.\nAs noted above, the same pro\fles were used for every con\fguration despite\nsigni\fcant di\u000berences in \f. Furthermore, including a realistic pro\fle for each\ncon\fguration would obscure some of the di\u000berences between the con\fgurations, which\nis the primary purpose of the results presented here.\nAnother limitation is the particle following algorithm is a guiding center algorithm\nand does not include \fnite gyro-orbits. Finite orbits for alpha particles can be large and\na full-orbit analysis between con\fgurations would help determine whether the alpha loss\nestimates are accurate. At increased machine size, the e\u000bects of \fnite orbits are smaller\nand the guiding center approximation gets increasingly better. Many other e\u000bects are\nalso not included, including any transport from Alfven Eigenmodes.\nThe con\fgurations presented here all rely on VMEC equilibria. VMEC describes the\nequilibria as having nested toroidal \rux surfaces without magnetic islands or stochastic\n\feld regions. This limitation is mitigated somewhat because the large orbits of alpha\nparticles may average over small regions of stochasticity. Calculations with more realistic\n\feld, which also includes the e\u000bects from \felds generated from coils are left for future\nwork.\nParticles are considered lost if they pass beyond the penultimate surface in the VMEC\nequilibrium. In reality, particles may leave the con\fned plasma and reenter. This e\u000bect\nmay be strongest in QA con\fgurations which have the longest connection lengths and\nthus the largets banana widths.\n7. Conclusions and Outlook\nThe analysis presented here shows that it is possible to optimize for stellarators to have\ngood energetic particle con\fnement, often by ensuring very high quasisymmetry, as was\ndone in the Ku5 and Simsopt equilibria. Post-hoc analysis of these two con\fgurations\nindicate that it may be possible to achieve acceptable levels of energetic particle loss\nby optimizing for \u0000 cinstead. Since \u0000 cis less restrictive than quasisymmetry, a largeEnergetic particle transport in optimized stellarators 17\ncon\fguration space is available, and it may be possible to \fnd a con\fguration that\nsatis\fes various other needs as well. Indeed, the Wistell-A con\fguration was optimized\nwith \u0000cand has both a vacuum magnetic well and a buildable coil set.\nUnfortunately, none of the con\fgurations presented here satisfy all of our needs\nfor a stellarator reactor, which requires not only energetic particle con\fnement, but\nperformance at high pressure, a buildable coil set, as well as other properties that are\nmore di\u000ecult to quantify, like a viable divertor solution and reduced turbulent transport.\nAs optimization algorithms and the physics metrics that feed into them improve, it is\nmore likely that con\fgurations which perform satisfactorily on all required axes will be\nfound.\nAcknowledgments\nThe authors would like to acknowledge Mike Zarnstor\u000b and Sam Lazerson for providing\nthe NCSX and ARIES-CS equilibria, Don Spong for providing the ITER equilibrium,\nand Joachim Geiger and Carolin N uhrenberg for providing the W7-X equilibrium. Work\nfor this paper was supported by DE-FG02-93ER54222, DE-FG02-00ER54546 and UW\n2020 135AAD3116. Matt Landreman was supported by Simons Foundation (560651,\nML)\nReferences\n[1] Mau T, Kaiser T, Grossman A, Ra\u000bray A, Wang X, Lyon J F, Maingi R, Ku L, Zarnstor\u000b M and\nTeam A C 2008 Fusion science and technology 54771{786\n[2] Grieger G, Lotz W, Merkel P, N uhrenberg J, Sapper J, Strumberger E, Wobig H, Burhenn R,\nErckmann V, Gasparino U et al. 1992 Physics of Fluids B: Plasma Physics 42081{2091\n[3] Reiman A, Fu G, Hirshman S, Ku L, Monticello D, Mynick H, Redi M, Spong D, Zarnstor\u000b M,\nBlackwell B et al. 1999 Plasma Physics and Controlled Fusion 41B273\n[4] N uhrenberg J and Zille R 1988 Physics Letters A 129113{117\n[5] Canik J, Anderson D, Anderson F, Likin K, Talmadge J and Zhai K 2007 Physical review letters\n98085002\n[6] Henneberg S, Drevlak M and Helander P 2019 Plasma Physics and Controlled Fusion 62014023\n[7] Bader A, Drevlak M, Anderson D, Faber B, Hegna C, Likin K, Schmitt J and Talmadge J 2019\nJournal of Plasma Physics 85\n[8] N uhrenberg J 2010 Plasma Physics and Controlled Fusion 52124003\n[9] Lotz W, Merkel P, Nuhrenberg J and Strumberger E 1992 Plasma physics and controlled fusion\n341037\n[10] Mynick H E 1983 The Physics of Fluids 261008{1017\n[11] Murakami S, Wakasa A, Maa\u0019berg H, Beidler C, Yamada H, Watanabe K, Group L E et al. 2002\nNuclear Fusion 42L19\n[12] Nemov V, Kasilov S, Kernbichler W and Heyn M 1999 Physics of plasmas 64622{4632\n[13] Nemov V, Kasilov S, Kernbichler W and Leitold G 2005 Physics of plasmas 12112507\n[14] Nemov V, Kasilov S, Kernbichler W and Leitold G 2008 Physics of plasmas 15052501\n[15] Bader A, Faber B, Schmitt J, Anderson D, Drevlak M, Du\u000b J, Frerichs H, Hegna C, Kruger T,\nLandreman M et al. 2020 Journal of Plasma Physics 86\n[16] Drevlak M, Beidler C, Geiger J, Helander P and Turkin Y 2018 Nuclear Fusion 59016010\n[17] Ku L and Boozer A 2010 Nuclear Fusion 51013004Energetic particle transport in optimized stellarators 18\n[18] Koniges A, Grossman A, Fenstermacher M, Kisslinger J, Mioduszewski P, Rognlien T, Strumberger\nE and Umansky M 2003 Nuclear fusion 43107\n[19] Mynick H E, Pomphrey N and Ethier S 2002 Physics of Plasmas 9869{876\n[20] Ku L, Garabedian P, Lyon J, Turnbull A, Grossman A, Mau T, Zarnstor\u000b M and Team A 2008\nFusion Science and Technology 54673{693\n[21] Mynick H, Boozer A and Ku L 2006 Physics of plasmas 13064505\n[22] Simsopt Optimization Group Simsopt URL https://github.com/hiddenSymmetries/simsopt\n[23] Grieger G, Beidler C and Maassberg H 1991 Physics and engineering studies for wendelstein 7-x\nPlasma physics and controlled nuclear fusion research 1990. V. 3\n[24] N uhrenberg C 1996 Physics of Plasmas 32401{2410\n[25] Tobita K, Nakayama T, Konovalov S and Sato M 2003 Plasma physics and controlled fusion 45\n133\n[26] Hirshman S P and Whitson J 1983 The Physics of \ruids 263553{3568\n[27] Drevlak M, Geiger J, Helander P and Turkin Y 2014 Nuclear Fusion 54073002\n[28] Albert C G, Kasilov S V and Kernbichler W 2020 Journal of Plasma Physics 86\n[29] Helander P 2014 Reports on Progress in Physics 77087001\n[30] Boozer A H 1982 The Physics of Fluids 25520{521\n[31] Garren D and Boozer A H 1991 Physics of Fluids B: Plasma Physics 32822{2834\n[32] Lobsien J F, Drevlak M, Jenko F, Maurer M, Navarro A B, N uhrenberg C, Pedersen T S, Smith\nH M, Turkin Y et al. 2020 Nuclear Fusion 60046012"    },    {        "title": "1406.3675v1.Electric_field_coupling_to_spin_waves_in_a_centrosymmetric_ferrite.pdf",        "content": "arXiv:1406.3675v1  [cond-mat.mes-hall]  14 Jun 2014Electric-field coupling to spin waves in a centrosymmetric f errite\nXufeng Zhang,1Tianyu Liu,2Michael E. Flatt´ e,2,∗and Hong X. Tang1,†\n1Department of Electrical Engineering, Yale University,\n15 Prospect St., New Haven, Connecticut 06511, USA\n2Optical Science and Technology Center and Department of Phy sics and Astronomy,\nUniversity of Iowa, Iowa City, Iowa 52242, USA\n(Dated: January 12, 2021)\nWe experimentally demonstrate that the spin-orbit interac tion can be utilized for direct electric-\nfield tuning of the propagation of spin waves in a single-crys tal yttrium iron garnet magnonic\nwaveguide. Magnetoelectric coupling not due to the spin-or bit interaction, and hence an order\nof magnitude weaker, leads to electric-field modification of the spin-wave velocity for waveguide\ngeometries where thespin-orbit interaction will notcontr ibute. A theoryof thephase shift, validated\nby the experiment data, shows that, in the exchange spin wave regime, this electric tuning can\nhave high efficiency. Our findings point to an important avenue for manipulating spin waves and\ndeveloping electrically tunable magnonic devices.\nPACS numbers: 75.30.Ds, 75.70.Tj, 75.85.+t, 85.75.-d\nInterest in magnonics, which focuses on collective spin\ncurrents, originates from the greater stability of the col-\nlective motion of spins (spin waves); their persistence for\nlonger distances and consumption of less energy com-\npared to spin-polarized current makes magnonics com-\npetitive for low loss integrated spintronics [1–4]. Par-\nticularly, the interaction between an electric field and\na spin wave provides fundamental insight into the cou-\npling between charge and spin degrees of freedom in a\nsolid. Detection of this interaction at room temperature\nin single-crystal yttrium iron garnet (Y 3Fe5O12, YIG),\na material of great interest for magnonic device design\nbecause of its exceptionally low damping rate for spin\nwaves [5] and rich linear and nonlinear properties [6–14],\nhas proved difficult due to the lack of spontaneous elec-\ntric polarization in YIG [15]. The presence of a center\nof inversion symmetry in single-crystal YIG prevents it\nfrom responding to applied electric fields via the same\nmechanism as materials such as frustrated magnets or\nmultiferroics [16–19]. So far only indirect electric tuning\nof YIG has been achieved, with the assistance of piezo-\nelectric materials [20–23].\nIn this Letter, we demonstrate direct electric field con-\ntrol of spin waves in a YIG magnonic waveguide via a\npredicted, but not previously observed, mechanism that\noccurs even in materials with a center of inversion sym-\nmetry. Our analysis shows that this effect mostly stems\nfrom a spin-orbit (SO) interaction with a minor contri-\nbution from a first-order magnetoelectric (ME) effect.\nThe SO interaction has recently attracted intense inter-\nest because it provides new approaches for manipulating\nelectron spins [24]. In ferromagnets it leads spin waves\nthat propagate in an applied electric field to acquire an\nAharanov-Casher (AC) phase[25]. To linear order of the\nelectric field this is equivalent to addinga Dzyaloshinskii-\nMoriya-like (DM-like) interaction between neighboring\nspins (Si,j) that takes the form [26, 27]: Hij=Dij·\n(Si×Sj), where |D| ∝E/ESO= 2mλ2\nSOE//planckover2pi12is the\nDM vector, mis the electron rest mass, /planckover2pi1is the reduced\nEYIG\nElectrodeGGGkB\nExcitation \ntransducerDetection\ntransducer\nElectrode\nFigure 1. (Color online). Schematic of the YIG magnonic\nwaveguide used in this experiment. B: bias magnetic field;\nE: electric field; k: wave vector.\nPlanck constant, and λSOis a characteristic length scale\nthat determines the SO interaction strength. Through\nthis effect the applied electric field adds an AC phase to\nthe spin waves [25, 28–30]. The SO interaction in YIG\nwas previously considered to be extremely small due to\nan assumption that λSO=λc(the reduced Compton\nwavelength). A recent theoretical study predicts that\nthe SO interaction can be orders of magnitude larger in\nYIG if one considers orbital hybridization, which yields\nλSO≫λc[31, 32]. Here we present experimental ob-\nservation of this SO interaction in a single-crystal YIG\nthin film. In addition, our experiments found an electric\ntuning of the ferromagnetic resonance (FMR) frequency\nwhichweattributetoafirst-orderMEeffect. Notingthat\nthe SO interaction depends on an orthogonality between\nthe applied electric field, the equilibrium magnetization\nand the wave vector of the spin waves, while the ME\neffect does not, we clearly identify the different contribu-\ntions from the two effects by applying the electric field\nout-of-plane and in-plane.\nFigure1shows the schematic of our device, containing\na narrow strip of YIG thin film as the magnonic waveg-\nuide, a pair of copper electrodes to apply electric fields\nacross the waveguide, and a pair of microstrip transduc-\ners to excite and detect the spin waves. The YIG strip (22\n-60-30\n3.4 3.5 3.6-400-2000 Magn. (dB) Phase (rad)\nFrequency f (GHz)\nRF SG RF PMSG\nRF Amp\nRF Amp LPFBHV Amp\nMixer\nLock-in \nAmp (a) (b)\n(c) 0 50 1003.53.6 Frequency f (GHz) \nWavevector k (cm-1)\nFigure 2. (Color online). (a) Dispersion relation of the spi n\nwave in the YIG magnonic waveguide. Red squares and blue\ncircles are the experimentally extracted dispersions with and\nwithout metal electrodes on the YIG surface, respectively.\nSolid lines are the theoretical calculations. Dashed black\nline is the linearized dispersion. (b) Vector network analy zer\ntransmission characterization oftheYIGmagnonic wavegui de\nwith magnitude response shown in the top panel and phase\nresponse shown in bottom panel. (c) Interferometry scheme\nfor measuring spin wave phase accumulation. SG: signal gen-\nerator; RF SG: radio frequency signal generator; Amp: am-\nplifier; PM: powermeter; LPF: low-pass filter; HV Amp: high\nvoltage amplifier.\nmm×40mm)iscutfroma5- µm-thickthinfilmofsingle-\ncrystalYIG epitaxiallygrownona0.5-mm-thickgadolin-\nium gallium garnet (Gd 3Ga5O12, GGG) substrate. To\navoid magnon reflection, the two ends of the YIG strip\nare terminated by 45◦angled cuts. The two microstrip\ntransducers are placed 30 mm apart over the two ends of\nthe magnonic waveguide. The excited spin waves propa-\ngate along the long axis of the magnonic waveguide. The\nelectrodes are attached onto the top and bottom surfaces\nof the device and cover20 mm length of the waveguide to\nprovide a sufficiently long interaction length. This leaves\na 5 mm gap between the electrode and the microstrip\ntransducer which is wide enough to avoidelectrical cross-\ntalk between transducers. As the SO interaction requires\nthe wave vector k, the magnetization Mand the elec-\ntric field Eto be orthogonal, we apply the bias magnetic\nfield in-plane and transverse to the wave propagation di-\nrection.\nIn this configuration the excited spin wave mode in\nthe magnonic waveguide is a magnetostatic surface spin\nwave (MSSW). Using methods provided in Refs.[33] and\n[34], we calculatethe dispersion ofthe MSSW taking intoaccount the effects of the GGG substrate and electrodes:\ne2kd=1−χ+κ−tanh(kt1)\n1−χ−κ+tanh(kt1)·1−χ−κ−tanh(kt2)\n1−χ+κ+tanh(kt2),\n(1)\nwhereχ=fBfM\nf2\nB−f2andκ=ffM\nf2\nB−f2withfbeing the fre-\nquency,fB=γB,fM= 4πγM0. Other parameters are:\nthe bias magnetic field B, the equilibrium magnetization\n4πM0, the gyromagnetic ratio γ, the wave vector k, YIG\nfilm thickness d, the gap between the YIG film and the\nupper (lower) electrode t1(t2). Note that t2is approxi-\nmately the thickness of the GGG layer.\nIn Fig.2(a) we present the calculated dispersions of\nthe waveguide with and without electrodes using Eq.( 1)\n(the solid red line versus the solid blue line). In both\ncases, the electric field is set at zero. The calculated\ndispersions agreewell with the experimental data (circles\nand squares). The presence of electrodes on the YIG\nsurface increases the group velocity of the spin waves.\nFor small kvalues (which is the case in our experiment\ndue to the limits of the transducers), the dispersion can\nbe linearized by expanding the original dispersion f=\nΩ(k) around k0to the first order of ( k−k0):\nf= Ω0+vg0(k−k0) =vg0k+fFMR,(2)\nwhere Ω 0= Ω(k0),vg0=∂kΩ(k0) is the group velocity\natk0, andfFMRis the FMR frequency obtained after\nthe linearization. The dashed line in Fig. 2(a) shows the\nlinearized dispersion expanded around k0= 60 cm−1and\nit replicatesthe completedispersionwithin the range k <\n70 cm−1.\nThe spin wave propagation along the waveguide is\ncharacterized using microwave transmission measure-\nment [Fig. 2(b)]. Under a bias magnetic field of 60.1\nmT the spin wave transmission band covers 3 .42−3.58\nGHz within which the spinwaveaccumulatesaverylarge\nphase after propagating through the waveguide owing to\nits small phase velocity. From the phase spectrum we\nextract the MSSW dispersion.\nWhen an external electric field is applied across the\nmagnonic waveguide as in Fig. 1, the spin wave phase\naccumulation is modified as a result of the SO interac-\ntion. Such phase changes can be precisely detected with\nour interferometry scheme [Fig. 2(c)]. One arm of the\ninterferometer is the magnonic waveguide, whereas the\nother arm is a reference signal originating from the same\nmicrowave source. The power sent into the magnonic\nwaveguide is kept below the nonlinear threshold of the\nMSSW to avoidundesired nonlineareffects. The electric-\nfield-induced phase is measured by comparing these two\narms at the phase detector, which consists of a mixer and\na low-pass filter. The measured phase is normalized by\nthe transmitted power and monitored by a RF powerme-\nter to eliminate the amplitude information. To increase\nthe measurement sensitivity and suppress system fluctu-\nations, the applied electric field is modulated at 7 kHz\nand a lock-in amplifier is used to detect the correspond-\ning phase modulation.3\n(a) (b)\n(c) (d) \n60 65 700510\n Theory\n Meas.×10-9 rad/(GHz ⋅V/m) ∂f,E \nMagnetic field B (mT)23.45 3.500.00.51.0\n0.20.8E (×106 V/m)\nfFMRϕ (10-3 rad)\nFrequency f (GHz)1.0\n0.6\n0.4\n0.2 0.4 0.6 0.8 1.0051015\n B = 60.1 mT \n B = 65.1 mT \n B = 70.1 mT  ∂fϕ  (10-3 rad/GHz )\nElectric field E (106 V/m)\n0.2 0.4 0.6 0.8 1.00.00.40.8\n×10-3 rad\n Theory\n Meas . \nϕFMR\nElectric field E (106 V/m)ϕ\nFigure 3. (Color online). (a) Measurement of the electric-\nfield-induced phase (symbols) at various electric fields (bi as\nmagnetic field B= 60.1 mT). Dashed lines show the linear\nfittings. (b) Dependence of ∂fϕon the electric field with\ndifferent bias magnetic fields. Solid lines show the theoreti cal\npredictions. (c) Dependence of ∂2\nf,Eϕon the magnetic field.\n(d) Phase induced by the first-order ME effect at the FMR\nfrequency (bias magnetic field B= 60.1 mT).\nFigure3(a) shows the measured phase signal induced\nby different electric fields with a bias magnetic field of\n60.1mT. At anapplied electricfield of ∼106V/m, the re-\nsulting phase (normalized to the propagationdistance) is\nofthe orderof10−5rad/mm. We notethat this valuecan\nbe drastically enhanced by decreasing the wavelength.\nEspecially, it is estimated that a π-phase shift can be\nachievedasthewavelengthapproachestheexchangelimit\n[32]. The phaseshift signalhasacleardependence on the\nelectric field, demonstrating the electric tuning origin.\nIntheACeffectpicture, theSOinteractionprovidesan\nelectric-field-dependent term f=fMλkto the dispersion\n[32], where λ= 2Ja5eE/µ0ESO/planckover2pi12γ2, withJbeing the\nexchange coefficient between neighboring lattices, athe\nlattice constant, ethe elementary charge, and µ0the vac-\nuum permeability. Since the magnetization of the YIG\nis not saturated under the applied magnetic field, Jhas\naBdependence and accordingly λcan be expressed as\nλ= (λ0+λBB)E, whereλ0andλBare constants deter-\nmined through the experiments. In another, equivalent,\ndescription, the spin wave gains an additional wave vec-\ntorkSOatagivenfrequency f, whichyields anadditional\nphaseϕSOafter the spin wave propagates a distance L.\nUsing the linear dispersion approximation we have:\nϕSO=L\nv2\ng0fM(f−f0\nFMR)(λ0+λBB)E,(3)\nwheref0\nFMRdenotes the FMR frequency in the absence\nof electric fields. This equation shows a clear linear de-pendence of the SO-interaction-induced phase on the fre-\nquency and the electric field, in agreement with the data\nshown in Fig. 3(a).\nHowever, Eq.( 3) also indicates a zero phase shift at\nthe FMR frequency, which deviates from our experimen-\ntal observation. We attribute this discrepancy to a first-\norder ME effect, which directly modifies the equilibrium\nmagnetizationand is inherent to magnetic materials. Be-\ncause of this ME effect, fMbecomes fM+pEin the pres-\nence of an applied electric field, where pis a constant.\nSubstituting the new expression into the linear disper-\nsion given by Eq.( 2) we have the total phase induced by\nboth SO and ME effects: ϕ=ϕSO+ϕME, where\nϕME=/parenleftbiggf−f0\nFMR\nv2\ng0v′\ng0+∂EΩ0−v′\ng0k0\nvg0/parenrightbigg\nLE,(4)\nwherev′\ng0=∂Evg0. Note that due to the existence of\nthe direct ME effect, vg0becomes a function of E. From\nEq.(4) we can see that there exists a nonzero phase at\nthe FMR frequency. In addition, the ME effect also con-\ntributes a f-dependent term.\nWe compareourmodel (the solidlines) with the exper-\niments (the dots) in Figs. 3(b)–(d) and a good agreement\nis achieved. The solid lines are obtained by taking into\naccount both the SO effect and the direct ME effect. As\npredicted in Eqs.( 3) and (4), the measured electric-field-\ninduced phase is linear in the frequency and increases\nwiththeelectricfield[Fig. 3(a)]. Thiselectricfielddepen-\ndence is shown in Fig. 3(b), where the partial derivative\n∂fϕis plotted as a function of the electric field at various\nbias magnetic fields. The magnetic field dependence of\nthe second derivative of the phase ( ∂2\nf,Eϕ) is plotted in\nFig.3(c). It can be seen that the effect of electric tuning\ncan be enhanced by increasing the electric field and the\nbias magnetic field. Figure 3(d) shows that the induced\nphase at f0\nFMRis indeed nonzero due to the direct ME\neffect.\nThegoodagreementbetweenthetheoryandtheexper-\nimental data supports our interpretation that the mea-\nsured electric tuning originates from the combined effect\nof the SO and ME interaction with dominant contribu-\ntion coming from the SO effect. In our model there are\nthree unknown parameters: λ0,λB, andp, while the rest\nof the parameters are all known constants. From the\nmeasurement data we obtain these unknown parameters\nthrough numerical fitting: λ0=−1.095×10−16m2/V,\nλB= 2.080×10−15m2/(V·T), andp= 2.34×10−3Hz\nV/m.\nAt a bias magnetic field of 60.1 mT and electric field\nof 1×106V/m, we obtain λ= 0.15˚A and accordingly\nλSO= 0.45˚A, which is indeed two orders of magnitude\nlarger than λc(3.85×10−3˚A).\nTofurther separatethe contributionsto the phase shift\nfrom the SO and ME effects, we examine their depen-\ndence on the direction of the applied electric field. By\nmoving the two electrodes to the side of the waveg-\nuide, we apply the electric field in the same direction\nas the magnetic field, as illustrated in the lower inset of4\n3.46 3.48 3.50 3.520.51.0 \nϕ  (10-3 rad)\nFrequency    (GHz)f\nEB k\nEB k(a) (b)\nE: out-of-plane \nE: in-plane \n0.0 0.5 1.0051015 ∂fϕ (a.u.)\nElectric field (106 V/m)\nFigure 4. (Color online). (a) The measured phase shift with\nthe electric field applied in the in-plane (blue squares) and\nout-of-plane (red circles) direction, respectively, unde r the\nsame bias magnetic field ( B= 60.1 mT). Dashed black lines\nare the linear fittings. (b) Dependence of ∂fϕon the elec-\ntric field for the in-plane (blue circles) and out-of-plane ( red\nsquares) electric field configuration, respectively. Solid lines\nshow the model predictions.\nFig.4(a). The SO interaction vanishes under this config-\nuration since it requires k,MandEto be all orthogonal.\nAs the first-orderME effect does not requiresuch orthog-\nonality, the phase shift for this electrode configuration\nwould arise solely from the ME effect.\nFigure4(a) compares the measured phase ϕfor in-\nplane (circles) and out-of-plane (squares) electric fields\nunder the same bias magnetic field ( B= 60.1 mT).\nTo obtain quantitative comparison between these two\ncurves, it is important to realize that the group velocities\nare different for these two cases because of the dispersion\nchangewhenremovingthecopperelectrodefromtheYIG\nsurface [squares versus circles in Fig. 2(a)]. In addition,\nthe obtainable electric field ranges are different due to\nthe large aspect ratio of the sample cross section. There-\nfore it is difficult to directly compare the effects at the\nsame electrical field. Nevertheless the change of slope or\npartial derivative of the phase ( ∂fϕ) truly differentiates\nthesetwoeffects. In theexperiments, wevarytheapplied\nelectric fields and normalize the measured ∂fϕwith the\ngroup velocity and the electric field [ ∂fϕ=∂fϕ·(v2\ng0/E)\nin Fig.4(b)]. The dramatically reduced slope signal in-\ndicates the greatly suppressed SO interaction for the in-\nplane electric field configuration. The theoretical predic-\ntion for the in-plane configuration, which only includes\nthe ME effect using parameters obtained from the out-\nof-plane configuration, shows good agreement with the\nexperiment data and validates our analysis.\nIn conclusion, we experimentally demonstrated the ex-\nistence of the SO interaction in single-crystal YIG. Such\ninteraction shifts the spin wave dispersion under exter-\nnal electric fields applied perpendicular to the magne-\ntization and wave propagation directions. As a result,electric-field-induced phase modulation of the propagat-\ning spin waves in a magnonic waveguide is achieved. On\nthe other hand, we found another effect, the first-order\nMEeffect, alsocontributestotheelectrictuningbymodi-\nfyingtheequilibriummagnetizationwithanelectricfield.\nThe latter effect can be separately measured by applying\nthe electric field in a direction parallel to the magnetiza-\ntion. A complete theoretical model including both effects\nis developed and is in agreement with the experimental\ndata. Theoretical calculations indicates that high tuning\nefficiency and low tuning voltage can be achieved by ex-\npanding to the exchange spin wave regime or by utilizing\ncompact on-chip magnonic waveguides. We anticipate\nthat further scaling the YIG devices to the micro- and\nnano-scale would allow on-chip electric field control of\nspin waves. 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We show that the power-flow \nlines of the microwave-cavity field interacting with a ferrite disk, in the proximity of its \nferromagnetic resonance, form whirlpool-like electromagnetic vortices. Such vortices are \ncharacterized by the dynamical symmetry breaking. The role of ohmic losses in waveguide walls \nand dielectric and magnetic losses in a disk  is a subject of our  investigations.  \n PACS: 41.20.Jb; 76.50.+g; 84.40.Az  \n1. Introduction \nA three-dimensional system of a rect angular-waveguide resonator with an  inserted thin ferrite disk is \na very attractive object for many physical aspects. Fi rst of all, such a struct ure gives an interesting \nexample of a nonintegrable system. Then, because of inserting a piece of a magnetized ferrite into \nthe resonator domain, a microwave resonator beha ves under odd time-reversal symmetry. A ferrite \ndisk may act as a topological defect causing i nduced vortices. Such vortices can appear due to \ngyrotropy of a ferrite material. Finally, the studies of this object may clarif y some important aspects  2concerning recent experiments of interaction of oscillating ma gnetic-dipolar modes (MDM) in \nferrite disks with microwav e electromagnetic fields. \n    Nonintegrable systems (such, for example, as Si nai billiards) are the subject for intensive studies \nin microwave experiments [1]. The results show th at microwave cavity experi ments can be an ideal \nlaboratory for studying the so-called quantum-classi cal correspondence, a centr al issue in quantum \nchaos. To get microwave billiards with broken tim e-reversal symmetry, ferrite strips and ferrite \ncylinders were introduced into the resonators [2]. A role of material gyro tropy, as a factor provoking \ncreation of vortices, was a subjec t of recent studies [3]. The vor tices are defined as lines of \npowerflow, i.e. lines to which the Poynting vector is tangential. It  is pointed out that in these \nexperiments, the gyrotropic effect cannot be consid ered separately from losses. At the same time, \ndue to just only inserted cylinders made of mate rials with losses (without any gyrotropic properties), \nvortices of the power flows can be obser ved in a rectangular billiard [4].  \n    Most of the investigated nonintegrable system s are considered as hard-wall billiards. However, \nfor the class of optical, or dielectric, noninteg rable systems the boundary manifests itself by a \nchange in the index of refracti on. The interplay of reflection a nd transmission at the different \ninterfaces gives rise to a rich  variety of wave phenomena [5]. The problem of non-hard-wall \ninclusions in microwave resonators suggests certai n questions about internal  fields inside these \ninclusions, especially when intrinsic material re sonances (such, for example, as FMR or plasmon \nresonance) occur. In papers [ 2, 3], ferrite inclusions with th e FMR conditions were used, but no \nproper analysis of the fi elds inside small ferrite samples were  made. These \"internal\" fields may \nexhibit, however, very unique properties. In partic ular, recent studies [6] shows that for a case of a \nnanoparticle illuminated by the electromagnetic field, the \"energy sink\" vortices  with spiral energy \nflow line trajectories are seen in the proximity of the nanoparticle's plasmon resonance. \n    The purpose of this paper is to study numerically the fields outside and inside a ferrite disk placed \nin a rectangular-waveguide cavity . This may call special attention in a view of a strong present  3interest in microwave noni ntegrable systems with broken time-re versal symmetry. The disk is very \nthin compared to the waveguide height. Such  a quasi two-dimensional nonintegrable system \ndistinguished from configurations an alyzed in [2, 3]. There is also a particular interest in such thin-\nfilm ferrite samples in a view of  recent studies of interaction of MDM oscillations with cavity EM \nfields. Experimental studies show unique propert ies of interaction of oscillating MDMs in small \nferrite disk resonators with microwave-cavity elec tromagnetic fields [7  10]. The character of the \nexperimental multi-resonance absorption spectra l eads to a clear conclusion that the energy of a \nsource of a DC magnetic field is absorbing by por tions, or discretely, in  other words [11]. The \nMDMs in a ferrite disk are characterized by dynami cal symmetry breaking [12]. In this paper, we \nput aside any detailed discussions  of MDM spectra in a disk, but  study just how the FMR and \ngeometry factors in a system (a microwave ca vity plus a gyrotropic disk) may effect on the \nelectromagnetic field patterns. It was assumed in [7   10], the cavity-field structure is not strongly \nperturbed by a ferrite sample. So it was supposed that  the acting RF field corresponds to the original \ncavity field in a point where a ferrite sample is pl aced. As we will show in this paper, even a small \nFMR-disk may strongly perturb the cavity field. The power-flow lines of the microwave-cavity field \ninteracting with a ferrite disk, in  the proximity of its ferromagneti c resonance, may form whirlpool-\nlike electromagnetic vortices. The role of ohmic losses in waveguide walls and dielectric and \nmagnetic losses in a disk is a subject of our inve stigations. We study the symmetry properties of the \nvortices. There are certain symmetry features of vortices when one reverses  the DC magnetic field \nand interchanges the RF s ource and receiver positions.  \n    Since the nonintegrable na ture of the problem precludes ex act analytical results for the \neigenvalues and eigenfunctions, numerical appro aches are required. Numeri cal studies of vortex \nformations in different electroma gnetic structures were a subject of numerous investigations. It \nconcerns, for example, vortices of the Poynting vector  field in the near zone of antennas [13], vortex \nformation near an iris in a rectangular wavegu ide [14], light transmissi on through a subwavelength  4slit [15], energy flow in photonic crystal waveguid e [16]. Using the HFSS (the software based on \nFEM method produced by ANSOFT Co mpany) and the CST MWS (the software based on FITD \nmethod produced by Computer Simulation Technol ogy Company) CAD simulation programs for 3D \nnumerical modeling of Maxwell equa tions, we are able to characterize the complete complex signal \nincluding the intensity of the si gnal and its phase relati ve to the incoming reference wave. In our \nnumerical experiments, both modulus and phase of the fields are determined. It allows \nreconstructing the Poynting vector at any point with in the resonator. The main results of numerical \nsimulations we obtained based on th e HFSS program since this program is more relevant for precise \nanalyzing high-resonant microwave objects. The CST program was used just as a test program for \nsome cases. All the below pictures correspond to the HFSS-program numerical results.   \n \n 2. Cavity field structure and vortices \nLet us consider a vacuum region of the cavit y space. For TE polarized (with respect to the y-\ndirection) electromagnetic waves, the singular features of the co mplex electric field component \n),(zxEy  can be related to those that will subsequently appear in the associated two-dimensional \ntime-averaged real-valued Poynting vector field ) ,(zxSr\n. \n    The transport of electromagnetic energy throu gh the resonator is described by the Poynting vector  \n                                              ( )( ) [] t  tRe Re4ω ω\nπi\nci\nc eH eEcSr r r\n× = ,                                                       (1) \nwhere cEr\n and cHr\n are complex amplitudes of the field v ectors. From the Maxwell equation in a \nvacuum one has \n                                                 () ( ) [ ] t  tIm Reω ω\nωi\nci\nc eEceHr r\n×∇ = .                                                      (2) \nSo Eq. (1) can be rewritten as  \n  5                                             () () () [] t  t2\nIm Re4ω ω\nπωi\nci\nc eE eEcSr r r\n×∇× = .                                               (3) \nWe take advantage now of the following v ector relation for two arbitrary vectors, ar and br\n. If one \nsupposes that these vectors have only one component (let it will be the y-component), one evidently \nhas:  ()y yb ab a⊥∇=×∇×r r r, where ⊥∇ is the differential operator with respect to the x and z \ncoordinates. For electromagnetic fields, which are invariant with respect to the y-direction, this gives \npossibility to represent a time-averag e part of the Poynting vector as \n                                                         ()y yE EcS⊥∇ =r r\n Im8*2\nπω,                                                        (4) \nwhere yE is a complex vector of the y-component of the electric field: ()ti\nyc y eE E ω≡ . \n    The fact that for electromagnetic fields inva riant with respect to a certain coordinate, a time-\naverage part of the Poynting vector can be approxim ated by a scalar wave f unction, allows analyzing \nthe vortex phenomena. For a TE pol arized field, we can write \n                                                   ),(  ),( ),( ),(zxi\ny ezx zx zxEχρ ψ = ≡ ,                                                (5) \nwhere ρ is an amplitude and χ is a phase of a scalar wave function ψ. One can rewrite Eq. (4) as  \n                                                         ),( ),(2zx zx S χ ρ⊥∇ =r\n.                                                         (6)                  \nThis representation of the Poynting vector in a quas i-two-dimensional system allows clearly define a \nphase singularity as a point ( x, z) where the amplitude ρ is zero and hence the phase χ is \nundefined. Such singular points of ) ,(zxEy correspond to vortices of the power flow Sr\n, around \nwhich the power flow circulates. A center is refe rred to as a (positive or negative) topological \ncharge. Since such a center occur in free space without energy absorption, it is evident that \n0= ⋅∇⊥Sr\n. Because of the phase singular ities of the free-space electromagnetic field, unique \nproperties of the power flow tran smission near a sub-wavelength slit can be demonstrated [15]. The \nfact that vortices of the free-space Poynting vector  in flat electromagnetic resonators appear as a  6consequence of the nontrivial to pological structure, make this  relevant for modeling quantum \nvortices [1-4].      It was shown in [3] that due to the ferrite  flat-boundary reflection eff ect, for TE polarized plain \nelectromagnetic wave, there are microwave vortices of the Poynting vector in a vacuum region. For \na waveguide structure, we reconstruct these result s of the Poynting vector distribution based on the \nCAD simulation program. The HFSS-program results are shown in Fig.1 for a lossless [the perfect-\nelectric-conductor (PEC) walls] r ectangular waveguide with an en closed lossless ferrite slab. A \nwaveguide is terminated by the PEC wall. A system is exited by the \n40TE  mode with the electric \nfield oriented along y axis. A ferrite [yttrium iron garnet (YIG)] is saturated ( G Ms  1880  4 = π ) by a \nDC magnetic fiel d directed along y axis. The working frequency (8.7 GHz) and a quantity of a bias \nmagnetic field (5030 Oe) correspond to necessary conditions  for a ferromagnetic resonance: a \ndiagonal component of the permeability tensor is 85.230=µµ  and an off-diagonal component is \nequal to 55.220=µµa  [17]. In Fig. 1, one sees that in a vacuum region of the cavity space the \nsingular points of Sr\n (the vortex cores) can be directly related to the topol ogical features of \n),(zxEy (where the electric field is zero). It is al so interesting to note that the vortex positions \ncorrespond to the regions with the homogeneous RF magnetic field. \n    Let us consider now a ferrite region of the cavity space. The transport of electromagnetic energy \nthrough this region is characteri zed by the Poynting vect or described by Eq. (1). However, one \ncannot express now the Poynting vector just only by the Er\n-field vector, like it was shown in Eq. (3). \nSo even if we suppose that there exists only one component of the electric field, ),(zxEy , we cannot \nexpect a priori to find any vortices of the power flow Sr\n at singular points of ) ,(zxEy . On the \nother hand, we can expect to find a vo rtex in a region of a maximal field ) ,(zxEy .  \n  73. The Poynting-vector vortices in a cavity with a ferrite disk \nThe main systems under our investigations are X-band rectangular-wavegui de cavities with an \nenclosed thin ferrite disk. The systems resemble ones used in experiments [8-10]. There are a short-\nwall cavity with an iris [Fig. 2] and a cavity with tw o irises [Fig. 3]. The disk is very thin compared \nto the waveguide height and is placed in the mi ddle of the waveguide height. The DC magnetic field \n( =0H 5030 Oe) is normal to the disk plane. As a star ting point of the study, we introduce in our \nCAD program the following material paramete rs of the disk: dielectric constant, 15=rε ; dielectric \nlosses, 01.0 tan =δ ; magnetic losses, Oe H 1.0=∆ .  Metal walls of a cavity are made of copper and \nare characterized by the conductivity of m Siemens / 108.57× =σ . The microwave power enters to a \ncavity through an iris. The cavity operates at the 104TE  mode. By virtue of such an elongated \nstructure one can clearly observe the power flow distribution in a system. The cavity resonances \nwere estimated via frequency dependent absorption peaks of the 11S and 21S parameters of the \nscattering matrix.  \n    A ferrite disk may act as a topological defect  causing induced vortices. Figs . 4 (a), (b) and Figs. 5 \n(a), (b) show the streamlines of the Poynting vect or in a short-wall cavity when a ferrite disk \n(diameter 6 mm, thickness 0.1 mm) is inserted in a maximum of the RF electric field ( 4λ=l ). One \ncan distinguish the clockw ise and counter-clockwise rotations of the power flow corresponding to \ntwo opposite orientations of a normal bias field.  The vortex center  the t opological singularity  is \nin a geometrical center of a disk. Fig. 6 shows th e Poynting vector distribu tion immediately inside a \nferrite disk. Figs. 7 and 8 show, respectively, th e electric and magnetic RF fields in a disk. An \nanalysis of the vortex pictures in  Figs. 4 and Fig. 6 gives a result worth special notice. One can see \nthat for the same direction of a bias magnetic fiel d, there are opposite directions of Poynting-vector \nrotations inside and outside a ferrite disk.      A character of the vortex picture, in genera l, is independent from a disk diameter. This is \nillustrated in Fig. 9 which shows the Poynting vector distribution fo r a \"big\" disk (diameter 12 mm,  8thickness 0.1 mm). A vortex in a ca vity with a ferrite disk takes a shape of a whirlpool. It becomes \nevident from a picture of the Poynting vector distribution immediately upper and below a ferrite \ndisk. Such a picture is shown in Fig. 10 for a \"big \" disk. One can see the \"energy-sink\" character of \nthe vortex with spiral energy flow  line trajectories in the proximity of the FMR disk. A role of \ngyrotropy in forming the vortex structure become s evident when one compares the above power \nflow characteristics with the power flow characteristics for a case of a simple lossy dielectric disk. \nFig. 11 demonstrates dist ribution of the Poynting ve ctor in a cavity with a lossy dielectric disk \n(diameter 6 mm, thickness 0.1 mm, dielectric constant, 15=rε ; dielectric losses, 01.0 tan =δ ) \ninside a rectangular-w aveguide cavity.  \n    The vortex pictures become essentially differe nt when one places a ferrit e disk in a maximum of \nthe short-wall-cavity magnetic field ( 2λ=l ). These pictures, for a \"small\" disk (diameter 6 mm, \nthickness 0.1 mm) are shown in Figs. 12 and 13. The pictures correspond to two opposite \norientations of a normal bias fiel d. In this situation one can dis tinguish two coupled vortices having \nthe same \"topological charge\" (the same direction of rotation for a given direction of the bias \nmagnetic field). The vortices center s are shifted from a geometrical cen ter of a disk and are situated \nnear the disk border.     A change of the microwave source and receiv er positions (at the same  direction of the bias \nmagnetic field) does not change th e vortex rotation direct ion. Fig. 14 (a) show s the Poynting vector \ndistribution in a two-irise cavity when a \"big\" (diameter 12 mm, thickness 0.1 mm) ferrite disk is \nplaced in maximum of the electric field. The power i nput is at the left-hand side of a system. If one \ninterchanges the microwave source and receiver posit ions, leaving fixed a direction of the bias \nmagnetic field, the vortex will have the same rotation direction. This is shown in Fig. 14 (b). It means that 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 \nmicrowave source and receiver positions, the vortex ch anges its rotation direct ion [compare Figs. 14  9(b) and 14 (c)]. It means that the vortex rotation dire ction is not invariant with respect to a combined \nsymmetry operation: mirror reflection and time reversal.   \n4. The role of material losses in forming vortices \nOne of the main points in our stud ies should concern a role of the material losses in a system in \nforming the ferrite-disk vortex stru ctures. An essential role of the material losses in forming the \nelectromagnetic vortices was stresse d in paper [6], where the spiral  energy flow line trajectories in \nthe proximity of the nanoparticle's plasmon resona nce were studied for different quantities of the \nimaginary parts of the permittivity. We found that  also in our case the vortex structure and the \ntopography of the field maps depend on the values of dissipation losses. \n    For a case when a ferrite di sk is placed in a maximum of the s hort-wall cavity electric field, we \nfound that in the regions distant fr om a ferrite disk there is a st rong vortex-picture  dependence on \nthe waveguide wall ohmic losses and the disk material losses. At the same time, the vortex structure \ninside a ferrite disk and in immediate proximity to  a ferrite disk remains practically the same when \none changes the wall conductivity and the disk losses. Fig. 15 (a) shows the Poynting-vector \ndistribution in a short-wall cavity for the PEC waveguide walls and disk losses parameters: \n01.0 tan =δ , Oe H 1.0=∆ . In comparison with Figs. 4 and 5,  one can see the difference in the \nPoynting-vector distributions in the cavity regions distant from a ferrite disk. Now, it appears an \nadditional vortex in a vacuum region of the cavity . A character of this additional vortex changes \nwhen one changes the disk losses parameters . Fig. 15 (b) corresponds  to parameters: PEC \nwaveguide walls, 0002 .0 tan,1.0 = =∆ δ Oe H  and Fig. 15 (c) is for PEC walls, \n01.0 tan,5 = =∆ δ Oe H . Further increase of magnetic losses parameter H∆ leads to elimination of \nan additional vortex. Regarding the vortex pictures shown in  Figs. 15 (a), (b), and (c) it is important \nto note that the cores of additional vortices are situated at a distance 2λ=l  from a short wall along \nz axis, evidently correspondi ng to regions where  0 ),( =zxEy .  10 \n6. Discussion and conclusion \nFrom a graphical representation of the Poynting vect or one sees that the absorption cross-section of \na ferrite particle can be much bigger than its ge ometrical cross-section. Wh en a vortex is created, \npower flow lines passing through th e ferrite particle gene rate the high energy lo sses associated with \nthe large microwave cross-section. The reason why a ferrite sample absorbs much more radiation \nthan that given by the geometrical cr oss-section can be explained by the fact that the field enters into \nthe particle not only from the front (with respect to a power  source) part of the surface, but also from \nthe back (\"shadow\") side. In ot her words, one can suppose that a disk absorbs incident energy \nthrough its whole surface. The featur e of the observed Poynting-vector  vortex is the fact that this \nvortex cannot be characterized by so me invariant, such as the flux of vorticity. So a vorticity thread \nmay not be defined for such a vortex.     The problem of microwave vor tices of the Poynting vector create d by ferrite disks looks as very \nimportant for many modern applications, e.g., for near-field microwave lenses, for field \nconcentration in patterned microwave metamaterials,  for high-Q resonance microwave devices. It is \nsupposed that these \"swirlin g\" entities can, in prin ciple, be used to carry  data and point to new \ncommunication systems. Application of such generi c ideas to microwave systems is of increasing \nimportance in numerous utilizations. For example, any work in the area of a \"vortex antenna\" is \nlikely to generate many unique microwave systems,  e.g., in direction findi ng and target ranging. \nWith respect to this aspect, an interesting phenomenon of the transition of a near-zone phase \nsingularity into a singularity of the radiat ion pattern has been shown recently [18]. \n \nReferences \n[1] E. Doron, U. Smilansky, and A. 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D: Appl. \nPhys. 35, 2484 (2002). \n[10] E.O. Kamenetskii, A.K. Sa ha, and I. Awai, Phys. Lett. A 332, 303 (2004). \n[11] E.O. Kamenetskii, Phys. Rev. E 63, 066612 (2001);  \n      E.O. Kamenetskii, R. Shavit, and M. Sigalov, Europhys. Lett. 64, 730 (2003);  \n      E.O. Kamenetskii, M. Sigalov, and R. Shavit, J. Phys.: Condens. Matter 17, 2211 (2005).  \n[12] E.O. Kamenetskii, Europhys. Lett. 65, 269 (2004); Phys. Rev. E 73, 016602 (2006); J.Magn. \nMagn. Mater. 302, 137 (2006). \n[13] F. Landstorfer, H. Meinke, an d G. Niedermair, Nachr. tech. Z. 25, 537 (1972). \n[14] R. W. Ziolkowski and J. B. Gran t, IEEE Trans. Microw. Theory Tech. MTT-34 , 1164 (1986). \n[15] H. F. Schouten, T. D. Visser, D.  Lenstra, and H. Blok, Phys. Rev. E 67, 036608 (2003). \n[16] T. Søndergaard and K. H. Dridi, Phys. Rev. B 61, 15688 (2000). \n[17] A. Gurevich and G. Melkov, Magnetic Oscillations and Waves  (CRC Press, New York, 1996).  12[18] H. F. Schouten, T.D. Visser, G. Gbur, D. Lenstra, and H. Blok, Phys. Rev. Lett. 93, 173901 \n(2004). \n \nFigure captions: \nFig. 1 . The 40TE  lossless waveguide structure with an encl osed lossless ferrite slab (crosses show \nthe vortex positions).  \n         (a) Distribution of the Poynting vector;           (b) Distribution of the electric field. Fig. 2.\n A short-wall cavity with an iris  and an enclosed ferrite disk. \nFig. 3.  A cavity with two irises a nd an enclosed ferrite disk. \nFig.4.  The cavity Poynting vector distribution. A ferrite  disk is placed in the maximum of the cavity \nelectric field. A bias field is directed along positive y  axis. (a)  A general view; (b) An \nenlarged picture of the vortex. \nFig.5.  The cavity Poynting vector distribution. A ferrite disk is placed in the maximum \nof the cavity electric field. A bias field is directed along negative y axis. \n         (a)  A general view;          (b) An enlarged picture of the vortex. Fig. 6.\n The Poynting vector distribu tion inside a ferrite disk. \nFig. 7.  Electric field vector inside a ferrite disk. \nFig. 8.  Magnetic field vector inside a ferrite disk. \nFig. 9. A vortex created by a big ferrite disk. \n           (a)  A general view;            (b) An enlarged picture of the vortex. Fig. 10.\n Illustration of the whirlpool-like character of the Poynting-vector vortex.  13Fig. 11 . The Poynting vector distributi on with a lossy dielectric disk in a maximum of the cavity  \nelectric field.         \nFig.12.  The cavity Poynting vector distribution. A ferr ite disk is placed in the maximum of the \ncavity magnetic  field. A bias field is directed along positive y axis. \nFig.13.  The cavity Poynting vector distribution. A ferr ite disk is placed in the maximum of the \ncavity magnetic  field. A bias fi eld is directed along negative y  axis.              \nFig. 14.  The Poynting-vector distribution in a two-iris cavity with a big ferrite disk in a maximum  \n             of the electric field.  \n            (a) The input is  at the left-hand side of a system. 0Hr\n in the positive y direction. \n            (b) The input is  at the right-hand side of a system. 0Hr\n in the positive y direction. \n            (c) The input is  at the left-hand side of a system. 0Hr\n in the negative y direction. \nFig. 15.  The Poynting-vector distributi on for the PEC waveguide walls a nd different disk    losses \nparameters.  \n            (a) 01 .0 tan,1.0 = =∆ δ Oe H ;   (b) 0002 .0 tan,1.0 = =∆ δ Oe H ; (c) 01 .0 tan,5 = =∆ δ Oe H  \n        \n                  \n   14\n \n \n \n   \nFig. 1.  The TE 40 lossless waveguide structure with an encl osed lossless ferrite slab (crosses  \nshow the vortex positions). \n             (a) Distribution of the Poynting vector \n             (b) Distribution of the electric field. \n \n  \n   15\n  \n \n \n \nFig. 2.  A short-wall cavity with an iris and an enclosed ferrite disk \n  \n \n \n \n  \n \n \n \n \nFig. 3.  A cavity with two irises and an enclosed ferrite disk. \n \n \n \n                            \n   \n                \n   16\n \nFig.4.  The cavity Poynting vector distribution. A fe rrite disk is placed in the maximum of the \ncavity electric field. A bias field is directed along positive y axis.                                         \n (a)  A general view; \n           (b) An enlarged picture of the vortex. \n  \n  \n \nFig.5.  The cavity Poynting vector distribution. A ferrite disk is placed in the maximum \nof the cavity electric field. A bias field is directed along negative y axis. \n               (a)  A general view; \n                     (b) An enlarged picture of the vortex. \n  17\n \n \n \nFig. 6.  The Poynting vector distribution inside a ferrite disk. \n \n   \n \n \n \n \nFig. 7.  Electric field vector inside a ferrite disk. \n  18\n \n \n \n \n \nFig. 8.  Magnetic field vector inside a ferrite disk. \n \n   \n \n  \nFig. 9.  A vortex created by a big ferrite disk. \n             (a)  A general view;              (b) An enlarged picture of the vortex.  19 \n \n  \n \n \n \nFig. 10.  Illustration of the whirlpool-like character of the Poynting-vector vortex. \n \n \n \n \n  \nFig. 11.\n The Poynting vector distribution with a lossy dielectric disk in a maximum of the \ncavity  electric field.         \n \n             \n  20\n \n \nFig.12.  The cavity Poynting vector distribution. A fe rrite disk is placed in the maximum of the \ncavity magnetic  field. A bias field is directed along positive y axis.                                         \n \n                             \n \n \nFig.13.  The cavity Poynting vector distribution. A fe rrite disk is placed in the maximum of the \ncavity magnetic  field. A bias field is directed along negative y  axis.                                         \n                             \n \n  \n     21 \n   \n \n \n  \nFig. 14.  The Poynting-vector distribution in a two-iris cavity with a big ferrite disk in a maximum  \n             of the electric field.  \n             a) The input is at the left-hand side of a system. 0Hr\n in the positive y direction. \n             b) The input is at the right-hand side of a system. 0Hr\n in the positive y direction. \n             c) The input is at the left-hand side of a system. 0Hr\n in the negative y direction. \n \n                 \n \n   \n \n  22\n \n \n   \nFig. 15.  The Poynting-vector distribution for the PEC waveguide walls and different disk    \nlosses parameters.  \n              (a) 01.0 tan,1.0 = =∆ δ Oe H ;  \n             (b) 0002.0 tan,1.0 = =∆ δ Oe H ;  \n             (c) 01.0 tan,5 = =∆ δ Oe H  \n                                     \n \n \n \n \n         \n    "    },    {        "title": "2012.07524v1.Significant_enhancement_of_magnetic_shielding_effect_by_using_the_composite_metamaterial_composed_of_mu_near_zero_media_and_ferrite.pdf",        "content": "Significant enhancement of magnetic shielding effect by using the \ncomposite metamaterial composed of mu -near -zero media and ferrite  \n \nXu Chen , Yuqian Wang , Zhiwei Guo ,a) Xian Wu , Yong Sun, Yunhui Li , Haitao \nJiang ,b) and Hong Chen  \n \nMOE Key Laboratory of Advanced Micro -Structured Materials, School of P hysics Science and Engineering,  Tongji \nUniversity, Shanghai 200092, China  \n \na)2014guozhiwei@tongji.edu.cn .  \nb)jiang -haitao@tongji.edu.cn  . \nThe magnetic shield plays an important role in magnetic near -field control. However, the requirements of \nefficient, ultrathin, lightweight and cheap are still the challenges. Here, we firstly propose a composite \nmetamaterial in which the mu -near-zero media is covered with a ferrite slab. We verify that this structure can \nenhance the shielding effectiveness in a small area. Furthermore, we optimize the magnetic path by changing \nthe bulk ferrite slab into a patterned slab. In this way, significant shielding effectiveness enhancement can be \nachieved in a large area. Experimental results show that the maximum shielding effectiveness (SE) of the \ncomposite metamaterial with a patterned ferrite is 20.56 dB, which is nearly 19 dB higher than that of a single \nferrite slab with the same thickness of the composite metamaterial. The results on the composite  metamaterial \nwould be very useful in the applications involving magnetic shielding.  \nMetamaterials have the ability to control electro -\nmagnetic waves that natural materials cannot possess  [1], \nsuch as negative refr action [2-4], imaging  [5, 6], cloaking [7, \n8] etc. The unique deep sub -wavelength characteristics \nenable metamaterials to achieve smaller size and work at  \nlower frequency [9-11]. Nowadays, metamaterials pla y an \nimportant role in controlling near -field electromagnetic \nwaves , which gives optical microcavities  or waveguides \nbroader application prospects [ 12-15]. At present, with the \ngradual development of near -field technology, a very \nimportant issue —magnetic sh ielding has attracted people's \nattention.  The global QI standard of Wireless Power \nConsortium  requires  that the frequency range of wireless \npower transfer (WPT ) is from 100 kHz to 205 kHz [ 28]. \nBased on this  fact, although metal can shield the magnetic \nfield, the generated eddy currents may cause serious heating \neffects. Therefore, in order to meet the requirements of the \ninternational radio frequency electromagnetic exposure \nstandard  [16-18], people usually use ferrite materials or \nferrites covered with a metal [ 20-23]. But this makes the \nequipment very bulky and expensive. Recently, mu-near-zero \n(MNZ ) media have been utilized  in magnetic near -field \ncontrol. For example, Ref.  [24] reported the use of MNZ \nmedia as absorbers at GHz frequencies. Ref. [25] prop osed \nthe MNZ media for magnetic shielding at MHz frequency, in \nwhich the MNZ media  is considered as a near -field reflector. \nHowever, MNZ medias can only achieve ideal shielding at \noblique incidence. Once at normal incidence, there is some \nmagnetic transmis sion between metamaterial units [ 25]. And \nthis phenomenon has been observed in the simulated \nmagnetic field results, which directly reduce the shielding \neffectiveness (SE ). \nIt is worth noting that diamagnetic materials or structure  \nwith negative susceptibility have the  magnetic  shielding effects. High permeability materials can control the magnetic \npath [26-28]. For example, a shell structure composed of \nYBCO and ferrite materials, which can be used for magnetic \ncloaking from dc to 250 kHz  [26], and the diamagnetic \ninterface of the active structure also achieves ideal magnetic \ncloaking [ 28]. On the other hand, a t room temperature , a \ncopper spherical shell structure with equivalent negative \nsusceptibility has similar diamagnetic prope rties to YBCO in \na specific frequency [ 27]. Similarly, if the MNZ media  \nsatisfies the equivalent negative susceptibility (𝜒<0), it can \nbe regarded as a diamagnetic interface. More  importantly , \none can provide an optimal magnetic path for the transmitted \nmagnetic field of the MNZ media by using high permeability \nmaterials  such as ferrite.  \nIn this work, we show that the MNZ media has \nequivalent diamagnetic shielding performance  at oblique \nincidence. To improve the SE at normal incidence,  we \npropose a  composite  metamaterial  which  combine the MNZ \nmedia with a ferrite to optimiz e the magnetic path  and reduce \nthe transmission of magnetic  field.  Compared with a single \nMNZ media  or a single  ferrite material, the significant \nenhancement of SE is realized by t he composite metamaterial \ncomposed of the MNZ media and a patterned ferrite.  And it \nhas a lighter weight compared with a single ferrite material . \nFurthermore, in the composite metamaterial , compared with \nthe bulk ferrite slab, the patterned ferrite slab not only has a \nhigher SE, but also greatly reduces the amount of ferrite used.  \nThis shows that the patterned ferrite can accurately provide \nan ideal magnetic path for the  transmitted magnetic field of \nthe MNZ media. Therefore , this kind of composite \nmetamaterials would be very useful in the applications of  \nmagnetic shielding.  2 \n  \nFig. 1 The s chematic diagram of the experimental sample s. (a) MNZ \nmedia, the total thickness is ℎ. (b) MNZ metamaterial unit. The side \nlength of the unit is  𝐿, and the radius of each single disc is 𝑟. The \nwidth of the coppe r is 𝑤, the gap is 𝑔. The pin a  and b of the \ncapacitor is connected to the terminal 2  and 3 by soldering, \nrespectively , and the other terminal 1 is connected to the terminal 4. \n(c) A single ferrite  slab. The side length of each bulk is 𝐿. The \nthickness of the slab is 2ℎ. (d) Composite metamaterials with a bulk \nferrite. For each ferrite bulk, the side length is 𝐿, the thickness is ℎ. \n(e) Composite metamaterials with a patterned ferrite. For patterned \nferrite, the outer length is  𝐿, the inner length 𝐿1, and the thickness is \nℎ. (f) Non-resonant system. The Tx coil is the same as the Rx coil , \nthe outer length is  𝐿2, the inner length  is 𝐿3. The distance between \nthem is 𝑆.  \nThe scheme of  the non-resonant near -field system in the \nexperiment  is shown in Fig. 1  (f). The Tx/Rx coils are made \nof litz  wire with a diameter of 6mm and the number of the \nturns is 10 . The outer length of the coil is 𝐿2= 150  𝑚𝑚, and \nthe inner length 𝐿3=70 𝑚𝑚. The two coils are placed \nsymmetrically along the axis, and the distance between each \nother is 𝑆=150  𝑚𝑚. Then, port 1 and port 2 are connected \nto the vector network analyzer (Agilent PNA Network \nAnalyzer E5071C ). Regarding the design of the metamaterial \nunit, we choose  a pre -pressed  0.1× 20 litz wire as the basic \nelement. As shown in Fig. 1(a), each unit cont ains four disks \nwith a total thickness h of 2mm. The details are shown in Fig. \n1(b). The side length L of the unit  is 53 𝑚𝑚. The thickness \nof the outer insulating  layer of a single wire  is 0.055  mm, thus, \nthe width w of the inner copper is 0.57 mm, and the gap g \nbetween the copper is 0.11 mm. Then, the wire is tightly \nwound to form a disc with 38 turns and a radius of 26  mm. \nNext, we glue the discs together with polyacrylate adhesive. \nHere,  we chose a CBB non -polar capacitor with a thickness \nof 2 mm and a capacitance of 8.2 𝑛𝐹 as the tuning element. \nIn this case, the thickness of the metamaterial is not enlarged. \nFinally,  the pin a  and b of the capacitor is connected to the \nterminal 2  and 3 by soldering, respectively, and the other \nterminal 1 is connected to the terminal 4 so that all \ncomponents form a closed LC resonant circuit (the circuit of \nthe unit is shown in Fig. 1 of Supplementary ). However, the diamagnetic effect comes from the mode (tangential wave \nvector) mismatch between the metamaterial and the \nenvironment. When  the incident angle is smaller than the \ncritical angle , the magnetic  field would transmit through the \nMNZ media s at normal incidence  [25]. To this end, we design \na composite metamaterial composed of the MNZ media and \na ferrite to improve SE (the parameters of the commercial  \nferrite are shown in Fig. 2 of supplementary ). As shown in \nFig. 1(d), the size of the bulk ferrite is 53 𝑚𝑚 × 53 𝑚𝑚 ×\n2 𝑚𝑚, where 𝐿=53 𝑚𝑚, ℎ= 2 𝑚𝑚. Therefore, the total \nside length of the shield ing is 159 mm and the total thickness \nis 4mm. This scheme can only shield a small area close to the \nback of the bulk ferrite slab  (Fig. 3(d)). In order to further \nachieve a wider range of magnetic shielding, we precisely \noptimize d the path of the transmitted magnetic field between \nthe units  by changing the bulk ferrite  of MNZ media into a \npatterned ferrite. In our design , commercially available \nferrite materials with the same parameters are used. The \npattern ed ferrite is composed of small ferrite pieces with t he \nsize of  20 𝑚𝑚 ×8 𝑚𝑚 × 2 𝑚𝑚. The outer side length is \n𝐿=53 𝑚𝑚, the inner side length is 𝐿1=41 𝑚𝑚, and the \nthickness is ℎ = 2 𝑚𝑚 (Fig. 1(e)). The total side length and \ntotal thickness of the two composite metamaterial s are the \nsame. Moreover, we also set up a single ferrite slab with the \nsame thickness as the composite metamaterial. The \nparameters of the ferrite material are the same as above . The \ntotal side length of the final shield ing body  is still 159 mm, \nand the to tal thickness is still 4 mm (Fig. 1(c)). Subsequently, \nthe model of the samples is established in the commercial \nCST Studio Suite simulation software. Generally , it is \ndifficult to model a wire with a multi -strand s structure in the \nsoftware  because of  a huge  amount  of calculation. Therefore, \na single strand of copper wire is used instead of multiple \nstrands of litz wire  (the simulation model of the sample s is \nshown in the Fig. 3 of Supplementary ).  \nAccording to the effective medium theory,  the \npermeability of the MNZ  metamaterials can be written as [ 19, \n25]: \n                               𝜇=1−𝐹𝜔2\n𝜔2−𝜔𝑟2+𝑗𝜔𝛾                       (1) \nwhere 𝐹 is the structural factor, 𝜔 is the frequency, 𝜔𝑟 is the \nresonance frequency of the metamaterial, and 𝛾 is the loss. Since \n𝜒=𝜇−1, the susceptibility needs to satisfy [ 27]: \n                               𝜒=−𝐹𝜔2\n𝜔2−𝜔𝑟2+𝑗𝜔𝛾<0                   (2) \nwhere 𝐹,𝜔,𝜔𝑟and 𝛾 are positive, s o long as 𝜔2−𝜔𝑟2+𝑗𝜔𝛾> \n0, that is, when the frequency is greater than resonance and the loss \nis negligible, the metamaterial is diamagnetic. Then , we obtain ed \nthe permeability of the metamaterial as shown in Fig. 2(a) according \nto the effective  medium  theory [ 30, 31 ]. Subsequently, we observe \nthat the structure has a negative susceptibility as shown by the gray \narea in Fig. 2(b). \n3 \n Fig. 2 (a) Permeability of the MNZ media. At 86.23 kHz, 𝜇′＝0.1,\n𝜇′′＝0.02 (zoom ed view box).  (b) Susceptibility of the MNZ media. \nAt 86.23 kHz, 𝜒′=−0.9,𝜒′′=0.02 (zoom ed view box). The gray \narea indicates that the diamagnetic properties start from 83 kHz.  \nNext,  we describe the equivalent diamagnetic mechanism of \nmetamaterials in detail by comparing the relationship between SE  \nand susceptibility. As the frequency increases from 83 kHz, the SE \nbegins to be greater than zero, as shown in Fig. 4(a).  Meanwhile, the  \nsusceptibility at 83 kHz is negative and the imaginary part is near \nzero, that is, 𝜒′=−1, 𝜒′′=0.05 (see the black dotted box of Fig. \n2(b)), which reflects the diamagnetic property. As the frequency \nincreases to 86.23 kHz, the real part of the susceptibi lity 𝜒′=−0.9 \nFig. 3 The schem e of the simulation and the magnetic field results. Since (a) \nand (c) are non -resonant, the same frequency as (b) was chosen to observe \nthe magnetic field. For other samples , the magnetic field at the frequency of \nmaximum SE i s shown. The left side is a schem e of each comparison . \nStarting from the gray d otted line (x = 0), three yoz slices are selected along \nthe x direction. The slices represent the three positions of the most central \nunit (center position of the unit: x=0 mm, 1/4 position of the unit: x=\n13.25 mm, boundary position of the unit: x=26.5 mm). and the imaginary part 𝜒′′= 0.02. The loss is further reduced and  \nthe SE reaches the maximum value of 17.23 dB, as shown in Fig. \n5(a). However, when the frequency is higher than 86.23 kHz, \nalthough the imaginary part is negligible, the real part of the \nsusceptibility gradually approaches zero. The diamagnetic \nperformance gradually weakens and the shielding effectiveness \ngradually decreases.  \nFig. 4(a) Experimental setup of non -resonant near -field system and \ncomposite metamaterial with a patterned ferrite. 4(b)  shows the \ndetails of composite metamaterial with patterned ferrite.  \nIn order  to more intuitively describe the magnetic shielding \nmechanism of different samples , four samples are positioned \nbetween a pair of Tx/Rx coils separated by a distance s and aligned \ncoaxially, respectively. As shown in the schematic diagram on the \nleft in Fig. 3. First, we layer the magnetic field in space, and then \nselect three field slices related to the location of the unit (The center \nposition of the unit 𝑥=0 𝑚𝑚, the 1/4 position of the unit 𝑥=\n13.25 𝑚𝑚, and the boundary position of the unit 𝑥= 26.5 𝑚𝑚). \nHere, for the non -resonant system and the single  ferrite slab, the \nselected observation frequency is 86.23 kHz . For the other samples, \nthe maximum SE was selected as the observation frequency, \nrespectively . As shown in Fig. 3(a), all the three slices show a strong \nmagnetic field near the Rx coil. When the metamaterial is added to \nthe system, as shown in Fig. 3(b), in the slice with x = 0 mm, the \nmagnetic field near the Rx coil is significantly  weaker than before. \nWith the change of the slice position ( 𝑥=13.25𝑚𝑚  and \n26.5 𝑚𝑚), a penetrating magnetic field appears between the units. \nIn order to make up for this shortcoming , considering that ferrite \nmaterials have the ability to control the magnetic path [ 26, 27 ], we \npropose a method to construct a composite  metamaterial  composed \nof the MNZ media and a bulk ferrite , as shown in Fig. 3(d). The \nmaximum SE of this composite  metamaterial is 15.92 dB at 75.8 \nkHz. On the contrary, compared with the MNZ media , the \nmaxim um SE is reduced by 1.3 dB ( see Fig. 5(a)). By comparing \nFig. 3(b) and Fig. 3(d), after covering the ferrite slab, although the \nmagnetic field is effectively suppressed in a small area near the  \nshield ing zone , the severe magnetic flux leakage occurred near the \nRx coil. Considering the actual application scenarios, the Tx/Rx \ncoils in the system cannot meet the requirements for accurate \nplacement, and the size of the coils cannot be determined. Therefore, \nit is more desirable to effectively shield a wider area away from the \nshielding body. However, using only ferrite for blocking or isolation \ncannot enhance magnetic shielding. Therefore, to achieve a precise \ncontrol  of the transmitted magnetic field of metamaterials, it is \nnecessary to construct interfaces with a permeability distribution  [27, \n28]. As a result , we design a pattern ed ferrite based on the structural \ncharacteristics of the metamaterial , as shown in  Fig. 3(e). It is a grid \nstructure with square holes . The ferrite part of the grid is aligned at \nthe boundary of the metamaterial unit, and the hole area corresponds \nto the position of the metamaterial unit. Compar ed with Fig. 3(b), it \n4 \n can be found that the penetrating magnetic field between the MNZ \nmedia un its disappears after the pattern ed ferrite  is introduced.  \nSimilarly,  compar ed with Fig. 3(d), the magnetic leakage at the \nposition of the Rx coil is perfectly suppressed  by adding a patterned \nferrite . This achieves a higher degree of freedom and a larger \nmagnetic shielding zone. As shown in Fig. 5(a), the maximum SE \nof the composite metamaterial with a pattern ed ferrite at 85.5 kHz is \n21.94 dB, which is about 4.7 dB higher than the MNZ media  and \nnearly 6 dB higher than the composite metamaterial with a bulk \nferrite . In addition, the SE of a single  ferrite slab of the same \nthickness as the composite metamaterial is always only 4 dB. The \nfield results in Fig. 3(c)  show that only a small part of the magnetic \nfield is blocked, and most of the magnetic field  transmits.  \n \nFig. 5 (a) The shielding effectiveness (SE) obtained by simulation. (b) The \nshielding effectiveness (SE) obtained by experiment.  \nSubsequently, the experiment al results show that the max -\nimum SE of the composite metamaterial with a pattern ed ferrite is \n20.56 dB at 82.25 kHz. This is even nearly 19  dB higher than the \nsingle  ferrite slab with  the same thickness. Compared with MNZ \nmedia , the introduction  of patterned ferrite can increase the  \nmaximum SE by 2.3  dB. In addition , the maximum SE of t he \ncomposite metamaterial with a pattern ed ferrite is 2.7 dB higher than  \nthat of the composite metamaterial with a bulk ferrite ( see Fig. 5(b)). \nThis fully shows that the  patterned ferrite can accurately provide an \nideal path for the transmitted magnetic field of the MNZ media.  \nFurther, by comparing Fig. 5(a) and Fig. 5(b), it can be seen that the \nSE of the simulation is slightly wider than the bandwidth obtained \nfrom the experiment. Obviously, litz wire makes the metamaterial \nhave lower loss char acteristics than the single -turn copper wire.  \nIn conclusion, we show that although the MNZ media has \nmagnetic shielding effect at oblique incidence, it does not work well \nat normal incidence. To overcome this shortcoming, we use ferrite \nmaterials with a high permeability to reduce  the magnetic field \ntransmission  of MNZ media. Particularly, we find that a patterned \nferrite instead of a bulk ferrite can provide an ideal magnetic path for the transmission magnetic field of the MNZ media . \nExperimental results show that , compared to  a single MNZ media \nor a single ferrite material , the maximum SE of the composite \nmetamaterial with  a patterned ferrite is 20.56 dB, which is nearly 2.3 \ndB higher than that of the MNZ media without ferrite. And this \nvalue is even 19  dB higher than that of a single ferrite slab with the \nsame thickness  of the composite metamaterial . Particularly, \nalthough  the patterned ferrite in the composite metamaterial is only \n37.6% of the bulk ferrite, the corresponding SE is significantly \nimproved by 2.7 dB. In summary, both simulation and experimental \nresults show that a composite metamaterial with patterned ferrite  \ncan significantly improve SE while greatly reducing the weight and \nvolume of the  ferrite  at the same time . 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"    },    {        "title": "1805.01591v1.The_effect_of_the_misfit_dislocation_on_the_in_plane_shear_response_of_the_ferrite_cementite_interface.pdf",        "content": "The effect of the misfit disloca tion on the in-plane shear resp onse of the \nferrite/cementite interface \nJaemin Kima, Keonwook Kangb,**, Seunghwa Ryua,* \na Department of Mechanical Engine ering & KI for the NanoCentury,  Korea Advanced \nInstitute of Science and Technology, Daejeon 34141, Republic of  Korea \nb Department of Mechanical Engine ering, Yonsei University, Seoul  03722, Republic of Korea \n* Corresponding author. Tel.: + 82-42-350-3019; fax: +82-42-350-3 059. \nEmail address: ryush@kaist.ac.kr (Seunghwa Ryu) \n** Co-corresponding author. Tel.: +82-2-2123-2825; fax: +82-2-312 -2159. \nEmail address: kwkang75@yons ei.ac.kr (Keonwook Kang) \n \nAbstract \n Although the pearlitic steel is on e of the most extensively st udied materials, there are \nstill questions unanswered about the interface in the lamellar structure. In particular, to deepen \nthe understanding of the mechanical behavior of pearlitic steel  with fine lamellar structure, it \nis essential to reveal the struc ture-property relationship of t he ferrite/cementite interface (FCI). \nIn this study, we analyzed the in-plane shear deformation of th e FCI using atomistic simulation \ncombined with extended atomically informed Frank-Bilby (xAIFB) method and disregistry \nanalyses. In the atomistic simulation, we applied in-plane shea r stress along twelve different \ndirections to the ferrite/cementite bilayer for Isaichev (IS), Near Bagaryatsky (Near BA) and \nNear Pitsch-Petch (Near PP) ori entation relationship (OR), resp ectively. The simulation results reveal that IS and Near BA ORs show dislocation-mediated plasti city except two directions, \nwhile Near PP OR shows mode II (in-plane shear) fracture at the  FCI along all directions. Based \non the xAIFB and disregistry anal ysis results, we conclude that  the in-plane shear behavior of \nthe FCI is governed by the magn itude of Burgers vector and core -width of misfit dislocations.  \nKeywords:  pearlitic steel, ferrite/cementite interface, in-plane shear r esistance, misfit \ndislocation, dislocation core-width. \n \n1. Introduction \n Pearlitic steel has been used in  many engineering applications  such as bridge cable, \nrail steel and tire cord for its high strength and good ductili ty. Pearlitic steel has a three-\ndimensionally interconnected lame llar structure composed of fer rite (α-Fe, body centered-\ncubic structure) and cementite (Fe 3C, orthorhombic structure) phas es with specific orientation \nrelationships (ORs) (Hillert 1962, Zhang, Esling et al. 2007). Extensive studies (Embury and \nFisher 1966, Langford 1970, Cooke and Beevers 1974, Hyzak and B ernstein 1976, Langford \n1977, Taylor, Warin et al. 1977, Sevillano 1991, Embury and Hir th 1994, Bae, Nam et al. 1996, \nJanecek, Louchet et al. 2000, Hono, Ohnuma et al. 2001, Hohenwa rter, Taylor et al. 2010, Li, \nChoi et al. 2011, Zhang, Godfrey et al. 2011, Li, Yip et al. 20 13, Baek, Hwang et al. 2014, \nIacoviello, Cocco et al. 2015, Hohenwarter, V olker et al. 2016,  Kapp, Hohenwarter et al. 2016) \non the pearlitic microstructure have been conducted to explain its unique mechanical properties. \nIn these works, various roles o f the ferrite/cementite interfac e (FCI) were reported such as a \ndislocation barrier (Langfor d 1970, Hyzak and Bernstein 1976, L angford 1977, Kapp, \nHohenwarter et al. 2016) and nucleation site (Sevillano 1991, E mbury and Hirth 1994, Janecek, \nLouchet et al. 2000, Guziewski, Coleman et al. 2018), sink to d efects (Inoue, Ogura et al. 1977, Choo and Lee 1982, Hong and Lee 1983, Languillaume, Kapelski et  al. 1997), crack nucleation \nsite (Cooke and Beevers 1974, Taylor, Warin et al. 1977) and pr opagation path (Taylor, Warin \net al. 1977, Bae, Nam et al. 1996). Nevertheless, most studies (Embury and Fisher 1966, \nLangford 1970, Hyzak and Bernstein 1976, Langford 1977, Sevilla no 1991, Embury and Hirth \n1994, Janecek, Louchet et al. 2000, Zhang, Godfrey et al. 2011,  Karkina, Karkin et al. 2015, \nKapp, Hohenwarter et al. 2016) consider the FCI as a mere barri er for the lattice dislocations \nto explain the hardening behavi or of pearlitic steel based on d islocation pile-up mechanism. \nPearlite colony is formed by diff usive eutectoid reaction from parent austenite (γ-Fe, \nface centered-cubic structure) phase into resultant ferrite and  cementite phase (Soffa and \nLaughlin 2014). During the phase transformation, ferrite and ce mentite form specific \ncrystallographic ORs and habit planes to minimize the energy ba rrier to phase transformation \n(Zhang and Kelly 1998, Zhang and Kelly 2005, Zhang, Esling et a l. 2007). For a few decades, \nIsaichev (IS), Bagariatsky (BA), and Pitsch-Petch (PP) ORs had been known as three major \nORs at the FCI (Isaichev 1947, Bagaryatsky 1950, Petch 1953, Pi tsch 1962, Andrews 1963, \nZhou and Shiflet 1991, Zhou and Shiflet 1992). However, recent high resolution experimental \nstudies (Zhang, Esling et al. 2007) found that three major ORs are IS, Near BA, and Near PP \nORs, and our recent computationa l study (Kim, Kang et al. 2016)  also confirmed that Near BA \nand Near PP ORs has lower interfacial energies than BA or PP OR s (see Supplementary Figure \nS1-4). The Near BA and Near PP correspond to near Bag and P-P I -1 ORs denoted by Zhang \net al. (Zhang, Esling et al. 2007) , respectively. Our previous study (Kim, Kang et al. 2016) \ncharacterized the misfit dislocations at the FCI for the five O Rs (IS, BA, PP, Near BA, and \nNear PP), and found that very different misfit dislocation netw orks are formed at the FCI in \nrelatively similar ORs (such as P P and Near PP, BA and Near BA) . In addition, there have been \nmany studies (Bowden and Kelly 1967, Morgan and Ralph 1968, Sha ckleton and Kelly 1969, \nDippenaar and Honeycombe 1973, Schastlivtsev and Yakovleva 1974 , Sukhomlin 1976, Mangan and Shiflet 1999, Zhang, Wang et al. 2012, Zhang, Zhang et al. 2012) to understand \nthe occurrence of specific ORs in  pearlitic steel. It has been reported that the occurrence of \nspecific OR in pearlitic steel can be controlled by carbon cont ents of the system (Dippenaar \nand Honeycombe 1973), heat treatment condition (Mangan and Shif let 1999) and applying \nmagnetic field during phase transformation (Zhang, Wang et al. 2012, Zhang, Zhang et al. \n2012).  \nA few experimental studies (I noue, Ogura et al. 1977, Languilla ume, Kapelski et al. \n1997) reported that dislocation density is high near the FCI wh ile it is low within the ferrite \nlayer, and suggested that the FCI may act as a strong sink of l attice dislocations existing at the \nvicinity of the FCI. This lattice dislocation trapping mechanis m in fine lamellar structure is \nknown to be caused by the interfa cial shear by the stress field  of the lattice dislocation \napproaching to the interface, and thus, the dislocation trappin g ability depends on the interfacial \nstrength. Once the lat tice dislocation is a bsorbed into the int erface by shearing the interface, \nthe lattice dislocation cannot e scape the interface easily beca use the core structure of lattice \ndislocation spreads along the in terface. This gives rise to ext remely high strength of multi-\nlayered metallic composite with a few nanometers of layer thick ness (Hoagland, Hirth et al. \n2006). Analogously, the strength of pearlitic steel with ultra- fine lamellar structure is attributed \nto the lattice dislocation trappi ng mechanism of the FCI. Moreo ver, we expect that the FCI \nstructure for each OR will show different lattice dislocation t rapping ability, because the \ncharacteristics of misfit dislocation for each OR is different (Kim, Kang et al. 2016) as well as \nthe interfacial strength for each OR is different as discussed in the later part of the present paper. \nThis suggests that FCI structure -property relationships can be varied depending on the \ncrystallographic OR.  \nIt was reported that pearlitic stel with a fine lamellar struct ure of a few tens to hundreds of nanometers can be formed by controlling the experimental con ditions during the diffusive \neutectoid reaction (Jaramillo, Babu et al. 2005, Wu and Bhadesh ia 2012). Such nanostructured \npearlites may have different mechanical strengths if different ORs dominate at the FCI, \nalthough their lamellar spacings are similar. In other words, i f we can fabricate pearlitic steels \nwith a fine lamellar structure with a desired OR, we can tune t he mechanical properties of them \nbased on the structure-property r elationship of the FCI for eac h OR. \nIn order to reveal the structure -property relationship of the F CI for each OR, we \ninvestigated the effect of misfit dislocation at the FCI in dif ferent ORs on the interfacial \nstrength under in-plane shear d eformation, by employing atomist ic simulations. We modeled \nthe lamellar structures of pear lite colony as ferrite/cementite  bilayer for IS, Near BA and Near \nPP OR, and characterized the misfit dislocations at the FCI by combining the xAIFB and \ndisregistry analysis. We conducted in-plane shear deformation a long twelve different directions \non the FCI to observe the in-plane shear response of the FCI fo r each OR. The simulation \nresults revealed that IS and Near  BA ORs show dislocation-media ted plasticity except two \ndirections, while Near PP OR show s mode II (in-plane shear) fra cture at the FCI along all \ndirections. Our results showed t hat the in-plane shear behavior  of the FCI is governed primarily \nby magnitude of Burgers vector and core-width of misfit disloca tion. \n \n2. Methods \n2.1.Modeling of the initial fe rrite/cementite interface \n In order to study the in-plane shear behavior of the FCI in pe arlitic steel, it is necessary \nto model the initial FCI. We idealized the lamellar structure o f pearlite colony as a flat \nferrite/cementite bilayer. In the atomic simulation, we used mo dified embedded-atom method \n(MEAM) potential developed by Liya nage (Liyanage, Kim et al. 20 14) to consider the directional interatomic interaction among Fe and C atoms. The l attice parameter of body-\ncentered cubic (BCC) ferrite is af = 2.851 Å and the lattice parameters of orthorhombic \ncementite are ac = 4.470 Å, bc = 5.088 Å and cc = 6.670 Å. To generate the initial configuration \nof ferrite/cementite bilayer, we determined the size of the per fect ferrite and cementite blocks \nto have minimal misfit strain for each OR. The detailed informa tion of each OR and its habit \nplane are listed in Table 1. For the sake of simplicity, we int entionally put the y-axis \nperpendicular to the habit plane. To assure the periodic bounda ry conditions in x and z-\ndirections, we allowed small misorientation within 0.5 degree a t most when necessary. Then, \nwe iteratively applied biaxial strain to perfect ferrite and ce mentite blocks separately until \nsatisfying both geometric compatibility and mechanical equilibr ium condition. Finally, we \nassembled the strained ferrite and cementite blocks into the un -relaxed ferrite/cementite bilayer. \nWhen we assemble the strained ferrite and cementite blocks, we considered all the shuffle \nplanes of cementite phase for each OR. To capture the initial s tate of the FCI, we carried out \nthe molecular statics (MS) simulation using the conjugate gradi ent method to find energy \nminimum state of the ferrite/cementite bilayer. In order to fin d global minimum state, we \nperformed the simulated annealing (SA) from 800 K to 10 K for 1 00 ps using Nosé-Hoover \nisobaric-isothermal (NPT) ensemble. After the SA procedure, we carried out MS simulation \nagain to find the energy minimum state at 0 K. Finally, we can obtain the initial (relaxed) \nferrite/cementite bilayer for each OR. The interface structure of un-relaxed and relaxed \nstructure of the FCI for Near PP OR is described in Figure 1-a and b. In addition, we computed \ninterface energies to choose the most stable interface structur e among all shuffle planes of \ncementite (see Supplementary Figure S1-3). For BA and IS OR, th ere are several atomistic \nsimulation studies on the most stable interface structure (Zhan g, Hickel et al. 2015, Guziewski, \nColeman et al. 2016, Zhou, Zheng et al. 2017) . The interface s tructures for BA and IS OR in \nthe references match well with our simulation results (see Supp lementary Figure S4 and S6). We note that the residual stress in ferrite and cementite block  induced by the biaxial strain can \naffect the interface energy and shear strength of the FCI calcu lated here, which would induce \nan intrinsic size effect. Indee d, because of the high computati onal cost of performing interfacial \nshear simulation, we used smaller simulation cells compared to those of our previous study  \n(Kim, Kang et al. 2016). However , the interface energy increase s at most by 4% upon the size \nreduction, and the characteristics of the misfit dislocation ne twork are almost identical. \nBecause the interfacial strength depends on the Burgers vector and core-width of the misfit \ndislocations, we expect that the qualitative findings in the pr esent study such as different failure \nmodes for different ORs do not c hange over the increase of simu lation cell. The detailed \ninformation such as size, misfit strain and the lowest interfac e energy can be found in the Table \n2 and schematic diagram of the initial ferrite/cementite bilaye r for each OR is presented in \nFigure 2-a.  \n2.2. In-plane shear deformation of the ferrite/cementite interf ace \n In order to investigate the in-p lane shear behavior of the FCI , we applied the simple \nshear deformation under force control at 300 K. Before applying  the shear deformation, we \nperformed equilibration procedure to the initial ferrite/cement ite bilayer using NPT ensemble \nat 300 K for 200 ps. To impose the simple shear condition to th e bilayer model, we assigned \nthe in-plane net force F (only with x and z-components) on the top and bottom slabs with same \nmagnitude but in opposite directions as described in Figure 2. Total 12 different in-plane shear \ndeformations with different direction angle θ were performed to capture the anisotropic shear \nresponse of the FCI for each OR. Where direction angle θ represents the angle between positive \nx-axis and direction of net for ce assigned on top slab around ne gative y-axis as depicted in \nFigure 2-a. The magnitude of the in-plane force applied to the top and bottom slabs was \ndetermined by desired shear stress at a given loading step. The  shear stress is monotonically increased by increment of shear stress, Δτ, at each loading ste p. For each loading step, we \nperformed NPT ensemble to eliminate all stress components excep t for τxy and τyz with given \nin-plane net force at 300 K for 200 ps.  \n2.3. Core-width of misfit dislocation at the ferrite/cementite interface \n In order to understand the anisotropic shear response of the F CI, we analyzed the \ncharacteristics of misfit dislocation at the FCI. To characteri ze the misfit dislocation of the FCI \nin terms of Burgers vector ( b = [be, bs]), line orientation ( ξ) and line spacing ( d), it is essential \nto determine a reference lattic e structure to both ferrite and cementite structures. We perform \nthe xAIFB analysis (Kim, Kang et al. 2016) on the initial FCI s tructures to find a reference \nlattice structure that make the net Burgers vector from the Fra nk-Bilby equation and Knowle’s \nequation equal to each other, and thus to determine the unique Burgers vector. In addition, \nPeierls-Nabarro (P-N) model (Peierls 1940, Nabarro 1947) indica tes that not only the \nmagnitude of Burgers vector ( b=|b|) but also the core-width (2 w) play an important role to \ndetermine the Peierls stress which represents the critical shea r stress to activate the glide of \nlattice dislocation in bulk. A nalogously, if the in-plane motio n of the FCI misfit dislocation is \nconsidered as glide of a lattice dislocation, how wide the core  spreads must have a significant \neffect on the in-plane shear str ength of the FCI. So, we perfor med the disregistry analysis (Kang, \nWang et al. 2012, Wang, Zhang et al. 2013, Wang, Zhang et al. 2 014, Kim, Kang et al. 2016) \nto measure the core-width of misfit dislocation at the FCI. To this end, we used disregistry rijrc, \nthe relative displacement from the reference structure to relax ed one. Each dislocation causes \na stepwise change in disregistry rijrc . The magnitude of Burgers vector and core-width \ndetermine the height of the step and transition from core to ad jacent core, respectively. In order \nto measure the core-width of disl ocation, we fitted the disregi stry rijrc to the following stepwise \nfunction:   rλrcሺqሻൌ൝bλβ\nπatan൭qെαpλെuλ\nwλβ൱n\nα=1ൡNdis\nβ=1vλ (1) \nwhere rλrc indicates the non-uniform displ acement from reference lattice structure to initial \n(relaxed) ferrite/cementite bila yer at the interface projected onto the s or t-axis for λ component. \ns and t are probe vectors perpendicular and paralleled to dislocation line ( ξ) on the interface, \nrespectively (see Figure 1-c and d). s can be determined as s =\ttൈn, where n is paralleled to \ny-axis. λ represents the component of each misfit dislocation an d the subscript e and s represent \nthe edge and screw components of each misfit dislocation, respe ctively. q represents position \nof given data point projected onto the probe axis s o r  q = x∙ s. bλβ  a n d  wλβ  r e p r e s e n t  λ  \ncomponent of Burgers vector and half-width of dislocation core for βth misfit dislocation, \nrespectively. pλ ,  uλ  a n d  vλ  represent the period, shift in s- and rλrc -axis, respectively. Ndis \nand n  represent the number of misfit dislocations and periodic func tion, respectively. The \nschematic diagram of disregistr y analysis using Equation 1 is g iven in Figure 3-a. There are \nfive unknowns ( bλβ, wλβ, pλ, uλ a n d  vλ) to define the shape of disregistry. In order to determine \nthe shape of disregistry, we performed the optimization procedu re to minimize the cost function, \nf, defined as \n  f൫bλβ,\twλβ,\tpλ,uλ,\tvλ൯ൌቄ rij, λrc, atom൫qα൯െrλrc൫qα൯ቅ2Natom\nα=1 (2) \nwhere Natom is the number of data points of disregistry obtained from atom istic simulation. \nrij, λrc, atom indicates the λ component of disregistry projected onto the lo cal axis ( s- or t-axis) for \nαth disregistry point. qα indicates the position projected onto s-axis for αth disregistry point. \nTo find the five unknown variable s with reasonable tolerance, w e used fmincon  function with \ntrust-region-reflective algorithm implemented in the MATLAB R20 16b. 2.4. Analysis of shear deformati on of the ferrite/cementite int erface \n Due to different shear moduli of ferrite and cementite, differ ent shear strains are \ndeveloped for each phase for a shear stress imposed and it acco mpanies ambiguity in \ndetermining the shear strain at the interface region during she ar deformation. Hence, it is \nnecessary to define a new measure of shear strain at the interf ace to analyze the shear behavior \nof the FCI. We assume that ferrite and cementite experience dif ferent uniform shear strains in \neach phase and there is no displacement in y-direction and that the in-plane displacement ( x \nand z-component) at a given point  linearly depends on the y-coordinate (see Figure 2-c and d). \nBased on these assumptions, we defined average in-plane shear s trains, γxy and γyz, as shown \nbelow \n uκൌmxκ∙ቀyκെypκቁupκ and wκൌmzκ∙ቀyκെypκቁwpκ, where κ ൌ f or c  \nγxyൌ൫uintcെuintf൯/h,γyzൌ൫wintcെwintf൯/h,hൌyintcെyintf (3) \nwhere κ represents the phase at a given y-coordinate and the superscripts f and c indicate the \nferrite and cementite phase, respectively. uκ  and wκ  represent the average in-plane \ndisplacement along the x and z-axis at yൌyκ for κ phase, respectively. upκ and wpκ a r e  t h e  \nin-plane displacement in x and z-directions, respectively, at the reference point at yൌypκ for \nκ phase. mxκ and mzκ are the slope of the in-plane displacement in x and z-directions for κ \nphase, respectively. We applied linear regression using multipl e reference points to compute \nthe mxκ and mzκ value. γxy and γyz are the average shear strain at the interface region and h \nrepresents the average interface spacing between topmost ferrit e layer and bottommost \ncementite layer in y-direction (see Figure 2). The average interface spacing h were 1.30, 1.33 \nand 1.24 Å for IS, Near BA and Near PP OR, respectively.  \n In order to analyze the in-plane shear deformation at the FCI,  we computed the disregistry rijr, which represents the difference between displacement of ith atom in ferrite (or \ncementite) and displacement of jth atom in cementite (or ferrite) for each loading step. The \ndisplacement of ith atom in κ  phase represents the displacement from the initial atomic \nposition to the deformed atomic position at the interface regio n. κ represents either ferrite or \ncementite phase. \n \n3. Results \n3.1. The characteristics of misf it dislocations for three orien tation relationships \n Having obtained the atomic structure of the FCI, we performed the xAIFB and \ndisregistry analysis to determine the characteristics of in-pla ne misfit dislocations such as λ \ncomponents of Burgers vector bλβ  and half-width of dislocation core wλβ , line orientation \nvector ξβ and line spacing dβ for βth misfit dislocation. For IS OR, we set the crystallographic \norientation as x = [010] c||[1ത11] f, y = ሺ1ത01ሻc||ሺ1ത2ത1ሻf and z = xൈy and generated the perfect \nferrite and cementite blocks with given orientation, separately . The subscript f and c represent \nthe ferrite and cementite phases, respectively. For IS OR, the dimensions of the initial \nferrite/cementite bilayer were Lx = 167.30 Å, Lyc = 81.69 Å, Lyf = 83.80 Å and Lz = 32.12 \nÅ. Lx and Lz represent the length of the simulation box for the ferrite/cem entite bilayer in x \nand z-directions, respectively. Lyc  a n d  Lyf  represent the length of the cementite and ferrite \nblock in y-direction, respectively. The mis fit strains between ferrite an d cementite block for IS \nOR were εxx = 9.08×10-6 and εzz = 2.17×10-3 in x- and z-directions, respectively. Habit plane \nof cementite phase has 8 different shuffle planes for IS OR. Am ong the 8 different shuffle \nplanes, the lowest interface energy of the FCI for IS OR was γcf = 501.9 mJ/m2. In the in-plane \nshear deformation, we used the FCI model which has the lowest i nterface energy (see supplementary Figure S1). In addition, recent experimental work  (Zhou, Zheng et al. 2017) \ncomputed most stable interface structure using density function al theory and directly observed \natomic configuration of the FCI f or IS OR. From the reference, we found that they only \nconsidered five shuffle planes out of eight shuffle planes of c ementite for IS OR (see \nsupplementary Figure S5). Even though they missed several shuff le planes of cementite, the \nreported atomic configuration at the interface is well matched with atomic configuration of the \nFCI model for IS OR computed by MEAM potential  (Liyanage, Kim et al. 2014) (see \nsupplementary Figure S6). In order to show the local coherency and geometry of misfit \ndislocations at the FCI, we visualized the atomic potential ene rgy maps for the interface atoms \nand presented the corresponding ge ometries of idealized misfit dislocation in Figure 4 . As \nshown in Figure 4-a, a single ar ray of edge dislocation was dev eloped in z-direction on the FCI \nbecause of the atomic mismatch along the x-axis. The xAIFB and disregistry analysis reveal \nthe characteristics of misfit dislocations. The edge component of Burgers vector was be1 = 2.53 \nÅ, and half-width of dislocation core was we1 = 13.21 Å as plotted in Figure 3-b. The line \norientation was ξ1 = 90° (paralleled to z-axis), and line spacing was d1 = 83.65 Å (see Figure \n4-a). \n F o r N ear B A  O R ,  w e set th e cry stallographic orientation as x = [010] c||[1ത11] f, y = \nሺ001ሻc||(5ത9ത4)f and z = xൈy. The dimensions of the initial ferrite/cementite bilayer for N ear \nBA OR were Lx = 167.18 Å, Lyc = 93.38 Å, Lyf = 94.47 Å and Lz = 54.22 Å. The misfit \nstrains between ferrite and cemen tite blocks for Near BA OR wer e εxx = 9.08×10-6 and εzz =  \n8.38×10-3 in x- and z-directions, respectively. Cemen tite block for Near BA OR has 6  different \nshuffle planes. Among these, the lowest interface energy of the  FCI for Near BA OR was γcf = \n539.1 mJ/m2 (see supplementary Figure S2). As shown in Figure 4-b, disloca tion network in \nbrick and mortar pattern was developed on the FCI with two edge  dislocations. The edge components of Burgers vector were be1  = 2.21 Å and be2  = 2.53 Å and the half-widths of \ndislocation core were we1 = 2.12 Å and we2 = 6.67 Å for each misfit dislocation, respectively \n(see Figure 3-c and d). The line orientations were ξ1 = 0° (paralleled to x-axis) and ξ2 = 90° \n(paralleled to z-axis), and the line spacings were d1 = 18.07 Å and d2 = 83.59 Å, respectively \n(see Figure 4-b).  \n F o r  N e a r  P P  O R ,  w e  p u t  t h e  c r y stallographic orientation as x = [010] c||[1ത13 ] f, y =  \n(001) c||(24 9 5) f and z = xൈy. The dimension of the initial ferrite/cementite bilayer for Ne ar \nPP OR were Lx  = 65.77 Å, Lyc  = 73.37 Å, Lyf  = 74.45 Å and Lz  = 44.99 Å. The misfit \nstrains between ferrite and cementite block with Near PP OR wer e εxx\t= 3.66×10-4 and εzz =  \n2.24×10-3 in x- and z-directions, respectively. Cemen tite phase for Near PP OR has 6  different \nshuffle planes. Among these, the lowest interface energy of the  FCI for Near PP OR was γcf = \n575.9 mJ/m2 (see supplementary Figure S3). As shown in Figure 4-c, the dis location network \ncomposed of two straight dislocation was formed at the FCI. As shown in Figure 3-e and f, the \nedge components of Burgers vector were be1 = 1.06 Å and be2 = 3.26 Å and the half-widths \nof dislocation core were we1 = 1.36 Å and we2 = 4.04 Å for each dislocation line, respectively. \nThe screw components of Burgers vector were bs1 = 2.18 Å and bs2 = 3.72 Å and the half-\nwidths of dislocation core were ws1  = 2.00 Å and ws2  = 4.37 Å for each dislocation line, \nrespectively. The line orientations were ξ1 = 0° (paralleled to x-axis) and ξ2 = 41.3°, and the \nline spacings were d1 = 22.35 Å and d2 = 43.41 Å, respectively (see Figure 4-c). The detailed \ninformation on the characteristic s of misfit disl ocation for th ree FCIs are summarized in Table \n3. \n3.2. In-plane shear response of the ferrite/cementite interface  \n Figure 5 presents the in-plane shear resistance maps for twelv e different shear directions and stress-strain curves for  θ = 0° for three ORs. The distance from each data point \nto origin represents the magnitude of the in-plane shear resist ance as depicted in Figure 5. From \nFigure 5-a, we noticed that in-plane shear resistance map shows  significant anisotropy for three \nORs. In-plane shear resistance map for IS OR has largest anisot ropy while Near PP has the \nsmallest. For IS OR, the minimum shear resistance among twelve directions is around 100 MPa \nand the maximum shear resistance is around 9.18 GPa. The maximu m in-plane shear resistance \nis out of the range of the Figure  5-a. Figure 6 contains the po tential energy map to identify the \ninitial geometry of misfit dislocation and disregistry rijr  maps for θ =  0 °  t o  i n s p e c t  t h e  \ndeformation behavior of the FC I. At low stress level for θ = 0° (τxy ≈ 50 MPa), the disregistry \nis relatively uniformly distribut ed along the mi sfit dislocatio n core region. At high stress level \nfor θ = 0° (τxy ≈ 100 MPa), the disregistry rijr is more concentrated near the core-region of \nmisfit dislocation (see Figure 5-b and 6-a). In addition, Figur e 7 shows a series of snapshot at \nthe FCI during in-plane shear deformation for θ = 0°. As shown in Figure 7-a, we observed the \nglide of the misfit dislocation on the interface plane at the F CI for IS (see supplementary Movie \n1). We also compute relative displacements of interface atoms i n x-direction under constant \nshear stress for θ = 0° (τxy ≈ 110 MPa for IS OR, τxy ≈ 570 MPa for Near BA OR and τxy ≈  \n1870 MPa for Near PP OR) to confirm if the deformation is produ ced by misfit dislocation \nmotion or interfacial fracture (see Figure 8). For easy compari son, we set the initial relative \ndisplacement as zero right after desired shear str ess is applie d. Relative displacement for IS OR \nlinearly increased with time because of the glide of misfit dis location, which corresponds to \nthe mobility law of dislocation in bulk metal under constant sh ear stress. The FCI for IS OR \nplastically deforms by misfit dislocation glide on the interfac e except for θ = 90° and 270°. \nUnlike other shear directions, the FCI for  θ = 90° and 270° shows mode II (in-plane shear) \nfracture at the interface. For Near BA and Near PP ORs, the minimum shear resistance among  twelve directions \nis around 380 MPa and around 1300 MPa, respectively, and the ma ximum shear resistance is \naround 4.66 GPa and 6.51 GPa, respectiv ely (see Figure 5-a and b). At low stress level for θ = \n0° (τxy ≈ 190 MPa for Near BA OR and τxy ≈ 560 MPa for Near PP OR), we observe that the \ndisregistry is locally initiate at the junction of two differen t misfit dislocation lines (see Figure \n6-b and c). At high stress level for θ = 0° (τ xy ≈ 380 MPa for Near BA OR and τ xy ≈ 1700 \nMPa for Near PP OR), the disregistry rijr  spreads out to adjacent core-regions of misfit \ndislocations (see Figure 6-b and c). As shown in Figure 7-b and  c, the FCI for Near BA OR \nshows glide of the misfit dislocation on the interface while th e FCI for Near PP OR shows \nmode II fracture for θ = 0° (see supplementary Movie 2 and 3). Figure 8 also shows th at relative \ndisplacement at the interface atoms of Near BA OR monotonically  increased with time because \nof glide of misfit dislocation on the FCI while relative displa cement of Near PP OR shows a \nsudden increase due to local fracture at the FCI. Moreover, sim ilar to in-plane shear behavior \nof the FCI for IS OR, the FCI for Near BA OR shows glide of mis fit dislocation on the interface \nexcept for θ = 90° and 270° where mode II frac ture is observed. Unlike the others, the FCI for \nNear PP OR shows mode II fracture  for all in-plane shear direct ions.  \n \n4. Discussion \n4.1. In-plane shear resistance of  Isaichev orientation relation ship (IS OR) \n For IS OR, in-plane shear resistance of the FCI shows highest anisotropy among the \nthree different ORs as plotted in Figure 5-a. The in-plane shea r resistance to the τxy component \nis nearly constant ( τxy ≈ 100 MPa) and τ yz component of in-plane shear resistance shows its \nhighest value of τyz  ≈ 9.18 GPa when θ = 270°. The extremely anis otropic in-plane shear response of the ferrite/cementite bilayer for IS OR can be expl ained by the magnitude of \nBurgers vector and core-width of  the misfit dislocation. The FC I contains an array of edge \ndislocations as depicted in Figure 4-a. The Peach-Koehler (P-K)  force  f acting on the misfit \ndislocation under in-plane shear is given as f = [be1τxy, 0, 0] where be1 and τxy represent the \nedge component of Burgers vector and xy-component of externally applied shear stress. Hence, \nwhen applied shear stress has  xy-component, the shear strength is estimated from the condition \nof τxy\t>\tτp where τp represents the Peierls-type c ritical stress. Although the conc ept of Peierls \nstress is originally applied to an infinitely straight single l attice dislocation gliding on the slip \nplane in bulk, in this  study, we adopt it to  estimate the requi red stress to activate the motion of \nthe misfit dislocation on the FCI . As the direction angle of in -plane shear stress is θ to the x-\naxis (i.e. τxy = τcosθ  where τ  represents the magnitude of a pplied in-plane shear stress), \ninterfacial strength is given as τp/c o sθ , which well describes the anisotropic interfacial \nstrength map of IS OR in Figure 5-a. According to the P-N model  (Peierls 1940, Nabarro 1947, \nLubarda and Markenscoff 2006, Lubarda and Markenscoff 2006), th e Peierls stress is expressed \nas τpൌ2μ/ab\tሾbe2/(1െν)bs2ሿ\texpሺെ4πwe/bሻ  where μ and ν represent effective shear \nmodulus and Poisson’s ratio, respectively. we, a a n d  b represent edge component of half-\nwidth of dislocation core, interatomic distance within the slip  plane in direction perpendicular \nto the dislocation line and magnitude of Burgers vector, respec tively. be and bs represent the \nedge and screw components of Burgers vector, respectively. P-N model indicates that the \nPeierls stress τp becomes smaller when we/b becomes larger. In other words, the wide core \nregion with small magnitude of Burgers vector gives rise to a s mall in-plane shear resistance \nof the FCI. In the P-N  model, the variable we/b of exponential function has a dominant effect \nin Peierls stress compared to other factors. So, we used the ra tio of half-width of dislocation \ncore to magnitude of Burgers vector, we/b, to explain the trend of in-plane shear resistance of the FCI in different ORs. The misfit dislocation developed at t he FCI in IS OR has the largest \nratio of dislocation core half-wi dth to Burgers vector magnitud e (we1 /b1ൌ  5.22), which \nexplains the smallest in-plane shear resistance of IS OR among three different ORs. On the \nother hand, the maximum in-plane  shear resistance among the thr ee different FCIs also can be \nfound in the FCI for IS OR. The IS OR shows 9.18 GPa of maximum  in-plane shear resistance \nwhen applied shear stress has only yz-component. The large in-plane shear resistance along yz-\ndirection is originated from the absence of P-K force along the  direction. Hence, the interfacial \nstrength is determined by the coherency of atoms at the FCI for  the case. We found that the \ndisregistry is concentrated at th e core-region of misfit disloc ation just before the fracture. It \nimplies that the fracture is initiated at the core-region due t o the incoherency near the core-\nregion. \nWe note that a recent experimental study (Zhou, Zheng et al. 20 17) predicts two sets \nof misfit dislocation arrays in both x and y directions with 25  and 50 nm line spacing, which is \ndifferent from our simulation results predicting a misfit dislo cation array along one direction. \nThe discrepancy may originate from the small errors in the pred icted lattice constants from the \nMEAM potential. In addition, the lattice constants at eutectoid  temperature are different from \nthose at room temperature while we construct the simulation cel l without considering the \ntemperature effect. Although one can resolve the discrepancy re lated to the lattice constants, a \nlarger simulation involving 50 nm spacing misfit dislocations i s too computationally expensive. \nSince the absence of misfit di slocation along one direction may  have a discernible effect on the \nresults of the Isaichev OR, depending on the characteristic of another misfit dislocation array, \nthe in-plane shear strength map in Figure 5-a may change if it is obtained with a larger \nsimulation. \n4.2. In-plane shear resistance of Near Bagaryatsky and Near Pit sch-Petch orientation relationships \n For Near BA and Near PP ORs, in-plane shear resistance of the FCI also shows \nsignificant anisotropy as depict ed in Figure 5-a. Unlike the FC I in IS OR, the FCIs for Near \nBA and Near PP ORs contain two types of misfit dislocations wit h different characteristics. \nFigure 4 shows that the multiple junctions of misfit dislocatio ns are developed at the FCI from \nthe interaction of two different misfit dislocations. In Figure  6-b and c, the disregistry for θ = \n0° is initiated at the junction of  the misfit dislocation at re latively low shear stress level ( τxy ≈  \n190 MPa for Near BA OR and τxy ≈ 560 MPa for Near PP OR). The localized initial shear \ndeformation can be understood from the potential energy map. Th e potential energy maps for \nNear BA and Near PP ORs show hig her potential energy near the j unction region (Figure 6-b \nand c), which implies that rel atively weak atomic bonding is pr esent at the junction formed by \nthe reaction of two m isfit dislocations. \nFor Near BA OR, the in-plane shear resistance map shows that th e in-plane shear \nresistance to the τxy component is nearly constant ( τxy ≈ 350 MPa) while the resistance to the \npure τyz  s t r e s s  h a s  t h e  h i g h e s t  v a l u e  ( τyz  ≈ 4.66 GPa) as plotted in Figure 5-a. From the \natomistic simulation results, we observed that the FCI for Near  BA OR shows glide of misfit \ndislocation on the interface except for θ = 90° and 270° where mode II fracture is observed. \nAccording to the P-K equation, fo rces exerted on the first and second misfit dislocations are f1 \n= [0, 0, be1τyz] and f2 = [be2τxy, 0, 0], respectively. τyz and τxy r e p r e s e n t  yz and xy-components \nof applied in-plane shear stress, and be1 and be2 represent the Burgers vectors for two edge \ndislocations. If both misfit dislocations are glissile, the she ar deformation will be initiated when \nif either τyz\t>\tτp1 o r  τxy > τp2 is satisfied. τp1 and τp2 represent the Peierls  stresses of first and \nsecond dislocations depicted in  Figure 4. The shear strength ca n be predicted as a function of the angle  θ ,  a s  m i n൫τp1 / sinθ, τp2 / cosθ൯ . The ratio of the half-width of dislocation core to \nmagnitude of Burgers vector for each misfit dislocation ( we1 /b1  = 0.96 and we2 /b2  = 2.64) \nimplies that the second misfit di slocation has significantly lo wer Peierls stress than the first \ndoes. In other words, the critical shear stress to activate the  motion of second misfit dislocation \nis significantly lower than that of the other, i.e. τp2 ≪ τp1. Hence, the interfacial shear strength \ncan be approximated as τ p2\t/ cos θ  unless θ is close to 90° or 270°, which explains nearly \nconstant τxy component of the in-pla ne shear resistance in Figure 5-a. Mode  II fracture at θ = \n90° and 270° is originated from large Peierls stress of the fir st misfit dislocations. We suspect \nt h a t  f r a c t u r e  o c c u r s  b e c a u s e  t h e  c r i t i c a l  s h e a r  s t r e s s  t o  p r o m o te the motion of first misfit \ndislocation is higher than the c ritical shear str ess to initiat e fracture at the FCI. \nFor Near PP OR, the in-plane shear strength is less anisotropic  and significantly larger \nthan IS or Near BA ORs, ranging from 1.30 to 6.51 GPa (Figure 5 -a). The disregistry analysis \nreveals that the ratio of the half-width of dislocation core to  magnitude of Burgers vector we/b \nare very small for both dislocations ( we1/b1 = 0.56 and we2/b2 = 0.82). We suspect that, for both \ndislocations, the critical shear stress to activate the glide o f misfit dislocation on the FCI is \nlarger than the critical shear stress to initiate fracture at t he FCI. Thus, the FCI for Near PP OR \nshows mode II fracture rather tha n dislocation-mediated plastic ity for all in-plane shear \ndirections. \nFrom the simulation results, we c onclude that the in-plane shea r responses of the FCIs \nin IS and Near BA ORs are simila r to each other in that shear d eformation is mainly governed \nby dislocation motion. Since in-plane shear strength of the FCI  for IS and Near BA  OR is \nconsiderably small (a few hundred  mega-pascals), the FCI may op erate as a strong sink to the \nlattice dislocation. The high dislocation density near the FCI region (Inoue, Ogur a et al. 1977, \nLanguillaume, Kapelski et al. 1997)  is likely attributed to the  weak interfacial shear strength of the FCI in IS or Near BA ORs. It is interesting to note that  the IS and Near BA OR have \nexactly same OR except different habit planes. Moreover, IS and  Near BA  OR are usually \nfound in the same pearlite colony while Near PP OR is found in different pearlite colony, as \nreported in the literature (Zhang, Esling et al. 2007). It mean s that the FCIs in the pearlite \ncolony in IS and Near BA OR has low in-plane shear resistance t o externally applied shear \nstress, while the FCI in the pearlite colony in Near PP OR has high in-plane shear resistance to \nexternally applied shear stress. Furthermore, the magnitude of the in-plane shear strength of \nt h e  F C I  f o r  I S  a n d  N e a r  B A  O R  i s  s i m i l a r  t o  t h e  P e i e r l s  s t r e s s  o f  t h e  f e r r i t e  w i t h  l a t t i c e  \ndislocation (a few hundred megapascals). Therefore, for IS and Near BA OR, in addition to \nglide of lattice dislocation in t he ferrite phase, glide of mis fit dislocation may contribute to the \nplastic response. For Near PP OR, since the in-plane shear resi stance of the FCI for Near PP \nOR (a few gigapascals) is much higher than the Peierls stress o f lattice dislocation in the ferrite \nlamellar, glide of lattice dislocation will govern the overall strain hardening behavior of the \npearlitic steel with fin e lamellar structure.  \nAs mentioned earlier, the occurrence of specific OR in pearliti c steel can be controlled \nby carbon contents of the system, heat treatment condition, and  application of the magnetic \nfield during the phase transformation stage. If we can fabricat e micro-pillar specimen from a \ncolony of such-fabricated pearlites, we can perform compression  tests of the specimen with \nspecific FCI ORs to assess the present predictions, following a n experimental procedure \ndescribed in the literature  (Kapp, Hohenwarter et al. 2016). \n \n5. Conclusions \n We study the in-plane shear response of the FCI for three ORs by using atomistic simulation methods combined with xAIFB and disregistry analyses . We find that the magnitude \nof Burgers vector and core-width  of misfit dislocation govern t he overall in-plane shear \nresponse of the FCI. We find that misfit dislocation is glissil e when the we/b ratio is large, but \nthey become sessile when the ratio is small. The anisotropy of in-plane shear strength can be \nunderstood by different we/b  ratios of misfit dislocations a long different directions. Whe n \ndislocation is sessile (first and second dislocations of Near P P OR and first dislocation of Near \nBA OR) or P-K force is zero for specific direction ( θ = 90° and 270° in IS OR), mode II fracture \nbehavior is observed. For Near BA and Near PP OR where two sets  of misfit dislocations \nintersect, we found that both pl astic deformation and fracture initiated at the junction of misfit \ndislocations because of the formation of weak bonds at the junc tion. As a future work, we plan \nto study the lattice dislocation trapping at the FCI for each O R by investigating the local stress \ndistribution near the FCI and st ress field induced by lattice d islocation approaching to the FCI. \nTo summarize. \n- The FCIs for IS, Near BA and Near PP ORs show different degrees  of anisotropy in \nthe in-plane shear resistance. \n- The FCI for IS and Near BA OR sho ws dislocation-mediated plasti city for all in-plane \nshear directions except mode II fracture for θ = 90° and 270°. \n- The FCI for Near PP OR shows mode  II fracture for all in-plane shear direction. \n- Glide of misfit dislocation at t he FCI for IS and Near BA  OR ca n be explained by \nmagnitude of Burgers vector and co re-width of misfit dislocatio n. \n- Mode II fracture for three ORs is initiated at the dislocation core-region because of the \nincoherency. \n- Despite the presence of misfit di slocation, mode II fracture at  the FCI can occur when the required shear stress to activate the glide of misfit dislo cation is higher than the \nrequired shear stress to initiate mode II fracture at the inter face. \n- The overall in-plane shear respons e of the FCI is governed by t he magnitude of Burgers \nvector and core-width of misfit dislocation. \n \nAcknowledgments \nThis research was supported by Basic Science Research Program ( 2016R1C1B2011979 and \n2016R1C1B2016484) through the National Research Foundation of K orea (NRF) funded by \nthe Ministry of Science, ICT & Future Planning.  \n \n \n  \nReferences \nAndrews, K. 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( 2 0 0 7 ) .  \" N e w  i n s i g h t s  i n t o  c ry s t a l l o g r a p h i c correlations between ferrite and \ncementite in lamellar eutectoid structures, obtained by SEM–FEG /EBSD and an indirect two-trace \nmethod.\" Journal of Applied Crystallography 40(5): 849-856. \n  \nZhou, D. S. and G. J. Shiflet (1991). \"Interfacial steps and gr owth mechanism in ferrous pearlites.\" \nMETALLURGICAL TRANSACTIONS A 22A: 1349-1365. \n  \nZhou, D. S. and G. J. Shiflet (1992). \"Ferrite: Cementite cryst allography in pearlite.\" METALLURGICAL \nTRANSACTIONS A 23A: 1259-1269. \n  \nZhou, Y. T., et al. (2017). \"Atomic structure of the Fe/Fe3C in terface with the Isaichev orientation in \npearlite.\" Philosophi cal Magazine: 1-12. \n  \n   Tables \nTable 1. The geometry of the initial ferrite/cementite bilayer and the interface energy for each \norientation relationship. \nName  Lx Lyc Lyf Lz εxx εzz γcf [mJ/m2]\nIS 167.30 81.69 83.80 32.12 9.08 × 10-6 2.17 × 10-3 501.9 \nNear BA  167.18 93.38 94.47 54.22 9.08 × 10-6 8.38 × 10-3 539.1 \nNear PP  65.77 73.37 74.45 44.99 3.66 × 10-4 2.24 × 10-3 575.9 \n*Unit: Å (Angstrom) \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n \n \n Table 2. Detailed information re garding misfit dislocation for each orientation relationship. \nName Notation IS Near BA Near PP \nMagnitude of edge and screw \ncomponent of Burgers vector [be1, bs1] [2.53, 0.00] [2.21, 0.00] [1.06, 2.18]\n[be2, bs2] N/A [2.53, 0.00] [3.26, 3.72]\nMagnitude of Burgers vector b1 2.53 2.21 2.42 \nb2 N/A 2.53 4.95 \nHalf-width of dislocation core for \nedge component we1 13.21 2.12 1.36 \nwe2 N/A 6.67 4.04 \nHalf-width of dislocation core for \nscrew component ws1 – – 2.00 \nws2 N/A – 4.37 \nHalf-width of dislocation core \n/magnitude of Burgers vector we1/b1 5.22 0.96 0.56 \nwe2/b2 N/A 2.64 0.82 \nLine orientation ξ1 (deg.) 90.00° 0.00° 0.00° \nξ2 (deg.)  N/A 90.00° 41.30° \nLine spacing d1 83.65 18.07 22.35 \nd2 N/A 83.59 43.41 \n*Unit: Å (Angstrom), line orientation represents the angle betw een positive x-axis and dislocation line around negative y-axis. \n*superscript e and s represents the component of the edge and s crew component for each dislocation line, respectively.  \n \n \n \n \n  \n \n \n \n \n \n  Figures \n \nFigure 1. (a) Un-relaxed, (b) relaxed structure, (c) 2D disregi stry map and (d) idealized misfit \ndislocations of the ferrite/cem entite interface for Near Pitsch -Petch (Near PP) orientation \nrelationship; Black rectangular  box in each figure represents t he simulation box. \n \n  \n \n \n \n \n \n \n \n \nFigure 2. Schematic representation of in-plane shear deformatio n of the ferrite/cementite \nbilayer for each orientation relationship \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3. (a) Schematic diagram of disregistry analysis using E quation 1, disregistry along the \nq-coordinate for (b) Isaichev (I S), (c-d) Near Bagaryatsky (Near  BA) and (e-f) Near Pitsch-\nPetch (Near PP) orientation rela tionship; Purple dashed line in dicates the data points used in \ndisregistry analysis for each graph. Black solid and dashed-dot ted line represent the fitted \nfunction for edge ( rerc) and screw ( rsrc) component of disregistry, respectively. Circle and square \nmarker represent the edge ( rij,erc,atom) and screw ( rij,src,atom)component of disregistry computed from \natomistic simulation, respectively. \n \nFigure 4. Potential energy maps (left) and idealized misfit dis location structures (right) of the \nferrite/cementite interface for (a) Isaichev (IS), (b) Near Bag aryatsky (Near BA) and (c) Near \nPitsch-Petch (Near PP) orientati on relationship; The dark regio n in potential energy map \nrepresent not only the local incoherency of the interface but a lso the core region of the misfit \ndislocation. For Isaichev orient ation relationship, there is sm all potential energy variation along \nthe x-axis, which implies that the  core of misfit  dislocation w idely spreads along the x-axis. \n \n \n \n \n \n \n \n \nFigure 5. (a) in-plane shear strength map and (b) stress-strain  curve for θ =  0 °  o f  t h e  \nferrite/cementite bilayer for Isaichev (IS, blue circle), Near Bagaryatsky (Near BA, red square) \nand Near Pitsch-Petch (Near PP , black triangle) orientation rel ationships \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 6. Potential energy maps o f initial (relaxed) ferrite/ce mentite interface and 2D disregistry ( rijr) maps of the ferrite/cementite bilayer for \n(a) Isaichev (IS), (b) Near Bagaryatsky (Near BA) and (c) Near Pitsch-Petch (Near PP) orienta tion relationships under low and high in-plane \nshear stresses for θ = 0°; Red and black dashed circles indicate the junction of th e misfit dislocations for Near Bagaryatsky (Near BA) and Near \nPitsch-Petch (Near PP) orientation relationships, respectively;  The green dashed lines represent the idealized misfit dislocat ion on the \nferrite/cementite interface for each orientation relationship. \n37 \n  \nFigure 7. A series of snapshot at the FCI under applied shear s tress τxy for (a) Isaichev (IS, τ xy \n≈ 110 MPa), (b) Near Bagaryatsky (Near BA, τ xy ≈ 570 MPa) and (c) Near Pitsch-Petch (Near \nPP, τ xy ≈ 1870 MPa) orientation relationships; Pink and blue atoms rep resent Fe and C atoms, \nrespectively. Gray and green atoms  represent the Fe atoms at th e interface region of cementite \nand ferrite, respectively. Black  dashed circle in Figure 7-c in dicates the fracture region. \n \n \n \n \n \n \n \n \n38 \n  \n \nFigure 8. Time versus relative displacement of interface atoms in x-direction for Isaichev (IS), \nNear Bagaryatsky (Near BA) and Near Pitsch-Petch (Near PP) orie ntation relationship; We \napplied constant in-plane shear stress for each orientation rel ationship for θ = 0° (τ xy= 110, 570 \nand 1870 MPa for IS, Near BA and Near PP orientation relationsh ip, respectively). For easy \ncomparison, we set the initial r elative displacement as zero ri ght after desired shear stress is \napplied. \n \n  \n39 \n Supplementary Information \n \nFigure S1. Atomic configurations  of un-relaxed interface struct ures and interface energy for \neach shuffle plane of Isaichev (IS) orientation relationship; # 1 shuffle plane is the most stable \ninterface structure among 8 different shuffle planes. \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n40 \n  \nFigure S2. Atomic configuration of un-relaxed interface structu re and interface energy for each \nshuffle plane of Near Bagariatsky (Near BA) orientation relatio nship; #2 shuffle plane is the \nmost stable interface structure among 6 different shuffle plane s. \n \n \n  \n \n \n \n \n \n \n \n \n \n \n41 \n  \n \nFigure S3. Atomic configuration of un-relaxed interface structu re and interface energy for each \nshuffle plane of Near Pitsch-Petch (Near PP) orientation relati onship; #5 shuffle plane is the \nmost stable interface structure among 6 different shuffle plane s. \n \n  \n \n \n \n \n \n \n  \n \n42 \n  \n \n \nFigure S4. Atomic configurations of un-relaxed interface struct ures and interface energies for \nBagaryatsky (left) and Pitsch-Pet ch (right) orientation relatio nship; Each figure represents the \nmost stable interface structure among 6 different shuffle plane s for each orientation relationship. \n \n \n \n  \n \n \n \n  \n \n \n \n43 \n  \n  \n \nFigure S5. The eight shuffle planes of cementite block for Isai chev (IS) orientation relationship. \nThe five shuffle planes (#1 to #5) are identical to the reporte d shuffle planes in the reference \n(Zhou, Zheng et al. 2017) but the three shuffle planes (#6 to # 8) were not reported.  \n \n \n \n  \n \n \n \n  \n44 \n  \nFigure S6. Atomic configurations of ferrite/cementite interface  (a) computed by MEAM \npotential in this study and reported by Zhou et al.  (Zhou, Zheng et al. 2017) for Isaichev (IS) \norientation relationship; Shuffle plane is identical but distan ce between Fe atoms indicated by \narrows are different. \n \n \n"    },    {        "title": "1411.2233v2.Gaussian_Beam_Transmission_through_a_Gyrotropic_Nihility_Finely_Stratified_Structure.pdf",        "content": "Chapter XX \nGaussian Beam Transmission through \na Gyrotropic-Nihility Finely-Stratified Structure \nVladimir R. Tuz1,2, Volodymyr I. Fesenko1,3 \n1Institute of Radio Astronomy of NASU, Kharkiv, Ukraine \n2School of Radio Physics, Karazin Kharkiv National University, Kharkiv, Ukraine, \ntvr@rian.kharkov.ua  \n3Lab. “Photonics”, Kharkiv National University  of Radio Electronics , Kharkiv, Ukraine, \nfesenko@kture.kharkov.ua ; fesenko_vladimir@mail.ru   \nAbstract  The three-dimensional Gaussian beam transmission through a ferrite-\nsemiconductor finely-stratified structure bein g under an action of an external static \nmagnetic field in the Faraday geometry is  considered. The beam field is repre-\nsented by an angular continuous spectrum of plane waves. In the long-wavelength \nlimit, the studied structure is described as a gyroelectromagnetic medium defined by the effective permittivity and effective permeability tensors. The investigations \nare carried out in the freque ncy band where the real parts of the on-diagonal ele-\nments of both effective permittivity and effective permeability tensors are close to zero while the off-diagonal ones are non-zero. In this frequency band the studied structure is referred to a gyrotropic-nihility medium. It is found out that a Gaus-sian beam keeps its parameters unchanged (beam width and shape) when passing through the layer of such a medium except of a portion of the absorbed energy. \nKeywords : Laser beam transmission, Magneto-optical materials, Metamateri-\nals, Electromagnetic theory, Propagation \n1.1 Introduction \nThe conception of nihility  was firstly introduced in the paper [1] for a hypothetical \nmedium, in which the following constitutive relations hold 0D=G\n, 0B=G\n. So, ni-\nhility is the electromagnetic nilpotent, and the wave propagation cannot occur in \nnihility, because 0E∇× =G\n and 0 H∇×=G\n in the absence of sources therein.  2  \nFurther, in [2], this conception of nihility was extended for an isotropic chiral \nmedium whose constitutive relations are: DE i Hερ=+GGG\n, BH i Eμρ=−GGG\n, where \nρ is the chirality parameter. Thus, a possible way for composing such a medium \nin the microwave band was proposed using canonical chiral wire particles. The ef-\nfective material parameters are calcula ted on the basis of the Maxwell-Garnett \nmixing rule, and in a certain narrow frequency band it is found out that the real parts of both effective permittivity and effective permeability become close to \nzero (\n0ε′≈, 0μ′≈) while the real part of the chirality parameter is maintained at \na finite value ( 0ρ′≠). It was revealed that in such an isotropic chiral-nihility  me-\ndium there are two eigenwaves with right (RCP) and left (LCP) circularly polar-\nized states, whose propagation constants depend only on the chirality parameter, \nand these propagation constants of the RCP ( γ+) and LCP ( γ-) waves are equal in \nmagnitude but opposite in sign to each other (0kγργ±=±= ± ). Thereby one of \nthese eigenwaves experiences the forward propagation while the other one experi-\nences the backward pr opagation. Here, the sign of th e chirality parameter, which \nin turn depends on the ch iral particles handedness, de termines which of the ei-\ngenwaves appears as a backward propagating one. In particular, this feature results \nin some exotic characteris tics in the wave transmission through and reflection \nfrom a single layer and multilayer system s which consist of such a chiral-nihility \nmedium [3, 4]. \nBesides chiral media, the circularly pola rized eigenwaves are also inherent to \nmagneto-optic gyrotropic materials (e.g. ferrites or semiconductors) in the pres-\nence of an external static magnetic field, when this field is biased to the specimen \nin the longitudinal geometry relative to the direction of wave propagation (in the \nFaraday configuration) [5]. Such gyrotr opic media are characterized by the per-\nmeability or permittivity tensor ˆ DEε=GG\n, ˆ BHμ=GG\n with non-zero off-diagonal \nelements (gyrotropic parameters). Apart from getting double-negative conditions \n[6-10], combining together gyromagnetic (ferrite) and gyroelectric (semiconduc-\ntor) materials into a certain unified gyroelectromagnetic structure [6] allows one to \nreach the gyrotropic-nihility  effect within a narrow frequency band [11]. In par-\nticular, in a finely stratified ferrite-sem iconductor structure such a condition is \nvalid in the microwave band nearly the frequencies of ferromagnetic and plasma \nresonances. In this case the real parts of  on-diagonal elements  of both effective \npermeability and effective permittivity tensors of such an artificial medium simul-\ntaneously acquire zero while the off-diagonal  ones are non-zero. It is revealed that \nin this medium the backward propagation can appear for one of the circularly po-\nlarized eigenwaves which leads to some u nusual optical features  of the system and \nprovides an enhancement of the polarization rotation, impedance matching to free \nspace, and complete light transmission. \nSince a gyrotropic-nihility medium with appropriate parameters can support \nbackward propagating eigenwaves and is impedance matched to free space , it be-\ncomes substantial to study the focusing pr operties of a finite thickness slab in \nwhich the gyrotropic -nihility condition holds [12]. It involves consideration of the \nfield in the form of a spatially finite wave beam, in particular, as a Gaussian beam 3 \nwhich is presented as a continual superposition of plane waves. On the other hand, \nit is also known that there are several b eam phenomena such as displacement of \nthe beam axis, beam splitting,  focal and angular shifts which are not found in the \nreflection and transmission of separate plane waves [13-16], and so they require \nparticular consideration. These studies are usually based on a two-dimensional \nbeam formulation, which is quite efficien t [17, 18]. Nevertheless, in gyrotropic \nmedia a three-dimensional model of beam representation should be considered to \ntake into account the polari zation effects and to predict the change in the ellipticity \nof the scattered beam [19, 20].  \nIn this chapter, we demonstrate the phenomenon of the three-dimensional \nGaussian beam transmission through a ferrite-semiconductor finely-stratified structure being under an action of an exte rnal static magnetic field biased along \nthe structure periodicity. The investigations are carried out for two different fre-\nquencies. The first one is chosen to be  far from frequencies of the ferromagnetic \nand plasma resonances and the second one is selected to be at the gyrotropic-\nnihility frequency. The main goal is to show  that such a finely-stratified structure \nis able to tunnel a Gaussian beam prac tically without any distortion of its form \nwhen the gyrotropic-n ihility condition holds. \n1.2 Problem Formulation and Methods of Solution \n1.2.1 Magnetic Multilayer Structure under Study \nA stack of N identical double-layer slabs (unit cells) which are arranged periodi-\ncally along the z axis is investigated (Fig. 1). E ach unit cell is composed of ferrite \n(with constitutive parameters \n1ε,1ˆμ) and semiconductor (with \nconstitutive parameters 2ˆε,2μ) \nlayers with thicknesses d1 and d2, \nrespectively. The structure's pe-riod is L = d\n1 + d 2, and in the x \nand y directions the system is in-\nfinite. We suppose that the struc-ture is finely-stratified, i.e. its characteristic dimensions d\n1, d2 \nand L are significantly smaller \nthan the wavelength in the corre-\nsponding layer 1dλ\u0013 , \n2dλ\u0013 , Lλ\u0013  (the long-wavelength limit). An external static magnetic field MG\n \nFig. 1.  A periodic stack of one-dimensional double-\nlayer ferrite-semiconductor structure under the Gaus-sian beam illumination. 4  \nis directed along the z-axis. The input 0z≤and output zN L≥  half-spaces are \nhomogeneous, isotropic and have constitutive parameters 0ε,0μ. \nWe use common expressions for constitutive parameters of normally magnet-\nized ferrite and semiconductor layers with taking into account the losses. They are defined in the form [21-23]: \n1\n11 1\n10\nˆ ,0 ,\n00T\nT\nf\nLi\niμα\nεε μ α μ\nμ⎛⎞−\n⎜⎟== ⎜⎟\n⎜⎟⎝⎠      2\n22 2\n20\nˆ 0, ,\n00T\nT\ns\nLi\niεβ\nεβ ε μμ\nε⎛⎞−\n⎜⎟==⎜⎟\n⎜⎟⎝⎠ (1) \nwhere for ferrite the auxiliary values are  11Ti μχ χ ′′′ =+ + , 2\n00mχω ωω′⎡= −⎣ \n221(1 )bDω−⎤ −−⎦, 22 2 1\n0 (1 )mbb Dχω ωωω−′′⎡⎤=− +⎣⎦, iα′′′=Ω+Ω , 2\n0 mωω ω′⎡ Ω=−⎣ \n221(1 )bDω−⎤ −+⎦, 21\n0 2mbDωωω−′′Ω= , 222 2 2 2 2\n00 (1 ) 4 Db bωω ω ω⎡⎤=−+ +⎣⎦, 0ω is \nthe Larmor frequency and b is a dimensionless damping constant; for semiconduc-\ntor layers the auxiliary values are  21\n20 1[ ( ) ] ,L\np i εε ω ω ω ν−⎡ ⎤ =− +⎣ ⎦ \n122 2\n0 (( ) ) ,pc c i βε ω ωω ων ω−⎡ ⎤ =+ −⎣ ⎦ 22 2 1\n20 1 ( )[ (( ) )] ,T\npc ii εε ω ω ν ω ω νω−⎡ ⎤ =− + +−⎣ ⎦ \n0ε is the part of permittivity attributed to the lattice, pω is the plasma frequency, \ncω is the cyclotron frequency and ν is the electron collision frequency in plasma. \n \n  \nFig. 2.  Frequency dependences of the perm eability and permittivity of ferrite ( a) and semicon-\nductor ( b) layers, respectively. We use typical parame ters for these materials in the microwave \nregion. For the ferrite layers, under satura tion magnetization of 2000 G, parameters are \n024 . 2ωπ=  GHz, 28 . 2mωπ=  GHz, 0.02b= , 5.5fε= . For the semiconductor layers, pa-\nrameters are: 24 . 5pωπ=  GHz, 24 . 0cωπ=  GHz , 20 . 0 5vπ=  GHz, 01.0ε= , 1.0sμ= . \nThe frequency dependences of the permeability and permittivity parameters \ncalculated using Eq. (1) are presented in  Fig. 2. Note that the values of 1 Im( )Tμ, \nIm( )α and 2 Im( )Tε, Im( )β are so close to each other that the curves of their fre-\nquency dependences coincide in the corresponding figures. 5 \n1.2.2 Gaussian Beam Representation \nThe auxiliary coordinate system xin, yin, zin (see, Fig. 1) is introduced to describe \nthe incident beam field [13, 14, 20]. In it, the incident field ,in in inEHψ=GG G is de-\nfined as a continued sum of the partial plane waves with the spectral parameter \ninκG (it has a sense of the transverse wave  vector of the partial plane wave): \n() ( ) ( )3 exp .in in in in in in in in Ui r a i z a d ψυκκ γ κ∞\n−∞=+ + + ⎡⎤⎣⎦ ∫∫GGGG G GG (2) \nIn Eq. (2) the vector υG is related to E (ineυ=GG) or H (inhυ=GG) field, respectively; \nin p in seP V b P V=− ×GGGG, in s in phP V b P V=+ ×GGGG\n where the vector 0 Pzn=×GGG describes \nthe field polarization. In the structure's coordinates x, y, z, the vector nG is charac-\nterized via the next components ( ) cos cos , cos sin , 0in in in inθϕ θϕ , where \n90in inθψ=−D; 0zG is the basis vector of z-axis, and the vector (cos cos ,in in inb θϕ =G\n \n2\n00 cos sin , cos )in in inθϕε μ θ−−  describes the direction of the incident beam \npropagation; ()in UκG is the spectral density of  the beam in the plane 0inz=; \n2\n0 ,in in in kγ κκ=− ⋅GG 2\n0 0a r g ( )in in kκκπ <− ⋅ <GG, and 12(, )inaa a=G. \nThe transformation from the structure's coordinate system { x, y, z } to the \nbeam's one { xin, yin, zin} can be realized in the sta ndard means [24]: by rotating \naround the z-axis on the angle inϕ; by rotating around the y-axis on the angle inψ; \nby shifting the point of origin to the point ()123,,aaa . Taking into account the \ncoordinate system transformation, the reflected and transmitted fields are obtained \nas follows: \n() []\n() []\n() ( )\n() ( )exp\nexp ,\nexp\nexp ,ref in in\nref in in\ntr in in\ntr in inPV U R i r i z d\nbP V U R i r i z d\nPV U T i r i z NL d\nbP V U T i r i z N L dυυ\nυ\nυυ\nυ\nυυ\nυ\nυυ\nυψκ κ γ κ\nκκ γ κ\nψκ κ γ κ\nκ κγ κ∞\n−∞\n∞′\n′−∞\n∞\n−∞\n∞′\n′−∞=⋅ − ±\n±× ⋅ −\n=⋅ + − ± ⎡⎤⎣⎦\n±× ⋅ + − ⎡⎤⎣⎦∫∫\n∫∫\n∫∫\n∫∫GGG G G G\nGG GG G G\nGGG G G G\nGG G GGG (3) \nwhere 2\n0 , kγ κκ=− ⋅GG2\n0 0a r g ( ) kκκπ <− ⋅ <GG; Rυυ, Rυυ′and Tυυ, Tυυ′ are the \ncomplex reflection and tran smission coefficients ( ,, sp υυ′= ) of the partial \nplane electromagnetic waves, respectively. They depend on the frequency of the \nincident field, angles inψ, inϕ and other electromagnetic and geometric parame-6  \nters of the structure. The coeffi cients with coincident indexes ( υυ) describe the \ntransformation of the incident wave of the perpendicular ( sυ=) or the parallel \n( pυ=) polarization into the co-polarized wave , and the coefficients with distinct \nindexes ( υυ′) describe the transformation of the incident wave into the cross-\npolarized wave at the structure output. The left and right indexes correspond to the \npolarization states of the incident and re flected (transmitted) waves, respectively. \nThe corresponding reflection and transmission coefficients are determined through the rigorous solution of the Cauchy problem related to the tangential field compo-nents on the structure's boundaries; the reader is referred here to Refs. [5, 11] for further details. \n1.2.3 Effective Medium Theory \nIn the long-wavelength limit, when the characteristic dimensions of the structure \n(d1, d2, L) are significantly smaller  than the wavelength in the corresponding layer \n(1dλ\u0013 , 2dλ\u0013 , Lλ\u0013 ), the interactions of electromagnetic waves with a pe-\nriodic gyromagnetic-gyroelect ric structure can be descri bed analytically using the \neffective medium theory. From the viewpoint of this theory, the periodic structure is represented approximately as an  anisotropic (gyroelectromagnetic) uniform  me-\ndium whose optical axis is directed along  the structure periodicity, and this me-\ndium is described with some effective permittivity and permeability tensors \nˆε \nandˆμ [11]. By this means, the investigation of the wave interaction with an in-\nhomogeneous periodic structure is reduced to the solution of the boundary-value \nproblem of conjugations of an equivalent homogeneous anisotropic layer with sur-rounding spaces. \nLet us consider a unit cell of the studied structure. It is made of two layers \n1 0zd≤≤  and 1dz L≤≤  of dissimilar materials whose constitutive relations are \nas follows: \n12\n11\n12ˆ0, .ˆDE DEzd d zL\nBH BHεε\nμμ⎫⎫==⎪⎪≤≤≤ ≤⎬⎬==⎪⎪⎭⎭GG GG\nGG GG  (4) \nIn general form, in the Cartesian coordinates, the system of Maxwell’s equations \nfor each layer has a form \n00\n00\n00ˆ ˆ () , ( ) ,\nˆ ˆ () , ( ) ,\nˆ ˆ () , ( ) ,yz zy j x y z z y j x\nz xx z j y z xx z j y\nx yy x j zx yy x j zik H H ik E ik E E ik H\nH ik H ik E E ik E ik H\nik H ik H ik E ik E ik E ik Hεμ\nεμ\nεμ−∂ =− −∂ =\n∂− = − ∂− =\n−= − −=G G\nG G\nG G (5) 7 \nwhere z z∂= ∂ ∂ , xk and yk are the wavevector tr ansverse components, \n0kcω=  is the free-space wavenumber, 1, 2j= , 1ˆε and 2ˆμ are the tensors with \n1ε and 2μ on their main diagonal and zeros elsewhere, respectively (11ˆ ˆ Iεε= , \n22ˆ ˆ Iμμ= , ˆI is the identity tensor). From six components of the electromagnetic \nfield EG\n and HG\n, only four are independent. Thus the components Ez and Hz can be \neliminated from the system (5) and derived a set of four first-order linear differen-tial equations related to the transversal field components inside each layer of the \nstructure [5]. For the ferrite (\n1 0zd≤≤) and semiconductor (1dz L≤≤) layers \nthese systems, respectively, are: \n22 2\n01 1 01\n22 2\n10 1 0 1\n0 22 2 2\n01 1 0 01\n22 2\n10 1 0 100\n00,00\n00T\nx x xy x\nT\ny y yx y\nz LL\nx x xy x\nLL\ny y yx yEE kk k i k k\nEE kk k kk iikHH kk k k k k\nHH kk k kkεα μ ε\nμε ε α\nμε μ\nεμ μ⎛⎞ +− ⎛⎞ ⎛⎞\n⎜⎟ ⎜⎟ ⎜⎟−+ − + ⎜⎟ ⎜⎟ ⎜⎟∂=⎜⎟ ⎜⎟ ⎜⎟ −− +⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟−⎝⎠ ⎝⎠ ⎝⎠\n(6) \n22 2\n02 2 02\n22 2\n20 2 0 2\n0 22 2\n02 2 02\n22 2\n20 2 0 200\n00.00\n00LL\nx x xy x\nLL\ny y yx y\nz T\nx x xy x\nT\ny y yx yEE kk k k k\nEE kk k kkikHH kk k i k k\nHH kk k kk iεμ ε\nμε ε\nμβ ε μ\nεμ μ β⎛⎞ − ⎛⎞ ⎛⎞\n⎜⎟ ⎜⎟ ⎜⎟−+ −⎜⎟ ⎜⎟ ⎜⎟∂=⎜⎟ ⎜⎟ ⎜⎟ −− − +⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟−−⎝⎠ ⎝⎠ ⎝⎠\n(7) \nThe sets of Eqs. (6) and (7) can be abbreviated by using a matrix formulation: \n()()()0 ,0 .zzi k z z z L∂Φ = Φ < < AGG\n (8) \nIn this equation, {},, ,T\nxy x yEEHHΦ=G\n is a four-component column vector \n(here upper index T denotes the matrix transpose operator), while the 4×4 matrix \nfunction ()zA  is piecewise uniform as \n()11\n21,0 ,\n,,zdzdz L<< ⎧=⎨<< ⎩AAA (9) \nwhere the matrices 1A and 2A correspond to Eqs. (6) and (7), respectively. 8  \nSince the vector ΦG\n is known in the plane z = 0, the Eq. (7) is related to the \nCauchy problem [25] whos e solution is straightforw ard, because the matrix ()zA  \nis piecewise uniform. Thus, the field co mponents referred to boundaries of the \ndouble-layer period of th e structure are related as1 \n()() ()()[][]()21 2 1 0 2 2 0 1 1 00 e x p e x p 0 , Ld i k d i k dΦ=Φ = Φ= Φ= Φ MM M A AGG G G G\nM  (10) \nwhere jM and M are the transfer matrices of th e corresponding layer and the pe-\nriod, respectively. \nSuppose that jγ is the eigenvalue of the corresponding matrix 0 jkA \n0 (det[ ] 0)jjkγ−= AI , j = 1, 2 and I is the 4 × 4 identity matrix. When \n|| 1jjdγ\u0013 (i.e., both layers in the period are electrically thin), the next long-\nwave approximations can be used [26] \n[][]02 2 01 1 01 1 02 2 exp exp .ik d ik d ik d ik d ++ AA I A A \u0011  (11) \nLet us now consider a single layer of effective permittivity ˆeε, effective per-\nmeability ˆeμ and thickness L. Quantity eA is defined in a way similar to (6), (7): \n22 2 0\n00\n22 2\n00\n22 2\n00\n22 2\n0000\n00\n,\n00\n00x\ny\nLT L z\nx x y eee x e\nTL L\nye y e x y e e\nLT L\nx xy e e e x e\nTL L\ny ey e x y e e\nx\nyE\nE\nikHk k k i k k\nHk k k k k i\nE kk k i k k\nE kk k kk i\nH\nHμβε μ\nεμ μ β\nεαμ ε\nμε ε α⎛⎞ ⎛\n⎜⎟ ⎜\n⎜⎟ ⎜∂=⎜⎟ ⎜ −− − +\n⎜⎟ ⎜⎜⎟ ⎜ −−⎝⎠ ⎝\n⎛⎞⎞ +−\n⎜⎟⎟−+ − + ⎜⎟⎟\n⎜⎟⎟\n⎜⎟⎟⎟⎜⎟⎠⎝⎠ (12) \nand (10): \n() ()()[]()0 00 e x p 0 .ee Li k LΦ=Φ= Φ= Φ MAGG G G\nM  (13) \n                                                           \n1 The series 11exp( )!m\nmm∞\n==+∑ XI X  converges for square matrices X, i.e. function \nexp( )X is defined for all square matrices [25].  9 \nProvided that eγ is the eigenvalue of the matrix 0 ekA 0 (det[ ] 0)eekγ−= AI  \nand || 1eLγ\u0013 (i.e., the entire composite layer is electrically thin as well), the \nnext approximation follows \n[]00 exp .ee ik L ik L + AI A\u0011  (14) \nEquations (11) and (14) permit us to establish the following equivalence be-\ntween bilayer and single layer: \n11 2 2 ,eff=+ AAA    jjfdL= . (15) \nIn the case when the directions of both wave propagation and static magnetic \nfield are coincident ( 0xykk== ), the following simple expressions for the effec-\ntive constitutive parameters of the homogenized medium can be obtained: \n11 2 2TT\nef f μμμ=+ ,  11 2 2TT\nef f εεε=+ ,  1 efαα= ,  2 efββ= . (16) \n \n  \nFig. 3.  Frequency dependences of ( a) effective permeability and ( b) effective permittivity of the \nhomogenized ferrite-semiconductor medium. Parameters of the ferrite and semiconductor layers are the same as in Fig. 2; d\n1 = 0.05 mm, d2 = 0.2 mm. The circles mark the situation when \nRe( )T\neμ and Re( )T\neε are close to zero while Re( ) 0eα≠, Re( ) 0eβ≠ and losses in the ferrite \nand semiconductor layers are small. \nThe effective constitutive parameters cal culated according to the formula (16) \nare given in Fig. 3. The whole frequency range can be divided into three specific \nbands where parameters of the tensors ˆeμand ˆeε acquire different properties. In \nthe first band, located between 2 GHz and 3 GHz, T\neμ, T\neε, eα and eβ have posi-\ntive values of their real parts and small imaginary parts. In the second band, be-\ntween 3 GHz and 4.5 GHz, the real parts of parameters vary from positive values \nto negative ones as the frequency increas es. These transitions occur at the fre-\nquencies of the ferromagnetic resonance of ferrite ( ffr = 4.2 GHz) and the cyclo-10  \ntron resonance of semiconductor fpr = 4.0 GHz), respectively. In this band the me-\ndium losses are very significant. Finally, in the third frequency band, located from \n4.5 GHz to 5.5 GHz, the real parts of parameters have a transition from negative to positive values while their imaginary parts are small. The latter band is given in the insets of Fig. 3 on a larger scal e. One can see that there is a frequency \nf\ngn ≈ 4.94 GHz where real parts of T\neμ and T\neεsimultaneously reach zero. It is sig-\nnificant that, by special adjusting ferrite and semiconductor type, external static \nmagnetic field strength and thicknesses of layers, it is possible to obtain the condi-\ntion when real parts of T\neμ and T\neε acquire zero at the same frequency. Exactly \nthis situation is marked in the insets of Fig. 3 with circles. Note that at this fre-\nquency, the real parts of eα and eβ are far from zero and the medium losses are \nsmall. \n1.2.4 Eigenvalue Problem \nThe formulation of the eigenvalue problem on the matrix eA (det[ ] 0eeη−= AI ), \nwhose coefficients are defined as (16), gi ves us the characteristic equation on the \neffective refractive index eη of the medium: \n() ()()()()22 2 2 4220TT TT T T\ne e ee e e ee ee ee e eη η εμ αβ εμ μβ εα αβ−+ + − − + = , (17) \nwhose solutions are \n()()2()TT\nee e e eηεβμ αεμ±± ±=± ±= . (18) \nHere the signs ‘ ±’ are related to two different eigenwaves with propagation con-\nstants 0 eekγη±±= . It is well known that in an unbounded gyrotropic medium they \nhave right (RCP, eγ+) and left (LCP, eγ−) circular polarizations [21]. \nEspecially interesting situation appears if the real parts of T\neμ and T\neε are close \nto zero and the medium losses are small. In this case there is || | |ee εμα β±±′′≈ , and \nthe propagation constants become as:  \n0||ee e e e k γγγ α β+−′′ =− = ≈ . (19) \nThus, the propagation constants of the RCP (eγ+) and LCP (eγ−) waves are equal \nin the magnitude but opposite in sign to each other, and the backward  propagation  \nappears for the RCP wave while for the LCP wave it is forward one (see also, 11 \nCh. IV of [27]) . Recall that the backward wave is the wave in which the direction \nof the Poynting vector is opposite to that of its phase velocity [28]. The similar pe-culiarity of the RCP and LCP waves propagation occurs also in the chiral-nihility media [2]-[4], so in the analogy with them, the condition (19) is related to the gy-\nrotropic-nihility  media [11]. The frequency band, at which the gyrotropic-nihility \ncondition is satisfied for the RCP wave, is de picted in Fig. 4. Particularly, the gy-\nrotropic-nihility frequency f\ngn is marked in the inset of this figure with circles. \n \n  \nFig. 4.  Frequency dependences of the material parameters of the equivalent gyrotropic medium \nfor the ( a) RCP and ( b) LCP eigenwaves. Parameters of the ferrite and semiconductor layers are \nthe same as in Fig. 2; d1 = 0.05 mm, d2 = 0.2 mm. \n1.3 Numerical Results: Reflecte d and Transmitted Fields \n1.3.1 Spectral and Angular Behaviors \nIt is anticipated that if the frequency of the electromagnetic wave which incidents \non a finite layer of such a composite medium is chosen to be nearly the frequency \nfgn of the gyrotropic-nihility condition, the transmitted and reflected fields will ac-\nquire some unusual properties. In order to demonstrate this, in the long-\nwavelength limit, the reflection and transmi ssion coefficients can equivalently be \ncalculated using the rigorous solution (10) or the approximate solution (13) of \nEq. (8) because these solutions  give the same result. In particular, we are inter-\nested here in the study of the energy relations between the transmitted and re-\nflected fields, i.e. the polar ization effects are not discussed in this chapter. Never-\ntheless we refer the reader to Ref. [11]  where some polarization features of the \nstudied structure are revealed.  \nSo, the transmittance is calculated as a function of the frequency and the angle \nof incidence (Fig. 5a). One can see that this  function exhibits an expanded flat area 12  \nof the transmittance at the frequency wher e the gyrotropic-nihility condition holds. \nAt this frequency the complete transmission of the partial plane monochromatic \nwaves takes a place almost in the entire ra nge of angles of incidence except the \nrange of glancing angles. Such a high tran smittance appears du e to the peculiari-\nties of the medium impedances ( Zμε± ±±= ) related to the RCP and LCP \nwaves. It is particularly remarkable that in the vicinity of the gyrotropic-nihility \nfrequency fgn, the parameters αe and βe are close in value to each other and their \nreal parts approach to unit wh ich can be clearly seen in Fi g. 3. It leads to the fact \nthat the medium becomes to be impedan ce matched to free space [11]. Directly at \nthe gyrotropic-nihility frequency, the impedances related to the RCP and LCP \nwaves become indistinguishable: || ||ee ZZ Z αβ+−′′ === . \nFig. 5.  (a) Transmittance as a function of the frequency and the angle of incidence of the plane \nmonochromatic wave for the equivalent gyrotropic layer with finite thickness ( NL = 2.5 mm). ( b) \nThe angular dependences of the transmittance, reflectance and absorption coefficient at the gyro-\ntropic-nihility frequency (top figure) and far from it (bottom figure). Parameters of the ferrite and \nsemiconductor layers are the same as in Fig. 2; d1 = 0.05 mm, d2 = 0.2 mm. \nThis feature is also confirmed by the data plotted in Fig. 5b where the reflec-\ntance, transmittance and absorption coeffi cient are calculated at two different fre-\nquencies for a comparison. Thus the first frequency is chosen at the gyrotropic-\nnihility condition and the second one is selected to be far from the frequencies of \ngyrotropic-nihility condition and the ferromagnetic and plasma resonances. At the \nfrequency of f = 10 GHz, the curves have typi cal form where the transmittance \nmonotonically decreases and the reflectance monotonically increases as the angle \nof incidence rises. On the other hand, at the gyrotropic-nihility frequency, the \ncurves of the transmittance and reflectance are different drastically from that ones \nin the first case. Thus, the level of the transmittance/reflectance remains to be in-\nvariable almost down to the glancing angles. At the same time, the reflectance is small down to the glancing angles because the medium is impedance matched to \nfree space. 13 \n1.3.2 Gaussian Beam Transmission \nSince the beam field is represented by an angular continuous spectrum of plane \nwaves, the transmitted beam distribution depends on the angular characteristic of the transmission coefficient of spatial plane monochromatic waves at a particular frequency. We consider an incident Gau ssian beam with the spectral density as-\nsigned due to the law \n2( ) exp[ ( ) 16] ( 2) ( 2)in in m xin x n yin yUw H k w H k wκκ=− ⋅GG G, \nwhere {, }x y ww w=G, wx and wy are the beam widths along xin and yin axis, respec-\ntively, ()Hυ⋅ is the Hermit polynomial of υ-th order ( ,mnυ= ). In this chapter \nwe restrict ourselves to the case of the zero-order ( m = n = 0) beam. The final dis-\ntribution of the transmitted beam is presented in Figs. 6, 7 in two- and three-dimensions. As before the results are obtained for two distinct frequencies. \n \n  \n \nFig. 6.  The two-dimensional distribution of the absolute value of the incident beam | Einc|2 and the \ntransmitted beam | Etr|2 for different ( a, c) angles of incidence of the primary beam and ( b, d) \nnumber of structure’s periods. Th e field distribution is normalized to the maximum value of the \nnormally incident beam. Parameters of the ferr ite and semiconductor layers are the same as in \nFig. 2. The incident beam parameters are: k0wx = k0h = 10, ϕin = 0 deg. Other structure parame-\nters are: d1 = 0.05 mm, d2 = 0.2 mm. 14  \n \n  \n \n  \nFig. 7.  The three-dimensional distribution of the absolute value of ( a) the incident beam | Einc|2 \nand ( b-d) the transmitted beam | Etr|2. The field distribution is normalized to the maximum value \nof the normally incident beam. Parameters of the ferrite and semiconductor layers are the same \nas in Fig. 2. The incident beam parameters are: k0wx = k0wy = k0h = 10, ϕin = 0 deg. Other struc-\nture parameters are: d1 = 0.05 mm, d2 = 0.2 mm. \nSo, at the frequency fgn where the gyrotropic-nihility condition holds, the stud-\nied gyroelectromagnetic medium is well impedance matched to free space, and spatial plane monochromatic waves can completely pass through the system up to the glancing angles. As a result, the tran smitted beam pattern does not acquire any \nsignificant distortion of its form nearly the frequency f\ngn, while this feature is not \ninherent to the pattern of the transmitted beam to be far from this frequency. It should be noted that this shape retention of the transmitted beam pattern remains unchanged even under the oblique incidence of the primary beam and this effect is polarization insensitive. \nIn conclusion, the peculiarities of the Gaussian beam interaction with a ferrite-\nsemiconductor finely-stratified structure bein g under an action of an external static \nmagnetic field in the Faraday geometry is presented in this chapter. In the long-wavelength limit, when the structure layers  are optically thin, the effective me-\ndium theory is developed, and the effective constitutive parameters of the equiva-15 \nlent uniform anisotropic medium are obtained analytically. On the basis of these \nparameters the peculiarities of the eige nwaves propagation are studied and the \npossibility of achieving the gyrotropic-nihility condition is predicted.  \nThe reflection, transmission and absorptio n of waves in the system are studied \nin vicinity of the gyrotropic-nihility frequency. It is found out that under the oblique incidence of the \nspatial  plane monochromatic wave on the studied struc-\nture, the level of the transmission/reflection remains to be invariable almost down to the glancing angles when the gyrotropi c-nihility condition is satisfied. As a re-\nsult, at the frequency of the gyrotropic-nihility cond ition the Gaussian beam can \npass through such a system keeping its \nparameters unchanged (beam width and \nshape) except of a portion of the absorbed energy even under the oblique inci-\ndence of the primary beam. \nAcknowledgments This work was partially supported (V.R. Tuz) by Ministry of Education \nand Science of Ukraine under the Program \"Electrodynamics of layered composites with \nchiral properties and multifunctional planar systems\", Project No. 0112 U 000561. \nReferences \n1. A. Lakhtakia, An electromagnetic trinity from “negative permittivity” and “negative \npermeability”, Int. J. 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Shmatko, Scattering of a wave \nbeam by inhomogeneous anisotropic chiral layer,  Int. Conf. Math. Methods EM The-\nory (MMET 2002), Sep. 10-13 , Kyiv, Ukraine,  566-568, (2002) \n14. T.M. Grzegorczyk, X. Chen, J. Pacheco Jr., J. Chen, B.-I. Wu, J.A. Kong, Reflection \ncoefficients and Goos-Hänchen shifts in anisotropic and bianisotropic left-handed \nmetamaterials, Prog. Electromagn. Res ., 51, 83–113, (2005) \n15. W.T. Dong, L. Gao, C.-W. Qi u, Goos-Hänchen shift at the surface of chiral negative \nrefractive media, Prog. Electromagn. Res. , 90, 255-268 (2009) \n16. R.-L. Chern, P.-H. Chang, Negative refraction and backward wave in pseudochiral \nmediums: Illustrations of Gaussian beams, Opt. Express , 21, 2657-2666, (2013) \n17. R.-L. Chern, P.-H. Chang, Negative refraction and backward wave in chiral mediums: \nIllustrations of Gaussian beams, J. Appl. Phys. , 113, 153504, (2013) \n18. K.M. Luk, A.L. Cullen, Three-dimensi onal Gaussian beam reflection from short-\ncircuited isotropic ferrite slab, IEEE. Trans. Antennas Propag.,  41, 962-966, (1993) \n19. V. Tuz, Three-dimensional Gaussian beam  scattering from a periodic sequence of bi-\nisotropic and material layers, Prog. Electromagn. Res.  B, 7, 53-73, (2008) \n20. A.G. Gurevich,  Ferrites at Microwave Frequencies  (Heywood, 1963) \n21. R.E. Collin, Foundation for Microwave Engineering (Wiley, 1992) \n22. F.G. Bass, A.A. Bulgakov, Kinetic and Electrodynamic Phenomena in Classical and \nQuantum Semiconductor Superlattices  (Nova Science, 1997) \n23. G.A. Korn, T.M. Korn, Mathematical Handbook for Scientists and Engineers: Defini-\ntions, Theorems, and Formulas for Reference and Review  (McGraw-Hill Book Co., \n1968) \n24. V.A. Jakubovich, V.H. Starzhinskij, Linear Differential Equations with Periodic Coef-\nficients (Wiley, New York, 1975) \n25. A. Lakhtakia, C.M. Krowne, Restricted e quivalence of paired epsilon-negative and \nmu-negative layers to a negative phase-veloc ity materials (alias left-handed material), \nOptik , 114, 305–307, (2003) \n26. V.G. Veselago, The electrodynamics of subs tances with simultaneously negative val-\nues of ε and μ, Sov. Phys. Usp ., 10, 509–514, (1968) \n27. V.V. Shevchenko, Forward and backward waves: Three definitions and their interrela-\ntion and applicability, Phys.-Usp ., 50, 287–290, (2007) \n "    },    {        "title": "2209.08979v1.Ferroelectric_Solitons_Crafted_in_Epitaxial_Bismuth_Ferrite_Superlattices.pdf",        "content": "   \n \n   \n Ferroelectric  Solitons  Crafted in Epitaxial Bismuth Ferrite Superlattices  \nV. Govinden1†, P. R. Tong2†, X. Guo3,4,5†, Q. Zhang1, S. Mantri6, S. Prokhorenko6, Y. Nahas6, \nY. Wu3, 5, L. Bellaiche6, H. Tian2*, Z. Hong3,5*, D. Sando1* and V. Nagarajan1* \n†- these authors contributed equally  \n*- To whom correspondence should be addressed  \nE-mail: hetian@zju.edu.cn , hongzijian100@zju.edu.cn , daniel.sando@unsw.edu.au , \nnagarajan@unsw.edu.au  \n \n \n1School of Materials Science and Engineering, University of New South Wales, Sydney,  \nNSW 2052, Australia  \n2Center of Electron Microscope, State Key Laboratory of Silicon Material, School of \nMaterial Science & Engineering, Zhejiang University, Hangzhou 310027, China  \n3School of Materials Science and Engineering, Zhejiang University, Hangzhou, Zhejiang \n310027, China  \n4Institute of Advanced Semiconductors & Zhejiang Provincial Key Laboratory of Power \nSemiconductor Materials and Devices, Hangzhou Innovation Center,  Zhejiang University, \nHangzhou, Zhejiang 311200, China  \n5Cyrus T ang Center for Sensor Materials and Applications, State Key Laboratory of \nSilicon Materials, Zhejiang University, Hangzhou, Zhejiang 310027, China  \n6Physics Department and Institute for Nanoscience and Engineering, University of  \nArkansas, Fayetteville AR 72 701, USA  \n \n \n     \n \n   \n  \nAbstract  \nIn ferroelectrics, complex interactions among various degrees of freedom enable the \ncondensation of topologically protected polarization textures. Known as ferroelectric solitons, \nthese particle -like structures represent a new class  of materials with promise for beyond -CMOS \ntechnologies due to their ultrafine size and sensitivity to external stimuli. Such polarization \ntextures have scarcely been reported in multiferroics. Here, we report a range of soliton \ntopologies in (BiFeO 3)m/(SrTiO 3)m superlattices. High -resolution piezoresponse force \nmicroscopy and Cs -corrected high -angle annular dark -field scanning transmission electron \nmicroscopy reveal a zoo of topologies, and polarization displacement mapping of planar \nspecimens reveals cent er-convergent/divergent topological defects as small as 3 nm. Phase \nfield simulations verify that some of these topologies can be classed as “bimerons”, with a \ntopological charge of ±1, and first -principle s-based effective Hamiltonian computations show \nthat the co -existence of such structures can lead to non -integer topological charges, a first \nobservation in a BiFeO 3-based system. Our results open new opportunities in multiferroic \ntopotronics.  \n  \n     \n \n   \n Ferroelectric s are known to exhibit  strong coupling of strain to electric polarization . This  \nphenomenon  enables the formation of exotic domain structures which can display emergent \nproperties useful for information storage and sens ing devices . Modern developments in \nfabrication techniques have enabled flexible tailoring  of strain , chemical  and electrical \nboundary conditions  to achieve  a new paradigm of nanoscale complex polarization textures, \nincluding topologically protected nontrivial states1-4. When such topologically protected \npolarization textures condense to ultra fine size dispersed in a parent medium, they can be \nconsidered as particle -like objects  known as ferroelectric solitons. An example of  such three -\ndimensional ferroelectric solitons  are spherical domain s (and their transitional states) , which  \npossess homogeneously polarized cores surrounded by a curling polarization forming a curved \nouter shell . The s trong  polarization and strain gradients  which exist  at the unit cell level within \nthese structures results in extremely high local crystalline anisotropy and even  symmetries  that \nare energetically unfavorable in the parent bulk . This has a  significant implication : the \npolarization curling dramatically raises the  internal energy , with two consequences : i) the size \nof such topological objects  is restricted to the nanoscale  (i.e., 1-10 nm), and ii) these objects  \nare highly sensitive to external stimulus . These virtues  make  ferroelectric solitons  prime \ncandidates  for low-energy  and high-density nanoelectronics5. \nFerroelectric solitons  have various forms  ranging from electrical bubbles6-8, to polar \nbubble skyrmions9, 10 and more recently  – thus far only theoretically predicted  – hopfions11. \nOther transitional topologies such as merons12, bimerons13 and disclinations13, 14 have also been \nobserved. Although spherical ferroelectric topologies were theoretically predicted more than \ntwo decades ago,15, 16 and flux closure domain structures studied previously17-19, the field has \narguably not seen such a dramatic surge in efforts20-22 since the demonstration of stable \npolarization vortex arrays in epitaxial lead titanate -strontium titanate superlattices  by Yadav23 \nand Tang1. Following these reports, it was shown  that ferroelectric/dielectric  superlattices \ncould  be tuned to fabricate skyrmion arrays10 with emergent chiral24, local negative \npermittivity25 and conduction properties26. However,  such topologies have also been  found in \nsimple ferroelectric sandwich heterostructures27, revealing that ultimately the delicate interplay \nbetween the electrical and mechanical  boundary conditions drives their formation16, 28, 29. \nWhilst  solitons with complex non -trivial topologies have been demonstrated in  pure \nferroelectrics7, 10 and type -II multiferroics30 (where the polarization is the secondary order \nparameter ), they have remained elusive in type-I multiferroics. These latter materials also \nharbor coexisting ferroelectric and magnetic orders , but here the ferroelectric polarization is    \n \n   \n the primary order parameter meaning that the local crystalline anisotropy (i.e ., strain ) energy \nplays a more dominant  role than is the case for  their type-II cousins . A major  type-I room \ntemperature multiferroic  candidate is bismuth ferrite (BiFeO 3 – BFO)  which boasts a plethora \nof appealing properties including multiferroic31, 32, photovoltaic33-35, piezoelectric36, domain -\nwall conduction37, 38 and optoelectronic responses39. In this context, t he observation of \nspherical  and transitional topologi es such as skyrmions, polar vortex arrays, merons and bubble \ndomains in BFO would undoubtedly have wide -reaching implications , both fundamentally and \npractically40. Although  epitaxial BFO heterostructures have been engineered  to demonstrate  \nwritable vortex cores41, cen ter convergent and quad -domain structures42, 43 or self-assembled \nflux closure  arrays44, the observation of topological solitons is still to be achieved . A natural \nquestion thus arises: how can one craft the hitherto evasive solitons  in BFO?  \nTo address  this, one must consider the requirements for the creation of such a  \npolarization configuration  in a polar material45. The challenge  is to engineer a ferroelectric on \nthe brink, where continuous polarization rotation is achieved  without pushing the system to \nform symmetry breaking Ising domain walls46. \nHere , we report the deterministic  stabilization of the complex topological phases  in \nepitaxial BiFeO 3 (BFO)–SrTiO 3 (STO) superlattices  fabricated on (001) -oriented  LaAlO 3 \n(LAO) substrates  using pulsed laser deposition . The superlattices display sharp interfaces as \nrevealed by Cs -corrected high -angle annular dark -field scanning transmission electron \nmicroscop y (HAADF -STEM) and atomic resolution energy dispersive spectroscopy (EDS) \nmapping . Piezoresponse force microscopy (PFM) reveal s a diverse range of non-trivial \nspherical domains , hinting  towards the coexistence of a zoo of ferroelectric solitons . Atomic -\nscale p olarization displacement mapping of planar HAADF -STEM specimens was thus carried \nout. The se measurements  confirmed  the existence of a range of topological structures, from \nbubbles to bimerons to possibly polar skyrmion s. These latter  defects  show both center -\nconvergent and center -divergent polar ization profiles w ith sizes as small as 3 nm . The \nexperimental results are verified with first-principle s-based effective Hamiltonian predictions \nand phase -field simulations , which find  that the origin  for the formation of these  non-trivial \ntopologies is  the competing electrical and mechanical boundary conditions . Critically , the \nsimulations find that the co -existing topological phases can each individually posses s either \nfractional or integer  topological charge . The discovery of such ultrafine topologically  protected \nstates in multiferroic BFO unlocks an uncharted  platform with new degrees of fr eedom, i.e., \ncontrol by both electric and magnetic fields.     \n \n   \n Figure 1A graphically depicts the basis of our approach. In \nferroelectric/dielectric/ferroelectric structures or ferroelectric/dielectric superlattices, the \nincorporation of a dielectric layer (often STO ) leads to the accumulation of bound charges at \nthe interfaces thus increasing the system’s free energy. The out -of-plane polarization is thus \nforced to curl in an in -plane direction to reduce the energy and can thus stabilize ferroelectric \nsolitons. In PbTiO 3 (PTO )-based systems, to achieve polarization curling, the aim is to force \nthe naturally  orient ed out-of-plane  pointing polarization to tilt towards the film plane. In BFO, \non the other hand, we use the opposite principle.  This is because bulk BFO crystallizes in \nrhombohedral symmetry with polarization along [111] direction . We thus need  to impose a \ncompressive strain to push the polarization vector towards the surface normal.  For this  reason , \nwe use LAO as our substrate, enabling us to impose an in -plane compressive stress that \nstabilizes an out -of-plane polarization towards  the [001] direction (i.e., the so -called T -phase). \nThis is counterbalanced by a depolarization field created  by the insertion of a STO spacer. The \nspacer has two key effects - (i) it breaks polarization and structural continuity and (ii) it provides \nthe mechanical compatibility at the interface to allow the polarization to curl47.  \nAlthough the principle seems simple, growing ultra -smooth layers of epitaxially \nstrained BFO to several tens of nm is not trivial. This is where our unique pulsed laser \ndeposition (PLD) chamber system – with a large substrate to target separation ( ~10 cm) – \ncomes  into play. First, this large distance results in low incident flux , enabling controlled ultra -\nslow layer -by-layer growth of BFO under compressive strain ( additional data available on \nrequest ). Second, the low flux at high temperatures can achieve self-regulated growth of \ntetragonal like (T -like) BFO to thicknesses up to 60 nm with no mixed phase formation48. This \nability to fabricate superlattices w herein the “long -range” in -plane compressive strain can be \nsustained ( i.e., canting the polarization towards  the [001] direction  of BFO ) – somewhat \nsurprisingly as we will see later – is a key first step towards the realization of curling \npolarization textures in thi s material.  \nThe next step is to identify the optimal superlattice design. Even a slight deviation from  \nthe optimized thickness induces relaxation mechanisms49.  The thickness of the BFO layer is \npivotal:  each layer must be thin enough  to maintain the imposed “macroscopic” strain  and \ndipolar coupling at the inter face50, and couple across the STO  spacers , but not to be locally  \ninfluenced  by intrinsic size effects (Fig. 1A). By carrying out a  detailed study on the influence \nof individual layer thickness  (additional data available on request ) we found the optimal  \nconfiguration for creating topological textures  to be  (BFO 7/STO 4)10.    \n \n   \n Structural  and chemical characterization of the above system is summarized in Fig. 1A. \nA representative cross -sectional STEM image within the heterostructure  (BFO 7/STO 4)10 and \nits corresponding atomic resolution  EDS map (see below) show atomically and chemically \nsharp  heterointerfaces with no interdiffusion  (Comprehensive HAADF - STEM  and EDS \nanalysis of the complete structure can be provided  on request) ). Since HAADF is sensitive to \nthe atomic number , the individual BFO and STO layers are  seen as  alternate bright  and dark \ncontrast  with a thickness of 7 and 4 unit -cells (u.c.) , respectively , in agreement  with the \nintended superlattice  design. The well-aligned atomic columns demonstrate  the high -quality  \nepitaxial configuration  with the absence of any defects. The EDS maps reveal distinct Ti \n(cyan ), Fe (green), Sr  (yellow) and Bi (red) atomic positions with no sign of interdiffusion \nconfirming  perfect coherent stacking of the BFO and STO  layers .  \nFigure 1B presents symmetrical XRD reciprocal space mapping (RSM) of a \n(BFO 7/STO 4)10 superlattice near the 002 reflection  of film and substate , revealing various peaks \nalong the out -of-plane direction (Q z). In addition to the 002 LAO substrate peak, the next \nbrightest spot corresponds to the out -of-plane periodicity of the superlattice with an average \nout-of-plane parameter of 4.00 ± 0.01 Å. Simulations of the diffraction pattern using a custom -\nmade MATLAB program51 show that the data are consistent with out -of-plane lattice \nparameters of BFO  (STO) of 4.07 Å (3.91 Å) (additional data available on request ). The sharp \ninterfaces between the BFO and STO layers lead to additional superlattice reflections such as \nthe SL -1, corresponding to the 11 u.c. repeat length . A pertinent feature of this dataset is the \nbreadth in the horizontal direction of the main film peak , which could arise from increased \nlocal mosa icity, defects, or strain gradients . Since we do not observe chemical defects or \ndislocations from STEM, we rule out the first two possible effects, implying that significant \nstrain gradients  exist within the superlattice structure. The origins of these strain gradients will \nbecome clear in Fig. 2. Finally, a peak with narrow horizontal width is detected at lower Q z \nvalues  which is  indexed as tetragonal -like T -BFO, likely stabilized in the layers closer to the \nsubstrate.  The average in-plane lattice parameter of the (BFO 7/STO 4)10 is 3.92 ± 0.02 Å, close \nto the value of bulk STO  (additional data available on request ). Thus,  the c/a ratio of the entire \nstack is ~1.04  implying that the BFO is not T -like, but more strained R -like due to  some degree \nof strain relaxation. We return to this key observation later whe n we discuss first -principle s-\nbased  computations.  Local measurements of the lattice parameters  using STEM are discussed \nlater.     \n \n   \n The critical role played by the LAO substrate now becomes  apparent. First, the imposed \nin-plane compressive strain favors the BFO layer’s polarization to be aligned predominantly \nout of plane. But at the same time, the STO spacer interrupts the polar continuity, inducing \nstrong polarization gradients near the interfaces to minimize the overall increased electrostatic \nenergy costs . These combined effects drive BFO into a state of absolute polar frustration: it can \nhave neither long range out -of-plane polarization, which is prohibited by the co st of \ndepolarization, nor full in-plane tilting, this being restricted by the underlying substrate.  \nNext,  the ferroelectric domain structure was imaged using high-resolution dual \namplitude resonance tracking  (DART)  PFM . The amplitude and phase PFM images (Fig . 1C) \ndemonstrate intricate nanoscale domain configurations .  Interestingly, we observe  topological \nfeatures with remarkable semblance to those found by Milde et al.  for magnetic skyrmions in \nFe1−xCoxSi (Ref. 52). A magnified scan (Fig . 1D) reveals the occurrence of sub-20 nm \nnanoscale domains , labelled (1) and outlined  by white dashed rectangles.  These nanodomains \nshow blurry amplitude contrast and a faint upward domain phase of the domain wall. \nPreviously, such features were ascribed to either bubble domains7, 53 or skyrmions8, 10. \nFurthermore, transitional topological defects such as bimeron s13  and disclination s13 (depicted \nas (2) and (3) respectively in Figs. 1, D and E, are also identified . Since the resolution of PFM \nlimits our ability to image any topological feature smaller th an ~20 nm, this technique alone \ncannot discern bubbles or skyrmions.  The main feature distinguishing  polar  skyrmions  from \nbubbles is an additional polarization vortex along the circumferential axis , which cannot be \nimaged in PFM . Hence, we next characterize these 3-dimensional structures at the atomic scale \nusing Cs-STEM . \nFigure 2A show the planar -sectional Cs -STEM of epitaxial (BFO 7/STO 4)10 \nsuperlattices which reveal arrays of circular feature  with typical diameters of ~3.5 nm, whose \nsize distribution is shown in Fig. 2 B. The application of a d isplacement vector -mapping \nalgorithm on both the cross -sectional and planar view HAADF -STEM images  provides  direct \nvisualization  of atomic -scale polarization  displacement  within  the superlattices  (Figs. 2, D and \nE) (additional data available on request ). The planar view HAADF -STEM image and \ncorresponding vector displacement mapping demonstrated a variety  of topological states. In \nFigs. 2D, Ⅰ and Ⅱ, we confirm a coexistence of in -plane center -divergent and center -\nconvergent polar textures, which take the form of circular domains  in low -magnification STEM \nimage s. The bimeron with both  center -divergent and anti -vortex  polar structures  is identified  \n(Fig. 2 D, Ⅲ), which appear s as a fusion of two bubble domain s in STEM image.  Polar regions    \n \n   \n with anti -parallel (up vs. down) polarizations are found in the cross -sectional HAADF -STEM \nvector displacement mapping (Fig. 2 E, Ⅰ), with polarization curling occurring near the \nBFO/STO interfaces. In Fig. 2 E, Ⅱ, we reveal a meron -type region  with a core consisting of \nout-of-plane polar ization, and as one moves away from the center , the polar ization  gradually \nchange s into in -plane di rections . The combination of these  out-of-plane and in -plane polar \nconfigurations  shows that we have three -dimensional polar solitons in our (BFO 7/STO 4)10 \nsuperlattices .   \nWhile this  observation is qualitatively  similar to the skyrmion s observed in  \n(PTO 16/STO 16)8 superlattices10, our system shows two important distinctions . First,  the objects \nin our (BFO 7/STO 4)10 are not completely confined within the BFO layers : their polar order also \npartially exists in the nominally paraelectric STO spacer.  We attribute this effect  to the  strong \nelectrostatic coupling between the thin paraelectric and ferroelectric layers  which  causes dipole \nrotation in the confined STO54. Second,  the smaller thickness of BFO  ferroelectric layer  (7 u.c.) \nleads to significantly  small er characteristic size of the topological objects (~3.5 nm)  (Fig. 2 B) \nas compared to the size observed in PTO  (16 u.c.) /STO superlattices ( 8 nm)10. Figure 2C show s \nthe statistics of c/a ratios of BFO and STO . The lattice parameters of unit cells were calculated \nby fitting each atom site by a spherical Gaussian using an algorithm in MATLAB . The average \nc/a ratio (1.0540) implies that the BFO is not T -like, fits with but is slightly larger than the \nvalue of the previously reported c/a ratio of R -phase BFO55. \nThe discrepancy between the size and type of topological features observed  by STEM \nvis-à-vis PFM could be attributed to various factors : (i) We have identified a whole range of \ndifferent topological structures, meaning that one should not be surprised that different \ntechniques see different topological objects . (ii) In PFM, t he trailing field effect of the tip18 \nwhere even the slight pressure and electric field enlarge the topological state, and/or  (iii) \nremoval/thinning of the substrate for making planar view STEM sample, thereby dramatically \naffecting the boundary conditions.  Arguably, case (iii) is less likely , as our previous work on \nbubbles does not find a difference in the bubble size for free -standing vs. constrained samples56.  \nMoreover,  no difference was observed between PFM and STEM for skyrmions in the PTO \nsystem. We have, o n the other hand , shown that bubbles are extremely sensitive  to applied \nscanning probe microscopy ( SPM ) pressure and scanning field7, 53. \nTo gain further insight into the topological polarization configuration and formation \nmechanism of the polar states observed experimentally in our superlattices, phase -field    \n \n   \n simulations were performed (additional data available on request ). Figure 3 shows the typical \npolar structures and topological feature s of (BFO 7/STO 4)8 superlattices. As seen in Fig. 3A, the \nin-plane polarizations show donut -like circular patterns, similar to the polar skyrmion  structure \nobserved in the PTO/STO system10. Meanwhile, the detaile d polar mapping demonstrates that \nthe observed structure in this system  is distinctly different from the polar skyrmions in the \nPTO/STO superlattice. As shown in Fig. 3B, the in -plane polarization forms a combination of \ncenter convergent dipole configurati on with anti -vortex like structure, which is the \ncharacteristic of a “bimeron” structure, consistent with the experimental observation by STEM \n(Fig. 2D). The topological feature is characterized by calculating the Pontryagin density, 𝑃𝑑=\n𝑃⃗ .(𝜕𝑃⃗ \n𝜕𝑥×𝜕𝑃⃗ \n𝜕𝑦). It can be observed that the bubble -like structure shows a circular distribution of \nthe Pontryagin density (Fig. 3C).  The magnified view of the Pontryagin density distribution \nfor one bimeron demonstrates a circular structure (Fig. 3D), while the surface integration of \nthe circle gives a topological structure of -1, confirming that such structures are  more like  \nbimerons. Furthermore, both center divergent and center convergent type structures are \nobserved in this system ( additional data on request ), in good agreement with the experimental \nobservation. The cross -section view is further plotted, as given in Fig s. 3, E and F . A 180o \ndomain wall like structure is observed, with alternating positive and negative out -of-plane \npolarization (Fig. 3E). The polarization vector is plotted, as demonstrated in Fig. 3F, showing \nthe formation of polar vortex like structure. While it is interesting to note that in the  BFO layer, \nboth R-like and T-like regions are identified, in agreement with the experimental observations.  \n Further computational approaches also give strong support for the formation of \ntopological objects under moderate compressive strain in our heterostructures, as explained \nnext.  A first -principle s-based effective Hamiltonian57, 58 was used to study thin film BFO. A \n16 u.c. thick (001) oriented BFO film with an initial 109 ° domain structure was placed under \nmechanical boundary condition of epitaxial substrate strain in the compressive range varying \nup to -5% with an open -circuit -like electrical boundary condition. Multidomain structures are \nknown to be p referred over monodoma ins under open circuit like electrical boundary \ncondition s16. The specific multidomain structure of  109° domain type  was chosen as it has been  \nthe most experimentally observed domain wall  in bulk BiFeO 3 and has the lowest domain wall \nenergy as predicted by first-principle s-based  calculations59-61. The film is modelled by a \nsupercell periodic in the x and y directions. Monte Carlo simulations were run for 50 ,000 steps \nat 10  K for each substrate strain. It was observed that the initial 109° domains changed into a    \n \n   \n topologically non -trivial structure (Fig. 4A), that had a vortex in the x-z plane with \nconvergent/divergent domain walls in the x-y plane located at the circumference of the vortices.  \nThese features seen in the computed polar structure agree  with HAADF -STEM \nexperiments.  Namely, effective Hamiltonian simulations reproduce the zig -zag vortex pattern \nobserved in cross -sectional HAADF -STEM images (pattern I in Fig. 2E) while \nconvergent/divergent lines at the BFO interfaces in Fig. 4A support the formation of center \nconvergent/divergent points observed in Fig. 2D. The vortex occurred in a moderate \ncompressive strain region of [ -3%, -2%] beyond which it is destroyed. T he c/a value for the \nsubstrate strains that resulted in a vortex lie s between 1.077 to 1.050  at the misfit strains of         \n-3% and -2%, respectively. To analy ze the topology of the resultant dipolar structure, we have \ncalculated the distribution of the Pontryagin’s charge density ( 𝜌𝑆𝑘=𝒏.(𝜕𝑥𝒏×𝜕𝑦𝒏),where n \ndenotes normalized electric dipoles)  in the x-z plane. The distribution of the resulting density \n(colored  plaquettes) and the normalized dipoles (arrows) are shown in Fig. 4B where red and \nblue circles indicate examples of meron and antimeron lines. These lines are found to carry a \nfractional  Skyrmion number. Particularly, the meron indicated by a red circle in Fig. 4B has a \nSkyrmion number of 0.54 while the anti -merons outlined in blue carry charges of -0.43 and      \n-0.5. Notably, despite the presence of both positively and negatively charged merons , the \nintegral Skyrmion number  (∫(𝑑𝑥 𝑑𝑦 𝜌𝑆𝑘) ) over the simulation supercell was found to be equal \nto one.  \n \nConcl usion  \nIn summary, we have crafted  polar soliton structures in epitaxial multiferroic BFO –STO \nsuperlattices. This demonstration of these previously elusive  topological states in this material \nsystem  is anticipated to have  far reaching implications on the topological landscape of \ntopological polar/spin textures  in multiferroics.  Given that the  various  Dzyaloshinskii -Moriya \ninteraction s – which govern the weak ferromagnetic moment and the spin cycloid in BFO  \n(Refs. 58, 62, 63 ) – are driven by strain and local symmetry, one can anticipate that solitons  in \nBFO could  reveal enhanced local ferromagnetic moments, depressed magnetic transition \ntemperatures, and/or enhanced electrical conductivity . Further, similar to how domain walls in \nmaterials such as TbMnO 3 can harbor exotic electronic and magnetic states64, our solitons may \nindeed constitute a fundamentally different multiferroic phase of BFO.  These findings are \nclearly  just the tip of the iceberg; we hope that our results will motivate practitioners and    \n \n   \n theorists in the field to dig deeper into these superlattice systems. Future work will require \nelucidating the specific role of the local symmetry changes within the solitons and how it \ninfluences the local polarization dynamics, weak ferromagnetic moment, optical  behavior,  and \ntransport responses - all functionalities that can be used in next generation nanoscale devices.  \nAcknowledgments  \n \nThe research at University of New South Wales (UNSW) is supported by DARPA Grant No. \nHR0011727183 -D18AP00010 (TEE Program), partially supported by the Australian Research \nCouncil Centre of Excellence in Future Low -Energy Electronics Technologies (project n umber \nCE170100039) and funded by the Australian Government. This work is also supported by the \nNational Natural Science Foundation of China (grant No. 92166104 & 12125407), the Zhejiang \nProvincial Natural Science Foundation (LD21E020002), the Joint Funds o f the National \nNatural Science Foundation of China (U21A2067), the National Key Research and \nDevelopment Program of China (No. 2021YFA1500800). ZH also gratefully acknowledge a \nstart-up grant from Zhejiang University. Q. Z. acknowledges the support of a Wo men in \nFLEET Fellowship. The research at University of Arkansas is also supported by the Vannevar \nBush Faculty Fellowship (VBFF) Grant No. N00014 -20-1-2834 from the Department of \nDefense and Arkansas Research Alliance, ARO Grant No. W911NF -21-1-0113 and AR O grant \nnumber W911NF -21-2-0162 (MURI -ETHOS). We also acknowledge the Arkansas High \nPerformance Computing Center (AHPCC).  \n \nBibliography  \n \n1. Y. L. Tang, Y. L. Zhu, X. L. Ma, A. Y. Borisevich, A. N. Morozovska , E. A. Eliseev, \nW. Y. Wang, Y. J. Wang, Y. 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(left) \nLAO substrate induces a compressive epitaxial strain favoring a tetragonal phase. (right) The \nintroduction of STO has two effects: (1) a polarization mismatch is introduced at BFO/STO interface \ncausing STO to become polar. (2) the polari zation discontinuity enhances the depolarization field \ncausing polarization to curl. ( B) Symmetrical x-ray diffraction r eciprocal space map near the (002) \nreflection, showing  superlattice peaks ( C) PFM Amplitude and PFM phase images. Scale bar s: 200 nm \n(D) Zoomed in PFM amplitude and phase images depicting topological structures such as (1) skyrmion \n(2) bimeron and (3) disclination.  Scale bar s: 100 nm . \n  \n   \n \n   \n  \n \nFigure 2.  Observation of ferroelectric solitons  in BFO -STO superlattices.  (A) Low-magnification \nplanar -view HAADF -STEM image of the (BFO 7/STO 4)10 superlattice. Scale bar, 5 nm. ( B) Histogram \nof size distribution of circular features. ( C) The statistics of c/a ratios of STO and BFO extracted from \nthe cross -sectional HAADF -STEM images . (D) Enlarged planar -view STEM -HAADF image and \ncorresponding polar vectors, showing center -divergent, center -convergent and bimeron polar textures. \nScale bar: 1 nm. ( E) Enlarged cross -sectional STEM -HAADF images and corresponding polar vectors, \nshowing a domain with anti -parallel (up –down) polarization and a trapezoidal domain with the \nconvergent polar configuration. Scale bar: 1 nm. ( F) Schematics of a planar -view HAADF -STEM \nimage (30 nm × 30 nm) overlaid with a cross -sectional HAADF -STEM image,  displaying an overview \nof the (BFO 7/STO 4)10 superlattice.  \n  \n   \n \n   \n  \n \nFigure 3.  Polar structures and Pontryagin density of (BFO 7/STO 4)8 superlattices as calculated \nfrom phase -field simulations.  (A) Planar view of the in -plane polarization magnitude. ( B) Magnified \nview of the in -plane polar vector plot, showing a bimeron structure. ( C) Corresponding plot of the \nPontryagin density. ( D) Magnified view of the Pontryagin density of the bimeron structure. Surface \nintegration of the Pontryagin density for the bi meron gives a topological charge of -1. (E) Cross -section \nview of the out -of-plane polarization, showing alternating positive and negative polarization. ( F) \nMagnified view of the out -of-plane polar vector plot, showing a vortex -like structure, with a mixtu re of \nR-like and T-like polar regions.  \n \n   \n \n   \n Figure 4. (A) Polar structure  (arrows)  of BiFeO 3 thin film supercell after relaxation from  a 109° domain \nstructure under -2.9% epitaxial strain at 10  K. The x -z plane is shown along with the top and bottom \nmost layer s of the film . (B) The distribution of the Pontryagin’s charge density (colored plaquettes) \nalong with  the normalized dipoles (arrows)  for the polar distribution shown in A.  \n \n"    },    {        "title": "1412.0097v1.Gyrotropic_nihility_state_in_a_composite_ferrite_semiconductor_structure.pdf",        "content": "Gyrotropic-nihility state in\na composite ferrite-semiconductor structure\nVladimir R Tuz1;2\n1Institute of Radio Astronomy of National Academy of Sciences of Ukraine, 4,\nChervonopraporna Street, Kharkiv 61002, Ukraine\n2School of Radio Physics, Karazin Kharkiv National University, 4, Svobody\nSquare, Kharkiv 61022, Ukraine\nE-mail: tvr@rian.kharkov.ua\nAbstract. Characteristics of the gyrotropic-nihility state are studied in a\n\fnely-strati\fed ferrite-semiconductor structure, which is under an action of an\nexternal static magnetic \feld. Investigations are carried out with the assistance\nof the e\u000bective medium theory, according to which the studied structure is\napproximated as a uniform gyroelectromagnetic medium. The theory of the\ngyrotropic-nihility state is developed in terms of the eigenwaves propagation in\nsuch gyroelectromagnetic medium. The frequency and angular dependencies of\nthe transmittance, re\rectance and absorption coe\u000ecient are presented. It turns\nout that in the frequency band around the frequency of gyrotropic-nihility state\nthe studied structure appears to be matched to free space with both the refractive\nindex and the wave impedance which results in its high transmittance almost in\nthe entire range of angles of the electromagnetic wave incidence.\nKeywords : electromagnetic theory, magneto-optical materials, e\u000bective medium\ntheory, metamaterials.\nPACS numbers: 42.25.Bs, 75.70.Ak, 78.20.Ci, 78.20.Ls, 78.67.Pt\nSubmitted to: J. Phys.: Condens. MatterarXiv:1412.0097v1  [physics.optics]  29 Nov 2014Gyrotropic-nihility state 2\n1. Introduction\nThe conception of \\nihility\" media [1] was \frstly introduced into electromagnetics in\nregard to the theory of metamaterials. It determines a distinctive exotic state of a\nhypothetical lossless medium whose material parameters happen to be zero quantities.\nIt corresponds to the ful\flment of the constitutive relations where the electric and\nmagnetic \rux densities are both equal to zero, ~D= 0 and~B= 0. Thereby, the nihility\nmedium exhibits a zero refractive index, so, there are r\u0002~E= 0 andr\u0002~H= 0,\nand the wave propagation in such extreme-parameter medium becomes forbidden in\nthe absence of sources therein. It means that the directionality of the phase velocity\nin relation to the wavevector is a non-issue for this medium. Although the nihility\nmedia determined in such a way are unachievable in nature, they can be approximately\nsimulated at a speci\fed frequency.\nAs such approximation the theory of \\epsilon-near-zero\" and \\mu-near-zero\"\nmaterials [2, 3, 4] can be considered since it is fundamentally identical with the\nconception of nihility media. In this theory the Drude medium model is consistently\nused to simulate a certain dispersive medium in which real parts of both complex\npermittivity and complex permeability simultaneously acquire zero at a speci\fed\nfrequency, and the refractive index of the resulting medium appears to be zero value,\ntoo. It means that while the zero refractive index is not matched to free space, the\nwave impedance is matched one. The most intriguing properties of such medium is\nthe nearly zero phase progression of propagating waves, which may be of interest in\nthe problems of transformation optics [5], and can be employed for the e\u000ecient energy\ntunneling [6] and for the design of the next generation of waveguides [7]. Furthermore,\na strong material dispersion leads to possibility of medium properties changing form\nbeing opaque \\metals\" to transparent dielectrics with a small variation of the material\nparameters around the epsilon-near-zero and mu-near-zero states.\nHowever, further mixing together electric and magnetic responses of the medium,\nfor example, by utilizing magneto-optic (gyrotropic) and/or bi-anisotropic e\u000bects,\nallows one to drastically change the peculiarities of the wave propagation in nihility\nmedia. For instance, in [8] it has been proposed a possible way to achieve a nihility\nstate in a bi-isotropic medium using canonical chiral wire particles. The dispersive\ncharacteristics of the e\u000bective constitutive parameters of the medium are calculated\non the basis of the Maxwell-Garnett mixing rule. It is found that, at a speci\fed\nfrequency, the real parts of both complex e\u000bective permittivity and complex e\u000bective\npermeability happen to be close to zero while the chirality parameter is maintained at\na \fnite value. This distinct extreme-parameter state is referred to as \\chiral-nihility\",\nand it turns out that, in contrast to the conventional nihility medium, in the chiral-\nnihility both the refractive index and the wave impedance appear to be matched to\nfree space, and the directionality of the phase velocity relative to the wavevector would\nnot be a non-issue any more. So, the wave propagation is allowed in chiral-nihility\nmedia, and, in particular, this wave propagation can be described in the term of\ntwo orthogonal circularly polarized eigenstates. It means that in this medium there\nare two eigenwaves with right- and left-circularly polarized states, and, noteworthy,\none of these eigenwaves experiences a backward propagation z. It results in some\nzWe consider a backward wave as a wave in which the direction of the Poynting vector is opposite\nto that of its phase velocity. We assume that the direction of the energy \rux is coincident with the\ndirection of exponential decay of the wave's \feld, caused by dissipation and absorption of the wave\nenergy in the medium [9, 10].Gyrotropic-nihility state 3\nexotic characteristics (e.g. wave tunneling and rejection) in the waves interaction\nwith a single layer and a multilayer system consisting of such chiral-nihility media\n[11, 12, 13, 14].\nMore recently, the extreme-parameter nihility state has been also found for\nthe other fundamental class of reciprocal bi-anisotropic media, namely, for omega\nmaterials [15]. The omega material is a reciprocal bi-anisotropic medium characterized\nwith an antisymmetric magnetoelectric dyad (e.g., it is a composite material formed\nby inserting conductive \n-shaped particles into a homogeneous dielectric host). In\ncontrast to chiral media where the chirality breaks the symmetry of the propagation\nconstants of the circularly polarized eigenwaves while the wave impedances are not\na\u000bected, in omega materials the properties are dual: the magneto-electric coupling\nbreaks the symmetry of the wave impedances while the propagation constants remain\nsymmetric. In this case, too, the extreme \\omega-nihility\" state of the omega\nmaterial appears at a speci\fed frequency, when real parts of both complex permittivity\nand complex permeability of the medium happen to be close to zero, and the\nmagnetoelectric parameter alone de\fnes the material response. Among other e\u000bects,\nthe omega-nihility material provides an extreme asymmetry in re\rection from a\nmaterial slab: the re\rection coe\u000ecients from the two opposite sides di\u000ber by sign, while\nthe transmission coe\u000ecient is symmetric as in any conventional reciprocal material\nslab.\nIt is a common knowledge that, besides chiral media, the circularly polarized\neigenstates are also inherent to circularly birefringent (gyrotropic) materials (e.g.\nplasmas, ferrites and semiconductors) in the presence of an external static electric\nor magnetic \feld, when this \feld is biased to the specimen in the longitudinal\ngeometry relative to the direction of the wave propagation (Faraday con\fguration).\nSuch gyrotropic media are characterized by the permittivity or permeability tensor\nwith non-zero o\u000b-diagonal elements (gyrotropic parameters). It is no wonder, that\nproperly combining together gyroelectric and gyromagnetic materials into a uniform\ngyroelectromagnetic structure [16] allows one to reach a \\gyrotropic-nihility\" state at a\nspeci\fed frequency. In particular, as it has been shown in [17], in a composite \fnely-\nstrati\fed ferrite-semiconductor structure the conditions of gyrotropic-nihility state\nare valid in the microwave band near the frequencies of ferromagnetic and plasma\nresonances. In this case real parts of diagonal elements of both complex e\u000bective\npermittivity and complex e\u000bective permeability tensors of such arti\fcial medium\nsimultaneously acquire zero, while the o\u000b-diagonal elements appear to be non-zero\nquantities. It is revealed that in this structure, in the certain frequency band, the\nbackward propagation takes a place for one of the circularly polarized eigenwaves\nwhich can lead to some unusual features of the system and provides an enhanced\npolarization rotation, impedance matching and complete light transmission.\nThe objective of this paper is a formalization of the theory of the gyrotropic-\nnihility state in terms of the eigenwaves propagation in such composite \fnely-strati\fed\nferrite-semiconductor structure.\n2. Constituents of a composite ferrite-semiconductor structure\nThe essence of any nihility medium is extreme characteristics of its constitutive\nparameters. These extreme characteristics appear in the small region near singular\npoints of dispersion curves where real parts of both complex permittivity and complex\npermeability simultaneously make transitions from negative to positive values or viceGyrotropic-nihility state 4\nversa. It is worth noting that media for which these transitions separately exist for\npermittivity or permeability may be directly found in nature. A well-known example\nis an electron gas, in which, due to the conduction current created by the drift of\nfree electric charges, may e\u000eciently interact with radiation as a continuous medium\ncharacterized by a Drude dispersion model, near the plasma frequency the real part of\npermittivity appears to be close to zero [18]. At infrared and optical frequencies some\nlow loss noble metals (e.g. silver, gold), semiconductors (e.g. indium antimonide),\nand some polar dielectrics (e.g. silicon carbide) may behave an epsilon-near-zero\nstate near their plasmas frequencies. At the same time, a mu-near-zero state is\ninherent to magnetic materials (e.g. ferromagnets, ferrimagnets) in the vicinity of\na gyromagnetic resonance [19]. Thereby, although any nihility medium does not exist\nin nature in its pure form, there is a possibility of obtaining nihility arti\fcially by\nmixing together materials which manifest epsilon-near-zero and mu-near-zero states\nin the same frequency range.\nIn particular, such an opportunity exists in the microwave part of spectrum where\na low temperature magnetized plasma can be obtained whose dispersion properties\nare simultaneously de\fned by tensors of e\u000bective permittivity ^ \"effand e\u000bective\npermeability ^ \u0016eff. Its constituents are micron ferrite grains (e.g. yttrium iron\ngarnet, YIG) admixed to the magnetized electron-ion plasma [20]. The dispersion\nof permeability is caused by the high frequency magnetization of the grain subsystem\nand is important in the vicinity of the frequency of a ferromagnetic resonance, which\ncoincides with the electron cyclotron frequency. Since the resulting medium is under\nan action of the static magnetic \feld, an e\u000bect of gyrotropy appears. We predict that\nthrough careful adjustment of the ratio of the magnetic grain fraction in the plasma\none can reach epsilon-near-zero and mu-near-zero states simultaneously at the same\nfrequency, and, thus, achieve a gyrotropic-nihility.\nAs an alternative, a layered heterostructure can be considered [17, 21, 22]. It is\nconstructed by periodically arranging into a certain uniform system of gyroelectric\n(semiconductor) and gyromagnetic (ferrite) layers, provided that these layers are\noptically thin and their underlying materials correspondingly have epsilon-near-zero\nand mu-near-zero states in the same frequency range. If the constitutive layers\nas well as the period of the \fnal structure are optically thin (i.e. it is a \fnely-\nstrati\fed one) the e\u000bective medium theory can be consistently applied in order to\nidentify its homogenized material parameters. It results in the consideration of the\n\fnely-strati\fed structure as a homogeneous gyroelectromagnetic medium described\nby tensors of e\u000bective permittivity ^ \"effand e\u000bective permeability ^ \u0016effwhich possess\nsome dispersion characteristics.\nThereby, further in this paper we consider a stack of Nidentical double-layer\nslabs (unit cells) periodically arranged along the z-axis (\fgure 1). Each unit cell is\nconstructed by juxtaposition together of ferrite (with constitutive parameters \"1, ^\u00161)\nand semiconductor (with constitutive parameters ^ \"2,\u00162) layers with thicknesses d1\nandd2, respectively. The structure's period is L=d1+d2, and along the xandy\ndirections the system is in\fnite. We suppose that the structure is a \fnely-strati\fed\none, i.e. its characteristic dimensions d1,d2andLare all much smaller than the\nwavelength in the corresponding layer d1\u001c\u0015,d2\u001c\u0015, and period L\u001c\u0015(the long-\nwavelength limit). An external static magnetic \feld ~Mis directed along the z-axis.\nThe inputz\u00140 and output z\u0015NLhalf-spaces are homogeneous, isotropic and have\nconstitutive parameters \"0,\u00160.\nWe use common expressions for underlying constitutive parameters of normallyGyrotropic-nihility state 5\nFigure 1. (Color online) A composite \fnely-strati\fed ferrite-semiconductor\nstructure, which is under an action of an external static magnetic \feld.\nmagnetized ferrite and semiconductor layers with taking into account the losses. For\nferrite layers the permittivity and permeability are de\fned in the form [23, 24]:\n\"1=\"f; ^\u00161=0\n@\u0016T\n1i\u000b 0\n\u0000i\u000b \u0016T\n10\n0 0\u0016L\n11\nA; (1)\nwhere\u0016T\n1= 1 +\u001f0+ i\u001f00,\u001f0=!0!m[!2\n0\u0000!2(1\u0000b2)]D\u00001,\u001f00=!!mb[!2\n0+!2(1 +\nb2)]D\u00001,\u000b= \n0+ i\n00, \n0=!!m[!2\n0\u0000!2(1 +b2)]D\u00001, \n00= 2!2!0!mbD\u00001,\nD= [!2\n0\u0000!2(1+b2)]2+4!2\n0!2b2,!0is the Larmor frequency and bis a dimensionless\ndamping constant.\nFor semiconductor layers the permittivity and permeability are de\fned as follows\n[25]:\n^\"2=0\n@\"T\n2i\f0\n\u0000i\f \"T\n20\n0 0\"L\n21\nA; \u0016 2=\u0016s; (2)\nwhere\"T\n2=\"l\u0002\n1\u0000!2\np(!+ i\u0017)[!((!+ i\u0017)2\u0000!2\nc)]\u00001\u0003\n,\"L\n2=\"l\u0002\n1\u0000!2\np[!(!+ i\u0017)]\u00001\u0003\n,Gyrotropic-nihility state 6\n\f=\"l!2\np!c[!((!+ i\u0017)2\u0000!2\nc)]\u00001,\"lis the part of permittivity attributed to the\nlattice,!pis the plasma frequency, !cis the cyclotron frequency and \u0017is the electron\ncollision frequency in plasma.\nFigure 2. Frequency dependencies of (a) permeability and (b) permittivity of\nferrite and semiconductor layers, respectively. We use typical parameters for\nthese materials in the microwave region. For the ferrite layers, under saturation\nmagnetization of 2000 G, parameters are: !0=2\u0019= 4:2 GHz,!m=2\u0019= 8:2 GHz,\nb= 0:02,\"f= 5:5. For the semiconductor layers, parameters are: !p=2\u0019=\n4:5 GHz,!c=2\u0019= 4:0 GHz,\u0017=2\u0019= 0:05 GHz,\"l= 1:0,\u0016s= 1:0.\nThe frequency dependencies of the real and imaginary parts of the underlying\npermeability and permittivity calculated using equations (1) and (2) are presented in\n\fgure 2. Note that imaginary parts of \u0016T\n1,\u000band\"T\n2,\fare so close to each other that\nthe curves of their frequency dependencies coincide in the corresponding \fgures.\n3. Homogenized material parameters of a gyroelectromagnetic medium\nIn the long-wavelength limit, in order to reveal the gyrotropic-nihility state conditions\nwhich are attainable in the studied structure, the e\u000bective medium theory can be\nengaged [17, 26]. From the viewpoint of this theory, the periodic structure is\napproximately represented as an anisotropic (gyroelectromagnetic) uniform medium\nwhose optical axis is directed along the structure periodicity, and this medium is\ndescribed with some tensors of e\u000bective permittivity ^ \"effand e\u000bective permeability\n^\u0016effwhich should be retrieved.\nLet us consider a unit cell of the studied structure. It is made of two layers\n0<z <d 1andd1<z <L of dissimilar materials whose constitutive relations are as\nfollows:\n~D=\"1~E\n~B= ^\u00161~H\u001b\n0<z<d 1;~D= ^\"2~E\n~B=\u00162~H\u001b\nd1<z<L: (3)\nIn general form, in Cartesian coordinates, the system of Maxwell's equations for eachGyrotropic-nihility state 7\nlayer has a formx\nikyHz\u0000@zHy=\u0000ik0(^\"j~E)x;ikyEz\u0000@zEy= ik0(^\u0016j~H)x;\n@zHx\u0000ikxHz=\u0000ik0(^\"j~E)y; @ zEx\u0000ikxEz= ik0(^\u0016j~H)y;\nikxHy\u0000ikyHx=\u0000ik0(^\"j~E)z;ikxEy\u0000ikyEx= ik0(^\u0016j~H)z;(4)\nwhere@z=@=@z ;kx=k0sin\u00120cos'0andky=k0sin\u00120sin'0are tangential\ncomponents of the wavevector ~k;\u00120and'0are polar and azimuthal angles,\nrespectively, which de\fne the electromagnetic wave propagation direction; k0=!=c\nis the free-space wavenumber; j= 1;2; ^\"1and ^\u00162are the tensors with \"1and\u00162\non their main diagonal and zeros elsewhere (i.e., ^ \"1=\"1^I, ^\u00162=\u00162^I, where ^Iis\nthe identity tensor). From six components of the electromagnetic \feld ~Eand~H,\nonly four are independent. Thus the components EzandHzcan be eliminated from\nsystem (4), and derived a set of four \frst-order linear di\u000berential equations related to\nthe transversal \feld components inside a layer of the structure [27]. The obtained sets\nof equations can be abbreviated by using a matrix notation:\n@z~\t(z) = ik0A(z)~\t(z);0<z<L: (5)\nIn this equation, ~\t =fEx;Ey;Hx;Hyg0is a four-component column vector (here the\nprime symbol denotes the matrix transpose operator), while the 4 \u00024 matrix function\nA(z) is piecewise uniform as\nA(z) =\u001aA1; 0<z<d 1;\nA2; d 1<z<L;(6)\nwhere the matrices A1andA2correspond to ferrite and semiconductor layers,\nrespectively.\nSince the vector ~\t is known in the plane z= 0, equation (5) is related to the\nCauchy problem [28] whose solution is straightforward, because the matrix A(z) is\npiecewise uniform. Thus, the \feld components referred to boundaries of the double-\nlayer period of the structure are related as k\n~\t(L) =M2~\t(d1) =M2M1~\t(0) = M~\t(0) = exp[i k0A2d2] exp[ik0A1d1]~\t(0);(7)\nwhere MjandMare the transfer matrices of the corresponding layer ( j= 1;2) and\nthe period, respectively.\nSuppose that \rjis the eigenvalue of the corresponding matrix k0Aj(det[k0Aj\u0000\n\rjI] = 0), and Iis the 4\u00024 identity matrix. When j\rjjdj\u001c1 (i.e., both layers in the\nperiod are optically thin), the next long-wave approximation can be used [26]\nexp[ik0A2d2] exp[ik0A1d1]'I+ ik0A1d1+ ik0A2d2: (8)\nLet us now consider a single layer of e\u000bective permittivity ^ \"eff, e\u000bective\npermeability ^ \u0016effand thickness L. Quantity Aeffcan be de\fned in a way similar to\n(7):\n~\t(L) =Meff~\t(0) = exp[i k0AeffL]~\t(0): (9)\nxHence, the \feld vectors will be supposed to contain a time factor of the form exp( \u0000i!t).\nkThe series exp( X) =I+ \u00061\nm=11\nm!Xmconverges for square matrices X, i.e. function exp( X) is\nde\fned for all square matrices [28].Gyrotropic-nihility state 8\nProvided that \reffis the eigenvalue of the matrix k0Aeff(det[k0Aeff\u0000\reffI] =\n0) andj\reffjL\u001c1 (i.e., the entire composite layer is optically thin as well), the next\napproximation follows\nexp[ik0AeffL]'I+ ik0AeffL: (10)\nEquations (8) and (10) permit us to establish the following equivalence between\nbilayer and single layer:\nAeff=f1A1+f2A2; (11)\nwherefj=dj=L, and the elements of the matrices A1,A2andAeffone can \fnd in\nAppendix A.\nFrom equation (11), in the particular case of coincidence of the directions of both\nthe wave propagation and the static magnetic \feld bias ( \u00120= 0), the following simple\nexpressions for the e\u000bective constitutive parameters of the homogenized medium can\nbe obtained:\n\u0016T\neff=f1\u0016T\n1+f2\u00162; \u000b eff=f1\u000b;\n\"T\neff=f1\"1+f2\"T\n2; \f eff=f2\f:(12)\nThe e\u000bective constitutive parameters calculated according to equalities (12) are\ngiven in \fgure 3. One can see that there is a frequency range where real parts of\nboth\u0016T\neffand\"T\neffreach zero. It is signi\fcant that, by special adjusting ferrite\nand semiconductor types, the external static magnetic \feld strength and the ratio\nof the layers' thicknesses, it is possible to obtain the condition where real parts of\nboth\u0016T\neffand\"T\neffsimultaneously acquire zero at the same frequency. Note well\nthat at this frequency, the parameters \u000beffand\feffare non-zero and the medium\nlosses are relatively small [see, \fgures 3(b) and (c)]. We de\fne exactly this situation\nas a gyrotropic-nihility state. In the structure under consideration such gyrotropic-\nnihility state appears at a particular frequency fgn\u00194.94 GHz which is distinguished\nin \fgure 3(a) by an arrow. It is expectable that in the general case when the directions\nof the static magnetic \feld bias and the wave propagation do not coincide ( \u001206= 0),\nthe gyrotropic-nihility state can acquire some changes, but nevertheless it should\nstay distinguishable. We discuss these changes further through the eigenwaves and\nstructure's transmittance analysis.\n4. Eigenwaves and gyrotropic-nihility state\nEvidently, the formulation of the eigenvalue problem on the matrix Aeff(det[Aeff\u0000\n\u0011I] = 0), whose coe\u000ecients are de\fned as (11), yields us the characteristic equation\non the e\u000bective refractive index \u0011of the medium. Thus, we arrive to the following\nbiquadratic equation with respect to \u00112[18, 20]:\nA\u00114+B\u00112+C= 0; (13)\nwhereA= 1;B=\u00002(\"T\neff\u0016T\neff+\u000beff\feff) + (\"T\neff=\"L\neff+\u0016T\neff=\u0016L\neff)(k2\nx+k2\ny)=k2\n0\nandC= [(\u0016T\neff)2\u0000\u000b2\neff\u0000(k2\nx+k2\ny)\u0016T\neff=k2\n0\"L\neff][(\"T\neff)2\u0000\f2\neff\u0000(k2\nx+k2\ny)\"T\neff=k2\n0\u0016L\neff]\nare known functions of frequency.\nOne can see that regarding the eigenwaves propagation in an unbounded\ngyroelectromagnetic medium, the problem acquires the cylindrical symmetry which\nresults in the independence of solutions of biquadratic equation (13) on the azimuthal\nangle'0, since there is ( k2\nx+k2\ny)=k2\n0= sin2\u00120. Thereby these solutions are\n\u00112\n\u0006(!;\u0012 0) =\u0000B\u0006p\nB2\u00004AC\n2A=\"\u0006\u0016\u0006: (14)Gyrotropic-nihility state 9\nFigure 3. (Color online) (a) Two surfaces depict behaviors of real parts of\ne\u000bective permeability (purple surface) and e\u000bective permittivity (orange surface)\nversus frequency and the ratio of the layers' thicknesses. The blue and red\ncurves plotted on the surfaces show the mu-near-zero and epsilon-near-zero states,\nrespectively. Curves at the bottom are just projections and are given for an\nillustrative purpose. Dispersion curves of the tensor components of (b) e\u000bective\npermeability and (c) e\u000bective permittivity at particular structure parameters\n(d1= 0:05 mm,d2= 0:2 mm) for which the gyrotropic-nihility state exists. Other\nparameters of the ferrite and semiconductor layers are the same as in \fgure 2;\nL= 0:25 mm.Gyrotropic-nihility state 10\nObtained root branches describe two e\u000bective refractive indexes \u0011+and\u0011\u0000related to\ntwo distinct eigenwaves with propagation constants \r+and\r\u0000(\r\u0006=k0\u0011\u0006). Those\nare the ordinary (+) and extraordinary ( \u0000) waves, respectively, as it is accepted\nin the optics community, and in the general case of \u001206= 0, these eigenwaves are\nelliptically polarized ones. Remarkably that each eigenwave propagates through its\nown medium with distinctive constitutive parameters \"+,\u0016+and\"\u0000,\u0016\u0000for ordinary\nand extraordinary eigenwave, respectively.\nFor lossless medium the electromagnetic waves can propagate through a medium\nonly if the condition \u00112\n\u0006(!;\u0012 0)>0 holds. Nevertheless, equation (14) does not specify\na sign of the refractive index. So the sign of \u0011must be chosen providing that the\nenergy carrying by the wave goes away from the source. Here the situation is possible\nwhen directions of the Poynting vector ~S= (c=8\u0019) Ref~E\u0002~H\u0003gand the wavevector ~k\ndo not coincide and so-called backward propagation appears. This problem has been\ndiscussed before in many papers related to the double-negative (left-handed) materials\nincluding those addressed on the circularly birefringent (gyrotropic) media (e.g., see\n[9, 20, 29, 30, 31]), and so we omit the details here. Howbeit, in our case the medium\nlosses should be taken into consideration, and the signs of the real and imaginary parts\nof the complex refractive index \u0011\u0006=\u00110\n\u0006+ i\u001100\n\u0006can be determined from the solution of\nthe next equation [32]\n\u0000\n\u00110\n\u0006+ i\u001100\n\u0006\u00012=\u0000\n\"0\n\u0006+ i\"00\n\u0006\u0001\u0000\n\u00160\n\u0006+ i\u001600\n\u0006\u0001\n; (15)\nwhich is further reduced to the set of two equations obtained by deriving the real and\nimaginary parts from (15)\n(\u00110\n\u0006)2\u0000(\u001100\n\u0006)2=\"0\n\u0006\u00160\n\u0006\u0000\"00\n\u0006\u001600\n\u0006;2\u00110\n\u0006\u001100\n\u0006=\"0\n\u0006\u001600\n\u0006+\"00\n\u0006\u00160\n\u0006: (16)\nThus, in order to ensure an electromagnetic wave damping as the wave propagates\nthrough the medium, the imaginary parts of permittivity, permeability and refractive\nindex all must be positive quantities ( \"00\n\u0006>0,\u001600\n\u0006>0, and\u001100\n\u0006>0). From these\nconditions it follows that according to the second equation in (16) the sign of \u00110\n\u0006is\nde\fned by the signs and absolute values of \"0\n\u0006and\u00160\n\u0006, and in the particular case of\nthe double-negative medium ( \"0\n\u0006<0,\u00160\n\u0006<0) the real part of refractive index \u00110\n\u0006\nmust be determined as a negative quantity [29].\nKeeping this in mind, the root branches (14) of equation (13) are properly chosen\nand plotted in \fgure 4 for two di\u000berent values of the polar angle \u00120. Each plot consists\nof four dispersion curves, from which we distinguish a pair of e\u000bective refractive indexes\n\u0011+and\u0011\u0000having positive imaginary parts [see, \fgures 4(b) and (c)]. It can be seen\nthat the curves within this pair demonstrate drastically distinctive features. While\nthe dispersion curve of the refractive index \u0011\u0000related to the extraordinary eigenwave\nundergos a small monotonic increasing with \u00110\n\u0000>1 and\u001100\n\u0000\u00190, the dispersion curve\nof the refractive index \u0011+related to the ordinary eigenwave experiences considerable\nchanging in which \u00110\n+has consistently gone through negative values to positive ones\nand\u001100\n+acquires some maximum. Thus, based on characteristics of \u0011+, the whole\nfrequency range of interest can be divided into three speci\fc bands. The \frst frequency\nband is located between 2 GHz and 4 GHz where \u00110\n+is a positive quantity and \u001100\n+\nis very signi\fcant. In the second band from 4 GHz to 6 GHz, \u00110\n+is a negative value\nwhich eventually acquires a transition to positive one on the band boundaries while \u001100\n+\ndecreases as the frequency rises. And \fnally, in the third frequency band which starts\nfrom 6 GHz, \u0011+and\u0011\u0000become comparable quantities. Besides, one can see that inGyrotropic-nihility state 11\nthe second frequency band there is a particular frequency fgn\u00194:94 GHz, where the\ncondition\u00110\n\u0000=\u0000\u00110\n+holds which is related to the gyrotropic-nihility state. The latter\nsituation is marked out in the inset of \fgure 4(a) with circles. Before proceeding to this\nparticular state, we should note that when \u001206= 0 all the mentioned frequency bands\ncan be still readily distinguished. However the condition \u00110\n\u0000=\u0000\u00110\n+of the gyrotropic-\nnihility state shifts toward higher frequencies and begins to hold in a certain frequency\nband rather than a particular frequency. Exactly this peculiarity is marked out in the\ninset of \fgure 4(c) with ellipses.\nFigure 4. (Color online) Dispersion curves of the refractive indexes \u0011\u0006for (a),\n(b)\u00120= 0 deg and (c), (d) \u00120= 25 deg. A pair of \u0011+and\u0011\u0000is distinguished\nin each plot with bright lines for which conditions \u001100\n+\u00150 and\u001100\n\u0000\u00150 hold.\nCurves for which conditions \u001100\n+<0 and\u001100\n\u0000<0 hold are plotted with pale lines.\nParameters of the ferrite and semiconductor layers are the same as in \fgure 2;\nd1= 0:05 mm,d2= 0:2 mm.\nFortunately in the case when the electromagnetic waves propagation is parallel\nto the magnetic \feld bias ( \u00120= 0) equation (13) has simple analytical solutions\n\u00112\n\u0006(!;0) = (\u0016T\neff\u0006\u000beff)(\"T\neff\u0006\feff); (17)\nthat allows us to analyze in more details the conditions at which the gyrotropic-\nnihility state occurs. At the same time, we recall that when \u00110\n\u0006(!;0)>0,\u0011+and\n\u0011\u0000in equation (17) are related to two eigenwaves which propagate along the positiveGyrotropic-nihility state 12\ndirection of the z-axis with right- and left-circular polarizations, respectively {.\nFrom \fgures 3(b) and (c) one can conclude that in the whole considered frequency\nband the imaginary parts of diagonal and o\u000b-diagonal components of both e\u000bective\npermeability and e\u000bective permittivity tensors are close values (( \u0016T\neff)00\u0019\u000b00\neff,\n(\"T\neff)00\u0019\f00\neff), while the real parts of diagonal elements of both tensors are greater\nthan the real parts of the corresponding gyrotropic parameters (( \u0016T\neff)0> \u000b0\neff,\n(\"T\neff)0> \f0\neff). Since all imaginary parts of the constitutive parameters must be\npositive quantities, the sign and absolute values of their real parts obviously de\fne\nthe sign of the refractive indexes related to the ordinary and extraordinary eigenwaves.\nThus, in accordance with the mentioned above characteristics of the complex\ne\u000bective constitutive parameters, it is appreciated that in the whole considered\nfrequency band the refractive index \u0011\u0000related to the extraordinary eigenwave is a\npositive quantity with the vanishingly small imaginary part. At once, the refractive\nindex\u0011+related to the ordinary eigenwave can be either positive or negative quantity\nin the corresponding frequency bands. In particular it becomes a negative quantity\nin the frequency bands where the next conditions hold: (i) ( \u0016T\neff)0<0, (\"T\neff)0<0;\n(ii)\u000b0\neff<0,\f0\neff<0 andj(\u0016T\neff)0j<j\u000b0\neffj,j(\"T\neff)0j<j\f0\neffj; (iii) (\u0016T\neff)0<0,\nj(\"T\neff)0j<j\f0\neffjorj(\u0016T\neff)0j<j\u000b0\neffj, (\"T\neff)0<0, and it is evident that in all these\nbands the imaginary part of \u0011+cannot be zero. From our estimations the \fgure of\nmerit \b\u0006=j\u00110\n\u0006j=\u001100\n\u0006is the order of tens and hundreds for ordinary and extraordinary\neigenwaves, respectively. It means that there is a di\u000berent damping of the ordinary and\nextraordinary eigenwaves as they propagate through the medium, which is a known\nmanifestation of the circular dichroism.\nRemarkably, from this set of conditions, the second one is unambiguously\nsatis\fed at the particular frequency of the gyrotropic-nihility state fgn, where\n(\u0016T\neff)0and (\"T\neff)0simultaneously appear to be close to zero. In this case the\nimaginary parts of \u0016T\neff,\u000beffand\"T\neff,\feffare relatively small, so we can write\nj\"\u0006\u0016\u0006j\u0019j\u000b0\neff\f0\neffj, which leads to the following approximate equalities related to\nthe eigenwaves propagation constants\n\r\u0000\u0019\u0000\r+\u0019k0q\nj\u000b0\neff\f0\neffj: (18)\nThus, the propagation constants of the ordinary and extraordinary eigenwaves are\nequal in the magnitude but opposite in sign to each other, and thus a backward\npropagation appears for the ordinary eigenwave while for the extraordinary eigenwave\nit is a forward one. So, it turns out that at the frequency of the gyrotropic-nihility\nstate, both ordinary \r+and extraordinary \r\u0000eigenwaves appear to be left-circularly\npolarized. This is because the wavevector ~kof the ordinary eigenwave now reverses its\ndirection and the handedness changes, accordingly, from left to right (see, \fgure 5).\nIt is particularly remarkable that at the gyrotropic-nihility state, the e\u000bective\ngyrotropic parameters \u000beffand\feffare close in value to each other and their real\nparts simultaneously approach unity which can be clearly seen in \fgure 3. It leads to\nthe fact that the studied medium appears to be matched to free space with both the\nrefractive index and the wave impedance. Directly at the gyrotropic-nihility frequency\nfgn, the normalized impedances related to the ordinary and extraordinary waves are\n{We use here the optical community convention on the handedness of the circular polarization.\nAccording to this convention right-circularly polarized light is de\fned as a clockwise rotation of the\nelectric vector when the observer is looking against the direction the wave is traveling.Gyrotropic-nihility state 13\nFigure 5. (Color online) Instantaneous electric \feld distribution for (a) ordinary\neigenwave and (b) extraordinary eigenwave at the particular frequency of the\ngyrotropic-nihility state fgn= 4:94. Parameters of the ferrite and semiconductor\nlayers are the same as in \fgure 2; \u00120= 0 deg,d1= 0:05 mm,d2= 0:2 mm.\nalmost indistinguishable and are both positive quantities because when calculating\nthe normalized impedance it is need to take the root branch as Z\u0006=p\n\u0016\u0006=\"\u0006>0\n[29]\nZ+\u0019Z\u0000\u0019s\nj\u000b0\neffj\nj\f0\neffj; (19)\nTherefore it turns out that this simultaneous matching of both the refractive index\nand the wave impedance to free space should inevitably result in the re\rectionless\ninteraction of electromagnetic waves when they impinge on the studied structure\nhaving a \fnite number of periods.\n5. Wave transmission through a gyrotropic-nihility layer\nIn order to \fnd the transmittance and re\rectance of the studied structure, we use the\nresults of [27] and write the solution of equation (5) in the form\n~\t(0) = ( MN)\u00001~\t(NL) =T~\t(NL); (20)\nwhere the \feld vector ~\t at the input and output structure's surfaces consists of the\nincident, re\rected and transmitted wave contributions as\n~\t(0) =~\tinc+~\tref;~\t(NL) =~\ttr: (21)\nThe vectors ~\tinc,~\trefand~\ttrare composed from tangential components of the\nelectromagnetic \feld, and in turn these components are determined by their complex\namplitudes. In the general case, the numerical solution of the Cauchy problem (5) for\nparticular structure's layers results in the matrices M1andM2, and then subsequent\nrising of their product M=M2M1to the power Nallows us to \fnd both the\ncoe\u000ecients of the transfer matrix Tand the complex amplitudes of the transmitted\nand re\rected \felds (we refer the reader to Ref. [27] here for further details on theGyrotropic-nihility state 14\ncalculation procedure). The ratios between the amplitudes of the transmitted \feld and\nthe incident \feld, and between the amplitudes of the re\rected \feld and the incident\n\feld establish the complex transmission and re\rection coe\u000ecients, respectively, which\ncan be calculated as functions of the frequency and angle of wave incidence, T(!;\u0012 0)\nandR(!;\u0012 0). Accordingly, the quantities jTj2,jRj2andW= 1\u0000jTj2\u0000jRj2are\nde\fned as the transmittance, re\rectance and absorption coe\u000ecient.\nIn \fgure 6(a) the transmittance calculated as a function of the frequency and\nangle of incidence is shown in the from of a surface plot. One can see that this surface\nis quite smooth on the distance from the ferromagnetic and plasma resonances. At\nonce, in the frequency band of interest (4.5 GHz { 5.5 GHz) there is a speci\fc ridge\non the surface where the transmittance reaches high values. This ridge runs through\nalmost the entire range of angles, and maximum of the transmittance appears near\nthe frequency of the gyrotropic-nihility state fgn= 4:94 GHz, which is distinguished\non the bottom contour by an arrow. Such high transmittance obviously appears due\nto the mentioned peculiarities of the refractive index and the wave impedance that\nare both matched to free space.\nThis feature is also con\frmed by the curves plotted in the bottom planes of\n\fgure 6, where the transmittance, re\rectance and absorption coe\u000ecient are presented\nfor two di\u000berent values of the frequency and polar angle. Here again, the frequency of\nthe gyrotropic-nihility state is marked by an arrow. From curves plotted in \fgure 6(b)\none can conclude that despite the fact that the angle of incidence rises the minimum of\nre\rectance remains to be nearly the frequency of the gyrotropic-nihility state wherein\na certain absorption in the medium exists.\nBesides, in \fgure 6(c) the \frst frequency is chosen at the gyrotropic-nihility state\nwhile the second one is selected to be far from the frequencies of the gyrotropic-nihility\nstate and the ferromagnetic and plasma resonances. At the frequency of f= 10 GHz,\nthe curves have typical form where the transmittance monotonically decreases and the\nre\rectance monotonically increases as the angle of incidence rises. On the other hand,\nat the frequency of the gyrotropic-nihility state, the curves of the transmittance and\nre\rectance are di\u000berent drastically from those of the discussed case. Thus, the level\nof the transmittance/re\rectance remains to be invariable almost down to the glancing\nangles. At the same time, the re\rectance is small down to the glancing angles because\nat this frequency the medium is matched to free space.\n6. Conclusions\nTo conclude, in this paper we study characteristics of the gyrotropic-nihility state in a\n\fnely-strati\fed ferrite-semiconductor structure which is under an action of an external\nstatic magnetic \feld applied in the Faraday con\fguration. In the long-wavelength\nlimit, when the structure's layers as well as its period are optically thin, with an\nassistance of the e\u000bective medium theory, the studied structure is approximated as a\nuniform gyroelectromagnetic medium de\fned with e\u000bective permittivity and e\u000bective\npermeability tensors. In general, the investigations of the eigenwaves propagation\nin such gyroelectromagnetic medium were carried out on the basis of numerical\ncalculations. At the same time, in the case, when directions of the electromagnetic\nwaves propagation and the static magnetic \feld bias are coincident, the components\nof e\u000bective permittivity and e\u000bective permeability tensors, e\u000bective refractive indexes\nand normalized wave impedances are obtained analytically.\nThe gyrotropic-nihility phenomenon is considered as some extreme-parameterGyrotropic-nihility state 15\nFigure 6. (Color online) (a) Transmittance as a function of the frequency\nand the polar angle of incidence of the plane monochromatic wave for a \fnely-\nstrati\fed structure with N= 10 periods which corresponds to the equivalent\ngyroelectromagnetic layer with thickness NL= 2:5 mm. (b), (c) The frequency\nand angular dependencies of the transmittance, re\rectance and absorption\ncoe\u000ecient for the same structure. Parameters of the ferrite and semiconductor\nlayers are the same as in \fgure 2; d1= 0:05 mm,d2= 0:2 mm.Gyrotropic-nihility state 16\nstate that appears in a small region near singular points of dispersion curves where\nreal parts of the diagonal components of both complex e\u000bective permittivity and\ncomplex e\u000bective permeability tensors simultaneously make transitions from negative\nto positive values while the o\u000b-diagonal components of the corresponding tensors\nremain to be non-zero quantities. On the basis of these constitutive parameters the\npeculiarities of the ordinary and extraordinary eigenwaves propagation are studied\nand the possibility of achieving a double-negative condition at a particular frequency\nof the gyrotropic-nihility state is predicted. In particular, it turns out that the\npropagation constants of the ordinary and extraordinary eigenwaves are equal in the\nmagnitude but opposite in sign to each other, and thus a backward propagation\nappears for the ordinary eigenwave while for the extraordinary eigenwave it is a\nforward one. Therefore, at the frequency of the gyrotropic-nihility state, both\nordinary and extraordinary eigenwaves appear to be left-circularly polarized because\nthe wavevector of the ordinary eigenwave reverses its direction and the handedness\nchanges, accordingly, from left to right.\nThe frequency and angular dependencies of the transmittance, re\rectance and\nabsorption coe\u000ecient are presented. It turns out that near the gyrotropic-nihility\nstate the studied structure is matched to free space with both the refractive index and\nthe wave impedance which results in its high transmittance almost in the entire range\nof angles of the electromagnetic waves incidence. We believe that this outcome can\nbe of great interest, particularly, in the problem of transformation optics.\nAppendix A.\nThe matrix Ain equation (5) can be written in the 2 \u00022 block representation [33]\nA=\u00120 A+\nA\u00000\u0013\n; (A.1)\nwhere 0is the 2\u00022 matrix with all its entries being zero, and for the ferrite ( A1),\nsemiconductor ( A2) and entire composite ( Aeff) layers corresponding matrices A\u0006\nare [17]:\nA+\n1=\u0012kxky=k2\n0\"1+i\u000b \u0016T\n1\u0000k2\nx=k2\n0\"1\n\u0000\u0016T\n1+k2\ny=k2\n0\"1\u0000kxky=k2\n0\"1+i\u000b\u0013\n; (A.2)\nA\u0000\n1=\u0012\u0000kxky=k2\n0\u0016L\n1\u0000\"1+k2\nx=k2\n0\u0016L\n1\n\"1\u0000k2\ny=k2\n0\u0016L\n1kxky=k2\n0\u0016L\n1\u0013\n; (A.3)\nA+\n2=\u0012kxky=k2\n0\"L\n2\u00162\u0000k2\nx=k2\n0\"L\n2\n\u0000\u00162+k2\ny=k2\n0\"L\n2\u0000kxky=k2\n0\"L\n2\u0013\n; (A.4)\nA\u0000\n2=\u0012\u0000kxky=k2\n0\u00162\u0000i\f\u0000\"T\n2+k2\nx=k2\n0\u00162\n\"T\n2\u0000k2\ny=k2\n0\u00162kxky=k2\n0\u00162\u0000i\f\u0013\n; (A.5)\nA+\neff=\u0012kxky=k2\n0\"L\neff+i\u000beff\u0016T\ne\u0000k2\nx=k2\n0\"L\ne\n\u0000\u0016T\neff+k2\ny=k2\n0\"L\neff\u0000kxky=k2\n0\"L\neff+i\u000beff\u0013\n; (A.6)\nA\u0000\neff=\u0012\u0000kxky=k2\n0\u0016L\neff\u0000i\feff\u0000\"T\neff+k2\nx=k2\n0\u0016L\neff\n\"T\neff\u0000k2\ny=k2\n0\u0016L\neffkxky=k2\n0\u0016L\neff\u0000i\feff\u0013\n: (A.7)Gyrotropic-nihility state 17\nAcknowledgments\nThis work was supported by National Academy of Sciences of Ukraine with Program\n`Nanotechnologies and Nanomaterials,' Project no. 1.1.3.17.\nReferences\n[1] Lakhtakia A. 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Available from: http://jpier.org/pierb/\npier.php?paper=11041802 ."    },    {        "title": "1608.03821v3.A_revisited_Johnson_Mehl_Avrami_Kolmogorov_model_and_the_evolution_of_grain_size_distributions_in_steel.pdf",        "content": "A REVISITED JOHNSON-MEHL-AVRAMI-KOLMOGOROV MODEL AND\nTHE EVOLUTION OF GRAIN-SIZE DISTRIBUTIONS IN STEEL\nD. H OMBERG, F. S. PATACCHINI, K. SAKAMOTO, AND J. ZIMMER\nAbstract. The classical Johnson-Mehl-Avrami-Kolmogorov approach for nucleation and growth\nmodels of di\u000busive phase transitions is revisited and applied to model the growth of ferrite in mul-\ntiphase steels. For the prediction of mechanical properties of such steels, a deeper knowledge of the\ngrain structure is essential. To this end, a Fokker-Planck evolution law for the volume distribution\nof ferrite grains is developed and shown to exhibit a log-normally distributed solution. Numerical\nparameter studies are given and con\frm expected properties qualitatively. As a preparation for\nfuture work on parameter identi\fcation, a strategy is presented for the comparison of volume dis-\ntributions with area distributions experimentally gained from polished micrograph sections.\nKeywords: grain-size distribution, Fokker-Planck equation, nucleation and growth, phase transi-\ntions.\n2010 Math Subject Classi\fcation: 35Q84, 35Q74, 35K10, 74H40.\n1.Introduction\nSteel is still the most important construction material in industrialised countries. Driven espe-\ncially by the goal of vehicle weight reduction in automotive industry, the last two decades have seen\nthe development of many new steel grades, such as dual, trip, or twip steels combining both strength\nand ductility. The production of these new steels requires a precise process guidance [25, 10]. It has\nturned out that best results are achieved if in addition to temperature measurements, the resulting\nmicrostructure is also monitored. Macroscopic phase transition models allowing for a coupling with\n\fnite element simulation of temperature evolution have thus gained increasing interest.\nA classical model to describe di\u000busive nucleation and growth is the Johnson-Mehl-Avrami-Kol-\nmogorov (JMAK) model, developed independently by [20], [2, 3, 4], and [21]. A review of the JMAK\nmodel can be found in [16]. Assuming constant nucleation and growth rates \u000band\u001a, respectively,\nit states that the volume fraction P(t) at some time tof a new phase growing from a parent phase\nby a nucleation and growth process is given by\nP(t) = 1\u0000exp\u0012\n\u0000\u0019\u000b\u001a3t4\n3\u0013\n: (1.1)\nThis JMAK equation is widely used in engineering literature due to its simplicity; in fact, many\nextensions of it to situations with non-constant nucleation and growth rates can be found; see for\nexample [1].\nThe \frst contribution of this paper is to revisit the classical nucleation and growth modelling\napproach of [20], [2, 3, 4], and [21]. Speci\fcally, in Section 2 we focus on the growth of ferrite\nphase from the high temperature austenite phase, which plays an important role, e.g., in dual\nphase and trip steels. Ferrite is a solid solution of carbon in face-centred cubic (f.c.c.) iron. Its\ntime-dependent growth is governed by the di\u000busion of carbon into the remaining austenite, thereby\nenriching its carbon content. The transition naturally ceases when the equilibrium fraction of\ncarbon in austenite is reached|this is the so-called soft impingement . The JMAK model employed\nDate : 28 April 2017.\n1arXiv:1608.03821v3  [cond-mat.mtrl-sci]  28 Apr 2017in this article has indeed been found to be in good agreement with soft impingement, and also with\nnon-random nucleation e\u000bects [11].\nAn important aim of material simulation is the prediction of mechanical properties. However,\nespecially in heterogeneous materials such as metals, a representative volume (or concentration)\napproach is not su\u000ecient to predict material properties if it does not account for the distribution of\ngrain (or nucleus) sizes. Furthermore, a macroscopic nucleation and growth model is not capable\nof resolving mesoscopic grain boundaries. The second contribution of this paper is to gather ad-\nditional information about the material heterogeneity by studying the grain-size distribution. As\nshown in [8], this can then be used by a stochastic homogenisation approach to derive mechanical\nproperties.\nAs a conserved quantity, the grain-size (volume) distribution is governed by a Fokker-Planck\nequation derived and solved in Section 3. Grain-size distributions in austenite and ferrite are of\nlog-normal type [22], as is the solution to the Fokker-Planck equation we study, up to a convolution\nwith the initial pro\fle. In fact, we rigorously obtain a log-normal solution for all times only if\nthe initial pro\fle is itself log-normal. Nevertheless, for any admissible inital datum the solution\npro\fle we \fnd is log-normal asymptotically in time in numerical tests. For a di\u000berent application\nof similar Fokker-Planck models; see for example [14]. One core aspect of our Fokker-Planck model\nis that it naturally couples the macroscopic scale (ferrite phase fraction) with the mesoscopic\nscale (ferrite grain-size distribution) via a \frst-moment constraint given in (3.3). In Section 4 we\npresent simulations and a numerical parameter study. In a forthcoming paper, we will discuss\nparameter identi\fcation issues for this Fokker-Planck model and compare it to real-world data.\nAs a preparation, we discuss in Section 5 how volume distributions can be compared with area\ndistributions drawn from polished micrograph sections using an approach by [19]. For this question\nwe also refer the reader to [26]. For results about the identi\fcation of temperature-dependent\ngrowth rates exploiting dilatometer experiments, we refer to [18, 17].\nThe promising feature of this approach is that it allows for an easy calculation of grain-size\ndistributions from a macroscopic level without explicit mesoscopic simulations as in the phase-\feld\napproach, opening up at least two interesting and obvious areas of further research. The \frst\none is the inclusion of thermal e\u000bects by a spatial two-scale model combining area space with\nthe macroscopic specimen space. The second one is the use of these grain-size distributions for a\ncomputation of homogenised mechanical properties.\nSomewhat similarly to our approach, the evolution of the absolute number of grains has also\nbeen modelled by [27] and [9] using a continuity equation of conservative type with a reaction\nterm. There too, the authors obtain an analytic expression for the solution which, remarkably, is\nasymptotically (as tgoes to in\fnity) log-normally distributed in one space dimension. In higher\nspace dimensions they obtain solutions which are not log-normal, but still qualitatively close to it.\nIn [27], the authors link their model to the JMAK model by choosing a speci\fc reaction term and\nspeci\fc nucleation and growth rates. In contrast, in the present paper we make this link through\nthe very de\fnition of the \frst moment of the grain-size distribution. We also remark that our\napproach can easily be adapted to arbitrary space dimensions and that obtaining a log-normally\ndistributed explicit solution does not depend on the choice of this dimension; indeed, the JMAK\nmodel can be extended to any dimension and then linked to our Fokker-Planck equation which\nis dimension-free. Let us mention as well that [28] uses a JMAK approach similar to the one of\nSection 2 and, using Fick's law, couples it to a di\u000busion equation for the new phase concentration.\nIt is noteworthy at this point that our approach should not be confused with grain boundary\ncharacter distribution evolution models of Fokker-Planck type as they have been investigated in\na series of papers by [6, 7]; see also the references therein. While these authors study coarsening\ne\u000bects in polycrystalline solids, i.e., a single phase situation, the present paper is concerned with the\nevolution of the grain-size distribution during an irreversible phase transition without coarsening.\n2Here, no grain can grow at the expense of others, no grain shrinks, and no grain can grow into\nothers when touching|this is the so-called hard impingement . Similarly, the approach taken in\nthis paper for nucleation and growth processes is di\u000berent from the Becker-D oring type models of\ncoagulating particles or droplets; see, e.g., [5] and [24].\n2.The revisited JMAK model\nConsider a bounded domain \n \u001aR3, whose volume is denoted by V=j\nj, composed exclusively\nof an austenite phase and a ferrite phase, and where austenite may transform into ferrite as time\nincreases. The sub-volume of austenite present at time tis denoted by VA(t), and the one of ferrite\nbyVF(t). By conservation of volume we have V=VA(t) +VF(t) for allt2[0;T], whereT >0 is a\n\fxed \fnal time. Then, the volume phase fraction of ferrite is de\fned by\nP(t) =VF(t)\nV: (2.1)\nTo derive our model, we assume that the phase transformation happens isothermally at temperature\n\u0012>0, although this can be easily generalised.\nWe assume that ferrite grains appear randomly in the austenite matrix \n with nucleation rate\n\u000b=\u000b(\u0012) (number of grains per unit time per unit volume) and grow isotropically, that is, as\nspheres, with growth rate \u001a(t;\u0012) =\u001a(t) (length per unit time). We suppose that when two growing\ngrains touch, these cannot grow into each other and thus only continue growing towards the \\free\"\ndirections, which is the hard impingement assumption. After two grains meet, they therefore do\nnot look as spheres anymore, but rather as the union of two intersected spheres. Let us point out\nthat hard impingement also describes the \\interaction\" between the grains and the boundary of the\ndomain \n when these touch. In this setting, the volume occupied at time tby an isolated ferrite\ngrain born at time \u001cis\n\u0017(t;\u001c) =4\u0019\n3\u0012Zt\n\u001c\u001a(s) ds\u00133\n: (2.2)\nConsider an extended volume, denoted by Vext(t), which is the total volume occupied by all ferrite\ngrains at time t, assuming temporarily that they may grow into each other. This gives, using (2.1),\nVext\nF(t) =V\u000bZt\n0\u0017(t;\u001c) d\u001c:\nInvoking the Avrami correction (see [2, 3, 4] for a derivation, and also [21]) to incorporate hard\nimpingement, we have\nVdP(t) = dVF(t) = (1\u0000P(t)) dVext\nF(t):\nWe remark that the Avrami correction is only an approximation, due to possible overgrowth of\nphantom nuclei [29]. By integrating the above equation, using (2.2) and supposing that P(0) = 0,\nwe get\n\u0000log (1\u0000P(t)) =4\u0019\u000b\n3Zt\n0\u0012Zt\n\u001c\u001a(s) ds\u00133\nd\u001c: (2.3)\nNote that, by assuming that \u001a(t) =\u001adoes not depend on time, and by taking the exponential of\nboth sides of (2.3), we recover (1.1). Equation (2.3) yields\n8\n><\n>:P0(t) = 4\u0019\u000b\u001a(t)(1\u0000P(t))Zt\n0\u0012Zt\n\u001c\u001a(s) ds\u00132\nd\u001c;\nP(0) = 0:(2.4)\nIn order to close the di\u000berential equation (2.4), we need now to choose a law for the evolution of\nthe growth rate \u001a. This is where we incorporate soft impingement into the model. This means\n3that the transformation ceases naturally when the actual carbon concentration in austenite, CA(t),\nreaches the equilibrium value Ceq\nA=Ceq\nA(\u0012), corresponding to an equilibrium volume Veq\nF=Veq\nF(\u0012)\nand equilibrium fraction Peq=Veq\nF=V. ThenCA(t) can be computed from mass conservation by\nassuming that the carbon concentration in ferrite is constant and equal to its equilibrium value\nCeq\nF=Ceq\nF(\u0012) (de\fned as the carbon concentration in ferrite when Ceq\nAis reached), i.e.,\nC=Ceq\nFP(t) +CA(t)(1\u0000P(t));\nwhereCis the overall carbon concentration in the steel sample \n. From this it follows that, if\nCeq\nA6=Ceq\nF(otherwise nothing happens), the equilibrium volume fraction of ferrite satis\fes\nPeq=Ceq\nA\u0000C\nCeq\nA\u0000Ceq\nFandPeq\u0000P(t)\n1\u0000P(t)=Ceq\nA\u0000CA(t)\nCeq\nA\u0000Ceq\nF;\nso that the ratio ( Peq\u0000P(t))=(1\u0000P(t)) equals the supersaturation . We then require the growth\nrate\u001a(t) to be proportional to Ceq\nA\u0000CA(t), and we make the choice\n\u001a(t) =\u001a\u0003\nct\rPeq\u0000P(t)\n1\u0000P(t);0\u0014\r <1; (2.5)\nwhere\u001a\u0003=\u001a\u0003(\u0012)>0 is some reference growth rate, and cis a constant with the same dimension\nast\u0000\r; for simplicity, we take c:= 1. The term t\rallows for the description of time-dependent\ngrowth rates, independently of soft impingement. In the case of classical di\u000busional growth, we\nhave\r= 0:5. This ansatz for the growth rate then results in the integro-di\u000berential equation model\nP0(t) = 4\u0019\u000b\u001a\u0003t\u0000\r(Peq\u0000P(t))Zt\n0\u0012Zt\n\u001c\u001a\u0003\ns\rPeq\u0000P(s)\n1\u0000P(s)ds\u00132\nd\u001c:\nNote that the equilibrium value Peqis only reached asymptotically. This equation can be dealt\nwith by transformation to a system of ODEs. To this end, we perform the substitutions\nz(t;\u001c) =Zt\n\u001c\u001a(s) ds; y (t) =\u000bZt\n0z(t;\u001c)2d\u001c; x (t) =\u000bZt\n0z(t;\u001c) d\u001c; w (t) =\u000bt: (2.6)\nAltogether we obtain(\nw0(t) =\u000b; x0(t) =\u001a(t)w(t); y0(t) = 2\u001a(t)x(t);\nP0(t) = 4\u0019(1\u0000P(t))\u001a(t)y(t);\nwithw(0) =x(0) =y(0) =P(0) = 0. We can \fnally introduce the number of grains born until\ntimetper unit volume\nN(t) =\u000bZt\n0\u0012\n1\u0000P(\u001c)\nPeq\u0013\nd\u001c: (2.7)\nThe expression (2.7) takes soft impingement into account as well by requiring that nucleation stops\nwhenPeqis reached.\n3.The grain-size distribution model\n3.1.Derivation of the governing equation. The volume distribution of ferrite grains\n\u001e(\u0017;t): (0;1)\u0002[t0;T]![0;1) (3.1)\ncounts, at time t, the number of grains of volume \u0017per unit volume, normalised by the total number\nof grains. This means that, for any \u00172>\u00171\u00150, the quantityR\u00172\n\u00171\u001e(\u0017;t) d\u0017is the relative number\nof grains with volumes in [ \u00171;\u00172], which implies that \u001e(\u0001;t) is a probability density on (0 ;1), that\nis,R1\n0\u001e(\u0017;t) d\u0017= 1. In (3.1), t0>0 is a small incubation time before which the notion of volume\ndistribution does not make physical sense. Indeed, there exists a small critical average grain volume\nbelow which we are unable to describe physically, or simply observe, the evolution of the ferrite\n4grain-size distribution in the specimen. The strictly positive incubation time t0is de\fned as being\nthe smallest time after which the average ferrite volume in the specimen has reached this critical\nvolume (see [30] for an account on the notion of incubation time). Therefore, while the JMAK\nmodel starts at t= 0 with zero volume fraction of ferrite as in (2.4)|which would correspond to\na volume distribution which is a Dirac mass at the origin|the ferrite volume distribution model\nthat we derive below is only meaningful for times t\u0015t0.\nSince\u001e(\u0001;t) has conserved unit mass over all times t2[t0;T], we assume that \u001esatis\fes the\ncontinuity equation \u001et+J\u0017= 0, with\nJ(\u0017;t) =\u00161(\u0017;t)\u001e\u0000(\u00162(\u0017;t)\u001e)\u0017;\nwhere the mobility terms \u00161and\u00162are assumed to be separable, \u00161(\u0017;t) =\u001611(t)\u001612(\u0017) and\n\u00162(\u0017;t) =\u001621(t)\u001622(\u0017). Here, we suppose that \u001612(\u0017) =\u0017and\u001622(\u0017) =\u00172. These choices are\njusti\fed a posteriori : they allow the derivation of an explicit solution for the volume distribution\nwhich is, up to a convolution with the initial datum, log-normally distributed (see Section 3.2);\nand this log-normal behaviour is experimentally observed. Also, we write u(t) :=\u001611(t) and\n\f(t) :=\u001621(t), where we assume \f(t) =f(u(t)) for some function f2C1(R) such that f(0) = 0\nandf(u)>0 for allu6= 0. The requirement that f(0) = 0 is physically justi\fed by the fact that the\nvolume distribution stops evolving as soon as the convection vanishes, and therefore the di\u000busion\nhas to vanish as well. The condition f(u)>0 for allu6= 0 is needed to avoid backward di\u000busion in\nthe case the convection velocity ubecomes negative. Indeed, as it becomes clearer in the following,\nthis may happen when the nucleation rate \\beats\" the grain growth and thus \\drags\" the volume\ndistribution pro\fle towards the left. In this paper, we choose f(u) =\f1u2, where\f1>0, although\nin Section 4.4 we show the appearance of in\fnite-time blow-up if we violate the condition f(0) = 0\nfor the special case f(u) =\f0+\f1u2with\f0>0. All in all, the volume distribution \u001eis assumed\nto satisfy the Fokker-Planck equation\n(\n\u001et=\u0000u(t)(\u0017\u001e)\u0017+\f1u(t)2(\u00172\u001e)\u0017\u0017;\n\u001e(\u0017;t0) =\u001e0(\u0017);for all (\u0017;t)2(0;1)\u0002(t0;T]; (3.2)\nwhere\u001e02C0(0;1)\\L1(0;1) is a probability density.\nAn essential feature of the present grain-size distribution model lies in the fact that we can\ndirectly link it to the revisited JMAK model developed in Section 2 using the natural moment\nrelationZ1\n0\u0017\u001e(\u0017;t) d\u0017=P(t)\nN(t)=:g(t) for allt2[t0;T]; (3.3)\nwhere we recall from (2.1) that Pis the volume phase fraction of ferrite and from (2.7) that Nis the\nrelative number of ferrite grains. The left-hand side of (3.3) is the \frst moment of \u001e(\u0001;t) at timet,\ni.e., the mean volume of the grains. This equation bridges the meso- and macroscopic scales, giving\nus another nice feature of the JMAK model, namely that it allows to compute the mean grain size\nwithout relying on further mesoscopic information. We refer to [13], [31] and [15] for mathematical\nstudies of Fokker-Planck/gradient \row equations with moment constraints. Equation (3.3) is a\nconstraint that is imposed by the JMAK model on the Fokker-Planck model (3.2); therefore the\nvolume distribution \u001esatis\fes the coupled system\n8\n><\n>:\u001et=\u0000u(t)(\u0017\u001e)\u0017+\f1u(t)2(\u00172\u001e)\u0017\u0017;R1\n0\u0017\u001e(\u0017;t) d\u0017=g(t);\n\u001e(\u0017;t0) =\u001e0(\u0017);for all (\u0017;t)2(0;1)\u0002(t0;T]: (3.4)\n5Constraint (3.3) also enforces a relation between gand the convection velocity u. Indeed,\ng0(t) =Z1\n0\u0017\u001et(\u0017;t) d\u0017=\u0000u(t)Z1\n0\u0017(\u0017\u001e)\u0017(\u0017;t) d\u0017+\f1u(t)2Z1\n0\u0017(\u00172\u001e)\u0017\u0017(\u0017;t) d\u0017\n=u(t)Z1\n0\u0017\u001e(\u0017;t) d\u0017\u0000\f1u(t)2Z1\n0(\u00172\u001e)\u0017(\u0017;t) d\u0017\n=u(t)Z1\n0\u0017\u001e(\u0017;t) d\u0017=u(t)g(t);\nwhere we implicitly need that\n8\n<\n:lim\n\u0017!0\u00172\u001e(\u0017;t) = lim\n\u0017!+1\u00172\u001e(\u0017;t) = 0;\nlim\n\u0017!0\u00173\u001e\u0017(\u0017;t) = lim\n\u0017!1\u00173\u001e\u0017(\u0017;t) = 0;(3.5)\nin order to carry out the integrations by parts. This gives\nu(t) =g0(t)\ng(t)= (log\u000eg)0(t) for allt2[t0;T]; (3.6)\nor, equivalently,\ng(t) =g0ea(t)for allt2[t0;T]; (3.7)\nwhereg0:=g(0) anda(t) :=Rt\nt0u(s) ds. Equation (3.6) tells us that the mesoscopic convection\nvelocityu(and therefore the di\u000busion coe\u000ecient \f, up to the multiplicative constant \f1) is de-\ntermined by the evolution of the macroscopic quantity ggiven to us by the model in Section 2.\nConversely, we can also see the convection velocity uas a measure of the evolution of g. Again, here,\nwe see the coupling between the meso- and macroscopic scales, and the system (3.4) is equivalent\nto8\n><\n>:\u001et=\u0000u(t)(\u0017\u001e)\u0017+\f1u(t)2(\u00172\u001e)\u0017\u0017;\nu(t) = (log\u000eg)0(t);\n\u001e(\u0017;t0) =\u001e0(\u0017);for all (\u0017;t)2(0;1)\u0002(t0;T]: (3.8)\nRemark 3.1.As already discussed at the beginning of this section, we cannot hope to describe\nphysically the evolution of the ferrite volume distribution before the incubation time t0>0 is\nreached. We observe, from a mathematical point of view, that (3.7) implies that if t0was taken\nto be zero, i.e., \u001e0was a Dirac mass at the origin according to the zero-fraction initial condition\nin (2.4), then any solution to (3.2) would stay equal to \u001e0for all times, that is, nothing would\nhappen. This re\rects the fact that, even mathematically, our Fokker-Planck model is unable to\ndescribe the evolution of \u001efor early times.\nRemark 3.2.In (3.2), as well as in (3.4) and (3.8), the volume domain, (0 ;1), is unbounded, which\nallows grains to grow instantaneously arbitrarily large. Our Fokker-Planck model can therefore be\nrigorously valid only for unbounded specimens and unbounded grain growth rates, although the\nJMAK model developed in Section 2 requires the specimen to be a bounded one and the growth\nrate (2.5) to be \fnite. Given that the grains are typically small relative to the size of the specimen,\none would expect our model to be a good approximation, which we can control by quantifying the\n\\portion\" of ferrite volume distribution \u001ehaving a larger volume than a time-dependent maximal\nvolume,\u0017max, imposed by the \fniteness of the specimen and the growth rate. Alternatively, we\ngive now a possible improvement of our model taking the boundedness of the specimen and growth\n6rate into account. We simply consider (3.2) and impose a boundary condition at \u0017max:\n8\n><\n>:\u001et=\u0000u(t)(\u0017\u001e)\u0017+\f1u(t)2(\u00172\u001e)\u0017\u0017;\n\u001e(\u0017max(t);t) = 0;\n\u001e(\u0017;t0) =\u001e0(\u0017);for all (\u0017;t)2(0;\u0017max(t))\u0002(t0;T]: (3.9)\nObtaining an exact value for this maximal volume \u0017max(t) may not be possible; nevertheless the\nJMAK model tells us that it has to satisfy \u0017max(t)\u0014min(PeqV;(4\u0019=3)z(t;0)3) for allt\u0015t0, where\nPeqis the equilibrium volume fraction of ferrite, Vis the volume of the specimen, and zis as\nin (2.6). The fact that \u0017max(t) is bounded by PeqVmakes sure that the equilibrium volume (and\nthus the volume of the specimen) is never exceeded, whereas the fact that \u0017max(t) is bounded by\n(4\u0019=3)z(t;0)3ensures that the maximal grain volume allowed by the growth rate at time tis not\nviolated. Unfortunately, there is no explicit solution formula for (3.9) akin to that for (3.2) derived\nin Section 3.2. Nevertheless, existence of a solution is known [12, Theorem 16.3.1] and we believe\nthat, qualitatively, such a solution is very similar to that of Section 3.2 for the unbounded case.\nOne way to support this would be to prove that if \u001e\u0017maxis solution to (3.9) and \u001e\u0017max!\u001e(in\nsome sense) as min t(\u0017max(t))!1 for some probability density \u001e, then\u001emust be solution to (3.2).\nBecause of the reasons just mentioned, we decide to focus in this paper on (3.2) only and we leave\nthe study of (3.9) to a future work. Note nonetheless that in Section 5 we actually derive a relation\nbetween volume and area distributions in the case of a bounded specimen; Section 5, however, is\nmostly independent of the rest of the paper and is mainly there to motivate a forthcoming paper.\nRemark 3.3.If one assumes that hard impingement in negligible (for example, if the \fnal time\nis very small or the nucleation and growth rates are very small), then grains are exact, non-\nintersected spheres and one may equivalently employ the radius distribution  in place of the\nvolume distribution \u001e. The radius distribution\n (r;t): (0;1)\u0002[t0;T]![0;1)\ncounts, at time t, the number of grains of radius rper unit radius, normalised by the total number\nof grains, which leads, as for the volume distribution, to  (\u0001;t) being a probability density on\n(0;1). Since grains are spheres, there is a direct one-to-one relation between \u001eand as the map\nA: [0;1)![0;1);r7!4\u0019r3=3, is a bijection. Indeed, this implies that, for all r2>r1\u00150,\nZr2\nr1 (r;t) dr=ZA(r2)\nA(r1)\u001e(\u0017;t) d\u0017=Zr2\nr1\u001e(A(r);t)A0(r) dr=Zr2\nr1\u001e\u00124\u0019r3\n3;t\u0013\n4\u0019r2dr;\nby a simple change of variable x!A(r). This equality being true for all r2>r1\u00150, we get\n (r;t) = 4\u0019r2\u001e\u00004\u0019\n3r3;t\u0001\nfor allr2(0;1):\nFrom the inverse transformation, one gets\n\u001e(\u0017;t) = (4\u0019)\u00001=3(3\u0017)\u00002=3 \u0010\u00003\u0017\n4\u0019\u00011=3;t\u0011\nfor all\u00172(0;1):\nRelation (3.3) then becomes\ng(t) =4\u0019\n3Z1\n0r3 (r;t) dr:\n3.2.A solution formula for the volume distribution. We now derive an explicit solution for\nthe Fokker-Planck equation (3.2), which we later couple to the moment constraint (3.3) as in (3.4)\nand (3.8). We introduce the transformation of variables\n\u0018:= log(\u0017) +b(t)\u0000a(t);\n\u001c:=b(t);\n7h(\u0018;\u001c) :=\u0017\u001e(\u0017;t);\nwithh(\u0018;\u001c):R\u0002[0;b(T)]![0;1) and where a(t) :=Rt\nt0u(s) dsandb(t) :=Rt\nt0\f(s) ds=\n\f1Rt\nt0u(s)2ds. The fact that bis increasing allows us to introduce the time change of variables\n\u001c=b(t); this justi\fes the requirement that f(u)>0 for allu6= 0 in\f(t) =f(u(t)) from a\nmathematical viewpoint. We see that h(\u0018;\u001c) is governed by the linear heat equation\n(\nh\u001c=h\u0018\u0018;\nh(\u0018;0) = e\u0018\u001e0(e\u0018);for all (\u0018;\u001c)2R\u0002(0;b(T)]:\nIt is well-known that his given by\nh(\u0018;\u001c) = (K(\u0001;\u001c)\u0003\b0)(\u0018) for all (\u0018;\u001c)2R\u0002(0;b(T)];\nwhere\u0003is the convolution and\nK(\u0018;\u001c) = (4\u0019\u001c)\u00001=2exp\u0000\n\u0000\u00182=(4\u001c)\u0001\nand \b 0(\u0018) = e\u0018\u001e0(e\u0018):\nTransforming back to the original variables \u0017andtwe \fnally obtain\n\u001e(\u0017;t) =\u0017\u00001(K(\u0001;b(t))\u0003\b0)(log(\u0017) +b(t)\u0000a(t)) for all ( \u0017;t)2(0;1)\u0002(t0;T]: (3.10)\nWe now see that the resulting solution is, up to a convolution with the initial distribution, log-\nnormal. As already mentioned, this justi\fes the choice of the mobility terms in the Fokker-Planck\nequation (3.2) made earlier, since experiments strongly suggest a log-normal shape for \u001e[22]. We\nnote that, in (3.10), \u001e(\u0001;t) is rigorously log-normally distributed for all times if \u001e0(\u0001) is too. Indeed,\nin this case \b 0is normal and thus the convolution in (3.10) is between two normal distributions and\nis therefore a normal distribution itself, evaluated at log( \u0017), which shows that \u001e(\u0001;t) is log-normal\nfor all times. Otherwise, if \u001e0(\u0001) is not log-normally distributed we can only infer from numerical\ntests that\u001e(\u0001;t) converges to a log-normal pro\fle as tincreases; see Section 4.3 where we simulate\nthe evolution of the solution for an initial datum which is not log-normal.\nNote that the decay conditions (3.5) are satis\fed by the solution in (3.10) as long as it holds\nthatR1\n0\u00172\u001e0(\u0017) d\u0017 <1; see Proposition 3.4 in [23]. Also, by Propositions 3.1 and 3.2 in [23], we\nhave that\u001eas given in (3.10) satis\fes \u001e(\u0001;t)!\u001e0(\u0001) ast!t0and\u001e2C1;0((0;1)\u0002[t0;1)).\n4.Numerical simulations\nWe study the general qualitative behaviour of the model derived in Section 2 and the Fokker-\nPlanck equations (3.4) and (3.8). As initial distribution, except in Section 4.3, we take the log-\nnormal pro\fle\n\u001e0(\u0017) = (\u0017\u001b0p\n2\u0019)\u00001exp(\u0000(log(\u0017)\u0000\u00160)2=(2\u001b2\n0)) for all\u00172(0;1); (4.1)\nwith\u00160= log(g0)\u0000\u001b2\n0=2; in fact,g0=R1\n0\u0017\u001e0(\u0017) d\u0017= exp(\u00160+\u001b2\n0=2). The standard variation\n\u001b0cannot be extracted from the model in Section 2 (unlike g0), and is therefore an additional\nparameter. Unless mentioned otherwise, the simulations below approximate (2.4) and plot (3.10)\nfor the parameters\n\u001a\u0003= 1; Peq= 0:45; \u000b = 0:001; \r = 0:5; \f 1= 0:01; t 0= 0:3387; \u001b 0= 0:4: (4.2)\nThe value of t0is arbitrary and is only chosen as above for convenience in the following simulations.\n4.1.The main quantities. From Figures 1a and 1b, we can see that the quantities g(t) andP(t)\nare sigmoid functions, reaching \\quickly\" values close to their equilibrium. The evolution of the\nlog-normal volume distribution of ferrite grains \u001e(\u0001;t) is given in Figure 1c.\n80 50 100 150\nt00.050.10.150.20.250.30.350.40.45P(t)Veryclosetoequilibrium\n(t'90)(a) Evolution of P(t)\n0 50 100 150\nt0510152025g(t) (b) Evolution of g(t) :=P(t)=N(t) =R1\n0\u0017\u001e(\u0017;t) d\u0017\n0 10 20 30 40 50 60\nν00.050.10.150.20.25φ(ν)t≃6\nt≃9\nt≃11\n20≤t≤150\n(c) Evolution of \u001e(\u0001;t) up tot= 150\nFigure 1. Evolution of the main quantities.\n4.2.Parameter study. Figures 2a and 2b show that the e\u000bect of increasing the reference growth\nrate\u001a\u0003or the equilibrium phase fraction Peqturns out to be to make the pro\fle \rat and drift to\nthe right more quickly. The e\u000bect of increasing the nucleation rate \u000bor the power \ris the opposite;\nsee Figures 2c and 2d. Increasing the di\u000busion coe\u000ecient \f1or the initial standard deviation \u001b0\nmakes the solution \ratten and shift to the left, as shown in Figures 2e and 2f.\nIn Figures 3a, 3b and 3c, one sees that the e\u000bect of increasing \u001a\u0003,Peqor\u000bis to make the solution\napproach the equilibrium faster. Figure 3d shows that increasing \rhas the contrary e\u000bect.\n90 10 20 30 40 50 60\nν00.010.020.030.040.050.060.07φ(ν)ρ* = 0.7\nρ* = 0.8\nρ* = 0.9\nρ* = 1\nρ* = 1.1\nρ* = 1.2\nρ* = 1.3(a) Variation of \u001a\u0003\n0 10 20 30 40 50 60\nν00.020.040.060.080.1φ(ν)Peq = 0.3\nPeq = 0.35\nPeq = 0.4\nPeq = 0.45\nPeq = 0.5\nPeq = 0.55\nPeq = 0.6 (b) Variation of Peq\n0 50 100 150\nν00.010.020.030.040.050.06φ(ν)α = 10-4\nα = 4*10-4\nα = 7*10-4\nα = 10-3\nα = 1.3*10-3\nα = 1.6*10-3\nα = 1.9*10-3\n(c) Variation of \u000b\n0 20 40 60 80 100\nν00.020.040.060.080.10.120.14φ(ν)γ = 0.2\nγ = 0.3\nγ = 0.4\nγ = 0.5\nγ = 0.6\nγ = 0.7\nγ = 0.8 (d) Variation of \r\n0 10 20 30 40 50 60\nν00.010.020.030.040.05φ(ν)β1 = 10-3\nβ1 = 4*10-3\nβ1 = 7*10-3\nβ1 = 10-2\nβ1 = 1.3*10-2\nβ1 = 1.6*10-2\nβ1 = 1.9*10-2\n(e) Variation of \f1\n0 10 20 30 40 50 60\nν00.010.020.030.040.05φ(ν)σ0 = 0.25\nσ0 = 0.3\nσ0 = 0.35\nσ0 = 0.4\nσ0 = 0.45\nσ0 = 0.5\nσ0 = 0.55 (f) Variation of \u001b0\nFigure 2. Volume distribution at t= 150 for di\u000berent parameter variations.\n100 50 100 150\nt00.050.10.150.20.250.30.350.40.45P(t)ρ* = 0.7\nρ* = 0.8\nρ* = 0.9\nρ* = 1\nρ* = 1.1\nρ* = 1.2\nρ* = 1.3(a) Variation of \u001a\u0003\n0 50 100 150\nt00.10.20.30.40.50.6P(t)Peq = 0.3\nPeq = 0.35\nPeq = 0.4\nPeq = 0.45\nPeq = 0.5\nPeq = 0.55\nPeq = 0.6 (b) Variation of Peq\n0 50 100 150\nt00.050.10.150.20.250.30.350.40.45P(t)α = 10-4\nα = 4*10-4\nα = 7*10-4\nα = 10-3\nα = 1.3*10-3\nα = 1.6*10-3\nα = 1.9*10-3\n(c) Variation of \u000b\n0 50 100 150\nt00.050.10.150.20.250.30.350.40.45P(t)\nγ = 0.2\nγ = 0.3\nγ = 0.4\nγ = 0.5\nγ = 0.6\nγ = 0.7\nγ = 0.8 (d) Variation of \r\nFigure 3. Ferrite volume phase fraction at t= 150 for di\u000berent parameter variations.\n4.3.The initial pro\fle. We start here with a di\u000berent initial pro\fle from (4.1), which is instead\ncompactly supported and therefore not log-normally distributed:\n\u001e0(\u0017) =(\ncexp\u0010\n\u00001\nk\u0000(\u0017\u0000\u00170)2\u0011\nfor all\u00172h\n\u00170\u0000p\nk;\u00170+p\nki\n;\n0 otherwise :(4.3)\nwherek= 0:1,\u00170=p\nk+ 0:1 andcis the normalising constant. The simulation in Figure 4\nshows that, qualitatively, the solution evolves into a shape very close to that in Figure 1c, although\nthe volume range is much larger. This supports the fact that our solution (3.10) is log-normal\nasymptotically in time, independently of the initial datum. The parameters used to obtain Figure 4b\nare those in (4.2), except for \u001b0which does not play a role in this case.\n4.4.In\fnite-time blow-up. We observe here the behaviour of the solution \u001ewhen the condition\nf(0) = 0 is violated in \f(t) =f(u(t)) in the special case f(u) =\f0+\f1u2for\f0= 0:005. Sinceg\ngoes to an equilibrium value as tincreases and u=g0=gby (3.6), then utends to 0 (if gdoes not\noscillate around its equilibrium value). Therefore f(u(t)) tends to \f06= 0 astincreases, and the\n110 0.2 0.4 0.6 0.8 1\nν01234567φ0(ν)(a) Compactly supported initial pro\fle (4.3)\n0 100 200 300 400\nν00.0050.010.0150.020.0250.030.0350.04φ(ν) (b) Evolution of \u001e(\u0001;t) up tot= 150\nFigure 4. The case of a compactly supported initial pro\fle.\nFokker-Planck equation (3.2) qualitatively becomes\n\u001et=\f0(\u00172\u001e)\u0017\u0017;\nwhose solution blows up in in\fnite time towards a Dirac mass at the origin, as illustrated in Figure 5.\nThe way this solution converges to a Dirac mass is in a very weak sense; indeed, one can check that\nits moments do not go to 0, but rather to a positive constant or to + 1.\nFrom Figure 5a, the solution \frst drifts to the right, until the di\u000busion takes over and makes the\nsolution drift to the left. In\fnite-time blow-up occurs; see Figure 5b.\n0 5 10 15 20 25 30\nν00.050.10.150.20.25φ(ν)t≃6\ntime arrow\n(a) Evolution of \u001e(\u0001;t) up tot= 150\n0 5 10 15 20\nν00.050.10.150.20.25φ(ν)\nt= 150time arrow (b) Evolution of \u001e(\u0001;t) up tot= 1000\nFigure 5. In\fnite-time blow-up for \f0= 0:005.\n5.Volume and area distributions: relation between model and experiments\nTo validate our grain-size model, the resulting volume distribution has to be related to experi-\nmental data which are typically derived from a two-dimensional micrograph section, under the form\n12of an area distribution. We here derive a relationship between these two distributions which can\npave the way to a quantitative validation with measurements in a forthcoming paper. We point\nout that the following derivation only holds in the setting of Remark 3.3; we therefore equivalently\ndeal with three-dimensional and two-dimensional radius distributions instead of volume and area\ndistributions, respectively.\nWe follow the approach of [19] and consider a cylindrically shaped steel specimen \n with base\nareaq, axially symmetric to the z-axis and of length L\u001dp\nq=\u0019. We want to relate the spherical\ngrains in \n with their two-dimensional counterparts, that is, with the discs resulting from the\nintersection of the spherical grains with the plane fz= 0g. Let us \fx a time t, and de\fne \u001f(\u0011;t)\nas the number of such intersection discs with radius \u0011per unit radius per unit surface. (Note\nthat, unlike  ,\u001fis not normalised by the total number of circular grains (intersection discs) in the\ncross-section \n\\fz= 0g, but it is rather a quantity per unit surface.) Due to the boundedness\nof the test specimen, we may assume that the radius of the spherical grains is bounded by some\nrmax\u0014p\nq=\u0019, so that 0\u0014\u0011\u0014rmax. Now let us choose \u00112[0;rmax) and \u0001\u0011 >0 small. Then the\nnumber of circular grains in the cross-section with radii in [ \u0011;\u0011+ \u0001\u0011] is given by qR\u0011+\u0001\u0011\n\u0011\u001f(\u0010;t) d\u0010.\nTo relate this to  , we \fx \u0001r>0 small and, for any spherical grain radius r2[\u0011+ \u0001\u0011;rmax\u0000\u0001r],\nwe infer that the centres of the spherical grains in the right part of the cylindrical specimen (i.e.,\nin \n\\fz\u00150g) with radii in [ r;r+ \u0001r] creating intersection discs with radii in [ \u0011;\u0011+ \u0001\u0011] lie in an\ninterval [~z;~z+\u000e], as shown in Figure 6.\nz z+ηη+η\nrr+r\n~~\nFigure 6. Position of spherical grain centres in right half of specimen.\nNote that\u000edepends on \u0011;\u0001\u0011;rand \u0001r; also, forr=\u0011+ \u0001\u0011, we have ~z= 0. As it is immediate\nfrom Figure 6, we have\n\u000e=\u000e(r) =p\n(r+ \u0001r)2\u0000\u00112\u0000p\nr2\u0000(\u0011+ \u0001\u0011)2: (5.1)\nNow we choose n2N, de\fne \u0001r= (rmax\u0000(\u0011+ \u0001\u0011))=nand introduce the equi-spaced partition\nri=\u0011+ \u0001\u0011+i\u0001r;0\u0014i\u0014n;\nand accordingly \u000ei=\u000e(ri); see (5.1). A \frst order Taylor expansion yields\n\u000ei=ci\u0011\u0001\u0011+ciri\u0001r+o(\u0001\u0011) +o(\u0001r);\n13withci:= (r2\ni\u0000\u00112)\u00001=2. Note that, by the boundedness of ri, the terms o(\u0001\u0011) ando(\u0001r) in the\nabove formula are uniform in i. In the limit \u0001 r!0, for any 0\u0014i\u0014n\u00001, the total number\nof spherical grains in \n \\fz\u00150gwith radii in [ ri;ri+1] contributing to circular grains in the\ncross-section with radii in [ \u0011;\u0011+ \u0001\u0011] isN(t)q\u000eiRri+1\nri (r;t) dr, withN(t) being the number of\nferrite grains per unit volume given in (2.7). We sum over all transversal cylindrical pieces of \nwith volumes q\u000ei, accounting also for those in the left part of the specimen \n \\fz\u00140g(hence, by\nsymmetry, the factor 2 in the computation below), and use the boundedness of  and rto obtain\nqZ\u0011+\u0001\u0011\n\u0011\u001f(\u0010;t) d\u0010= 2N(t)qn\u00001X\ni=0\u000eiZri+1\nri (r;t) dr+\"(\u0001r)\n= 2N(t)qn\u00001X\ni=0\u000ei\u0010\n\u0001r (ri;t) +o(\u0001r)\u0011\n+\"(\u0001r)\n= 2N(t)qn\u00001X\ni=0\u0010\nci\u0011\u0001\u0011+ciri\u0001r+o(\u0001\u0011) +o(\u0001r)\u0011\u0010\n\u0001r (ri;t) +o(\u0001r)\u0011\n+\"(\u0001r)\n= 2N(t)qn\u00001X\ni=0 (ri;t)\u0011\u0001\u0011\u0001rq\nr2\ni\u0000\u00112+o(\u0001\u0011) +\"(\u0001r);\nwhere\"(\u0001r)!0 as \u0001r!0. By letting \u0001 r!0 in the above computation, we get\nqZ\u0011+\u0001\u0011\n\u0011\u001f(\u0010;t) d\u0010= 2N(t)qZrmax\n\u0011+\u0001\u0011\u0011\u0001\u0011 (r;t)p\nr2\u0000\u00112dr+o(\u0001\u0011):\nThen, dividing by q\u0001\u0011and passing to the limit with \u0001 \u0011!0, we \fnally obtain\n\u001f(\u0011;t) = 2N(t)\u0011Zrmax\n\u0011 (r;t)p\nr2\u0000\u00112drfor all\u00112(0;rmax]: (5.2)\nThis equation relates the three-dimensional radius distribution of a given specimen to the two-\ndimensional one in a cross-section of this specimen; note that this is independent of the area qof\nthe cross-section. In Figure 7 we give an example of comparison between  and\u001fforrmax= 3,\naccording to (5.2); there we are given a radius distribution  which is log-normal, as well as cut\no\u000b and normalised in the range (0 ;rmax], and then \u001fis computed thanks to (5.2) and normalised\nto have mass one in (0 ;rmax].\nLet us \fnally point out that an easy calculation shows that the surface fraction of ferrite over\nthe cross-section actually coincides with its volume fraction in the specimen. Call Psthe surface\nfraction of ferrite, i.e., the total surface of ferrite present on the cross-section normalised by q, and\nuse formula (5.2) to get, for all t2[t0;T],\nPs(t) =\u0019Zrmax\n0\u00112\u001f(\u0011;t) d\u0011= 2\u0019N(t)Zrmax\n0\u00113Zrmax\n\u0011 (r;t)p\nr2\u0000\u00112drd\u0011\n= 2\u0019N(t)Zrmax\n0 (r;t) Zr\n0\u00113\np\nr2\u0000\u00112d\u0011!\n|{z}\n=2r3=3dr=4\u0019N(t)\n3Zrmax\n0r3 (r;t) dr=P(t):\n140 0.5 1 1.5 2 2.5 3\nr, η00.20.40.60.811.21.4ψ(r), χ(η)3d radius distribution\n2d radius distributionFigure 7. Three-dimensional radius distribution against two-dimensional one.\nAcknowledgements\nD. H omberg gratefully acknowledges a sabbatical stay at the University of Bath where the\nresearch that led to this work was initiated. J. Zimmer gratefully acknowledges partial funding by\nthe EPSRC through project EP/K027743/1, the Leverhulme Trust, RPG-2013-261, and a Royal\nSociety Wolfson Research Merit Award. 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Alloys Compd. ,\n348(1):189{194, 2003.\n[29] M. Tomellini and M. Fanfoni. Why phantom nuclei must be considered in the johnson-mehl-avrami-kolmogoro\u000b\nkinetics. Phys. Rev. B , 55:14071{14073, Jun 1997.\n[30] G. E. Totten and M. A. H. Howes. Steel heat treatment handbook . CRC Press, Boca Raton, FL, 1997.\n[31] A. Tudorascu and M. Wunsch. On a nonlinear, nonlocal parabolic problem with conservation of mass, mean and\nvariance. Comm. Partial Di\u000berential Equations , 36(8):1426{1454, 2011.\nWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany\nand Department of Mathematical Sciences, NTNU, Alfred Getz vei 1, 7491 Trondheim, Norway.\nE-mail address :hoemberg@wias-berlin.de\nDepartment of Mathematics, Imperial College London, South Kensington Campus, London SW7\n2AZ, UK.\nE-mail address :fsp13@imperial.ac.uk\nMathematical Science & Technology Research Lab., Advanced Technology Research Laboratories,\nTechnical Development Bureau, Nippon Steel & Sumitomo Metal Corporation, 20-1 Shintomi, Futtsu-\nCity, Chiba, 293-8511, Japan.\nE-mail address :sakamoto.a2c.kenichi@jp.nssmc.com\nDepartment of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.\nE-mail address :zimmer@maths.bath.ac.uk\n16"    },    {        "title": "1512.06160v1.Cross_spectrum_Measurement_of_Thermal_noise_Limited_Oscillators.pdf",        "content": "Cross -spectrum Measurement of Thermal -noise Limited \nOscillators  \n \nA. Hati, C. W. Nelson  and D. A. Howe  \nTime and Frequency Division  \nNational Institute of Standards and Technology  \nBoulder, CO/USA  \n \n(Dated: December 18, 2015)  \n \nAbstract:  Cross -spectrum analysis is a commonly -used technique for the d etection of phase \nand amplitude noise of a signal in the presence of interfering  noise . It extract s the desired \ncorrelated noise from two time series in t he presence of uncorrelated interfering noise. \nRecently, we demonstrated that the phase -inversion  (anti -correlation) effect due to AM \nnoise leakage can c ause complete or partial collapse of the cross -spectr al function  [1], [2] . In \nthis paper , we discuss the newly discovered effect of anti-correlat ed thermal noise that \noriginates from the common -mode power divider ( splitter ), an essential component in a \ncross -spectrum noise measurement system. We studied this effect for different power \nsplitters and discuss its influence on the  measurement of thermal -noise  limited oscillator s. \nAn oscillator whose thermal noise is primarily set by the 50  source resistance is referred  \nto as a thermally -limited oscillator. We provide theory,  simulation an d experimental results.  \nIn addition , we expand this study to reveal  how the presence of ferrite -isolators and \namplifiers at the output ports of the power splitters can affect t he oscillator noise \nmeasurement s. Finally, we discuss a possible solution to ove rcome this problem.  \n \nKeywords —anti-correlation; cross -spectrum; collapse; isolators;  oscillator; phase \ninversion; power spectral density ; thermal noise  \n \nI.  INTRODUCTION  \n Oscillators enable much of our modern technology, including smart phones, GPS \nreceivers,  radar/surveillance/imaging systems, electronic test and measurement \nequipment , and much more. System designers and manufacturers need oscillators with \nthe lowest possible phase noise (timing jitter or spectral purity), especially for high \nperformance appl ications. However, the measurement of phase noise of many state -of-\nthe-art oscillators at current noise  levels  is challenging;  commercial phase noise \nmeasurement systems give results varying by more than  a factor of 10, often severely \nunder -reporting phase  noise  [3].  \n The cros s-spectrum  technique is a common  tool used  for the  measurement of low-\nphase and amplitude noise  oscillators  [4]–[14]. It uses two independent  channels  as \ndiscussed in Section II ; each consist s of a reference oscillator and a phase detector  (PD) \nthat simultaneously  measure s the noise of the oscillator under test. Computing the cross -\nspectral density of voltage fluctuations between two channels improves the spectral \nresolution of the noise measurements by reducing the effect of uncorrelated noise sources \nin each channel by √ m, where m is the number of averages of the  fast Fourier Transform \n(FFT). If t wo channels are statistically independent, the average cross spectrum \nconverges to th e DUT noise spectrum . Until  very recently, it was believed that the cross -\nspectrum method always over -estimates the measurement of DUT noise in the presence of correlated but unwanted and uncontrolled noise phenomen a affecting both channels  \n(DUT AM noise, vibration  induced noise , EMI  etc.)  However, it was demonstrated in [1] \nand [2] that if two time series, each composed of the summation of two fully independent \nsignals, are correlated in the first time signal and anti -correlated (pha se inverted) in the \nsecond, and have the same average spectral magnitude, the cross -spectrum power density \nbetween two time series is annihilated and collapses to zero. This effect can lead to \ndramatic under -reporting of the DUT noise. These conditions may  occur only at localized \noffset frequencies or over a wide range of frequency of the cross -spectrum . Significant  \npartial annihilation can occur if the interfering noise is within 10 dB of the desired noise. \nSuch interfering signal s can either be correlated  to the DUT or completely uncorrelated.  \nIn  [2], the anti -correlation collapse  mainly due to AM noise leakage  was discussed. More \nrecently a different source of anti -correlation in a cross -spectrum measurement has been \nidentified ; the o rigin is from the common -mode power splitter ( reactive Wilkinson  or \nresistive ).  The correlated thermal noise of the power splitter appears equally but in \nopposite phase in two channels of the cross -spectrum system. This  new source of phase -\ninverted interf ering noise was first addressed by Joe Gorin  [15].  As early as the y ear \n2000, anomalously low -noise in a cross -spectrum measurement system was reported by \nIvanov and Walls and it was interpreted that it is possible to measure the additive noise of \na device with an effective temperature much lower than the ambient temperatu re [16], \n[17]. Those results were in reality an observation of anti -correlated cross -spectrum \nthermal noise measurements . In this paper, we will discuss the influence of anti -\ncorrelated  thermal noise of the power splitter on the thermally -limited oscillator noise \nmeasurements. When we say thermally -limited, we mean that the white signal to noise \nratio of the  oscillator is at or near the level generated by the thermal  noise of a 50  \nsource resistor.  We will provide theory, simulation and experimental results and also \ndiscuss solutions to overcome this problem.   \nII. BRIEF OVERVIEW  OF COLLAPSE OF THE CROSS -SPECTRUM  \nA detailed theory and simulations of the positive correlation and anti -correlat ion \n(collapse) of the cross -spectral function are discussed in  [12] and [2] . In this section, w e \nbriefly revisit these two cases of cross -spectrum. Let us first consider  two signals x(t) and \ny(t), each composed of four statistically independent, ergodic and random processes a(t), \nb(t), c(t) and d(t)  such that  \n( ) ( ) ( ) ( ),\n( ) ( ) ( ) ( ).x t a t c t d t\ny t b t c t d t  \n  \n   (1) \nHere, c(t) and d(t) are the desired signal s that we wish to recover, and a(t) and b(t) are the \nuncorrelated interfering signals. If d(t) is correla ted in both x(t) and y(t), then the cross -\nspectrum , Syx(f) converges to the sum of the average power spectral densities  (PSD)  of \nc(t) and d(t)   \n ** 1( ) ( ) ( ).yx c d T S f CC f DD f S f S f    \n (2) \nThe cross -spectrum  Syx(f) is calculated  from the ensemble average of the product of \ntruncated  Fourier transform of time series x(t) and complex conjugate of Fourier \ntransform y(t).  T is the measurement time normalizing th e PSD to 1 Hz.   However, when  c(t) is correlated in x(t) and y(t) and d(t) is anti -correlated (phase inverted) in x and y as in    \n(3) such that  \n \n( ) ( ) ( ) ( ),\n( ) ( ) ( ) ( ),x t a t c t d t\ny t b t c t d t  \n           (3) \nthen the corresponding cross -PSD is represented as  \n    ** 1.yx c d T S f CC f DD f S f S f    \n  (4)  \n   \n(a) \n \n(b) \nFig. 1: Mathworks simulation results  of the cross -spectrum collapse when x(t) = c(t) + d(t), and y(t) = c(t) -  \nd(t): (a) for the addition of two completely independent noise sources, c(t) and d(t), each with power \nspectral density of -153 dB/Hz relative to unity.  (b) for two independent noise sources, c(t) and d(t), with \ndifferent frequency dependence are added. Signal Sc( f ) has a power spectral density of −153 dB/Hz \nrelative to unity. Signal Sd( f ) has a f −1 slope and intersects signal Sc( f ) at a frequency of 0.16 Hz.  \nThe cross -spectrum in (4) collapses to zero when c(t) equals d(t). In this paper we will \nmainly discuss the noise measurement conditions that are of type  (4). Mathworks \nSimulink simulation results for two different categories of the cross -spectrum collapse \nare depicted. Beginning with  Fig. 1a, a collapse over a wide range of offset frequencies \noccurs whe n two completely independent white noise sources, c(t) and d(t), each with \nequal power spectral density , are anti -correlated in x(t) and y(t). Second, a localized \ncollapse occurs ( Fig. 1b) due to the interaction of  two different sloped noise types, this \nappears as a notch in the magnitude of the cross -spectrum as well as 1800 change in its \nargument.  For this simulation we use the biased magnitude estimator\n yxmSf , as well \nas its pair  argument\n   yxmSf  for describing the amplitude and phase relationships  [12]. \n \nFig. 2: Plot of \n   yxmSf when two independent noise sources, c(t) and d(t), with different frequency \ndependence are added.   \nThe detection of a cross -spectrum collapse is diffic ult when noise slope s of the desired \nand interfering signal s are the same. However, in a case when two noise types intersect; \nthe appearance of a notch in the magnitude  of the cross -spectrum is a  clear indication of \nthe problem. This notch will also have a n associated  phase change of 180 degrees  in the \nargument  of the cross -spectrum as in Fig. 1b. In rectangular coordinates , this will be \nobserved as a change in sign of the real part of the cross -spectrum,  \n   yxmSf as shown \nin Fig. 2 .  \nIII. CHALLENGE S OF CROSS -SPECTRUM  NOISE MEASUREMENT OF A THERMAL LY-\nLIMITED OSCILLATOR   \nThe configuration of a cross -spectrum phase noise measurement is shown in  Fig. 3. \nHere  the comp onent noise contribution s from each parallel signal path within the dotted \ndashed blue box appear uncorrelated  and are rejected by the cross -spectrum while the \nnoise contributions of components in the red box appear  correlated and are retained in  the \noutpu t of the cross -spectrum . In addition to the DUT  noise , thermal noise of the power \nsplitter (PS in  Fig. 3) will also be correlated in both channels.  We will show later that the \nnoise from the green dotted box es with question mark and the PDs can appear as \ncorrelated if the PS doesn’t have high isolation  between the two outputs . The \nmeasurement of white PM or AM noise of most typical oscillators is not near the thermal  \nlimit  and therefore not significantly biased by the thermal noise of the common -mode -\npower splitter. However, recently several commercial ultra -low phase noise (ULPN) \noscillators have been introduced that are now reaching the thermal limit. In this new class \nof oscillators , the bias, either positive or n egative, from the power splitter thermal noise \nplays a dominant role . Repeatable and reproducible noise measurements of these ultra -\nlow noise thermally -limited oscillators have become difficult due to the effect of anti -\ncorrelated thermal noise or iginating  from the power splitter. An example of this problem \nis demonstrated in Fig. 4; where  the shaded band between the red and black curves \nrepresents the range of different phase noise measur ement results of the same U LPN \noscillator at 100 MHz.  Measurement are made using the cross -spectrum technique ; each  \nwith a slightly different configuration or components. For instance, the phase noise is \nmeasured either with different power splitters types (such as Wilkinson, resis tive 2 -R or \n3-R) or different components between the power splitter and the phase detector (such as \nattenuator/isolator/amplifier).  In addition to the thermal noise of the power splitter, the \nresults shown in Fig. 4 might also have been affected by uncontrolled AM noise leakage, \nground loops  or electromagnetic interference  (EMI).  Good metrology relies on method \nvalidation ; the results of different ly calibrated methods and measurement configurations \nshould match wit hin the measurement uncertainty [18], [19] . Fig. 4 clearly shows the \nresults var ying by more than a factor of 10 , either over or severely under -reporting the \nmeasured phase noise . \nRef #1\nLPF IF AMP\nFFT \nAnalyzerPLL\nRef #2DUT\nLPFIF AMP\nPLL1\n2ΣPD\nPD3 dB?\n?\n?\nCross-spectrum Measurement SystemPSPPS POUTPower \nSplitter\nIMHFIMHF\nIMHF ´ N ´ N S Offset voltage\nS \nOffset voltage´ N \n´ N \n \nFig. 3: Block diagram of a cross -spectrum phase noise measurement system. The green dotted box with “ ?” \nin each channel contains any one component of three shown inside the gray dotted box.   The frequency \nmultiplication factor ‘N’   1.  IMHF – Impedance matching and harmonic filtering, LP F – Low pass filter, \nPD – Phase Detector, FFT - Fast Fourier Transform, PLL – Phase L ocked Loop  \n \n \nFig. 4: Variation in the phase noise of a 100 MHz thermally -limited oscillat or measured with cross -\nspectrum system.  More than 10 dB  difference  in the phase noise is observed  either by changing the \ncommon -mode power splitter type or the measurement configuration . The bottom  noise plot is  limited by \nthe number of FFT averages m; for offset frequencies above 10 kHz , m = 100 ,000. The bottom black curve \nand the top red curve correspond to a Wilkinson power splitter with and without the isolation resistor (R i) \nrespectively . \nIV. EFFECT OF POWER SPLITTER THERMAL NOISE ON THE CROSS -SPECTRUM \nMEASUREMENT  \nIn the following sub -sections , we primari ly discuss the theory and simulation  studies \non the effect of thermal noise of various power -splitter  types on the cross -spectrum \nanalysis. Theoretical f indings are supported with experimental results.  \nA. Theory  \nThe schematic  representation  of a few power splitters [20]–[22] such as the \nconventional Wilkinson power splitter (CWPS), the modified Wilkinson power splitter \n(MWPS ), and the resistive 3 -R, 2-R and 1 -R power splitter  configuration s are shown in \nFig. 5a, b, c, d, e and f , respectively .  For the ideal case, the inser tion loss in both types of \nWilkinson power splitters (Fig. 5a and b) is 3 dB and isolation of each is  infini te. On the \nother  hand, for the resi stive power splitters 3 -R, 2-R and 1 -R the loss  is 6 dB  and the \ncorrespond ing isolation is 6 dB , 12 dB , and 2.5 dB , respectively . Also, a terminated 3 -R \nsplitter (Wye or Delta) present s a 50  impedance looking into any of the  three ports .  \nThe 2-R and 1-R power splitter s both have 50   input impedances, while present ing \n83.33   and 30  impedan ces, respectively , at their output ports.  \n \nR2\nR1\nR312\n3\nR1 = R2 = R3  = Rx = 16.67  l/4\nRi = 100  12\n3l/4l/4\n12\n3l/4\n(a) (b) (c) R = 50  Isolator\nIsolator\nR12\nR23\nR1312\n3\nR12 = R13 = R23  = Rx = 50  \n(d) R12\nR1312\n3\nR12 = R13 = Rx = 50  \n(e) R12\n3\n R = 25  for 50  load \n(f) R = 50  \n \nFig. 5: (a) Conventional Wilkinson power splitter, (b) Modified Wilkinson power splitter , (c) Resistive 3 -R \n(Wye configuration), (d) Resistive 3 -R (Delta  configu ration) , (e) Resistive 2 -R and (f) Resistive 1 -R. \n \nFor the analysis of thermal noise of power splitter s, we will consider the 3-R power \nsplitter in delta configuration since it closely resembles the CWPS. The equivalent circuit \nto the delta 3-R power split ter with thermal  voltage  noise sources for each resistor is \nshown in Fig. 6. The voltage source \nvS  corre sponds to the DUT source noise , and 12 13 23 1v  , v  , v  , v  n n n nLand \n2vnL  are respectively the thermal noise of the power splitter \nand the load resistances R 12, R13, R23, RL1 and R L2. Assuming a 50  system, a ll resistors \nin Fig. 6 are equal to 50  and the corresponding node volt ages \n1v, \n2v,  and \n3v  at port 1, \n2 and 3 can be written as  \n \n12 13 2 3\n1\n12 23 2 3\n2\n13 23 2 3\n32v + v v v vv = ,4\nv v v 2v vv = ,4\nv v v v 2vv = .4S n n nL nL\nS n n nL nL\nS n n nL nL  \n   \n   \n   (5) \nR12 = R13 = R23  = R = 50  \nRS = RL2= RL3 = 50  RL2\nvn23\nRL3R12\nR1312\n3R23\n+\n-vn12\nvn13\n+-\n+-RSv2\nVS+\n- +\n-+\n-\nvnL3vnL2\nv3v1\n \nFig. 6: Equivalent circuit of the delta 3-R power splitter including thermal  noise  sources for each resistor . \n  \nThe expectation of the cross -PSD between two output signals \n2v  and \n3v is represented by  \n \n *\n23 2 3\n* * * *\n23 23 2 2 3 31= V V\n11V V V V 2V V 2V V ,16S S n n nL nL nL nLS f f fT\nT   \n  (6) \nwhere, the Fourier transforms of  \n12 13 23 2 v , v , v , v , v  s n n n nL and \n3vnL  are represented by \nthe corresponding capitalized variables . From  (5) we see  that the source noise  is present \nequally and in same phase  at the output ports 2 and 3 in contrast to the thermal no ise of the resistor  R23 which appear s 1800 out of phase  between outputs 2 and 3.  The same \neffect occurs for the 100  isolation resistor used i n the CWPS .  The implication of this \nin the cross -spectrum (6) is that the expected value of thermal noise of the R23 is \nsubtracted from the noise in \nVS . It is also i mportant to note that due to the limited \nisolation of the resistive splitter, the thermal noise of the load resistors also appear \ncorrelated in both output channels .  In a perfect 50  system  the noise of the source \nVS  \nwill cancel o ut with \n23Vn , leaving only the thermal noise of the load resistors in  the \noutput cross -spectrum.  \nB. Simulation   \nThe propagation  of thermal noise in different types of power splitters w as simulated \nin the Advanced Design System ( ADS ) software . In addition to noise from the source  and \npower splitter , the thermal noise contribution of isolator s and load resistors was also \nanalyzed. Each thermal noise source was modeled a s a unique single sideband from the \ncarrier .  In this way each individu al noise source could be observed and its contribution to \nthe final cross -spectrum easily determined. A Circuit Envelope  method  of simulation was \nchosen to be able to include frequency dependent effects of the complex terminations,  \nreactive splitters , and to enable the inclusion of isolators in the analysis .  The contribution \nof the various circuit n oise sources to the cross -spectrum between the two outputs was \ndetermined.  Simulations for various power splitter configurations were tested . The block \ndiagram  for the simulation is shown in Fig. 7. \n \nRL1\nRL21\nDUT: RSRISO1\nRISO2Ch-1\nCh-2x(t)\ny(t)32 Power \nSplitterOptional \nIsolators\n \nFig. 7: The main sources of thermal noise used for ADS simulation. Rs represents the thermal noise of the \nsource or the device under test (DUT).  Total number of resistors in the power splitter varies from 0 to 3 \ndepending on the configuration. Option al isolators, with thermal resistances are indicated by R ISO1 and \nRISO2. The load resistors R L1 and R L2 represent the thermal noise of the measurement system.   \n \nTable 1 tabulates the results of thermal noise contribution of the individual  component to \nthe output cross -spectrum as a fraction of the noise from  Rs. The simulation is performe d \nfor load and  source impedance s equal to 50  and at 300 K  temperature . The values \nreported in the table are from the expected value of the cross -spectrum.  All uncorrelated \ncross -terms, which reside in the imaginary component of the cross -spectrum , have \naveraged to zero and the result is an entirely real  component .  Table 1: 2-way Power Splitter  (PS): Source impedance  (ZS) = Load Impedance  (ZL) = 50 ,  \nT = 300 K , Isolator :  Insertion L oss =  0 dB , Isolation =  \n \n1 2 3 4 5 6 7 8 9 10 \nCase \n# Type of \nPower \nsplitter (PS)  Relative cross -spectrum of individual  component  \n 21s ch ch R S f S f\n Total Noise  \nRs Power Splitter   \nISO#1 \n  \nISO#2  \n  \nRL#1 \n  \nRL#2 \n Without \nRs   All \nComponents   \nR1  \nR2  \nR3 \n1 Wilkinson  \nRi = 100  1 -1 - - 0 0 -1 0 \n2 Wilkinson  \nRi  =  1 - - - -3/2 -3/2 -3 -2 \n3 Wilkinson  \nRi  = , \nIsolators  1 0 -0.5 -0.5 0 0 -1 0 \n4 3-R Wye  \nRx = ~17  1 1/3 -2/3 -2/3 - - 2 2 3 4 \n5 3-R Wye,  \nRx = ~17 , \nIsolators  1 1/3 -2/3 -2/3 0 0 0 0 -1 0 \n6 3-R Delta  \nRx = 50  1 0 -1 0 - - 2 2 3 4 \n7 3-R Delta,  \nRx = 50  \nIsolators  1 0 -1 0 0 0 0 0 -1 0 \n8 2-R \nRx = 50  1 0 -3/4 -3/4 - - 5/4 5/4 1 2 \n9 2R , \nRx = 50 , \nIsolator  1 0 -3/4 -3/4 1/4 1/4 0 0 -1 0 \n10 1-R, \nR = 25  \n 1 1/2 - - - - 9/5 9/5 5 6 \n11 1-R, \nR = 25 , \nIsolator  1 1/2 - - -3/4 -3/4 0 0 -1 0 \nHere, R i and Rx respectively correspond to the isolation resistor  and the resistor s for 2-R and 3 -R power \nsplitters . \n \nReferring to Table 1, column 1, let  us first consider case #1 for the CWPS  where the \nisolating resistor (R i) is 100 .  The power splitter  noise (Ri, column 4)  is equal in \nmagnitude to the source noise  (Rs, column 3)  but 1800 out of phase .  A negative real \nportion of the cross -spectrum indicates an anti -correlated cross -spectrum .   Column s 5 \nand 6 are blank because no isolators were used for this case. The noise contribution of the \nload resistors in columns 7 and 8 is zero due to  the large  isolation  between two output \nports of the CWPS . The summed noise contribution of all the individual components \nexcept R s is presented in column 9.  For case #1, the noise of the isolating resistor (R i) is equal and anti -correlated (indicated by a negative real quantity) to the R s source resistor \nnoise.  In the final column  10 the total noise of all components is shown.  The total noise \nof all components is zero for case #1, a clear indication of a complete cross -spectrum \ncollapse.  Note that all the power splitting configurations shown in Ta ble 1 exhibit either a \ncomplete cross -spectral collapse or  are limited by the noise of the load resistances: none \ncan measure the noise of R s. \n \nAdditional simulations, with realistic isolator parameters (isolation of 10 to 30 dB , \ninstead of ) produced various intermediate levels of partial correlation collapse. The \nresults from these simulation s for various power splitter s are shown in  Table 2. \n \nTable 2:  2-way Power Splitter (PS): Source impedance (Z S) = Load Impedance (Z L) = 50  ,  \nT = 300 K, Iso lator:  Insertion Loss = 0 .5 dB, Isolation = 15 dB \n1 2 3 4 5 6 7 8 9 10 \nCase \n# Type of \nPower \nsplitter (PS)  Relative cross -spectrum of individual  component  \n 21s ch ch R S f S f\n Total Noise  \nRs Power Splitter   \nISO#1 \n  \nISO#2  \n  \nRL#1 \n  \nRL#2 \n Without \nRs   All \nComponents   \nR1  \nR2  \nR3 \n1 Wilkinson  \nRi  = , \nIsolators  1 - -0.62 -0.62 -0.21 -0.21 -1.66 -0.66  \n2 3-R Wye,  \nRx = ~17 , \nIsolators  1 1/3 -2/3 -2/3 0.35 0.35 0.38 0.38 0.46 1.46 \n3 3-R Delta,  \nRx = 50  \nIsolators  1 0 -1 0 0.35 0.35 0.38 0.38 0.46 1.46 \n4 2-R , \nRx = 50 , \nIsolator  1 0 -3/4 -3/4 0.40 0.40 0.20 0.20 -0.30 0.70 \n5 1-R, \nR = 25 , \nIsolator  1 1/2 - - -0.14 -0.14 0.54 0.54 -1.30 2.30 \n \n \nIn addition to the power splitter s discussed in Table 1, other device s such as \ndirectional couplers, 900 and 1800 hybrids, and N-way power splitters were tested. They \nall introduced phase -inverted  thermal noise  between two channels.  \n \nThe conclusion s of the simulation are as follows:  \n1. Resistive power splitters  \na. Resistive power splitters do not have sufficient isolation to allow a cross -\nspectrum measurement to overcome the loss of signal to noise in each \nindividual channel.  They cannot be used to accurately measure a thermally \nlimited source because the dominating n oise of the load to the power splitter \nappears correlated in both channels  and cannot be rejected.  \nb. 3-R (Delta or Wye) and 2 -R splitters produce anti -correlated thermal noise between the outputs.   \n2. Reactive splitters (Wilkinson, Hybrid -90, Hybrid -180, coupl ers) \na. The isolation resistor produces anti -correlated thermal noise which is equal \nin magnitude to that of the source resistor.  \nb. While removal of the isolation resistor (i.e. , MWPS) eliminates the anti -\ncorrelated thermal noise, it destroys the isolation and prevents the \nmeasurement of a thermally limited source.  \n3. Isolators, which are circulators with the third port terminated with 50 , present \nthe thermal noise of the isolating resistor at their input.  This makes them \nessentially useless for improving the pe rformance of a splitter with low \nisolation in terms of thermal noise.  \na. Patching any of the splitters configurations (MWPS, 3 -R, 2-R or 1 -R) with \nan ideal isolator produces a complete thermal noise correlation collapse.  \nb. Simulation with realistic isolator par ameters (isolation of 10  to 30 dB) \nproduces various intermediate levels of partial correlation collapse.  \n \nC. Experimen tal Result s  \nFor the experimental verification  we chose an ultra -low-phase noise oscillator  at \n100 MHz . The schematic of the oscillator is sh own in Fig. 8a, it contains a high -Q clean -\nup filter  and 3 dB attenuator  at the output.  The output impedance of this oscillator is \nfrequency dependent , it presents 50  at the resonant frequency but a non -50   \nimpedance at Fourier frequenc ies away from the resonance   [22].  Measured s-parameter  \nS11  and the smith chart for this oscillator are also  shown in Fig. 8b and c .  \n \n \nFig. 8: (a) Schematic of th e 100 MHz oscillator with cleanup filter , (b) Measured data of  S11 in dB and \n(c) Smith chart .  \nThe measurement  set-up as  shown in  Fig. 3 is used for the phase noise  measurement of \nthis oscillator . A variable dc offset voltage was added at the input of the PLL integrator  to \noptimize  the rejection of the DUT AM noise. With a few exceptions, the AM  noise  of the \nDUT was rejected by more than 30 dB to minimize the effect of anti -correlation collapse \ndue to  the AM nois e leakage. The phase noise of the oscillator was first measured with a \nCWPS . Assuming a 50  system, and taking into account the DUT power loss in the \nimpedance matching and harmonic filtering ( IMHF ) circuit  in the common path, the \ntheoretical  noise should  be -189.5 dBc/Hz  i.e., -177 - PPS. As shown in  Fig. 9, a complete  \n1\n2Σ\nDUT3 dBPower splitter\n(b) (c) (a)collapse (limited by the number of FFT averages) of the noise spectrum was observed \ndue to the anti -correlated thermal noise of the CWPS.  Initially , it was thought that the \nproblem of thermal noise of the CWPS could be resolve d by removing the 100  \nisolating resistor and the required isolation restored with the introduction of ferrite  \nisolators at the o utput s of the power divider.  This modification of the power splitter is \nrepresented as MWPS and shown in Fig. 5b. The MWPS  does provide good isolation \nbetween port 2 and 3, however, the thermal noise of the 50  termination of these \nisolators appears via port 1  and a ½ -wave transmission line  to the other channel again \ncausing an anti-correlation  collapse . The phase relation of the isolators between two \nchannels can be seen in case #3 of Table 1 and the resulting measured  PM noise of the \nDUT affe cted by the anti -correlated thermal noise of the MWPS is shown  in Fig. 9. \n \nFig. 9: Phase noise of a 100 MHz oscillator measured with a conventional Wilkinson power s plitter  \n(CWPS ) and a modified WPS . Theoretical noise  of this oscillator  referenced to  the input power of \ncommon -mode power splitter , PPS is -189.5  dBc/Hz calculated from (-177  – PPS) assuming a 50  \nsystem . The far -from -the-carrier noise in both cases are limited by the maximum FFT number, N = \n100 ,000  available on the analyzer but there is  clear  indication of a spectrum collapse .  \n \nWe also measured the phase noise of the same 100 M Hz oscillator using resistive 1 -R, \n2-R and 3 -R (Wye configuration) power spli tters. We observed large variation s in the \nmeasured phase noise. For each splitter type, three different measurement configurations \nwere used: ( a) a direct connection between power splitter and the phase detector, ( b) a \nferrite isolator was introduced betw een the power splitter and the phase detector in each \nchannel, and ( c) the isolators were replaced with amplifiers (as shown in the inset of Fig. \n10). For configurations ( a) and ( b), the measured and the simulated thermal phase noise \ndid not agree  because the simulations  were performed with  an ideal 50   load \nimpedance . However, in actual practice  the power splitter is connected to  the reactive \nload of the double balanced mixer used as a phase detector.  When the i solator is replaced \nby an amplifier ( c) it provide s higher isolation and a better impedance match out of the \npower splitter. This configuration resulted in a closer agreement between the simulation \nand the experimental results. Fig. 10 shows the experimental result of phase noise \nmeasured with amplifiers  as well as a strong anti -correlation collapse li mited only by the \nnumber of FFT averages. It is also observed that the amount of anti -correlation collapse \nincreases  with higher isolation between the power splitter and the phase detector.  \n \n  \nFig. 10: PM noise of the 100 MHz oscillator measured with resistive 2-R and 3 -R power splitter s. The \nmeasurement configuration is depicted in the inset. T he measured thermal  phase  noise  is average limited \nbut again there is an indication of noise spectrum collapse due to the anti-correlation effect. The measured \nnoise is  significantly lower than the theoretical thermal no ise.   \nSimilar tests were performed for AM measurements  of the same oscillator using the \nconfiguration shown in Fig. 11. The power splitter is directly connected to  an AM \ndetector in each channel  whose input impedance is almost a perfect  50 .  AM no ise was \nmeasured with resistive and reactive power splitters and the thermal AM matches the \nsimulation result s for each power splitter as shown in Fig. 12. The simulation results \ncorrespond to case #1 0, #8, #4, and #1 in Table 1. For resistive splitter s there is  a \npositive -correlation  and the noise is higher than the theoretical thermal noise . This is due \nto the lack of isolation between the AM detectors ’ input impedance noi ses.  On the other \nhand, t he AM noise measured with the CWPS  leads to an  anti-correlation  collapse  as \nexpected from the simulation.   \nIF AMP\nFFT \nAnalyzer\nIF AMPAM \nDetector\nAM \nDetectorDUT\n1\n2Σ3 dB\nPSPPS POUTPower \nSplitter\nIMHF \nFig. 11: Block diagram of a dual -channel cross -spectrum system used for  measuring AM noise of the DUT.   \nIMHF – Impedance matching and harmonic filtering  \n \n \nFig. 12:  AM noise of the thermally limited 100 MHz oscillator measured with resistive 1 -R, 2-R and 3 -R \nand Wilkinson power splitters. There is a cl ose agreement between experiment and the simulation results.  \nThe simulation results correspond to case # 1 0, #8, #4 and #1 in Table 1. \n \nV. SUMMARY  \nWe discussed the difficult challenge of PM and AM noise measurement of oscillators  \nat or near the thermal limit  of the source impedance and also discussed the limitations of \nthe cross -spectrum system widely used for such measurements. Our conclusions from \ndifferent simulation and experimental results are as follows:  \n \n1. While r eactive power splitters such as the Wilkinson have sufficient isolation to \nmeasure the thermal noise of source resistance ( Rs), the thermal noise of the isolation \nresistor (Ri) appears anti -correlated and is subtracted from the R s noise in the cross -\nspectr um.  This produces a complete collapse in a perfect 50  system.    \n2. Resistive power splitters do not have sufficient isolation to allow a cross -spectrum \nmeasurement to overcome the loss of signal to noise in each individual channel.  \nThey cannot be used to accurately measure a thermally -limited source because the \ndominating noise of the load to the power splitter appears correlated in both channels.  \n3. Patching any of the low -isolation spl itters configurations (MWPS, 3 -R, 2-R or 1 -R) \nwith an ideal isolator also produces a complete correlation collapse of thermal noise.  \nSimulation with realistic isolator parameters (isolation of 10  to 30 dB) produces \nvarious intermediate levels of partial correlation collapse.    \n4. In practical measurements, the delicate balance be tween correlated terms and anti -\ncorrelated  terms that cause these partial or complete collapses are subject to \nenvironmental and circuit variations that make the measurement of noise near the \nthermal limit of R s extremely difficult to do with any confidenc e.  \n \nIn conclusion, all room temperature power splitter configurations we tested, reactive or \nresistive , introduce  either positive or negative correlation  biases for heterodyne cross -\nspectrum measurements near the thermal limit . Any measurement within 10 d B of the \nthermal limit will have significant  bias.   \nOne possible solution to mitigate this problem is to cool the power splitter to cryogenic \ntemperatures. If the power splitter is cooled to a liquid helium temperature (4 K), then its \nthermal noise will d ecrease by 19 dB compared to room temperature (300 K). In the near \nfuture, we will test a cryogenic Wilkinson splitter to measure the noise of an ultra -low-\nthermal - noise limited oscillator.  However, the non -50  output impedance of such \noscillators may ca use problems in that they degrade the isolation of the Wilkinson splitter \nand may cause measurement limitations even when the isolation resistor noise is \neliminated.  We also will perform a similar analysis on power splitter configurations for \nresidual hom odyne methods.  \n \nACKNOWLEDGEMENT  \nAuthor s thank Joe Gorin of Keysight Technologies for discovering the important aspect \nof phase inversion (anti -correlation) in the power splitters that helped to explain the \nunrealistically low noise level of the thermally l imited oscillators.  \n \nREFERENCES  \n \n[1] C. W. Nelson, A. Hati, and D.  A. Howe, “Phase inversion and collapse of cross -\nspectral function,” Electronics Letters , vol. 49, no. 25, pp. 1640 –1641, Dec. 2013.  \n[2] C. W. Nelson, A. Hati, and D. A. Howe, “A collapse of the cross -spectral function in \nphase noise metrology,” Review of Scientific Instruments , vol. 85, no. 2, p. 024705, \nFeb. 2014.  \n[3] A. K. Poddar, U. L. Rohde, and A. M. Apte, “How Low Can They Go?: Oscil lator \nPhase Noise Model, Theoretical, Experimental Validation, and Phase Noise \nMeasurements,” IEEE Microwave Magazine , vol. 14, no. 6, pp. 50 –72, Sep. 2013.  [4] R. Vessot, L. Mueller, and J. Vanier, “The specification of oscillator characteristics \nfrom mea surements made in the frequency domain,” Proceedings of the IEEE , vol. \n54, no. 2, pp. 199 –207, 1966.  \n[5] F. L. Walls, S. R. Stein, J. E. Gray, and D. J. Glaze, “Design Considerations in State -\nof-the-Art Signal Processing and Phase Noise Measurement Systems ,” in \nProceedings of the 30th Annual Symposium on Frequency Control , 1976, pp. 269 –\n274.  \n[6] F. L. Walls and E. Ferre -Pikal, “Measurement of Frequency, Phase noise and \nAmplitude noise,” in Wiley Encyclopedia of Electrical and Electronics Engineering , \n1st ed ., vol. 12, 24 vols., Wiley -Interscience, 1999, pp. 459 –473.  \n[7] D. Fest, J. Groslambert, and J. -J. 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Vernotte, “The cross -spectrum experimental method,” arXiv e -\nprint 1003.0113, Mar. 2010.  \n[13] U. L. Rohde and A. K. Poddar, “Phase noise measurement techniques, associated \nuncertainty, and limtations,” in European Frequency and Time Forum International \nFrequency Control Symposium (EFTF/IFC), 2013 Joint , 2013, pp. 438 –441.  \n[14] A. K. Poddar and U. L. Rohde, “The pursuit for low cost and low phase noise \nsynthesized signal sources: Theory amp; optimization,” in Frequency Control \nSymposium (FCS), 2014 IEEE International , 2014, pp. 1 –9. \n[15] J. Gorin, “Cross -spectrum L(f) workshop,” presented at the 2015 Joint \nConference of the IEEE International Frequency Control Symposium & European \nFrequency an d Time Forum, Denver,USA, Apr -2015.  \n[16] E. N. Ivanov and F. L. Walls, “Effect of anomalously low voltage noise in \nmeasurement systems with cross -correlation signal processing,” in Conference on \nPrecision Electromagnetic Measurements. Conference Digest. CP EM 2000 (Cat. \nNo.00CH37031) , Sydney, NSW, Australia, pp. 447 –448.  \n[17] E. N. Ivanov and F. L. Walls, “Interpreting anomalously low voltage noise in two -\nchannel measurement systems,” IEEE Transactions on Ultrasonics, Ferroelectrics \nand Frequency Control , vol. 49, no. 1, pp. 11 –19, 2002.  \n[18] BIPM, “Evaluation of measurement data — Guide to the expression of \nuncertainty in measurement ,” 2008.  [19] B. N. Taylor and C. E. Kuyatt, “Guidelines for Evaluating and Expressing the \nUncertainty of NIST Measurement Resu lts,” NIST Technical Note 1297 , 1994.  \n[20] D. M. Pozar, Microwave Engineering , 4 edition. Hoboken, NJ: Wiley, 2011.  \n[21] “Wiley: Microwave and Wireless Synthesizers: Theory and Design - Ulrich L. \nRohde.” [Online]. Available: \nhttp://www.wiley.com/WileyCDA/ WileyTitle/productCd -0471520195.html. \n[Accessed: 30 -Oct-2015].  \n[22] J. Breitbarth, “Source impedance influence on cross -correlation phase noise \nmeasurements,” in European Frequency and Time Forum International Frequency \nControl Symposium (EFTF/IFC), 2013 J oint, 2013, pp. 434 –437.  \n \n "    },    {        "title": "2104.11817v2.Plasticity__localization__and_damage_in_ferritic_pearlitic_steel_studied_by_nanoscale_digital_image_correlation.pdf",        "content": "1  Vermeij & Hoefnagels, Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation, Scripta Mater. (2021) Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation  T. Vermeij1 & J.P.M. Hoefnagels1,* 1 Dept. of Mechanical Engineering, Eindhoven University of Technology, The Netherlands *Corresponding author: j.p.m.hoefnagels@tue.nl   https://doi.org/10.1016/j.scriptamat.2021.114327  Abstract The evolution of deformation from plasticity to localization to damage is investigated in ferritic-pearlitic steel through nanometer-resolution microstructure-correlated SEM-DIC (µ-DIC) strain mapping, enabled through highly accurate microstructure-to-strain alignment. We reveal the key plasticity mechanisms in ferrite and pearlite as well as their evolution into localization and damage and their relation to the microstructural arrangement. Notably, two contrasting mechanisms were identified that control whether damage initiation in pearlite occurs and, through connection of localization hotspots in ferrite grains, potentially results in macroscale fracture: (i) cracking of pearlite bridges with relatively clean lamellar structure by brittle fracture of cementite lamellae due to build-up of strain concentrations in nearby ferrite, versus (ii) large plasticity without damage in pearlite bridges with a more “open”, chaotic pearlite morphology, which enables plastic percolation paths in the interlamellar ferrite channels. Based on these insights, recommendations for damage resistant ferritic-pearlitic steels are proposed.  Graphical Abstract \n  Keywords: Ferritic-pearlitic steel, nanoscale digital image correlation, plasticity, damage, localization   \n2  Vermeij & Hoefnagels, Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation, Scripta Mater. (2021) Ferritic-pearlitic steels are strong and formable through the combined effects of ferrite grains and pearlite colonies, comprising alternating lamellae of randomly distributed and oriented cementite within a ferritic matrix [1,2]. Deformations in ferritic-pearlitic steels intrinsically arise from the nanometer scale arrangements of cementite and ferrite, while the ferritic-pearlitic distribution determines how plasticity and damage will propagate and coalesce, ultimately leading to failure. In the literature, challenges have arisen from the intrinsic multi-scale nature of deformations in ferritic-pearlitic steel, inhibiting identification of deformation mechanisms and potential experimental-numerical comparisons [3–10].  Deformations in ferritic-pearlitic steels can be observed through in-situ mechanical testing in a scanning electron microscope (SEM) [3,6–8]. For example, pearlite plasticity was found to be controlled by ferrite deformation, with damage occurring through cementite cracking, likely caused by dislocation pile-up at the ferrite-cementite interfaces [3,7]. Previously, we employed digital image correlation (DIC) on SEM micrographs to measure strain fields in banded microstructures in dual-phase (ferrite-martensite) and ferritic-pearlitic steels, wherein pearlite appeared to deform plastically [6]. However, the SEM microstructure maps that were employed as DIC patterns limited the spatial resolution of the strain fields to several micrometers, which was insufficient to resolve deformations at lamellar scales. Such studies (and others) reveal limited and contrasting conclusions on the role of cementite lamellae on pearlite deformation, moreover, the full evolution (or lack thereof) of plasticity to damage is rarely studied. More recently, advances in DIC speckle patterning techniques enables SEM-DIC (µ-DIC) for highly robust measurements of localized strains at micrometer and even nanometer scales [11–14]. Such high spatial strain resolutions permit experimentalists to resolve plasticity between (or inside) cementite lamellae and provides the opportunity to simultaneously monitor damage initiation in the microstructure [12,13]. Damage and failure mechanisms at the micro- and nanoscale are notoriously complex and can only be unraveled by relating them to the preceding unique plastic deformation mechanisms, in order to provide proper understanding to enable accurate model prediction and/or material design optimization. Plasticity and damage have been studied for the same material (e.g. [9]), however, not both at the same location in the microstructure. In fact, to our knowledge, the complete full-field evolution from preceding microscale plasticity to resulting damage has yet to be studied for ferritic-pearlitic steels. Here, we measure strains with high-resolution and robust SEM-DIC all the way up to damage and local fracture. Careful alignment between ferrite-pearlite-cementite morphology and nanometer resolution strain fields reveals how plasticity localizes in relation to ferritic-pearlitic interfaces and (individual) cementite-ferrite lamellae. Upon continued deformation, damage occurs at critical microstructure locations. These are unraveled on the basis of the preceding strain evolution, providing insights in damage mechanisms. Finally, we discuss potential strategies for increased robustness to early damage and failure in ferritic-pearlitic steels. A ferritic-pearlitic steel plate (0.17C-0.16Si-1.27Mn-0.014P-0.012Cr-0.037Al-0.038Ni wt.%) was retrieved after 45 minutes austenization at 900 °C and subsequent cooling in the furnace. Samples were cut and metallographically prepared to a final mechanical polish (OPS colloidal silica particles), followed by a 5-seconds, 2%-Nital etching step. Mapping of the microstructural arrangement of ferrite, pearlite and cementite was performed in a Tescan Mira 3 SEM using backscattered electron (BSE) imaging. Figure 1(a) shows the chosen region of interest with ~20 µm ferrite grains and pearlite with ~225 nm lamellar spacing.  The crystallographic orientation of all ferrite phase was mapped with EBSD (Edax Digiview 2 camera) at 50 nm pixelsize, in order to identify the ferrite grain orientations and grain boundaries as well as the orientation of the ferrite matrix within the pearlite. Subsequently, to enable high-resolution SEM-DIC strain measurements at lamellar scales, a well-controlled, highly dense, nanometer sized speckle pattern was applied using a recently proposed novel patterning method, i.e. single-step sputter deposition of a low temperature solder alloy that naturally forms nanoscale islands during deposition [13]. Specifically, In52Sn48 was sputtered at 15 mA, 1E-2 mbar, 25 °C, 180 s, according to (c) in Table 1 of Hoefnagels et al. [13], resulting in a random, dense, high-quality pattern with features ~25-100 nm in size as shown in Figure 1(b). Thereafter, the sample was deformed through 3  Vermeij & Hoefnagels, Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation, Scripta Mater. (2021) uniaxial tension using a micro-tensile stage (Kammrath & Weiss) in the SEM. To capture the strain fields, the DIC pattern was imaged intermittently using 5 kV in the region of interest at 15 nm pixel size, according to guidelines given in [13]. SEM images of 4096 by 4096 pixels resulted in a 60´60 µm2 field of view.  DIC was performed with VIC-2D software using the following main parameters: subset size of 21 pixels (325 nm), step size 1 pixel (15 nm) and linear subset shape functions, see Hoefnagels et al. [13] and the DIC guide [15] for more details. This small step size appears to yield the highest possible spatial strain resolution, which is critical to distinguish the high strain levels in the ferrite lamellae in pearlite from the low strain levels in the adjacent cementite lamellae. The Green-Lagrange strain tensor 𝜺=!\"[\t%∇#''''⃗𝑢'⃗*$+∇#''''⃗𝑢'⃗+%∇#''''⃗𝑢'⃗*$∙∇#''''⃗𝑢'⃗\t] is calculated from the DIC displacement field, 𝑢'⃗(𝑥⃗), through the (numerically computed) displacement gradient tensor, ∇%𝑢'⃗. We use the 2D equivalent von Mises strain measure, defined as 𝜀&'(𝑥⃗)=√𝟐*2%𝜀++−𝜀,,*\"+𝜀++\"+𝜀,,\"+6𝜀+,\", as it is a good indicator for (local) plasticity [16]. Highly accurate alignment of the strain fields to the ferritic-pearlitic microstructural morphology is required to attribute the domains of high/low strains to, respectively, large/small plasticity in the ferrite/cementite lamellae. Therefore, micrographs were captured at 20 kV (before tensile testing) to image both the DIC speckle pattern and the underlying ferritic-pearlitic morphology (not shown) for the microstructure to strain fields alignment. Linear interpolation was employed through several selected homologous points in both microstructure datasets (before and after pattern application), yielding a “distortion” field with which the full dataset was aligned. In contrast to SEM images, the EBSD map contains severe spatial distortions [17], therefore, a novel procedure based on global-DIC [18] was used to align the EBSD map to the microstructure, which enables to plot the ferrite grain boundaries as overlay to all figures in this work. To intuitively link strains to the ferritic-pearlitic morphology, we plot the aligned microstructure maps over the strain fields as an overlay with variable transparency, such that only the cementite lamellae are clearly visible (using a Jet colormap that is cut off to distinguish the cementite), see Figure 1(c). Note that all presented strain fields are plotted in the deformed configuration, by forward-deforming the microstructure and strain field datasets (from their reference configuration) using the DIC displacement field.  To enable nanoscale damage analysis, (post-mortem) BSE scans were acquired at 20 kV, after conclusion of the in-situ test, such that the electron beam protrudes the nanometer thickness InSn DIC pattern. The post-mortem 20 kV BSE scan is aligned to the (last increment of the) forward-deformed 5kV SEM scan of the microstructure, by selection of homologous points (such as clearly recognizable undamaged cementite lamellae). After all processing steps, the aligned data was imported in MTEX [19] for consistent plotting requiring no manual adjustments, such as cropping, alignment, etc. 4  Vermeij & Hoefnagels, Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation, Scripta Mater. (2021)  Figure 1: Overview of microstructure and deformations of the complete region of interest: (a) BSE microstructure map, showing ferrite grains, pearlite colonies and cementite lamellae. (b) DIC pattern on the same region of interest. (c) SEM-DIC equivalent strain field at a global von Mises strain of 𝜀!\"\"\"\"\"\"=0.19. The ferritic-pearlitic morphology is mapped as a transparent overlay, with stronger transparency for ferrite than cementite and grayscale inverted. (d) Post-mortem BSE image shows damage (at 𝜀!\"\"\"\"\"\"=0.36), with a pronounced damage event circled in red. For each dataset, a magnified inset of the same area is included to highlight the alignment, spatial resolution and quality of the data; note also the small scalebar of 500 nm. Grain boundaries of ferrite, derived from a spatial-distortion corrected EBSD scan, have been overlayed as red (a,b,d) and black (c) lines. In insert (e) strain and microstructure data is plotted over a line profile taken from the magnified inset of (c), showing nanoscale strain partitioning between cementite and ferrite lamellae.   \n5  Vermeij & Hoefnagels, Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation, Scripta Mater. (2021) The evolving equivalent (𝜀&') strain fields are computed for 4 subsequent global deformation levels of: 𝜀&'55555 = 0.07, 𝜀&'55555 = 0.12, 𝜀&'55555 = 0.19 and 𝜀&'55555 = 0.36. The global von Mises strain 𝜀&'55555 is defined as the 2D equivalent von Mises strains averaged over the FOV. Therefore, we do not rely on the nominal strain, derived from the clamp displacements, as it is unreliable due to the heterogeneous deformations over the gauge section and slip in the clamps. The global engineering stresses at these deformation steps were 250 MPa, 475 MPa, 525 MPa and 550 MPa, respectively. The strain field at 𝜀&'55555\t=\t0.19 in Figure 1(c) shows how the ferritic-pearlitic microstructure deforms highly heterogeneously. As expected, strains proceed through the microstructure under angles of approximately ±45° with respect to the horizontal loading direction, predominantly propagating through ferrite channels between pearlite colonies. Indeed, ferritic-pearlitic strain partitioning is the dominant deformation mechanism which appears to evolve into ferritic-pearlitic interface “decohesion”, as observed in the center of Figure 1(d), indicated with the red ellipse. However, detailed analysis of the BSE contrast, which shows darker regions in the ferrite grains near the pearlite colonies in Figure 4(b5) and (c5), indicates these to be so-called “distributed nano-damage” (similar to high-resolution observations of ferritic-martensitic steels, identifying distributed nano-damage in ferrite near martensite as the main damage mechanism [20]). For both types of steel, however, interface “decohesion” between ferrite and the ‘hard’ phase (pearlite or martensite) is typically reported as the main damage mechanism, which is likely caused by the scales of observation as distributed nano-damage can only be recognized at high magnification. Next, we focus on the high spatial resolution of the strain fields. Particularly, strain partitioning in regions with interlamellar spacing <100 nm shows that we can resolve strains at these small dimensions. To clarify the actual spatial strain resolution and the effectiveness microstructure-to-strain alignment, Figure 1(e) shows a line plot of microstructure and strain data taken over the line shown in the inset of Figure 1(c). The dashed black line shows the SEM image intensity, corresponding to cementite at high values and ferrite at low values. The equivalent strain data is shown through the dotted red line, with several steps having a width of ~40 nm as marked by the blue boxes, giving an indication of the resolution [21]. Furthermore, the effectiveness of microstructure-to-strain alignment is validated here by the occurrence of strain peaks between cementite lamellae, as expected. Even though the spatial strain resolution appears to be as low as ~40 nm, one should be careful in assessing the strain magnitude in the lamellae for lamellar spacings of similar magnitude as the strain resolution, which is not always taken into account in literature. Next, we investigate how plasticity propagates through pearlite colonies. Typically, plasticity initializes in ferrite as hotspots near pearlite. Upon increased loading, these hotspots need to be connected, often requiring percolation through pearlite colonies. In Figure 2, two smaller areas (taken from the larger region of interest in Figure 1) show two contrasting pearlite deformation modes. For percolation paths aligned with the pearlite lamellar direction, see Figure 2(a), the plasticity hotspots can connect by protruding between the cementite lamellae, through well aligned or chaotic (“open”) lamellae structures. The deformation is localized predominantly in interlamellar ferrite, only occasionally crossing single cementite lamellae, resulting in a strong cementite-ferrite strain partitioning. In contrast, Figure 2(b) highlights examples of deformation percolation paths perpendicular to the pearlite lamellar direction, where the deformation is inhibited by continuous, non-chaotic cementite lamellae. In this case, as the cementite lamellae block the formation of localization bands inside the interlamellar ferrite, the deformation is smeared out over wide, intermittent, bands with much lower cementite-ferrite partitioning. Analysis of the principal strain fields (not shown) reveals that these bands deform approximately by the globally applied deformation, which is uniaxial tension, in contrast to the percolation paths in Figure 2(a). In essence, the clear contrast between these two configurations can be loosely interpreted as the difference in deformation of a parallel chain (Figure 2(a)) versus a series chain (Figure 2(b)) under tension. 6  Vermeij & Hoefnagels, Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation, Scripta Mater. (2021)  Figure 2: Equivalent strain maps (with transparent microstructure overlay) showing progression of plasticity through pearlite at 𝜀!\"\"\"\"\"\" = 0.19, for two contrasting cases: (a) pearlite percolation bands along the lamellar direction, showing clear evidence of cementite-ferrite strain partitioning with sharp, continuous plasticity bands localized between the cementite lamellae, as highlighted by the solid ellipses, and (b) pearlite percolation bands perpendicular to the lamellar direction, showing smeared out, interrupted plasticity bands in pearlite, as highlighted by the dashed ellipses. Ferrite grain boundaries from EBSD are overlayed as black lines.   Upon further deformation, damage events are observed in the microstructure. We focus on a selection of pearlite “bridges” (i.e. narrow pearlite areas surrounded by ferrite) that show significant plasticity and damage, as these are prone to act as percolation pathways between deformation hotspots in nearby ferrite grains. When these pearlite bridges undergo (early) local failure, it enables meso-scale localization bands to form, that are likely responsible for early damage propagation leading up to global failure. Two distinct deformation modes are identified: high plasticity without pearlite cracking (Figure 3) and clear pearlite cracking/fracture events (Figure 4). The nanometer level strain resolution enables detailed analysis of the evolution of strain localization and/or strain partitioning that does, or does not, lead to damage. In both figures, subfigure (x1) shows the microstructure, subfigures (x2, x3, x4) show the SEM-DIC equivalent strain fields, and subfigure (x4) shows damage through post-mortem BSE micrographs. Figure 3 shows three examples of pearlite bridges that deform significantly without showing clear fracture. Indeed, in Figure 3(a) and (b) there are no signs of any damage in the BSE micrographs, while plasticity proceeds mostly through the interlamellar ferrite, allowed by the “open” pearlite arrangement. In Figure 3(c), the localization crosses the pearlite bridge through the vertical interlamellar ferrite channel, by overcoming the location where two cementite lamellae touch (indicated by the black arrow). These touching lamellae move apart, causing localized plasticity (Figure 3(c3) and (c3)), ultimately causing local damage (Figure 3(c5)), without however resulting in cracking of the full pearlite bridge. The plastic localization across pearlite bridges (inhibition of damage) seems to be uniquely tied to the two-phase (ferritic-pearlitic) microstructure, e.g. in fully pearlitic steel, plastic localization across pearlite colonies does not seem to be relevant [3]. \n7  Vermeij & Hoefnagels, Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation, Scripta Mater. (2021)  Figure 3: Three examples of pearlite “bridges” that do not show clear pearlite cracking, investigated at high spatial resolutions (a-b-c). (a1,b1,c1) The microstructure with bright cementite lamellae, with the ferrite grain boundaries from EBSD overlayed as red lines; SEM-DIC equivalent strain fields (with transparent overlaid microstructure) at (a2,b2,c2) 𝜀!\"\"\"\"\"\"=0.07, (a3,b3,c3) 𝜀!\"\"\"\"\"\"=0.12 and (a4,b4,c4) 𝜀!\"\"\"\"\"\"=0.36; (a5,b5,c5) post-mortem BSE image, showing damage at 𝜀!\"\"\"\"\"\"=0.36. The arrows in (c) mark a location where two touching cementite lamellae move apart. The ferrite grain boundaries (black or red lines) are not perfectly aligned and are for illustrative purposes only.  In contrast, Figure 4 shows three examples of clear fracture of pearlite bridges with apparently more regular and cleaner lamellar configurations than those in Figure 3. In Figure 4(a) and (b), plasticity evolves in ferrite, at two sides of the eventually cracking pearlite as indicated by arrows, while the strains inside the pearlite remain rather low as a few parallel cementite lamellae block the localization band. Thus, this pearlite cracking mechanism appears to originate from (rather) brittle fracture of multiple cementite lamellae, forced by increasing levels of surrounding (ferrite) deformation, which agrees well with observations in coarse fully pearlitic steel [3]. Figure 4(c) shows similar behavior, although the localization band first propagates through an interlamellar ferrite channel in the pearlite bridge that is blocked by 2-3 remaining cementite lamellae (Figure 4(c3)), which again abruptly crack in a collective manner at the final deformation stage. These observations reveal that the interaction between ferrite and pearlitic is critical, resulting in key microscale deformation and damage mechanisms in ferritic-pearlitic steel that do not occur in fully ferritic or fully pearlitic microstructures. Deformation may initialize in ferrite grains and channels, but will inevitably interact with pearlite colonies, activating various deformation mechanisms and ultimately leading to damage and failure. \n8  Vermeij & Hoefnagels, Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation, Scripta Mater. (2021)  Figure 4: Three examples of pearlite “bridges” that clearly show pearlite cracking, investigated at high spatial resolutions (a-b-c). (a1,b1,c1) The microstructure with bright cementite lamellae, with the ferrite grain boundaries from EBSD overlayed as red lines; SEM-DIC equivalent strain fields (with transparent overlaid microstructure) at (a2,b2,c2) 𝜀!\"\"\"\"\"\"=0.07, (a3,b3,c3) 𝜀!\"\"\"\"\"\"=0.12 and (a4,b4,c4) 𝜀!\"\"\"\"\"\"=0.36; (a5,b5,c5) post-mortem BSE image, showing damage at 𝜀!\"\"\"\"\"\"=0.36. The arrows mark locations of strain localization outside of a (partial) pearlite bridge, resulting in catastrophic pearlite cracking. The ferrite grain boundaries (black or red lines) are not perfectly aligned and are for illustrative purposes only.  Finally, we discuss potential improvements for the design of future ferritic-pearlitic steels. Pearlite bridges delay easy percolation of plasticity through the compliant ferrite grains and thereby provide increased strength to the material, however, subsequent cracking of these pearlite bridges needs to be suppressed to increase global ductility. Therefore, we recommend tailoring of the microstructure morphology by tuning the processing conditions.  Cementite lamellae in pearlite bridges should be configured to allow percolation of localized plasticity, e.g. through an “open” pearlite arrangement as in Figure 3(a) and (b) (could be attempted by slower furnace cooling) or by interlamellar ferrite channels that cross the pearlite bridges as in Figure 3(c). Alternatively, the pearlite bridges can be designed to be longer (for example by increasing ferrite grain size) to significantly increase the average number of bridge-crossing percolation paths. Note, however, that the width of these paths should remain small enough to enforce localized ferrite plasticity and hardening, in order to preserve the global strength. Additionally, refinement of the pearlite has been reported to improve ductility [3] and could therefore also be worth exploring. Interestingly, these recommendations align with the suggestions for damage inhibition strategies in other multi-phase microstructures (ferritic-martensitic [22,23], martensitic-austenitic [24] and Ti-Al-V-Fe [25] alloys), wherein the ideal response of the ‘hard’ phase combines a high yield strength (thus high instead of low phase contrast) with an internal plastic mechanism that can prolong plasticity. In summary, we successfully employed nanometer-resolution microstructure-correlated SEM-DIC strain mapping of the full evolution of strain localization to damage at lamellar scales in ferritic-pearlitic steel. This high-quality strain quantification could only be achieved with: (i) a high-quality SEM-DIC speckle pattern yielding accurate determination of the high levels of plastic deformation occurring just \n9  Vermeij & Hoefnagels, Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation, Scripta Mater. (2021) before and around damage events, (ii) successful high-eV BSE characterization of damage without hindrance from the DIC pattern, and (iii) careful alignment of the sub-micrometer microstructure morphology, strain fields, and damage maps in the deformed configuration.  Using this novel approach, a variety of deformation micro-mechanisms was observed. Notably, two contrasting mechanisms were identified that control whether damage initiation in pearlite occurs and, through connection of localization hotspots in ferrite grains, potentially results in macroscale fracture: (i) cracking of pearlite bridges with relatively clean lamellar structure by brittle fracture of cementite lamellae due to build-up of strain concentrations in nearby ferrite, versus (ii) large plasticity without damage in pearlite bridges with a more “open”, chaotic pearlite morphology, which enables plastic percolation paths in the interlamellar ferrite channels. Based on these insights, recommendations for damage resistant ferritic-pearlitic steels are proposed. Acknowledgements The authors thank Marc van Maris for experimental support. This research was carried out under project number S17012b in the framework of the Partnership Program of the Materials innovation institute M2i (www.m2i.nl) and the Technology Foundation TTW (www.stw.nl), which is part of the Netherlands Organization for Scientific Research (http://www.nwo.nl). References [1] Zelin, M., Acta Mater. 50 (2002) 4431–4447. [2] Raabe, D., Choi, P., Li, Y., Kostka, A., Sauvage, X., Lecouturier, F., Hono, K., Kirchheim, R., Pippan, R., Embury, D., MRS Bull. 35 (2010) 982–991. [3] Porter, D.A., Easterling, K.E., Smith, G.D.W., Acta Metall. 26 (1978) 1405–1422. [4] Elwazri, A.M., Wanjara, P., Yue, S., Mater. Sci. Eng. A 404 (2005) 91–98. [5] Izotov, V.I., Pozdnyakov, V.A., Luk’yanenko, E. V., Usanova, O.Y., Filippov, G.A., Phys. Met. Metallogr. 103 (2007) 519–529. [6] Tasan, C.C., Hoefnagels, J.P.M., Geers, M.G.D., Scr. Mater. 62 (2010) 835–838. [7] Sidhom, H., Yahyaoui, H., Braham, C., Gonzalez, G., J. Mater. Eng. Perform. 24 (2015) 2586–2596. [8] Zhang, X., Wang, Y., Yang, J., Qiao, Z., Ren, C., Chen, C., Opt. Lasers Eng. 85 (2016) 24–28. [9] Teshima, T., Kosaka, M., Ushioda, K., Koga, N., Nakada, N., Mater. Sci. Eng. A 679 (2017) 223–229. [10] Isavand, S., Assempour, A., Int. J. Mech. Sci. (2021) 106441. [11] Di Gioacchino, F., Quinta da Fonseca, J., Exp. Mech. 53 (2013) 743–754. [12] Yan, D., Tasan, C.C., Raabe, D., Acta Mater. 96 (2015) 399–409. [13] Hoefnagels, J.P.M., van Maris, M.P.F.H.L., Vermeij, T., Strain 55 (2019). [14] Shafqat, S., Hoefnagels, J.P.M., Exp. Mech. (2021). [15] Jones, E., Iadicola, M., Digit. Image Correl. Soc. (2018). [16] Tasan, C.C., Diehl, M., Yan, D., Zambaldi, C., Shanthraj, P., Roters, F., Raabe, D., Acta Mater. 81 (2014) 386–400. [17] Nolze, G., Ultramicroscopy 107 (2007) 172–183. 10  Vermeij & Hoefnagels, Plasticity, localization, and damage in ferritic-pearlitic steel studied by nanoscale digital image correlation, Scripta Mater. (2021) [18] Vermeij, T., Verstijnen, J.A.C., Hoefnagels, J.P.M., (2021). [19] Bachmann, F., Hielscher, R., Schaeben, H., Ultramicroscopy 111 (2011) 1720–1733. [20] Hoefnagels, J.P.M., Tasan, C.C., Maresca, F., Peters, F.J., Kouznetsova, V.G., J. Mater. Sci. 50 (2015) 6882–6897. [21] Lorusso, G.F., Joy, D.C., Scanning 25 (2003) 175–180. [22] Ghadbeigi, H., Pinna, C., Celotto, S., Yates, J.R., Mater. Sci. Eng. A 527 (2010) 5026–5032. [23] Liu, L., Maresca, F., Hoefnagels, J.P.M., Vermeij, T., Geers, M.G.D., Kouznetsova, V.G., Acta Mater. 205 (2021) 116533. [24] Oh, H.S., Kang, J., Tasan, C.C., Scr. Mater. 192 (2021) 13–18. [25] Wei, S., Zhu, G., Tasan, C.C., Acta Mater. 206 (2021) 116520.  "    },    {        "title": "0705.2883v3.The_beta_Phase_of_Multiferroic_Bismuth_Ferrite_and_its_beta_gamma_Metal_Insulator_Transition.pdf",        "content": "arXiv:0705.2883v3  [cond-mat.mtrl-sci]  13 Jul 2007R. Palai et al.\nTheβPhase of Multiferroic Bismuth Ferrite and Its γ-βMetal-Insulator Transition\nR. Palai1, R.S. Katiyar1, H. Schmid2, P. Tissot2,S. J. Clark3, J. Robertson4, S.A.T. Redfern5and J. F. Scott5\n1Department of Physics, University of Puerto Rico, San Juan, PR 00931-3343, USA\n2Department of Inorganic, Analytical and Applied Chemistry ,\nUniversity of Geneva, CH-1211 Geneva 4, Switzerland\n3Department of Physics, Durham University, Durham DH1 3LE, U K\n4Department of Engineering, University of Cambridge, Cambr idge CB2 1PZ, UK and\n5Department of Earth Science, University of Cambridge, Camb ridge CB2 3EQ, UK\n(Dated: October 27, 2018)\nWe have carried out extensive experimental studies, includ ing differential thermal analysis, polarized\nand high-temperature Raman spectroscopy, high-temperatu re X-ray diffraction, optical absorption and\ndomain imaging, and show that epitaxial (001) thin films of mu ltiferroic bismuth ferrite (BiFeO 3) are\nmonoclinic at room temperature instead of tetragonal or rho mbohedral like bulk as reported earlier. We\nreport a orthorhombic order-disorder β-phase between 820 and 950 ( ±5)◦C contrary to the earlier re-\nport. The transition sequence rhombohedral-orthorhombic transition in bulk (monoclinic-orthorhombic\nin (001)BiFeO 3thin film) resembles that of BaTiO 3or PbSc 1/2Ta1/2O3. The transition to the cu-\nbicγ-phase causes an abrupt collapse of the bandgap toward zero ( insulator-metal transition) at the\northorhombic-cubic β-γtransition around 950◦C. Our band structure models confirm this metal-insulator\ntransition, which is similar to the metal-insulator transi tion in Ba 0.6K0.4BiO3.\nPACS numbers: 64. 70. Kb, 71. 30. +h, 78. 30. - j, 77. 55. + f, 77. 80. Bh\nMagnetoelectric(ME) multiferroics aretechnologically\nand scientificallyimportantbecauseoftheirpotential ap-\nplications in data storage, spin valves, spintronics, quan-\ntum electromagnets, microelectronic devices etc. [1, 2, 3]\nand the novel mechanism that gives rise to electromag-\nnetic coupling. Ferroelectricity originates from off-center\nstructural distortions (d0electrons) and magnetism is in-\nvolved with local spins (dnelectrons), which limits the\npresence of off-center structural distortion [4]. These two\nare quite complementary phenomena, but coexist in cer-\ntainunusualmultiferroicmaterials. BiFeO 3(BFO)isone\nof the most widely studied multiferroic material because\nof its interesting ME properties i.eferroelectricity with\nhighCurietemperature( Tc≈810-830◦C)[5]andantifer-\nromagneticpropertiesbelow TN≈370◦C [5, 6]. The bulk\nBFO single crystal shows rhombohedral ( a= 5.58˚A and\nα= 89.50) crystal structure at room temperature (RT)\nwiththespacegroupR3cand G-typeantiferromagnetism\n[6, 7]. If BFO were an ordinary antiferromagnet, the\nspacegroupwouldbeR3m[7]. Thestructureandproper-\nties ofbulk BFO havebeen studied extensively[5, 6, 7, 8]\nand although early values of polarization were low due to\nsample quality, Pr= 40µC/cm2is now found in bulk by\nseveral different groups [9]. The weak ferromagnetism\nat room temperature occurs due to the residual moment\nfrom the canted spin structure [6]. It is very difficult to\ngrow a high quality (defect free and stoichiometric) bulk\nsingle crystal with low leakage, which is detrimental to\nthe practical applications of this material.\nIt has been found that thin films of BFO grown on\n(100) SrTiO 3(STO) substrates show very high values\nofPr(∼55, 86, and 98 µC/cm2for the (001), (101),\nand (111) BFO films, respectively) and magnetization\n(Mr∼150 emu/cc) [1, 10, 11]. (Very recently, Ricinschi\net al. [12] have claimed Prof 150µC/cm2for the poly-\ncrystalline BFO films grown on Si substrates, but thisis probably an artifact due to leakage and charge injec-\ntion.) This makes BFO as one of the potential materials\nfor the novel device applications, although the mecha-\nnism(s) behind the huge polarization claimed by some\ngroups is not yet fully understood. Some experimental\nresults [1, 10] and theoretical reports [13] suggest that\nthe epitaxial strain might be the cause of such high value\nofPrandMr. However, a recent study showed that the\nepitaxial strain does not enhance Mrin BFO thin films\n[14]. It is believed that the heteroepitaxy induces sig-\nnificant and important structural changes in BFO thin\nfilms, which may lead to very high values for Pr.\nThere are some controversies in literature about the\ncrystalstructure of (001)epitaxial thin films. There have\nbeen several reports claiming tetragonal [1, 15], rhombo-\nhedral [11, 16], and monoclinic [17] structure of (001)\nBFO films on STO substrates. Therefore, the sequence\nof transitions is poorly understood, and there is no un-\nderstandingofthe overallphysicsofthe phasetransitions\ninvolved; more structural analysis of thin films is neces-\nsary for better understanding of engineered epitaxial and\nhetrostructure BFO thin films. In the present work we\nused a variety of techniques which combined show a se-\nquence of transitions resembling the well-known 8-site\nmodel of barium titanate.\nABO3oxide perovskites which are rhombohedral at\nlow temperatures, such as LaAlO 3, PrAlO 3, or NdAlO 3\n[18, 19] have ferroelastic instabilities at the A-ion site\nthat induce displacive phase transitions directly to cu-\nbic; but those which haveB-site instabilities instead have\norder-disorder transitions to cubic that involve two or\nmore steps. This has been successfully described [20, 21]\nby an eight-site model in which the B-ion displacements\nare always locally toward a [111] axis, but thermally av-\nerage via hopping over [111], [ ¯111], [1¯11], and [11 ¯1] to\ngive orthorhombic, tetragonal, or cubic time- and space-2\nTABLE I: Selection rules for the Raman active modes for rhom-\nbohedral ( R), tetragonal ( T) and, monoclinic ( M) crystal struc-\ntures in different polarization configurations with total nu mber\nof normal ( N) Raman modes. The notation ” <001>up” means\nunpolarizedspectraalongthepseudo-cubic <001>direction per-\npendiculartothesubstrate. The notationZ (along <001>direc-\ntion) and ¯Z are the directions of the incident and backscattered\nlight, respectively.\nScattering R(R3c) T(P4mm) M(Bb)\ngeometry (C 3v) (C 4v) (C s)\nN(Raman) 4 A1+ 9E3A1+B1+ 4E13A′+ 14A′′\n<001>up 4A1+ 9E3A1+B1 13A′\nZ(XX)¯ZA1andE A 1andB1 A′\nZ(XY)¯Z E No modes A′\nY(ZZ)¯Y A1 A1 A′\nY(ZX)¯Y E E A′′\nglobal averages. In the present work we show that this\nmodel describes BFO, contrary to conventional wisdom\n[22] but in agreement with NMR, which shows some B-\nsite disorder [23].\nA. The α-phase\nThe rhombohedral (R3c), tetragonal ( P4mm) [1], and\nmonoclinic ( Bb)[24] structures of BFO give rise to 13,\n8, and 27 distinct Raman-active modes, respectively, as\nlisted in the Table I. For the orthorhombic distortion in\ntheβ-phase, as discussed below, the tetragonal entries\nwill remain correct, with only a small splitting of the\nE-modes.\nThe Raman data for 300 nm thick (001)BFO thin\nfilm on the (100)STO substrate in the parallel, Z(XX) ¯Z\nand perpendicular, Z(XY) ¯Z polarization configurations\nreveal strong peaks at 74, 140, 172 and 219 cm−1, while\nweaker peaks were observed at around 261, 370, 406,\n478, 529, 609, 808 (very weak), 946, 1093 cm−1. All of\nthese peaks are due to the BFO normal modes of vibra-\ntions and none of them are arising from the substrate,\ncontrary to the earlier reported Raman measurements\n[15, 25]. We verified our results using target materials\nand also growing (001)BFO films on different substrates.\nThe existence of 13 identical peaks in both the Z(XX) ¯Z\nand Z(XY) ¯Z polarizationconfigurationsconfirmsthe Ra-\nman selection rules for the monoclinic structure (see Ta-\nble I) instead of tetragonal or rhombohedral as reported\nearlier[11, 15, 16]. This observation verifies the very\nrecent report of monoclinic structure for the epitaxial\nBFO films grown on (001)STO substrates by Xu et al.\n[17] studied via synchrotron radiation. As the spectra\nalong Y(ZZ) ¯Y and Y(ZX) ¯Y were heavily dominated by\nthe scattering from the STO substrate, the contribu-\ntion from the BFO film could not be separated. We\nlooked very closely near the phase transition Tempera-tures. Two noticeable changes have been observed in the\nRaman spectra: disappearance of all stronger peaks (74,\n140, 171and 220cm−1) at∼820◦C with the appearance\nof a few new peaks, and complete disappearance of spec-\ntra around 950◦C. This temperature behavior implies\nthat the BFO maintains its room-temperature structure\nup to∼820◦C indicating the ferroelectric-paraelectric\n(FE-PE) phase transition, in agreement with the earlier\ninvestigations on BFO bulk single crystal and polycrys-\ntalline [5, 22] samples. No evidence of soft phonon modes\nimplies that the BFO has an order-disorder, first-order\nferroelectric transition, unlike PbTiO 3. No decomposi-\ntion was observed above 810◦C contrary to the earlier\nstudies [8], suggesting that our samples had fewer defects\nand dislocations.\nB. The β-phase\nThe Raman spectra show that four lines ( ∼213, 272,\n820 and 918 cm−1) persist above 820◦C. In the cubic\nperovskite phase no first-order Raman lines are allowed;\nall ions sit at inversion centers, and all long wavelength\nphonons are of odd parity. The data show that the\nbeta phase from 820◦C to 950◦C cannot be cubic as re-\nported earlier [22]. Since our backscattering geometry\nwith incidence radiation along Z-axis favors A1andB1\nphonons, four Raman modes (3 A1+B1) are predicted in\nthe tetragonal (or orthorhombic) perovskite phase (Ta-\nble I), in agreement with experiment. Small orthorhom-\nbic splitting offour unobserved Emodes is predicted, but\nthese modes are unobserved in the backscattering geom-\netry. We observe no soft modes in these studies but some\nmerely line-width increases, suggesting that the α-βand\nβ-γtransitions are both order-disorder, compatible with\nthe eight-site model originally developed by Comes et al.\n[20], and developed in detail by Chaves et al[21].\nC. Thermodynamics of phases\nThe existence of a β-phase below the cubic γ-phase\n∼930◦C and below the decomposition point at 960◦C has\nbeen knownforsomefortyyears[26], but the earlySoviet\nwork is rarely cited, and was sometimes considered not\nto be single-phase material. Using differential thermal\nanalysis (DTA) in conjunction with high-temperature\nreflected polarized light microscopy, the BiFeO 3-Fe2O3\nphase diagram of Speranskaya et al. [26] has been refined\n(Fig. 1). DTA thermograms shown in Fig. 2 illustrate\nthe phase transformation sequence for both BiFeO 3sin-\nglecrystals(smallcrusheddendrites) andthin film target\nmaterials.\nThese show four sharp endothermic peaks at around\n823, 925, 933, and 961◦C. They can be interpreted as:\n(a) a first-order α-βtransition; (b) a β-γtransition; (c)\nperitectic decomposition of the cubic phase into flux and\nBi2Fe4O9; and (d) decomposition of Bi 2Fe4O9into flux3\n0 20 40 60 80 100 600 700 800 900 1000 1100 \n723 oC777 oC961 oC\n933 oCBi40 Fe 2O63 \nBi2Fe 4O9\nFe 2O3   Temperature ( oC) \nMole % Fe 2O3BiFeO 3α\nBi2O3βγ\n825 oCLiquid \nMixed decomposition Metastable state \n(1) (2) \nFIG. 1: Phase diagram of BiFeO 3. Open circles show the data\npoints obtained by DTA. The DTA peaks were reversible below\nthe solid line (line (1)) on cooling, while metastable state s above\nthe dotted line (line (2)). The α-phase is monoclinic in (001)\nthin film (rhombohedral in bulk), while the βandγphases are\northorhombic and cubic, respectively.\n \nFIG. 2: DTA studies on crushed single crystal (dotted points)\nand thin film target material (solid line) of BiFeO 3. Signal has\nbeen inverted and offset for clarity.\nand Fe 2O3. The experimental values obtained on pow-\nders of crushed dendrites may be different for thin films,\nwhere transformations may be controlled by the build up\nstrain energy. These DTA data and others for which the\nBi/Fe ratio is varied produce the phase diagram shown\nin Fig.1.\nHigh-temperature X-ray study of BFO powder (Fig.3)\nshowed that the rhombohedral bulk structure has a\nstrongly first-order transition near 825( ±5)◦C to a\nP2mm orthorhombic structure. The unit cell volume\nshrinksby morethan 1%upon heatingthrough this tran-\nsition from 64.05 (per formula unit) in the rhombohedral\nphase to 62.91 ˚A3in cubic. At around 925( ±5)◦C there\nis a nearly second-order transition from orthorhombic to\na cubic Pm3m phase. All of the phases have one formula\ngroupper unit cell, asin BaTiO 3. The rhombohedralcell\nparameteris a=4.0011 ˚A. The orthorhombiccell param-\neters at 825( ±5)◦C area= 3.9765(8) ˚A;b= 3.9860(6) ˚A\nandc= 3.9687(8) ˚A; and at 905◦Ca= 3.9892(13) ˚A;\nb= 3.9926(9) ˚A andc= 3.9848(9) ˚A. In the cubic0 200 400 600 800 1000 3.960 3.965 3.970 3.975 3.980 3.985 3.990 3.995 4.000 4.005 \n  Lattice parameters (A o)\nTemperature ( oC) a, Rhombo \n c, Ortho \n b, Orth \n a, Ortho \n a, Cubic \nFIG. 3:Temperature variation of lattice parameters for rhombo-\nhedral (pseudo-cubic setting), orthorhombic and cubic pha ses of\nBiFeO 3. At 825◦C the lattice constant a= 4.0011 ˚A splits into\na triplet (open circles, squares, and triangles), which com bines\nagain at ca. 925( ±5)◦C in the cubic phase (solid square), where\na= 3.9916 ˚A.\nphasea= 3.9916(1) ˚Aat around 925( ±5)◦C. The short-\nened bond length in the P2mm and Pm3m phases, com-\npared with those in the rhombohedral phase, favors a\nmetallic state. The sequence is similar to the other per-\novskite metal-insulator systems such as, NdNiO 3and re-\nlated rare-earth nickelates showing a sharp decrease of\nthe unit-cell volume exactly at the metal-insulator tran-\nsition [27].\nD. Domains structures in BiFeO 3single crystal\nThe Raman spectra do not discriminate between or-\nthorhombic and tetragonal structures for the β-phase,\nbut domain structures do. The orthorhombic domains\nthat are symmetry-forbidden in tetragonal structures are\ntoo week to reproduce in Fig. 4, but in pesudo-cubic (pc)\nnotion. They unambiguously rule out tetragonal struc-\nture for the beta-phase. The β-BFO phase has been rec-\nognized to be ferroelastic and non-cubic on the basis of\nmicroscopical observation of ferroelastic domains using\nreflected polarized light. The β-phase produces recti-\nlinear ferroelastic domains and a clear phase boundary.\nOn the basis of the ferroelastic domain pattern of the\nβ-phase on a (111)pc-cut, showing rectilinear traces of\nboth (100)pc and (110)pc walls, orthorhombic symmetry\nwith axes parallel to <110>and<001>(analogous to\nBaTiO 3) is deduced, whereas a tetragonal phase would\nallow only (110)pc walls. The postulation of a cubic γ-\nphase in a very narrow temperature interval is based\nupon the reflected polarized light optical observation\nthat the ferroelastic domains of the β-phase disappear at\nabout 925◦C, leaving the sample optically isotropic up to\nthe decomposition point. Both types of domains can be\nseen in the photographs taken, but one cannot see recti-\nlinear domain walls in the β-phase on this photo (Fig. 4).\nThe ferroelasticnature of the α-βphasetransition is nec-4\n(a)  \n(b) \nFIG. 4: Evolution of domain structure of β-phase of BFO with\ntemperature; (a) at room temperature; (b) at 825◦C with fer-\nroelastic domains.\nessary and sufficient to satisfy the Toledano [28] require-\nment that changes in crystal class are required for ferroe-\nlastic transitions (this treats rhombohedral and trigonal\nas a single super-class). Thus, for example, triglycine\nsulphate (TGS) with its monoclinic-monoclinic C 2h-C2\ntransition cannot be ferroelastic.\nE. The γ-phase\nVisual observation shows [29] that bismuth ferrite sin-\nglecrystalsatroomtemperatureareyellowandtranspar-\nent. ThebandgapofBFOiscalculatedtobe Eg=2.8eV\n[30]andusuallymeasuredexperimentallyas2.5eV.How-\never, it has a large bandgap shift at the α-βphase tran-\nsition temperature and turns deep red. If we define the\nband edge as where the absorption is 100 in 10 µm, then\nthe experimental gap is 2.25, 2.18, 2.00 and 1.69 eV at\n20, 160,and300◦C,respectively. Butit shiftsabruptlyto\n1.69 eV at 500◦C, above which the Egvalues are not yet\nknown. There has been some controversy about its leak-\nagecurrent, with Clark et al.[30] showingthat the ambi-\nent bandgap is too large for intrinsic mechanisms. Opti-\ncal absorption shows that the bandgap decreases slowly\nand linearly with temperature in the αandβphases,\nfrom 2.5 eV to ca. 1.5 eV, but then drops abruptly to\nnear zero at the β-γtransition near 930◦C. Our obser-\nvation is thus that Egdecreases significantly by the α-\nβtransition temperature, and hence conduction in the\nhigh-temperature β-phase can be intrinsic. In the cu-\nbicγ-phase it is black and opaque. That is compatible\nwith our band structure calculations (following section)\nthat a non-cubic distortion is required to give it a finite\nbandgap.\nThus we regard β-BFO phase as semiconducting with\nsmall gap (calculated ∼1.5 eV) and γ-BFO phase from\n∼930◦C to∼960◦C as cubic and metallic; above 960◦C\ndecomposition occurs. The insulator-metal transitionupon entering the cubic phase of BFO is very similar\nto that occurring in Ba( 0.6)K0.4)BiO3[31, 32] where the\nmaterialbecomescubicandmetallic. Thisimpliesineach\ncase a strong electron-phonon interaction. The similar-\nity with high- Tcsuperconducting cuprates in their nor-\nmal states has been discussed [33]. The data graphed\nin this figure were obtained in two different ways: Be-\nlow 830 K they were obtained by conventional absorp-\ntion spectroscopy at a fixed temperature; but between\n830 K and 1230 K they were obtained at a fixed wave-\nlength (632.8 nm He-Ne) by slowly varying temperature\nand using the Urbach equation to relate absorption coef-\nficient (a) to bandgap E g(T):log a(T) = (E−Eg)/Eg+\nconstant .\n-8 -6 -4 -2 0246Energy (eV) \nΓ               X                 R       M                      Γ                        R c-BiFeO 3(a) \n-8 -6 -4 -2 0246Energy (eV) \nΓ               X                 R       M                      Γ                        R o-BiFeO 3 (b) \nFIG. 5: Bandgap energy calculation for BiFeO 3using screened\nexchange method for different structures; (a)cubic; (b) or-\nthorhombic. Fermi level is at 0 eV in (a) and the valence band\nmaximum is at 0 eV in (b).\nThe differences in transition temperatures between\nbulk and thin film, and single crystal, and also between\nthe different studies drawn together in this paper, show\nthat the transitions in the thin film are at higher temper-\natures than in the bulk material. The effect of pressure\non these sorts of transitions is known, they have a nega-\ntive dp/dT slope, as pressure stabilizes the higher sym-\nmetry phases in BaTiO 3. The thin film data would be\nconsistent with this if you regard any surface relaxation\nor epitaxial strains in the thin films as imposing a nega-\ntive pressureorlowerdensity on the thin films. The large\nnegativechangeinvolumeobservedattherhombohedral-\northorhombic transition in BFO is also consistent with\nthe transition having a strong negative Clapeyron slope,5\nsosurfacerelaxationmightbe expected tomovethe tran-\nsition to higher temperature in the thin film.\nF. Bandstructure model: Bandgap Collapse and\nMetal-Insulator Transition\nObtaining a band gap of zero is a common artifact\nin theoretical models using the local density approxima-\ntion (LDA). Therefore, in the present study we circum-\nvent this by using the screened exchange (sX) method\n[34]. This method is a density functional method based\non Hartree-Fock, which includes the electron exchange\nvia a Thomas-Fermi screened exchange term. It gives\nthe correct band gap of many oxides, including anti-\nferromagnetic NiO and rhombohedral BiFeO 3[30]. We\nfind that the band gap of the cubic phase of BFO is in-\ndeed zero, as shown in Fig.5(a). The Fermi level lies\nwithin the Fe 3d states. It is interesting that in the cubic\nphase there is a direct gap at any point in the zone, and\nthis opens up into a true band gap of 0.8 eV (calculated)\nin the orthorhombic phase as seen in Fig.5(b). Note that\nat RT the BFO has has an indirect band gap, but the\ndirect gap lies only ca. 0.05 eV above it, and the va-\nlence band is very flat. Thus, with increasing T, as the\nconduction band descends, the gap can become direct.\nG. Conclusion\nIn conclusion, high quality epitaxial (001)BFO films\nhavebeengrownon(100)STOsubstratesusingPLD.The\nXRD studies showed that films are c-axis oriented with\nhigh degree of crystallinity. The RT polarized Raman\nscattering of (001)BFO films showed monoclinic crystal\nstructure contrary to the rhombohedral and tetragonal\nas reported earlier. The results obtained from DTA,\nhigh-temperature XRD, optical absorption, and polar-\nized optics studies of domains were consistence. We ob-\nserved the FE-PE structural phase transition at around\n820◦C no softening of Raman modes was observed at\nlow frequencies. An intermediate β-BiFeO 3phase be-\ntween 820-950◦C has been observed and recognized to\nbeorthorhombic for the first time. The sequence of\nmonoclinic(film)-orthorhombic-cubic phases or rhombo-\nhedral (single crystal)-orthorhombic-cubic in bulk; the\nphase sequence is extremely similar to that in BaTiO 3.\nMoreover, the transitions appear to be order-disorder\nfrom the Raman data, suggesting that the eight-site\nmodel of Comes et al. [20] and Chaves et al. [21] is\napplicable. We note that the high-T csuperconductor\nBa1−xKxBiO3is also a perovskite oxide which becomes\nsimultaneously cubic and metallic at x=0.4 [35]. This\nsuggests a similar electron-phonon coupling; however, in\nthe latter material the Bi-ion is at the B-site, whereas in\nBiFeO 3it is at the A-site.Acknowledgement\nWethankW.PerezandDr. M.K.Singhforexperimen-\ntal help. This work was supported by the DoD W911NF-\n06-0030 and W911NF-05-1-0340 grants and by an EU-\nfunded project ”Multiceral” (NMP3-CT-2006-032616)at\nCambridge.\nH. Method\nBFO thin films of 300 nm thick were grown by pulsed\nlaser deposition (PLD) using a 248 nm KrF Lambda\nPhysik laser. Films were grown on STO(100) substrates\nofarea(5mm)2with∼25nmthickSrRuO 3(SRO)buffer\nlayer. The growth parameters were as follows: substrate\ntemperature of 700◦C, oxygen pressure of 10 mTorr,\nlaserenergydensityof2.0Jcm−2atapulserateof10Hz,\nand a target-substrate distance of 50 mm. After the de-\nposition, the chamber was vented with 0.4 atm of oxygen\nand then cooled at a rate of 30◦C/min to room tem-\nperature with an intermediate holding at 500◦C for 20\nmin. The orientation, crystal structure and phase purity\nof the films were examined using Siemens D5000 X-ray\ndiffractometer. The Jobin Yvon T64000 micro-Raman\nmicroprobe system with Ar ion laser ( λ= 514.5 nm)\nin backscattering geometry was used for polarized and\ntemperature depended Raman scattering. The laser ex-\ncitation power was 2.5 mW and the acquisition time was\n10 min per spectrum. The single crystal samples were\nmade from crushed dendrites. The high-temperature X-\nray data were collected from powdered bulk material\nusing a Bruker D8-ADVANCE diffractometer with Cu-\nradiation, G¨ obel Mirror, and fast Vantec (linear psd) de-\ntector. Scans from 20 to 80o2θwere obtained every 15o\nfrom 40oto the decompostion point of BFO with an ac-\nquisition time of 8 min per pattern and a heating rate of\n30◦C/min between scans.6\n[1] Wang, J. et al. Epitaxial BiFeO 3multiferroic thin film\nhetrostructures. Science299, 1719 (2003).\n[2] Fiebig, M., Lottermoser, T., Fr¨ ohlich, D., Goltsev, A. V.\n& Pisarev, R. V. Observation of coupled magnetic and\nelectric domain. Nature419, 818 (2002);\n[3] Tokura, T. Multiferroic as quantum electromagnet. Sci-\nence312, 1481 (2006).\n[4] Hill, N. A. Why are there so few magnetic ferroelectrics.\nJ. Phys. Chem. B 104, 6694 (2000).\n[5] Fischer, P., Polomska, M., Sosnowska, I. & Szyma´ nski,\nM. Temperature depedence of the crystal and magnetic\nstructures of BiFeO 3.J. Phys. C 13, 1931 (1980).\n[6] Smolenskii, G. A. & Chupis, I. Ferroelectromagnets. Sov.\nPhys. Usp. 25, 475 (1982).\n[7] Kubel, F. & Schmid, H. Structureof ferroelectric and fer -\nroelastic monodomain crystal of the perovskite BiFeO 3.\nActa Crystallogr. Sect. B: Struct. Sci. 46, 698 (1990).\n[8] Bucci, J. D., Robertson, B. K. & James, W. J. The preci-\nsion determination of the lattice parameters and the co-\nefficients of thermal expansion of BiFeO 3.J. Appl. Cryst.\n5, 187 (1972).\n[9] Shvartsman, V. V., Kleemann, W., Haumont, R. &\nKrisel, J. Large bulk polarization and regular domain\nstructures in ceramic. Appl. Phys. Lett. 90, 2115 (2007).\n[10] Li, J. et al. Dramatic enhanced polarization in (001),\n(101), and (111) BiFeO 3thin films due to epitaxial-\ninduced transitions. Appl. Phys. Lett. 84, 5261 (2004).\n[11] Das, R. R. et al. Synthesis and ferroelectric properties of\nepitaxial BiFeO 3films grown by sputtering. Appl. Phys.\nLett.88, 242904 (2004).\n[12] Ricinschi, D., Yun, K. & Okuyama, M. A mechanism\nfor the 150 µC cm−2polarization of BiFeO 3films based\non first-principle calculations and new structural data. J.\nPhys: Condens. Matter. 18, L97 (2006).\n[13] Ederer, C. & Spaldin, N. A. Influence of strain and oxy-\ngen vacancies on the magnetoelectric properties of mul-\ntiferroics. Phys. Rev. B 71, 224103 (2005).\n[14] Eerenstein, W., Morrison, F. D., Dho, J., Blamire,M. G. ,\nScott, J. F. & Mathur, N. D. Comment on ”Epitaxial\nBiFeO 3thin film hetrostructures”. Science 307, 1203a\n(2005).\n[15] Singh, M. K., Ryu, H. & Jang, H. M., Polarized Raman\nscattering of multiferroic BiFeO 3thin films with pseudo-\ntetragonal symmetry. Phys. Rev. B 72, 132101 (2005).\n[16] Qi, X. Dho, J., Tomov, R., Blamire, M. G. & J.L.\nMacManus-Driscoll, Greatly reduced leakage current and\nconduction mechanism in aliovalent-ion-doped BiFeO 3.\nAppl. Phys. Lett. 86, 062903 (2005).\n[17] Xu, G. et al.Low symmetry phase in (001) BiFeO 3\nepitaxial constrained thin films. Appl. Phys. Lett. 86,\n182905 (2005).\n[18] Scott, J.F. Raman study of trigonal-cubic phase tran-\nsitions in rare-earth aluminates. Phys. Rev. 183, 823\n(1969);\n[19] Geller, S. & Raccah, P.M. Phase transitions in per-ovskitelike compounds of the rare earths. Phys. Rev. B\n2, 1167 (1970).\n[20] Comes, R., Lambert, M. & Guinier, A. Chain structure\nof BaTiO 3andKNbO 3.Solid State Comm. 6, 715(1968).\n[21] Chaves, A. S., Barreto, F. C. S., Nogueira, R. A. & Z˜ eke,\nB. Thermodynamics of aneight-site order-disorder model\nfor ferroelectrics. Phys. Rev. B 13, 207 (1976).\n[22] Haumont, R., Kreisel, J., Bouvier, P. & Hippert, F.\nPhonon anomalies and the ferroelectric phase transition\nin multiferroic BiFeO 3.Phys. Rev. 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Theory of the ferroelastic transition i n\nbarium sodium niobate Phys. Rev. B 12, 943 (1975).\n[29] Tabares-Munoz, C., Rivera, J. P. & Schmid, H. Ferro-\nelectric domains, birefringence and absorption of single\ncrystal of BiFeO 3.Ferroelectrics 55 , 235 (1984);\n[30] Clark, J. S. & Robertson, J. Band gap and Schottky bar-\nrier heights of multiferroic BiFeO 3.Appl. Phys. Lett. 90,\n132903 (2007)\n[31] Cava, R. J., et al., Superconductivity near 30 K without\ncopper: The Ba (0.6)K(0.4)BiO3perovskite Nature332,\n814 (1988).\n[32] Tajima, S., Yoshida M., Koshizuka N., Sato H., and\nUchida S., ”Raman-scattering study of the metal-\ninsulator transition in Ba (1−x)K(x)BiO3,”Phys. Rev. B\n46, 1232 (1992).\n[33] Sharifi, F., Pargellis A. & Dynes, R. C., Tunneling den-\nsity of states in the lead-bismuth-oxide superconductors\nPhys. Rev. Lett. 67, 509 (1991).\n[34] Robertson, J, Xiong, K. & Clark, J. S. Band structure of\nfunctionaloxidesbyscreenedexchangeandweightedden-\nsity approximation Phys. Stat Solidi b 243, 2054 (2006)\n[35] Anderson, P.W., New Physics of Metals: Fermi Surfaces\nwithout Fermi Liqwuids Proc. Natl. Acad. Sci. USA 92,\n6668 (1975)."    },    {        "title": "1812.08297v4.Size_dependent_bistability_in_multiferroic_nanoparticles.pdf",        "content": "Size-dependent bistability in multiferroic nanoparticles\nMarc Allen,1, 2Ian Aupiais,3Maximilien Cazayous,3and Rog\u0013 erio de Sousa1, 2\n1Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada\n2Centre for Advanced Materials and Related Technology,\nUniversity of Victoria, Victoria, British Columbia V8W 2Y2, Canada\n3Laboratoire Mat\u0013 eriaux et Ph\u0013 enom\u0012 enes Quantiques, UMR 7162 CNRS,\nUniversit\u0013 e Paris Diderot, B^ atiment Condorcet 75205 Paris Cedex 13, France\nMost multiferroic materials with coexisting ferroelectric and magnetic order exhibit cycloidal\nantiferromagnetism with wavelength of several nanometers. The prototypical example is bismuth\nferrite (BiFeO 3or BFO), a room-temperature multiferroic considered for a number of technological\napplications. While most applications require small sizes such as nanoparticles, little is known\nabout the state of these materials when their sizes are comparable to the cycloid wavelength. This\nwork describes a microscopic theory of cycloidal magnetism in nanoparticles based on Hamiltonian\ncalculations. It is demonstrated that magnetic anisotropy close to the surface has a huge impact on\nthe multiferroic ground state. For certain nanoparticle sizes the modulus of the ferromagnetic and\nferroelectric moments are bistable, an e\u000bect that may be used in the design of ideal memory bits\nthat can be switched electrically and read out magnetically.\nI. INTRODUCTION\nMultiferroic materials display coexisting ferroic orders\nat the same temperature [1]. An important class is\nthe magnetoelectric multiferroics, with coexisting ferro-\nelectricity and magnetism, usually antiferromagnetism\n[2]. The impact of ferroelectricity on the magnetic\nstate occurs through the spin-orbit interaction, more\nspeci\fcally through the spin-current contribution of the\nDzyaloshinskii-Moryia (DM) interaction. This interac-\ntion induces spiral magnetism of the cycloidal type,\nwith cycloid period \u0015= 2\u0019=Q much larger and incom-\nmensurate with the material's lattice spacing a[3{5].\nConversely, cycloidal magnetism induces ferroelectricity\n[6, 7].\nA notable example of magnetoelectric multiferroic is\nbismuth ferrite (BiFeO 3or BFO), one of the few room-\ntemperature multiferroics [8, 9] with potential for techno-\nlogical applications such as electrically-written magnetic\nmemories [10{13] or photoelectricity [14]. Bulk BFO is\nferroelectric at temperatures below 1100 K, with a record\nhigh polarization P\u0019100\u0016C=cm2at room temperature.\nBelow 640 K the Fe spins form a nearly cubic antiferro-\nmagnetic lattice with cycloidal spin ordering of period\nequal to\u0015Bulk = 630 \u0017A [15, 16], much larger than the\nlattice parameter a= 3:96\u0017A.\nMemory applications require miniaturized multifunc-\ntional devices with multiferroic size approaching \u0015Bulk.\nSo far only a few studies have considered the impact of\n\fnite size on BFO's ferroelectric and magnetic properties.\nIt was shown experimentally that BFO nanoparticles re-\nmain ferroelectric and antiferromagnetic at room tem-\nperature, but with decreased Curie and N\u0013 eel transition\ntemperatures, and enhanced ferromagnetism [17{23].\nThe presence of a depolarizing electric \feld in \fnite-\nsized ferroelectrics is known to reduce their Curie tem-\nperature and polarization P. There exists a critical size\nbelow which the nanoparticle ceases to be ferroelectric\n[24, 25]. This e\u000bect was measured in free-standing BFOnanoparticles [19], where it was shown that P\u0019PBulk\nfor sizes down to 30 nm, with Preduced to 0 :75PBulkfor\nsize 13 nm. Extrapolating to even smaller nanoparticles\nsuggested a critical size of approximately 9 nm.\nDemagnetizing \felds play a similar role in ferromag-\nnetic nanoparticles, favouring the formation of magnetic\nvortex states [26]. However, vortex states do not occur\nin antiferromagnetic nanoparticles, even when they have\na weak ferromagnetic moment arising from spin canting\nor uncompensated spins at the surface. The small ferro-\nmagnetic moment leads to a demagnetizing energy that\nis several orders of magnitude smaller than the antifer-\nromagnetic exchange energy. As a result, the spin-spin\ndipolar interaction can be neglected in models for anti-\nferromagnetic nanoparticles [27]. However, the impact\nof \fnite size and the role of surface interactions on the\nspin texture of cycloidal multiferroics has not yet been\nexplored.\nNanoparticles di\u000ber qualitatively from the bulk due\nto their larger surface-to-volume ratios. Here it is ar-\ngued that the magnetic order of multiferroic nanopar-\nticles is greatly in\ruenced by magnetic interactions at\nthe surface. The most important of these interactions is\nsingle-ion anisotropy [28, 29], which originates from two\nlarge spin-orbit contributions of opposite sign, both as-\nsociated to the location of the Bi ion in BFO [30, 31]. At\nthe surface of the nanoparticle reduced symmetry means\nthat the two contributions to single-ion anisotropy no\nlonger cancel each other out, leading to large magnetic\nanisotropy at or nearby the surface. Reduced symme-\ntry at the surface has also been thought to increase sur-\nface anisotropy due to factors including broken exchange\nbonds and interatomic distance variation [32, 33].\nStrain also increases single-ion anisotropy and this ex-\nplains why the magnetic cycloid order is destroyed in\nBFO thin \flms grown on top of substrates with large\nrelative strain [34]. The present research article focuses\non unstrained nanoparticles, i.e., those that are either\nfreestanding or grown on top of a lattice-matched sub-arXiv:1812.08297v4  [cond-mat.mtrl-sci]  11 Jul 20192\nstrate. Experiments show that substrates with relative\nstrain smaller than 0 :5% preserve the cycloid order of\nBFO [34]. Below it is shown that in these unstrained\nnanoparticles the combination of cycloidal spin order and\nsurface magnetic anisotropy leads to multiferroic bistabil-\nity.\nII. MODEL FOR MULTIFERROIC\nNANOPARTICLES\nIn this article a model for the impact of surface\nanisotropy on a multiferroic nanoparticle is proposed. As\na starting point, consider the Hamiltonian describing an-\ntiferromagnetism in a magnetoelectric multiferroic [4{6],\nH0=1\n2X\ni;^ vh\nJSi\u0001Si+^ v+D^P\u0001^ v\u0002(Si\u0002Si+^ v)i\n:(1)\nThe classical vectors Si= (Six;Siy;Siz) represent the\nithspin in a hypercubic lattice (dimension d= 1;2;3)\nwithout periodic boundary conditions. For example, the\nd= 2 case has each spin located at Ri=a(ix;iy), with\neachi\u000b= 1;:::;N , etc (\u000b=x;y;z and total number of\nspins isNd). The unit vectors ^ vlink nearest neighbours\ncoupled by exchange energy J >0 (antiferromagnetism).\nThe spins are a\u000bected by the ferroelectric moment\nPvia the spin-current energy D. The second term in\nEq. (1) can be interpreted as \u0000a3P\u0001Espin\ni, with Espin\ni\nthe spin-induced electric \feld at site Ri. This local \feld\nchanges the polarization according to \u0001 Pi=\u001fEspin\ni,\nwith\u001fan electric susceptibility [35], which averaged over\nall sites leads to the spin-induced ferroelectric moment\nPspin=\u0000D\u001f\n2Pa3NdX\ni;^ v^ v\u0002(Si\u0002Si+^ v): (2)\nThe ground state of Hamiltonian (1) occurs for spins\nlying in the ^Q^Pplane,\nSi= (\u00001)P\n\u000bi\u000b\u0010\nsin(\u001ei)^Q+ cos(\u001ei)^P\u0011\n; (3)\nwith cycloid unit vector ^Qsimultaneously perpendicular\nto^Pand parallel to one of the nearest neighbour direc-\ntions ^ v. WhenN!1 (the bulk limit) the angle \u001eiis\nsimply given by [3{5]\n\u001ei=\u001e0+Q\u0001Ri; (4)\nwith\u001e0an arbitrary phase slip, and Qa constant cycloid\nwavevector withjQj=QBulk= arctan (D=J)=a. Such a\nstate has Pspin/\u0000sin (QBulka)^P.\nConsider the Hamiltonian for surface anisotropy,\nHS=\u0000KSX\ni2surfaces(Si\u0001^ n)2; (5)\nwhereKSis the extra anisotropy energy arising due to\nthe reduced symmetry either at the nanoparticle/air sur-\nface or the nanoparticle/substrate interface (in case thenanoparticle is on top of a substrate). The surface unit\nvector ^ npoints perpendicular to the surface, with spins\nlying at the intersection of n0surfaces appearing n0times\nin the sum. There is an important di\u000berence between the\nsurfaces with ^ nk^Pand the ones that are not. The for-\nmer necessarily has Q?^ n, so that the surface is made\nup of cycloid chains, with allspins subject to anisotropy.\nIn this case,jKSj>0 greatly reduces the surface Qand\nthe cycloid is destroyed ( Q= 0) forjKSj> D2=J[36].\nThis reduction in Qpropagates a distance close to \u0015Bulk\ntowards the interior of the nanoparticle due to the prox-\nimity e\u000bect . For surfaces with ^ n6=\u0006^P,KSa\u000bects only\nthe edge of the cycloid chains penetrating into the ma-\nterial. These chains can adjust their Qto minimize the\nimpact of surface anisotropy as it is shown below. This\nwill be referred to as the edge e\u000bect .\nIII. THE EDGE EFFECT\nConsider a spin chain along xwith total length equal\nL= (N\u00001)a, and take ^P=^ z. The edges of the chain\ndescribed by i= 1 andi=Nare the only ones subject\nto surface anisotropy in Eq. (5). The total Hamiltonian\nH0+HSwas minimized numerically for each size N, with\n\u0015Bulk=a= 40 remaining \fxed (corresponds to D=J =\n0:15708). The numerical minimization was done using\nthe Nelder-Mead method with several random starting\npoints (NMinimize function in Mathematica ). The en-\nergy minima has the form of Eq. (4) with \u001e0pinned to\ncertain \fxed values and Q=Q^ xwithQstrongly depen-\ndent on size L. Figure 1 shows the results of the min-\nimization for Q=Q Bulk versusL=\u0015 Bulk, forNeven and\nsurface anisotropy KS=J= 0;\u00060:1;\u00061. The results\nfor easy axis ( KS>0) were identical to the ones for easy\nplane (KS<0). WhenKS= 0 theQwas independent\nof sizeLand equal to QBulk, as expected. The intro-\nduction of edge anisotropy caused asymptotic behaviour\ninQ. Its value became proportional to 1 =Lwith jump\ndiscontinuities at Ln= (2n+ 1)\u0015Bulk=4 (n= 0;1;2;:::).\nThe origin of this behavior is the necessity to make\nthe two edge spins perpendicular ( KS>0) or parallel\n(KS<0) to the surface, at the same time that the angle\nbetween every spin is kept constant. This can only be\nachieved by increasing or decreasing the angle between\neach spin to accommodate an integer multiple of \u0019=Q =\n\u0015=2. More insight is gained by looking at the winding\nnumberQL=(2\u0019): This is the number of 2 \u0019revolutions\ninside the chain. The inset of Fig. 1 shows that winding\nnumber versus Lhas well-de\fned plateaus at half-integer\nvalues.\nAt the locations of the discontinuities ( L=Ln), the\nequilibrium value of Qis bistable { the energy landscape\nis a double well with global minima at two di\u000berent values\nofQ. Later it will be shown that this bistability in Q\nimplies bistability of the modulus of the total electric\nand magnetic moments of the nanoparticle.\nThe drastic variability of Qas a function of size shown3\n��������������\n�� �� �� �� ���������\n�����������\n������\n��������\n�� �� �������\nFIG. 1. (colour online) Ground state cycloid wavevector Q\nas a function of system size Lfor the one-dimensional model\nwith edge anisotropy KS=J= 0;\u00060:1;\u00061. The quantities\nQandLare normalized by QBulkand\u0015Bulk, whereQBulk=\n2\u0019=\u0015 Bulk is the cycloid wave vector for the in\fnite system.\nInset: Winding number QL=2\u0019versusL=\u0015 Bulk.\nin Fig. 1 can be directly observed in experiments prob-\ning the cycloid in an ensemble of nanoparticles. A direct\nprobe ofQis to measure the cycloidal magnons using Ra-\nman [37] or TeraHertz spectroscopy [38]. The cycloidal\nmagnons lead to optical resonances at frequencies ap-\nproximately proportional to integer multiples of Q [39].\nFor an ensemble of nanoparticles, the variability in Q\nwill lead to inhomogeneous broadening of these optical\nresonances.\nIV. COMPETITION BETWEEN THE EDGE\nAND PROXIMITY EFFECTS\nTo see what happens in the presence of four surfaces\n(two sides with ^ n=\u0006^ x?^Pand top/bottom with ^ n=\n\u0006^ zk^P) consider a d= 2 platelet of spins oriented along\nthexzdirection, with spin labels i= (ix;iz). As a check,\nthe \frst calculation was done with KS6= 0 only for spins\nlocated on the side surfaces, at i= (1;iz) andi= (N;iz).\nThe optimal Qvalues were all equal for di\u000berent iz's.\nWith no surface anisotropy along the top and bottom of\nthe platelet, the spins behaved as in the d= 1 case, as\nexpected.\nIncluding surface anisotropy along all four surfaces\nmade the cycloid anharmonic, in that the optimal Qde-\npended on the index iz. Figure 2 shows results for N\neven (compensated) and KS=J= 0:1 (easy axis) on all\nsurfaces. The values of Qare listed according to their\nposition with respect to the z-direction, e.g. Qbottom is\nfor the bottom row ( iz= 1) andQ1=2is for the mid-\ndle row (iz=N=2). Note the mirror symmetry (e.g.\nQtop=Qbottom , andQ1=4=Q3=4), and how the value of\nQincreases gradually as the spin location moves towards\nthe centre. The competing interactions impose a \\prox-\nimity e\u000bect\" for surface anisotropy that a\u000bects spins well\n0.0 0.5 1.0\nL/Bulk\n0.81.01.21.4Q/QBulk\nQbottomQ1/4Q1/2Q3/4Qtop|M|\n0.000.250.500.751.00\n|M|\nFIG. 2. (colour online) Cycloid wavevector Qversus length\nfor a compensated two-dimensional platelet of side L, oriented\nalong the ^Q^Pplane (xz), with KS=J=\u00000:1. The values of\nQdepend on the spin location along z; the subscripts in Q\nindicate the value of z, e.g. Q3=4corresponds to z= 3L=4, etc.\nAtL=\u0015 Bulk\u00190:5 there is a jump discontinuity and the values\nofQare bistable. Also shown is the value of ferromagnetic\nmomentjMjcalculated from Eq. (7), in units of D0=J. Note\nhowjMjscales proportionally to QforL<0:5\u0015Bulk.\ninto the centre of the platelet.\nRemarkably, for L\u00190:5\u0015BulkallQ's become bistable\nin theKS=J=\u00000:1 case (note the jump discontinuity\nin Fig. 2). This is a surprising result, in view of the\nfact that the presence of several di\u000berent wavelengths\n\u0015(z) = 2\u0019=Q(z) does not allow the \ftting of odd in-\nteger multiples of a single \u0015=4 insideL. The bistabil-\nity occurs for several other values of surface anisotropy.\nFor example, when KS=J= 0:1 the bistability occurs\natL\u00190:8\u0015Bulk and 1:2\u0015Bulk. The results for Nodd\n(uncompensated) were quite similar, with the bistabil-\nity happening at a slightly di\u000berent L. The minimum\nenergy con\fguration had unpaired spins in each chain\naligning antiparallel to each other, leading to additional\ncontribution to the ferromagnetic moment per spin jMj\napproximately equal to 1 =L2, quite similar to the un-\npaired moment in non-cycloidal antiferromagnets in two\nand three dimensions [27].\nThe dependence of Qon nanoparticle size and KSis\ndepicted in Fig. 3a, the phase diagram for the modulus\nof the spin-induced ferroelectric moment jPspinjwhich is\nproportional tohsin (Qa)i.\nV. FERROMAGNETISM IN MULTIFERROIC\nNANOPARTICLES\nThe reduction of Qtowards the surfaces ^ nk^Pdramat-\nically impact the ferromagnetism of nanoparticles. To see\nthis, the spin-canting contribution of the DM interaction\nis considered,\nHDM=D0\n2X\ni;v(\u00001)P\n\u000bi\u000b^ z\u0001Si\u0002Si+v: (6)4\n0.2 0.4 0.6 0.8 1.0 1.2\nL/Bulk\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00KS/J\n0.020.060.100.140.180.220.260.30\n(a) Spin-induced ferroelectric moment.\n0.2 0.4 0.6 0.8 1.0 1.2\nL/Bulk\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00KS/J\n0.000.160.320.480.640.800.96 (b) Spin-canting-induced ferromagnetic moment.\nFIG. 3. (colour online) Phase diagram for the ferroelectric and ferromagnetic properties of a compensated L\u0002Lnanoparticle\nplatelet, as a function of size Land surface anisotropy KS. (a) Ferroelectric polarization per spin jPspinjcalculated from\nEq. (2), in units of D\u001f=(Pa3). (b) Ferromagnetic moment per spin jMjcalculated from Eq. (7), in units of D0=J. The black\nlines mark jump discontinuities in both jPspinjandjMj. At these points the nanoparticle is bistable with respect to both\nelectric and magnetic properties.\nForD0\u001cJthis interaction results in nanoparticle fer-\nromagnetic moment per spin given by\nM=1\nNdSX\niSi=D0\n2JNdSX\ni(\u00001)P\n\u000bi\u000b^ z\u0002Si:(7)\nFor bulk cycloids the argument of the sum in Eq. (7) is\na sine wave with period \u0015Bulkpointing perpendicular to\nthe cycloid plane. Thus Maverages out over distances\nL\u001d\u0015Bulk[40]. In nanoparticles with size L.\u0015Bulkthe\nferromagnetic moment does not average out. Measured\nin several experiments, it was interpreted to arise from\nuncompensated antiferromagnetism [17{23].\nFigure 3b shows the phase diagram for jMjin units of\nD0=Jfor compensated samples. Quite remarkably, the\nnanoparticles have sizable ferromagnetism in a large pa-\nrameter range. Here the ferromagnetism arises close to\nthe surfaces perpendicular to ^P, whereQis greatly re-\nduced so the spin canting contribution to Mdoes not\naverage out. This result shows that spin canting at the\nsurface provides an additional mechanism for nanoparti-\ncle surface ferromagnetism, scaling as jMj\u00181=Lin three\ndimensions, in agreement with experiments [18]. Even for\nsmall values of D0=Jthis can be much larger than the\nmoment arising from unpaired spins in uncompensated\nsurfaces [27].\nFigure 3 demonstrates a rich magnetoelectric phase\ndiagram as a function of particle size and surface\nanisotropy. There are several lines of bistability for Pspin.\nEach time this happens, there are four possible states for\nM(\u0006MwithjMjassuming two di\u000berent values).\nThe bistability in PspinandMcan be used as a mem-\nory where either PspinorMencodes information. With\nPspin\u00183\u0016C/cm2[35] the bit is switchable electricallywith electric \felds of the order of 102V/cm (see endnote\n27 in [30]). With M\u00180:1\u0016B=Fe [40] corresponding to\na local \feld of 200 G, it can be read out magnetically us-\ning usual hard drive read heads, or with state of the art\noptical read heads based on diamond NV-center magne-\ntometry [41]. Altogether such a memory bit corresponds\nto the \\ideal memory\" that is electrically written and\nmagnetically read envisioned in [10].\nVI. CONCLUSIONS\nThe considerations above allows general predictions\nabout the magnetoelectric behaviour of nanoparticles of\narbitrary shape. For example, in d= 3 consider a\nnanoparticle shaped as a cube with surfaces perpendic-\nular to the x;y;z axes. ForKS<0 (easy plane) the\nspin con\fguration that minimizes energy consists of xz\nspin planes stacked next to each other, each with spin\ncon\fguration identical to the d= 2 platelets described in\nFig. 3. Note that such con\fguration is a planar cycloid\nwithQ=Q(z)^ x, leading to anisotropy energy equal to\nzero for the two surfaces with ^ n=\u0006^ y.\nThe case of KS>0 (easy axis) in d= 3 warrants addi-\ntional calculations. Minimizing the ^ n=\u0006^ x;\u0006^ ysurface\nenergies leads to a twist con\fguration for the Qvector.\nConsider a cylindrical geometry with radius Rin the xy\nplane and axis length Lz!1 . Numerical minimization\nwithQ=Q^\u001a(the \\Q-monopole\") shows that Q(\u001a)\u001c\nQBulk for\u001a.\u0015Bulk andQ(\u001a)\u0019QBulk for\u001a&\u0015Bulk,\nwith jump discontinuities in QatR= (2n+ 1)\u0015Bulk=4\ndue to the edge e\u000bect. For Lz\fnite theQ0sare fur-\nther reduced due to the proximity e\u000bect of the ^ n=\u0006^ z\nsurfaces.5\nThese considerations show that nanoparticles of arbi-\ntrary shape with KS>0 will have smaller Pspinand\nlarger Mthan shown in Fig. 3 within an interior volume\n.\u00153\nBulk. 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Mart\u0013 \u0010nez,\nS. Chouaieb, K. Garcia, C. Carr\u0013 et\u0013 ero, A. Barth\u0013 el\u0013 emy,\nP. Appel, P. Maletinsky, J.-V. Kim, J. Y. Chauleau,\nN. Jaouen, M. Viret, M. Bibes, S. Fusil, and V. Jacques,\nReal-space imaging of non-collinear antiferromagnetic or-\nder with a single-spin magnetometer, Nature 549, 252\n(2017)."    },    {        "title": "1703.10903v1.Spin_Seebeck_effect_in_Y_type_hexagonal_ferrite_thin_films.pdf",        "content": "arXiv:1703.10903v1  [cond-mat.mtrl-sci]  31 Mar 2017Spin Seebeck effect in Y-type hexagonal ferrite thin films\nJ. Hirschner,1K. Kn´ ıˇ zek,1,∗M. Maryˇ sko,1J. Hejtm´ anek,1R. Uhreck´ y,2\nM. Soroka,2J. Burˇ s´ ık,2A. Anad´ on Barcelona,3and M. H. Aguirre3\n1Institute of Physics ASCR, Cukrovarnick´ a 10, 162 00 Prague 6, Czech Republic.\n2Institute of Inorganic Chemistry ASCR, 250 68 ˇReˇ z near Prague, Czech Republic.\n3Instituto de Nanociencia de Arag´ on, Universidad de Zaragoz a, E-50018 Zaragoza, Spain\nSpin Seebeck effect (SSE) has been investigated in thin films o f two Y-hexagonal ferrites\nBa2Zn2Fe12O22(Zn2Y) and Ba 2Co2Fe12O22(Co2Y) deposited by a spin-coating method on\nSrTiO 3(111) substrate. The selected hexagonal ferrites are both f errimagnetic with similar magnetic\nmoments at room temperature and both exhibit easy magnetiza tion plane normal to c-axis. Despite\nthat, SSE signal was only observed for Zn2Y, whereas no signi ficant SSE signal was detected for\nCo2Y. We tentatively explain this different behavior by a pre sence of two different magnetic ions in\nCo2Y, whose random distribution over octahedral sites inte rferes the long range ordering and en-\nhances the Gilbert damping constant. The temperature depen dence of SSE for Zn2Y was measured\nand analyzed with regard to the heat flux and temperature grad ient relevant to the SSE signal.\nKeywords:\nI. INTRODUCTION\nSpintronics is a multidisciplinary field which involves\nthe study of active manipulation of spin degrees of free-\ndom in solid-state systems [1]. Thermoelectricity con-\ncerns the ability of a given material to produce voltage\nwhen temperature gradient is present, thus converting\nthermal energy to electric energy [2]. The emerging re-\nsearch field of spin caloritronics, which may be regarded\nas interconnection of spintronics and thermoelectricity,\ncombines spin-dependent charge transport with energy\nor heat transport. One of the core elements of spin\ncaloritronics is the spin-Seebeck effect discovered in 2008\nby Uchida et al.[3]. The spin-Seebeck effect (SSE) is\na combination of two phenomena - the generation of a\nspin current by a temperature gradient applied across a\nmagnetic material, and a conversion of the spin current\nto electrical current by means of the inverse spin Hall\neffect (ISHE) [4] in the attached metallic thin layer. A\nnecessarycondition for the observationof SSE is that the\ndirections of the spin current, magnetic moments of the\nmagnetic material, and electrical current in the metal-\nlic layer, are mutually perpendicular. Since the resulting\nelectric field is related to temperature gradient, it is pos-\nsible in the regime of linear response to define a spin\nSeebeck coefficient SSSE=EISHE/∇T.\nAs regards the magnetic material as a source of the\nspincurrent,itismoreconvenienttouseinsulatorsrather\nthan conductors, in order to avoid parasitic signals such\nas a planar or anomalous Nernst effect [5]. There are\nthree main types of magnetic insulators possessing crit-\nical temperature TCabove the room temperature: gar-\nnets, spinels, and hexagonal ferrites. So far, most of the\nSSE experiments employed iron-based garnet because of\ntheir very low Gilbert damping constant, i.e.slow decay\n∗corresponding author: knizek@fzu.czof spin waves, since this decay limits the thickness of the\nmagnetic layer that actively generates the spin flow.\nIn this work we have focused on Y-hexaferrites as\nmagnetic material, namely Ba 2Zn2Fe12O22(Zn2Y) and\nBa2Co2Fe12O22(Co2Y). Their mass magnetizations at\nroomtemperatureare42.0emu/gforZn2Yand34emu/g\nfor Co2Y [6], which are higher than 27.6 emu/g of yt-\ntrium ferrite garnet Y 3Fe5O12[7]. Since a positive cor-\nrelation between SSE and the saturation magnetization\nhas also been proposed [8], Y-hexaferrites appear to be\na suitable material for the spin current generation in the\nspin-Seebeck effect.\nThe crystal structure of Y-hexaferrites belongs to the\ntrigonalspace group R3mand is composed ofalternating\nstacksofS (spinel Me 2Fe4O8, Me =Zn or Coin our case)\nand T (Ba 2Fe8O14) blocks along the hexagonal c-axis.\nThe magnetic configuration of Y-hexaferrites is usually\nferrimagnetic, with spin up orientation in octahedral 3 a,\n3band 18hsites and spin down in tetrahedral 6 cT, 6cS\nand octahedral 6 csites, see Fig. 1.\nMagnetocrystallineanisotropyis observedin all hexag-\nonal ferrites, which means that their induced magnetisa-\ntion has a preferred orientation within the crystal struc-\nture, either with an easy axis of magnetisation in the\nc-direction or with an easy plane of magnetisation per-\npendicular to c-direction, the latter being the case of the\nselected Y-hexaferrites. Due to their direction of easy\ngrow lying in ab-plane, hexaferrites inherently tend to\ngrow with their c-axis perpendicular to the film plane\nwhen deposited as thin films. Since the magnetization\nvector in SSE element should lie in parallel to the film\nsurface, the hexaferrites with an easy plane of magneti-\nsation are more suitable for the SSE experiment.\nThe principal difference between Zn2Y and Co2Y\ncomes from a different site preferences and magnetic\nproperties of Zn2+and Co2+. Zn2+ion is non-magnetic\n(d10) and occupies preferentially the tetrahedral sites.\nSince both Fe3+in tetrahedral sites have spin down ori-\nentation, thesubstitutionofZn2+tothesesitesmaximize2\n    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n T \nS /g3 /g3 /g127/g374/g1006/g122/g3 /g18/g381/g1006/g122/g3\n/g1007/g258/g3/g3/g894/g381/g272/g410/g895/g3 → /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0001/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286\u0002/g127/g374/g3 /g38/g286/g3\n/g1010/g272/g3/g3/g894/g381/g272/g410/g895/g3 ← /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1005/g1012/g346/g3/g894/g381/g272/g410/g895 /g3→→→ /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0003/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286/g853/g127/g374/g3 /g38/g286/g3\n/g1007/g271/g3/g3/g894/g381/g272/g410/g895/g3 → /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0000/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286/g853/g127/g374/g3 /g38/g286/g3\n/g1005/g1012/g346/g3/g894/g381/g272/g410/g895 /g3→→→ /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272/g3/g3/g894/g381/g272/g410/g895/g3 ← /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0004/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286/g853/g127/g374/g3 /g38/g286/g3\n/g3 /g3 /g3 /g3\n/g3\nFIG. 1: One formula unit of Ba 2Zn2Fe12O22or\nBa2Co2Fe12O22structure with alternating structural\nblocksSandT. Shown are Fe, Co and Zn polyhedra and Ba\ncations (magenta bullets). The description includes Wycko ff\npositions, types of polyhedra (tetrahedral or octahedral) ,\narrows indicating spin direction of the collinear ferrimag netic\nstructure, and the preferential occupation of sites.\ntheoverallmagneticmomentandthesaturationmagneti-\nsation at low temperature reaches 18.4 µB(theoretical\nlimit considering 5 µBperFe3+would be 20 µB). How-\never, because of the relatively low critical temperature\nTC∼130◦C, the magnetization at room temperature is\nonly about 10.7 µB(42 emu/g). Co2+ion is in the low\nspin state (LS, t6\n2ge1\ng) and occupies preferentially the oc-\ntahedral sites. The resulting magnetic moment depends\non the actual distribution of Co between octahedral sites\noccupied by Fe3+with spin up or spin down orienta-\ntion, nevertheless generally will be much lower than in\nZn2Y and the typical saturation magnetization is around\n10µB. On the other hand, since the critical temperature\nTC∼340◦C of Co2Y is higher, the magnetic moment\natTroomaround 8.6 µBis not so different from that of\nZn2Y, see e.g.the review paper [6].\nSpin-SeebeckeffectinY-hexaferritewasstudiedforthe\ncompound of stoichiometry Ba 2−xSrxZn2Fe12O22(x=\n1.5) [9]. In this study it was observed, that the magni-\ntude of SEE is proportional to bulk magnetization even\nthrough the successive magnetic transitions among vari-\nous helimagnetic and ferrimagnetic phases. M-type hex-\naferrite BaFe 12O19was studied in [10]. Since M-type\nhexaferrite have strong anisotropy with an easy axis of\nmagnetisation in the c-direction, a proper substrate and\ndeposition procedure must be selected in order to grow\nthe thin films with the c-axis oriented parallel to the sur-\nface. The advantage of M-type is its high coercive field,\nwhich makes the resulting SSE element self-biased, thus\nproducing SEE signal even without presence of magnetic\nfield. Spin-Seebeck effect was also studied in Fe 3O4with\nspinel structure, which may be in some context consid-\nered as the simplest structural type of hexagonal ferrites.\nLarge coercive fields and high saturation magnetisationISHEEr\nMr/g115\nsJrΔ/g100/g460/g410/g460/g87/g410\n/g122/g882/g346/g286/g454/g258/g296/g286/g396/g396/g349/g410/g286\n/g94/g396/g100/g349/g75/g1007/g400/g437/g271/g400/g410/g396/g258/g410/g286\n/g4/g367/g69 /g400/g286/g393/g258/g396/g258/g410/g381/g396\n/g346/g286/g258/g410/g286/g396\nFIG. 2: Schema of the longitudinal experimental configura-\ntions. Directions of temperature gradient ( ∇T), magnetiza-\ntion (M), spin current ( Js) and electrical field resulted from\ninverse spin Hall effect ( EISHE) are shown. The meaning of\nparameters Vx,tz,dxand△Tzused in eq. 2 is also indicated.\nmakes Fe 3O4promising magnetic material for the inves-\ntigation of self-biased SSE elements [11–16]\nCurrent researchdescribes the SSE using typical quan-\ntity of spin Seebeck coefficient with unit of µV/K, which\nis in conventional thermoelectric materials used for eval-\nuating the effectiveness of the process. However, in most\nof the experimental setups the temperature sensors mea-\nsuring the temperature difference △Tare attached to\nthe measure cell itself. This implies that △Tdescribes\nnot only the thermal characteristics of the studied ma-\nterial, but the whole measurement cell instead, making\nthe quantity in unit of µV/Kphysically irrelevant to the\nspin Seebeck effect itself. This issue was studied in de-\ntails in Ref. [17]. The authors pointed out, that when\nusing the setup dependent △Tas independent variable\nthe determined SSE can be hardly comparable between\nlaboratories. In order to solve this problem, the authors\ndesigned a measurement system with precise measure-\nment of the heat flux through the sample and proposed\nusing heat flux or thermal gradient at the sample as the\nindependent variable.\nIn this work we have followed this approach and man-\nifested, that the total temperature difference △Tis not\nsuitable independent variable even for measuring within\none setup if the temperature dependent experiment is\nperformed, since the temperature evolution of thermal\nconductivity of the whole setup may be different from\nthat of the sample material itself.\nII. EXPERIMENTAL\nThin films of Ba 2Zn2Fe12O22(Zn2Y) and\nBa2Co2Fe12O22(Co2Y) were prepared by spin-coating\ntechnique on (111)-oriented, epitaxially polished3\nSrTiO 3(STO) single crystals with metalorganic\nprecursor solutions. Commercial 2-ethylhexanoates\nMe(CH 3(CH2)3CH(C2H5)COO) n(n= 2 for Me = Ba,\nCo, Zn; n = 3 for Me = Fe, ABCR, Germany) were used\nas precursors. Calculated amounts of metal precursors\nwere dissolved in iso-butanol, mixed and heated for\nseveral hours at 80◦C to accomplish homogenization.\nSubsequently a suitable amount of 2,2-diethanolamine\n(DEA) used as a modifier was added. The modifier to\nalkali earth metal molar ratio was n(DEA)/n(alkali earth\nmetal) = 2. Prior to the deposition the stock solutions\nwere usually diluted with iso-butanol to obtain films of\ndesired thickness. All reactions and handling were done\nunder dry nitrogen atmosphere to prevent reaction with\nair humidity and preliminary formation of alkaline earth\ncarbonates in solutions. Single crystals of STO were\nwashed in acetone combined with sonication and then\nannealed at 1200◦C in air for 24 hours to heal up the sur-\nface damage caused during polish treatment. Prior the\ndeposition they were treated with plasma (Zepto Plasma\ncleaner, Diener Electronic, Germany). After the drying\nat 110◦C for several minutes and pyrolysis of gel films at\n300◦C for 5 minutes, crystallization annealing was done\nat 1000◦C for 5 minutes in conventional tube furnace\nunder open air atmosphere. The deposition-annealing\ncycle was repeated ten times to obtain the desired film\nwith approximately 300 −350 nm of thickness. Final\nannealing was done in tube furnace under open air\natmosphere at 1050◦C for 5 min (Zn2Y) or 1000◦C for\n60 min (Co2Y).\nSpin Seebeck effect was measured using home-made\napparatus. A longitudinal configuration was used, in\nwhich the directions of the spin current, magnetic mo-\nments and electrical current are mutually perpendicular\n[18], see Fig. 2. AlN plate with high thermal conduc-\ntivity was used to separate the heater and the sample\nin order to uniformly spread the heat flux over the sam-\nple area. The thermal barriers between individual parts\nof the call were treated by appropriate greases (Apiezon\ntype N, Dow Corning Varnish, Ted Pella silver paste).\nThe width of the measured sample was 2 mm, the\nlengthwas7mmandcontactdistancewasapprox. 5mm.\nThickness of the Zn2Y-hexaferrite layers was between\n300−350 nm, the thickness for Co2Y-hexaferrite layers\nranged between 150 −300 nm. Pt layer was deposited\nusing K550X Quorum Technologies sputter coater. The\nthickness of the layer was determined by internal FTM\ndetector (Tool factor 4.7), the final Pt deposition thick-\nness was ∼8 nm. The resistanceof the Pt-layermeasured\nby a 2-point technique was within the range 350 −650 Ω\nat room temperature and linearly decreased by 10 −15%\ndown to 5 K, whereas the resistance of the Y-hexaferrite\nthin layer itself was more than GΩ. Therefore, the con-\ntributionfromthe anomalousNernst effect(ANE) canbe\nconsidered as negligible due to the resistivity difference\nbetween Y-hexaferrite and Pt layers.\nThe magnetic hysteresis loops were measured within\nthe range of magnetic field from −25 to 25 kOe at room25 30 35 45 50 55 60 650100020003000400050006000\n-0.5 0.0 0.5-0.5 0.0 0.5\n00120015\n0027\n0030Co2Y\nZn2Y\n2theta (Cu, K α) counts per sec.\nomega (°) counts per sec.(0012)\n±0.45°counts per sec.\nomega (°) \n(0012)\n±0.58°\nFIG. 3: X-ray diffraction of the Ba 2Zn2Fe12O22(black line,\nZn2Y) and Ba 2Co2Fe12O22(blue line, Co2Y) thin film. The\ninsets show rocking-curve measurements. The diffraction\npeak (111) of the SrTiO 3substrate is skipped.\nFIG. 4: AFM images of surface topography of (a)\nBa2Zn2Fe12O22(calculated roughness r.m.s. = 27 nm) and\n(b) Ba 2Co2Fe12O22(r.m.s = 30 nm).\ntemperature using a SQUID magnetometer (MPMSXL,\nQuantum Design)\nThe phase purity and degree of preferred orientation\nof the thin films was checked by X-ray diffraction over\nthe angular range 10 −100◦2θusing the X-ray pow-\nder diffractometer Bruker D8 Advance (CuK α1,2radia-\ntion, secondary graphite monochromator). Atomic force\nmicroscopy AFM (Explorer, Thermomicroscopes, USA)\nwas used to evaluate surface microstructure of the thin\nfilms.\nIII. RESULTS AND DISCUSSION\nThe X-ray diffraction confirmed single phase purity of\nthe thin film and c-axis preferred orientation, quantified\nby the full-width at the half-maximum (FWHM) of the\nrocking curve as 0.45◦for Zn2Y and 0.58◦for Co2Y, see\nFig. 3. The c-lattice parameters 43.567(7) ˚A for Zn2Y4\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s50/s45/s49/s48/s49/s50/s86\n/s83/s83/s69/s32/s40 /s86/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s32 /s84/s32/s61/s32/s49/s75\n/s32 /s84/s32/s61/s32/s50/s75\n/s32 /s84/s32/s61/s32/s51/s75\n/s32 /s84/s32/s61/s32/s52/s75\n/s32 /s84/s32/s61/s32/s53/s75/s84/s32/s61/s32/s51/s48/s48/s75\n/s116/s32/s61/s32/s51/s48/s48/s32/s110/s109\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s80/s101/s114/s112/s101/s110/s100/s105/s99/s117/s108/s97/s114\n/s32/s80/s97/s114/s97/s108/s108/s101/s108\nFIG. 5: Spin Seebeck signal (upper panel), and in plane and\nout of plane magnetization (lower panel), in dependence on\nmagnetic field at 300 K for Ba 2Zn2Fe12O22.\nand 43.500(9) ˚A for Co2Y, calculated using cos θ/tanθ\nextrapolation to correct a possible off-centre position of\nthe film during XRD measurement, are in good agree-\nment with literature values [19].\nFig. 4 shows AFM images of surface topography of\nZn2Y and Co2Y. Platelets with hexagonal shape can be\nidentified in both images with similar shape and size.\nCalculated roughness (r.m.s.) values are around 27 −\n30 nm.\nThe magnetic properties ofthe Y-hexaferritesthin lay-\ners werecharacterizedby magnetization curvesmeasured\nat room temperature. The magnetic moment of Zn2Y at\nTroomdetermined from the saturatedvalue of magnetiza-\ntionin parallelorientationis11 µB, seethelowerpanelof\nFig. 5, which is comparable with the expected value [6].\nThe measurement confirms that Zn2Y is a soft magnet\nwith negligible hysteresis. The saturation in the orien-\ntation parallel with the thin layer is attained already at\nlow field, whereas the saturation in the out of plane ori-\nentation, i.e.along the c-direction, is achieved at higher\nfield above 1 ∼T, in agreement with the abeasy plane\norientation.\nThe Spin Seebeck signal of Zn2Y at room tempera-\nture is displayed in the upper panel of Fig. 5 for various/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s86\n/s83/s83/s69/s32/s40 /s86/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s116/s32/s61/s32/s51/s48/s48/s32/s110/s109\n/s32/s116/s32/s61/s32/s49/s53/s48/s32/s110/s109/s84/s32/s61/s32/s51/s48/s48/s32/s75\n/s84/s32/s61/s32/s53/s32/s75\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s80/s101/s114/s112/s101/s110/s100/s105/s99/s117/s108/s97/s114\n/s32/s80/s97/s114/s97/s108/s108/s101/s108\nFIG. 6: Spin Seebeck signal (upper panel) for 2 selected\nthin layers, and in plane and out of plane magnetization\n(lower panel), in dependence on magnetic field at 300 K for\nBa2Co2Fe12O22.\ntemperature gradients applied across the thin layer. The\nmeasuredvoltageispositiveinpositiveexternalmagnetic\nfield, in agreement with the positive spin Hall angle of Pt\n[20], and changes sign when switching the polarity of the\nmagneticfield. Thedependence onthe magneticfield has\nthe same shape with negligible hysteresis as the magne-\ntization in parallel orientation. The data clearly show\nlinear dependence on temperature gradient.\nThe magnetic moment of Co2Y at Troomdetermined\nfrom the saturated value of magnetization in parallel ori-\nentation is 10 µB/f.u., see the lower panel of Fig. 6.\nThis value is slightly higher than the expected moment\n[6], presumably due to relatively higher structural pref-\nerence of Co for spin down sites in the case of our thin\nfilms. The difference between the saturation in parallel\nandperpendicularorientationisbiggerinagreementwith\nhigher magnetocrystalline anisotropy of Co2Y compared\nto Zn2Y.\nHowever, despite the similar magnetic properties, the\nSSE signal for Co2Y was not observed, see the upper\npanel of Fig. 6. To explain this different behavior of\nZn2Y and Co2Y, we have considered the difference in\nthe cation distributions of the transition metal cations\nover the structure, see for details the Introduction sec-5\n/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s83/s83/s69/s32/s40 /s86/s47/s75/s41/s32/s84/s32/s61/s32/s53/s75 /s32/s84/s32/s61/s32/s49/s48/s48/s75/s83/s83/s69/s32/s40 /s86/s47/s75/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s84/s32/s61/s32/s50/s48/s48/s75\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s84/s32/s61/s32/s51/s48/s48/s75\nFIG. 7: Spin Seebeck signal (SSE) dependence on magnetic\nfield, divided by the total temperature difference △T, for\nBa2Zn2Fe12O22at selected temperatures.\ntion. Since Zn2+ion is a non-magnetic, the structure of\nBa2Zn2Fe12O22only contains one type of magnetic ion.\nZn cation preferentially substitutes Fe3+in two tetrahe-\ndral sites with the same direction of spin polarization,\ntherefore the total spin polarization of the unit cell does\nnot significantly fluctuate across the material. In dis-\ntinction, Ba 2Co2Fe12O22contains two types of magnetic\nions,i.e.Co2+in low spin state in addition to Fe3+\nin high spin state. Co substitute Fe3+in all octahedral\nsites, where Fe may have both directions of spin polar-\nization, without strong preference for a particular sites.\nWe tentatively propose, that this random distribution of\nCo2+over octahedral sites with various spin polariza-\ntions interferes the long range magnetic ordering across\nthe material, enhancesthe Gilbert damping constant and\npossibly results in suppressing of the SSE signal.\nSSE loops of Ba 2Zn2Fe12O22was measured at several\ntemperatures down to 5 K, see the measurements at se-\nlected temperatures 5, 100, 200 and 300 K in Fig. 7. The\noutput power of the heater was the same for all temper-\natures. The character of the loops is not changed with\nloweringtemperature, the magnitude of the signalis only\nvarying.\nIn order to investigate the temperature dependence of\nthe SSE signal of Zn2Y in more details, we have per-\nformed measurementdown to low temperature with 10 K\nstep. The output power of the heater was also kept con-\nstantduringthismeasurement. ThevalueoftheSSEwas\ndetermined by switching the magnetic field to ±0.4 T at\neach temperature and calculating the difference\nSSE=V+0.4T−V−0.4T\n2(1)\nThe resulting temperature dependence is displayed in\nthe Fig. 9 in three ways. In the upper part of the figure,\nFig. 9a, the SSE signal is divided by the total temper-\nature difference △Tdetermined over the whole measur-0 50 100 150 200 250 3000.00.51.01.52.0\n0 100 200 300050100150200250 Thermal conductivity (W/K/m)\nTemperature (K) AlN\n SrTiO3 W/K/m\nT (K)\nFIG. 8: Thermal conductivity of bulk Ba 2Zn2Fe12O22. Inset:\nthermal conductivity of AlN and SrTiO 3.\ning cell. The temperature evolution of △Tshown in the\ninset revealed, that △Tincreased several times during\ncooling. Since the output power of the heater was kept\napproximately constant, this increase should be related\nto a decrease of the thermal conductivity of the materi-\nals between the temperatures probes. To verify this as-\nsumption, we have measured thermal conductivity of the\nrelevant materials, i.e.the bulk sample Ba 2Zn2Fe12O22\nsynthesizedfromtheprecursorsusedforthethinlayerde-\nposition and compacted by isostatic pressing, AlN plate\nused to separate the heater and the sample, and the\nSrTiO 3substrate, see Fig. 8. However, thermal conduc-\ntivities of these materials weighted by their thickness in\nthe measuring cell cannot explain the evolution of △T.\nIt is obvious, that in order to explain the observed tem-\nperature dependence of △T, the thermal resistance of\nthe thermal barriers between the attached parts of the\ncell must be taken into account. We have calculated,\nthat the thermal resistance of the barriers at room tem-\nperature represents more than 50% of the total thermal\nresistance of the cell, and its percentage increases with\ntemperature.\nSince the value of the total temperature difference △T\ncannot be used as independent variable in different mea-\nsurement setups among various laboratories, another less\nsetup dependent parameter should be used instead, in\norder to normalize the measured SSE signal. We used\nthe heat flux through the sample, as it was proposed in\nRef [17]. SSE signal divided by the heat flux through\nthe sample is displayed in Fig. 7b, the corresponding\nheat flux corrected for the heat losses due to radiation,\nis shown in the inset.\nIn order to extract quantity comparable over differ-\nent measurement setups including the geometry of the\nsample, an expression for spin Seebeck effect related to\nsample dimensions and temperature difference over the\nsample itself was defined [17, 21]\nSSSE=Vxtz\ndx△Tz(2)\nwhereVxis the voltage measured, tzis the thickness6\n0.00.20.40.60.8\n0 100 200 30002040\n020406080100\n0 50 100 150 200 250 3000.00.10.20.30.40.50 100 200 300110120130\n0 100 200 3000.00.51.01.5SSE (µV/K)\na)ΔT (K)\nT (K)\nb) SSE (µV/ W)\nc) SSSE (µV/K)\nTemperature (K) Heat flux\n(mW)\nT (K)\n ΔTz(10-3K)\nT (K)\nFIG. 9: Spin Seebeck signal (SSE) dependence on tempera-\nture for Ba 2Zn2Fe12O22, (a) divided by overall temperature\ngradient, inset: temperature difference. (b) divided by hea t\nflux, inset: heat fluux. (c) calculated according to eq. 2, ins et:\ntemperature difference △Tz.\nof the magnetic material, dxis the electric contact dis-\ntance, and △Tzis the temperature difference at the mag-\nnetic material along the thickness tz, see Fig 2. With the\nknowledge of the heat flux and the thermal conductivity\nof the sample material Ba 2Zn2Fe12O22we were able to\ncalculate SSSEaccording to eq. 2, see the temperature\ndependence in Fig. 9c, the evolution of △Tzis displayed\nin the inset.\nThe correct normalization of SSE signal is important\nnot only for comparing among various measurement se-\ntups, but also for the correct determination of the tem-\nperature dependence, as it is evident by comparison of\nvarious temperature evolutions of SSE shown in Fig. 7.\nThe SSE related to the total temperature difference △T\n(Fig. 7a) shows incorrect temperature dependence influ-\nenced by the temperature dependence of the total ther-\nmal conductivity of the measuring setup. We propose\nthat the correct temperature dependence is determined\nby relating SSE to heat flux (Fig. 7b) or to temperature\ndifference at the sample △Tz(Fig. 7c). In this case, SSE\nis almost linearly increasing with lowering temperature.\nThe almost 5 ×increase of SSE at low temperaturecompared to room temperature can be partially ex-\nplained by the increased magnetization (almost 2 ×), but\nthe decrease of Gilbert damping factor αshould be of\ngreater influence in this regard. It was determined in\nthe study of the temperature dependence of SSE signal\nin Y3Fe5O12garnet (YIG) [22], that the effective prop-\nagation length of thermally excited magnons ξis pro-\nportional to T−1, and since at the same time α∼ξ−1\n[23, 24], it means that Gilbert damping factor α, which is\nexpected to suppress the SSE signal, is linearly decreas-\ning with temperature.\nIn distinction to temperature dependence of SSE in\nYIG [22], where a maximum in SSE was observed and\nexplained by the interplay of the increase of magnon ef-\nfective propagationlength and decrease ofthe total num-\nber of thermally excited magnons, we observed no max-\nimum down to low temperature. We ascribe it to the\nlower dispersion of acoustic branches in magnon spectra\nof Y-hexaferrite, which makes the influence of increasing\ntotal number of thermally excited magnons less impor-\ntant.\nFor the confrontation of the normalized room val-\nues between Y-hexaferrite and garnet, we have deter-\nmined values 21 µV/W and SSSE= 0.11µV/K for\nBa2Zn2Fe12O22, which are lower in comparison with\n46.6µV/W and 0.28 µV/K for Y 3Fe5O12[17], despite\nthe higher magnetic moment of Zn2Y. We presume, that\nit is due to the lower Gilbert damping constant αand\nhigher dispersion of acoustic branches in magnon spectra\nof Y3Fe5O12.\nIV. CONCLUSIONS\nSpin Seebeck effect (SSE) has been investigatedin thin\nfilms of two Y-hexagonal ferrites Ba 2Zn2Fe12O22(Zn2Y)\nand Ba 2Co2Fe12O22(Co2Y) deposited by spin-coating\nmethod on SrTiO 3(111) substrate. The SSE signal was\nobservedfor Zn2Y, whereasno significant SSE signal was\ndetected for Co2Y. This can be explained by a pres-\nence of two different magnetic ions in Co2Y, whose ran-\ndom distribution over octahedral sites interferes the long\nrange ordering and enhances the Gilbert damping con-\nstant. The magnitude of spin-Seebeck signal of Zn2Y\nnormalized to the temperature difference at the inves-\ntigated layer and sample dimensions ( SSSE) is compa-\nrable to the results measured on yttrium iron garnet\nY3Fe5O12.SSSEofZn2Yexhibitsmonotonicallyincreas-\ning behaviour with decreasing temperature, as a result\nof the simultaneous increase of the magnetization and\nmagnon effective propagation length.\nAcknowledgement . This work was supported by\nProject No. 14-18392S of the Czech Science Foundation\nand SGS16/245/OHK4/3T/14 of CTU Prague.7\n[1] Y. Xu, D. D. Awschalom, and J. Nitta, Handbook of\nSpintronics (Springer Netherlands, 2015).\n[2] D. M. Rowe, Thermoelectrics Handbook: Macro to Nano\n(Taylor & Francis, 2005).\n[3] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. 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Algarabel, L. Morellon, S. Maekawa, E. Saitoh, and\nM. R. Ibarra, Unconventional scaling and significant en-\nhancementofthespinSeebeckeffectinmultilayers, Phys.Rev. B92, 220407 (2015).\n[14] A. Anadon, R. Ramos, I. Lucas, P. A. Algarabel,\nL. Morellon, M. R. Ibarra, and M. H. Aguirre, Char-\nacteristic length scale of the magnon accumulation in\nFe3O4/Pt bilayer structures by incoherent thermal ex-\ncitation, Appl. Phys. Lett. 109, 12404 (2016).\n[15] A. J. Caruana, M. D. Cropper, J. Zipfel, Z. X. Zhou,\nG. D. West, and K. Morrison, Demonstration of poly-\ncrystalline thin film coatings on glass for spin Seebeck\nenergy harvesting, Phys. Status Solidi-Rapid Res. Lett.\n10, 613 (2016).\n[16] R. Ramos, A. Anadon, I. Lucas, K. Uchida, P. A. Al-\ngarabel, L. Morellon, M. H. Aguirre, E. Saitoh, and\nM. R. Ibarra, Thermoelectric performance of spin See-\nbeck effect in Fe3O4/Pt-based thin film heterostructures,\nAPL Mater. 4, 104802 (2016).\n[17] A. Sola, M. Kuepferling, V. Basso, M. 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Robert Odette  b and Dane Morgan  a*  \n \na Department of Materials Science and Engineering, University of Wisconsin -Madison, Madison, WI 53706, USA  \nb Materials Department, University of California, Santa Barbara, CA 93106, USA  \n* Corresponding author. E -mail address: ddmorgan@wisc.edu  \n \nAbstract  \n Mn-Ni-Si precipitates (MNSPs) are known to be responsible for irradiation- induced hardening and \nembrittlement in structural alloys used in nuclear reactors. Studies have shown that precipitation of the \nMNSPs in 9 -Cr ferritic -martensitic (F -M) alloys, such as T91, i s strongly associated with heterogeneous \nnucleation on dislocations, coupled with radiation- induced solute segregation to these sinks. Therefore it \nis important to develop advanced predictive models for Mn- Ni-Si precipitation in F -M alloys under \nirradiatio n based on an understanding of the underlying mechanisms. Here we use a cluster dynamics \nmodel, which includes multiple effects of dislocations, to study the evolution of MNSPs in a commercial \nF-M alloy T91. The model predictions are calibrated by data fro m proton irradiation experiments at 400 \n°C. Radiation induced solute segregation at dislocations is evaluated by a continuum model that is \nintegrated into the cluster dynamics simulations, including the effects of dislocations as heterogeneous \nnucleation s ites. The result shows that MNSPs in T91 are primarily irradiation -induced and, in particular, \nboth heterogeneous nucleation and radiation- induced segregation at dislocations are necessary to \nrationalize the experimental observations.  \n \nKeywords: Ferritic -martensitic steel, heterogeneous nucleation , dislocation, precipitation,  cluster \ndynamics  model  \n  2 \n 1 Introduction \nThe extremely harsh  environment in advanced fission and fusion reactor s requires the development \nof structural materials with superior performance and stability. Ferritic -martensitic (F-M) steels are a \nleading candidate structural alloy because of  their attractive and well -established properties  including  \nreduced activation , swelling  resistance and irradiation stability  [1]. Nevertheless, irradiation drives  \nmicro structural, micro chemical and  precipitate evolutions  in F-M alloys  [2-5], which can degrade the \nperformance and  safe lifetime limits  of advanced reactor  structural components . For example , \nradiation -induced precipitates cause hardening and embrittlement by acting as obstacles  to dislocation  \nglide. Thus understanding  and predicting precipi tate development  as a function of the alloy composit ion \nand starting microstructure , as well as the irradiation conditions , are critical to a wide range of nuclear \nenergy technologies .  \nIn this work we focus on the alloy T91, a commercial F -M steel with ~9% Cr and small additions of \nother micro -alloying element s such as C, Ni, Mn, Si, V , Mo, etc.  Special attention is given to the  \nprecipit ation of MNSPs  that have been shown to cause  high fluence embrittlement of reactor pressure \nvessel  (RPV) steels  [6-9]. Radiation -induced segregation (RIS) , solute clustering and precipitation of \nminor elements in T91 have been  investigated  experimentally in a number of recent studies . Jiao and Was \n[3] and Wharry et al . [4] reported segregation of Cr, Ni, and Si at defect sinks such as dislocation \nlines/loops , grain boundaries  and precipitate/matrix interfaces  in T91 under proton and heavy -ion \nirradiation. Wharry and Was  [10] studied the temperature dependence of RIS at grain boundaries and \nfound that enrichment of  minor element (Ni, Si, Cu) RIS peaked at 400 -500 °C. Both MNSPs  and \nCu-rich precipitates  (CRPs) have  been observed  by atom probe tomography (APT)  following \nproton- irradiated T91 at 400 and 500 °C  [4]. However, Jiao et al . [11] noted that no MNSPs  were \nobserved in T91 irradiated at less than  1 dpa  (displacement s per atom)  at 400 °C  under proton irradiation  3 \n at a dose rate of 10-5 dpa/s , suggesting significant kinetic limits on the ir formation . In both MNSP studies  \nin T91 [4, 11], dislocations were found to be the preferred MNSP  nucleation site . Wharry et al. [4] also \nsuggested that the strong segregation of Si at dislocation s due to RIS may correlate with the evolution of \nthe MNSPs .  \nSimulation studies of MNSPs  in steels have been carried out by  calculat ion of phase diagrams \n(CALPHAD)  methods  [12], lattice Monte Carlo simulations  [13, 14] and very recently by  cluster \ndynamics  (CD) model ing [7, 15]. A recent Monte Carlo study  showed that dislocation loops can act as \nheterogeneous nucleation sites for solute clusters  in RPV steels  [16]. Solute clusters  with enriched Ni, \nSi, P and Cr  were found in irradiated Fe -Cr model alloys  [17-20]. However , previous  modeling studies \nhave not focused on MNSPs  in F-M alloy s, including  T91. Additionally , RIS and dislocation effect s on \nthe density and size evolution of MNSPs  are not well understood. Here we utilize the CD model to \npredict  the evolution of MNSPs in a commercial F -M alloy T91 and compare with available experimental \ndata. The model considers multiple effects of dislocation, including heterogeneous nucleation and \nradiation -induced solute segregation at dislocations.  \n \n2 Methods  \nWe emp loy thermodynamic analysis and CD  model ing [21] to study the stability and evolution of \nMNSPs  in T91. The CD method has been utilized successfully to study nucleation -growth processes of \ndiffusional phase transformations, and recent applications include Cu precipitation in α -Fe [22], oxide \nprecipitation in nanostructured ferritic alloys [ 23], and MNSPs  in RPV steel s under irradiation [7, 15]. \nGoverning equations and parameters used in the CD  model are summarized in the Supplementary \nInformation (SI Section S1). The CD  simulations includes an embedded radiation -enhanced diffusion  \n(RED) model developed by Odette et al . [24]. The thermodynamic driving force of MNSP  nucleation is 4 \n calculated based on the TCAL3 database. To account for the multiple effects of dislocations, we \ncombine the CD model  with the theory of heterogeneous nucleation at dislocations  [25], as well as an \nRIS model  [26, 27] used as a semi -empirical approach for  tracking  the dose (and dose rate) -dependent \nsolute enric hment  near dislocations .  \n \n2.1 Heterogeneous nucleation on dislocations  \nDislocations are known to possess a catalytic effect on nucleation and act as favorable nucleation \nsites if the process releases their  excess free energy. We assume that the nucleation of MNSPs  follows \nthe incoherent nucleation theory first developed  by Cahn [ 25] and extended by Gomez -Ramirez and \nPound [28]. The theory allows the nucleation of clusters to replace the region of dislocation core s while \ncreating an incoherent interface with the matrix. T he replaced dislocation segment is assumed to be \nsmeared out as a continuous distribution of infinitesimal interfacial dislocations  so as to reduce any \nstrain energy contribution to near zero [ 28]. Thus the  energy released by nucleation includes the \ncontributions of dislocation core and strain energy inside the precipitate region. By applying t his model \nand assuming the dislocation line passes through the center of the spherical MNSPs , the released excess \nfree energy can be calcul ated. Addit ionally, dislocatio ns provide  a fast pipe  diffusion pathway  to the \nprecipitates for solutes segregated to the dislocation lines. Thus we consider the rate controlling process \nfor solute transport to be both 3D diffusion to the precipitates and 2D diffusion to the attached \ndislocation segments . Detailed calculations and parameters used in the model a re described in SI \nSection S1.1 and S2.  \nWe make the approximation  that the number density of available heterogeneous nucleation sites at  \ndislocations is equal to the number density of atoms along  the dislocation line , and we assume that no \nnew clusters nucleate within a distance  of 5 nm  from any evolving cluster  to avoid precipitate volume  5 \n overlap . This distance of 5 nm is chosen to be close to as small as possible  given that the typical \nprecipitate radius is ≈  2 nm. Note that the separation distance, along with the dislocation density, sets the \nultimate number density of precipitates. Thus for any value near 5 nm the number density predicti ons, for \na specified dislocation density, will be relatively similar , and the exact predicted value is not particularly \nsignificant. Note we do not consider precipitate coherency strains, since t he radiation -induced MNSP \n(G-phase)  precipitates have a cube -on-cube orientation relationship with ferrite matrix with small lattice \nmismatch  [29]. Any such strains woul d likely be compensated for by reductions in strains for a \ndislocation outside the precipitate region. In the present study we did not include heterogeneous \nnucleation at grain boundaries because  of the lower nucleation site densities  compared to dislocati ons, as \ndetailed  in SI Section S2. The recent APT studies of T91 strongly suggest that dislocations are the major \nnucleation site [ 4, 11]. More rapid nucleation is also expected due to the stronger RIS at dislocations than \nat grain boundaries in T91 [ 3, 4]. \n \n2.2 Radiation- enhanced diffusion \nThe RED  model developed by Odette et al . [24] was used to calculate X vDv and scale thermal \ndiffusion coefficients. Under irradiation X v is much greater than the equilibrium vacancy concentration, \nXve, resulting in RED. Odette’s model treats the eff ect of dpa -rate-dependent solute vacancy trapping that \nenhances r ecombination with self -interstitial atoms (SIA), reducing X vDv, relative to the condition when \nall the diffusing vacancies and SIA annihilate at sinks (no recombination) [ 24]. The effect of solute trap \nenhanced recombination is to increase the dpa required to reach a specified amount of precipitation under \nRED. In the RED calculation, we consider the highly concentrated Cr atoms in T91 as a solute traps that \nenhance with a binding energy of 0.094 eV taken from ab- initio  calculati on in Ref. [30]. Detailed X vDv \ncalculations are presented in SI Section S1.2. 6 \n  \n2.3 Radiation- induced segregation \nHeterogeneous nucleation at dislocations was treated by combining the local microalloy time \n(dose) -dependent RIS and cluster dynamic model s. The RIS model is based on the formulations \nproposed by Wiedersich et al. [26] and Wolfer [27], that  consider both the contributions of the inverse \nKirkendall effect and vacancy drag  as detailed in SI Section S3. The underlying mechanism driving Si \nRIS in our model is vacancy solute drag based on recent ab initio  evaluation of transport coefficients in \ndilute bcc Fe- Ni and Fe -Si alloys [ 31]. Note, in contrast  to Ni and Si, Mn segregation to dislocations is \nnot observed in T91 [3]. Thus here only Ni and Si segregation was modeled. In both RIS calculations \nwe did not include the contribution of grain boundaries as sinks due to the relatively  much higher value \nof dislocation density. The grain boundary sink strength [ 32] of the 1- μm grain (~1013 m-2) is about 2 \norders of magnitude smaller than the dislocation sink strength.  \n \n2.4 Description of simu lations  \nWe first employ  the thermodynamic simulations showing the equilibrium phase fraction of MNSPs \nin T91 at 400 °C , followed by the cluster dynamics simulations of Mn -Ni-Si precipitate formation \ninduced by proton irradiation at 400 °C and 10-5 dpa/s. Figure 1 shows the flow chart of the cluster \ndynamics simulations, which involve the quantitative calculations of radiation and dislocation effects. \nThe cluster dynamics model is calculated by a series of master equations of cluster distribution functions  \n(Eq. (S1)) in which their evolution is determined by the rate coeff icients and formation energy of clusters. \nThe former includes the contributions of thermal diffusion, radiation- induced diffusion by the diffusion \npaths through the matrix and dislocation lines , which are detailed in SI Section S . The latter is calculated \nby Eq. (S 15) which considers  the chemical free energy, interfacial ene rgy, and catalytic effect of 7 \n dislocations. It is noted that for the simulation of heterogeneous nucleation at dislocations, the free \nenergy contribution is calculated by using the RIS composition at the segregated and microalloyed region \nnear dislocations.  \nThe CD model integrates the RIS and RED calculation results of proton- irradiated alloy T91  at 400 \n°C and a dose rate of 10-5 dpa/s . The model considers multiple  dislocation effects, including the density \nof heterogeneous nucleation sites and the decrease of nucleation free energy barrier. The nucleation \nbarrier  was calculated based on the free energy reduction driving phase separation in the RIS microalloy, \nthe precipitate -matrix interfacial energy and the energy release associated with annihilation of the \ndislocation segment inside the precipita te. The dpa -dependent local RIS compositions (enrichment) of Ni \nand Si scaled in proportion to the instantaneous solute concentrations in matrix. The bulk solute contents \ndecreased with dpa in proportion to the Mn, Ni, Si precipitation. The cluster dynamic s model was used \nto simulate the evolution of the number density ( N), mean radius (< r>) and mole fraction ( f) of MNSPs \nat 400°C up to 100 dpa. \n \n3 Results and discussion  \n3.1 Thermodynamic calculations  \nThe thermodynamic state in T91 was evaluated  first by the Thermo -Calc software using the TCAL 3 \ndatabase [33]. Three alloy compositions  were considered: the original T91 ( a); the RIS composition \nmeasured by Wharry et al . [4] at grain boundaries (b); and, the composition measured by Jiao et al.  [3] at \ndislocations  (c). The measured compositions were 0.45 at% Mn, 0.58 at% Ni  and 0.95 at% Si  at grain \nboundaries , and 0.37 at% Mn, 0.78 at% Ni, and 4.01 at% Si at dislocations . Figure 2 shows the calculated \nphase fraction of the equilibrium bulk MNSPs  (no interface effects) as a function of temperature. Only \nthe T3 (Mn 6Ni16Si7) phase (also called G -phase) is stable. In the case of the bulk composition prior to 8 \n irradiation , MNSPs  are not able to form at temperatures  higher than 327 °C. However, for the  RIS \ncompositions , the maximum temperature for the formation of MNSPs  increases to 410 °C at grain \nboundaries and 505 °C at dislocations. The corresponding thermodynamic phase fractio n is 0.12% and \n1.0% at 400 °C at grain boundaries and dislocations , respectively . Note that the phase  fraction of \nprecipitate predicted here would only occur in the  RIS enhanced region, which can be considered a local \n“microalloy”,  and not throughout the bulk of the alloy  [34]. These results suggest that the formation of \nthe MNSPs  is radiation -induced by the segregation of solutes a t defect sinks, which increase s the local \ndriving force of precipitate nucleation and growth . The thermodynamic prediction is consistent with the \nexperimental observation [ 4] showing  MNSPs  in proton- irradiated T91 at 400 °C . Note that in this \nthermodynamic analysis, the T6 ( Mn(Ni,Si) 2) phase was included in the calculation but did not  form  \neven for the T91 local dislocation  RIS composition. Therefore, the precipitation modeling of MNSPs  in \nthis study focuses on  the T3 or G -phase,  which are also observed in RPV steels [ 6-9] and HT-9 F-M \nalloy  [35, 36] .  \n \n3.2 Result of radiation- induced segregation  \nFigure 3 shows the predicted RIS as a function of dpa dose at 400 °C for the experimental condition \nof ≈ 10-5 dpa/s  for the proton irradiations  in Ref. [3]. The solid lines are simple fits to the small filled \nsquare computed data points, while the large open symbols are the experimentally observed solute \nconcentration at dislocations. RIS increases up to a steady state local concentration  of Ni ≈ 7×10-3\n at.% at \nabout 4 dpa, while the Si segregation does not saturate at less than  10 dpa , where it reaches the observed \nconcentration of ≈ 4.5% . The fitting to  experiment al data [4] was done by adjusting the pre-exponential \nfactors to find the consistent magnitude of RIS integrated through a distance of 2 nm from the \ndislocation core. The RIS model is  used to estimate  the local microalloy composition at dislocations, 9 \n hence the chemical driving force of precipitation , as a function of dpa . We note that t he RIS model is \nvery approximate and cannot be used for quantitative prediction outside t he conditions studied here \n(please see SI Section 3 for more details).   \n \n3.3 Cluster dynamic simulation results  \nThe result of RIS and RED calculations are integrated in the CD model for the simulation of \nMNSPs in proton- irradiated alloy T91  at 400 °C and a dose rate of 10-5 dpa/s , including the evolution of \nthe number density ( N), mean radius ( <r>) and mole fraction ( f) of MNSPs  up to 100 dpa .  \nThe predicted  N, <r> and f , respectively  are shown in Figure 4 (a), (b) and (c) . Figure 4 also shows \nthe corresponding experiment al data reported by  Jiao et al.  [11]. Note  that to enable a direct comparison \nwith the G -phase MNSPs  in the model , the radii from experiments were estimated based only on the  Mn, \nNi, and Si atom ic fractions  and volumes  in a precipitate , and any nominal contributions from Fe were \nremoved. This approach is consistent with evidence that the Fe in the MNSPs is an artifact of the APT \n[11]. The volume  is estimated from  the crystal structure and number of atoms in a unit cell of G -phase. \nOnly the clusters with sizes larger than 65 atoms were counted in the  calculations of number density and \nmean radius, which is consistent with the typical resolution limit of the APT. The result in Figure 4 \ndemonstrates excellent consistency with the experimental observation [ 11] in number density and mean \nradius, which were characterized as 1.27 ×1023 m-3 and 1.6 nm , respectively . The model predicts  that \nrapid nucleation starts at doses between 0.2 and 0.5 dpa, and the volume fraction grows continuously \nafter about 1 dpa. This  evolution of the MNSPs  is also in qualitative agreement with the description by  \nJiao et al.  [11] who reported the observable MNSPs  started to appear  at doses between 1 and 7 dpa. \nHowever it is again important to note that the choice of the minimum precipitate spacing and the \nadjusting of parameters in the RED model  (see S.I. Sec. 1.2) ensure that the predictions are consistent 10 \n with the proton data if the underlying thermodynamic assumptions and segregation estimates are valid.  \nTo develop a clear er understanding of dislocation and radiation effects on G -phase precipitation, \nwe explore the separate effects of dislocations, RIS, and RED on the evolution of precipitation based \nthe same framework of model and parameters . Figure 5 (a) and (b) shows  respectively  the simulation \nresults of N, <r> and f for various combinations of mechanisms , including the full calculations (with \ndislocations, RIS and RED) and without RIS  or RED. The results show that as long as  dislocation s are \nincluded as a  favorable nucleation site, the n N ≈ 3.5×1021 m-3 at 7 dpa. The absence of RIS lowers N  \nrelative to the full model, but the more significant effect of the absence of RIS is that the precipitates do \nnot grow  larger than 0.59  nm in  mean  radius  (an average of just 70 atoms) , which is much smaller than \nthe reported mean radius (1.6 nm) . Indeed in this case the number of atoms in the precipitates  is less \nthan a single G -phase unit cell (116 atoms)  with a lattice parameter of 1.117 nm [ 37], hence, are better \ndescribed as slowly evolving clusters rather than precipitates. These small clusters are stabilized by the \nnominal energy gain associated with annihilation of the dislocation core, do not grow in absence of the \nRIS. Given that the model makes other significant approximations with respect to very small clusters \non the dislocations, e.g., ignoring any thermal segregation and assuming complete annihilation of the \ndislocation core, the  model prediction might deviate significantly from actual c luster behavior in this \nsituation. However, regardless of the detailed accuracy of the model in this somewhat artificial limit of \nno RIS , the result s indicate that segregation is  necessary to provide sufficient driving force for MNSP \nnucleation  and growth. In contrast to RIS, RED simply  accelerates the precipitate evolution, shift ing it \nto a lower dpa.  \nWhile the present model has been successful in  its goal of determining the qualitative mechanisms \ncontrolling precipitation of MNSPs  in T91  under proton irradiation, it cannot yet  provide quantitative \npredictions  for other alloy and irradiation conditions . Refinements that might lead to more  generally 11 \n quantitative predictions , including m ore accurate modeling of RIS and other rel evant physics like \ndislocation loop evolution. Additionally, Wharry [4] have  reported that  about 76% of the MNSPs are \nassociated with  CRPs  in T91. These authors hypothesize that  CRPs  formed after the MNSPs  since some \nof the MNSPs were not associated with  CRPs . However, s tudies of neutron -irradiated Cu -Ni-Mn-Si RPV \nsteels show that CRPs  form much earlier than MNSPs . The CRPs  enhance the nucleation and growth of \nMNSPs by providing sites to form co -precipitate appendages [ 7, 38, 39]. Notably at low supersaturations \nCRPs  also form preferentially on dislocations , as illustrated in the APT reconstruction in SI Section S5 . \nThe synergi stic effect s of dislocations and CRPs  on MNSP evolution should be the focus of future work .  \n \n4 Conclusions  \nIn summary, w e have developed a model to study the evolution of MNSPs  in F-M alloy T91  under \nproton irradiation. The approach  is based on a CD model with heterogeneous nucleation on dislocations, \nincluding  the effects of RIS . Dose -dependent  local microalloy solute concentrations are es timated based \non a fitted continuum RIS model . The local solute concentrations are used in the C D calculation. In the \nabsence of  dislocation effect s on heterogeneous nucleation and RIS , the CD model underestimates the  \nnumber density and size of the MNSPs  compared to the proton irradiation results . In contrast, including \nthe catalytic effect of dislocations  and RIS, with all fitted parameters in physically reasonable ranges , the \nCD model is consistent with observations , although this is partially imposed, like the saturated N . The \nmodel developed here can be used to qualitatively explore the effects of temperature and dose rate on \nMNSPs precipitation . As detailed in SI Section S6, decreasing the dpa rate by a factor of 100 at 400  °C, \ncloser to in -service neutron irradiation conditions, decreases the MNSP  volume fraction by ~3 5%. \nDecreasing the temperature to 300 °C at a dpa rate of 10-5 dpa/s increases  the volume fraction by ~43% . \nHowever, a number of improvements in the model are needed for more quantitatively reliable predictions , 12 \n including more physics as discussed above and bett er constrained fitting parameters  as described in SI \nSection S4 . The physically reasonable values obtained for all the fitting parameters suggests  that the \nmodel is representative of the dominant physics in the problem, but the extensive fitting to limited data \nmeans that the model cannot be used for more than qualitative guidance.  \nThe model supports the hypothesis that MNSPs  in T91 are controlled by the combination of G -phase \nthermodynamics, RIS and dislocation enhanced heterogeneous nucleation rates , consistent with previous \nobservations and interpretations  [4, 11]. The approach in this study provides a framework for integrating \nRIS and dislocation effects into more general modeling  of precipitat ion in under -saturated  alloys with \nrealistic treatment of their microstructures.  \n \nAcknowledgements  \nThis work at the University of Wisconsin -Madison and UC Santa Barbara (UCSB) is funded by the US \nDepartment of Energy (DOE) , Office of Nuclear Energy  (NE), Integrated Research Project ( IRP) titled \n“High Fidelity Ion Beam Simulation of High  Dose Neutron Irradiation” (DE -NE0000639) . Partial \nsupport at UCSB was provided by the DOE Office of Fusion Energy Sciences Grant \nDE-FG03 -94ER54275.  \n \nReferences  \n[1] S.J. Zinkle, J.T. Busby, Structural materials for fission & fusion energy, Mater Today 12(11) (2009) \n12-19. \n[2] G.S. Was, Z. Jiao, E. Getto, K. Sun, A.M. Monterrosa, S.A. Maloy, O. Anderoglu, B.H. 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Brailsford, R. Bullough, The theory of sink strengths, Philosophical Transactions of the Royal \nSociety of London A: Mathematical, Physical and Engineering Sciences 302(1465) (1981) 87 -137. 15 \n [33] J.O. Andersson, T. Helander, L.H. Hoglund, P.F. Shi, B. Sundman, THERMO -CALC & DICTRA, \ncomputational tools for materials science, Calphad -Computer Coupling of Phase Diagrams and \nThermochemistry 26(2) (2002) 273- 312. \n[34] P.J. Maziasz, C.J. McHargue, Microstructural evol ution in annealed austenitic steels during neutron \nirradiation, International Materials Reviews 32(1) (1987) 190- 219. \n[35] O. Anderoglu, J. Van den Bosch, P. Hosemann, E. Stergar, B.H. Sencer, D. Bhattacharyya, R. \nDickerson, P. Dickerson, M. Hartl, S.A. Ma loy, Phase stability of an HT -9 duct irradiated in FFTF, \nJournal of Nuclear Materials 430(1–3) (2012) 194 -204. \n[36] B.H. Sencer, J.R. Kennedy, J.I. Cole, S.A. Maloy, F.A. Garner, Microstructural analysis of an HT9 \nfuel assembly duct irradiated in FFTF to 155 dpa at 443 °C, Journal of Nuclear Materials 393(2) (2009) \n235-241. \n[37] X. Yan, A. Grytsiv, P. Rogl, V . Pomjakushin, X. Xue, On the crystal structure of the Mn–Ni –Si \nG-phase, Journal of Alloys and Compounds 469(1) (2009) 152- 155. \n[38] M.K. Miller, K.A. Powers, R.K. Nanstad, P. Efsing, Atom probe tomography characterizations of \nhigh nickel, low copper surveillance RPV welds irradiated to high fluences, Journal of Nuclear Materials \n437(1–3) (2013) 107 -115. \n[39] J.M. Hyde, G. Sha, E.A. Marquis, A. Morley, K .B. Wilford, T.J. Williams, A comparison of the \nstructure of solute clusters formed during thermal ageing and irradiation, Ultramicroscopy 111(6) (2011) \n664-671. \n  16 \n FIGURE CAPTION  \n \nFigure 1.  Flow chart showing the cluster dynamics simulation of M NSPs formation in T91 under \nirradiation.  \n \nFigure  2. Phase fraction of the T3 or G -phase as a function of temperature calculated by TCAL3 \ndatabase. The black, blue and orange curves show the calculated results of T91 original and \nRIS compositions at grain boundaries (GB) and dislocations (DL), respectively. No other \nMNSPs are predicted to be stable.  \n \nFigure 3. Calculation result showing the evolution of Ni and Si RIS compositions at dislocations as a \nfunction of dose. The symbols show the experimental values reported in Ref. [ 3]. \n \nFigure 4. Calculation result of the cluster dynamics model showing the (a) number density, (b) mean \nradius, and (c) volume fraction of MNSPs  as a function of irradiation dose (dpa). The dose rate \nand temperature are 10-5 dpa/s and 400 °C, respectively. The model inc ludes the effect of \nheterogeneous nucleation at  dislocations as well as the evolution of RIS shown in Figure 2 . The \nsymbols show the values reported by the experiment [ 11]. The error bar on the experimental \nradius indicates the standard error in the mean radius (𝜎𝜎 〈𝑟𝑟〉) determined by the formula 𝜎𝜎〈𝑟𝑟〉=\n𝜎𝜎𝑟𝑟√𝑁𝑁⁄ , where 𝜎𝜎𝑟𝑟 is the standard deviation of the precipitate size of all the APT -identified particles \nand N = 43 is the number of particles that were measured.  \n \nFigure  5. Calculation result of the cluster dynamics model showing the (a) number density and (b) mean \nradius  of MNSPs as a function of irradiation dose (dpa) under various conditions, including the 17 \n full calculations (with dislocations, RIS and RED) and calculations without RIS or RED. The \ndose rate and temperature are 10-5 dpa/s and 400 °C, respectively. The symb ols show the values \nreported by the experiment [ 11].  1 \n  \n \nFigure 1 . Flow chart showing the  cluster dynamics simulation of M NSPs formation in T91 under \nirradiation.  \n \n  \n2 \n  \n \n \nFigure 2. Phase fraction of the T3 or G -phase  as a function of temperature calculated by  TCAL3 \ndatabase. The black, blue and orange  curves show the calculated results of T91 original and RIS \ncompositions  at grain boundaries  (GB)  and dislocations  (DL), respectively. No other MNSPs  are \npredicted to be stable.  \n \n3 \n   \n \nFigure  3. Calculation result showing the evolution of Ni and Si RIS compositions at dislocations as a \nfunction of dose. The symbols show the experimental values reported in Ref. [ 3]. \n \n \n4 \n  \nFigure 4. Calculation result of the cluster dynamics model showing the (a) number density, (b) mean \nradius, and (c) volume fraction of MNSPs  as a function of irradiation dose (dpa). The dose rate and \ntemperature are 10-5 dpa/s and 400 °C, respectively. The model inc ludes the effect of heterogeneous \nnucleation at dislocations as well as the evolution of RIS shown in Figure 2. The symbols show the \nvalues reported by the experiment [ 11]. The error bar on the experimental radius indicates the standard \nerror in the mean radius ( 𝜎𝜎〈𝑟𝑟〉) determined by the formula 𝜎𝜎 〈𝑟𝑟〉=𝜎𝜎𝑟𝑟√𝑁𝑁⁄ , where 𝜎𝜎𝑟𝑟 is the standard \ndeviation of the precipitate size of all the APT -identified particles and N  = 43 is the number of particles \nthat were measured.  \n  \n5 \n  \nFigure 5. Calculation result of the cluster dynamics model showing the (a) number density and (b) mean \nradius of MNSPs as a function of irradiation dose (dpa) under various conditions, including the full \ncalculations (with dislocations, RIS and RED) and calculations without RIS or RED. The dose rate and \ntemperature are 10-5 dpa/s and 400 °C, respectively. The s ymbols show the values reported by the \nexperiment [ 11].  \n \n \n \n1 \n Supplementary Information  \n \nCluster dynamics modeling of Mn -Ni-Si precipitates in ferritic- martensitic steel under irradiation  \n \nJia-Hong Ke  a, Huibin Ke  a, G. Robert Odette  b and Dane Morgan  a*  \n \na Department of Materials Science and Engineering, University of Wisconsin -Madison, Madison, WI 53706, USA  \nb Materials Department, University of California, Santa Barbara, CA 93106, USA  \n* Corresponding author. E -mail address: ddmorgan@wisc.edu  \n \nS1. Cluster dynamics model ing of Mn -Ni-Si-rich precipitates   \nS1.1 Cluster dynamics model  \nWe followed the kinetic  equation s developed by Slezov and Schmelzer  [1-4] to describe \nnucleation -growth processes of diffusional phase transformations in multicomponent systems. The \nmodel assumes that the Mn -Ni-Si-rich phases can be treated as pure Mn -Ni-Si phases, with no other \nalloying elements. With the assumption that only monomers can migrate, the discrete cluster size \ndistribution s are governed by the coupled master equation s: \n ()\n1,\nnnf ntJJt−∂= −∂  (S0) \nwhere f is the distribution function of clusters with n atoms at time t  and nJ is the flux of clusters \nbetween cluster sizes n and n + 1.  The flux can be determined by  \n ()()()(), 1 1, , 1,n nn n n J fn t fn tωω+−\n++= −+   (S1) \nwhere ()\n,1nnω+\n+ is the absorption coefficient or the rate at which a cluster of size n absorbs a single atom \nand grow to size n +1, and similarly , ()\n1,nnω−\n+ is the em ission coefficient  or the rate at which a cluster of \nsize n + 1 emit s a single particle  and shrinks  to size n. The symbol s (+) and (−) correspond to absorption \nand emission of an atom or monomer, respectively. It is noted that in the present study, we adopt the \ntreatment from Slezov and Schmelzer [ 4] by assuming that the evolution of clusters follows  the trajectory 2 \n of minimum free energy in composition space, and accordingly the cluster size distribution considered \nhere is a function of the total number of atoms in the cluster (instead of the number of each  components  in \nthe cluster) . The absorption c oefficient can be determined directly by the macroscopic growth kinetics of \nthe cluster. With the assumptions that the growth of clusters is diffusion -limited and spherical in shape, \nthe effective absorption coefficient can be expressed as  [4] \n () 13\n, 1 eff 4d\nnn caD nβa ωπ+\n+=   (S2) \nand \n 2\neff1i\nd\ni ii D xDa\nβν=∑  (S3) \n \nwhere cβ and xiβ are the total volume concentration of the particles and molar fraction of the component i  \nin the ambient phase  β, respectively , and Di is the diffusion coefficient of species i  in the ambient phase . \nA fixed atomic fraction in the clusters of precipitates  is assumed, and the parameter  νiα is the atomic \nfraction  of the component i  in the precipitate phase α .  \n The absorption coefficient in Eq. (S3) is derived under the assumption that  the growth is controlled \nby volume diffusion and the long -range matrix composition is located far away from the cluster \ninterphase boundaries. The assumption will not be valid for a  microstructure  affected by other diffusion \npathways or local solute s egregation. Modification of Eq. ( S3) is necessary  for the precipitation  of \nMn-Ni-Si-rich phase in T91 under proton irradiation, which was found to be strongly associated  with \ndislocations and radiation- induced segregation  [5]. Particularly, because of  the extremely fast diffusion \nalong dislocation pipes, dislocations provide an additional diffusion pathway facilitating the nucle ation \nand growth of precipitates . We assume that the diffusion flux along dislocations is controlled by the \nlong-range diffusion into the dislocation lines , and that any solutes arriving at the dislocation are \nimmediately added to the precipitate . Therefore, t he absorption coefficient should also consider this 3 \n additional dislocation transport contribution, which is cylindrical in symmetry owing to the  diffusion \ntoward the straight dislocation lines. By considering pathway s of volume diffusion and transport  along \ndislocation pipes, t he modified absorption coefficient  is derived  as \n ()()\n()()\n()eff core\n0 core\n,1 22\ncore 0\neff eff core\n0 0 core22                                                       for  ln\n24 2   for  lnn\np dn\np\nnnn n\np pp d dn\np nn\nppdrcD r rrr\nrr r nr d rcD cD r rr r r rrβ\nββπ\nω\nππ+\n+ −< \n=\n−  −+> −   (S3) \nwhere n\npr is the radius  of a cluster with n atoms , r0 is the radius of the microalloy region with solute \nsegregation, rcore (0.4 nm [6]) is the dislocation core radius, and d is the distance which avoids the overlap \nof precipitate volume on dislocations , which is 5 nm in this study . A recent study of radiation -induced \nsegregation (RIS) in T91 shows that the RIS -affected microalloyed region is approximated 5 nm in size  \n[7], so we take r 0 = 2.5 nm in Eq. (S 5). For precipitates with  a radius smaller than the dislocation core, \ndiffusion along dislocation pipes dominates  (Eq. (S5 , top)), whereas for that precipitates  larger than the \ndislocation core, both volume diffusion and transport along dislocation pipes ( (Eq. (S5 , bottom) ) \ncontribute to the growth of precipitates. The first term in Eq. (S5, bottom ) reflects the cluster growth  by \nvolume diffusion and the ratio 22\np cpr rr− remove s the overlapped contribution of the surface area \nintersecting  with dislocation cores. The second term corresponds to the  cylindrical diffusion into the \ndislocation lines , which takes effect for precipitates  on dislocations  with any given size .  \nThe em ission coefficient can be expressed by the given relation:  \n ()()()()\n1, , 11expn n nnGn Gn\nkTωω−+\n++∆ + −∆=  (S4) \nwhere ∆G(n) is the formation free energy of a cluster containing n atoms. Note that , consistent with the \nassumption of very fast dislocation transport used above, the dislocation is in equilibrium with the 4 \n surrounding bulk and we need only consider changes in Gibbs free energy associated with emission \ninto the bulk region near the dislocations  in Eq. (S6) . If only the chemical free energy and interfacial \nenergy are considered, ∆G(n) can be expressed as  \n ()( )23344p\np ii inGn ng x µπσπΩ∆=− + ∑  (S5) \ngp is the free energy per atom of the precipitate phase, μ i is the chemical potential of component i  in the \nmatrix,  σ is the interfacial energy  of the precipitate  per unit area , and Ω p is the atomic volume of the \nprecipitate phase. For dilute components such as the Mn, Ni, and Si additions in the alloy T91  as listed in  \nTable S1, p ii igxµ−∑  can be simplified to  the expression of solute product s iixx\nii iicc∏∏ , in which  \nthe bar  on the solute product indicates the value in thermodynamic equilibrium and can be obtained from \nthermodynamic  database. In our thermodynamic model based on the TCAL3  database t he only Mn- Ni-Si \nphase we find stable for T91 (at normal bulk or RIS compositions)  is the T 3 (or G -phase), with a \ncomposition of Mn 6Ni16Si7 [8]. We therefore f ocus on this phase in the cluster dynamics modeling. The  \nvalue of  ix\niic∏  for the Mn -Ni-Si phase  in T91 at 400 °C  can be obtained as 6.26 ×10-3 according to the \nTCAL3  database, where xi for Mn, Ni, and Si are respectively 0.21, 0.55, and 0.24. The interfacial energy \nwas assumed to be 0.19 J m-2 according to the fitting result for the G -phase in  RPV steels  [9].  \n \nTable S 1. Chemical composition of T91 in at% and wt%  [10].  \n Cr Ni Mn Si C P Cu V Mo S N Nb Al \nat.% 8.90 0.20 0.45 0.55 0.46 0.016  0.15 0.23 0.52 0.005  0.19 0.005  0.045  \nwt.%  8.37 0.21 0.45 0.28 0.1 0.009  0.17 0.216  0.9 0.003  0.048  0.008  0.022  \n \nS1.2 Radiation- enhanced diffusion \nWe apply the radiation -enhanced diffusion model developed by Odette et al. [11] to calculate X vDv \nand scale thermal diffusion coefficients . The radiation enhanced diffusion coefficients is given as  5 \n  th\nrad th r i\niV i e\nVDDX DX= +   (S6) \nwherer\nVX  is the non-equilibrium  vacancy concentration unde r irradiation , e\nVX is the vacancy \nconcentration at thermodynamic equilibrium.th\niDis the diffusion coefficient of the solute  i under  the \ncondition of thermal annealing . Under the steady state when defect production is balanced by \nannihilation at sinks  as well as recombination at matrix  and solute trapped vacancies , the vacancy \nconcentration can be expressed as  \n dpa s r\nV\nVtgXDSξσφ=   (S7) \nwhere  σdpa is the displacement -per-atom (dpa) cross -section , ξ is the cascade efficiency or the fraction of \nvacancies and self -interstitial atom (SIA)  created per dpa, S t is the  total sink strength, and r v is the \nSIA-vacancy recombination radius. The total sink strength inc ludes the contribution of dislocations ( Sd) \nand vacancy clusters ( Sc). The former can be characterized by the dislocation density and the latter can \nbe evaluated by  \n 4/c cc c aSrπ σ φτ= Ω   (S7) \nwherecr,cσandcτare the recombination radius, production cross -section and annealing time for \nvacancy clusters. gs is the vacancy surviva l fraction which can be obtained by solving  the steady -state \nequation [ 11]: \n () ()\n( )222\ndpa dpa\n2 2\ndpa10\n1r s t tt s\ns\ni vt t s tt tR g RX ggDDS S g RSξσφ ξσφτ\nξσφτ− −− =\n+  (S7) \nRr and Rt are respectively the matrix and trap recombination radii , Xt is the trap density, and  tτ is the \nannealing time for trapped vacancies  which can be expressed as  [11] 6 \n  2bH\nkT\ntVd Deτ−=   (S8) \nwhere Hb is the binding energy for trapped vacancies  and d is the nearest neighbor distance of bcc Fe \nlattice (2.48 ×10-10 m). Here we consider the highly concentrated Cr atoms in T91 (9%) as the main \ntrapping site . Table S 2 lists the parameters used in the calculation of radiation -enhanced di ffusion.  \n \nTable S 2. Parameters  used in the RED model for calculating radiation -enhanced diffusion. * The \nvariables marked with a *, which include “ Vacancy cluster recombination radius”  \n and “Vacancy cluster annealing time” were fit to match the experimental onset of MNSP after 1 dpa \nand the observed size and number density at 7 dpa, as reported in Ref . [5]. \n \nSIA – vacancy recombination radius  vr 0.57 nm [9, 11] \n*Vacancy cluster recombination radius  rc 5.0×10-10 m \nDisplacement -per-atom cross -section  dpaσ 1.5×10-25 m2 [9, 11] \nVacancy cluster production cross -section  cσ 4.5×10-25 m2 \n*Vacancy cluster annealing time  cτ 8.1×10-12/e(-1.85/kT) \nCascade efficiency  ξ 0.9 [12] \nAtomic volume  aΩ 1.18×10-29 m3 [9] \nVacancy diffusion coefficient at 400  °C VD 5.79×10-13 m2 s-1 [9] \nDislocation sink strength (dislocation density)  dS 6.25×1014 m-2 [13] \nFe self-diffusivity  at 400  °C th\nFeD 5.33×10-23 m2 s-1 [9] \nMn diffusivity in Fe at 400  °C th\nMnD 1.02×10-22 m2 s-1 [9] \nNi diffusivity in Fe at 400  °C th\nNiD 1.18×10-23 m2 s-1 [9] \nSi diffusivity in Fe at 400  °C th\nSiD 8.36×10-23 m2 s-1 [9] \nMatrix recombination radius  Rr 5.7×10-10 m [11] \nTrap recombination radius   Rt 5.7×10-10 m [11] \nTrap concentration  Xt 0.09 \nBinding energy for trapped vacancies  Hb 0.094 eV  [14] \n \nS2. Consideration of h eterogeneous nucleation at dislocations  in cluster dynamics model  \nFollowing the model by Cahn [ 15] and Gomez -Ramirez  [16] and assuming the dislocation line 7 \n passes through the center of the spherical Mn- Ni-Si precipitate or cluster, the  released excess free \nenergy associated with the nucleation of a cluster at a dislocation can be given as: [15] \n ()core core\ndisl 2\n0\ncore core core[                                ]      p recipitate on core, \n[ ln( ( ) )]      precipitate on core, 4p\np\np\npr\npr\npr\nprE dl r r\nGrAaE r l r dl r rπ−\n−<∆= \n +>∫\n∫  (S8) \nrcore and Ecore are respectively the dislocation core radius and core energy, whose values we take from the \nestimations by Marian et al . [17] and Dudarev et al . [6]. rp is the cluster radius, r  is the distance between \na point on the precipitate interface and the dis location line, l  is the distance from the center of a \nprecipitate along the dislocation line, a 0 is the lattice constant (2.87 ×10-10 m), and A is a factor that \ncorresponds to the anisotropic elastic strain energy of dislocations  [18]:  \n 22\n0 A Kb a=   (S8) \nwhere b is the dislocation Burgers vector  and K is the energy factor which depends on the elastic \nconstant cij , dislocation type and direction ( l). The factor A  can be calculated  by following Ref. [18] for \nvarious types of dislocations and typical values are summarized in Table S3 . In this study we consider \nthe [001] type pure edge dislocation for all of our CD calculations with a value A  = 104.2 GPa. It is \nnoted that the variation of A  from 57.7 to 134.2 GPa does not have significant influence on the growth \nof MNSPs in both number density and size . The calculation results using the different  A factor s (57.7, \n134.2, and 104.2 GPa)  are shown in Section S7, and  indicate that larger A causes only slightly earlier \nnucleation due to the lar ger energy release by dislocation strain energy. The insensitiv ity of the \nprecipitate evolution to the  A value is reasonable in this study  because as RIS develop s and increases with \ndose and time, the nucleation barrier becomes much lower and the energy release by the dislocation strain \nenergy produces relatively minor impact.  \nIt should be noted that the two lines in Eq. ( S13) are, from top to bottom, for clusters that are on \nthe dislocation and have sizes smaller than the dislocation core and for clusters that are on the 8 \n dislocation and that have size larger than the core  radius , respectively.  \n \nTable S 3. The factor  A (Eq. (14))  for various types of dislocation in bcc Fe .  \nDislocation type  l (direction)  0ba  A (GPa) \nScrew  [001]  [001]  116.0 \nEdge in {100}  [001]  [100]  104.2  \nMixed in {110}  [001]  1/2[111]  81.1 \nMixed in {100}  [101]  [100]  91.2 \nEdge in {110}  [101]  [010]  134.2  \nMixed in{110}  [101]  1/2 [111]  57.7 \nEdge in {112}  [101]  1/2 [1¯11] 100.7  \n \nBy considering the catalytic effect of dislocation on nucleation, t he total formation free energy of a \ncluster in the cluster dynamics model then becomes : [16] \n ()( ) ()23\ndisl344p ii p inGn ng x G r µπσπΩ∆ = − + +∆∑   (S8) \nThe last term corresponds to the released excess free energy associated with the nucleation of a cluster \nat a dislocation as given in Eq. ( S13). The dislocation core radius and core energy  were estimated by \nMarian et al. [17] and Dudarev et al . [6]. The structural information of dislocation cores together with the \nreported dislocation density allow s us to calculat e nucleation on dislocations semi -quantitatively.  Table \nS4 lists the parameters used in the calculations. Note that these parameters are for nominally pure bcc \nFe as we a re not aware of values for T91.  \nWe note that heterogeneous nucleation at grain b oundaries can be modeled in a similar manner as at \ndislocations, but it was not considered in the present study because recent APT studies strongly suggest \nthat dislocations are the major nucleation site [ 5, 19]. This is expected, since the grain boundaries provide \nmany fewer sites for nucleation than the dislocations. More specifically, based on t he measured \ndislocation density of ρ d = 6.25×1014 m-2 and an effective grain size of 1.49 μm [13], the heterogeneous 9 \n nucleation site densities can be estimated as 3.3 ×1024 m-3 and 6.3×1023 m-3 for dislocations and grain \nboundaries, respectively. Here we have assumed cubic grains with side dimensions equal to the grain size \nand that nucleation sites are separated by 5 nm (same as above) on a square grid with cell dimensions of \n5 nm. Thus the heterogeneous nucleation site density is about 5 times larger for dislocations than grain \nboundaries. More rapid nucleation is also expected due to the stronger RIS at dislocations than at g rain \nboundaries in T91 [7, 19] .  \n \nTable S 4. Parameters for bcc Fe that are used in the cluster dynamics model of  heterogeneous  nucleation.  \nrcore 0.4 nm  [6] \nEcore 0.937 eV/ Å [6] \nb = a0 0.287 nm  [20] \nΩa 1.18×10-29 m-3 [20] \nc11 231.5 GPa  [21] \nc12 135.0 GPa  [21] \nc44 116.0 GPa  [21] \nρd 6.25×1014 m-2 [13] \n \nS3. Continuum modeling of radiation -induced segregation  \nWe implement the  theory of RIS developed by Wiedersich  et al . [22] and Wolfer [ 23], which  \nconsiders both the contributions of the inverse Kirkendall effect  and vacancy drag . We adapt ed the codes \nthat were applied successfully to study the  Cr segregation behavior of the 9 wt% Cr F-M steel  [24]. The \ntime evolution of the solute and defect concentrations are respectively given by  \n \n AAdC dJ\ndt dx=−   (S8) \nand  \n 0dd\nIVdC dJK RC Cdt dxξ= −+−   (S8) 10 \n where the subscript A  denotes  the alloy component , d is the type of defects such as vacancies and \ninterstitials, J is the diffusion flux, K 0 is the dpa  rate of irradiation ,ξis the displacement  efficiency , and \nCI and CV are concentrations of interstitials and vacancies . By following the treatment of Ref. [ 23] and \nassuming a dilute system with negligible dependence of of thermodynamic factor and vacancy \nformation energy  on alloy compositi on, the atom flux is expressed as  \n Wind WindAV AV VV\nV BB AA\nA VA V\nAB ABGD GD DDJ CC CCC CC = − − ∇+ − ∇    (S8) \n II\nI AA\nA IA I\nAADDJ CC CCC= − ∇− ∇   (S8) \n VI\nAAAJJJ= +   (S8) \n ()VV\nV ABJ JJ= −+   (S8) \n II\nI ABJJJ= +   (S8) \nD is the diffusion coefficient defined by ()0expm D E kT−  where D0 is the  pre-exponential factor and \nEm is the migration energy. WindAG  is the vacancy wind factor.  The recombination  coefficient  R is given \nby \n 4VI\nrec\naDDRdπ+=Ω  (S8) \nwhere d rec is the recombination distance .  \nThe equations are numerically solved by implementing SUNDIALS (SUite of Nonlinear and \nDIfferential/ALgebraic Equation Solvers)  [25]. The initial condition of each concentration is \ndetermined by the nominal composition in T91 or defect formation energy ( Ef). The size of 1D \ncomputational supercell is determined by the dislocation density  by (ρd)-0.5 = 40 n m with s ymmetry \nboundary conditions imposed at the two boundaries. The fitting with experiment data [ 19] was done by 11 \n adjusting the pre -exponential factors to find the consist ent magnitude of RIS  integrated through 2 nm \nfrom the dislocation core . Table S5  lists the physical parameters  of the RIS model.  It is noted that some  \npre-exponential factor s are not available in literature, so in this study we consider  them as fitting \nparameters with reasonable values that were  determined by comparing with  the experimental  RIS \nmeasurement s at dislocations by Jiao and Was [ 19], in which the RIS at 7 dpa was measured. Due to \nlimited data available for fitting parameters we assume th at the pre -exponential for interstitial diffusion \nfor Ni and Si are the same, yielding only two  parameters to be fit, as shown in Table S5 . It is noted that \nbecause the RIS calculation here is quite simple the fit values a re likely not transferable to significantly \ndifferent temperatures or fluxes. More accurate modeling requires full evaluation of interstitial and \nvacancy Onsager coefficients as a function of composition, and experimental comparisons over a wide \nrange of conditions. However, the RIS modeling presented here provides approximate evaluation of \ndose-dependent solute enrichment at different time scales .  \n \nTable S 5. Physical parameters used in the RIS model.  * The variables marked with a *, which include \nthe pre -exponential factors of interstitial diffusivities of Ni and Si and the pre -exponential factor of \nvacancy diffusivity of Si are determined by fitting with the experimental RIS measurement at 7 dpa  \nfrom Ref. [ 7]  \n \nPre-exponential factor of Fe interstitial diffusivity  \n0,Int\nFeD 6.59×10-7 m2 s-1 [24] \n*Pre-exponential factor of Ni interstitial diffusivity  \n0,Int\nNiD 1.25×10-7 m2 s-1  \n*Pre-exponential factor of Si interstitial diffusivity  \n0,Int\nSiD 0.92×10-7 m2 s-1  \nPre-exponential factor of Fe vacancy diffusivity  \n0,Vac\nFeD 1.02×10-4 m2 s-1 [26] \nPre-exponential factor of Ni vacancy diffusivity  \n0,Vac\nNiD 2.3×10-4 m2 s-1 [27] \n*Pre-exponential factor of Si vacancy diffusivity  \n0,Vac\nSiD 1.7×10-4 m2 s-1  \nMigration  energy for Fe interstitial diffusivity  \n,Int\na FeE 0.36 eV  [24] \nMigration  energy for Ni interstitial diffusivity  \n,Int\na NiE 0.45 eV  [28] \nMigration energy for Si interstitial diffusivity  \n,Int\na SiE 0.52 eV  [28] \nMigration energy for Fe vacancy diffusivity  \n,Vac\na FeE 0.55 eV  [29] 12 \n Migration energy for Ni vacancy diffusivity  \n,Vac\na NiE 0.50 eV [27] \nMigration energy for Si vacancy diffusivity  \n,Vac\na SiE 0.51 eV [30] \nVacancy wind factor for Ni in Fe  Ni\nWindG  -1.6 [30] \nVacancy wind factor for Si in Fe  Si\nWindG  -1.8 [30] \nVacancy formation energy  Vac\nfE 2.00 eV [29] \nInterstitial formation energy  Int\nfE 3.64 eV [24] \nCascade efficiency  ξ 0.9 [12] \nRecombination distance  drec 5.7×10-10 m [11] \n \nWe note that the RIS model is extremely approximate, and really should be considered no more than \na physics -based interpolation of segregation as a function of fluence between zero in the unirradiated \ncondition up to the local enrichment concentrations at dislocations in T91 at 7 dpa measured by Jiao and \nWas [19]. The model therefore cannot be expected to be accurate for irradiation conditions that are \ndifferent from those used in Jiao and Was [ 19], such as variations in temperature, flux, fluence, \nirradiation type and Mn, Ni, Si alloy solute contents. Further, the method used to calculate the point \ndefect concentration in the RIS model is based on a standard lattice recombination mechanism. This \ndiffers somewhat from the defect concentrations in the RED model that treats recombination at solute \ntrapped vacancies [ 11]. This is not an issue in the present proton irradiation case, but a more accurate and \nself-consistent RIS model will be developed in future research for more general applications. Future \nwork will also add other new physics to standard RIS models, such as consideration of enriched solute \ninteractions and co -segregation. However, these improvements are beyond the scope of this work, which \nis focused on MNSPs in T91 and not on developing highly accurate RIS models. \n \nS4. Summary of fitted values and their implications  \nThe physical parameters used in the present study are listed in  Table S2 -S4. The vacancy cluster \nrecombination radius, vacancy cluster annealing time, and pre -exponential factors of solute  interstitial 13 \n diffusion and that of vacancy diffusivity of Si  have not been evaluated by experiments and simulations \npreviously, so in the present modeling of precipitate evolution, we treated those parameters by applying \nphysical and reasonable values  in an attempt to generate simulation results  in quantitative  agreement \nwith experiments. For the pre -exponential factors of Ni and Si interstitial diffusion in bcc ferrite, we \nassumed the values to be at the order of  10-7 m2/s, which is a reasonable magnitude  for interstitial \ndiffusion. The dislocation density in i rradiated T91 has been reported in the range of  1014 to 1015 m-2 [31]. \nA value of 6.25×1014 m-2 was chosen  in this study  according to the measurement by Penisten [ 13]. In \ncalculating the heterogeneous nucleation sites on dislocation lines, we also impose a distance of 5 nm \nfrom any evolving cluster to avoid precipitate volume overlap. Although our value of 5 nm is a \nreasonable lower bound considering th e size of the precipitate, the model can be improved by the \nevaluation of better constrained fitting parameters.  14 \n S5. APT reconstruction showing nucleation of Cu- rich precipitates on dislocations  [32] \n \nFigure S1. APT reconstruction of the neutron -irradiated LI alloy (see composition in Table S6 ) showing \nthe preferential precipitation of Cu -rich partic les on dislocations  [32]. The sample w as irradiated under \nthe condition at 310 °C  at the flux 3.4×1011 n cm-2 s-1. The fluence is 1.6×1019 n cm-2. The measured \nnumber density and mean radius of Cu- rich precipitates are 8.9 ×1022 m-3 and 1.76 nm, respectively.  \n \nTable S 6. Chemical composition of LI in wt% [ 32] \n C Si Mn P Ni Cu Mo \nwt.%  0.2 0.24 1.37 <0.005  0.74 0.2 0.55 \n \nS6. Exploration of precipitation kinetics at different temperatures  and dpa rate s \nWe utilize the cluster dynamics model integrated with the abovementioned RED and RIS \ncalculations to explore the effects of temperature and dpa rate on the precipitation kinetics of the MNSP \nat dislocations.  Note that due to the limited physics in this model and fitting of multiple parameters to \nvery limited data these calculations  for temperature and flux outside the experimental conditions used in \n15 \n fitting are useful only for qualitative guidance and should not be taken to provide quantitati ve pred ictions . \nFigure S 2 shows the evolution of number density, mean radius and volume fraction at 400, 350 and 300 \n°C under the dose rate of 1×10-5 dpa/s. The result shows that decreasing temperature slows down the \noverall nucleation kinetics due to the l ower diffusion and longer time for solute enrichment at \ndislocations. At all temperatures the number density reached to the magnitude of 1023 m-3, which is \ndetermined by the dislocation density . Nevertheless, lower temperature provides larger chemical driving \nforce of precipitation, which corresponds to the larger size and volume fraction after higher dose of \nirradiation. The volume fraction at 300 °C  after 100 dpa  is able to reach 0.33%, which is larger  than that \nof 0.23% at 400 °C.  \n \nFigure S 2. Calculation result of the cluster dynamics model showing the (a) number density, (b) mean \nradius, and (c) volume fraction of MNSPs  as a function of irradiati on dose (dpa). The solid lines show the \nresults at 300, 350, and 400 °C under the dose rate of 1×10-5 dpa/s. The symbols show the values reported \nby the experiment [ 5] at 400 °C . \n \n16 \n  Figure S 3 shows evolution of number density, mean radius and volume fraction of  MNSP at 400 °C \nunder the dose rate of 10-5 and 10-7 dpa/s , where the  latter is the typical magnitude of  dpa rate under fast \nneutron irradiation. The comparison shows that under lower -flux irradiation of 10-7 dpa/s, the onset of \nnucleation is reached at a lower dose compared to irradiation at 10-5 dpa/s. Both the size and number \ndensity during lower -flux irradiation saturates at a  lower dpa than that under higher -flux irradiation. It is \nnoted that the saturated volume fraction and size at 10-7 dpa/s are both smaller than that under 10-5 dpa/s . \nThe difference is because of the less  RIS level under low -flux irradiation, which cannot produce larger \ndefect con centration gradient around dislocations  to drag solute atoms .  \n \nFigure S 3. Calculation result of the cluster dynamics model showing the (a) number density, (b) mean \nradius, and (c) volume fraction of MNSPs  as a function of irradiation dose (dpa). The solid lines show the \nresults under the dose rate s of 10-5 and 10-7 dpa/s  at 400 °C . The symbols show the values reported by the \nexperiment [ 5] at 400 °C . \n \n17 \n S7. Effect of A factor on the Mn -Si-Ni precipitation kinetics in proton irradiated T91 \n \nFigure S4. Simulation results showing the effect of A  factor (Eq. 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Ols son, Exact ab initio transport coefficients in bcc \nFe-X (X=Cr, Cu, Mn, Ni, P, Si) dilute alloys, Phys Rev B 90(10) (2014).  \n[31] G. Gupta, Z. Jiao, A.N. Ham, J.T. Busby, G.S. Was, Microstructural evolution of proton irradiated \nT91, Journal of Nuclear Materi als 351(1- 3) (2006) 162- 173. \n[32] G.R. Odette, personal communication. \n \n  "    },    {        "title": "1811.07948v1.Photovoltaic_effect_in_multi_domain_ferroelectric_perovskite_oxides.pdf",        "content": "Photovoltaic e\u000bect in multi-domain ferroelectric perovskite oxides\nYing Shi Teh and Kaushik Bhattacharya\nDivision of Engineering and Applied Science\nCalifornia Institute of Technology\nPasadena, CA 91125 USA\nAbstract\nWe propose a device model that elucidates the role of domain walls in the photovoltaic e\u000bect in multi-\ndomain ferroelectric perovskites. The model accounts for the intricate interplay between ferroelectric\npolarization, space charges, photo-generation and electronic transport. When applied to bismuth ferrite,\nresults show a signi\fcant electric potential step across both 71\u000eand 109\u000edomain walls, which in turn\ncontributes to the photovoltaic (PV) e\u000bect. We also \fnd a strong correlation between polarization and\noxygen octahedra tilts, which indicates the nontrivial role of the latter in the PV e\u000bect. The domain\nwall-based PV e\u000bect is further shown to be additive in nature, allowing for the possibility of generating\nabove-bandgap voltage.\n1 Introduction\nIn conventional photovoltaics, electron-hole pairs are created by the absorption of photons that are then\nseparated by an internal \feld in the form of heterogeneous junctions such as p-n junctions. Less than a\ndecade ago, Yang et al. [1] reported large photovoltages generated in thin \flms of multi-domain bismuth\nferrite (BFO), and suggested a new mechanism where electrostatic potential steps across the ferroelectric\ndomain walls drives the photocurrent. This discovery has since revitalized the \feld of photoferroics. Among\nmany ferroelectric oxides, BFO has particularly attracted considerable interest due to its high ferroelectric\npolarization and relatively small bandgap.\nMany novel experiments were subsequently devised to investigate the role of domain walls in the observed\nphotovoltaic (PV) e\u000bect in ferroelectric perovskites. Alexe and Hesse [2] performed measurements of the local\nphotoelectric e\u000bect using atomic force microscopy (AFM). They found that the photocurrent is essentially\nconstant across the entire scanned area, hence indicating the absence of the domain wall (DW) e\u000bect. The\nnanoscale mapping of generation and recombination lifetime using a method combining photoinduced tran-\nsient spectroscopy (PITS) with scanning probe microscopy (SPM) points to a similar conclusion [3]. This\nled to the hypothesis that the bulk photovoltaic (BPV) e\u000bect, which arises from the noncentrosymmetry of\nperovskites, is the key mechanism instead [4, 5]. However further recent studies focused on characterizing\nboth the BPV and DW e\u000bects show that the latter e\u000bect is much more dominant [6, 7]. The lack of clear\nconcensus among the scienti\fc community on the key mechanism in the PV e\u000bect in perovskites as well as\non the role of domain walls could be understood from the inherent di\u000eculties in the experimental techniques.\nThe nanoscale order of ferroelectric domain walls makes it di\u000ecult to probe into and separate the e\u000bects\nfrom the bulk domains and the domain walls. Other issues such as defect formation and grain boundaries in\nperovskite crystals further complicate the analysis.\nFirst-principles calculations have provided a detailed understanding of the structure of domain walls\n[8, 9, 10], and have established the drop in electrostatic potential across it. However, they are limited to a\nfew nanometers, and cannot examine the interaction of domain walls with other features. On the other hand,\nmodels at the device scale provide understanding at the scale [11], but assume a priori the polarization and\nother aspects of the domain wall. Finally, various phase \feld models provide understanding of the domain\npattern [12, 13], but in the absence of space charge and photocurrent. Thus, there is a gap in our modeling\nof the intricate interplay between space charge, ferroelectric polarization and electronic transport.\nThis paper seeks to \fll this gap by building on prior work of Xiao et al. [14, 15] and Suryanarayana and\nBhattacharya [16] who developed a continuum theory of semiconducting ferroelectrics including electron and\n1arXiv:1811.07948v1  [cond-mat.mtrl-sci]  19 Nov 2018Figure 1: Crystal structure of bulk BFO. The two O 6octahedra rotate out-of-phase about the polarization\naxis marked by the dotted line.\nhole transport. We extend their work to include photogeneration due to illumination and study photovoltaic\ne\u000bect in ferroelectric perovskite oxides. We investigate the photovoltaic response of BFO \flms with di\u000berent\ndomain wall con\fgurations by solving the model at the device scale. At a 71\u000eor 109\u000edomain wall, we\nobserve a change in the component of polarization perpendicular to the domain wall. This in turn results in\na relatively large electrostatic potential step across the wall which allows for separation of photogenerated\nelectron-hole pairs. Thus this model supports the hypothesis of domain walls contributing to the photovoltaic\ne\u000bect.\nWe emphasize that this model does not a priori assume the domain wall structure or the electrostatic\npotential step across it. Instead, this is a prediction of the model that is based on well-established Devonshire-\nLandau models of ferroelectrics and lumped band models of semiconductors.\nThe rest of the paper is organized as follows: In Section 2, we brie\ry review the structure of BFO and\ndiscuss the classi\fcations of ferroelectric domain walls in BFO. We develop the theoretical framework in\nSection 3. We apply the theory to examine PV e\u000bect in ferroelectrics with domain walls in Section 4. We\nconclude with a brief discussion in Section 5.\n2 Bismuth Ferrite\nIn this work, we focus on bismuth ferrite (BFO), though we note that the same framework can be applied to\nother ferroelectrics and the results are expected to be similar qualitatively.\nAt room temperature, BFO has a rhombohedral phase with space group R3c(Figure 1). The displace-\nments of the atoms from the ideal cubic structure in this phase lead to a spontaneous polarization pointing\nin the [111] pseudocubic direction. Another distintive feature is the network of O 6octahedra surrounding\nthe Fe ions that rotate or tilt out-of-phase about the polarization axis. This is also commonly known as the\nantiferrodistortive (AFD) mode and has been found to play an important role in the ferroelectric phase of\nthe material [10].\nElectric polarization in rhombohedral BFO can take one of the eight variants of the [111] pseudocubic\ndirection which gives possible domain wall orientations of 71\u000e, 109\u000eand 180\u000e. On each domain, there can be\ntwo possible orientations of oxygen octahedra. We follow Lubk et al. 's method of classifying oxygen octahedra\ntilts (OTs) across domain walls as either continuous ordiscontinuous [9]. In the continuous case, the direction\nof oxygen octahedra tilt remains the same along the polarization vector \feld. In the discontinuous case, the\ndirection reverses across the domain wall.\n3 Theory\nWe consider a metal-perovskite-metal (MPM) structure that is connected to an external voltage source to\nform a closed electrical circuit (see Figure 1). The multi-domain ferroelectric perovskite \flm occupying the\n2Figure 2: Schematic of a device model in a metal-perovskite-metal con\fguration.\nspace \n is subjected to light illumination. The two metal-perovskite interfaces are denoted by @\n1,@\n2\n2@\n. All the processes are assumed to occur at constant temperature T. We present the equations and their\nphysical meanings here. Readers may refer to Appendix A for the thermodynamically consistent derivation.\n3.1 Charge and electrostatic potential\nThe total charge density ( x2\n) is given by\n\u001a=q(pv\u0000nc+zdN+\nd\u0000zaN\u0000\na); (1)\nwhere q is the electronic charge, ncis the density of electrons in the conduction band, pvis the density of\nholes in the valence band, N+\ndis the density of ionized donors, N\u0000\nais the density of ionized acceptors, zdis\nthe valency number of donors, and zais the valency number of acceptors. The polarization and space charge\nin the ferroelectrics together generate an electrostatic potential. This is determined by Gauss' equation\nr\u0001(\u0000\"0r\u001e+p) =\u001a; (2)\nwhere\"0is the permittivity of free space, subject to appropriate boundary conditions.\n3.2 Transport equations\nIn the presence of light illumination, an incident photon may be absorbed in the semiconductor to promote an\nelectron from the valence band to the conduction band, thus generating an electron-hole pair in the process\nof photogeneration. The reverse may also occur such that an electron and a hole recombine. Electrons and\nholes may also move from one point to another point, as represented by the electron and hole density \rux\nterms, JnandJp. With conservation of electrons and holes, we can relate the time derivatives of densities\nof electrons and holes to the aforementioned processes via the following transport equations,\n_nc=\u0000r\u0001Jn+G\u0000R; (3)\n_pv=\u0000r\u0001Jp+G\u0000R; (4)\nwhereGis the rate of photogeneration which can be taken to be proportional to the intensity of light\nillumination, while Ris the recombination rate. Here we assume that the only form of recombination present\nis radiative recombination, which involves the transition of an electron from the conduction band to the\nvalence band along with the emission of a photon. Rtakes the form of B(ncpv\u0000N2\ni) with the intrinsic\ncarrier density being given by Ni=pNcNvexp(\u0000Ec\u0000Ev\n2kBT) [17]. The radiative recombination coe\u000ecient Bis\na material property, independent of the carrier density.\nThe electron and hole \ruxes Jn;Jpare taken to be proportional to the gradient in its electro-chemical\npotential [17] via\nJn=\u00001\nq\u0017nncr\u0016n; (5)\nJp=\u00001\nq\u0017ppvr\u0016p; (6)\nwhere\u0017nand\u0017pare the electron and hole mobilities respectively.\nIn this work, the di\u000busion of donors and acceptors are neglected.\n33.3 Free energy\nThe free energy of the ferroelectric is postulated to be of the form\nW=WDL(p;\u0012) +WG(rp;r\u0012) +Wnc(nc) +Wpv(pv) +WNd(N+\nd) +WNa(N\u0000\na): (7)\nThe various terms are currently explained.\nWDLrefers to the Devonshire-Landau free energy of bulk ferroelectrics. In addition to the typical primary\norder parameter of electric polarization p, we include a second order parameter | oxygen octahedral tilts \u0012.\nWe adopt the following energy form for BFO [12]. The corresponding coe\u000ecients can be found in Table 1.\nWDL=a1(p2\n1+p2\n2+p2\n3) +a11(p4\n1+p4\n2+p4\n3) +a12(p2\n1p2\n2+p2\n2p2\n3+p2\n1p2\n3)\n+b1(\u00122\n1+\u00122\n2+\u00122\n3) +b11(\u00124\n1+\u00124\n2+\u00124\n3) +b12(\u00122\n1\u00122\n2+\u00122\n2\u00122\n3+\u00122\n1\u00122\n3)\n+c11(p2\n1\u00122\n1+p2\n2\u00122\n2+p2\n3\u00122\n3) +c12[p2\n1(\u00122\n2+\u00122\n3) +p2\n2(\u00122\n1+\u00122\n3) +p2\n3(\u00122\n1+\u00122\n2)]\n+c44(p1p2\u00121\u00122+p1p3\u00121\u00123+p2p3\u00122\u00123):(8)\nThe energy stored in the ferroelectric domain walls is accounted for through the gradient or Ginzburg\nenergy term WGwhich includes the energy cost associated with rapid change in polarization and octahedral\ntilts.\nWG(rp;r\u0012) =1\n2a0jrpj2+1\n2b0jr\u0012j2: (9)\nHere we assume that the gradient terms are isotropic for simplicity but can easily be modi\fed.\nWnc;Wpv;WNd;WNain equation (7) are the free energies of electrons in the conduction band, holes in the\nvalence band, donors and acceptors respectively. The explicit expressions of these energies can be determined\nby considering each system as a canonical ensemble in the framework of statistical mechanics[16]\nWnc(nc) =ncEc+kBT[\u0000NclogNc+nclognc+ (Nc\u0000nc) log(Nc\u0000nc)]; (10)\nWpv(pv) = (Nv\u0000pv)Ev+kBT[\u0000NvlogNv+pvlogpv+ (Nv\u0000pv) log(Nv\u0000pv)]; (11)\nWNd(N+\nd) =(Nd\u0000N+\nd)Ed\u0000(Nd\u0000N+\nd)kBTlog(2zd)\n+kBT[\u0000NdlogNd+N+\ndlogN+\nd+ (Nd\u0000N+\nd) log(Nd\u0000N+\nd)];(12)\nWNa(N\u0000\na) =N\u0000\naEa\u0000(Na\u0000N\u0000\na)kBTlog(2za)\n+kBT[\u0000NalogNa+N\u0000\nalogN\u0000\na+ (Na\u0000N\u0000\na) log(Na\u0000N\u0000\na)]:(13)\n3.4 Polarization and tilt equations\nThe polarization and tilt evolve according to the (time-dependent) Landau-Ginzburg equations\n1\n\u0017p_p=r\u0001@WG\n@rp\u0000@WDL\n@p\u0000r\u001e; (14)\n1\n\u0017\u0012_\u0012=r\u0001@WG\n@r\u0012\u0000WDL\n@\u0012: (15)\nwhere\u0017p;\u0017\u0012are the respective mobilities. They are subject to natural boundary conditions\n^ n\u0001@Wg\n@rp= 0; (16)\n^ n\u0001@Wg\n@r\u0012= 0: (17)\n43.5 Electrochemical potentials\nThe electrochemical potentials are obtained from the energy to be\n\u0016n=@Wnc\n@nc\u0000q\u001e; (18)\n\u0016p=@Wpv\n@pv+q\u001e; (19)\n\u0016N+\nd=@WNd\n@N+\nd+qzd\u001e; (20)\n\u0016N\u0000\na=@WNa\n@N\u0000a\u0000qza\u001e: (21)\nAt thermal equilibrium, \u0016n=\u0000\u0016p=\u0000\u0016N+\nd=\u0016N\u0000\na=EFmwhere the magnitude of EFmis the workfunction\nof the metal electrode. Further, using equations (10) to (13) we can invert the relations to obtain\nnc=Nc\n1 + exp(Ec\u0000EFm\u0000q\u001e\nkBT);\npv=Nv\n1 + exp(EFm\u0000Ev+q\u001e\nkBT);\nN+\nd=Nd\u0014\n1\u00001\n1 +1\n2zdexp(\u0000EFm+Ed\u0000q\u001ezd\nkBT)\u0015\n;\nN\u0000\na=Na\u0014\n1 + 2zaexp(\u0000EFm+Ea\u0000q\u001eza\nkBT)\u0015\u00001\n;(22)\nconsistent with Fermi-dirac distribution [18]. Finally, assuming Nc>>ncandNv>>pv, equations (5) and\n(6) become\nJn=\u0000\u0017nkBT\nqrnc+\u0017nncr\u001e; (23)\nJp=\u0000\u0017pkBT\nqrpv\u0000\u0017ppvr\u001e: (24)\nEquations (23) and (24) show that each of JnandJpcan be resolved into two contributions: (1) a di\u000busion\ncurrent, driven by concentration gradient of carriers, and (2) a drift current, driven by an electric \feld. By\napplying the Einstein relation which relates di\u000busion constant Dto mobility \u0017viaD=\u0017kBT=q, we recover\nthe equations that are typically written to describe the \row of electrons and holes in solar cells.\n3.6 Ohmic boundary conditions\nWe prescribe ohmic boundary conditions at the contacts with metal electrodes following [19] for convenience.\nWe have\nnc=Nce\u0000(Ec\u0000Efm)=kBT\npv=Nve\u0000(Efm\u0000Ev)=kBT)\non@\n1[@\n2:\nThis is equivalent to assuming that the Fermi level in the semiconductor aligns with that of the metal, hence\ngiving rise to electron and hole densities that are independent of the applied voltage.\n53.7 Steady-state model\nAt steady state, all the \felds of interest do not vary with respect to time. Further, we are interested in domain\nwalls, and therefore can assume that things are invariant parallel to the domain wall. This means that we\nhave one independent space variable which we denote r. We denote the components of polarization and tilt\nparallel (respectively perpendicular) to the domain wall to be ps;\u0012s(respectively pr;\u0012r). Withzd=za= 1,\nwe have a coupled system of di\u000berential equations for region x2(0;L), whereLis the length of the \flm.\na0d2pr\ndr2\u0000@WDL\n@pr\u0000d\u001e\ndr= 0; (25)\na0d2ps\ndr2\u0000@WDL\n@ps= 0; (26)\nb0d2\u0012r\ndr2\u0000@WDL\n@\u0012r= 0; (27)\nb0d2\u0012s\ndr2\u0000@WDL\n@\u0012s= 0; (28)\n\u0000\"0d2\u001e\ndr2+dpr\ndr=q(pv\u0000nc+N+\nd\u0000N\u0000\na); (29)\n\u0000dJn\ndr+G\u0000B(ncpv\u0000N2\ni) = 0; (30)\n\u0000dJp\ndr+G\u0000B(ncpv\u0000N2\ni) = 0; (31)\nJn=\u0000\u0017nkBT\nqdnc\ndr+\u0017nncd\u001e\ndr; (32)\nJp=\u0000\u0017pkBT\nqdpv\ndr\u0000\u0017ppvd\u001e\ndr; (33)\nwhere\nN+\nd=Nd\u0014\n1\u00001\n1 +1\n2exp(\u0000EFm+Ed\u0000q\u001e\nkBT)\u0015\n;\nN\u0000\na=Na\u0014\n1 + 2 exp(\u0000EFm+Ea\u0000q\u001e\nkBT)\u0015\u00001\n;\nNi=p\nNcNvexp(\u0000Ec\u0000Ev\n2kBT);\nwith boundary conditions\ndpr\ndr(r= 0) =dpr\ndr(r=L) = 0;\ndps\ndr(r= 0) =dps\ndr(r=L) = 0;\nd\u0012r\ndr(r= 0) =d\u0012r\ndr(r=L) = 0;\nd\u0012s\ndr(r= 0) =d\u0012s\ndr(r=L) = 0;\n\u001e(r= 0) = 0; \u001e(r=L) = 0;\nnc(r= 0) =nc(r=L) =Nce\u0000(Ec\u0000Efm)=kBT;\npv(r= 0) =pv(r=L) =Nve\u0000(Efm\u0000Ev)=kBT:\n6Symbols Values Units\na1\u00001:19\u0002109V m C\u00001\na11 9:93\u0002108V m5C\u00003\na12 3:93\u0002108V m5C\u00003\nb1\u00001:79\u00021010V m\u00003C\nb11 1:14\u00021011V m\u00003C\nb12 2:25\u00021011V m\u00003C\nc11 1:50\u00021010V m C\u00001\nc12 7:50\u0002109V m C\u00001\nc44\u00001:60\u0002101\u0000V m C\u00001\nTable 1: Coe\u000ecients of Laudau-Devonshire energy for BFO\n3.8 Numerical issues\nThe model derived above comprises of di\u000berential equations that are nonlinear and coupled, which can prove\ntroublesome numerically. So we non-dimensionalize the problem as in Appendix B. Further, we notice that\nthe coupling between the \frst \fve governing equations and the rest of the model is weak. This is especially\nso when the length of the simulated device is much smaller than the Debye length, or when the dimensionless\nquantity\u000eis small, which is generally the case in the simulations in this paper. Therefore we treat them\nas two subproblems, that are then solved self-consistently until convergence occurs. Each subproblem is\nconstructed within the \fnite di\u000berence framework, and the resulting system of nonlinear equations is solved\nusing the trust-region dogleg method.\n4 Application to Bismuth Ferrite\n4.1 Material constants\nThe coe\u000ecients of the Devonshire-Landau energy for BFO in equation (8) are presented in Table 1. They\nare derived to match the values of spontaneous polarization, tilt angles, and dielectric constant [20, 10, 21].\nOther material parameters including band structure information [22] and carrier mobility values [23] are\nlisted in Table 2. The values of a0andb0are chosen to match a ferroelectric domain wall width of 2 nm.\nTypically BFO exists as a n-type semiconductor due to oxygen vacancies. It can also become p-type with Bi\nde\fciency. Here we restrict our simulations to n-type semiconductors.\n4.2 Two-domain ferroelectrics\n4.2.1 71\u000eand 109\u000edomain walls\nWe consider a device comprising of a BFO \flm with two ferroelectric domains separated by either a 71\u000eor\n109\u000edomain wall, with continuous or discontinuous oxygen octahedra rotations across the DW. This gives a\ntotal of four di\u000berent cases, as illustrated in Table 3.\nFigures 3 and 4 show the variation of various \feld quantities when the perovskite \flm is exposed to light\nillumination and shorted. Notice that in all cases, the perpendicular component of the polarization pris not\nconstant in the vicinity of the domain wall. In other words, the polarization is not divergence free, and we\nsee a voltage drop across the domain wall. The polarization pro\fle (i.e. pr) of 71\u000eand 109\u000edomain walls\nwith continuous OT are qualitatively similar to those obtained from \frst-principles calculations [9]. This\nvoltage drop across the domain wall leads to charge separation of photogenerated electron-hole pairs, and a\nnon-zero photocurrent. This is evident in the current-voltage plots shown in Figure 5 and is consistent with\nthe mechanism proposed by Yang et al. [1].\nFigure 5 shows that the magnitude and direction of photocurrent due to the domain wall e\u000bect hinge\ngreatly upon the changes in the crystallographic structure across the domain wall. The case with 109\u000eDW\nand continuous OT gives a positive short-circuit current, which is in the same direction as net polarization\n7Parameters Symbols Values Units\nElectron mobility \u0016n 2\u000210\u00005m2V\u00001s\u00001\nHole mobility \u0016p 1\u000210\u00005m2V\u00001s\u00001\nEnergy of conduction band Ec\u00003:3 eV\nEnergy of valence band Ev\u00006:1 eV\nDonor level Ed\u00003:7 eV\nAcceptor level Ea\u00005:8 eV\nE\u000bective density of states for conduction band Nc 1\u00021024m\u00003\nE\u000bective density of states for valence band Nv 1\u00021024m\u00003\nDonor concentration Nd 1\u00021020m\u00003\nAcceptor concentration Na 0 m\u00003\nPolarization gradient coe\u000ecient a0 9\u000210\u000010V m3C\u00001\nAFD gradient coe\u000ecient b0 2\u000210\u00009V m\u00001C\nRate of photogeneration G 1\u00021027m\u00003s\u00001\nRadiative recombination coe\u000ecient B 1\u000210\u00009m3s\u00001\nThickness of \flm L 100 nm\nTemperature T 300 K\nWork function of Pt \u0000EFm 5:3 eV\nTable 2: Material and simulation parameters\nPolarization: [111] \u0000![111] Polarization: [111] \u0000![111]\n(a) (b) (c) (d)\nContinuous OT Discontinuous OT Continuous OT Discontinuous OT\nh111i\u0000!h 111i h 111i\u0000!h 111i h 111i\u0000!h 111i h 111i\u0000!h 111i\nEDW= 0:53 J m\u00002EDW= 0:63 J m\u00002EDW= 0:53 J m\u00002EDW= 0:45 J m\u00002\nJsc=\u00000:22 A m\u00002Jsc=\u00000:84 A m\u00002Jsc= 0:98 A m\u00002Jsc=\u00001:0 A m\u00002\nVoc= 6:8 mV Voc= 29 mV Voc=\u000070 mV Voc= 38 mV\n* [\u0001]\u0000![\u0001] andh\u0001i\u0000!h\u0001i denote the directions of electric polarization and oxygen octahedra\ntilt (OT), respectively, on two neighboring domains. JscandVocare the short-circuit current\ndensity and open-circuit voltage obtained from our device model simulations. EDWrefers to\nthe domain wall energy calculated at thermal equilibrium in the absence of light illumination.\nTable 3: Device models with di\u000berent types of domain walls.\n8Figure 3: Spatial variation of \feld quantities (polarization components, OT tilt angles, electric potential and\ncarrier densities) along the 71\u000eDW device with either continuous or discontinuous OT at short circuit under\nlight illumination\nin the \flm while the rest show negative currents. Importantly, this DW-based photovoltaic e\u000bect shows that\nthe direction of current \row does not necessarily correlates with the direction of net polarization consistent\nwith experimental observations [6].\nFurthermore, we observe strong coupling between polarization and oxygen octahedra tilt (OT). This is\nevident from the vastly di\u000berent results (including the change in current direction) obtained when changing\nthe OT pro\fles without changing the type of domain walls. In actual experiments, we may only observe one\ntype of oxygen octahedra rotation for each domain wall type. We compute the domain wall energy for each\ncase as in Table 3, and \fnd that it is energetically more favorable to have 71\u000edomain wall with continuous\nOT and 109\u000edomain wall with discontinous OT. This is consistent with previous \frst-principles calculations\n[10, 12].\nAs the perovskite \flm is \frst exposed to light illumination, which is simulated in terms of an increase in\nphotogeneration rate of electron-hole pairs, the magnitude of short-circuit current density generated increases\nrapidly initially as shown in Figure 6. The increase slows down at higher illumination (or photogeneration\nrate) due to recombination of the excited electrons and holes.\nFinally we consider changing the order of the domains in a two-domain device. Figure 7 shows that doing\nso does not pose any di\u000berence to the pro\fles of other quantities such as prand\u001e. This implies that current\n\rows in a single direction irrespective of the order of the domains. If we were to stack the di\u000berent domains\nto form a device with periodic domain pattern (i.e. alternating domains), the photovolatic e\u000bect would be\nadditive and would not cancel out. This is exactly what we observe in Section 4.3.\n9Figure 4: Spatial variation of \feld quantities (polarization components, OT tilt angles, electric potential and\ncarrier densities) along the 109\u000eDW device with either continuous or discontinuous OT at short circuit under\nlight illumination\n10Figure 5: Current-voltage plot in dark ( G= 0) and under light illumination ( G= 1027m\u00003s\u00001) for BFO\ndevices with a 71\u000eor 109\u000eDW with continuous or discontinuous OT.\nFigure 6: Short-circuit current density versus photogeneration rate for a two-domain ferroelectrics separated\nby a 71\u000eDW with continuous OT.\n11Figure 7: Spatial variation of \feld quantities along a two-domain device continuous (71\u000eDW) at short circuit.\nThe two cases are identical except the order of the two domains is reversed.\n12Figure 8: Spatial variation of \feld quantities for ferroelectrics with ten 71\u000eDWs and continuous OTs at\nshort circuit.\n4.2.2 180\u000edomain walls\nIn the case of the 180\u000edomain walls, with either continuous or discontinuous OTs, there is no visible\ndisturbance to the polarization component normal to the domain wall at the domain wall. With a lack of\nsymmetry breaking, the photovoltaic e\u000bect fails to be generated. The \fgures are omitted for brevity.\n4.3 Ferroelectrics with multiple domain walls\nWe now examine the case with multiple domain walls. We keep the width of the perovskite \flm constant and\nuniformly place a number of domain walls parallel to the metal electrodes within the \flm. The distribution\nof polarization, oxygen octahedra tilts (OTs) and other \feld quantities for a shorted device with ten 71\u000e\ndomain walls are presented in Figure 8. Polarization and OTs are periodic and electric potential varies in a\nzig-zag manner but with a slope.\nFigure 9 shows that the magnitudes of both short-circuit current and open-circuit voltage increase with\nthe density of domain walls in the device. The additive e\u000bect becomes smaller at higher domain wall number\nas it is in\ruenced by the boundary.\n13Figure 9: Current-voltage plot for multi-domain BFO devices with 71\u000eDWs and continuous OTs\n4.4 E\u000bect of varying doping and width of ferroelectric \flm\nNext, we investigate the e\u000bect of doping and width on the ferroelectric response using a two-domain example.\nAll the previous simulations are run using a small donor doping density of Nd= 1020m\u00003and a width\nof 100 nm, which corresponds to the state of complete depletion. Typically a depletion layer forms at a\nmetal-semiconductor interface and the width of the depletion layer is related to the Debye length which is\ndependent on the dopant density and dielectric constant. Complete depletion occurs when the Debye length\nof the material is much larger than the width of the device. Otherwise there is partial or local depletion.\nFigure 10 shows the short-circuit distributions for two di\u000berent perovskite widths of 100 nm and 500 nm at\na low dopant density level of Nd= 1020m\u00003and a high dopant density of Nd= 1022m\u00003. At a small width\nof 100 nm, changing the doping level from low to high does minimal changes to the \feld pro\fles and the\nphotovolatic response, with its short-circuit current density staying almost constant at -0.22 A m\u00002. On the\nother hand, at a larger width of 500 nm, increasing the doping level changes the state of the perovskite \flm\nfrom complete depletion to partial depletion, and at the same time, raises the short-circuit current density\nfrom -0.011 A m\u00002to -0.033 A m\u00002. The resulting electric potential pro\fle can be viewed as the superposition\nof two contributions: domain walls and depletion. This illustrates the feature of the model in combining the\nferroelectric and semiconductor behavior of the material.\n5 Conclusions and Discussion\nIn this paper, we have proposed a thermodynamically consistent continuum device model to study photo-\nvoltaic e\u000bect in multi-domain ferroelectric perovskites accounting systematically for the interactions among\nspace charge, polarization and oxygen octahedra tilts. The model has been successfully implemented numer-\nically. Our results show that there is an electric potential step across each 71\u000eor 109\u000edomain wall, and that\nthis produces a PV e\u000bect. There is no electric potential step across a 180\u000edomain wall, and correspondingly\nno PV e\u000bect. Further, the model shows that the direction of current depends on the nature of the domain\nwall and not the orientation of domains. Therefore, the PV e\u000bect becomes additive across multiple domain\nwalls with alternating domains.\nWe note that the presence of electric potential step across non-180\u000eor the lack of such a step for a\n180\u000edomain wall is a generic feature. Consider a generic Devonshire-Landau energy landscape shown in\nFigure 11. Further, consider a non-180\u000edomain wall that separates two ferroelectric domains, one with\npolarization Land the other with polarization R, as marked in the same \fgure. These two polarizations\nvectors are spontaneous polarizations, and are thus energy minima of the Devonshire-Landau energy. Now,\nas the polarization changes from the value Lto the value Ror vice versa across the domain wall, it will do\nso along the low energy valley as shown by the dashed line. This path necessarily involves a change in the\ncomponent of polarization normal to the domain wall. Therefore, there will indeed be a electric potential\nstep across this domain wall. Note that the electric potential step depends on the path connecting the two\npolarizations, and this is unchanged if the domains are swapped. This argument is generic because there is\nno reason in symmetry for the low energy valley to go in a straight line from LtoR. A similar argument\n14(a) Perovskite width of 100 nm\n(b) Perovskite width of 500 nm\nFigure 10: Low dopant level: Nd= 1020m\u00003, high dopant level: Nd= 1022m\u00003\n15p1(pr)p2(ps)LRLRFigure 11: The energy landscape of a non-180\u000edomain wall.\nshows the lack of such a step for a 180\u000edomain wall is also a generic feature. While the presence or absence\nof the step is a generic feature, its magnitude and direction depend on the speci\fc energy landscape.\nThe model and results presented here support the hypothesis that non-180\u000edomain walls contribute\nto the photovoltaic e\u000bect. Importantly, this model does not a priori assume the domain wall structure or\nthe electrostatic potential step across it. Instead, this is a prediction of the model that is based on well-\nestablished Devonshire-Landau models of ferroelectrics and lumped band models of semiconductors. This\nmodel is agnostic about the bulk photovoltaic e\u000bect. We could modify our model to include it by coupling\nthe photogeneration to the polarization, but we chose not to do so since this would not be predictive.\nWe note that it remains an open question as to why Alexe and Hesse [2] do not see any di\u000berence in\nphotocurrent in their AFM-based measurement. One possibility is that the AFM-tip created a depletion\nzone around it which dominated over the potential step across the domain wall. This can be investigated\nfurther by the proposed model but it requires a multi-dimensional numerical implementation that is the topic\nof current work.\nFinally, we hope that the model presented here as well as the multi-dimensional numerical implementation\nof it will prove useful for future device designs.\nAcknowledgement\nYing Shi Teh gratefully acknowledges the support of the Resnick Institute at Caltech through the Resnick\nGraduate Research Fellowship.\nAppendix A Derivation of a thermodynamically consistent theory\nWe outline a derivation of the model in Section 3, and show that it is thermodynamically consistent. Since we\nconsider only isothermal processes, the second law of thermodynamics (Clausius-Duhem inequality) requires\nthat the rate of dissipation be non-negative. This rate of dissipation is given by\nD=F\u0000dE\ndt(34)\nwhereFis the rate of external work done on the system\nF=Z\n\n\u0016nGnetdV+Z\n\n\u0016pGnetdV+d\ndtZ\n@\n1[@\n2\u001e\u001bdS\u0000Z\n@\n\u0016nJn\u0001^ ndS\u0000Z\n@\n\u0016pJp\u0001^ ndS: (35)\nandEis the energy stored in the system\nE=Z\n\n\u0010\nW+\"0\n2jr\u001ej2\u0011\ndV; (36)\n16The \frst two terms in Equation (35) denote the rate of work done by incident photons in generating electron-\nhole pairs. Here Gnet=G\u0000Rdenotes the net rate of photogeneration. The third term refers to the work\ndone by the external voltage and \u001b=J\u0000\"0r\u001e+\u001fpK\u0001^ nis the surface charge density where J\u0001Kindicates a\njump in the respective quantity ( \u0001) and ^ nis a unit vector normal to the surface. The \fnal two terms are the\nenergy carried into the systems by electron and hole \ruxes at the boundary. The total energy consists of the\nfree energy and the electrostatic energy stored in the electrostatic \feld.\nApplying the divergence theorem and transport equations (3), (4), we rewrite equation (35) as\nF=d\ndtZ\n@\n1[@\n2\u001e\u001bdS\u0000Z\n\nr\u0016n\u0001JndV\u0000Z\n\nr\u0016p\u0001JpdV\u0000Z\n\nr\u0016N+\nd\u0001JN+\nddV\u0000Z\n\nr\u0016N\u0000\na\u0001JN\u0000\nadV\n+Z\n\n\u0016n_ncdV+Z\n\n\u0016p_pvdV+Z\n\n\u0016N+\nd_N+\nddV+Z\n\n\u0016N\u0000\na_N\u0000adV:\n(37)\nFrom equations (36) and (7), the rate of change of the total energy of the system can be expressed as\ndE\ndt=Z\n\n\u0012\n\u0000r\u0001@WG\n@rp+@WDL\n@p\u0013\n\u0001_pdV+Z\n\n\u0012\n\u0000r\u0001@WG\n@r\u0012+WDL\n@\u0012\u0013\n\u0001_\u0012dV +d\ndt\u0012\"0\n2Z\nR3jr\u001ej2dV\u0013\n+Z\n\n\u0012@Wpv\n@pv_pv+@Wnc\n@nc_nc+@WNd\n@N+\nd_N+\nd+@WNa\n@N\u0000a_N\u0000a\u0013\ndV\n+Z\n@\n\u0012\n^ n\u0001@WG\n@rp\u0013\n\u0001\u000epdS+Z\n@\n\u0012\n^ n\u0001@WG\n@r\u0012\u0013\n\u0001\u000e\u0012dS:(38)\nFollowing [16], we can show\nd\ndt\u0012\"0\n2Z\n\njr\u001ej2dV\u0013\n=Z\n@\n1[@\n2\u001e_\u001bdS +Z\n\nr\u001e\u0001_pdV+Z\n\n\u001eq(zd_N+\nd\u0000za_N\u0000a\u0000_nc+ _pv)dV: (39)\nTherefore,\nD=Z\n\n\"\u0012\nr\u0001@Wg\n@rp\u0000@WGLD\n@p\u0000r\u001e\u0013\n\u0001_p+\u0012\nr\u0001@Wg\n@r\u0012\u0000WGLD\n@\u0012\u0013\n\u0001_\u0012+\u0012\n\u0016n\u0000@Wnc\n@nc+q\u001e\u0013\n_nc\n+\u0012\n\u0016p\u0000@Wpv\n@pv\u0000q\u001e\u0013\n_pv+\u0012\n\u0016N+\nd\u0000@WNd\n@N+\nd\u0000qzd\u001e\u0013\n_N+\nd+\u0012\n\u0016N\u0000\na\u0000@WNa\n@N\u0000a+qza\u001e\u0013\n_N\u0000a\n\u0000r\u0016n\u0001Jn\u0000r\u0016p\u0001Jp\u0000r\u0016N+\nd\u0001JN+\nd\u0000r\u0016N\u0000\na\u0001JN\u0000\na#\ndV\n+Z\n@\n\"\u0012\n^ n\u0001@Wg\n@rp\u0013\n\u0001_p+\u0012\n^ n\u0001@Wg\n@r\u0012\u0013\n\u0001_\u0012#\ndS:(40)\nEach of the above terms is a non-negative product of a generalized force and a generalized velocity or rate.\nAssuming either over-damped dynamics or equilibrium gives rise to the equations in Section 3.\nAppendix B Non-dimensionalization and scaling\nBefore solving for the model numerically, we introduce dimensionless variables through appropriate scalings\nas follows:\npr=p0pr\nps=p0ps\nr=L0r\n17WDL=W0WDL\n\u001e=\u001e0\u001e\nnc=N0nc\npv=N0pv\nN+\nd=N0N+\nd\nN\u0000\na=N0N\u0000\na\nNi=N0Ni\nJn=J0Jn\nJp=J0Jp\nJtotal=qJ0Jtotal\nG=G0G\n,\nwhere\nfp0;\u00120g= arg min\np;\u0012(WDLjjp1j=jp2j=jp3j=p;j\u00121j=j\u00122j=j\u00123j=\u0012)\nN0= 1\u00021021m\u00003\nW0=ja1jp2\n0\nL0=p0ra0\nW0= 3:9609\u000210\u00008m\nJ0=G0L0\n\u001e0=a0p0\nL0\n.\nThe steady-state non-dimensionalized equations are\nd2pr\ndr2\u0000@WDL\n@pr\u0000d\u001e\ndr= 0 (41)\nd2ps\ndr2\u0000@WDL\n@ps= 0 (42)\nb0d2\u0012r\ndr2\u0000@WDL\n@\u0012r= 0 (43)\nb0d2\u0012s\ndr2\u0000@WDL\n@\u0012s= 0 (44)\n\"0d2\u001e\ndr2\u0000dpr\ndr+\u000e(\u0000nc+pv+N+\nd\u0000N\u0000\na) = 0 (45)\nJn=\u0000Kn(dnc\ndr\u0000\f\u001e0ncd\u001e\ndr) (46)\nJp=\u0000Kp(dpv\ndr+\f\u001e0pvd\u001e\ndr) (47)\n\u0000d\ndrJn+G\u0000B(ncpv\u0000N2\ni) = 0 (48)\n18\u0000d\ndrJp+G\u0000B(ncpv\u0000N2\ni) = 0; (49)\nwhere\n\"0=\"0\u001e0\nL0p0\n\u000e=qN0L0\np0\nKn=\u0017nN0\nq\fL 0J0\nKp=\u0017pN0\nq\fL 0J0\nB=BN2\n0\nG0\n.\nReferences\n[1] S. 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Science , 315(5814):954{959, 2007.\n20"    },    {        "title": "2009.13619v6.Ferrite__A_Judgmental_Embedding_of_Session_Types_in_Rust.pdf",        "content": "Ferrite: A Judgmental Embedding of Session\nTypes in Rust\nRuo Fei Chen /envelope/orcid\nIndependent Researcher\nStephanie Balzer /envelope\nCarnegie Mellon University\nBernardo Toninho /envelope/orcid\nUniversidade Nova de Lisboa and NOVA LINCS\nAbstract\nSession types have proved viable in expressing and verifying the protocols of message-passing systems.\nWhile message passing is a dominant concurrency paradigm in practice, real world software is written\nwithout session types. A limitation of existing session type libraries in mainstream languages is their\nrestriction to linear session types, precluding application scenarios that demand sharing and thus\naliasing of channel references.\nThis paper introduces Ferrite, a shallow embedding of session types in Rust that supports both\nlinearandsharedsessions. The formal foundation of Ferrite constitutes the shared session type\ncalculus SILL S, which Ferrite encodes via a novel judgmental embedding technique. The fulcrum of\nthe embedding is the notion of a typing judgment that allows reasoning about shared and linear\nresources to type a session. Typing rules are then encoded as functions over judgments, with a valid\ntyping derivation manifesting as a well-typed Rust program. This Rust program generated by Ferrite\nserves as a certificate , ensuring that the application will proceed according to the protocol defined\nby the session type. The paper details the features and implementation of Ferrite and includes a\ncase study on implementing Servo’s canvas component in Ferrite.\n2012 ACM Subject Classification Theory of computation →Linear logic; Theory of computation →\nType theory; Software and its engineering →Domain specific languages; Software and its engineering\n→Concurrent programming languages\nKeywords and phrases Session Types, Rust, DSL\nDigital Object Identifier 10.4230/LIPIcs.ECOOP.2022.22\nFunding Stephanie Balzer : National Science Foundation Award No. CCF-1718267\nBernardo Toninho : FCT/MCTES grant NOVALINCS/BASE UIDB/04516/2020\n1 Introduction\nMessage-passing is a dominant concurrency paradigm, adopted by mainstream languages such\nas Erlang, Scala, Go, and Rust, putting the slogan “Do not communicate by sharing memory;\ninstead, share memory by communicating” [11] into practice. In this setting, messages are\nexchanged along channels, which can be shared by several senders and receivers. Type\nsystems in such languages typically allow channels to be typed, specifying and constraining\nthe types of messages they may carry (e.g. integers, strings, sums, references, etc.).\nAn aspect inherent to message-passing concurrency that is not captured in mainstream\ntype systems, however, is the idea of a protocol. Protocols dictate the sequencing and types\nof messages to be exchanged. To express and enforce such protocols, session types [13,14,15]\nwere introduced. Session typing disciplines assign types to channel endpoints according to\ntheir intended usage protocols in terms of sequencing of input/output actions (e.g. “send\nan integer and, afterwards, receive a string”) and branching/selection actions (e.g. “receive\neither a buy message and process the payment; or a cancellation message and abort the\n©R. F. Chen, S. Balzer, and B. Toninho;\nlicensed under Creative Commons License CC-BY 4.0\n36th European Conference on Object-Oriented Programming (ECOOP 2022).\nEditors: Karim Ali and Jan Vitek; Article No.22; pp.22:1–22:47\nLeibniz International Proceedings in Informatics\nSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, GermanyarXiv:2009.13619v6  [cs.PL]  31 May 202222:2 Ferrite: A Judgmental Embedding of Session Types in Rust\ntransaction”), ensuring the action sequence is followed correctly and thus, adherence to the\nprotocol. Thanks to their correspondence to linearlogic [4,49,48,47,28,5] session types\nenjoy a strong logical foundation and ensure, in addition to protocol adherence ( session\nfidelity), the existence of a communication partner ( progress). Session types have also been\nextended with safe sharing[1,2,3] to accommodate multi-client scenarios that are rejected\nby exclusively linear session types.\nDespite these theoretical advances, session types have not (yet) been adopted at scale.\nWhile various session type embeddings exist in mainstream languages such as Java [ 17,16],\nScala [43], Haskell [ 42,38,22,29], OCaml [ 36,21], and Rust [ 23,27,6,7], all of these\nembeddings lack support for multi-client scenarios that mandate controlled aliasing in\naddition to linearity. This paper introduces Ferrite, a shallow embedding of session types in\nRust. In contrast to prior work, Ferrite supports bothlinear and shared session types, with\nprotocol adherence guaranteed statically by the Rust compiler.\nFerrite’s underlying theory is based on the calculus SILL Sintroduced in [ 1], which develops\nthe logical foundation of shared session types. As a matter of fact, Ferrite encodes SILL S\ntyping derivations as Rust functions, through a technique we dub judgmental embedding .\nThrough our judgmental embedding, a type-checked Ferrite program yields a Rust program\nthat corresponds to a SILL Styping derivation and thus the proofof protocol adherence.\nIn order to faithfully encode SILL Styping in Rust, this paper further makes several\ntechnical contributions to emulate advanced typing features, such as higher-kinded types,\nby a skillful combination of traits (type classes) and associated types (type families). For\nexample, Ferrite supports recursive (session) types in this way, which are limited to recursive\nstructs of a fixed size in plain Rust. A combination of type-level natural numbers with\nideas from profunctor optics [ 37] are also used to support named channels and labeled\nchoices. We adopt the idea of lenses[9] for selecting and updating individual channels\nin an arbitrary-length linear context. Similarly, we use prismsfor selecting a branch out\nof arbitrary-length choices. Whereas session-ocaml [36] has previously explored the use of\nn-ary choice through extensible variants in OCaml, we are the first to connect n-ary choice\nto prisms and non-native implementation of extensible variants. Remarkably, the Ferrite\ncodebase remains entirely in the safe fragment of Rust, with no direct use of unsafe features.\nGiven its support of both linear and shared session types, Ferrite is capable of expressing\nanysessiontypedprograminRust. Wesubstantiatethisclaimbyprovidinganimplementation\nof Servo’s production canvas component with the communication layer entirely within Ferrite.\nWe report on our findings, including benchmarks in Section 7.\nIn summary, this paper makes the following contributions:\ndesign and implementation of Ferrite, an embedded domain-specific language (EDSL) for\nwriting session-typed programs in Rust;\nsupport of both linearandsharedsessions, guaranteed to be observed by type checking;\na noveljudgmental embedding of custom typing rules in a host language with the resulting\nprogram carrying the proof of successful type checking;\nan encoding of arbitrary-length choice in terms of prisms and extensible variants in Rust;\nanempirical evaluation based on a full implementation of Servo’s canvas component in\nFerrite.\nOutline. Section 2 gives a brief account of session types and sharing, as found in the\nSILL Scalculus [ 1]. Section 3 tours through the key ideas underlying Ferrite, which are refined\nin subsequent sections. Section 4 introduces the technical aspects of Ferrite’s type system,\nfocusing on the judgmental embedding and enforcement of linearity. Section 5 explains howR. F. Chen, S. Balzer, and B. Toninho 22:3\nTable 1Overview of session types and terms in SILL Stogether with their operational meaning.\nSubscripts Land Sdenote linear and shared sessions, resp., where m,n∈{L,S}.\nSession type Process term\ncurrent cont current cont Description\ncL:⊕{l:AL}cL:ALhcL.lh;P P provider sends label lhalongcL\ncasecLofl⇒Q Q hclient receives label lhalongcL\ncL:N{l:AL}cL:ALh casecLofl⇒P P hprovider receives label lhalongc\ncL.lh;Q Q client sends label lhalongcL\ncL:Am⊗BLcL:BL sendcLdm;P P provider sends channel dm:AmalongcL\nym←recvcL;QymQdmclient receives channel dm:AmalongcL\ncL:Am(BLcL:BLym←recvcL;PymPdmprovider receives channel dm:AmalongcL\nsendcLdm;Q Q client sends channel dm:AmalongcL\ncL:1 - closecL - provider sends “ end” alongcL\nwaitcL;Q Q provider receives “ end” alongcL\ncL:↓S\nLAScS:AScS←detachcL;PxSPcSprovider sends “ detachcS” alongcL\nxS←releasecL;QxSQcSclient receives “ detachcS” alongcL\ncS:↑S\nLALcL:ALcL←acquirecS;QxLQcLclient sends “ acquirecL” alongcS\nxL←acceptcS;PxLPcLprovider receives “ acquirecL” alongcS\ncm:Amcm:Amzn←X←dm;PznPznspawn (\"cut\") Xalongzn:Bnwithdm:Dm\ncm:Am - fwdcmdm - forward to channel dm:Amand terminate\nFerrite addresses Rust’s limited support of recursive data types to allow for arbitrary recursive\nand shared session types. Section 6 describes the implementation of n-ary choice using prisms\nand extensible variants. Section 7 provides an evaluation of Ferrite via a re-implementation\nof the Servo canvas component. Section 8 reports on related and future work.\nAn anonymized version of Ferrite’s source code with examples is provided as an artifact.\nAll typing rules and their encoding as well as further materials of interest to an inquisitive\nreader are provided in the appendix.\n2 Background\nThis section gives a brief tour of linear and shared session types. The presentation is based\non the intuitionistic session-typed process calculus SILL S[1], which Ferrite builds upon. We\nconsider the protocol governing the interaction between a queue and its client:\nqueueA=N{enq:A(queueA,deq:⊕{none :1,some :A⊗queueA}}\nTable 1 provides an overview of the types used in the example. Since SILL Sis based on a\nCurry-Howard correspondence between intuitionistic linear logic and the session-typed π-\ncalculus [ 4,5] it uses linear logic connectives ( ⊕,N,⊗,(,1) as session types. The remaining\nconnectives concern shared sessions, a feature we remark on shortly. A crucial—and probably\nunusual—characteristic of session-typed processes is that a process changesits typing along\nwith the messages it exchanges. As a result, a process’ typing always reflects the current\nprotocol state. Table 1 lists state transitions inflicted by a message exchange in the first and\nsecond column and corresponding process terms in the third and fourth column. The fifth\ncolumn provides the operational meaning of a type.\nConsulting Table 1, we gather that the above polymorphic session type queueAimposes\nthe following recursive protocol: A client may either send the label enqordeqto the queue,\nECOOP 202222:4 Ferrite: A Judgmental Embedding of Session Types in Rust\ndepending on whether the client wishes to enqueue or dequeue an element of type A, resp.\nIn the former case, the client sends the element to be enqueued, after which the queue recurs.\nIn the latter case, the queue indicates to the client whether it is empty ( none) or not ( some),\nand proceeds by either terminating or sending the dequeued element and recurring, resp.\nA linear typing discipline is beneficial because it immediately guarantees session fidelity—\neven in the presence of perpetual protocol change—by ensuring that a channel connects\nexactly two processes. Unfortunately, linearity also rules out various practical programming\nscenarios that demand sharing and thus aliasing of channel references. For example, the\nabove linear session type queueAis limited to a singleclient. To support safe sharing of\nstateful channel references while upholding session fidelity, SILL Sextends linear session types\nwith shared session types ( ↓S\nLAS,↑S\nLAL). These two connectives mediate between shared and\nlinear sessions by requiring that clients of shared sessions interact in mutual exclusion from\neach other. Concretely, a type ↑S\nLALmandates a client to acquirethe process offering the\nshared session. If the request is successful, the client receives a linear channel to the acquired\nprocess along which it must proceed as detailed by the session type AL. A type↓S\nLAS, on the\nother hand, mandates a client to releasethe linear process, relinquishing ownership of the\nlinear channel and only being left with a shared channel alias to the now shared process at\ntypeAS.\nUsing these connectives, we can turn the above linear queue into a shared one, bracketing\nenqueue and dequeue operations within acquire-release:\nsqueueAS=↑S\nLN{enq:AS(↓S\nLsqueueAS,deq:⊕{none :↓S\nLsqueueAS,some :AS⊗↓S\nLsqueueAS}}\nIn contrast to the linear queue, the above version recurs in the nonebranch and thus keeps\nthe queue alive to serve the next client. For convenience, SILL Sallows the connectives ⊗and\n(to be used to transport both linear and shared channels along a linear carrier channel.\nTo provide a flavor of session-typed programming in SILL S, we briefly comment on the\nbelow processes emptyand elem, which implement the shared queue session type as a\nsequence of elemprocesses, ended by an emptyprocess. A process implementation consists of\nits signature (first two lines) and body (after =). The first line indicates the typing of channel\nvariables used by the process (left of /turnstileleft) and the type of the providing channel variable (right\nof/turnstileleft). The second line binds the channel variables. In SILL S,←generally denotes variable\nbindings. We leave it to the reader to convince themselves, consulting Table 1, that the code\nin the body of the two processes executes the protocol defined by session type squeueAS.\n·/turnstileleftempty ::q:squeueAS\nq←empty←·=\nq/prime←acceptq;\ncaseq/primeof\n|enq→x←recvq/prime;\nq←detachq/prime;\ne←empty ;q←elem←x,e\n|deq→q/prime.none ;\nq←detachq/prime;\nq←emptyx:AS,t:squeueAS/turnstileleftelem ::q:squeueAS\nq←elem←x,t=\nq/prime←acceptq;\ncaseq/primeof\n|enq→y←recvq/prime;\nt/prime←acquiret;\nt/prime.enq;sendt/primey;\nt←releaset/prime;q←detachq/prime;\nq←elem←x,t\n|deq→q/prime.some ;sendq/primex;\nq←detachq/prime;fwdqt\nImposing acquire-release not only as a programming methodology but also as a typing\ndiscipline has the advantage of recovering session fidelity for shared sessions. To this\nend, shared session types in SILL Smust be strictly equi-synchronizing [1,3], imposing theR. F. Chen, S. Balzer, and B. Toninho 22:5\ninvariant that an acquired session is released to the type at which previously acquired. For\nexample, the shared session type squeueASis strictly equi-synchronizing whereas the type\ninvalid =↑S\nLN{left:↓S\nL↑S\nL⊕{yes:↓S\nLinvalid,no:1},right :↓S\nLinvalid}is not.\nIt is instructive to review the typing rules for acquire-release:\n(T-↑S\nLL)\nΨ,xS:↑S\nLAL; ∆,yL:AL/turnstileleftQyL:: (zL:CL)\nΨ,xS:↑S\nLAL; ∆/turnstileleftyL←acquirexS;QyL:: (zL:CL)(T-↑S\nLR)\nΨ;·/turnstileleftPyL:: (yL:AL)\nΨ/turnstileleftyL←acceptxS;PyL:: (xS:↑S\nLAL)\n(T-↓S\nLL)\nΨ,xS:AS; ∆/turnstileleftQxS:: (zL:CL)\nΨ; ∆,yL:↓S\nLAS/turnstileleftxS←releaseyL;QxS:: (zL:CL)(T-↓S\nLR)\nΨ/turnstileleftPxS:: (xS:AS)\nΨ;·/turnstileleftxS←detachyL;PxS:: (yL:↓S\nLAS)\nDue to its foundation in intuitionistic linear logic, SILL S’ typing rules are phrased using a\nsequent calculus , leading to leftandrightrules for each connective. Left rules describe the\ninteraction from the point of view of the client, right rules from the point of view of the\nprovider. The typing judgments Ψ; ∆/turnstileleftP:: (xL:AL)andΨ/turnstileleftP:: (xS:AS)read as \"process\nPoffers a session of type Aalong channel xusing sessions offered along channels in Ψ(and\n∆).\" The typing contexts Ψand∆provide the typing of shared and linear channels, resp.\nWhereas Ψis a structural context, ∆is a linear context, forbidding channels to be dropped\n(weakened) or duplicated (contracted). In contrast to linear processes, shared processes must\nnot use any linear channels, a requirement crucial for type safety. The notions of acquire\nand release are naturally formulated from the point of view of a client, so these terms appear\nin the left rules. The right rules use the terms acceptanddetachwith the meaning that an\naccept accepts an acquire and a detach initiates a release. The rules are read bottom-up,\nwhere the premise denotes the next action to be taken after the message exchange.\n3 Key Ideas\nThis section introduces the key ideas underlying Ferrite. Subsequent sections provide further\ndetails.\n3.1 SILL R– A stepping stone from SILL Sto Ferrite\nIn Section 2, we reviewed SILL Sand its typing judgment. Our goal with Ferrite is to\nfaithfully and compositionally encode SILL Styping derivations in Rust. However, when\nviewed under the lens of a general purpose programming language, most readers will find\nSILL Sa prohibitively austere formalism, lacking most facilities needed to write realistic\nprograms (e.g. basic data types, pattern matching, etc.) and provided by a convenient and\nusable programming language like Rust. From an ergonomics standpoint alone it would\nbe unreasonably prohibitive for our embedding to forbid the use of Rust features such as\nfunctions, traits and enumerations, only for the sake of precisely mirroring SILL S. Moreover,\nto realize such an embedding we must be able to account for both SILL S’ linear session\ndiscipline (i.e. the linearcontext ∆) and shared session discipline (i.e. the structural context\nΨ) within Rust’s usage discipline. Since Rust’s typing discipline is essentially affine, its\ntreatment of variable usage is neither linear nor purely structural, and so both shared and\nlinear channels must be treated explicitly in the encoding.\nECOOP 202222:6 Ferrite: A Judgmental Embedding of Session Types in Rust\nTable 2Overview of SILL Rtypes and terms and their encoding in Ferrite. Note that SILL Ruses\nτ /triangleleft A Landτ ⊿ A Lfor shared channel output and input, resp., and /epsilon1for termination.\nType Terms ( SILL R)\nFerrite SILL R provider client\nInternalChoice<Row> ⊕{li:ALi} offerli;K casea{li:Ki}\nExternalChoice<Row> N{li:ALi} offer_choice{li:Ki} chooseali;K\nSendChannel<A,B> AL⊗BL send_channel_froma;K a←receive_channel_fromf a;K\nReceiveChannel<A,B> AL(BLa←receive_channel ;K send_channel_tof a;K\nSendValue<T,A> τ /triangleleft A L send_valuex;K x ←receive_value_fromax;K\nReceiveValue<T,A> τ ⊿ A Lx←receive_value ;K send_value_toax;K\nEnd /epsilon1 terminate wait a;K\nSharedToLinear<A> ↓S\nLAS detach_shared_session ;Ksrelease_shared_sessiona;Kl\nLinearToShared<A> ↑S\nLAL accept_shared_session ;Kla←acquire_shared_sessions;Kl\nThe two points above naturally lead us to the language SILL Ras a formal stepping stone\nbetween SILL Sand our embedding, Ferrite. SILL Ris, in its essence, a pragmatic extension of\nSILL Swith Rust (type and term) constructs, allowing us to intersperse Rust code with the\ncommunication primitives of SILL S. In SILL Rwe use the judgment\nΓ; ∆/turnstileleftexpr ::A,\ndenoting that expression exprhas session type A, using the sessions tracked by Γand∆.\nThis judgment differs from that of SILL Sin its context region Γand term expr, with the\nlatter permitting arbitrary Rust expressions in addition to SILL Scommunication primitives.\nWhereas SILL S’s structural context Ψexclusively tracks shared channels, SILL R’sΓtracks\nbothshared sessions (subject to weakening and contraction) and plain Rust (affine) variables.\nA shared channel type in both SILL Rand SILL Sis always of the form ↑S\nLA, so there is no\nconfusion among the affine and shared contents of Γ. As we discuss in Section 5.2, the\ndistinction between a plain Rust variable, which is treated as affine, and a shared channel,\nwhich is treated structurally, is modelled in Ferrite by making shared channels implement\nRust’s Clonetrait.\nTable 2 provides an overview of SILL Rtypes and terms and their Ferrite encoding. SILL R\ntypes stand in direct correspondence with SILL Stypes (see Table 1), apart from shared\nchannel output and input. The SILL Stypes for sending and receiving shared channels ( AS⊗AL\nandAS(AL) correspond to SILL Rtypes for sending and receiving values ( T /triangleleftAandT ⊿A,\nresp.), which support bothRust values and shared channels. Their typing rules are:\n(T/triangleleftR)\nΓ ; ∆/turnstileleftK::A\nΓ, x:τ; ∆/turnstileleftsend_valuex;K::τ /triangleleft A(T/triangleleftL)\nΓ,x:τ; ∆,a:A/turnstileleftK::B\nΓ ; ∆,a:τ ⊿ A/turnstileleftx←receive_value_froma;K::B\nRuleT/triangleleftRindicates that the value bound to variable xof typeτwill be sent, after which\nthe continuation Kwill execute, offering type A. Dually, rule T/triangleleftLstates that using such a\nprovider bound to awill bindxof typeτin continuation K, which must now use the channel\nbound toaaccording to A.\n3.2 Judgmental Embedding\nHaving introduced the SILL Rtyping judgment and illustrated some of its typing rules, we\ncan now clarify the idea behind our notion of judgmental embedding , which enables the RustR. F. Chen, S. Balzer, and B. Toninho 22:7\nTable 3Judgmental embedding of SILL Rin Ferrite.\nSILL R Ferrite Description\nΓ ;·/turnstileleftA Session<A> Typing judgment for top-level session (i.e. closed program).\nΓ ; ∆/turnstileleftA PartialSession<C, A> Typing judgment for partial session.\n∆ C: Context Linear context; explicitly encoded.\nΓ - Shared / Affine context; delegated to Rust.\nA A: Protocol Session type.\ncompiler to typecheck SILL Rprograms by encoding typing derivations as Rust programs.\nThe basic idea underlying this encoding can be schematically described as follows:\nΓ ; ∆ 2/turnstileleftcont ::A2\nΓ ; ∆ 1/turnstileleftexpr;cont ::A1fnexpr<...>\n( cont: PartialSession<C2, A2> )\n-> PartialSession<C1, A1>\nOn the left we show a SILL Rtyping rule and on the right its encoding in Ferrite. Ferrite\nencodes a SILL Rtyping judgment Γ; ∆/turnstileleftexpr ::Aas a value of Rust type PartialSession<\nC, A>, where Cencodes the linear context ∆and Athe session type A, standing for any of\nthe Ferrite types of Table 2. Ferrite then encodes a SILL Rtyping rule for an expression expr\nas a Rust function exprthat accepts a PartialSession<C2, A2> and returns a PartialSession\n<C1, A1>, where exprstands for any of the SILL Rterms of Table 2. The encoding makes\nuse ofcontinuation passing style (arising from the sequent calculus-based formulation of\nSILL R), with the return type being the conclusion of the rule and the argument type being\nits premise. Table 3 summarizes the judgmental embedding; Section 4.1 provides further\ndetails. Whereas Ferrite explicitly performs a type-level encoding of the linear context ∆, the\nrepresentation of the shared and affine context region Γis achieved through Rust’s normal\nbinding structure, with the obligation that shared channels implement Rust’s Clonetrait to\npermit contraction. To type a closed program, Ferrite defines the type Session<A> , which\nstands for a SILL Rjudgment with an empty linear context.\nAdopting a judgmental embedding technique for implementing a DSL delivers the benefits\nof proof-carrying code: the PartialSession<C1, A1> returned from a well-typed Ferrite expris\nthe typing derivation of the corresponding SILL Rterm. In case the SILL Rterm is a SILL S\nterm, its typing derivation certifies protocol adherence by virtue of the type safety proof of\nSILL S[1]. In case the SILL Rterm includes Rust code, its typing derivation certifies protocol\nadherence modulo the possibility of a panic raised by the Rust code.\n3.3 Recursive and Shared Session Types in Ferrite\nRust’s support for recursive types is limited to recursive struct definitions of a known size. To\ncircumvent this restriction and support arbitrary recursive session types, Ferrite introduces\na type-level fixed-point combinator Rec<F>to obtain the fixed point of a type function F.\nSince Rust lacks higher-kinded types such as Type→Type, we usedefunctionalization [40,51]\nby accepting any Rust type Fimplementing the trait RecAppwith a given associated type\nF::Applied , as shown below. Section 5.1 provides further details.\ntrait RecApp<X> { type Applied; }\nstruct Rec<F: RecApp<Rec<F>>> { unfold: Box<F::Applied> }\nRecursive types are also vital for encoding shared session types. In line with [ 3], we restrict\nshared session types to be recursive, making sure that a shared component is continuously\navailable. To guarantee type preservation, recursive session types must be strictly equi-\nsynchronizing [1, 3], requiring an acquired session to be released to the same type at which\nECOOP 202222:8 Ferrite: A Judgmental Embedding of Session Types in Rust\nit was previously acquired. Ferrite enforces this invariant by defining a specialized trait\nSharedRecApp which omits an implementation for End:\ntrait SharedRecApp<X> { type Applied; } trait SharedProtocol { ... }\nstruct SharedToLinear<F> { ... } struct SharedChannel<S: SharedProtocol> { ... }\nstruct LinearToShared<F: SharedRecApp<SharedToLinear<LinearToShared<F>>>> { ... }\nFerrite achieves safe communication for shared sessions by imposing an acquire-release\ndiscipline [ 1] on shared sessions, establishing a critical section for the linear portion of the\nprocess enclosed within the acquire and release. SharedChannel denotes the shared process\nrunning in the background, and clients with a reference to it can acquirean exclusive linear\nchannel to communicate with it. As long as the linear channel exists, the shared process\nis locked and cannot be acquired by any other client. With the strictly equi-synchronizing\nconstraint in place, the now linear process must eventually be released ( SharedToLinear ) back\nto the same shared session type at which it was previously acquired, giving turn to another\nclient waiting to acquire. Section 5.2 provides further details on the encoding.\n3.4 N-ary Choice and Linear Context\nFerrite implements n-ary choicesand linear typing contexts as extensible sumsandproducts\nof session types, resp. Ferrite uses heterogeneous lists [ 24] to annotate a list of session types\nof arbitrary length. The notation HList ![A0,A1,...,A N−1]denotes a heterogeneous list of\nN session types, with Aibeing the session type at the i-th position of the list. The HList !\nmacro acts as syntactic sugar for the heterogeneous list, which in its raw form is encoded as\n(A0,(A1,(...,(AN−1,())))). Ferrite uses the Rust tuple constructor (,)forHCons, and unit\n()forHNil. The heterogeneous list itself can be directly used to represent an n-ary product.\nUsing an associated type, the list can moreover be transformed into an n-ary sum.\nOne disadvantage of using heterogeneous lists is that its elements have to be addressed by\nposition rather than a programmer-chosen label. To recover labels for accessing list elements,\nwe use optics [ 37]. More precisely, Ferrite uses lenses[9] to access a channel in a linear\ncontext and prismsto select a branch of a choice. We further combine the optics abstraction\nwithde Bruijn levels and implement lenses and prisms using type level natural numbers.\nGiven an inductive trait definition of natural numbers as zero ( Z) and successor ( S<N>), a\nnatural number Nimplements the lens to access the N-th element in the linear context, and\nthe prism to access the N-th branch in a choice. Schematically, the lens encoding can be\ncaptured as follows:\nΓ ; ∆, ln:B2/turnstileleftK::A2\nΓ ; ∆, ln:B1/turnstileleftexprln;K::A1fnexpr<...>\n( l: N, cont: PartialSession<C1, A2> )\n-> PartialSession<C2, A1>\nwhere N: ContextLens<C1, B1, B2, Target=C2>\nThe index Namounts to the type of the variable lthat the programmer chooses as a name for\na channel in the linear context. Ferrite handles the mapping, supporting random access to\nprogrammer-named channels. Section 4.2 provides further details, including the support of\nhigher-order channels. Similarly, prisms allow choice selection in constructs such as offer_case\nto be encoded as follows:\nΓ; ∆/turnstileleftK::An\nΓ; ∆/turnstileleftoffer_caseln;K::⊕{..., ln:An,...}fnoffer_case<N, Row, C, A>\n( l: N, cont: PartialSession<C, A> )\n-> PartialSession<C, InternalChoice<Row>>\nwhere N: Prism<Row, Elem=A>, ...\nFerrite maps a choice label to a constant having the singleton value of a natural number\nN, which implements the prism to access the N-th branch of a choice. In addition to prisms,\nFerrite implements a version of extensible variants [30] to support polymorphic operationsR. F. Chen, S. Balzer, and B. Toninho 22:9\non arbitrary sums of session types representing choices. Finally, the define_choice! macro\nis used as a helper to export type aliases as programmer-friendly identifiers. Details are\nreported in Section 6.\n4 Ferrite – A Judgmental Embedding of SILL R\nHaving introduced some of the key concepts to the implementation of Ferrite, we now cover\nin detail the implementation of Ferrite’s core constructs, building up the knowledge required\nfor Section 5 and Section 6. Ferrite, like any other DSL, has to tackle the various technical\nchallenges encountered when embedding a DSL in a host language. In doing so, we take\ninspiration from the range of embedding techniques developed for Haskell and adjust them\nto the Rust setting. The lack of higher-kinded types, limited support of recursive types,\nand presence of weakening, in particular, make the development far from trivial. A more\nconceptual contribution of this work is thus to demonstrate how existing Rust features can\nbe combined to emulate many of the missing features that are beneficial to DSL embeddings\nand how to encode custom typing rules in Rust or any similarly expressive language. The\ntechniques described in this and subsequent sections also serve as a reference for embedding\nother DSLs in a host language like Rust.\n4.1 Encoding Typing Rules via Judgmental Embedding\nA distinguishing characteristic of Ferrite is its propositions as types approach, yielding a\ndirect correspondence between SILL Rnotions and their Ferrite encoding. This correspondence\nwas introduced in Section 3.2 (see Table 3) and we now discuss it in more detail. To this\nend, let’s consider the typing of value input. We remind the reader of Table 2 in Section 3,\nwhich provides a mapping between SILL Rand Ferrite session types. Interested readers can\nfind a corresponding mapping on the term level in Table 5 in the supplement.\nΓ,a:τ; ∆/turnstileleftK::A\nΓ ; ∆/turnstilelefta←receive_value ;K::τ ⊿ A(T⊿R)\nThe SILL Rright rule T⊿Rtypes the expression a←receive_value ;Kas the session type\nτ ⊿ Aand the continuation Kas the session type A, whereais now in scope with type\nτ. Following the schema hinted in Section 3.2, Ferrite encodes this rule as the function\nreceive_value , parameterized by a value type T(τ), a linear context C(∆), and an offered\nsession type A.\nfnreceive_value<T, C:Context, A:Protocol>(cont: impl FnOnce (T) -> PartialSession<C, A>)\n-> PartialSession<C, ReceiveValue<T, A>>\nThe function yields a value of type PartialSession<C, ReceiveValue<T, A>> , i.e. the con-\nclusion of the rule, given an (affine) closure of type T→PartialSession<C, A> , encoding the\npremise of the rule. Notably, Ferrite uses plain Rust binding (through function types) to\nencode the contents of Γ, as illustrated for the received value above. The use of a closure\nreveals the continuation-passing-style of the encoding, where the received value of type Tis\npassed to the continuation closure. The affine closure implements the FnOncetrait, ensuring\nthat it can only be called once.\nThe type PartialSession is a core construct of Ferrite that enables the judgmental\nembedding of SILL R. A Rust value of type PartialSession<C, A> represents a Ferrite program\nthat guarantees linear usage of session type channels in the linear context Cand offers the\nlinear session type A, corresponding to the SILL Rtyping judgment Γ; ∆/turnstileleftexpr ::A. The type\nECOOP 202222:10 Ferrite: A Judgmental Embedding of Session Types in Rust\nparameters Cand Aare constrained to implement the traits Contextand Protocol – two other\nFerrite constructs representing a linear context and linear session type, resp.:\ntrait Context { ... } trait Protocol { ... }\nstruct PartialSession<C: Context, A: Protocol> { ... }\nFor each SILL Rsession type, Ferrite defines a corresponding Rust struct that implements\nthe trait Protocol, yielding the listing shown in Table 2. Implementations for /epsilon1(End) and\nτ ⊿ A(ReceiveValue<T, A> ) are shown below. When a session type is nested within another\nsession type, such as in the case of ReceiveValue<T, A> , the constraint to implement Protocol\nis propagated to the inner session type, requiring Ato also implement Protocol:\nstruct End { ... } struct ReceiveValue<T, A> { ... }\nimpl Protocol for End { ... }\nimpl <A: Protocol> Protocol for ReceiveValue<T, A> { ... }\nThus, while Ferrite delegates the handling of the shared/structural context Γto Rust,\nthe encoding of the linear context ∆is explicit. Being affine, the Rust type system permits\nweakening, a structural property rejected by linear logic. Ferrite encodes a linear context as a\nheterogeneous (type-level) list [ 24] of the form HList![A 0, A1, ..., A N−1], with all its type\nelements Aiimplementing Protocol. Internally, the HListmacro desugars the type-level list\ninto a nested tuple (A0, (A 1, (..., (A N−1, ())))) . The unit type ()is used as the empty\nlist ( HNil) and the tuple constructor (,)is used as the HConsconstructor. The implementation\nforContextis defined inductively as follows:\nimpl Context for () { ... } impl <A: Protocol, C: Context> Context for (A, C) { ... }\nTo represent a closed program, i.e. a program without free channel variables, we define a\ntype alias Session<A> forPartialSession<C, A> , with Crestricted to the empty context:\ntype Session<A> = PartialSession<(), A>;\nA complete session type program in Ferrite is thus of type Session<A> and amounts to\ntheSILL Rtyping derivation proving that the program adheres to the defined protocol. Below\nwe show a “hello world”-style program in Ferrite:\nlet hello_provider = receive_value(|name| {\nprintln! (\"Hello, {}\", name); terminate() });\nThe Ferrite program hello_provider has an inferred Rust type Session<ReceiveValue<String\n, End>>. It offers the type ReceiveValue<String, End> by first receiving a string value using\nreceive_value , binding it to namein the continuation closure. Upon receiving the name string,\nIt prints out the name with a \"Hello\"greeting, and terminates using terminate() .\n4.2 Manipulating the Linear Context\nContext Lenses\nThe use of a type-level list to encode the linear context has the advantage of allowing contexts\nof arbitrary length. However, the list imposes an order on the context’s elements, disallowing\nexchange. To allow exchange, we make use of the concept of lenses[9] to define a ContextLens\ntrait, which is implemented using type-level natural numbers.\n#[derive(Copy)] struct Z; #[derive(Copy)] struct S<N>(PhantomData<N>);\ntrait ContextLens<C: Context, A1: Protocol, A2: Protocol> { type Target: Context; ... }\nThe ContextLens trait defines the read and update operations on a linear context, such\nthat given a sourcecontext C = HList![..., A N, ...], the source element of interest, ANR. F. Chen, S. Balzer, and B. Toninho 22:11\nat position N, can be updated to the target element Bto form the targetcontext Target =\nHList![..., B, ...] , with the remaining elements unchanged. We use natural numbers to\ninductively implement ContextLens at each position in the linear context, such that it satisfies\nall constraints of the form:\nN: ContextLens<HList![..., A N, ...], A N, B, Target=HList![..., B, ...]>\nThe implementation of natural numbers as context lenses is done by first considering the\nbase case, with Zused to access the first element of any non-empty linear context:\nimpl <A1: Protocol, A2: Protocol, C: Context> ContextLens<(A1, C), A1, A2>\nfor Z { type Target = ( A2, C ); ... }\nimpl <A1: Protocol, A2: Protocol, B: Protocol, C: Context, N: ContextLens<C, A1, A2>>\nContextLens <(B, C), A1, A2> for S<N> { type Target = (B, N::Target); ... }\nIn the inductive case, for any natural number Nimplementing the context lens for a context\nHList![A 0, ..., A N, ...], it’s successor S<Z>implements the context lens for HList![A−1,\nA0, ..., A N, ...], with a new element A−1appended to the head of the linear context.\nUsing context lenses, we can encode the SILL Rleft rule T⊿Lshown below, which types sending\nan ambient value xto a channel ain the linear context that expects to receive a value.\nΓ ; ∆,a:A/turnstileleftK::B\nΓ, x:τ; ∆, a:τ ⊿ A/turnstileleftsend_value_toax;K::B(T⊿L)\nIn Ferrite, T⊿Lis implemented as the function send_value_to , which uses a context lens Nto\nsend a value of type Tto the N-th channel in the linear context C1. This requires the N-th\nchannel to have type ReceiveValue<T,A> . A continuation contis then given with the linear\ncontext C2, which has the N-th channel updated to type A.\nfnsend_value_to<N, T, C1: Context, C2: Context, A: Protocol, B: Protocol>\n( n: N, x: T, cont: PartialSession<C2, B> ) -> PartialSession <C1, B>\nwhere N: ContextLens<C1, ReceiveValue<T, A>, A, Target=C2>\nChannel Removal\nThe above definition of a context lens is suited for updating channel types in a context.\nHowever, we have not addressed how channels can be removed oraddedto the linear context.\nThese operations are required to implement session termination and higher-order channel\nconstructs such as ⊗and(. To support channel removal, we introduce a special Empty\nelement to denote the absenceof a channel at a given position in the linear context:\nstruct Empty; trait Slot { ... }\nimpl Slot for Empty { ... } impl <A: Protocol> Slot for A { ... }\nTo allow Emptyto be present in a linear context, we introduce a new Slottrait and make\nboth Emptyand Protocol implement Slot. The original definition of Contextis then updated\nto allow types that implement Slotinstead of Protocol.\nΓ ; ∆/turnstileleftK::A\nΓ ; ∆, a:/epsilon1/turnstileleftwaita;K::A(T1L)Γ ;·/turnstileleftterminate ; ::/epsilon1(T1R)\nUsing Empty, it is straightforward to implement SILL R’s session termination. Rule T1Lis\nencoded via a context lens that replaces a channel of session type Endwith the Emptyslot. The\nfunction waitshown below does not really remove a slot from a linear context, but merely\nreplaces the slot with Empty. The use of Emptyis necessary, because we want to preserve the\nposition of channels in a linear context in order for the context lens for a channel to work\nacross continuations.\nECOOP 202222:12 Ferrite: A Judgmental Embedding of Session Types in Rust\nfnwait<C1: Context, C2: Context, A: Protocol, N>\n( n: N, cont: PartialSession<C2, A> ) -> PartialSession<C1, A>\nwhere N: ContextLens<C1, End, Empty, Target=C2>\nWith Emptyintroduced, an empty linear context may now contain any number of Empty\nslots, such as HList![Empty, Empty] . We introduce a new EmptyContext trait to abstract\nover the different forms of empty linear contexts and provide an inductive definition as its\nimplementation:\ntrait EmptyContext: Context { ... } impl EmptyContext for () { ... }\nimpl <C: EmptyContext> EmptyContext for (Empty, C) { ... }\nGiven the empty list ()as the base case, the inductive case (Empty, C) is an empty linear\ncontext, if Cis also an empty linear context. Using the definition of an empty context, the\nSILL Rright rule T1Rcan then be easily encoded as the function terminate , which works\ngenerically for all contexts that implement EmptyContext as shown below:\nfnterminate<C: EmptyContext>() -> PartialSession<C, End>\nChannel Addition\nThe Ferrite function waitremoves a channel from the linear context by replacing it with\nEmpty. Dually, the function receive_channel , adds a new channel to the linear context. The\nSILL RruleT(Rfor channel input is shown below. It binds the received channel of session\ntypeAto the channel variable aand adds it to the linear context ∆of the continuation.\nΓ ; ∆,a:A/turnstileleftK::B\nΓ ; ∆/turnstilelefta←receive_channel ;K::A(B(T(R)\nTo encode T(R, an append operation on contexts is defined via the AppendContext trait:\ntrait AppendContext<C: Context>: Context { type Appended: Context; ... }\nimpl <C: Context> AppendContext<C> for () { type Appended = C; ... }\nimpl <A: Slot, C1: Context, C2: Context, C3: Context> AppendContext<C2>\nfor (A, C1) where C1: AppendContext<C2, Appended=C3> { type Appended = (A, C3); ... }\nThe AppendContext trait is parameterized by a linear context Cand an associated type\nAppended. If a linear context C1implements the trait AppendContext<C2> , it means that context\nC2canbeappendedto C1, with C3=C1::Appended beingtheresultoftheappendoperation. The\nimplementation of AppendContext is defined inductively, with the empty list ()implementing\nthe base case and the cons cell (A, C)implementing the inductive case.\nUsing AppendContext , a channel Bcan be appended to the end of a linear context C, ifC\nimplements AppendContext<HList![B]> . The new linear context after the append operation\nis given in the associated type C::Appended . We then observe that the position of channel\nBinC::Appended is the same as the length of the original linear context C. In other words,\nthe context lens for channel BinC::Appended can be generated by obtaining the length of C.\nIn Ferrite, the length operation is implemented by adding an associated type Lengthto the\nContexttrait. The implementation of Contextfor()and (A, C)is updated correspondingly.\ntrait Context { type Length; ... } impl Context for () { type Length = Z; ... }\nimpl <A: Slot, C: Context> Context for (A, C) { type Length = S<C::Length>; ... }\nThe SILL Rright rule T(Ris then encoded as follows:\nfnreceive_channel<A: Protocol, B: Protocol, C1: Context, C2: Context>(\ncont: impl FnOnce (C1::Length) -> PartialSession<C2, B>) ->\nPartialSession<C1, ReceiveChannel<A, B>> where C1: AppendContext<(A, ()), Appended=C2>R. F. Chen, S. Balzer, and B. Toninho 22:13\nThe function receive_channel is parameterized by a linear context C1implementing\nAppendContext to append the session type AtoC1. The continuation argument contis a closure\nthat is given a context lens C::Length , and returns a PartialSession with C2=C1::Appended as\nits linear context. The function returns a PartialSession with linear context C1, offering\nsession type ReceiveChannel<A, B> .\nIt is worth noting that in the type signature of receive_channel , the type C1::Length is\nnot shown to have any ContextLens implementation. However when C1::Length is instantiated\ninto the concrete types Z,S<Z>, etc in the continuation body, Rust will use the appropriate\nimplementations of ContextLens so that they can be used to access the appended channel in\nthe linear context.\nThe use of receive_channel is illustrated with the hello_client example below:\nlet hello_client = receive_channel(|a| {\nsend_value_to(a, \"Alice\".to_string(), wait(a, terminate())) });\nThe hello_client program is inferred to have the Rust type Session<ReceiveChannel<\nReceiveValue<String, End>, End>> . It is written to communicate with the hello_provider pro-\ngram defined earlier in Section 4.1. The interaction is achieved by having hello_client offering\nthe session type ReceiveChannel<ReceiveValue<String, End>, End> . In its body, hello_client\nuses receive_channel toreceivechannel aoftype ReceiveValue<String, End> from hello_provider\n. The continuation closure is given an argument a:Z, denoting the context lens generated by\nreceive_channel for accessing the received channel in the linear context. The context lens a:Z\nis then used for sending a string value, after which we waitforhello_provider to terminate.\nWe note that the type Zof channel a(i.e. the channel position in the context) is automatically\ninferred by Rust and not exposed to the user.\n4.3 Communication\nAt this point we have defined the necessary constructs to build and typecheck both\nhello_provider and hello_client , but the two are separate Ferrite programs that are yet\nto be linked with each other and executed.\nΓ ; ∆ 1/turnstileleftK1::A Γ ; ∆ 2,a:A/turnstileleftK2::B\nΓ ; ∆ 1,∆2/turnstilelefta←cutK1;K2::B(T-cut)Γ ;a:A/turnstileleftforwarda::A(T-fwd)\nInSILL R, rule T-cutallows two session-typed programs to run in parallel, with the\nchannel offered by K1added to the linear context of program K2. Together with the forward\nruleT-fwd, we can use cut twice to run both hello_provider and hello_client in parallel,\nand have a third program that sends the channel offered by hello_provider tohello_client .\nThe program hello_main would have the following pseudo code in SILL R:\nhello_main :/epsilon1= f←cuthello_client ;\na←cuthello_provider ;\nsend_channel_tof a;\nforward f\nTo implement cutin Ferrite, we need a way to split a linear context C=∆1,∆2into two\nsub-contexts C1=∆1and C2=∆2so that they can be passed to the respective continuations.\nMoreover, since Ferrite programs use context lenses to access channels, the ordering of\nchannels inside C1and C2must be preserved. We can preserve the ordering by replacing the\ncorresponding slots with Emptyduring the splitting. Ferrite defines the SplitContext trait to\nimplement the splitting as follows:\nenum L {} enum R {}\ntrait SplitContext<C: Context> { type Left: Context; type Right: Context; ... }\nECOOP 202222:14 Ferrite: A Judgmental Embedding of Session Types in Rust\nWe first define two (uninhabited) marker types Land R. We then use type-level lists\nconsisting of elements Land Rto implement the SplitContext trait for a given linear context C.\nThe SplitContext implementation contains the associated types Leftand Right, representing\nthe contexts C1and C2after splitting. As an example, the type HList![L, R, L] would\nimplement SplitContext<HList![A1, A2, A3]> for any slot A1,A2and A3, with the associated\ntype Leftbeing HList![A1, Empty, A3] and Rightbeing HList![Empty, A2, Empty] . We omit\nthe implementation details of SplitContext for brevity. Using SplitContext , the function cut\ncan be implemented as follows:\nfncut<XS, C: Context, C1: Context, C2: Context, C3: Context, A: Protocol, B: Protocol>\n( cont1: PartialSession<C1, A>,\ncont2: impl FnOnce (C2::Length) -> PartialSession<C3, B> ) -> PartialSession<C, B>\nwhere XS: SplitContext<C, Left=C1, Right=C2>, C2: AppendContext<HList![A], Appended=C3>\nThe function cutworks by using the heterogeneous list XSthat implements SplitContext\nto split a linear context Cinto C1and C2. To pass on the channel Athat is offered by cont1to\ncont2,cutuses a similar technique to receive_channel to append the channel Ato the end of\nC2, resulting in C3. Using cut, we can write hello_main in Ferrite as follows:\nlet hello_main: Session<End> =\ncut::<HList![]>(hello_client, |f| { cut::<HList![R]>(hello_provider, |a| {\nsend_channel_to(f, a, forward(f)) }) });\nDue to ambiguous instances for SplitContext , the type parameter XShas to be annotated\nexplicitly for Rust to know in which context a channel should be placed. In the first use of\ncut, the context is empty, so we call cutwith the empty list HList![]. We pass hello_client\nas the first continuation to run in parallel, and name the channel offered by hello_client as\nf. In the second use of cut, the linear context would be HList![ReceiveValue<String, End>] ,\nwith one channel f. We then have cutmove fto the right side using HList![R] . On the left\ncontinuation, we have hello_provider run in parallel, and name the offered channel as a. In\nthe right continuation, we use send_channel_to to send channel atof. Finally, we forward\nthe continuation of f, which now has type End.\nAlthough cutprovides the primitive way for Ferrite programs to communicate, its use\ncan be cumbersome and requires a lot of boilerplate. For simplicity, Ferrite provides a\nspecialized construct apply_channel that abstracts over the common pattern usage of cut\ndescribed earlier. apply_channel takes a client program foffering session type ReceiveChannel\n<A, B>and a provider program aoffering session type A, and sends atofusing cut. The\nuse of apply_channel is akin to regular function application, making it more intuitive for\nprogrammers to use:\nfnapply_channel<A: Protocol, B: Protocol>(\nf: Session<ReceiveChannel<A, B>>, a: Session<A>) -> Session<B>\n4.4 Executing Ferrite Programs\nTo actually executea Ferrite program, the program must offer some specific session types. In\nthe simplest case, Ferrite provides the function run_session for running a top-level Ferrite\nprogram offering End, with an empty linear context:\nasync fn run_session(session: Session<End>) { ... }\nFunction run_session executes the session asynchronously using Rust’s async/await infra-\nstructure. Internally, the struct PartialSession<C, A> implements the dynamic semantics of\nthe Ferrite program, which is only accessible by public functions such as run_session . Ferrite\ncurrently uses the tokio[46] runtime for asynchronous execution, as well as the one shotR. F. Chen, S. Balzer, and B. Toninho 22:15\nchannels from tokio::sync::oneshot to implement the low-level communication of Ferrite\nchannels.\nSince run_session accepts an argument of type Session<End> , this means that programmers\nmust first use cutorapply_channel to fully link partial Ferrite programs with free channel\nvariables, or Ferrite programs that offer session types other than Endbefore they can be\nexecuted. This restriction ensures that all linear channels created by a Ferrite program\nare consumed. For example, the programs hello_provider and hello_client cannot be ex-\necuted individually, but the linked program resulting from composing hello_provider with\nhello_client can be executed:\nasync fn main() { run_session(apply_channel(hello_client, hello_provider)).await; }\nWe omit the implementation details of the dynamics of Ferrite, which use low-level\nprimitives such as Rust channels while carefully ensuring that the requirements and invariants\nof session types are satisfied. Interested readers can find more details in Appendix B.\n5 Recursive and Shared Session Types\nManyrealworldapplications, suchaswebservicesandinstantmessaging, implementprotocols\nthat are recursive in nature. As a result, it is essential for Ferrite to support recursive session\ntypes. In this section, we first report on Rust’s limited support for recursive types and how\nFerrite addresses this limitation. We then discuss how Ferrite encodes sharedsession types,\nwhich are recursive.\n5.1 Recursive Session Types\nConsider a simple example of a counter session type, which sends an infinite stream of integer\nvalues, incrementing each by one. To write a Ferrite program that offers such a session type,\nwe may attempt to define the counter session type as follows:\ntype Counter = SendValue<u64, Counter>;\nIf we try to use the type definition above, the compiler will emit the error \"cycle detected\nwhen processing Counter\". The problem with the above definition is that it is a directly\nself-referential type alias, which is not supported in Rust. Rust imposes various restrictions\non the legal forms of recursive types to ensure that the memory layout of data is known at\ncompile-time.\nType-Level Fixed Points\nTo address this limitation, we implement type-level fixed points using defunctionalization [40,\n51]. This is done by introducing a RecApptrait that is implemented by defunctionalized types\nthat can be “applied” with a type parameter:\ntrait RecApp<X> { type Applied; } type AppRec<F, X> = <F asRecApp<X>>::Applied;\nstruct Rec<F: RecApp<Rec<F>>> { unfold: Box<AppRec<F, Rec<F>>> }\nThe RecApptrait is parameterized by a type X, which serves as the type argument to\nbe applied to. This makes it possible for a Rust type Fthat implements RecAppto act\nas if it has the higher-kinded type Type→Type, and be “applied” to type X. We define\na type alias AppRec<F, X> to refer to the associated type Appliedresulting from “applying”\nFtoXvia RecApp. Using RecApp, we can now define a type-level recursor Recas a struct\nparameterized by a type Fthat implements RecApp<Rec<F>> . The body of Reccontains a boxed\nvalue Box<AppRec<F, RecApp<Rec<F>>>> to make it have a fixed size in Rust.\nECOOP 202222:16 Ferrite: A Judgmental Embedding of Session Types in Rust\nFerrite implements RecAppfor all Protocol types, with the type Zused to denote the\nrecursion point. With that, the example Countertype would be defined as follows:\ntype Counter = Rec<SendValue<u64, Z>>;\nThe type Rec<SendValue<T, Z>> is unfolded into SendValue<T, Rec<SendValue<T, Z>>> . This\nis achieved by having the following generic implementations of RecAppforSendValue and Z:\nimpl <X> RecApp<X> for Z { type Applied = X; }\nimpl <X, T, A: RecApp<X>> RecApp <X> for SendValue <T, A> {\ntype Applied = SendValue<T, AppRec<A, X>; }\nInside RecApp,Zsimply replaces itself with the type argument X.SendValue<T, A> delegates\nthe type application of XtoA, provided that the session type Aalso implements RecAppforX.\nThe session type Counteris iso-recursive, as the rolled type Rec<SendValue<u64, Z>> and\nthe folded type SendValue<u64, Rec<SendValue<u64, Z>> are considered distinct types in Rust.\nAs a result, Ferrite provides the constructs fix_session and unfix_session for converting\nbetween the rolled and unfolded versions of a recursive session type.\nNested Recursive Session Types\nThe use of RecAppis akin to emulating the higher-kinded type (HKT) Type→Typein Rust.\nAs of this writing, HKTs are only available in the nightly (unstable) version of Rust through\ngeneric associated types . However even with support for HKTs, our defunctionalization-based\napproach via RecAppallows us to generalize to nestedrecursive types.\nTo account for a recursive type with multiple recursion points, we introduce a recursion\ncontext Ras a type-level list of elements (c.f. the linear context of Section 4.2). The type-level\nnatural numbers Z,S<Z>, etc. are now used as de Bruijn indices to unfold to the elements\nin the recursion context. The type-level fixed point combinator Recis redefined as RecX,\ncontaining the recursion context:\nstruct RecX<R, F: RecApp<(RecX<R, F>, R)>> { unfix: Box<AppRec<F, (RecX<R, F>, R)>> }\ntype Rec<F> = RecX<(), F>;\nimpl <R, F: RecApp<(RecX<R, F>, R)>> RecApp<R> for RecX<(), F> {\ntype Applied = RecX<R, F>; }\nA recursive session type is defined starting with an empty recursion context. Since nested\nrecursive session types allow a RecXto be embedded inside another RecX, we have RecXalso\nimplement RecApp, provided it has an empty recursion context. When unfolded from another\nrecursion context R,RecXsimply saves Ras its own recursion context and does not unfold\nfurther in F. The inner type Fis only unfolded once with the full recursion context after all\nsurrounding RecXtypes are unfolded.\nThe recursive marker Zis modified to unfold to the first element of the recursion context.\nWe then implement S<N>to unfold to the (N+1)-th position in the recursion context:\nimpl <A, R> RecApp<(A, R)> for Z { type Applied = A; }\nimpl <A, R, N: RecApp<R>> RecApp<(A, R)> for S<N> { type Applied = N::Applied; }\n5.2 Shared Session Types\nIn the previous section we explored a recursive session type Counter, which is defined using\nRecand Z. Since Counteris defined as a linear session type, it cannot be shared among\nmultiple clients. Shared communication, however, is essential to implement many practical\napplications. For instance, we may want to implement a simple counter web-service, to send\na unique count for each request. To support such shared communication, we introduce shared\nsession types in Ferrite, enabling safeshared communication in the presence multiple clients.R. F. Chen, S. Balzer, and B. Toninho 22:17\nShared Session Types in Ferrite\nAs introduced in Section 2, the SILL S(and SILL R) notion of shared session types is recursive\nin nature, as a shared session type must offer the same linear critical section to all clients\nthat acquire a shared resource. For instance, a shared version of the Countertype in SILL Ris:\nSharedCounter =↑S\nLInt/triangleleft↓S\nLSharedCounter\nThe linear portion of SharedCounter in between↑S\nL(acquire) and↓S\nL(release) amounts to\na critical section. When a SharedCounter isacquired, it offers a linear session type Int/triangleleft\n↓S\nLSharedCounter , willing to send an integer value, after which it must be releasedto become\navailable again as a SharedCounter to the next client.\nThe recursive aspect of shared session types in SILL Rmeans that we can reuse the\nimplementation technique that we use for recursive session types. The type SharedCounter\ncan be defined in Ferrite as follows:\ntype SharedCounter = LinearToShared<SendValue<u64, Release>>;\nCompared to linear recursive session types, the main difference is that instead of using\nRec, a shared session type is defined using the LinearToShared construct. This corresponds\nto↑S\nLinSILL R, with the inner type SendValue<u64, Release> corresponding to the linear\nportion of the shared session type. At the point of recursion, the type Releaseis used in\nplace of↓S\nLSharedCounter . As a result, the type LinearToShared<SendValue<u64, Release>> is\nunfolded into SendValue<u64, SharedToLinear<LinearToShared<SendValue<u64, Release>>>> after\nbeing acquired. Type unfolding is implemented as follows:\ntrait SharedRecApp<X> { type Applied; } trait SharedProtocol { ... }\nstruct SharedToLinear<F> { ... } struct LinearToShared<F> { ... }\nimpl <F> Protocol for SharedToLinear<LinearToShared<F>>\nwhere F: SharedRecApp<SharedToLinear<LinearToShared<F>>> { ... }\nimpl <F> SharedProtocol for LinearToShared<F>\nwhere F: SharedRecApp<SharedToLinear<LinearToShared<F>>> { ... }\nThe struct LinearToShared is parameterized by a linear session type Fthat implements the\ntrait SharedRecApp<SharedToLinear<LinearToShared<F>>> . It uses the SharedRecApp trait instead\nof the RecApptrait to ensure that the session type is strictly equi-synchronizing [3], requiring\nan acquired session to be released to the same type at which it was previously acquired.\nFerrite enforces this requirement by omitting an implementation of SharedRecApp forEnd,\nruling out invalid shared session types such as LinearToShared<SendValue<u64, End>> . We\nnote that the type argument to F’sSharedRecApp is another struct SharedToLinear , which\ncorresponds to↓S\nLinSILL R. A SharedProtocol trait is also defined to identify shared session\ntypes, i.e. LinearToShared .\nOnce a shared process is started, a shared channel is created to allow multiple clients to\naccess the shared process through the use of shared channel:\nstruct SharedChannel<S: SharedProtocol> { ... }\nimpl <S> Clone for SharedChannel<S> { ... };\nThe code above shows the definition of the SharedChannel struct. Unlike linear channels,\nshared channels follow structural typing, i.e. they can be weakened or contracted. This\nmeans that we can delegate the handling of shared channels to Rust, given that SharedChannel\nimplements Rust’s Clonetrait to allow contraction. Whereas SILL Sprovides explicit constructs\nfor sending and receiving shared channels, Ferrite’s shared channels can be sent as regular\nRust values using Send/ReceiveValue .\nOn the client side, a SharedChannel serves as an endpoint for interacting with a shared\nprocess running in parallel. To start the execution of such a shared process, a corresponding\nECOOP 202222:18 Ferrite: A Judgmental Embedding of Session Types in Rust\nFerrite program has to be defined and executed. Similar to PartialSession , we define\nSharedSession as shown below to represent such a shared Ferrite program.\nstruct SharedSession<S: SharedProtocol> { ... }\nfnrun_shared_session<S: SharedProtocol>(session: SharedSession<S>) -> SharedChannel<S>\nJustas PartialSession encodeslinearFerriteprogramswithoutexecutingthem, SharedSession\nencodes shared Ferrite programs without executing them. Since SharedSession does not\nimplement the Clonetrait, the shared Ferrite program is itself affine and cannot be shared.\nTo enable sharing, the shared Ferrite program must first be executed with run_shared_session .\nThe function run_shared_session takes a shared Ferrite program of type SharedSession<S> and\nstarts it in the background as a shared process. Then, in parallel, the shared channel of\ntype SharedChannel<S> is returned to the caller, which can then be sent to multiple clients for\naccess to the shared process.\nThe details of each shared Ferrite construct are described in Appendix A.2.11. Below\nwe demonstrate how a program with a shared session can be defined and used by multiple\nclients:\ntype SharedCounter = LinearToShared<SendValue<u64, Release>>;\nfncounter_producer(current_count: u64) -> SharedSession<SharedCounter> {\naccept_shared_session( async move {\nsend_value(current_count, detach_shared_session(\ncounter_producer(current_count + 1))) }) }\nfncounter_client(counter: SharedChannel<SharedCounter>) -> Session<End> {\nacquire_shared_session(counter, move | chan | {\nreceive_value_from(chan, move | count | {\nprintln! (\"received count: {}\", count);\nrelease_shared_session(chan, terminate()) }) }) }\nThe recursive function counter_producer creates a SharedSession program that, when\nexecuted, offers a shared channel of session type SharedCounter . On the provider side, a\nshared session is defined using the accept_shared_session construct, with a continuation given\nas an async thunk that is executed when a client acquires the shared session and enters\nthe linear critical section (of type SendValue<u64, SharedToLinear<SharedCounter>> ). Inside the\nclosure, the producer uses send_value to send the current count to the client and then uses\ndetach_shared_session to exit the linear critical section. The construct detach_shared_session\noffers the linear session type SharedToLinear<SharedCounter> and expects a continuation that\noffers the shared session type SharedCounter to serve the next client. We generate the\ncontinuation by recursively calling the counter_producer function.\nThe counter_client function takes a shared channel of session type SharedCounter and\nreturns a session type program that acquires the shared channel and prints the received\ncount value to the terminal. A linear Ferrite program can acquire a shared session using\nthe acquire_shared_session construct, which accepts a SharedChannel object and adds the\nacquired linear channel to the linear context. In this case, the continuation closure is given\nthe context lens Z, which provides access to the linear channel of session type SendValue\n<u64, SharedToLinear<SharedCounter>> in the first slot of the linear context. It then uses\nreceive_value_from to receive the value sent by the shared provider and then prints the value.\nOn the client side, the linear session of type SharedToLinear<SharedCounter> must be released\nusing the release_shared_session construct. After releasing the shared session, other clients\nwill then be able to acquire the shared session.\nasync fn main () {\nlet counter1: SharedChannel<SharedCounter> = run_shared_session(counter_producer(0));\nlet counter2 = counter1.clone();\nlet child1 = task::spawn( async move { run_session(counter_client(counter1)).await; });R. F. Chen, S. Balzer, and B. Toninho 22:19\nlet child2 = task::spawn( async move { run_session(counter_client(counter2)).await; });\njoin!(child1, child2).await; }\nTo illustrate a use of SharedCounter , we have a mainfunction that initializes a shared produ-\ncer with an initial value of 0 and then runs the shared provider using the run_shared_session\nconstruct. The returned SharedChannel is then cloned, making the shared counter accessible\nvia aliases counter1 and counter2. It then uses task::spawn to spawn two async tasks that run\ncounter_client twice. A key observation is that multiple Ferrite programs that are executed\nindependently can access the same shared producer through a reference to the shared channel.\n6 N-ary Choice\nSession types support internal andexternal choice, leaving the choice among several options\nto the provider or the client, resp. (see Table 2). When restricted to binary choice, the\nimplementation is relatively straightforward, as shown below by the two right rules for\ninternal choice in SILL R. The offer_leftand offer_rightconstructs allow a provider to offer\nan internal choice A⊕Bby offering either AorB, resp.\nΓ ; ∆/turnstileleftK::A\nΓ ; ∆/turnstileleftoffer_left;K::A⊕B(T⊕2LR)Γ ; ∆/turnstileleftK::B\nΓ ; ∆/turnstileleftoffer_right;K::A⊕B(T⊕2RR)\nIt is straightforward to implement the two versions of the right rules by writing the two\nrespective functions offer_left and offer_right :\nfnoffer_left<C: Context, A: Protocol, B: Protocol>\n( cont: PartialSession<C, A> ) -> PartialSession<C, InternalChoice2<A, B>>\nfnoffer_right < C: Context, A: Protocol, B: Protocol >\n( cont: PartialSession<C, B> ) -> PartialSession<C, InternalChoice2<A, B>>\nHowever, this approach does not scale if we want to generalize choice beyond two options.\nTo support N-ary choice, the functions would have to be explicitly reimplemented N times.\nInstead, we implement a single offer_case function which allows selection from n-ary branches.\n6.1 Prisms\nIn Section 4.2, we explored heterogeneous list to encode the linear context, i.e. products of\nsession types of arbitrary lengths. We then implemented context lensesto access and update\nindividual channels in the linear context. Observing that n-ary choices can be encoded as\nsumsof session types, we now use prismsto implement the selection of an arbitrary-length\nbranch. Instead of having a binary choice type InternalChoice2<A, B> , we can define an n-ary\nchoice type InternalChoice<HList![...]> , with InternalChoice<HList![A, B]> being the special\ncase of a binary choice. To select a branch out of the heterogeneous list, we define the Prism\ntrait as follows:\ntrait Prism<Row> { type Elem; ... }\nimpl <A, R> Prism<(A, R)> for Z { type Elem = A; ... };\nimpl <N, A, R> Prism<(A, R)> for S<N> where N: Prism<R> { type Elem = N::Elem; ... }\nThe Prismtrait is parameterized over a row type Row=HList![...] , with the associated\ntype Elembeing the element type that has been selected from the list by the prism. We then\ninductively implement Prismusing type-level natural numbers, with the number Nused for\nselecting the N-th element of the heterogeneous list. The definition of Prismis similar to\nContextLens , with the main difference being that we only need Prismto support extraction\nand injections operations on the sum types that are derived from the heterogeneous list.\nUsing Prism, a generalized offer_case function is implemented as follows:\nECOOP 202222:20 Ferrite: A Judgmental Embedding of Session Types in Rust\nfnoffer_case<C: Context, A: Protocol, Row, N: Prism<Row, Elem=A>>\n(n: N, cont: PartialSession<C, A>) -> PartialSession<C, InternalChoice<Row>>\nThe function accepts a natural number Nas the first parameter, which acts as the prism\nfor selecting a session type ANout of the row type Row=HList![..., A N, ...]. Through the\nassociated type A=N::Elem ,offer_case forces the programmer to provide a continuation that\noffers the chosen session type A.\n6.2 Binary Branching\nWhile offer_case is a step in the right direction, it only allows the selection of a specific\nchoice, but not the provision of allpossible choices. The latter, however, is necessary to\nencode the SILL Rleft rule of internal choice and right rule of external choice. To illustrate\nthe problem, let’s consider the right rule of a binary external choice, TN2R:\nΓ ; ∆/turnstileleftKl::A Γ ; ∆/turnstileleftKr::B\nΓ ; ∆/turnstileleftoffer_choice_2KlKr::ANB(TN2R)\nThe offer_choice_2 construct has two possible continuations KlandKr, with only one of\nthem being executed, depending on the selection by the client. In a naive implementation,\nwe can define the construct to accept two continuations as follows:\nfnoffer_choice_2<C: Context, A: Protocol, B: Protocol>\n( cont_left: PartialSession<C, A>, cont_right: PartialSession<C, B> )\n-> PartialSession<C, ExternalChoice2<A, B>>\nWhile the above implementation works in most languages, it is not adequate in Rust.\nSince Rust’s type system is affine, variables can only be captured by one of the continuation\nclosures, but not both. As far as the compiler is aware, both closures can potentially be\ncalled, and we cannot state that one of the branches is guaranteed to never run.\nIn order for offer_choice_2 to work in Rust’s affine typing, it has to accept only one\ncontinuation closure and have it return either PartialSession<C, A> orPartialSession<C, B> ,\ndepending on the client’s selection. It is not as straightforward to express such behavior as a\nvalid type in a language like Rust. If Rust supported dependent types, offer_choice_2 could\nbe implemented along the following lines:\nfnoffer_choice_2<C: Context, A: Protocol, B: Protocol>\n( cont: impl FnOnce (first: bool) ->\niffirst { PartialSession<C, A> } else { PartialSession<C, B> } )\n-> PartialSession<C, ExternalChoice2<A, B>>\nThat is, the return type of the contclosure depends on the whether the valueof the first\nargument is true or false. However, since Rust does not support dependent types, we emulate\na dependent sum in a non-dependent language, using a CPS transformation:\nfnoffer_choice_2<C: Context, A: Protocol, B: Protocol>\n( cont: impl FnOnce (InjectSum2<C, A, B>) -> ContSum2<C, A, B> )\n-> PartialSession<C, ExternalChoice2<A, B>>\nThe function offer_choice_2 accepts a continuation function contthat is given a value of\ntype InjectSum2<C, A, B> and returns a value of type ContSum2<C, A, B> . We will now look at\nthe definitions of ContSum2 and InjectSum2 . First, we observe that the different return types\nfor the two branches can be unified with a type ContSum2:\nstruct ContSum2<C: Context, A: Protocol, B: Protocol> { ... }\nasync fn run_cont_sum<C: Context, A: Protocol, B: Protocol>(cont: ContSum2<C, A, B>)R. F. Chen, S. Balzer, and B. Toninho 22:21\nThe type ContSum2 contains the necessary data for executing either a PartialSession<C, A>\nor a PartialSession<C, B> , together with the runtime data for the linear context C. For brevity,\nthe implementation details of ContSum2 are omitted, with the private function run_cont_sum\nprovided as an abstraction for Ferrite to execute the continuation.\nWe then define InjectSum2 as a sum of boxed closures that would construct a ContSum2\nfrom either a PartialSession<C, A> or a PartialSession<C, B> :\nenum InjectSum2<C, A, B> {\nInjectLeft(Box< dyn FnOnce (PartialSession<C, A>) -> ContSum2<C, A, B>>),\nInjectRight(Box< dyn FnOnce (PartialSession<C, B>) -> ContSum2<C, A, B>>) }\nWhen the contpassed to offer_choice_2 is given a value of type InjectSum2<C, A, B> , it\nhas to branch on it and match on whether the InjectLeft orInjectRight constructors are\nused. Since the return type of contisContSum2<C, A, B> and the constructor for ContSum2 is\nprivate, there is no other way for contto construct the return value other than to call either\nInjectLeft orInjectRight with the appropriate continuation.\nThe use of InjectSum2 prevents the programmer from providing the wrong branch in the\ncontinuation by keeping the constructor private. However a private constructor alone cannot\nprevent two uses of InjectSum2 to bedeliberately interchanged, causing a protocol violation.\nTo fully ensure that there is no way for the user to provide a ContSum2 from elsewhere, we\ninstead use a technique from GhostCell [ 52] that uses higher-ranked trait bounds (HTRB) to\nmark a phantom invariant lifetime on both InjectSum2 and ContSum2:\nfnoffer_choice_2<C: Context, A: Protocol, B: Protocol>\n( cont: for </quotesingle.ts1r>impl FnOnce (InjectSum2< /quotesingle.ts1r, C, A, B>) -> ContSum2< /quotesingle.ts1r, C, A, B> )\n-> PartialSession<C, ExternalChoice2<A, B>>\nThe use of HRTB ensures that each call of offer_choice_2 would generate a unique lifetime\n/quotesingle.ts1rfor the continuation. Using that, Ferrite can ensure that a value of type InjectSum2< /quotesingle.ts1r1,\nC, A, B> cannot be used to construct the return value of type ContSum2< /quotesingle.ts1r2, C, A, B> , if the\nlifetimes </quotesingle.ts1r1>and </quotesingle.ts1r2>are different. An example use of offer_choice_2 is as follows:\nlet choice_provider: Session<ExternalChoice2<SendValue<u64, End>, End>>\n= offer_choice_2(| b | { match b {\nInjectLeft(ret) => ret(send_value(42, terminate())),\nInjectRight(ret) => ret(terminate()) } });\nThe example code above requires some boilerplate code to call the session injector ret\nto wrap around the continuation expression. To free the programmer from writing such\nboilerplate, Ferrite also provides a macro offer_choice that translates into the underlying\npattern matching syntax, which is explained in the next section.\n6.3 N-ary Branching\nTo generalize offer_choice_2 to n-ary choices, Ferrite has its own version of polymorphic\nvariants implemented in Rust. Our implementation specifically targets Rust, and is based on\nsimilar work by [30] and [31]. The base variant types are as follows:\nenum Bottom {} enum Sum<A, B> { Inl(A), Inr(B) }\ntrait TypeApp<A> { type Applied; } trait SumApp<F> { type Applied; }\ntype App<F, A> = <F asTypeApp<A>>::Applied;\ntype AppSum<Row, F> = <Row asSumApp<F>>::Applied;\nimpl <F> SumApp<F> for () { type Applied = Bottom; }\nimpl <A, F: TypeApp<A>, R: SumApp<F>> SumApp<F> for (A, R) {\ntype Applied = Sum<F::Applied, R::Applied>; }\nSimilar to RecAppdescribed in Section 5.1, TypeAppis used to represent a Rust type\nemulating the kind Type→Typefor non-recursive usage. Furthermore, the SumApptrait is\nECOOP 202222:22 Ferrite: A Judgmental Embedding of Session Types in Rust\nused to represent a Rust type emulating the kind (Type→Type)→Type. The type alias\nApp<F, A> is used to extract the associated type Appliedwhen Fis applied to Avia TypeApp.\nThe type alias AppSum<Row, F> is used to extract the associated type Appliedwhen a row type\nRowis applied to a type constructor F, which implements TypeApp<A> for all A.1\nUsing SumApp, we map an heterogeneous list to nested sums such that AppSum<HList![A0,\nA1, ...], F> = Sum![App<F, A0>, App<F, A1>, ...] , with the macro Sum!used to expand the\nmacro arguments into nested sums, i.e. Sum![A0, A1, ...] = Sum<A0, Sum<A1, ..., Bottom>> .\nWe then define the n-ary versions of InjectSum2 and ContSum2 as follows:\nstruct InjectSessionF< /quotesingle.ts1r, Row, C> {} struct InjectSession< /quotesingle.ts1r, Row, C, A> { ... }\nstruct ContSum< /quotesingle.ts1r, Row, C: Context> { ... }\nimpl </quotesingle.ts1r, Row, C: Context> TypeApp<A> for InjectSessionF<C> {\ntype Applied = InjectSession<C, A>; }\nimpl </quotesingle.ts1r, Row, C: Context, A: Protocol> FnOnce (PartialSession<C, A>)\n-> ContSum< /quotesingle.ts1r, Row, C> for InjectSession< /quotesingle.ts1r, Row, C, A> { ... }\nThe type InjectSessionF< /quotesingle.ts1r, Row, C> serves as a marker type for TypeApp, such that when\napplied to a type A, we get the struct InjectSession< /quotesingle.ts1r, Row, C, A> . Conceptually, the struct\nimplements the trait FnOnce (PartialSession<C, A>) -> ContSum< /quotesingle.ts1r, Row, C> , so that we can\napply a PartialSession<C, A> to it and get back a ContSum< /quotesingle.ts1r, Row, C> .2The composed type\nAppSum<Row, InjectSessionF< /quotesingle.ts1r, Row, C> represents a row of InjectSession , with Rowbeing a\nheterogeneous list in the form HList![A 0, A1, ..., A N−1]. For example, the type AppSum<\nHList![A, B], InjectSessionF< /quotesingle.ts1r, Row, C>> evaluates to Sum![InjectSession< /quotesingle.ts1r, Row, C, A>,\nInjectSession< /quotesingle.ts1r, Row, C, B>] , which is isomorphic to the type InjectSum2<C, A, B> that we\ndefined for the binary case. Using the row constructs, we can define n-ary version offer_choice\nas follows:\nfnoffer_choice<C: Context, Row>(cont1 : impl for </quotesingle.ts1r>\nFnOnce (AppSum<Row, InjectSessionF< /quotesingle.ts1r, Row, C>>) -> ContSum< /quotesingle.ts1r, Row, C>\n) -> PartialSession<C, ExternalChoice<Row>>\nwhere Row: SumApp<InjectSessionF< /quotesingle.ts1r, Row, C>>>, ...\nWith the n-ary version of offer_choice available, we can re-implement binary choice as\na specialized version. To do that, we only need a few type aliases and struct definitions to\nmake the syntax more pleasing:\nenum EitherSum<A, B> { Left(A), Right(B) }; type Either<A, B> = HList![A, B];\nconst LeftLabel: Z = Z::new(); const RightLabel: S<Z> = <S<Z>>::new();\nimpl <A, B> std::convert:: From <Sum![A, B]> for EitherSum<A, B> { ... }\nWe first define an EitherSum enum, and a std::convert:: Frominstance that converts an\nunlabeled nested sum Sum![A, B] into the labeled sum EitherSum<A, B> . The conversion\nallows users to use a flat list of labeled match arms during branching, and give meaningful\nlabels Leftand Rightto each branch. We also define Either<A, B> as a type alias to the\nrow type HList![A, B] , to give a meaningful name to the choice protocol. Finally we define\nthe constants LeftLabel and RightLabel to refer to the prisms Zand S<Z>, resp. Ferrite also\nprovides a helper macro define_choice! to help users define custom choice protocols that look\nsimilar to the above. This is used in conjunction with macros such as offer_choice! , which\n1For brievity, we omit some details that the types AppandAppSumare actually implemented as structs that\nareisomorphic to<F as TypeApp<A>>::Applied and <Row as SumApp<F>>::Applied , resp. The main\ndifference is that the actual structs hidethe propagation of the trait bound requirements of TypeApp\nandSumAppfrom their callers, resulting in much cleaner code. This does not affect the understanding of\nthe core concepts introduced in this section.\n2Technically, Rust does not allow custom implementation of FnOnce, so Ferrite defines a custom trait\nwith the same behavior.R. F. Chen, S. Balzer, and B. Toninho 22:23\n1enum CanvasMsg { Canvas2d(Canvas2dMsg, CanvasId), Close(CanvasId), ... }\n2enum Canvas2dMsg { LineTo(Point2D), GetTransform(Sender<Transform2D>),\n3 IsPointInPath(f64, f64, FillRule, IpcSender<bool>), ... }\n4enum ConstellationCanvasMsg { Create { id_sender: Sender<CanvasId>, size: Size2D } }\n5struct CanvasPaintThread { canvases: HashMap<CanvasId, CanvasData>, ... }\n6impl CanvasPaintThread { ...\n7 fnstart() -> (Sender<ConstellationCanvasMsg>, Sender<CanvasMsg>) {\n8 let (msg_sender, msg_receiver) = channel();\n9 let (create_sender, create_receiver) = channel();\n10 thread::spawn( move || { loop {select! {\n11 recv(canvas_msg_receiver) -> { ...\n12 CanvasMsg::Canvas2d(message, canvas_id) => { ...\n13 Canvas2dMsg::LineTo(point) => self .canvas(canvas_id).move_to(point),\n14 Canvas2dMsg::GetTransform(sender) =>\n15 sender.send( self .canvas(canvas_id).get_transform()).unwrap(), ... }\n16 CanvasMsg::Close(canvas_id) => canvas_paint_thread.canvases.remove(&canvas_id) }\n17 recv(create_receiver) -> { ...\n18 ConstellationCanvasMsg::Create { id_sender, size } => {\n19 let canvas_id = ...; self .canvases.insert(canvas_id, CanvasData::new(size, ...));\n20 id_sender.send(canvas_id); } } } } });\n21 (create_sender, msg_sender) }\n22 fncanvas(& mut self , canvas_id: CanvasId) -> & mut CanvasData {\n23 self .canvases.get_mut(&canvas_id).expect(\"Bogus canvas id\") } }\nFigure 1 Message-passing concurrency in Servo’s canvas component (simplified for illustration\npurposes).\ncleans up the boilerplate required to enable different match branches to return different types.\nUsing the macros, users can define the same Eitherprotocol and write an external choice\nprovider as follows:\ndefine_choice!{ Either<A, B>; Left: A, Right: B }\n// Inferred type: Session<ExternalChoice<Either<SendValue<u64, End>, End>>>\nlet provider = offer_choice!{\nLeft => send_value(42, terminate()), Right => terminate() };\nFor convenience, Ferrite exports the choice definition for Eitherfor the anonymous\ndeclaration of binary choice in session types.\n7 Evaluation\nThe Ferrite library is more than just a research prototype. It is designed for practical\nuse in real world applications. To evaluate the design and implementation of Ferrite, we\nre-implemented the communication layer of the canvas component of Servo [ 32] entirely in\nFerrite. Servo is an under development browser engine that uses message-passing for heavy\ntask parallelization. Canvas provides 2D graphic rendering, allowing clients to create new\ncanvases and perform operations on a canvas such as moving the cursor and drawing shapes.\nThe canvas component is a good target for evaluation as it is sufficiently complex and\nalso very demanding in terms of performance. Canvas is commonly used for animations in\nweb applications. For an animation to look smooth, a canvas must render at least 24 frames\nper second, with potentially thousands of operations to be executed per frame.\nThe changes we made are fairly minimal, consisting of roughly 750 lines of additions and\n620 lines of deletions, out of roughly 300,000 lines of Rust code in Servo. The sources of\nour implementation are provided as an artifact. To differentiate the two versions of code\nsnippets, we use blue for the original code, and green for the code using Ferrite.\n7.1 Servo Canvas Component\nFigure 1 provides a sketch of the main communication paths in Servo’s canvas compon-\nent [33]. The canvas component is implemented by the CanvasPaintThread , whose function\nstartcontains the main communication loop running in a separate thread (lines 10–20). This\nECOOP 202222:24 Ferrite: A Judgmental Embedding of Session Types in Rust\nloop processes client requests received along canvas_msg_receiver and create_receiver , which\nare the receiving endpoints of the channels created prior to spawning the loop (lines 8–9).\nThe channels are typed with the enumerations ConstellationCanvasMsg and CanvasMsg , defining\nmessages for creating and terminating the canvas component and for executing operations on\nan individual canvas, resp. When a client sends a message that expects a response from the\nrecipient, such as GetTransform and IsPointInPath (lines 2–3), it sends a channel along with\nthe message to be used by the recipient to send back the result. Canvases are identified by\nan id, which is generated upon canvas creation (line 19) and stored in the thread’s canvases\nhash map (line 5). If a client requests an invalid id, for example after prior termination and\nremoval of the canvas (line 16), the failed assertion expect(\"Bogus canvas id\") (line 23) will\nresult in a panic!, causing the canvas component to crash and subsequent calls to fail.\nThe code in Figure 1 uses a clever combination of enumerations to type channels and\nownership to rule out races on the data sent along channels. Nonetheless, Rust’s type system\nis not expressive enough to enforce the intended protocol of message exchange and existence\nof a communication partner. The latter is a consequence of Rust’s type system being affine,\nwhich permits “dropping of a resource”. The dropping or premature closure of a channel,\nhowever, can result in a proliferation of panic!and thus cause an entire application to crash.\nIn fact, while refactoring Servo to use Ferrite, we were able to uncover a protocol violation in\nServo, caused by one of the nested match arms of the provider doing an early return before\nsending back any result to the client.\n7.2 Canvas Protocol in Ferrite\nIn the original canvas component, the provider CanvasPaintThread accepts messages of type\nCanvasMsg , made up of a combination of smaller sub-message types such as Canvas2dMsg . We\nnote that the majority of the sub-message types have the following trivial form:\nenum CanvasMsg { Canvas2d(Canvas2dMsg, CanvasId), Close(CanvasId), ... }\nenum Canvas2dMsg { BeginPath, ClosePath, Fill(FillOrStrokeStyle), ... }\nThe trivial sub-message types such as BeginPath ,Fill, and LineTodo not require a response\nfrom the provider, so the client can simply fire them and proceed. Although we can offer\nall sub-message types as separate branches in an external choice, it is more efficient to keep\ntrivial sub-messages in a single enum. In our implementation, we define CanvasMessage to\nhave similar sub-messages as Canvas2dMsg , with non-trivial messages such as IsPointInPath\nmoved to separate branches.\nenum CanvasMessage { BeginPath, ClosePath, Fill(FillOrStrokeStyle), ... }\ndefine_choice! { CanvasOps; Message: ReceiveValue<CanvasMessage, Release>, ... }\ntype Canvas = LinearToShared<ExternalChoice<CanvasOps>>;\nWe use the define_choice! macro described in Section 6 to define an n-ary choice CanvasOps .\nThe first branch of CanvasOps is labelled Message, and the only action is for the provider to\nreceive a CanvasMessage . The choices are offered as an external choice, and the session type\nCanvasProtocol is defined as a shared protocol that offers the choices in the critical section.\nThe original design of the CanvasPaintThread would be sufficient if the only messages being\nsent were trivial messages. However, Canvas2dMsg also contains non-trivial sub-messages, such\nasGetImageData and IsPointInPath , demanding a response from the provider:\nenum Canvas2dMsg { ..., GetImageData(Rect< u64>, Size2D< u64>, IpcBytesSender),\nIsPointInPath( f64,f64, FillRule, IpcSender< bool >), ... }\nTo obtain the result from the original canvas, clients must create a new inter-process\ncommunication (IPC) channel and bundle the channel’s sender endpoint with the message.\nIn our implementation, we define separate branches in CanvasOps to handle non-trivial cases:R. F. Chen, S. Balzer, and B. Toninho 22:25\nTable 4MotionMark Benchmark scores in fps (higher is better)\nBenchmark Name Servo Servo/Ferrite Firefox Chrome\nArcs 12.21±6.75% 11.83±11.49% 52.61±32.88% 46.00±9.00%\nPaths 43.76±10.66% 40.98±18.94% 55.59±28.80% 59.50±14.90%\nLines 7.48±7.06% 11.47±12.74% 14.35±6.65% 32.43±6.48%\nBouncing clipped rects 18.43±7.06% 18.23±11.00% 34.82±7.76% 58.07±19.85%\nBouncing gradient circles 8.02±7.74% 7.72±12.63% 58.79±21.03% 59.77±10.07%\nBouncing PNG images 7.97±5.91% 6.31±10.26% 24.61±6.35% 59.94±13.04%\nStroke shapes 10.60±3.95% 10.35±10.96% 51.21±11.25% 59.38±16.87%\nPut/get image data 60.01±3.81% 32.08±10.83% 59.66±20.16% 60.00±5.00%\ndefine_choice! { CanvasOps; Message: ReceiveValue<CanvasMessage, Release>,\nGetImageData: ReceiveValue<(Rect< u64>, Size2D< u64>), SendValue<ByteBuf, Release>>,\nIsPointInPath: ReceiveValue<( f64,f64, FillRule), SendValue< bool , Release>>, ... }\nThe original GetImageData accepts an IpcBytesSender , which sends raw bytes back to the\nclient. In Ferrite, we translate the use of IpcBytesSender to the type SendValue<ByteBuf, Z> ,\nwhich sends the raw bytes wrapped in a ByteBuftype. We discuss possible performance\npenalties of this approach in Section 7.3.\nAside from the Canvasprotocol, we also redesign the use of ConstellationCanvasMsg into\nits own shared protocol, ConstellationCanvas :\ntype ConstellationCanvas = LinearToShared<ReceiveValue<Size2D,\nSendValue<SharedChannel<Canvas>, Release>>>;\nTo create a new canvas, a client first acquires the shared channel of type SharedChannel<\nConstellationCanvas> . Afterwards, the client sends the Size2Dparameter to specify the canvas\nsize. The constellation canvas provider then spawns a new canvas shared process through\nrun_shared_session and sends back the shared channel of type SharedChannel<Canvas> as a\nvalue. Finally, the session is released, allowing other clients to acquire the shared provider.\n7.3 Performance Evaluation\nTo evaluate the performance of the canvas component, we use the MotionMark benchmark\nsuite [50]. MotionMark is a web benchmark that focuses on graphics performance of web\nbrowsers. It contains benchmarks for various web components, including canvas, CSS, and\nSVG. As MotionMark does not yet support Servo, we modified the benchmark code to make\nit work in the absence of features that are not implemented in Servo.\nWe provide the modified benchmark source code along with instructions for running\nit as an artifact. Appendix D is also provided to highlight some of the implementation\nchallengesinportingServotouseFerrite, inparticularonthelatencyincurredbyinter-process\ncommunication, and our workaround to compensate the complication.\nFor the purpose of this evaluation, we focused on benchmarks that target the canvas\ncomponent and skipped benchmarks that fail in Servo due to missing features. We ran\neach benchmark in a fixed 1600x800 resolution for 30 seconds, on a Core i7 Linux desktop\nmachine. We ran the benchmarks against the original Servo, modified Servo with Ferrite\ncanvas (Servo/Ferrite), Firefox, and Chrome. Our performance scores are measured in the\nfixed mode version of MotionMark, which measures frames per second (fps) performance of\nexecuting the same set of canvas operations per frame.\nThe benchmark results are shown in Table 4, with the performance scores in fps (higher\nfps is better). It is worth noting that a benchmark can achieve at most 60 fps. Our goal in\nthis benchmark is to keep the scores of Servo/Ferrite close to those of Servo, notto achieve\nbetter performance than the original. This is shown to be the case in most of the benchmarks.\nECOOP 202222:26 Ferrite: A Judgmental Embedding of Session Types in Rust\nThe only benchmark with a large difference between Servo and Servo/Ferrite is Put/get\nimage data , with Ferrite performing 2x worse. This is because in Servo/Ferrite, we use\nByteBufto transfer the images as raw bytes within the same shared channel. In contrast,\nServo uses a specialized structure IpcBytesSender for transferring of raw bytes in parallel to\nother messages. As a result, the communication in Servo/Ferrite is congested during the\ntransfer of the image data, while the original Servo can process new messages in parallel to\nthe image data being transmitted.\nWe also observe that there are significant performance differences in the scores between\nServo and those in Firefox and Chrome, indicating that there exist performance bottlenecks\nin Servo unrelated to communication protocols.\n8 Related and Future Work\nSession type embeddings exist for various languages, including Haskell [ 38,22,29,34],\nOCaml [ 36,21], Java [17,16], and Scala [ 43]. Functional languages like ML, OCaml, and\nHaskell, in particular, are ideal host languages for creating EDSLs thanks to their advanced\nfeatures (e.g. type classes, type families, higher-rank and higher-kinded types and GADTs).\n[38] first demonstrated the feasibility of embedding session types in Haskell, with refinements\ndone in later works [ 22,29,34]. Similar embeddings have also been contributed in the context\nof OCaml by FuSe[36] and session-ocaml [21].\nAside from Ferrite, there are other implementations of session types in Rust, including\nsession_types [23],sesh[27], and rumpsteak [6,7].session_types were the first implementation\nto make use of affinity to provide a session type library in Rust. seshemphasizes this aspect\nby embedding the affine session type system Exceptional GV [ 10] in Rust. Both session_types\nand seshadopt a classical perspective, requiring the endpoints of a channel to be typed with\ndual types. rumpsteak develops an embedding of multiparty session types by generating Rust\ntypes derived from multiparty session types defined in Scribble [53].\nDue to their reliance on Rust’s affine type system, neither session_types nor seshprevents\na channel endpoint from being dropped prematurely, relegating the handling of such errors\nto the runtime. rumpsteak uses some type-level techniques similar to Ferrite to enforce a\nchannel’s linear usage in the continuation passed to the try_session function. This ensures\nthat a linear channel in rumpsteak is always fully consumed, if it is ever consumed. However,\nprior to the call to try_session , the linear channel exist as an affine value, which may be\ndropped by the Rust program without being consumed at all, thereby causing deadlock. In\ncomparison, Ferrite enforces linearity at all level, including safe linking of multiple linear\nprocesses using cut.\nIn terms of concurrency, session_types ,sesh, and rumpsteak all require the programmer to\nmanually manage concurrency, either by spawning threads or async tasks. This introduces\npotential failure when the code fails follow the requirement to spawn all processes. On the\nother hand, the simplicity of such a model allows relatively few threads or async tasks to be\nspawned, thereby allowing the underlying runtime to execute the processes more efficiently.\nIn comparison, Ferrite offers fully managed concurrency, without the programmer having to\nworry about how to spawn the processes and execute them in parallel.\nIn terms of performance, the downside of Ferrite’s concurrency approach is that it\naggresively spawns new async tasks in each use of cut. Although async tasks in Rust are\nmuch more lightweight than OS threads, there is still a significant overhead in spawning and\nmanaging many async tasks, especially in micro-benchmarks. As a result, Ferrite tends to\nperform slower than alternative Rust implementations in settings where only a fixed smallR. F. Chen, S. Balzer, and B. Toninho 22:27\nnumber of processes need to be spawned. Nevertheless, it is worth noting that the async\necosystem in Rust is still relatively immature, with many potential improvements to be made.\nIn practice, the overhead of the async runtime may also be negligible when compared to\nthe core application logic. In such cases, Ferrite would also allow applications to scale more\neasily by allowing many more processes to be spawned and managed concurrently without\nrequiring additional effort from the programmer.\nIn terms of DSL design, Ferrite is more closely related to the embeddings in OCaml and\nHaskell, as it fully enforces a linear treatment of session type channels and thus statically rules\nout any panics arising from dropping a channel prematurely. Ferrite also differs from other\nlibraries in that it adopts intuitionistic typing [ 4], allowing the typing of a channel rather\nthan its two endpoints. On the use of profunctor optics, our work is the first to connect n-ary\nchoice to prisms, while prior work by session-ocaml [22] has only established the connection\nbetween lenses, the dual of prisms, and linear contexts. FuSe[36] and session-ocaml [21] have\npreviously explored the use of n-ary (generalized) choice through extensible variants available\nonly in OCaml. Our work demonstrates that it is possible to encode extensible variants, and\nthus n-ary choice, as type-level constructs using features available in Rust.\nA major difference in terms of implementation is that Ferrite uses a continuation-passing\nstyle, whereas Haskell and OCaml embeddings commonly use (indexed) monads and do-\nnotation style. This technical difference amounts to a key conceptual one: a direct corres-\npondence between the Rust programs generated from Ferrite constructs and the SILL Rtyping\nderivation. As a result, the generated Rust code can be viewed as carrying the proof of\nprotocol adherence.\nThe embeddings of ESJ[16] and lchannels [43] also adopt a continuation-passing style,\nbut do not faithfully embed typing derivations (i.e. they do not statically enforce linearity).\nThese approaches follow an encoding of session types using linear types [ 8] first proposed by\nKobayashi [ 25] in the setting of π-calculus. Type systems for message-passing in π-calculus\nhave a long history, dating back to the work of Kobayashi and Igarashi [ 18,19,20]. These\nsystems often focus on (but are not limited to) deadlock-freedom and lock-freedom [ 26] by\nenforcing a partial order on matching communication. This approach has been studied for the\nlinearπ-calculus [ 35] and in the presence of interrupts [ 44] or unbounded process networks [ 12].\nWhile session types are generally less powerful than the approaches of Kobayashi et al., they\nprovide a useful compromise between expressiveness and simplicity, being more amenable to\nembeddings in general-purpose language constructs and type systems.\nIn terms of expressiveness, Ferrite contributes over all prior session-based works in its\nsupport for shared session types [ 1], allowing it to express real-world protocols, as demon-\nstrated by our implementation of Servo’s canvas component. Shared session types reclaim the\nexpressiveness of the untyped asynchronous π-calculus in session-typed languages [ 2], at the\ncost of deadlock-freedom. Recent extensions of classical linear logic session types contribute\nanother approach to softening the rigidity of linear session types to support multiple client\nsessions and nondeterminism [ 39] and memory cells and nondeterministic updates [ 41], resp.\nOur technique of a judgmental embedding opens up new possibilities for embedding\ntype systems other than session types in Rust. Although we have demonstrated that the\njudgmental embedding is sufficiently powerful to encode a type system like session types, the\nembedding is currently shallow, with the implementation hardcoded to use the channels and\nasync run-time from tokio. Rust comes with unique features such as affine types and lifetimes\nthat makes it especially suited for implementing concurrency primitives, as evidenced by the\nwealth of channel and async run-time implementations available. As discussed in Section 7,\none of our future goals is to explore the possibility of making Ferrite a deepembedding of\nECOOP 202222:28 Ferrite: A Judgmental Embedding of Session Types in Rust\nsession types in Rust, so that users can choose from multiple low-level implementations.\nAlthough deep embeddings have extensively been explored for languages like Haskell [ 45,29],\nit remains a open question to find suitable approaches that work well in Rust.\nReferences\n1Stephanie Balzer and Frank Pfenning. Manifest sharing with session types. Proceedings of the\nACM on Programming Languages (PACMPL) , 1(ICFP):37:1–37:29, 2017.\n2Stephanie Balzer, Frank Pfenning, and Bernardo Toninho. A universal session type for untyped\nasynchronous communication. 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In Martín Abadi and Alberto Lluch-Lafuente, editors, Trustworthy Global Computing\n- 8th International Symposium, TGC 2013, Buenos Aires, Argentina, August 30-31, 2013,\nRevised Selected Papers , volume 8358 of Lecture Notes in Computer Science , pages 22–41.\nSpringer, 2013. doi:10.1007/978-3-319-05119-2\\_3 .\nECOOP 202222:32 Ferrite: A Judgmental Embedding of Session Types in Rust\nTable 5Overview of session terms in SILL Rand Ferrite.\nSILL R Ferrite Term Description\n/epsilon1 End terminate ; Terminate session.\nwaita;K Wait for channel ato close.\nτ ⊿ A ReceiveValue<T, A> x←receive_value ;K Receive value xof typeτ.\nsend_value_toax;K Send value xof typeτtoa.\nτ /triangleleft A SendValue<T, A> send_valuex;K Send value of type τ.\nx←receive_value_fromax;KReceive value of type τfrom\nchannela.\nA(B ReceiveChannel<A,\nB>a←receive_channel ;K Receive channel aof session type\nA.\nsend_channel_tof a;K Send channel ato channel fof\nsession type A(B.\nA⊗B SendChannel<A, B> send_channel_froma;K Send channel aof session type\nA.\na←receive_channel_fromf a;KReceive channel afrom channel\nfof session type A⊗B.\n↑S\nLA LinearToShared<A> accept_shared_session ;Kl Accept an acquire, then continue\nas linear session Kl.\na←acquire_shared_sessions;KlAcquire shared channel sas lin-\near channel a.\n↓S\nLS SharedToLinear<S> detach_shared_session ;Ks Detach linear session and con-\ntinue as shared session Ks.\nrelease_shared_sessiona;Kl Release acquired linear session.\nANB ExternalChoice<\nEither<A, B>>offer_choice_2KlKr Offer either continuation Klor\nKrbased on client’s choice.\nchoose_lefta;K Choose the left branch offered\nby channel a\nchoose_righta;K Choose the right branch offered\nby channel a\nA⊕B InternalChoice<\nEither<A, B>>offer_left;K Offer the left branch\noffer_right;K Offer the right branch\ncase_2aKlKr Branch to either KlorKrbased\non choice offered by channel a.\nN{li:Ai}ExternalChoice<Row> offer_choice{li:Ki} Offer continuation Kiwhen the\nclient selects li.\nchooseali;K Choose the libranch offered by\nchannela\n⊕{li:Ai}InternalChoice<Row> offerli;K Offer thelibranch\ncasea{li:Ki} Branch to continuation Kiwhen\nchannelaoffersli.\n- Rec<F> fix_session(cont) Fold session type F::Applied\noffered by cont.\nunfix_session(a, cont) Unfold channel ato session type\nF::Applied incont.\nA Typing Rules\nA.1 Typing Rules for SILL R\nFollowing is a list of inference rules in SILL R.\nCommunication\nΓ ; ∆ 1/turnstilelefta::A Γ ; ∆ 2,a/prime:A/turnstileleftb::B\nΓ ; ∆ 1,∆2/turnstilelefta/prime←cuta;b::B(T-cut)Γ ;·/turnstilelefta::A Γ ; ∆,a/prime:A/turnstileleftb::B\nΓ ; ∆/turnstilelefta/prime←includea;b::B(T-incl)R. F. Chen, S. Balzer, and B. Toninho 22:33\nΓ ;·/turnstileleftf::A(B Γ ;·/turnstilelefta::A\nΓ ;·/turnstileleft apply_channelf a::B(T-app)Γ ;a:A/turnstileleftforwarda::A(T-fwd)\nTermination\nΓ ;·/turnstileleftterminate ; ::/epsilon1(T1R)Γ ; ∆/turnstileleftK::A\nΓ ; ∆, a:/epsilon1/turnstileleftwaita;K::A(T1L)\nReceive Value\nΓ,x:τ; ∆/turnstileleftK::A\nΓ ; ∆/turnstileleftx←receive_value ;K::τ ⊿ A(T⊿R)Γ ; ∆,a:A/turnstileleftK::B\nΓ, x:τ; ∆, a:τ ⊿ A/turnstileleftsend_value_toax;K::B(T⊿L)\nSend Value\nΓ ; ∆/turnstileleftK::A\nΓ, x:τ; ∆/turnstileleftsend_valuex;K::τ /triangleleft A(T/triangleleftR)Γ,a:τ; ∆,a:A/turnstileleftK::B\nΓ ; ∆,a:τ ⊿ A/turnstileleftx←receive_value_froma;K::B(T/triangleleftL)\nReceive Channel\nΓ ; ∆,a:A/turnstileleftK::B\nΓ ; ∆/turnstilelefta←receive_channel ;K::A(B(T(R)Γ ; ∆,f:A2/turnstileleftK::B\nΓ ; ∆, f:A1(A2, a:A1/turnstileleftsend_channel_tof a;K::B(T(L)\nSend Channel\nΓ ; ∆/turnstileleftK::B\nΓ ; ∆,a:A/turnstileleftsend_channel_froma;K::A⊗B(T⊗R)Γ ; ∆,f:A2, a:A1/turnstileleftK::B\nΓ ; ∆, f:A1⊗A2/turnstilelefta←receive_channel_fromf;K::B(T⊗L)\nShared Session Types\nΓ ;·/turnstileleftK::A\nΓ ;·/turnstileleftaccept_shared_session ;K::↑S\nLA(T↑S\nLR)\nΓ ;·/turnstileleftK::S\nΓ ;·/turnstileleftdetach_shared_session ;K::↓S\nLS(T↓S\nLR)Γ ; ∆, a:A/turnstileleftK::B\nΓ,s:↑S\nLA; ∆/turnstilelefta←acquire_shared_sessions;K::B(T↑S\nLL)\nΓ,s:S; ∆/turnstileleftK::B\nΓ ; ∆, a:↓S\nLS/turnstilelefts←release_shared_sessiona;K::B(T↓S\nLL)\nExternal Choice (Binary)\nΓ ; ∆/turnstileleftKl::A Γ ; ∆/turnstileleftKr::B\nΓ ; ∆/turnstileleftoffer_choiceKlKr::ANB(TN2R)Γ ; ∆,a:A1/turnstileleftK::B\nΓ ; ∆, a:A1NA2/turnstileleftchoose_lefta;K::B(TN2L)\nΓ ; ∆,a:A2/turnstileleftK::B\nΓ ; ∆, a:A2NA2/turnstileleftchoose_righta;K::B(TN2L)\nInternal Choice (Binary)\nΓ ; ∆/turnstileleftK::A\nΓ ; ∆/turnstileleftoffer_left;K::A⊕2B(T⊕2R)\nΓ ; ∆/turnstileleftK::B\nΓ ; ∆/turnstileleftoffer_right;K::A⊕B(T⊕2R)Γ ; ∆,a:A1/turnstileleftKl::B\nΓ ; ∆,a:A2/turnstileleftKr::B\nΓ ; ∆, a:A1⊕A2/turnstileleftcaseaKlKr::B(T⊕2L)\nExternal Choice\nΓ ; ∆/turnstileleftKi::Ai\nΓ ; ∆/turnstileleftoffer_choice{li:Ki}::N{li:Ai}(TNR)Γ ; ∆,a:Ai/turnstileleftK::B\nΓ ; ∆, a:N{li:Ai}/turnstileleft chooseali;K::B(TNL)\nInternal Choice\nΓ ; ∆/turnstileleftK::A\nΓ ; ∆/turnstileleftofferli;K::⊕{li:Ai}(T⊕R)Γ ; ∆,a:Ai/turnstileleftKi::B\nΓ ; ∆, a:⊕{li:Ai}/turnstileleft casea{li:Ki}::B(T⊕2L)\nECOOP 202222:34 Ferrite: A Judgmental Embedding of Session Types in Rust\nA.2 Typing Constructs in Ferrite\nFollowing is a list of function signatures of the term constructors provided in Ferrite.\nA.2.1 Forward\nfnforward<N, C, A>(_: N) -> PartialSession<C, A>\nwhere\nA: Protocol,\nC: Context,\nN::Target: EmptyContext,\nN: ContextLens<C, A, Empty>\nA.2.2 Termination\npub struct End;\nimpl Protocol for End { ... }\nfnterminate<C>() -> PartialSession<C, End>\nwhere\nC : EmptyContext\nfnwait<N, C, A>(\n_ : N,\ncont : PartialSession<N::Target, A>\n) -> PartialSession<C, A>\nwhere\nC : Context,\nA : Protocol,\nN : ContextLens<C, End, Empty>\nA.2.3 Communication\nfncut<X, C, C1, C2, A, B>(\ncont1 : PartialSession<C1, A>,\ncont2 : impl FnOnce (C2::Length) -> PartialSession<C2::Appended, B>\n) -> PartialSession<C, B>\nwhere\nA : Protocol,\nB : Protocol,\nC : Context,\nC1 : Context,\nC2 : Context,\nX : SplitContext<C, Left = C1, Right = C2>,\nC2 : AppendContext<(A, ())>\nfninclude_session<C, A, B>(\nsession : Session<A>,\ncont : impl FnOnce (C::Length) -> PartialSession<C::Appended, B>\n) -> PartialSession<C, B>\nwhere\nA : Protocol,\nB : Protocol,\nC : Context,\nC : AppendContext<(A, ())>\nfnapply_channel<A, B>(\nf : Session<ReceiveChannel<A, B>>,\na : Session<A>,R. F. Chen, S. Balzer, and B. Toninho 22:35\n) -> Session<B>\nwhere\nA : Protocol,\nB : Protocol\nA.2.4 Receive Value\nstruct ReceiveValue<T, A> { ... }\nimpl <T, A> Protocol for ReceiveValue<T, A>\nwhere\nT:Send +/quotesingle.ts1static ,\nA: Protocol\n{ ... }\nfnreceive_value<T, C, A>(\ncont : impl FnOnce (T) -> PartialSession<C, A> + Send +/quotesingle.ts1static\n) -> PartialSession<C, ReceiveValue<T, A>>\nwhere\nT : Send +/quotesingle.ts1static ,\nA : Protocol,\nC : Context\nfnsend_value_to<N, C, A, B, T>(\n_ : N,\nval : T,\ncont : PartialSession<N::Target, A>\n) -> PartialSession<C, A>\nwhere\nA : Protocol,\nB : Protocol,\nC : Context,\nT : Send +/quotesingle.ts1static ,\nN : ContextLens<C, ReceiveValue<T, B>, B>\nA.2.5 Send Value\nstruct SendValue<T, A> { ... }\nimpl <T, A> Protocol for SendValue<T, A>\nwhere\nT:Send +/quotesingle.ts1static ,\nA: Protocol\n{ ... }\nfnsend_value<T, C, A>(\nval : T,\ncont : PartialSession<C, A>\n) -> PartialSession<C, SendValue<T, A>>\nwhere\nT : Send +/quotesingle.ts1static ,\nA : Protocol,\nC : Context\nfnreceive_value_from<N, C, T, A, B>(\n_ : N,\ncont : impl FnOnce (T) -> PartialSession<N::Target, B> + Send +/quotesingle.ts1static\n) -> PartialSession<C, B>\nwhere\nA : Protocol,\nB : Protocol,\nC : Context,\nT : Send +/quotesingle.ts1static ,\nN : ContextLens<C, SendValue<T, A>, A>\nECOOP 202222:36 Ferrite: A Judgmental Embedding of Session Types in Rust\nA.2.6 Receive Channel\npub struct ReceiveChannel<A, B> { ... }\nimpl <A: Protocol, B: Protocol> Protocol for ReceiveChannel<A, B> { ... }\nfnreceive_channel<C, A, B>(\ncont : impl FnOnce (C::Length) -> PartialSession<C::Appended, B>\n) -> PartialSession<C, ReceiveChannel<A, B>>\nwhere\nA : Protocol,\nB : Protocol,\nC : Context,\nC : AppendContext<(A, ())>\nfnsend_channel_to<N1, N2, C, A1, A2, B>(\n_ : N1,\n_ : N2,\ncont : PartialSession<N1::Target, B>\n) -> PartialSession<C, B>\nwhere\nC : Context,\nA1 : Protocol,\nA2 : Protocol,\nB : Protocol,\nN2 : ContextLens<C, A1, Empty>,\nN1 : ContextLens<N2::Target, ReceiveChannel<A1, A2>, A2>\nA.2.7 Send Channel\nstruct SendChannel<A, B> { ... }\nimpl <A: Protocol, B: Protocol> Protocol for SendChannel<A, B>\nfnsend_channel_from<C, A, B, N>(\n_ : N,\ncont : PartialSession<N::Target, B>\n) -> PartialSession<C, SendChannel<A, B>>\nwhere\nA : Protocol,\nB : Protocol,\nC : Context,\nN : ContextLens<C, A, Empty>\nfnreceive_channel_from<C1, C2, A1, A2, B, N>(\n_ : N,\ncont_builder : impl FnOnce (C2::Length) -> PartialSession<C2::Appended, B>\n) -> PartialSession<C1, B>\nwhere\nA1 : Protocol,\nA2 : Protocol,\nB : Protocol,\nC1 : Context,\nC2 : AppendContext<(A1, ())>,\nN : ContextLens<C1, SendChannel<A1, A2>, A2, Target = C2>\nA.2.8 External Choice\nstruct ExternalChoice<Row> { ... }\nimpl <Row> Protocol for ExternalChoice<Row>\nwhere\nRow: ToRow + Send +/quotesingle.ts1static\n{ ... }R. F. Chen, S. Balzer, and B. Toninho 22:37\nfnoffer_choice<C, Row1, Row2>(\ncont1: impl for </quotesingle.ts1r>FnOnce (\nAppSum< /quotesingle.ts1r, Row2, InjectSessionF< /quotesingle.ts1r, Row1, C>>,\n) -> ContSum< /quotesingle.ts1r, Row1, C>\n+Send\n+/quotesingle.ts1static\n) -> PartialSession<C, ExternalChoice<Row1>>\nwhere\nC: Context,\nRow1: Send +/quotesingle.ts1static ,\nRow2: Send +/quotesingle.ts1static ,\nRow1: ToRow<Row = Row2>,\nRow2: RowCon,\nRow2: SumFunctor\nfnchoose<N, M, C1, C2, A, B, Row1, Row2>(\n_: N,\n_: M,\ncont: PartialSession<C2, A>\n) -> PartialSession<C1, A>\nwhere\nC1: Context,\nC2: Context,\nA: Protocol,\nB: Protocol,\nRow2: RowCon,\nRow1: Send +/quotesingle.ts1static ,\nRow2: Send +/quotesingle.ts1static ,\nRow1: ToRow<Row = Row2>,\nN: ContextLens<C1, ExternalChoice<Row1>, B, Target = C2>,\nM: Prism<Row2, Elem = B>\nA.2.9 Internal Choice\nstruct InternalChoice<Row> { ... }\nimpl <Row> Protocol for InternalChoice<Row>\nwhere\nRow: ToRow + Send +/quotesingle.ts1static\n{ ... }\nfnoffer_case<N, C, A, Row1, Row2>(\n_: N,\ncont: PartialSession<C, A>\n) -> PartialSession<C, InternalChoice<Row1>>\nwhere\nC: Context,\nA: Protocol,\nRow1: Send +/quotesingle.ts1static ,\nRow2: Send +/quotesingle.ts1static ,\nRow2: RowCon,\nRow1: ToRow<Row = Row2>,\nN: Prism<Row2, Elem = A>\nfncase<N, C1, C2, B, Row1, Row2>(\n_: N,\ncont1: impl for </quotesingle.ts1r>FnOnce (\nAppSum< /quotesingle.ts1r, Row2, ContF< /quotesingle.ts1r, N, C2, B>>,\n) -> ChoiceRet< /quotesingle.ts1r, N, C2, B>\n+Send\n+/quotesingle.ts1static\n) -> PartialSession<C1, B>\nwhere\nB: Protocol,\nC1: Context,\nECOOP 202222:38 Ferrite: A Judgmental Embedding of Session Types in Rust\nC2: Context,\nRow1: Send +/quotesingle.ts1static ,\nRow2: Send +/quotesingle.ts1static ,\nRow1: ToRow<Row = Row2>,\nN: ContextLens<C1, InternalChoice<Row1>, Empty, Target = C2>\nA.2.10 Recursive Session Types\nfnfix_session<R, F, A, C>(\ncont: PartialSession<C, A>\n) -> PartialSession<C, RecX<R, F>>\nwhere\nC: Context,\nR: Context,\nF: Protocol,\nA: Protocol,\nF: RecApp<(RecX<R, F>, R), Applied = A>\nfnunfix_session<N, C1, C2, A, B, R, F>(\n_n: N,\ncont: PartialSession<C2, B>\n) -> PartialSession<C1, B>\nwhere\nB: Protocol,\nC1: Context,\nC2: Context,\nF: Protocol,\nR: Context,\nF: RecApp<(RecX<R, F>, R), Applied = A>,\nA: Protocol,\nN: ContextLens<C1, RecX<R, F>, A, Target = C2>\nA.2.11 Shared Session Types\nstruct LinearToShared<F> { ... }\nstruct SharedToLinear<F> { ... }\nstruct Lock<F> { ... }\nimpl <F> SharedProtocol for LinearToShared<F>\nwhere\nF: Protocol,\nF: SharedRecApp<SharedToLinear<LinearToShared<F>>>,\nF::Applied: Protocol\n{ ... }\nimpl <F> Protocol for SharedToLinear<LinearToShared<F>>\nwhere\nF: SharedRecApp<SharedToLinear<LinearToShared<F>>> + Send +/quotesingle.ts1static\n{ ... }\nimpl <F> Protocol for Lock<F>\nwhere\nF: Protocol,\nF: SharedRecApp<SharedToLinear<LinearToShared<F>>>,\nF::Applied: Protocol\n{ ... }\nA detail we omitted in the main text is that we introduced a special linear session type\ncalled Lock, internal to the library. The Locktype holds the underlying shared Rust channel\nthat connects to the corresponding endpoint held by SharedChannel . This allows multiple usesR. F. Chen, S. Balzer, and B. Toninho 22:39\nofaccept_shared_session and detach_shared_session to all access the same underlying Rust\nchannel without having to rely on global state.\nAn additional role of the linear session type Lockis that it also enforces the equi-\nsynchronizing constraint of shared session type, by requiring all use of accept_shared_session\nto always be accompanied by detach_shared_session with the same shared session type. This\nprovides the same functionality of enforcing the equi-synchronizing constraint as specified in\nSection 3.3 in [1].\nfnaccept_shared_session<F>(\ncont: impl Future<Output = PartialSession<(Lock<F>, ()), F::Applied>>\n+Send\n+/quotesingle.ts1static\n) -> SharedSession<LinearToShared<F>>\nwhere\nF: Protocol,\nF: SharedRecApp<SharedToLinear<LinearToShared<F>>>,\nF::Applied: Protocol\nThe accept_shared_session constructisparameterizedoverasharedsessiontype LinearToShared\n<F>. The type Fis required to implement SharedRecApp<SharedToLinear<LinearToShared<F>>> ,\nwhich unfolds the shared session type by applyig the type SharedToLinear<LinearToShared<F>>\ntoF. The continuation is an asyncblock with PartialSession result that offers the linear\nsession type F::Applied . It also has an internal session type Lock<F>, which is described next.\nThe construct returns a shared session type program of type SharedSession<LinearToShared<F\n>>. This needs to be passed to run_shared_session to execute the program and get back a\nshared channel of type SharedChannel<LinearToShared<F>> .\nfndetach_shared_session<F, C>(\ncont: SharedSession<LinearToShared<F>>\n) -> PartialSession<(Lock<F>, C), SharedToLinear<LinearToShared<F>>>\nwhere\nF: Protocol,\nF: SharedRecApp<SharedToLinear<LinearToShared<F>>>,\nF::Applied: Protocol,\nC: EmptyContext\nThe detach_shared_session constructisparameterizedbyalinearsessiontype LinearToShared\n<F>and an empty linear context C. The type Fis required to implement SharedRecApp<\nSharedToLinear<LinearToShared<F>>> to unfold Frecursively. This is required for LinearToShared\n<F>to satisfy the SharedProtocol constraint. The construct accepts a SharedSession continu-\nation with the offered shared session type LinearToShared<F> . Note that this is the only\ncontinuation that is notaPartialSession . It is also notaSharedChannel , as this is a shared\nFerrite program that is yet to be executed. The construct returns a PartialSession that offers\nthe linear session type SharedToLinear<F> . It also has a linear context with Lock<F>being the\nfirst linear channel, and the tail Cbeing an empty linear context of arbitrary length.\nfnacquire_shared_session<C, F, A>(\nshared: SharedChannel<LinearToShared<F>>,\ncont1: impl FnOnce (C::Length) -> PartialSession<C::Appended, A> + Send +/quotesingle.ts1static\n) -> PartialSession<C, A>\nwhere\nC: Context,\nF: Protocol,\nA: Protocol,\nF::Applied: Protocol,\nF: SharedRecApp<SharedToLinear<LinearToShared<F>>>,\nC: AppendContext<(F::Applied, ())>\nThe acquire_shared_session construct is parameterized over a shared session type\nLinearToShared<F> , a linear context C, and an offered session type A. The type Fis required\nECOOP 202222:40 Ferrite: A Judgmental Embedding of Session Types in Rust\nto implement SharedRecApp<SharedToLinear<LinearToShared<F>>> , which unfolds the shared ses-\nsion type by applying the type SharedToLinear<LinearToShared<F>> toF. The unfolded session\ntype F::Applied is a linear session type implementing Protocol, and it is appended to the end\nofCusing AppendContext , with C::Appended being the result.\nThe first argument to acquire_shared_session is a cloneable SharedChannel of (shared)\nsession type LinearToShared<F> . The second argument is the continuation closure. It is\ngiven the context lens C::Length , which implements the context lens to access the linear\nchannel F::Applied inC::Appended . The continuation closure returns a PartialSession with\nC::Appended being the linear context, and Abeing the offered session type. The construct\nreturns a PartialSession that works with the original linear context C, and offers the session\ntype A.\nfnrelease_shared_session<N, C1, C2, A, B>(\n_n: N,\ncont: PartialSession<C2, B>,\n) -> PartialSession<C1, B>\nwhere\nA: Protocol,\nB: Protocol,\nC1: Context,\nC2: Context,\nA: SharedRecApp<SharedToLinear<LinearToShared<A>>>,\nN: ContextLens<C1, SharedToLinear<LinearToShared<A>>, Empty, Target = C2>\nThe release_shared_session constructisparameterizedoveralinearsessiontype SharedToLinear\n<A>, a linear context C, a context lens Nfor accessing SharedToLinear<LinearToShared<A>> from\nC, and an offered session type B. The continuation is a PartialSession with N::Target being\nthe linear context Cwith SharedToLinear<LinearToShared<A>> removed, and offers the session\ntype B. The construct returns a PartialSession with the original linear context C, and offers\nthe session type B.\nB Dynamics\nSection 4 introduced the type system of Ferrite, based on the constructs End,ReceiveValue , and\nReceiveChannel . This section revisits those constructs and fills in the missing implementations\nto make the constructs executable, amounting to the dynamic semantics of Ferrite.\nB.1 One-shot Channels\nInternally, Ferrite uses tokio’soneshot[46] channels as the primitive building block for session-\ntyped channels. A one-shot channel with a payload type Pis consist of a pair of sender and\nreceiver, of type Sender<P> and Receiver<P> , resp., denoting the two endpoints of the channel.\nThe channel is one-shot in the sense that at most one value of type Pcan be sent across the\nchannel. However since the one-shot channel is affine, it is also possible to have no value\nbeing sent over the channel.\nThe one-shot channel can be used directly by Rust programmers to emulate simple session\ntypes. As an example, the session type ReceiveValue<i32, End> can be implemented using\none-shot channels as follows:\nuse tokio::{task, try_join};\nuse tokio::sync::oneshot::{channel, Sender, Receiver};\nasync fn receive_int_provider(value_receiver: Receiver<(i32, Sender<()>)>) {\nlet (value, end_sender) = value_receiver.await.unwrap()\nprintln! (\"provider received value: {}\", value);R. F. Chen, S. Balzer, and B. Toninho 22:41\nend_sender.send(()).unwrap();\n}\nasync fn receive_int_client(value_sender: Sender<(i32, Receiver<()>)>) {\nlet (end_sender, end_receiver) = channel();\nvalue_sender.send((42, end_sender));\nend_receiver.await.unwrap();\n}\nasync fn main() {\nlet (value_sender, value_receiver) = channel();\nlet child1 = spawn( async move {\nreceive_int_provider(value_receiver).await;\n});\nlet child2 = spawn( async move {\nreceive_int_client(value_sender).await;\n});\n}\nThe code above defines the receive_int_provider and receive_int_client functions to\nexecute the provider and client processes corresponding to the session type ReceiveValue<i32,\nEnd>, resp. On the provider side, it needs to first receive an i32value and then send back\nan end signal to the client when it is terminating. This corresponds to the one-shot channel\ntype Receiver<(i32, Sender<()>)> , with the Sender<()> used to send a unit ()as termination\nsignal. On the receiver side, the polarity of the one-shot channel is switched and become\nSender<(i32, Receiver<()>)> . This indicates that the client first sends an i32value, together\nwith a Receiver<()> for the provider to send back the termination signal.\nB.2 Protocol Definitions\nThe above example demonstrates that even a simple session type like ReceiveValue<i32, End>\nrequires non-trivial effort to be implemented manually using one-shot channels. To automate\nthis in Ferrite, we need to derive the one-shot channel types Receiver<(i32, Sender<()>)> and\nSender<(i32, Receiver<()>)> from the session type ReceiveValue<i32, End> . This is achieved\nby defining some associated types and methods in the Protocol trait:\ntrait Protocol {\ntype ProviderEndpoint;\ntype ClientEndpoint;\nfncreate_endpoints() -> ( Self ::ProviderEndpoint, Self ::ClientEndpoint);\n}\nThe associated types ProviderEndpoint and ClientEndpoint are used to define the one-shot\nchanneltypesfortheproviderendandconsumerend, resp. Thetraitmethod create_endpoints\nis used to create a channel pair which connects both the provider and client endpoints.\nFollowing the previous example, the implementation should derive the type <ReceiveValue<i32\n, End>>::ProviderEndpoint to be Receiver<(i32, Sender<()>)> , and <ReceiveValue<i32, End>>::\nClientEndpoint to be Sender<(i32, Receiver<()>)> . This is implemented by first implementing\nProtocol forEnd:\nimpl Protocol for End\n{\ntype ProviderEndpoint = Sender<()>;\ntype ClientEndpoint = Receiver<()>;\nfncreate_endpoints() -> ( Self ::ProviderEndpoint, Self ::ClientEndpoint)\n{\nchannel()\n}\n}\nECOOP 202222:42 Ferrite: A Judgmental Embedding of Session Types in Rust\nIn the implementation of the Endprotocol, the provider end is the party that needs to send\nthe termination signal ()to the client end. Hence its ProviderEndpoint type is Sender<()> ,\nand vice versa for the client end. The implementation of the create_endpoints method is to\nsimply call channel() to create the one-shot channel pair.\nTo implement Protocol for a session type ReceiveValue<T, A> , we would need to make use\nof the Protocol implementation for the continuation session type A:\nimpl <T, A> Protocol for ReceiveValue<T, A>\n{\ntype ProviderEndpoint = Receiver<(T, A::ProviderEndpoint)>;\ntype ClientEndpoint = Sender<(T, A::ProviderEndpoint)>;\nfncreate_endpoints() -> ( Self ::ProviderEndpoint, Self ::ClientEndpoint)\n{\nlet (sender, receiver) = channel();\n(receiver, sender)\n}\n}\nThe provider end is given a receiver for the value T, together with its continuation endpoint\nforA. Given that the continuation for the provider also needs the provider endpoint, and it\nhas to be extracted from the receiver, the provider would need to receive A::ProviderEndpoint\nalongside with the value T. Hence the associated type <ReceiveValue<T, A>>::ProviderEndpoint\nbecomes Receiver<(T, A::ProviderEndpoint)> . On the client side, the value Tneeds to be\nsent alongside with A::ProviderEndpoint , hence the associated type <ReceiveValue<T, A>>::\nClientEndpoint isSender<(T, A::ProviderEndpoint)> . Notice that both the ProviderEndpoint\nand ClientEndpoint associated types for ReceiveValue<T, A> contains A::ProviderEndpoint , but\nnot A::ClientEndpoint .\nIn the implementation of create_endpoints forReceiveValue , the ordering of the sender\nand receiver pair returned from calling channel() is flipped. This is because create_endpoints\nalways return the provider endpoint first followed by the client endpoint. And since the\nprovider endpoint is a receiver in this case, it needs to be returned in the first position.\nWith the Protocol definitions of both Endand ReceiveValue , we can follow that the\nassociated types and channel creation for ReceiveValue<i32, End> matches the channel types\nand behavior of the example at the beginning of this section.\nB.3 Linear Context\nThe linear context of a Ferrite program comprises the client endpoints for the session types.\nConceptually, Ferrite needs to derive from a session type list HList![A0, A1, ...] into a list of\nclient endpoint list HList![A0::ClientEndpoint, A1::ClientEndpoint, ...] . However a linear\ncontext may also contain the special Emptyelement, which do not implement Protocol. To\nallow the transformation of the linear context, we need to first add an associated type to the\nSlottrait as follows:\ntrait Slot {\ntype Endpoint: Send ;\n}\nimpl <A: Protocol> Slot for A {\ntype Endpoint = A::ClientEndpoint;\n}\nimpl Slot for Empty {\ntype Endpoint = ();\n}\nWe define the associated type Endpoint inSlotsuch that if a type Aimplements Protocol,\nthen A::Endpoint is simply A::ClientEndpoint . We also define the special case for Empty, whichR. F. Chen, S. Balzer, and B. Toninho 22:43\nthe Endpoint associated type is ()to represent the absence of a client endpoint. With that,\nwe can extend the Contexttrait to include the Endpoints associated type:\ntrait Context {\ntype Endpoints;\n}\nimpl Context for () {\ntype Endpoints = ();\n}\nimpl <A: Slot, C: Context> Context for (A, C) {\ntype Endpoints = (A::Endpoint, C::Endpoints);\n}\nFor the base case of an empty list ()(HList![]), the result Endpoints is also an empty list.\nFor the inductive case, if the tail Cof a linear context (A, C) implements Context, and the head\nAimplements Slot, then the associated type (A, C)::Endpoints is(A::Endpoint, C::Endpoints) .\nB.4 Session Dynamics\nFerrite generates session type programs by composing PartialSession objects generated\nby constructs such as receive_value . To enable execution of the Ferrite program, the\nPartialSession struct contains an internal executor field that is defined as follows:\nstruct PartialSession<C: Context, A: Protocol> {\nexecutor: Box<\ndyn FnOnce (\nC::Endpoints,\nA::ProviderEndpoint,\n) -> Pin<Box< dyn Future<Output = ()> + Send >>\n+Send ,\n>\n}\nThe executor field contains an FnOnceclosure that accepts two arguments – the endpoints\nfor the linear context C::Endpoints , and the provider endpoint for the offered session type\nA::ProviderEndpoint . When called, the closure executes asynchronously by returning a\nfuturewith the type Pin<Box< dyn Future<Output = ()> + Send >>. The boilerplate signature is\nrequired, as Rust has not stabilized the syntactic sugar for async closures. Conceptually,\nthe closure signature is equivalent to the async function signature async fn (C::Endpoints, A::\nProviderEndpoint) .\nFerrite keeps the executor field private within the library to prevent end users from\nconstructing new PartialSession values or running the executor closure. This is because\nthe creation and execution of PartialSession may be unsafe. We demonstrate two simple\nexamples of unsafe (i.e. non-linear) usage of PartialSession .\nBelow shows an example Ferrite program p1of type Session<SendValue<String, End>> is\nconstructed, but in the executor closure both the client endpoints and the provider endpoint\nare ignored. As a result, p1violates the linearity constraint of session types and never sends\nany string value or signal for termination.\nlet p1: Session<SendValue<String, End>>\n= PartialSession { executor: Box::pin( async |_ctx, _provider_end| { }) };\nBelow shows an example client, which calls a Ferrite program p2of type ReceiveValue<\nString, End> by directly running its executor. The client creates an endpoint pair but drops\nthe client endpoint. It then executes p2with the provider endpoint. However because the\nclient endpoint is dropped, p2fails to receive any value, and the program results in a deadlock.\nlet p2: Session<ReceiveValue<String, End>> = ...;\nlet (provider_end, _client_end) = <ReceiveValue<String, End>>::create_endpoints();\n(p2.executor)((), provider_end).await;\nECOOP 202222:44 Ferrite: A Judgmental Embedding of Session Types in Rust\nFrom the examples above we can see that direct access to the executor field is unsafe.\nThe PartialSession is used with care within Ferrite to ensure that linearity is enforced in\nthe implementation. Externally, the run_session function is provided for executing Ferrite\nprograms of type Session<End> , as only such programs can be executed safely without\nadditional safe guard.\nC Rust as a Host Language\nIn this section, we address some common questions arise from the choice of using Rust as a\nhost language for Ferrite.\nC.1 Benefits of Affine Type System\nThe affine type system in Rust helps Ferrite to better verify the correctness of its underlying\nimplementation. Internally, Ferrite uses one-shot Rust channels to implement the communic-\nation. The affine property in Rust helps us guarantee that our underlying implementation\ncannot accidentally send two payloads through the one-shot channels.\nFerrite user programs also benefit from the affine type system in Rust. Ferrite constructs\naccept continuation closures with the FnOncetrait bound, to guarantee that the continuation\ncannotbecalledmorethanonce. Asaresult, Rustvaluescanbemovedinsidethecontinuation\nclosures and work more efficiently without requiring copies to be made. Similarly, the\nsend/receive value constructs works with the affine type system in Rust, so values such as\nbyte arrays can be sent efficiently in Ferrite without requiring copying.\nIn comparison, while previous works in Haskell and OCaml are able to enforce the linear\nusage in session type programs, the structural semantics of these languages may impose\nchallenge on the compiler from being able to optimize the use of linear resources inside the\nprogram. In particular, the indexed monad that encapsulates the session type program is\nitself copyable. As a result, continuations cannot guarantee that the variables they capture\ncannot be used more than once.\nC.2 Support for Lifetime\nAt the moment, Ferrite requires the continuations to have /quotesingle.ts1staticlifetime. This is due to the\nunderlying async implementations requiring spawned async tasks to have /quotesingle.ts1staticlifetime.\nWe plan to overcome this limitation in the future by finding ways to spawn async tasks with\na scoped lifetime. Once that limitation is overcome, it will also be possible to access mutable\nreferences inside scoped Ferrite programs.\nC.3 Type Errors\nType error messages in Ferrite are expressed in terms of the structs and traits of Ferrite. As\na result it is not difficult for users to read and understand the error messages, provided they\nare familiar with the basic terminology used by Ferrite.\nConsider the example hello_client from section 4\nlet hello_client: Session<\nReceiveChannel<ReceiveValue<String, End>, End>>\n= receive_channel(| a | {\nsend_value_to(a, \"Alice\".to_string(),\nwait(a, terminate())\n) });R. F. Chen, S. Balzer, and B. Toninho 22:45\nIf we were to forget to wait for channel aand terminate immediately, the following error\nis generated:\nlet hello_client: Session<\nReceiveChannel<ReceiveValue<String, End>, End>>\n= receive_channel(| a | {\nsend_value_to(a, \"Alice\".to_string(),\n// the trait /grave.ts1EmptyContext /grave.ts1is not implemented for /grave.ts1(End, ()) /grave.ts1\nterminate()\n) });\nThis indicates that the linear context (End, ()) is not empty, and as a result the terminate\nconstruct cannot be used.\nIf we try to wait for ato terminate before sending a value to a, we get a different error:\nlet hello_client: Session<\nReceiveChannel<ReceiveValue<String, End>, End>>\n= receive_channel(| a | {\n// the trait /grave.ts1ContextLens<(ReceiveValue<String, End>, ()), End, Empty> /grave.ts1\n// is not implemented for /grave.ts1Z/grave.ts1\nwait(a, terminate())\n});\nThe error message indicates an invalid use of a context lens to update a channel of\nthe wrong session type in the linear context. Recall from section 3.2 that the constraint\nZ: ContextLens<(ReceiveValue<String, End>, ()), End, Empty> would require the first channel\n(Z) in the linear context ( (ReceiveValue<String, End>, ()) ) to be of session type End, but here\nthe session type of the first channel in the linear context is ReceiveValue<String, End> .\nError messages such as the above are commonly generated by non-linear use of channels\nor a mismatch in session types. While they require some understanding of the concepts such\nas linear context and context lenses, the error messages are not too difficult to decipher.\nC.4 Hole Driven Development\nAside from designing readable error messages, we recommend a hole-driven approach of\nwriting Ferrite programs to minimize the chance of users encountering complex type errors.\nIn this approach, the user would implement a Ferrite program in small steps, with the\ncontinuation filled with todo!()as a placeholder. We demonstrate this by showing how a\nnew user would implement the hello_provider program in section 4:\nlet hello_provider: Session<SendValue<String, End>> = todo!();\nThe todo!()macro allows us to put a placeholder in unfinished Rust code so that we can\ntry and compile the code and see if there is any type error. By writing our code step by step\nand filling the blank with todo!(), we can narrow down the potential places where our code\nis incorrect. At this stage, we should be able to compile our program with no error. This\nshows that the protocol that we have defined, SendValue<String, End> , is a valid session type.\nIf we have gotten a compile error otherwise, it could have been caused by us trying to write\nan invalid protocol like SendValue<String, String> .\nWe can try to compile our code again, and Rust will accept the code we have written.\nHowever the use of todo!()does not tell us how we should continue our program. In Rust,\nwe could use the unit type ()to deliberately cause a compile error:\nlet hello_provider: Session<SendValue<String, End>> =\nsend_value(\"Hello World!\".to_string(), ());\nNow if we compile our code, we would get a compile error from Rust:\nECOOP 202222:46 Ferrite: A Judgmental Embedding of Session Types in Rust\nerror[E0308]: mismatched types\n|\n| send_value(\"Hello World!\".to_string(), ());\n| ^^ expected struct PartialSession, found ()\n|\n= note: expected struct PartialSession<(), End>\nfound unit type ()\nWith this compile error, we can know that we are supposed to fill in the hole with Rust\nexpression that has the type PartialSession<(), End> . Sometimes we may also intuitively\nthink of a type that should be in a hole. In such case, we can also use the todo!() asT\npattern to verify if our intuition is correct. So we can for example write:\nlet hello_provider: Session<SendValue<String, End>> =\nsend_value(\"Hello World!\".to_string(), todo!() asSession<End>);\nAnd our code will compile successfully. If we were to annotate it with an invalid type,\nsuch as todo()! asSession<ReceiveValue<String, End>> again, Rust will also return a compile\nerror. Now that we know the continuation needs to have the type Session<End> , we can then\nfill in the blank with terminate() and complete our program.\nD Challenges in Using Ferrite on Servo\nWe report on some of the challenges that we faced when implementing the Servo canvas\ncomponent in Ferrite in Section 7, and how the challenges are addressed.\nD.1 Interprocess Communication\nAs a browser rendering engine, Servo puts much emphasis on security, using sandboxing to\nensure that malicious web applications cannot easily compromise a user’s computer. A main\ndesign outcome of this emphasis is that the provider and client are executed in separate\nOS processes. Regular Rust channels cannot be used for communication between different\nprocesses, because the underlying implementation requires a common address space. As a\nresult, Servo uses the ipc_channel crate to create inter-process communication (IPC) channels\nfor communication between the provider and client of its components. The IPC channels\nin Servo create a local file socket and serialize the Rust messages to send them over the\nsocket as raw bytes. This requires the channel payload types to implement the Serialize and\nDeserialize traits for them to be usable in the IPC channels. IPC channels are themselves\nserializable, so it is possible to send an IPC channel over another IPC channel.\nSince Ferrite internally makes use of tokiochannels for communication, this presents\nchallenges since they cannot be serialized and sent through Servo’s IPC channels. For the\npurpose of the evaluation, we implemented our own serialization of SharedChannel . Our\nserialization involves creating a bidirectional pair of opaque (untyped) IPC channels, and\nforwards all communication from the regular Rust channels to the IPC channels. This\napproach works, albeit inefficiently, as there needs to be two background tasks in the provider\nand client processes to perform the actual serialization and forwarding. We benchmarked\nthe performance of our implementation, revealing a decrease of about a factor of ten. We\nhave not spent much effort on fine-tuning our serialization implementation because the\nprimary purpose of this study is to show that the message protocols underlying Servo’s\ncanvas component can be made explicit and verified in Ferrite.R. F. Chen, S. Balzer, and B. Toninho 22:47\nD.2 Latency in Acquire-Release\nServo’s canvas component has very high performance demands, requiring the sending of\nthousands of messages in a few milliseconds. In our initial implementation, we found the\nFerrite implementation to be lacking in performance, despite not saturating the CPU usage. A\ncloser inspection revealed that the bottleneck was in the latency caused by the acquire-release\ncycle introduced in the implementation of shared session types. In Ferrite, the client of a\nshared channel needs to first send an acquire to the shared provider and then wait for the\nacknowledgment before it can start communicating through the acquired linear channel. This\nround trip latency becomes significant if the communication frequency is high. Consider\ntwo canvas messages being sent right after each other. In the original design, the second\nmessage can be sent immediately after the first message has been sent. In the Ferrite\nimplementation, on the other hand, the two messages are sent in two separate acquire-release\ncycles, interspersing additional acquire and release messages and possibly delays because of\nblocking acquires.\nThe latency is aggravated by the use of IPC channels. Since IPC channels are mapped to\nfile sockets, efficient parallel communications must be multiplexed among a small number\nof channels. For the case of Ferrite shared channels, the multiplexing currently is done by\nqueuing and forwarding the requests in serial, which can be inefficient. As a workaround,\nwe batch messages on the client side, such that trivial messages like LineToare stored in a\nlocal Vec<CanvasMessage> buffer before being sent to the provider in a new Messages branch\ninCanvasOps . The buffered messages are sent in batch every ten milliseconds, or when a\nnon-trivial protocol such as GetImageData is called. With batching, we have gained enough\nperformance to render complex canvases smoothly.\nECOOP 2022"    },    {        "title": "1108.6256v1.Catalytic_effect_of_the_spinel_ferrite_nanocrystals_on_the_growth_of_carbon_nanotubes.pdf",        "content": "1 \n Catalytic effect of the spinel ferrite nanocrystals  on the growth of \ncarbon nanotubes  \nR. Hosseini Akbarnejad, V. Daadmehr*, F. Shahbaz Tehrani, F. Aghakhani, and S. Gholipour  \nMagnet a nd Superconducting Research Lab , Department of Physics, Alzahra University, Tehran 19938 , IRAN  \nAbstract  \nWe prepared three ferrite nanocatalysts : (i) copper ferrite ( CuFe 2O4), (ii) ferrite where  cobalt was  \nsubstituted by nickel ( NixCo1-xFe2O4, with x= 0, 0.2, 0.4, 0.6) , and (iii) ferrite where  nickel was  \nsubstituted by zinc ( ZnyNi1-yFe2O4 with y= 1, 0.7, 0.5, 0.3 ), by the sol -gel method. The X-ray \ndiffraction patterns show that the ferrite samples have been crystallized in the cubic spinel structural \nphase.  We obtained the grain size by FE -SEM images in the range of 10-70 nm , and the ir magnetic \nproperties  by VSM. Ne xt, carbon nanotubes were grown on these nanocatalysts by the CCVD method. \nWe show  that the catalytic effects of these nanocrystals on the carbon nanotube growth depend on \ncation distributions in the octahedral and tetrahedral sites , structural isotropy and catalytic power due \nto cations . Our study can have applications in finding a suitable candidate of doped f errite nanocrystals  \nas catalysts  for carbon nanotube growth. More interestingly , the yield of the fabrication of carbon \nnano tubes  can be considered as an indirect tool to study catalytic activity of ferrites.   \nKey words: carbon nanotubes 61.48.De, catalysis 81.16.Hc, chemical vapor deposition 81.15.G h, \nferrites 75.50.Gg, sol -gel processing 81.20.Fw, superparamagnetic 75.20. -g \n1. Introduction  \n      In the spinel ferrites of MFe 2O4, the metallic cations M2+ and Fe3+ can occupy octahedral \nand tetrahedral sites. If the M2+ cations occupy tetrahedral sublattices in the cubic closed - \npacked O2- lattice, the spinel ferrite is a normal spinel , otherwise , the ferrite  is an inverse \nspinel . If both of  the sublattices contain  M2+ and Fe3+ cations , the ferrite  is a mixed spinel. The \noccupations of cations at these sites have an important effect on the properties of spinels , such \nas magnetic behavior, conductivity and catalytic activity [1-3]. Mixed nickel ferrites with \ndifferent magnetization and various cation distributions form an important class of magnetic \nmaterials  [4]. \n     Ni/Zn ferrites have the mixed spinel structure with the unit cell consisting of eight unit s of \nthe form [Zn x2+ Fe1-x3+]tet [Ni1-x2+ Fe1+x3+]oct O42-. The Zn2+ cations preferably occupy the \ntetrahedral sites and  the Ni2+ cations always occupy the octahedral sites  [5]. In Ni /Co ferrites 2 \n the cation distribution of Co2+ depends on heat treatment [6, 7]. As an application, spinel \nferrites can acts as good catalyst  [8]. The catalytic power of these materials  was studied in \nsome chemical reactions such as decomposition of hydrogen peroxide [9], oxidation of carbon \nmonoxide  [10], and oxid ative dehydrogenation of but ene [11]. Chemical composition, crystal \nstructure, electr onic, electrochemical , and micro  structural  factors have been found to \ncontribute to the overall activity of such catalysts [9]. Since the nano -sized catalysts  with non -\nzero magnetic moment  are widely used for growth of carbon nanotubes  (CNTs),  it is of \nparamount  practical and theoretical importance to investigate the catalytic effect of spinel \nferrite  nanocrystals  on the growth of CNTs.   \n     In this paper , we stud y the catalytic effect of Ni /Co ferrites (NixCo1-xFe2O4 with x= 0, 0.2, \n0.4, 0.6 ), Ni/Zn ferrites (ZnyNi1-yFe2O4 with y= 0.3, 0.5, 0.7, 1 ), and copper ferrite (CuFe 2O4) \non the growth of CNTs by catalytic chemical vapor deposition  (CCVD) method. Specifically , \nwe investigate the occupa tion effect of  ferromagnetic ions Fe3+, Co2+, Ni2+ and non -magnetic \nions such as Cu2+ and Zn2+ in the tetrahedral and octahedral sites of fe rrospinels on the ir \ncatalytic activity  for growth of CNTs.  \n \n2. Experiment  \n2.1. Preparation of catalyst  \n     The sol -gel method is one of the best procedures for fabricat ing the ferrite nanocrystals.  \nThus in our experiment, Ni/Co, Ni /Zn, and Cu ferrites were  all prepared  by this method. In \nthis way, with the sto ichiometric laws  (depending on  the combination ), we prepare d 0.5M \nsolutions of Fe(NO 3)3.9H 2O (98%) , Co(NO 3)2.6H 2O (98%) , Ni(NO 3)2.6H 2O (99%) , \nZn(NO 3)2.3H 2O (99%) , and Cu(NO 3)2.3H2O (99%) ; and add ed these solutions to 0.5 M \nsolution of citric acid with 1  : 1 mol ratio for nitrates  : citric acid. The pH value of the \nsolution was adjusted to 1 by ethylenediamine , in order to make the environment more \nconducive for generating fine particles . The prepared solution was baked in 70°C to form  a \nbrown gel. The obtain ed gel was dried in 135°C during 24 hours and was ground into a fine \npowder. Finally , the sample was calcined in 450 -600°C  range (depending on the doping value \nof the ferrite s) for 4 hours.  We characterized the structure and grain size of nanocrystals by \nPhilips® PW1800 X-ray Diffraction (XRD)  with Cu  Kradiation  (λ=1.54056Å)  and Hitachi® \nS4160 Field Emission Scanning Electron Microscopy (FE -SEM). The magnetic properties of 3 \n the samples  were measured by Meghnatis Daghigh Kavir Co .® Vibrating Sample \nMagnetometer (VSM) in the room temperature (~25ºC) .  \n2.2. Growth of carbon nanotubes  \n     The CNTs were grown by the CCVD  method in a quartz reactor with a programmable \nfurnace . The carbon source was acetylene  (C2H2) with argon  (Ar) as the carrier gas. To \nsynthesize, an alumina boat containing 0.05g of catalyst was placed in the hot zone of the \nquartz reactor. A mixture of C 2H2 and Ar with 1:8 volume ratios was passed over the catalys t \nin atmospheric pressure . The temperature was risen with the rate of 5°C/min to a specific \ntemperature depending on  the crystallization temperature  of each catalyst and the reaction \ntime was 45  minutes. Since  the cation distribution of ferrites depends on  heat treatment  [6, 7], \nthe growth temperatures were selected equal the calcination  temperatures. The amount of \ncarbon  was evaluated by  the mass of the fabricated  samples. These sample s contain  \namorphous carbon, CNT s, and catalyst. In order to remove  the amorphous carbon s, we \noxidized the samples in the air at 400°C for 1 hour .  \n3. Results and discussion  \n    We studied the crystalline structure of  the ferrite nano catalysts with the XRD analysis. \nFigure 1 shows the XRD pattern of the prepared nanocrystals.  These patterns were compared \nwith the Joint Committee on Powder Diffraction Standards ( JCPDS ). The presence of (220), \n(311), (400), (422), (511) , and (440) major lattice planes revealed the cubic spinel phase with \nFd3m space group. In addition,  the minor lattice planes of (111) and (222) are present . These \nresults emphasize the presence of only spinel phase without any  significant impurities.  The \nXRD pattern of samples was refined by using the MAUD software and Reitveld ’s method for \nstructural analysis, cation distribution , and lattice parameter calculation. Crystallographic \nproperties of the  samples  were obtained from this calculation and are listed in Table 1. This \nstudy indicates  the inverse spinel structure for Ni /Co a nd copper ferrite s. This means that in \nthe Ni/Co ferrites, the Ni2+ and Co2+ cations occupy the octahedral sites , and the Fe3+ cations \noccupy the octahedral and tetrahedral sites equally; and , in the copper ferrite  the Cu2+ cations \nfill only the half of  the octahedral sites. This study shows  as well  the normal spinel structure \nfor the zinc ferrite and mixed spinel structure for the Ni/Zn ferrites because  the Ni2+ cations \noccupy the octahedral and the Zn2+ cations occupy the tetrahedral sites. Notice that these \nresults are consistent with Ref. [5]. 4 \n      The l attice constant of the samples  were  obtained from the formula a=d (h2+k2+l2)1/2, \nwhere d is the interplane spacing and is calculated from the position of the highest peak in the \nXRD pattern  (here (311) plane ) by Bragg's formula  (and refined  later by using the MAUD \nsoftware ). In Table 1 , we observe that the lattice constant  increase s with the Co content in the \nNi/Co ferrites and the Zn content in the Ni /Zn ferrites  increase . This can be attributed to the \nhigher ionic radii of Co2+ (0.79 Å)  and Zn2+ (0.82 Å)  compared to Ni2+ (0.69 Å).  \n     The magnetic properties of the nanocrystals were measured by VSM at the room \ntemperature  (~25ºC) . the c oercivity (H c) and the saturation magnetization (M S) of the \nnanocrystals are listed in Table 1. H c of the Ni/Co ferrites decrease s inversely with the  nickel \ncontent  since the coercivity of a magnetic material is a measure of magneto -crystalline \nanisotropy . This decrease  is attributed to the lower magneto -crystalline anisotropy of nickel as \ncompared to cobalt, which in turn leads to a lower coercivity.  Similarly,  the decreas e in M S is \nattributed to the smaller magnetic moment of the Ni2+ as compared to the Co2+. MS of \nCoFe 2O4 is more than CuFe 2O4 as a result of the higher magnetic moment of Co2+ (3µ B) than \nCu2+ (1µB). The Ni /Zn ferrites as a result of near zero Hc show  superparamagnetic behavior s \nat the room temperature . The c ation distribution of  the zinc ferrite, zero magnetic moment of \nthe Zn2+ cations , and the anti-ferromagnetic interactions between the Fe3+ cations in the \noctahedral sites cause the magnetic moment for each ZnFe 2O4 formula  to vanish ; but a low \nmagnetic moment for these nanocrystals was obtained because of the small size of the \nparticles and the presence of  ions on the ir surface. The cation distribution change s by the  \nsubstitution of Zn2+ by Ni2+cations  and occupation of the octahedral sites with Ni2+ cations ; \nthus this substitution  transfer s part of the Fe3+ cations to the tetrahedral sites, and accordingly \nmakes  the MS of the nanocrystals  vary. This is  consistent with  Neel's ferrimagneti sm theory  \n[12]. This behavior of the Ni/Zn ferrite nanocrystals for the different values of M S is similar to \nthat of bulk samples [5]. It is important  to notic e that the maximum M S is found in the \nZn0.5Ni0.5Fe2O4 nanocrystals.   \n     The samples were characterized by XRD after the growth of CNTs.  Figure 2 shows the se \nXRD pattern s. The peaks  attributed  to the cubic spinel structure are present in these patterns \nand represent that  the nanocatalysts did  not change  during the growth of the CNTs.  The (002), \n(100),  (101), (004) , and (110)  peaks are related to the  presence of the  CNTs. The presence of \nthe (002) plane between 2θ=26 -26.5° is due to the presence of multiwall CNTs  (MWNTs) \n[13]. According to  Ref. [14], the (110) and (100) peaks are in the (hk0) group peaks and \ndisplay  an asymmetric shape due to the curvature of the CNT . The (004) and (101) reflections  5 \n are also due to flat graphitic layers, residual carbon particles , and the defect of  the CNTs  [15, \n16]. \n    The FE-SEM images of the nanocatalyst s are shown in Fig. 3. We observe  that the \nprepared nanocrystals have a spherical morphology , and cohesion of particles is due to the \nmagnetic attraction.  The catalytic power  of the spinel ferrite nanocrystals  for the growth of the \nCNTs is evaluated by the rate of production of the CNTs in the surface unit of all \nnanocatalysts . The average grains  diameters of the nanocrystals were  obtained from the FE-\nSEM  images and are listed in Table 2. In addition,  the surfaces/ gram  ratio of the nanocatalysts \nwere  found from dividing the average surface of a particle by its mass. The catalytic powers \nof these catalysts were  obtained by normaliz ing the amount of the samples to unit time , \nsurface , and mass of the catalysts  in the crystallization  temperature of each catalyst . \n      In Table 2 , we observe that the catalytic power of the Ni/Co ferrites increase s with the  \nnickel content –as a result of higher catalytic power of Ni in comparison to  Co. Besides,  in the \nNi/Zn ferrites it is observed that Ni0.5Zn0.5Fe2O4 has the highest catalytic power  while  \nZnFe 2O4 comes next . The grown CNTs on ZnFe 2O4 are more pure than CNTs grown on \nZn0.5Ni0.5Fe2O4. As mentioned above, i n the ZnFe 2O4 nanocatalyst, Zn2+ cations occupy the \ntetrahedral sites and all of these sites are in the same conditions . Thus  the octahedral sites  are \nisotropic . This structural isotropy enhances the catalytic power. With the decrease in the Zn2+ \ncontent and entrance of the Ni2+ cations in to the structure, the Ni2+s occupy the octahedral \nsites and transfer part of Fe3+ to the tetrahedral sites. These changes cause a decrease  in the \nstructural isotropy and the catalytic power of Ni 0.3Zn0.7Fe2O4 and Ni 0.7Zn0.3Fe2O4 \nnanocatalysts . Because  of the occupation of the half of tetrahedral sites with Zn2+ and \nremainder  with Fe3+, the Ni0.5Zn0.5Fe2O4 nanocatalyst has a structural isotropy , and th us the \ncatalytic power  increases . This isotropy exists completely in the tetrahedral sites of the \ninverse spinels . This fact makes  the catalytic power of the copper ferrite nanocrystals  increase \ncompare d to zinc ferrite  - see Ref. [11], where the oxidati ve dehydrogenation  of butene  in the \npresence of ferrospinel catalysts  have been considered . Moreover,  the catalytic power of the \nNi/Co ferrites is more than the copper ferrite because  the catalytic power due to Ni  and Co in \nthe growth of CNTs is high in comparison to Cu . \n     Figure 4 shows the FE -SEM images of the CNTs obtained on these nano catalysts. The \nparticles of the nano catalysts are observed in the top of the CNTs. The CNTs are not \nuniformly straight because  of catalyst particle movements during the growth process  [17]. \nThese movements induce  structural defects that were observed in the XRD patterns by the \n(101)  and (004)  peaks.  6 \n 4. Conclusion  \n     In summary, we prepared Ni /Co, Ni /Zn, and Cu ferrites by the sol-gel method. XRD \nshowed the cubic spinel structure for all of the se ferrites . The MAUD analysis on the XRD \npatterns confirm ed the inverse spinel structure for  the Ni/Co and Cu ferrites, the normal spinel \nfor Zn ferrite , and the mixed spinel for the Ni/Zn ferrites.  Magnetic properties of these \nnanocrystals were  measured by VSM  at the room temperature . M S and H c of the Ni/Co ferri tes \nwere shown to decrease with the increase of the n ickel content because the magnetic moment \nof Ni2+ as compared to Co2+ and magneto -crystalline anisotropy of Ni  as compared to Co are \nlower. The behavior of  the Ni/Zn ferrite nanocrystals with different values of MS appeared \nsimilar to that in  bulk samples . The maximum M S was found in the Zn 0.5Ni0.5Fe2O4 \nnanocrystals . The catalytic power s of these nanocatalysts were obtained from the growth of \nthe CNTs on them.  We found that the catalytic power of the spinel ferrites is relate d to the \nstructural isotropy, the cation distribution , and the catalytic power of cations. Hence, the \ncatalytic power of the Ni/Co ferrites increase s with the increase in the Ni content due to \nhigher catalytic power of Ni compared to  Co. In the Ni/Zn ferrites , the structural isotropy is \nan effective factor for their catalytic power . The catalytic power of Cu ferrite is higher than \nthe Ni/Zn ferrites since it is inverse spinel , and less than Ni/Co ferrites as a result of the \ncatalytic power due to cations.   \n     Briefly , we have obtained that the catalytic power has the follow ing order Ni 0.6Co0.4Fe2O4 \n> Ni 0.4Co0.6Fe2O4 > Ni 0.2Co0.8Fe2O4 > CoFe 2O4 > CuFe 2O4 > Zn 0.5Ni0.5Fe2O4 > ZnFe 2O4 > \nNi0.7Zn0.3Fe2O4 > Ni 0.3Zn0.7Fe2O4. \nAcknowledgement  \n     The authors acknowledge the  Iranian Nano Technology Initiative Council and Vice Chair \nfor research of Alzahra University.  We like to extend our thanks and appreciation to Dr. A. \nRezakhani  for useful discussion.  \nReferences  \n[1] I.B. Bersuker , Electronic  Structure and Properties of Transition Metal Compounds: Introduction to the Theory  (New \nYork: Wiley)  (1996 ). \n[2] R. Burns , Mineralogical Applications of Crystal Field Theory (Cambridge: Cambridge University Press) vol 5 (1993 ). \n[3] R.J. Borg , and G.J. Dienes , physical chemistry of solids (Academic: San Diego, CA)  (1992 ). \n[4] A.M.  El-Sayed , Electrical conductivity of nickel –zinc and Cr substituted nickel –zinc ferrites , Mat. Chem. phys. 82 \n(2003)583 -587. \n[5] A. Goldman,  Modem Ferrite Technology , 2nd edition, Pittsburgh, PA, USA , (1987 ), p.63 -71, ISBN 10: 0 -387-29413 -9. \n[6] T.A.S. Ferreira , J.C. Waerenborgh, M.H.R.M.  Mendonca ,  M.R.  Nunes , and F.M. Costa , Structural and morphological \ncharacterization of FeCo 2O4 and CoFe 2O4 spinels prepared by a coprecipitation method , Sol. State Sci. 5(2003 ) 383-392. 7 \n [7] P.J. Murray, J.W. Linn ette, Cation distribution in the spinels CoxFe3−xO4, J. Phys. Chem. Solids  37(1976)  1041 -1042 . \n[8] P. Lahiri, S.K. Sengupta, Spinel ferrites as catalyst: A study on catalytic effect of coprecipitated ferrites on hydrogen \nperoxide decomposition, Can. J. Chem . 69 (1991)  33-36. \n[9] J.R. Goldste in, A.C.C.  Tseung , The kinetics of hydrogen peroxide decomposition catalyzed by cobalt -iron oxides , J. \nCatal ysis, 32(1974) 452-465. \n[10] K.R. Krishnamurthy, B. Viswanathan, M.V.C.  Sastri,  Catalytic  activity of transition metal spinel type ferrites: Structure -\nactivity correlations in the oxidation of CO,  J. Res. Inst. Catalyst, Hokkaido Univ. , 24(1976)  219-226. \n[11] F.E. Massoth, P.A. Scarpiello,  Catalyst characterization studies on the Zn ---Cr---Fe oxide system , J. Catalysis , 21 \n(1971)  294-302. \n[12] L. Néel, Magnetic properties of fe rrites: ferrimagnetism  and antiferromagnetism,  Ann. Phys. Paris 3(1948 )137-98. \n[13] T. Belin, F. Epron,  Characterization methods of carbon  nanotubes: a review,  Mat. Sci. Eng. B 119 (2005) 105 –118. \n[14] P. Lambin, A. Loiseau, C. Culot, L. Biro, Structure of carbon nanotubes  probed by local and global probes , Carbon  40 \n(2002) 1635 -1648 . \n[15] M. Liu, J. Cowley,  Structures of the helical carbon nanotubes , Carbon  32 (1994) 393 -403. \n[16] D. Bernaerts, S. Amelinckx, P. Lambin, A. Lucas, The diffraction space of circular and polygonized multishell \nnanotubules , Appl. Phys. A 67 (1998) 53 -64. \n[17] J.L. Figueiredo , J.J.M. Orfao, A.F. Cunha,  Hydrogen production via methane decomposition on Raney -type catalysts,  \nInt. J. Hydrogen Energy 35(2010)9795 -9800.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 8 \n  \n \nFigures:  \nFig. 1. The XRD pattern of the ferrite nanocatalysts  \nFig. 2. XRD patterns of the CNTs obtained on (a) ZnFe 2O4 (b) CoFe 2O4 and (c) CuFe 2O4 (F: peaks are relate d to \nferrites)  \nFig. 3. FE -SEM images from the nanocatalysts (a) Ni 0.4Co0.6Fe2O4 (b) CuFe 2O4 (c) ZnFe 2O4 (d) CoFe 2O4 \n \nFig. 4. FE -SEM images from the grown CNTs on (a) Ni 0.4Co0.6Fe2O4 (b) CuFe 2O4 (c) ZnFe 2O4 (d) CoFe 2O4 \n \n \n \n \n \n \nTables:  \nTable  1. Magnetic and crystallographic properties of ferrite nanocatalysts  \n \nTable  2. Dimensions and cat alytic power of ferrite nanocatalysts  \n \n \n \n \n \n \n \n \n \n \n 9 \n  \n                Fig.1. XRD pattern of the ferrite nano catalysts  \n  \nFig.2. XRD patterns of the CNTs obtained on (a) ZnFe 2O4 (b) CoFe 2O4 and (c) CuFe 2O4 (F: peaks are related  to ferrites)  \n \n \nCatalytic effect of the spinel ferrite nanocrystals on the growth of carbon nanotubes  \nR. Hosseini Akbarnejad, V. Daadmehr*, F. Shahbaz Tehrani, F. Aghakhani, and S. Gholipour  \n11 \n   \n  \n \nFig.3. FE -SEM images from the nanocatalysts  (a) Ni 0.4Co0.6Fe2O4 (b) CuFe 2O4 (c) ZnFe 2O4 (d) CoFe 2O4 \n \n \n \n \n \n \n \n \n \n \n \n \nCatalytic effect of the spinel ferrite nanocrystals on the growth of carbon nanotubes  \nR. Hosseini Akbarnejad, V. Daadmehr*, F. Shahbaz Tehrani, F. Aghakhani, and S. Gholipour  \n \na b \nc d 11 \n   \n  \n \nFig.4. FE-SEM images from the grown CNTs on (a) Ni 0.4Co0.6Fe2O4 (b) CuFe 2O4 (c) ZnFe 2O4 (d) CoFe 2O4 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nCatalytic effect of the spinel ferrite nanocrystals on the growth of carbon nanotubes  \nR. Hosseini Akbarnejad, V. Daadmehr*, F. Shahbaz Tehrani, F. Aghakhani, and S. Gholipour  \n \na b \nc d 12 \n  \nTable  1. Magnetic and crystallographic properties of the ferrite nano catalysts  \nFerrite  Cation distribution  Degree of \ninversion  a (Å)  MS (emu/g)  Hc (Oe) \nCoFe 2O4 [Fe3+]tet [Co2+,Fe3+]oct 1 8.389(9)  69.86  1186  \nNi0.2Co0.8Fe2O4 [Fe3+]tet [Ni2+\n0.2, Co2+\n0.8 ,Fe3+]oct 1 8.376(8)  58.05  1013  \nNi0.4Co0.6Fe2O4 [Fe3+]tet [Ni2+\n0.4, Co2+\n0.6 ,Fe3+]oct 1 8.356(0)  47.96  877 \nNi0.6Co0.4Fe2O4 [Fe3+]tet [Ni2+\n0.6, Co2+\n0.4 ,Fe3+]oct 1 8.355(8)  27.12  400 \nZnFe 2O4 [Zn2+]tet [Fe3+,Fe3+]oct 0 8.430(0)  1.65 3.31591  \nNi0.3Zn0.7Fe2O4 [Zn2+\n0.7, Fe3+\n0.3]tet [Ni2+\n0.3,Fe3+\n1.7]oct 0.3 8.400(2)  28.43  0.31242  \nNi0.5Zn0.5Fe2O4 [Zn2+\n0.5, Fe3+\n0.5]tet [Ni2+\n0.5,Fe3+\n1.5]oct 0.5 8.389(9)  34.83  0.14973  \nNi0.7Zn0.3Fe2O4 [Zn2+\n0.3, Fe3+\n0.7]tet [Ni2+\n0.7,Fe3+\n1.3]oct 0.7 8.370(0)  25.75  0.12476  \nCuFe 2O4 [Fe3+]tet [Cu2+,Fe3+]oct 1 8.372(2)  14.83  168.156  \n \n \n \nTable  2. Dimensions and catalytic power of the ferrite nano catalysts  \nFerrite  Dave (nm)  S (m2/g) Catalytic power \n(m-2min-1) × 10-3 \nCoFe 2O4 71.91  15.81  2.8286  \nNi0.2Co0.8Fe2O4 53.77  21.04  7.2975  \nNi0.4Co0.6Fe2O4 56.49  19.92  8.7056 \nNi0.6Co0.4Fe2O4 57.08  19.68  10.442 \nZnFe 2O4 34.57  16.23  1.6607  \nNi0.3Zn0.7Fe2O4 42.46  13.19  1.1979 \nNi0.5Zn0.5Fe2O4 25.81  21.73  2.2207 \nNi0.7Zn0.3Fe2O4 24.60  22.77  1.3370 \nCuFe 2O4 37.62  14.72  2.6185 \n \n \n \n \n \n \n \nCatalytic effect of the spinel ferrite nanocrystals on the growth of carbon nanotubes  \nR. Hosseini Akbarnejad, V. Daadmehr*, F. Shahbaz Tehrani, F. Aghakhani, and S. Gholipour  \n "    },    {        "title": "1507.05814v1.Frequency_dependent_effective_permeability_tensor_of_unsaturated_polycrystalline_ferrites.pdf",        "content": "arXiv:1507.05814v1  [cond-mat.mtrl-sci]  21 Jul 2015Frequency-dependent effective permeability tensor of unsa turated\npolycrystalline ferrites\nPascal Thibaudeau1,a)and Julien Tranchida1,2,b)\n1)CEA DAM/Le Ripault, BP 16, F-37260, Monts,\nFRANCE\n2)CNRS-Laboratoire de Math´ ematiques et Physique Th´ eoriqu e (UMR 7350),\nF´ ed´ eration de Recherche ”Denis Poisson” (FR2964),\nD´ epartement de Physique, Universit´ e de Tours, Parc de Gra ndmont,\nF-37200, Tours, FRANCE\n(Dated: July 22, 2015)\nFrequency-dependent permeability tensor for unsaturated poly crystalline fer-\nrites is derived through an effective medium approximation that comb ines\nboth domain-wall motion and rotation of domains in a single consistent scat-\ntering framework. Thus derived permeability tensor is averaged on a distri-\nbution function of the free energy that encodes paramagnetic st ates for an-\nhysteretic loops. The initial permeability is computed and frequency spectra\nare given by varying macroscopic remanent field.\nPACS numbers: 75.50.Gg, 75.60.Ch, 75.78.-n\na)Electronic mail: pascal.thibaudeau@cea.fr\nb)Electronic mail: julien.tranchida@cea.fr\n1I. INTRODUCTION\nThe determination of the frequency-dependent permeability tens or of a mag-\nnetic unsaturated gyrotropic medium such as ferrites materials is a long stand-\ning problem1–3. It has been demonstrated useful to describe precisely this tens or\nin frequency to obtain novel radiation and scattering characteris tics of microstrip\nantennas4. To the physical point of view in dense polycrystalline materials, the\nfrequency behavior of the permeability in the range from 100kHz to 20GHz is under-\nstood as a superposition of magnetization changes due to domain-w alls motion and\ncoherent rotations of magnetic domains1,5. However, Grimes6,7has suggested that\nthe observed spectra may be also interpreted as multiple scatterin g of electromag-\nneticwaves inthesepolycrystalline materialsbybothrandommagnet icanddielectric\nhomogeneous spheres. Unfortunately this deduced interpretat ion has not been tested\nwhen magnetic materials, subject to external dc uniform magnetic field present an\nhysteresis behavior on the magnetization and changes the derived permeability. A\nconvergence of these two alternate mechanisms was first recogn ized by Schl¨ omann8\nwho’s derived a relationship between the isotropic permeability in a com pletely de-\nmagnetized state by computing the scattering of magnetic fields in t he static limit\nby out-of-phase and concentrical cylinders of gyromagnetic sing le-domains. The ex-\ntension of this idea to unsaturated magnetic states was conducte d by Bouchaud\nand Z´ erah9but the bridge between the local remanent magnetization, included as\nfractions of magnetic volumes of scatterers and the observed av erage distribution of\nmagnetization directions of domains in the tridimensional space remin ded elusive.\nTo get into account the anisotropic nature of this scattering prob lem, Stroud10was\npioneer to generalize an old effective-medium approximation (EMA) fo r the con-\nductivity tensor of a randomly inhomogeneous medium to treat mate rials consisting\nof crystallites of arbitrary shape and conductivity tensors of arb itrary symmetry.\n2Derivation of non diagonal permeability tensors for magnetized gra nular composites\nhas been then reported11and several models for the calculation of complex perme-\nability of magnetic composite materials have been proposed12,13. By using the EMA,\neven the influence of porosity induced by non-magnetic inclusions in m agnetized het-\nerogeneous materials has been computed14. However, even in the quasi-static limit\nof frequency, the problem of taking into account dynamically the ma gnetic multiple-\nscattering of both domain-walls and single domains simultaneously in a c onsistent,\nanisotropic formulation is not achieved yet. This is the main purpose o f this paper.\nII. EFFECTIVE MEDIUM APPROXIMATION FOR\nUNSATURATED FERRITES\nBecause electrical non-conducting media are considered, the fer rites are treated\nas uniform from the dielectric point of view. The amplitude of an ac mag netic\nfield applied is sufficiently low to let first the transverse permeability te nsor being\nindependent of it and to generate a small local transverse oscillatin g magnetic field\nonly.\nSaturated ferrites exhibit anisotropic permeability described by th e well known\nPolder tensor15, resulting in the nonreciprocal behavior of microwaves into magne-\ntized ferrite used for the design of circulators and isolators16. As any reference axis\nmay be chosen to project the third component of an arbitrary mag netization vec-\ntor, thez-axis is then taken for that. So in a cartesian frame and for any arb itrary\ndirection, the magnetization vector is represented by only 2 angles , a polar one θ\nand an azimutal one φ. Then in this arbitrary frame, the gyromagnetic permeability\ntensor can be expressed in a general form17,18. Such saturated ferrite is thus char-\nacterized by only three circular permeabilities which come from the re sponse of the\nmagnetization to a rotating ac magnetic field. This corresponds to d iagonalize the\n3general form of the gyromagnetic permeability tensor as first rec ognized by Tyras\na long time ago17. It is always possible to select a frame that puts the saturation\nmagnetization vector along the z-axis and in that case, one of the third component\nof the permeability is equal to 1 and the ferrite is characterized by o nly 2 remaining\npermeabilities.\nFor an unsaturated polycristalline ferrite, the spectral value of t he 3 eigenvalues\nas a function of a given reduced remanent magnetization state m=Mz/Ms≡cos(θ)\npointingalongthe z-axis, isnotsoeasilyfound. Onemaygetthiseffective tensorasa\nstatistical average of the Polder tensor in a non-interacting magn etic grains picture.\nThis strategy was investigated19. Because the distribution of the magnetization\ndirection(themagnetictexture) hastobegiven, andcanchangea safunctionof m, it\nresults that the non-interacting assumption is strictly satisfied fo r a system consisted\nin an assembly of single-crystal spheres situated in a non-magnetic medium far from\none another, a condition rarely observed in dense soft ferrites20,21. No interaction\nbetween domains and domain-walls can be considered in such picture a nd this also\nneglects the influence of the shape of the magnetic domains and por es. So a two\nsteps mechanism is introduced. The magnetostatic interaction bet ween domains has\nto be treated first and then the resulting composite medium has to b e statistically\naveraged.\nFor any value of m, consider that the effective 3 ×3 permeability tensor µecan\nbe diagonalized and expressed in a circular frame by the following diago nal tensor\n(see Appendix A for the notations):\nµe=\nµe+κe0 0\n0µe−κe0\n0 0 µez\n, (1)\nwhereµeandκeare the diagonal and off-diagonal components of the permeability\n4expressed in a cartesian frame17.µezis the permeability value which connects both\nthe magnetization and the ac field along the z-axis.\nA. Two-phase model\nBy using the same argument to the scale of a magnetic ”grain”, let us consider\n”up” and ”down” borderless single domain grains which differ only by th e direction\nof their magnetization vector and thus by a change of a sign. Their r espective\npermeability tensors are\nµ1=\nµ+κ0 0\n0µ−κ0\n0 0 µz\nandµ2=\nµ−κ0 0\n0µ+κ0\n0 0 µz\n.(2)\nNow, one consider that each borderless domain can be represente d with Polder ex-\npressions in frequency. Such analytical expressions of µ,κandµzare determined\nfrom the solution of the Landau-Lifchitz equation in the small signal approximation\nsuch as :\nµ= 1+η−ıαΩ\n(η−ıαΩ)2−Ω2, (3)\nκ=Ω\n(η−ıαΩ)2−Ω2, (4)\nµz= 1, (5)\nwithη=Hk/Msthe reduced anisotropy field assuming an uniaxial symmetry, Ω =\nω/γµ0Msthe reduced pulsation with γthe gyromagnetic ratio, Msthe saturation\nmagnetization, µ0the permeability of free space and αis the damping constant.\nIf the domain structure (”up” and ”down”) is distributed such as o ne cannot dis-\ntinguish any direction in the perpendicular plane to the magnetization on a macro-\nscopic scale, then the permeability tensor must be diagonal with res pect to such a\n5rotatingfield8. To getthat, onemayconsider domainsintheshapeofinfinitecircula r\ncylinders8. The demagnetizing tensor for a single cylinder embedded in a effectiv e\nanisotropic medium in the quasi-static approximation is then\nΓ =\n−1\n2µe0 0\n0−1\n2µe0\n0 0 0\n. (6)\nEven when the medium is characterized by an anisotropic tensor, Eq .(A12) yields a\ndiagonal Γ tensor with Γ 11= Γ22for both cylindrical and spherical crystallites. In\nour situation, Γ is invariant through any cylindrical rotation around the z-axis and\npreserves this structure in the rotating frame.\nNowsome EMAprocedurehastobeapplied inorder togettheeffectiv e quantities\n(µe,κe,µez) as functions of ( µ,κ,µ z) andm. It is well known that the EMA depends\non the choice of the reference permeability22. Among these choices, the strong-\ncoupling interaction between domains is assumed and the symmetric s elf-consistent\nsituation is so retained. In our situation, this corresponds to let eq uation (A15) to\nbe zero. For each two-phase grain, the local magnetization of the composite medium\nism= (v1−v2)/(v1+v2) wherev1(resp.v2) is the volume occupied by the portion\nof the grain where the magnetization is pointing in the z-direction (resp. opposite\nto). Because the total volume is v1+v2=V, one can evaluate v1=V(1+m)/2 and\nv2=V(1−m)/2. Thus Eq.(A15) reads\n(1+m)[1−Γ(µ1−µe)]−1(µ1−µe)+(1−m)[1−Γ(µ2−µe)]−1(µ2−µe) =0.(7)\nBy substituting matrices (1), (2) and (6) into equation (7), µe,κe,µezare connected\nwithµ,κ,µzandmas\nµ2\ne=µ2µ2−κ2\nµ2−m2κ2(8)\n6κe=mκµe\nµ(9)\nµez=µz (10)\nRemarkably when m=±1,µe=µ,κe=κandµez= 1, which implies that the\nPolder tensor is recovered to the saturation limit. Equation (8) and (9) agree with\nequations (3a) and(3b)of reference9which state that forany cylindrically symmetric\nbutotherwise arbitraryconfiguration, thisEMAproceduredoesn otdependuponthe\ndetailsofthedomainconfiguration8,23. Equation(9)alsoagreeswithseveral works1–3\nwhich relies linearly the static magnetization along the z-axis with the value of the\noff-diagonal permeability, at least for small values of m.\nAs a consequence, it is worth noting that for an effective two-comp onents medium\nand any diagonal surface tensor Γ, the demagnetized situation m= 0 always gives\nκe= 0 and the following implicit equation for µe:\nκeµe2Γ22(µe,µez)+(1−2µΓ22(µe,µez))µe+Γ22(µe,µez)(µ2−κ2)−µ= 0,(11)\ndepends only on the single component Γ 22= Γ11. This result is a generalization of\nEq.(8) to any geometry and anisotropic embedded medium resulting in a diagonal\nsurface tensor. By inserting Eq.(6) into Eq.(11), the equation (8) is recovered.\nB. Three-phase model\nNeglecting any magnetic after-effect or magnetic viscosity24, the permeability\nspectra are described by two types of magnetizing processes: gy ration of domains\nand domain-wall motion25which are generally analyzed separately26. In order to\ntake into account the lower frequency response of the previously introduced biphasic\nmedium, the permeability response for a domain-wall has to be consis tently added.\nIts spectral response is commonly assumed to be modeled by an isot ropic expres-\nsion of permeability27–31that follows from a stiff and dampened 180odomain-wall\n7motion25:\nµdw= 1+χdwη2\ndw\nη2\ndw−Ω2−ıαdwηdwΩ. (12)\nHereηdw=Hdw/Msis the reduced domain-wall resonance field, αdwis a damping\nfactorand χdwrepresents itsstaticsusceptibility. Toallowthespectralreprese ntation\nofsuch permeability tobereduced totwo parametersonly, itsfreq uency permeability\nmay be approximated as a single uniaxial magnetic domain distribution in the case\nof small damping and if χdw��1/ηdw. In order to let the static susceptibility of\nthe domain-wall to be higher than the value of a magnetic domain, it is a ssumed\nthatηdwreturns a smaller value of the anisotropy field in the domain-wall than the\nanisotropy field ηcharacterizing a domain.\nThe goal is to treat on the same footing the dynamics of interacting domains and\ndomain-wall motion for a given unsaturated value of magnetization m. An EMA\nis thus constructed with a three components system. ”Up” and ”d own” domains\nare considered supplemented by a domain-wall, each component emb edded in an\nanisotropic effective medium with a cylindrical geometry. The homoge nization pro-\ncedure is depicted for a composite medium made up of uniaxial anisotr opy particles\non Fig. (1). This gives the following implicit equation for µe\n3/summationdisplay\ni=1vi/bracketleftig\n1−Γi(µi−µe)/bracketrightig−1\n(µi−µe) =0, (13)\nwhereµ1andµ2are the permeability tensors of domains and µ3≡µdw1. Here vi\nare the volume fraction of each component. As a consequence of t he cylindrical\ngeometry, the same surface matrix Eq.(6) is taken into account fo r all the embedded\nmedia. The fraction volumes have to be normalized according to lim m→±1v3= 0\nand one assumes\nv1≡1+m\n2(1+a(1−m2)),v2≡1−m\n2(1+a(1−m2)),v3≡a(1−m2)\n1+a(1−m2),(14)\n8«Up» domain «Down» domain Domain-WallEffective medium\n0=v1 +v2 +v3\nFigure 1. Schematization of the homogenization procedure f or a composite medium made\nup of uniaxial anisotropy domains and domain-wall.\nwhereadescribes some proportionof domain-walls in the whole system. When a= 0\nthe previous two-phase situation is recovered and the limit case a→ ∞provides\nµe=µdw. Because Γ 33is here zero, the zz-component of equation (13) gives exactly\nµezand reads\nµez=µz+aµdw(1−m2)\n1+a(1−m2). (15)\nOne shows that µez=µzwhena= 0 orm=±1 as expected. In the demagnetized\nstatem= 0,µezis the volume addition of both the permeability of the domain\nand domain-wall by reason of the superposition of the z-component of the magnetic\ninduction in all media. When κe= 0,µeis the solution of a single equation in (13)\n9which is written directly in a third-order polynomial form\nµe3+µdw(1−a)+2aµ\n1+aµe2−(µ2−κ2)(1−a)+2aµdwµ\n1+aµe−µdw(µ2−κ2) = 0.(16)\nThe 3-body composite medium appears isotropic but the value of µedepends sur-\nprisingly on the off-diagonal κterm. Among all the three complex-valued solutions\nof equation (16), the unique root with the positive imaginary part is f ollowed con-\ntinuously in frequency to get the physical picture of a lossy materia l.\nForm/negationslash= 0, the xxandyy-components of equation (13) expand as\n\n\n(1+m)µ−κ−µe+κe\nµe+µ−κ+κe+(1−m)µ+κ−µe+κe\nµe+µ+κ+κe+(1−m2)2a(µdw−µe+κe)\nµe+µdw+κe= 0\n(1+m)µ+κ−µe−κe\nµe+µ+κ−κe+(1−m)µ−κ−µe−κe\nµe+µ−κ−κe+(1−m2)2a(µdw−µe−κe)\nµe+µdw−κe= 0\n(17)\nand have to be solved simultaneously to get µeandκe. These equations are identical\nby interchanging κ↔ −κandκe↔ −κewhich guaranties an exhibition of the\nquantities κ2,κ2\neandκκe. A bi-dimensional root-finding numerical algorithm in the\ncomplex plane using a variant ofthe Newton procedure is developed t o get the couple\n(µe,κe) for any values of m,a,µ,κandµdw. As for the two-phase situation, Eqs.(15)\nand (17) show that if m=±1 thenµe=µ,κe=κandµez= 1. This is also true\nby construction of the three-phase model. This implies that µegoes to the spectral\nPolder tensor to the saturation limit.\nC. Anhysteretic texture\nThe presence of a large amount of such composite media not necess arily dis-\ntributed at random, has to be assessed by a space dependent fluc tuation in the\nmagnetization direction to represent its magnetic texture. The st atistical average\nof any function U(θ,φ) of the direction angles θandφis defined as the following\n10weighted normalized integral19:\n/angb∇acketleftU/angb∇acket∇ight ≡/integraldisplay2π\n0dφ/integraldisplayπ\n0dθsinθf(θ,φ)U(θ,φ), (18)\nwheref(θ,φ) is a normalized texture function that distribute the angles on the u nit-\nsphere at equilibrium and is a function of external parameters such as applied dc\nmagnetic field, constrains etc.\nWhen a full demagnetized state is desired at the macroscopic level, t hen the\naverage remanent field /angb∇acketleftm/angb∇acket∇ightshould remain zero but on a statistical distribution of\nthelocaldirectionofthemagnetization. Itisparticularlytruefort heuniformangular\ndistribution. It is noted that even if κe/negationslash= 0, one could also realized /angb∇acketleftκe/angb∇acket∇ight= 0 for any\nangular distribution f(θ,φ) of the composite medium by satisfying\n/angb∇acketleftκe/angb∇acket∇ight=/integraldisplay2π\n0dφ/integraldisplayπ\n0dθsinθf(θ,φ)/parenleftbiggµez−µe\n2sin2θsin2φ+κecosθ/parenrightbigg\n= 0,(19)\nwhich is true when fis uniform. This means that if desired, a locally anisotropic\ncomposite medium which behaves statically isotropic may be construc ted, once their\ncomposite media are properly distributed.\nFor an uniform distribution of composite domains (2- or 3-boby) in th e demagne-\ntized state, one has /angb∇acketleftκe/angb∇acket∇ight= 0 and\n/angb∇acketleftµe/angb∇acket∇ight=/integraldisplay2π\n0dφ/integraldisplayπ\n0dθsinθ1\n4π/parenleftbig\nµe+(µez−µe)sin2θcos2φ/parenrightbig\n=2µe+µez\n3,(20)\n/angb∇acketleftµez/angb∇acket∇ight=/integraldisplay2π\n0dφ/integraldisplayπ\n0dθsinθ1\n4π(µez−(µez−µe)sin2θ) =2µe+µez\n3,(21)\nfor biphasic cylindrical inclusions uniformly distributed into space and embedded\nin a gyrotropic effective medium. So the statistical average effectiv e tensor of a\ndemagnetized ferrite is\n/angb∇acketleftµe/angb∇acket∇ight(m= 0) =1\n3Tr(µe)1, (22)\nwhich is the result found by Schl¨ omann8.\n11For an unsaturated medium when mincreases in the z-direction, the distribu-\ntion of the local magnetization angles cannot be uniform and is distrib uted around\nthez-axis. Moreover whatever the value of mmay take, /angb∇acketleftκe/angb∇acket∇ight= 0 for an uniform\ndistribution, which is in contradiction to experiments.\nThis problem can be circumvent by recalling that the probability that t he local\nmagnetization occupies an infinitesimal solid angle in a given direction is t herefore\nstrongly dependent on the magnetic free energy in this direction. L et\n/angb∇acketleftm/angb∇acket∇ight=/integraldisplay2π\n0dφ/integraldisplayπ\n0dθsinθf(θ,φ)cosθ, (23)\nthe statistical average of the remanent magnetization. So when a ll the collective me-\ndia are in the paramagnetic state driven by the exterior constant in duction field µ0H\npointing in the z-direction, the function fis then given by an equilibrium Boltzmann\ndistribution f(θ,φ) =Xexp(Xcos(θ))/4πsinh(X) whereX=AsMsµ0Hwhich nat-\nurally breaks the space isotropy when H/negationslash= 032. The local magnetic moments of\nthe composite medium do not follow nor instantly nor spatially, the dire ction of\nthe external induction by coherent rotations but instead in defoc using their local\nmagnetization about the average direction. The free energy is sca led by the Arm-\nstrong’s parameter Aswhich narrows statistically the distribution fwhen both the\nnature and degree of disorder are known. An estimation of this par ameter is given\nby equating linearly /angb∇acketleftm/angb∇acket∇ighttoHfor small value of Xand one has in our case\nAs≈3/angb∇acketleftµez/angb∇acket∇ight−1\nµ0M2s, (24)\na result also found previously33.\nNow the statistical average of µehas to be derived. By regrouping all the terms,\none can link exactly /angb∇acketleftµe/angb∇acket∇ight,/angb∇acketleftκe/angb∇acket∇ightand/angb∇acketleftµez/angb∇acket∇ightwith/angb∇acketleftm/angb∇acket∇ightto read\n/angb∇acketleftµe/angb∇acket∇ight=µe+/angb∇acketleftm/angb∇acket∇ight\nL−1(/angb∇acketleftm/angb∇acket∇ight)(µez−µe) (25)\n12/angb∇acketleftκe/angb∇acket∇ight=/angb∇acketleftm/angb∇acket∇ightκe (26)\n/angb∇acketleftµez/angb∇acket∇ight=µez−2/angb∇acketleftm/angb∇acket∇ight\nL−1(/angb∇acketleftm/angb∇acket∇ight)(µez−µe) (27)\nwhereL(x) = coth( x)−1/xistheLangevinfunctionand L−1(x)isitsinversefunction\nsuch asL−1◦L(x) =L◦L−1(x) =x. These three equations constitute the central re-\nsult ofthis paper. Onerecognizes that /angb∇acketleftm/angb∇acket∇ight=L(Asµ0MsH)is thewell known param-\nagnetic result. Moreover the external field is modulated by the Arm strong parameter\nanditscontributiontorotatetheaveragemagnetizationdepends onthesusceptibility\nofthematerial. Inageneral situation, both µeandκehave tobeevaluatedbyconsid-\nering a direct map between the local remanent state mof the composite medium and\nthe macroscopic remanent magnetization /angb∇acketleftm/angb∇acket∇ightin a some sort of mean-field approx-\nimation, i.e. m=/angb∇acketleftm/angb∇acket∇ight. For practical applications, an approximation of the inverse\nLangevin function is taken34and one has /angb∇acketleftm/angb∇acket∇ight/L−1(/angb∇acketleftm/angb∇acket∇ight)≈(1−/angb∇acketleftm/angb∇acket∇ight2)/(3−/angb∇acketleftm/angb∇acket∇ight2) for\nall values of /angb∇acketleftm/angb∇acket∇ight. One observes that Tr( /angb∇acketleftµe/angb∇acket∇ight) = Tr(µe) is an invariant quantity what-\never the value of /angb∇acketleftm/angb∇acket∇ightmay take. Moreover when /angb∇acketleftm/angb∇acket∇ight= 0 then /angb∇acketleftm/angb∇acket∇ight/L−1(/angb∇acketleftm/angb∇acket∇ight) = 1/3,\n/angb∇acketleftµe/angb∇acket∇ight= (2µe+µez)/3,/angb∇acketleftµez/angb∇acket∇ight= (2µe+µez)/3 and/angb∇acketleftκe/angb∇acket∇ight= 0 as expected. This is\nprovided by the fact that lim H→0f(θ,φ) = 1/4π.\nIn the previous description of the effective permeability tensor, th e magnetization\nratio/angb∇acketleftm/angb∇acket∇ightis a fixed parameter that states the magnetization along the z-axis. How-\never the effective anisotropy field, which is characterized by the va lue ofηis assumed\nto be independent of such a state. When the material is probed by in creasing an\nexternal dc field H, a continuous variation of /angb∇acketleftm/angb∇acket∇ightoccurs. Neglecting any supplemen-\ntary demagnetizing fields coming from the geometry enforced on th e domains and\ndomain-walls, the effective gyromagnetic resonance frequency of the magnetic do-\nmain shifts linearly with the amplitude of Hduring this process. This experimental\nphenomenon has to be reproduced and to take it into account, the local effective field\ninside each uniform media has to be carried out from a magnetization la w. Several\n13strategies have been derived to get an anhysteretic or hysteric m agnetization law,\nfromcoherent rotationmechanism14,35tomoreelaboratedmodels36,37. Asafirststep,\nfor each branches along the hysteresis loop, a local equilibrium func tionf(θ,φ) has\nto be evaluated to derive the hysteretic unsaturated spectral p ermeabilities. Even\nin the anhysteretic situation, one has at least to replace in the calcu lation of the\nlocal permeability tensor of domains and domain-wall, the anisotropy field by an\neffective field that rotates statically the magnetic moments from th e easy axis of the\ncorresponding medium. This is done in our case by adding H/Msto the anisotropy\nconstants, hence expressed as a function of /angb∇acketleftm/angb∇acket∇ightonly one has\nη→η+L−1(/angb∇acketleftm/angb∇acket∇ight)\n3(/angb∇acketleftµez/angb∇acket∇ight−1), (28)\nηdw→ηdw+L−1(/angb∇acketleftm/angb∇acket∇ight)\n3(/angb∇acketleftµez/angb∇acket∇ight−1). (29)\nBy reason of the dependance of /angb∇acketleftµez/angb∇acket∇ightin the anisotropy fields, equation (27) has to\nbe solved self-consistently. This procedure is initiated first by cons idering known the\ncase/angb∇acketleftm/angb∇acket∇ight= 0 as a guess of /angb∇acketleftµez(Ω = 0)/angb∇acket∇ightand secondly, that an infinitesimal increment\nof/angb∇acketleftm/angb∇acket∇ightdoesnotaffectthepreviousvalueof /angb∇acketleftµez(Ω = 0)/angb∇acket∇ighthencefound. Thisprocedure\ngenerates the desired permeability as a function of /angb∇acketleftm/angb∇acket∇ightby consecutive steps such as\n/angb∇acketleftµez(Ω = 0,/angb∇acketleftm/angb∇acket∇ight+δ/angb∇acketleftm/angb∇acket∇ight)/angb∇acket∇ight ≈ /angb∇acketleftµez(Ω = 0,/angb∇acketleftm/angb∇acket∇ight)/angb∇acket∇ight.\nIII. RESULTS\nSoft NiZnferrites have been synthesized andthe effective permea bility component\nin a coaxial wave guide at the APC7 standard have been acquired38. These ferrites\nhave been demagnetized by proper thermal treatment above the Curie temperature.\nIt has been verified by measuring hysteresis loops along and perpen dicular direction\nto the hollow of the cylinder that the average remanent field /angb∇acketleftm/angb∇acket∇ightdoes not exceed\n1410−3. This spectral measurement reported on figure (2) allows to fit mo del param-\neters from equation (25), which are collected on table I. The fitted values (η,α)\n100 k 1 M 10 M 100 M 1 G 10 G0100200300400500600(a)\n100 k 1 M 10 M 100 M 1 G 10 G\nfrequency (Hz)050100150200250300(b)\nFigure 2. (Color online) Experimental real part (a) and imag inary part (b) of the effective\npermeability /angb∇acketleftµe/angb∇acket∇ightmeasured in a coaxial wave guide at the APC7 standard for a dem agne-\ntized sintered NiZn ferrite (points). The black curve comes from the calculated effective\npermeability described in the text using parameters found i nto table I.\nfor domains are in agreement with reference9. The ratio between domain-wall and\ndomain frequencies is ranged from (2 −5)10−2in MgFe ferrites25which is compatible\nwith our fitted values. It has been reported a strong dependance on composition38,\nporosity39, grain size5,40and applied stresses41in the low frequency regime, where\n15aη α η dwαdw\n5.0E-1 6.6E-3 2.4E-1 4.0E-4 9.6E-1\nTable I. Parameters of the described model for demagnetized NiZn ferrites\ndomain-wall motion is the supposed dominant mechanism. However, t he reported\ndependancies may be interpreted more as a consequence of the EM A that mimic\nthe effective permeability tensor than a change of the intrinsic value s of the param-\neters of the magnetic inclusions. For example, the static effective p ermeability of a\ndemagnetized medium is given by\n/angb∇acketleftµ′\ne(0)/angb∇acket∇ight=1+aµdw+(µ−µdw)(1−a)+/radicalbig\n((µ−µdw)(1−a))2+4(1+a)2µµdw\n3(1+a)\n(30)\nwhereµ= 1 + 1/η,µdw= 1 + 1/ηdware the corresponding static permeabilities.\nIt contains an equal contribution of the domain-walls and uniform sp in rotation in\ncontradiction to the classical interpretation given by Snoek25which links the initial\npermeability to domain-wall susceptibility only. This formula also conta ins the frac-\ntion of the domain-walls aas a continuous parameter. For large value of a, the initial\npermeability is well described by the permeability of the domain-wall in a greement\nwith the classical interpretation. On the other side, when a→0 this is the static\npart of the domain permeability that drives the initial permeability. Th e classical\ninterpretation has already been questioned42and the assumption of a perfect addi-\ntion of the two types of magnetic processes as described in refere nce43, reveals itself\npuzzling with a varying remanent field.\nThe behavior of the effective diagonal and off-diagonal permeability with an in-\ncreasing value of the remanent magnetization /angb∇acketleftm/angb∇acket∇ightis now investigated. For the same\nparameters in the table I, the spectra are depicted on figure 3. Th e magnetic behav-\nior of these spectra is strongly similar to those measured from refe rence28, once given\n16Figure 3. (Color online) Spectral real (straight line) and i maginary (dotted line) part\nof the effective permeability /angb∇acketleftµe/angb∇acket∇ight(sub-figure a), effective off-diagonal susceptibility /angb∇acketleftκe/angb∇acket∇ight\n(sub-figure b) and effective zz permeability /angb∇acketleftµez/angb∇acket∇ight(sub-figure c) as a function of several\nmacroscopic remanent magnetization values /angb∇acketleftm/angb∇acket∇ight.\nthe magnetization law that connects /angb∇acketleftm/angb∇acket∇ightto the external uniform magnetic field H.\nBecause the domain-wall permeability tensor does not carry any off -diagonal expres-\nsion, the resulting effective off-diagonal permeability is small at very low frequency\nand simply dominated by the susceptibly coming from the domains. This is in agree-\nment with first direct observation of domain rotation in ferrites44and by accurate\nmeasurements given by Green et al.45.\n17IV. CONCLUSION\nEffective permeability tensor for unsaturated polycrystalline ferr ites are derived\nthrough an EMA and combines domain-wall motion and rotation of dom ains in a\nsingle consistent framework. The dispersion of the local magnetiza tion axis that\nencodes the polycrystalline character is taken into account by ave raging the free\nenergy to restore the magnetic anhysteretic behavior. The initial permeability is\ngiven as a mixture formula of magnetic inclusions and fraction of doma in-wall which\ngives a picture of the low-frequency permeability spectra as a magn etic scattering\nin geometrically arrangement of domains and domain-walls instead of v arying mate-\nrial properties. It is envisioned that this theory can be extended t o treat multiple\nscattering of electromagnetic fields in such cylindrical geometry to include multiply\npeaked spectra observed in several uniform, polycrystalline mate rials.\nACKNOWLEDGMENTS\nJT acknowledges financial support through a joint doctoral fellow ship “CEA-\nR´ egion Centre”.\nAppendix A: EMA formalism\nThe double overbars indicate a tensor notation of rank 2, such as t hei-th anj-th\ncomponent of a tensor Ais [A]ij≡Aij. A medium of volume V, bounded by a\nsurfaceSis characterized by a permeability tensor µas a function of frequency and\nwhose values are random in position. It is random in a sense that the c onfiguration\naverage of µ(/vector r), denoted /angb∇acketleftµ/angb∇acket∇ight, is independent of /vector r. It is also implicitly supposed\nthat this configuration average is equivalent to a volume average, w hich is the zero\n18wave-vector of the spatial Fourier transform of µ(/vector r), such as /angb∇acketleftµ/angb∇acket∇ight ≡V−1/integraltext\nVµ(/vector r)d/vector r.\nThe measurable permeability tensor µeis the constant of proportionality between\napplied magnetic field /vectorH0such as/angb∇acketleft/vectorH/angb∇acket∇ight ≡/vectorH0and the average induction /angb∇acketleft/vectorB/angb∇acket∇ight ≡µe/angb∇acketleft/vectorH/angb∇acket∇ight.\nConsider that the discussion is restricted to the quasi-static limit wh end/λ≪1;\ndis a typical dimension of the scatterer and λis the wavelength of the incident\noscillating magnetic field for that only the lowest order of scattering wave mode is\nevaluated for high-resistive ferrite46. In this limit, the magnetic field equations can\nbe approximated by magneto-static equations only\n/vector∇·(µ(/vector r)/vectorH(/vector r)) = 0\n/vector∇×/vectorH(/vector r) =/vector0(A1)\nwhich are combined up to a given gauge, to find a magnetic potential φ(/vector r) inside the\nmedium satisfying\n/vector∇·(µ(/vector r)/vector∇φ(/vector r)) = 0. (A2)\nNowamagnetictensor µ0isconsideredtodependonthemagneticfieldintheaverage\nmedium. In the quasi-static limit, the boundary conditions of a unifor m applied field\nare imposed and µ0does not exhibit any spatial variations. The local permeability\ntensor, thus decomposed as µ(/vector r) =µ0+δµ(/vector r), leads to the following boundary-value\nproblem\n/vector∇·(µ0/vector∇��(/vector r)) =−/vector∇·(δµ(/vector r)/vector∇φ(/vector r)) in V ,\nφ(/vector r) =−/vectorH0·/vector ron S.(A3)\nWith the introduction of the two-points Green’s function g(/vector r,/vectorr′) defined by\n/vector∇·(µ0/vector∇g(/vector r,/vectorr′)) =−δ(/vector r−/vectorr′) in V,\ng(/vector r,/vectorr′) = 0 /vectorr′on S,(A4)\nthe potential φ(/vector r) admits a formal solution as\nφ(/vector r) =−/vectorH0·/vector r−/integraldisplay\nVg(/vector r,/vectorr′)/vector∇′·(δµ(/vectorr′)/vector∇′φ(/vectorr′))d/vectorr′ (A5)\n19and the magnetic field /vectorH(/vector r) =−/vector∇φ(/vector r), deriving of it, is given by\n/vectorH(/vector r) =/vectorH0−/integraltext\nV(δµ(/vectorr′)/vectorH(/vectorr′))/vector∇g(/vector r,/vectorr′)d/vectorr′,\n=/vectorH0+/integraltext\nVG(/vector r,/vectorr′)(δµ(/vectorr′)/vectorH(/vectorr′))d/vectorr′,(A6)\nwith\nGαβ(/vector r,/vectorr′) =∂2g(/vector r,/vectorr′)\n∂r′\nα∂rβ. (A7)\nBy combining these equations, the problem thus reduces to the tas k of computing a\nsusceptibility tensor /angb∇acketleftχ/angb∇acket∇ightsuch asδµ(/vector r)/vectorH(/vector r) =χ(/vector r)/vectorH0and\nχ(/vector r) =δµ(/vector r)/parenleftbigg\n1+/integraldisplay\nVG(/vector r,/vectorr′)χ(/vectorr′)d/vectorr′/parenrightbigg\n, (A8)\nwhich finally gives\nµe=µ0+/angb∇acketleftχ(/vector r)/angb∇acket∇ight. (A9)\nIf/vector rlies in a medium labeled i, of volume vi, equation (A8) decomposes itself as\nχ(/vector r) =δµi/parenleftbigg\n1+/integraldisplay\nviG(/vector r,/vectorr′)χ(/vectorr′)d/vectorr′+/integraldisplay\nV−viG(/vector r,/vectorr′)χ(/vectorr′)d/vectorr′/parenrightbigg\n,\nwithδµi≡µi−µ0, andµiis the permeability tensor of the medium i, assumed\nspatially uniform and known. The last integral is approximated by\n/integraldisplay\nV−viG(/vector r,/vectorr′)χ(/vectorr′)d/vectorr′≈/integraldisplay\nV−viG(/vector r,/vectorr′)/angb∇acketleftχ(/vectorr′)/angb∇acket∇ightd/vectorr′ (A10)\nonceneglectinghigherordercorrelationterms. Thislastexpressio nclosestheintegral\nequation for χ(/vector r). To see this, this last integral is substituted, integrated by part s\nand by imposing the boundary condition one obtains\nχi=/parenleftig\n1−δµiΓi/parenrightig−1\nδµi/parenleftig\n1−Γi/angb∇acketleftχ/angb∇acket∇ight/parenrightig\n(A11)\nwhereχistands for χ(/vector r) for/vector r∈vi. HereΓiis a surface integral, also called depolar-\nizationmatrixandcontains g(/vector r,/vectorr′)whichgoesovertothefree-spaceGreen’sfunction,\n20satisfying the differential equation (A4) and the boundary conditio ng(/vector r,/vectorr′)→0 as\n/ba∇dbl/vector r−/vectorr′/ba∇dbl → ∞. By reason of the translational invariance, it becomes a function o f\n/vector r−/vectorr′to finally reduces to a constant in space variables. For example, the field inside\nellipsoids is uniform and χ(/vector r) becomes independent of the position. In such case,\neach cartesian component of this matrix is given by\nΓαβ\ni=−/contintegraldisplay\nS′∂g(/vector r−/vectorr′)\n∂rαn′\nβd2/vectorr′, (A12)\nwheren′\nβis the component of the unit normal outward from the surface S′. By\ntaking the average of χi, one has\nµe=µ0+/angb∇acketleft(1−δµΓ)−1/angb∇acket∇ight−1/angb∇acketleft(1−δµΓ)−1δµ/angb∇acket∇ight (A13)\nwhere\n/angb∇acketleft(1−δµΓ)−1/angb∇acket∇ight ≡lim\nV→∞/summationdisplay\nivi(1−δµiΓi)−1(A14)\n/angb∇acketleft(1−δµΓ)−1δµ/angb∇acket∇ight ≡lim\nV→∞/summationdisplay\nivi(1−δµiΓi)−1δµi (A15)\nNow, a two-phase medium of magnetic objects labeled by i= 1,2, each fully charac-\nterized by their respective position and field independent permeabilit y tensors µiis\nconsidered. 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Soc. 203A, 385 (1904).\n24"    },    {        "title": "1210.0356v1.Effect_of_Site_disorder__Off_stoichiometry_and_Epitaxial_Strain_on_the_Optical_Properties_of_Magnetoelectric_Gallium_Ferrite.pdf",        "content": " Page 1 of 18 Effect of Site-disorder, Off-stoich iometry and Epitaxial Strain on the 1 \nOptical Properties of Magnet oelectric Gallium Ferrite  2 \n 3 \nAmritendu Roy1, Somdutta Mukherjee2, Surajit Sarkar2, Sushil Auluck3, Rajendra 4 \nPrasad2, Rajeev Gupta2, 4 and Ashish Garg1*  5 \n 6 \n1Department of Materials Science & Engineering, Indian Institute of Technology, 7 \nKanpur-208016, India 8 \n2Department of Physics, Indian Institute of Technology, Kanpur - 208016, India 9 \n3National Physical Laboratory, Dr. K. S.  Krishnan Marg, New Delhi-110012, India 10 \n4Materials Science Programme, Indian Institute of Technology, Kanpur - 208016, India 11 \n 12 \nABSTRACT 13 \n 14 \nWe present a combined experimental-theoretical study demonstrating the role of site 15 \ndisorder, off-stoichiometry and strain on th e optical behavior of magnetoelectric gallium 16 \nferrite. Optical properties (band-gap, refrac tive indices and dielectric constants) were 17 \nexperimentally obtained by performing ellipsometric studies over the energy range 0.8 – 18 \n4.2 eV on pulsed laser deposited epitaxial thin  films of stoichiometric gallium ferrite with 19 \nb-axis orientation and the data was compared w ith theoretical results. Calculations on the 20 \nground state structure show that the optical activity in GaFeO 3 arises primarily from O2p- 21 \nFe3d transitions. Further, inclusion of site disorder and epitaxial strain in the ground state 22 \nstructure significantly improves the agreement between the theory and the room 23 \ntemperature experimental data substantiating the presence of site-disorder in the 24 \nexperimentally derived strained GaFeO 3 films at room temperature. We attribute the 25 \nmodification of the ground state optical behavior upon inclusion of site disorder to the 26 \ncorresponding changes in the electronic band structure, especially in Fe3d states leading 27 \nto a lowered band-gap of the material. 28 \n 29 \n 30 \nKeywords:  Gallium ferrite, thin films, optical prop erties, ellipsometry, first-principles 31 \ncalculation 32 \n 33 \n 34 \nPACS No.:  78.20.-e, 77.55.Nv, 71.15.Mb  35 \n 36 \n 37 \n38 \n                                                 \n* Corresponding author, Tel: +91-512-2597904; FAX - +91-512-2597505, E-mail: ashishg@iitk.ac.in    Page 2 of 18 I. Introduction 1 \n 2 \nOptical properties of piezoelectric and ferroelectric oxides have been of interest for a 3 \nvariety of applications [1-4] ranging from optical waveguides, [2] photocatalysis, [5] and 4 \ninfrared detection [3, 4] to mo re recently photovolta ics. [6, 7] As an additional degree of 5 \nfreedom in many of these materials such as magnetoelectrics and multiferroics, 6 \nmagnetism often yields a reduced band gap [8, 9], enhancing their suitability for 7 \nphotovoltaic applications. Further, tunabilty of the band gap vis-à-vis  electronic structure 8 \nin these materials could be accomplished by tailoring the crystal structure where external 9 \nperturbations such as doping [10] and strain [11] play crucial role in determining 10 \nstructural symmetry as well as physical properties e.g. optical properties. To fabricate 11 \nefficient optical devices, it is essential to understand the microscopic effects of such 12 \nperturbations vis-à-vis  their contributions to the electronic structure as well as optical 13 \nproperties. In this regard, first-principles density functional theory (DFT) based studies 14 \nhave been quite successful in predicting and analyzing the ground state as well as 15 \nelectronic structure and optical properties of complex oxide systems. [12, 13] The 16 \ndisadvantage of underestimation of electronic band gap by conventional LDA and GGA 17 \nfunctionals [14, 15] of DFT could be avoided by suitably scaling the calculated results 18 \nwith the experimental observations.  19 \nWe have chosen to study the optical  properties of gallium ferrite (GaFeO 3 or 20 \nGFO) which is a prospective room temperature magnetoelectric with comparatively small 21 \nband gap (~2.5-3 eV) [14, 16] and therefore, attractive for potential photovoltaic 22 \napplications. Further, GFO exhibits a number of exciting optical phenomena such as 23 \noptical magnetoelectric effect, [17, 18] an d magnetization induced non-linear second 24 \nharmonic Kerr effect. [19, 20] The observed ferrimagnetism, which can be tuned by 25 \ntailoring the Ga:Fe ratio [21, 22] and processing conditions, [21-23] is believed to be the 26 \nmanifestation of inherent cation site disorder [21, 24] emanating from almost similar 27 \nsizes of Fe and Ga. Previous optical studies on GFO single crystals [25] and thin films 28 \n[16] using ellipsometry and absorption studies showed red-shift of the fundamental 29 \nabsorption edge with increasing Fe content.  However, microscopic origin of such red- 30 \nshift is rather generalized and one needs to d ecouple the possible contributions of external 31 \nfactors such as Fe content, cationic site disorder and epitaxial strain to completely 32 \nunderstand the optical response of the material. In this context, systematic first-principles 33 \ncalculations along with appropriate experimental data could provide an atomistic insight 34 \ninto optical response of the material. 35 \nIn this paper, we present the results of a combined experimental and theoretical 36 \nstudy of the linear optical properties of GFO.  Epitaxial thin films of GFO were chosen 37 \nspecifically to understand the combined effects of  inherent cation site disorder as well as 38 \nepitaxial strain on its optical properties which were decoupled using first-principles’ 39 \ncalculations. Our calculations show that the in ter-band transitions, responsible for optical 40 \nactivities of GFO are primarily due to O2 p-Fe3d transitions. More importantly, we 41 \nclearly observe that incorporation of cation si te disorder and epitaxial strain into the 42 \nground state structure yields a much improved agreement between the theoretical 43 \npredictions and experimental observations subs tantiating the role of cation site disorder 44 \nwhich is explained in terms of modification of the Fe 3d bands. Subsequent parts of the 45 \npaper are organized as follows: section II discusses the experimental techniques and 46  Page 3 of 18 calculation methodologies used in the study, section III presents the results of both 1 \nellipsometric measurements and first-principles calculations on the ground state structure 2 \nalong with external effects such as disorder, strain, off-stoichiometry and hydrostatic 3 \npressure and section IV summarizes the manuscript. 4 \n 5 \n 6 \nII. Experimental and Calculation Details 7 \n 8 \nGFO thin films were grown using pulsed laser deposition technique on (100) oriented 9 \ncubic yttria stabilized zirconia (YSZ) substrates from a stoichiometric target of GFO 10 \n(Ga:Fe = 1). Film growth was carried out using KrF excimer laser ( =248 nm) in an 11 \noxygen ambient (pO 2 ~ 0.53 mbar) at a substrate temperature of 800°C using a laser 12 \nfluence of 2 J/cm2 at a laser repetition rate of 3 Hz. As-grown films were subsequently 13 \ncooled slowly to room temperature under the same ambient pressure. X-ray diffraction of 14 \nthe films was performed using a high resoluti on PANalytical X’Pert PRO MRD thin film 15 \ndiffractometer using CuK α radiation. Ellipsometric measurements were carried out using 16 \nHORIBA JOBIN-YVON spectros copic ellipsometer (SE) over the energy range of 0.8- 17 \n4.2 eV with an incidence angle of 70°.  18 \nFor calculations of the optical properties of the ground state structure of GFO, [14, 19 \n24] we employed density functional theory (DFT+U) [26] using pseudopotential based, 20 \nVienna Ab-initio  simulation package (VASP) [27] and applied the projector augmented 21 \nwave method (PAW) [28]. Kohn-Sham equation [29] was solved using the generalized 22 \ngradient approximation (GGA+U) method (U = 5 eV, J = 1 eV) with the optimized 23 \nversion of Perdew-Burke-Ernzerhof functional for solids (PBEsol). [30] For calculations, 24 \nwe included three valence electrons of Ga ( 4s24p1), eight for Fe ( 3d74s1) and six for O 25 \n(2s22p4) ions. A plane wave energy cut-off of 550 eV was used. We used Monkhorst- 26 \nPack [31] 4×4×4 mesh in our calculations. To check the robustness of our calculations, 27 \nwe also repeated some of our calculations using LSDA+U with identical U and J values 28 \nused in GGA+U calculations. Further, we employed full-potential based WIEN2k code 29 \nusing TB-mBJ functional [32] to substantia te our pseudopotential based calculations. 30 \nRecently developed TB-mBJ functional [32] has been reported to reproduce the 31 \nexperimental band gap quite accurately in a number of systems [32]. 32 \n 33 \nResults and Discussion: 34 \n 35 \n(a) Ellipsometric determination of the optical properties of GaFeO 3 thin films  36 \nFig. 1(a) shows the XRD spectrum of an as-grown GFO film over the 2  range from 15° 37 \nto 85° showing (010)-type reflections of orthorhombic structure of GFO (shown in the 38 \nright inset), indicating epitaxial nature of the film. Absence of any other peaks in the 39 \nXRD spectra suggests that the film is free of  any impurity phase. Th e out-of-plane lattice 40 \nparameter ( b = 9.3973 Å) estimated from the peak positions shows a close agreement 41 \nwith the previously reported XRD data on single crystal GFO ( b = 9.3950 Å) [23] 42 \nindicating that the film is fully relaxed along b-direction. However, in-plane lattice 43 \nparameters are strained by ~ 1.63 % due to a mismatch with the substrate lattice 44 \nparameters. Presence of large strain is also depicted by noticeably large full width half 45  Page 4 of 18 maximum (FWHM) of the rocking curve analysis of (040) peak, as shown in the left inset 1 \nof Fig. 1(a).  2 \nEllipsometric measurements provide a relative change of the amplitude and phase 3 \nof linearly polarized monochromatic light reflected from the sample surface, with respect 4 \nto the incident light. Ellipsometric parameters, ψ and Δ are related to sample’s optical and 5 \nstructural properties by: tan .p i\nsReR  where, Rp and Rs are the coefficients of reflection 6 \nof polarized light parallel and perpendicular to  the plane of incidence, respectively. [33] 7 \nEllipsometric parameters, ψ and Δ can further be used to describe two intensity 8 \nparameters termed as:  9 \n                                                 sin 2 sin\ncos 2 cosIS\nIC\n \n                                                     (1)                                        10 \nIn the present work, we have used a thre e layer model, as shown in the inset of 11 \nFig. 1(b), to analyze the ellipsometric data. The layer, labeled as L2 represents the actual 12 \nfilm while L1 and L3 take into account the substrate-film interface and film roughness, 13 \nrespectively. The dispersion in L1 consists of 50 % film and 50 % substrate while that in 14 \nand L3 consists of 50 % film and 50 % void. In order to derive the complex dielectric 15 \nfunction and other optical properties from our ellipsometry data, we used Tauc-Lorentz 16 \n(TL) model [34] in which the imaginary part of the dielectric function is given by: 17 \n                              2\n0\n22 2 2\n0() 1.   () ( )\n     0     g\ng\ngAE E Efor E EEE E E\nfor E E \n                                     (2) 18 \nwhere A is amplitude factor which is a function of material density and the momentum 19 \nmatrix element, E0 is peak transition energy corresponding to Penn gap, Γ is broadening 20 \nparameter related to crystallite size [35] and Eg is the band gap energy. To take into 21 \naccount the substrate effect, we assigned a three oscillator Tauc-Lorentz model [34] 22 \nwhich described the dispersion of a bare substrate satisfactorily. Fig. 1(b) shows the 23 \nexperimental data (symbols) and corresponding fits (solid lines) of IS and IC of the GFO 24 \nfilm. Our ellipsometry data show ed a surface roughness of ~ 21 nm and a film thickness 25 \nof ~ 85 nm which were consistent with our atomic force microscopy (AFM) and surface 26 \nprofilometer measurements. The fitting parameters are listed in the Fig. 1(b) 27 \n 28 \n(b) Comparison of ellipsometric data wi th the ground state properties with no 29 \nexternal perturbation 30 \nEllipsometry data along with the results of th e density functional calculations of real ( ) 31 \nand imaginary ( ) components of dielectric function are plotted in Fig. 2 (a). Simulation 32 \nof the ellipsometric data usi ng Tauc-Lorentz (TL) model [34]  yielded an energy band gap, 33 \nEg, of ~ 2.28 ± 0.08 eV, consistent with our ground state electronic structure calculations 34 \n[14, 24] but lower than the previous experimental data. [16] The plots of ε versus photon 35 \nenergy show that absorption in our samples begins at ~ 2-2.5 eV. For a clear visualization 36 \nof the absorption edge, absorption coefficient ( α) is plotted semi-logarithmically, as 37 \nshown in Fig. 2(b). Further, initial part of the absorption spectra beyond the band gap 38 \nshows a quadratic dependency on incident photon energy (solid line in of Fig. 2(b)) 39 \nindicating GFO to be an indirect band gap semiconductor. From the fitting of the 40  Page 5 of 18 absorption spectra, the indirect band gap was estimated as E g ~ 2.28 ± 0.02 eV which is 1 \nin excellent agreement with the one obtained from the simulation of the ellipsometric data 2 \nusing TL model. However, this is in contrast with our earlier calculations [14] on the 3 \nground state structure where we found that direct and indirect band gaps in GFO are 4 \nidentical. We find that below the band edge, the values of ε″ and k are zero while ε and 5 \nrefractive index ( n) possess dispersive behavior, as a function of photon energy, as shown 6 \nin the inset of Fig.2 (b). To understand the origin of the experimental optical behavior in 7 \nGFO, we performed first-principles studies using both pseudopotential (GGA+U and 8 \nLSDA+U) and full-potential (TB-mBJ) based approaches. A comparison, as shown in Fig. 9 \n2(a), demonstrates that while our experimental data is consistent with a previous report 10 \non single crystal GFO, [25] our LSDA+U, GGA+U and TB-mBJ calculations do not 11 \nreproduce the experimental data very well. While LSDA+U and GGA+U underestimate 12 \nthe band gap (red shift of the absorption edge with respect to the experiment) TB-mBJ 13 \nyields a good agreement of the band gap with the experiment. The difference in the 14 \nintensities between the experimental data and the calculated profiles could be attributed 15 \nto a number of factors namely, sample quality, temperature, difference between 16 \nexperimental and the ground state crystal and magnetic structures, type of approximation 17 \nscheme used in the first-principles calculations and the type of broadening used in the 18 \nexperimental and calculated data. [36]  19 \nOverall, since our GGA+U profile matched best (among all the calculated results) 20 \nwith the experimental data, subsequent discussions are limited to GGA+U results only. 21 \nWe first calculated the ground state dielectric properties, ε and ε″, along the three 22 \nprincipal crystallographic directions as plotted in Fig. 2(c). Here, we observe that the 23 \noptical constants of GFO are anisotropic in nature, a feature consistent with the 24 \northorhombic symmetry of the unit cell and also supported by previous report on GFO 25 \nsingle crystal. [25] Further, we also identified the major features (peaks) in the ε″ plot 26 \nwhich, in case of insulators like GFO, orig inate primarily from the inter-band transitions, 27 \ni.e., from valence (VB) to conduction (CB) bands. We computed the electronic band 28 \nstructure and density of states using GGA+U to  identify the transitions responsible for the 29 \noptical activities in GFO and the results are shown in Fig. 3 (a) and (b). The allowed 30 \noptical transitions are labeled in the band stru cture. The density of states plot shows that 31 \nthe upper most part of the VB, -2 eV to 0 eV, is dominated by O 2p states with rather 32 \nsuppressed Fe 3d and Ga 4p states. On the other hand, the lower part of CB is mostly 33 \noccupied by Fe 3d bands with a subdued presence of O 2p and Ga 4s states. Thus, we can 34 \nconclude that the optical transitions labeled in  Fig. 2(c) and Fig. 3(a) involve transitions 35 \nfrom O 2p to Fe 3d states which is further corroborated by the experimental work of 36 \nKalashnikova et al. [25] Small occupation of O 2p states near the conduction band 37 \nminimum indicates that GFO has a significant ionic character which is substantiated by 38 \nour previous charge density and electron localization function (ELF) calculations [14]  39 \ninterpreting negligible ELF values at Fe sites as a signature of complete charge transfer 40 \nbetween Fe and O.  41 \nO2p-Fe3d electronic transition could further be elucidated using crystal field 42 \ntheory. Within the regular FeO 6 octahedral crystal environment with Oh point symmetry, 43 \nfive Fe 3d and 18 O 2p atomic orbitals construct Fe 3d–O 2p bonding and antibonding 44 \nmolecular orbitals (MO), e g and t 2g, respectively. In addition, O2p π - O2p π hybridization 45 \nleads to oxygen nonbonding orbitals are t 1g(π), t2u(π), t1u(π), t1u(σ) and a1g(σ). [37] The 46  Page 6 of 18 relative energy states of these orbitals play  a pivotal role in determining O 2p - Fe 3d 1 \ntransitions responsible for the observed optical activity of GFO.  Selection rule allows six 2 \ntransitions in the strong ab sorption region (energy level ≥ 3.0 eV): 6A1g→6T1u related to 3 \none electron transition between t 2u(π), t 1u(π), t 1u(σ) and t 2g and e g levels, [38, 39]. 4 \nHowever, MOs in the Oh point symmetry further split due to non-cubic (D 2h) crystal field 5 \ndistortion in GFO and such symmetry lowering in the actual crystal environment would 6 \nlift some of the restrictions of the transitions leading to the appearance of many more 7 \ntransitions, as shown by several peaks in Fig. 2(c). 8 \n 9 \n(c) Effect of external perturbations on the ground state optical properties 10 \nThe discussion in the previous section was based on the assumption that GFO has 11 \nantiferromagnetically ordered bulk structure with no site disorder at 0 K. These 12 \nassumptions are often challenged because materials in thin film forms experience 13 \nsubstantial substrate induced strain leading to structural distortion. Consequently, such 14 \ndistortions give rise to modifications in th e inter-ionic bond spacing and angles affecting 15 \nthe electronic structure and materials properties. For example, a number of ferroelectric 16 \noxides have been reported to demonstrate large variations in the polarization upon 17 \napplication of epitaxial strain. [40] Moreover, experimental structure of GFO is shown to 18 \npossess significant cation site disorder amon g Ga and Fe sites driven by their similar 19 \nionic sizes [21, 23] which is also ignored in the ground state calculation. In the following 20 \nsections, we introduce these structural changes in the ground state structure and compare 21 \nthe results of GGA+U calculations with the experiments.  22 \n 23 \n(i) Effect of epitaxial strain 24 \nFirst, we analyze effect of epitaxial strain on the optical properties and electronic band 25 \nstructure of GFO. The range of strain chosen is on the basis of present and past 26 \nexperiments [16] where choice of substrate leads to a misfit strain of the order of ~1-3 %. 27 \nHere, first we plot the real and imaginary parts of dielectric constant ( ε and ε″) as a 28 \nfunction of incident photon energy (Fig. 4(a)) with varying magnitudes of strain. We find 29 \nthat the nature of the ε plot remains almost identical to that of ground state structure for 30 \nstrain = ± 1 %. Further ε plot shows that the peak at ~ 3.8 eV remains similar for the 31 \nground state and -1% strain and it splits in to two peaks for +1% strain with splitting 32 \nfurther getting pronounced upon increasing the strain to +3%. However, the low energy 33 \nregions of the ε'' spectra remain identical with no no ticeable shift of the absorption edge 34 \nwith the application of strain. These observatio ns indicate that the application of epitaxial 35 \nstrain on the ground state structure alone does not improve the agreement between the 36 \nexperimental and calculated dielectric spectra in GFO.  37 \nIn addition, the refractive index (at 3 eV ) increased linearly with increasing 38 \napplied strain. To identify the origin of such strain dependent optical activity, we 39 \ncompared the electronic structure obtained at 0% and +3% strains.  These showed that the 40 \nPDOS of Fe 3d states slightly shifted towards higher energy in the conduction band with 41 \nthe application of tensile strain which is attributed to the reduction of some of the Fe-O 42 \nbond lengths resulting in overlapping wavefunctions and consequent hybridization. 43 \n 44 \n 45 \n 46  Page 7 of 18 (ii) Effect of cation site disorder 1 \nSo far, our calculations were limited to the ground state antiferromagnetic structure of 2 \nGFO assuming that there was no cation site disorder. However, the actual structure of 3 \nGFO always contains cation site disorder driven by quite similar ionic sizes of Ga and Fe. 4 \n[21, 23] Our previous work demonstrated that site disorder between Fe2 and Ga2 sites is 5 \nmost probable followed by Fe1 and Ga1 sites. [24] We incorporated these Ga-Fe site 6 \ndisorders, one at a time, to study their effe ct on the optical response in GFO in the 7 \nunstrained structure. Since the structure of GFO contains four equivalent ions of each 8 \ncation, exchange of ionic sites between one Fe1/Fe2 to one Ga1/Ga2 would lead to a 25% 9 \nsite disorder in the structure. The degree of disorder, particularly Fe2-Ga2 disorder 10 \nconceived here is similar to the experimental structure. [21] Calculated dielectric 11 \nconstants of GFO consisting of cation site disorder are plotted in Fig. 4(b) along with 12 \nthose obtained on thin film samples, and the ordered ground stat e structure. A close 13 \ninspection of the ε″ spectra near the absorption edge reveals that while Fe1 to Ga1 site 14 \ninterchange does not affect the position of the absorption edge with respect to that of the 15 \nground state structure, Fe2 to Ga2 site interchange imparts a leftward shift indicative of a 16 \nreduction in the band gap. Further, we also find that with Fe1-Ga1 site interchange, the 17 \npeak in the ε″ plot at ~ 2.86 eV (peak A in Fig. 2(c)) is suppressed whereas it almost 18 \nvanishes for Fe2-Ga2 interchanged structure. On the other hand, the peak at ~ 2.70 eV in 19 \nthe ε spectra of the ordered ground state structure is effectively flattened for Fe2-Ga2 site 20 \ninterchange resulting in a remarkably closer  resemblance of the calculated with the 21 \nexperimental data reported by Kalashnikova et al. [25] and a much improved match with 22 \nour experimental results. The observed similarities between the experimental and Fe2- 23 \nGa2 site interchanged spectra further substantiate the fact that GFO has inherent cation 24 \nsite disorder with a predominant Fe2-Ga2 site exchange.  25 \nSubsequent calculations of electronic band structure and site projected density of 26 \nstates (Fig. 3(c) and (d)) reveal that the evolution of band structure upon incorporating the 27 \nsite disorder differs significantly from that of the ground state band structure. We find 28 \nthat the band gap of site-interchanged GFO is of indirect type (Eg ~ 1.82 eV), consistent 29 \nwith our ellipsometry measurement. The difference in magnitude could be attributed to 30 \nthe GGA method used for the band structure calculation.  Site projected DOS plots show 31 \nthat there is a shift of Fe2 3d states towards lower energy which is translated into the 32 \ndown shift of the bands in the band structure and is responsible for the observed red shift 33 \nof the absorption edge in Fig. 4(b). The reduction of the band gap upon Fe2-Ga2 site 34 \ninterchange is related to the reduction in some of the Fe2-O bond lengths (in the 35 \ndisordered structure) with respect to the corresponding Ga2-O bonds (in the ground state 36 \nstructure) and variation of crystal environment upon imparting the disorder. Reduction of 37 \nthe bond length would induce stronger hybridization and consequently widen the band 38 \ndispersion leading to a reduction in the band gap.  39 \nDisorder induced variation of crystal environment could further be studied by a 40 \ncomparison of electron localization function (ELF) of the ground state structure and the 41 \nFe2-Ga2 site interchanged structure (not shown here). While finite ELF values between 42 \nGa1-O and Ga2-O indicate significant covalency, complete charge transfer between Fe2 43 \nand O sites is evident from zero ELF value at Fe2 site and across Fe2-O bonds. Thus site 44 \ndisorder modifies the ELF mapping within the unit cell and consequently the crystal 45  Page 8 of 18 environment. Such variation of the crystal environment is believed to be responsible for 1 \nthe evolution of electronic structure and consequent optical spectra in the disordered GFO.  2 \n 3 \n(iii) Combined effects of site -disorder and epitaxial strain 4 \nFinally, to completely mimic the experimental  scenario in GFO epitaxial thin films, we 5 \napplied tensile epitaxial strain to Fe2-Ga2 site disordered structure (as discussed in 6 \nsection (ii)) and calculated the optical properties and then compared the results with our 7 \nexperimental data of thin films (Fig. 4(c)). Here we observe that the agreement between 8 \nthe experiment and the calculations further improves significantly upon application of 9 \ntensile strain on the structure with 25% Fe2- Ga2 cation site disorder. While the band gap 10 \ncalculations suffer from inherent limitatio n of LDA and GGA techniques and hence are 11 \nunreliable, the calculated spectra obtained upon applying tensile strain on Fe2-Ga2 site 12 \ndisordered GFO is remarkably similar to the experimentally obtained dielectric function 13 \nas shown in Fig. 4(c). The difference in th e absolute intensities can be attributed to 14 \nvarious extrinsic parameters such as the sample quality, temperature etc. The electronic 15 \nstructure of the modified structure of GFO would have contributions from the above two 16 \neffects described in 3(b) and 3(c) where disorder drives the Fe 3d bands of the Fe2 ion at 17 \nthe Ga2 site to shift downward while tensile strain alters the cation-oxygen bond-lengths 18 \nand angles which in turn alter the positions of the electronic states. We further compared 19 \nthe experimental optical constants with these calculations over the experimental 20 \nmeasurement domain as shown in Fig. 5. In Fig 5(a), we show the calculated and 21 \nexperimental results for n and k which match well with each other. It is found that 22 \nexperimentally determined n starts with a finite value ( n ~2.1) and then slowly increases 23 \nwith energy showing a peak ( n ~ 2.5) at ~ 3 eV. On the other hand, the calculated spectra 24 \nshow the position of this peak at slightly lower energy due to underestimation of the band 25 \ngap by GGA+U method. The peak in n spectra can be attributed to the beginning of 26 \nabsorption in the material triggered by transition from valence band O 2p to conduction 27 \nband Fe 3d state.  On the other hand, the result for k starts with zero value and then 28 \nbegins to increase at energy corresponding to the band gap. Since our calculated gap is 29 \nlower compared to the experimental gap, the experimental curve starts to rise at higher 30 \nenergy. However, calculated spectra could be appropriately scaled (shifted by a constant 31 \nenergy), to yield minimum qualitative mismatch of the overall profile between calculated 32 \nand experimental spectra. Magnitudes of n and k show a clear departure for the 33 \nexperiment and calculations, attributed to the factors such as sample  quality, temperature 34 \netc. as mentioned before. Reflectivity ( R) spectra in Fig. 5(b), shows an initial gradual 35 \nincrease followed by a peak ( R ~ 0.19) near the absorption edge which is consistent with 36 \nthe absorption spectra, also shown in Fig. 5(b). Again, the difference between the 37 \ncalculated and the experimental R spectra could be attributed to the band gap 38 \nunderestimation by GGA+U calculations. Optical conductivity, as shown in Fig. 5(c), 39 \ndemonstrates an onset above 2 eV for both experiment (~ 2.5 eV) and calculation (~ 2.1 40 \neV) and level off to values, 7500 -1cm-1  and 13000 -1cm-1 at 4 eV for experimental 41 \nand calculated spectra, respectively. 42 \n 43 \n(iv) Effect of off-stoichiometry 44 \nSince GFO has large compositional tolerance, [22] it would be of interest to examine the 45 \neffect of off-stoichiometric Ga:Fe ratio on the optical properties and consequent changes 46  Page 9 of 18 in the electronic structures. This is importan t because it results in an imbalance in the 1 \ncation site occupation between Fe and Ga and its effects are manifested in the magnetic 2 \nbehavior of GFO [22]. In this context, we have considered two cases: first, considering 3 \nsubstitution of one Ga ion at Ga2 site by one Fe ion at Fe2 site and second, considering 4 \nsubstitution of two Ga ions at Ga2 sites by two Fe2 site ions. These scenarios resemble 5 \ntwo Fe-excess compositions x = 1.125 and x = 1.25 in Ga 2-xFexO3, well within the 6 \nexperimentally obtained single phase domain of GFO. [22] Subsequently, we relaxed the 7 \ntwo structures and computed the optical properties and the electronic structures. Here, we 8 \nhave not included the site diso rder allowing us to exclusively investigate the effect of off- 9 \nstoichiometry.  10 \nIt was found that the electronic band structure and the density of states (plots not 11 \nshown) calculations of these off-stoichiometric compositions show similar effects as 12 \nobserved for the disordered GFO. It was observed that with increasing Fe content, Fe ions 13 \nsubstituting Ga2 sites would have Fe 3d states at increasingly low energies resulting in a 14 \nmonotonic decrease in the band gap. Lowering of crystal symmetry due to doping further 15 \ninduces band splitting of Fe 3d band in these cases. Resulting modifications in the 16 \nelectronic structures thus, affect the optical spectra for compositions having excess Fe 17 \ncontent. Fig. 6(a) shows the yy component of real and imaginary parts of dielectric 18 \nconstant tensors with different stoichiometry viz., x = 1.0, x = 1.125 and x = 1.25.  We 19 \nobserve that with increasing Fe content th e fundamental absorption edge shifts towards 20 \nlower energy consistent with the previous experimental observations on single crystal 21 \n[25] and our band structure calculations which show a reduction in the band gap with 22 \nincreasing Fe content ( x). Moreover, the intensity of dielectric function increases with 23 \nincreasing Fe content.  24 \n 25 \n(v) Effect of hydrostatic pressure  26 \nFinally, we take into account the structural distortion which can induce the instability 27 \nleading to phase transformation in a few systems with significant magneto-structural 28 \ncoupling. [41, 42] Such distortion which can be brought about by applying hydrostatic 29 \nstress has also been reported to alter the magnetic behavior in GFO owing to the presence 30 \nof magneto-structural coupling. [24] Here, we study the evolution of optical constants of 31 \nGFO as a function of distortion induced by hydrostatic stress. The evolution of yy 32 \ncomponent of real and imaginary parts of dielectric constants as a function of application 33 \nof hydrostatic pressure is shown in Fig. 6(b). The figure shows that with increasing 34 \nhydrostatic pressure, the position of the first peak in the ε″ spectra remains almost 35 \nidentical, while the peak at 3.78 eV tends to shift towards higher energy. An additional 36 \npeak also appears which is marked with the vertical arrow beyond 20 GPa. The evolution 37 \nof such optical behavior has its origin in the electronic structure as we explain below. 38 \nWith increasing hydrostatic pressure, since the structure is distorted as the bond lengths, 39 \nin general, are decreased. As a result, there is a growing tendency of the wave functions 40 \nof the adjacent ions to overlap with each other. Consequently, the energy levels shift in a 41 \nrepulsive manner which is reflected in the observed optical behavior. Such modification 42 \nof electronic structure of GFO upon application of pressure has been explained in our 43 \nprevious work. [24] The inset of Fig. 6 (b) shows yy component of the refractive index 44 \n(n) as a function of applied pressure  exhibiting a gradual fall at the incident photon 45  Page 10 of 18 energy of 3 eV. Thus, this study qualitative ly demonstrates the presence of a coupling 1 \nbetween the optical and the structural parameters.  2 \n 3 \nIV. Conclusions 4 \n 5 \nIn summary, we have performed ellip sometry studies on epitaxial GaFeO 3 thin films and 6 \nhave compared the dielectric response and other optical constants with our density 7 \nfunctional calculations using different approximation schemes with GGA+U showing the 8 \nbest agreement with the experiments. The origin  of optical activities in GFO is identified 9 \nas transition from O 2p to Fe 3d states. We find that the inclusion of site disorder, off- 10 \nstoichiometry, epitaxial strain and hydrostatic pressure influence the optical properties 11 \ndue to shifting of Fe 3d state. We observe that incorporation of the cation site disorder 12 \ninto GFO lattice renders it to become an indirect band gap semiconductor, consistent with 13 \nthe experimental observations. Further, the cation site disorder also brings about a 14 \nsignificant reduction in the electronic band gap with respect to that of the ground state 15 \nstructure of GFO. Interestingly, we find that inclusion of site disorder and epitaxial strain 16 \ninto the ground state structure significantly improves the agreement between calculated 17 \nand experimental results clearly illustrating that gallium ferrite contains inherent cationic 18 \nsite-disorder.  19 \n 20 \nAcknowledgements 21 \nThe work was supported by Department of Science and Technology, Govt. of India 22 \nthrough project number SR/S2/CMP-0098/2010. Authors thank Prof. Y.N. Mohapatra, 23 \nSA thanks NPL for the J C Bose Fellowship.  24 \n 25 \nReferences 26 \n 27 \n[1] Teowee G, Boulton JM, Uhlmann DR 1998 Int. Ferroelectrics  20 39. 28 \n[2] Murphy TE, Chen D, Phillips JD 2004 Appl. Phys. Lett.  85 3208. 29 \n[3] Muralt P 1996.  ISAF '96., Proceedings of the Tenth IEEE International 30 \nSymposium  1 145. 31 \n[4] Polla DL 1992 ISAF '92., Proceedings of the Eighth IEEE International 32 \nSymposium 127. 33 \n[5] Li S, Lin Y-H, Zhang B-P, Nan C-W, Wang Y 2009 J. Appl. Phys.  105 056105. 34 \n[6] Brody P, Crowne F 1975 J. Electron. Mater.  4 955. 35 \n[7] von Baltz R, Kraut W 1980 Phys. Lett. A 79 364. 36 \n[8] Pintilie L, Alexe M, Pi ntilie I, Botila T 1996 Appl. Phys. Lett. 69 1571. 37 \n[9] Yang SY, Martin LW, Byrnes SJ, Conr y TE, Basu SR, Paran D, Reichertz L, 38 \nIhlefeld J, Adamo C, Melville A, Chu YH, Yang CH, Musfeldt JL, Schlom DG, 39 \nAger JW, Iii, Ramesh R 2009 Appl. Phys. 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Lett. 92 9 \n047401. 10 \n[20] Kundaliya DC, Ogale SB, Dhar S, McDonald KF, Knoesel E, Osedach T, Lofland 11 \nSE, Shinde SR, Venkatesan T. 2006 J. Magn. Magn. Mater.  299 307. 12 \n[21] Arima T, Higashiyama D, Kaneko Y, He JP, Goto T, Miyasaka S, Kimura T, 13 \nOikawa K, Kamiyama T, Kumai R, Tokura Y 2004 Phys. Rev. B 70 064426. 14 \n[22] Mukherjee S, Ranjan V, Gupta R, Garg A. 2012 Solid State Commun.  152 1181. 15 \n[23] Mukherjee S, Garg A, Gupta R 2011 J. Phys.: Condens. Matter 23 445403. 16 \n[24] Roy A, Prasad R, Auluck S, Garg A 2012 J. Appl. Phys.  111 043915. 17 \n[25] Kalashnikova A, Pisarev R, Bezmaternykh L, Temerov V, Kirilyuk A, Rasing T.  18 \n2005  JETP Lett . 81 452. 19 \n[26] Jones RO, Gunnarsson O 1989 Rev. Mod. Phys. 61 689. 20 \n[27] Kresse G, Joubert D 1999 Phys. Rev. B 59 1758. 21 \n[28] Blöchl PE 1994  Phys. Rev. B 50 17953. 22 \n[29] Kohn W, Sham LJ 1965  Phys. Rev. 140 A1133. 23 \n[30] Perdew JP, Ruzsinszky A, Csonka G,  aacute, bor I, Vydrov OA, Scuseria GE, 24 \nConstantin LA, Zhou X, Burke K 2008 Phys. Rev. Lett.  100 136406. 25 \n[31] Monkhorst HJ, Pack JD 1976  Phys. Rev. B 13 5188. 26 \n[32] Tran F, Blaha P 2009  Phys. Rev. Lett.  102 226401. 27 \n[33] Azzam RMA, Bashara NM. Ellipsometry and Polarized Light. Amsterdam: 28 \nNorth-Holland, 1977. 29 \n[34] Jellison JGE, Modine FA 1996 Appl. Phys. Lett. 69 371. 30 \n[35] Feng GF, Zallen R 1989  Phys. Rev. B  40 1064. 31 \n[36] Skorodumova NV, Ahuja R, Simak SI, Abrikosov IA, Johansson B, Lundqvist BI 32 \n2001  Phys. Rev. B  64 115108. 33 \n[37] Pisarev RV, Moskvin AS, Kalashnikova AM, Rasing T 2009  Phys. Rev. B 79 34 \n235128. 35 \n[38] LikhtenshteÏn AI, Moskvin AS, Gubanov VA 1982 Sov. Phys. Solid State 24 2049. 36 \n[39] Usachev P, Pisarev R, Balbashov A, Kimel A, Kirilyuk A, Rasing T 2005 Phys. 37 \nSolid State  472292. 38 \n[40] Ederer C, Spaldin NA 2005 Phys. Rev. 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B  57 88. 41 \n 42 \n43  Page 12 of 18 Figure Captions  1 \n 2 \nFig. 1 (a) XRD spectrum of a pulsed laser deposited GFO thin film showing (010) 3 \norientation (the left inset shows the rocking curve of (040) peak and the right inset shows 4 \na schematic of the orthorhombic unit cell of GFO); (b) fitting of ellipsometry data using 5 \nTauc-Lorentz model with fit parameters listed inside the plot (inset shows the three layer 6 \nmodel used for simulation).   7 \n 8 \nFig.2 (a) Real ( ε′) and imaginary ( ε″) parts of dielectric function determined 9 \nexperimentally and theoretically and compared with the literature; (b) experimentally 10 \ndetermined absorption coefficient ( α) showing the absorption edge (inset plots dispersion 11 \nof experimentally computed refractive index ( n) and extinction coefficient ( k)) and (c) ε′, 12 \nε″ spectra along principal crystallographic directions corresponding to the ground state 13 \nstructure of GFO calculated using GGA+U method, plotted as a function of incident 14 \nphoton energy 15 \n 16 \nFig.3 (a) Electronic band structure of the ground state structure of GFO with arrows 17 \nindicating the interband transitions responsi ble for the evolution of the peaks in the ε″ 18 \nspectra shown in Fig.1 (c); (b) total and partial density of states of ground state structure 19 \nof GFO; (c) comparison of band structures of the ground state and Fe2-Ga2 site disorder 20 \nstructures and (d) total and partial density of states of Fe2-Ga2 site disordered structure. 21 \n 22 \nFig.4 Real ( ε′) and imaginary ( ε″) components of dielectric function plotted as a function 23 \nof incident photon energy obtained experimentally and theoretically showing effect of (a) 24 \nepitaxial strain, (b) site disorder and (c) epitaxial strain and Fe2-Ga2 site disorder. 25 \n 26 \nFig.5 Comparison of experimentally determined optical constants with Fe2-Ga2 site 27 \ninterchanged structure with tensile strain of 3 %. 28 \n 29 \nFig.6 Effect of (a) off-stoichiometry and (b) hydrostatic pressure on the ε′, ε″ spectra of 30 \nGFO, inset shows the variation of refractive index at 3 eV as a function of applied 31 \nhydrostatic pressure. 32 \n 33 \n 34 \n 35 \n36  Page 13 of 18  1 \n 2 \n 3 \n 4 \n 5 \nFig. 1 Roy et al 6 \n 7 \n 8  Page 14 of 18 \n 1 \nFig. 2 Roy et al 2 \n 3  Page 15 of 18 \n 1 \n 2 \n 3 \n 4 \nFig.3 Roy et al 5 \n 6 \n 7 \n 8 \n 9 \n 10 \n 11 \n  12 \n 13 \n 14  Page 16 of 18 \n 1 \n 2 \nFig.4 Roy et al 3 \n 4 \n 5 \n 6 \n 7 \n 8 \n 9 \n 10 \n 11 \n 12 \n     13 \n 14 \n 15  Page 17 of 18  1 \n 2 \n 3 \nFig.5 Roy et al 4 \n 5  Page 18 of 18 \n 1 \n 2 \n 3 \n 4 \nFig. 6 Roy et al 5 \n 6 \n 7 \n 8 \n 9 "    },    {        "title": "2302.10711v1.Per_grain_and_neighbourhood_stress_interactions_during_deformation_of_a_ferritic_steel_obtained_using_three_dimensional_X_ray_diffraction.pdf",        "content": "Per-grain and neighbourhood stress interactions during deformation of a ferritic steel\nobtained using three-dimensional X-ray di ffraction\nJames A. D. Balla,b, Anna Kareerc, Oxana V . Magdysyukb, Stefan Michalikb, Thomas Connolleyb, David M. Collinsa\naSchool of Metallurgy and Materials, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom\nbDiamond Light Source Ltd., Harwell Science and Innovation Campus, Didcot, OX11 0DE, United Kingdom\ncDepartment of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, United Kingdom\nAbstract\nThree-dimensional X-ray di ffraction (3DXRD) has been used to measure, in-situ, the evolution of ∼1800 grains in a single phase\nlow carbon ferritic steel sample during uniaxial deformation. The distribution of initial residual grain stresses in the material was\nobserved to prevail as plasticity builds, though became less pronounced, and therefore less influential as strain increased. The initial\nSchmid factor of a grain was found to be strongly correlated to the intergranular stress change and the range of stresses that are\npermissible; a grain well aligned for easy slip is more likely to exhibit a range of stresses than those orientated poorly for dislocation\nmotion. The orientation path of a grain, however, is not only dependent on its initial orientation, but hypothesised to be influenced\nby its stress state and the stress state of its grain environment. A grain neighbourhood e ffect is observed: the Schmid factor of\nserial adjoining grains influences the stress state of a grain of interest, whereas parallel neighbours are much less influential. This\nphenomenon is strongest at low plastic strains only, with the e ffect diminishing as plasticity builds. The influence of initial residual\nstresses becomes less evident, and grains rotate to eliminate any orientation dependent load shedding. The ability of the BCC ferrite\nto exhaust such neighbourhood interactions, which would otherwise be detrimental in crystal structures with lower symmetric and\nfewer slip systems, is considered key to the high ductility possessed by these materials.\nKeywords: High-energy X-ray di ffraction, Crystallographic texture, 3D characterization, Neighbour orientation, Steel\n1. Introduction\nTo fully understand the response of a polycrystalline mate-\nrial to deformation, it is necessary to understand the factors that\ngovern the onset of plastic deformation for individual grains.\nThe behaviour of individual grains must be influenced by any\nlong-range stress acting over the whole sample or component,\nreferred to as Type I stresses, but additionally from the presence\nof short /medium-range internal stresses at the microstructure\nscale—these are referred to as the Type II (grain average, inter-\ngranular) or Type III (intragranular, varying stress within a grain)\nstresses [ 1]. An important source of these inter- and intragranu-\nlar stresses in a single phase material arises from dislocations\nand dislocation structures that form during plasticity [ 2,3,4,5].\nFor an idealised polycrystal, the resolved shear stress (RSS) is\nthe degree of shear stress applied to a slip plane of an individual\ngrain when an external load is applied [ 6]. The RSS is a function\nof the applied stress and the orientation of the grain relative to\nthe loading axis of the polycrystal. When the RSS exceeds a\ncritical value (the critical resolved shear stress or CRSS), for a\nslip plane within a grain, slip will initiate on that plane. The\nRSS value presupposes, for an ideal polycrystal, that the onset\nof plastic strain within each grain is entirely governed by the\norientation of the grain relative to the tensile axis. Recent stud-\nies, however, suggest that even microstructurally simple alloys\nEmail address: D.M.Collins@bham.ac.uk (David M. Collins)may deviate from these Taylor law predictions at the meso- and\nmacroscale. In austenitic stainless steels, Juul et al. [7]found\ndeviations from predictions in both grain lattice rotations and the\ndependence of stress state on grain orientation. Greeley et al. [8]\nfound inconsistencies between measured and predicted resolved\nshear stresses in a simple Mg-Ndalloy; such studies reinforce\nthe need for new experiments to reveal these fundamental aspects\nof deformation.\nThe requirement for understanding the governing agents of\ndeformation is certainly true of high-performance multiphase\nsteels, possessing complex microstructures and processing meth-\nods; a complete understanding of the micro- and macromechan-\nical mechanisms that govern plastic deformation is far from\ntrivial. However, there is evidence that even microstructurally\nsimple steel alloys respond to deformation in unexpected ways.\nRecent analyses of single phase ferritic steels have shown strain-\npath dependent deformation, in combination with initial texture,\ninfluences hardening [ 9,10]. The yield stress of interstitial-free\nsteels has also been shown to depend on the distribution of dislo-\ncations, even with near-identical dislocation densities [ 11]. This\ndemonstrates an evolving understanding of even simple steel\nalloys and how they perform under deformation.\nOne factor that can modify the stress state (and therefore the\nonset of slip) for individual grains within a polycrystal is its im-\nmediate local neighbourhood. Kocks et al. [12] found deviations\nin grain strain rates from Taylor model predictions, which were\nattributed to grain neighbourhood e ffects. For a grain of interest,\nPreprint submitted to Acta Materialia February 22, 2023arXiv:2302.10711v1  [cond-mat.mtrl-sci]  21 Feb 2023neighbourhood grains can generate stress concentrations that\nmodify the stress state of the grain, depending on the orientation\nrelationship between the grain and its neighbourhood grains.\nSubsequent simulations by Raabe et al. [13] on the generation\nof orientation gradients within grains during loading found a\nsignificant dependence of orientation gradient strength on inter-\nactions between grains and their immediate neighbours. Further\nsimulation studies have noted grain neighbourhood interactions\nwhich significantly modify grain strain values [ 14], leading to\nneighbourhood dependencies in dynamic strain aging [ 15], void\ngrowth [16], and fatigue life [17].\nGrain neighbourhood e ffects have also been directly ob-\nserved experimentally. Using electron back-scatter di ffraction\n(EBSD) and digital image correlation (DIC) methods, grain\nneighbourhood e ffects have been found to significantly influence\ndeformation twinning in twinning-induced plasticity (TWIP)\nalloys [ 18], strain localization in Ni-based superalloys [ 19], and\ngrain boundary sliding in Alalloys [ 20]. These e ffects have\nalso been studied in-situ using three-dimensional techniques.\nSynchrotron X-ray studies have demonstrated a strong influence\nof grain neighbourhood on grain stress states in hexagonal close\npacked (HCP) polycrystals [ 21]. These e ffects are partially at-\ntributed to the elastic and plastic anisotropies inherent to HCP\nsystems. However, grain neighbourhood e ffects may have been\nobserved in-situ in cubic systems by Neding et al. [22], who\nfound stacking faults being generated in grains that were poorly\norientated for fault formation according to their Schmid factor.\nAnother significant parameter that influences the response\nof a grain to an external load is the residual stress of the grain.\nThese residual stresses can be desirable—residual compressive\nstresses induced by shot peening, for example, can significantly\nimprove resistance to fatigue crack initiation [ 23]. However,\nthey can be deleterious to material performance: for example\nby driving creep cracking at elevated temperatures in stainless\nsteels [ 24]. Residual stresses can be generated by many material\nprocessing operations, such as welding [ 25,26], forging [ 27]\nand casting [ 28]. Significant e fforts have been made to simulate\nthe formation of residual stresses during such operations [ 26,\n28,29], and to explore the e ffects of initial residual stresses\non further deformation [ 30,24]. Although direct observation\nof residual stresses in three-dimensional components is non-\ntrivial, synchrotron radiation has been used to quantify them in\ntwo-dimensional projections [25, 30].\nIt is evident that direct in-situ measurements of individual\ngrain responses to deformation are required to further explore\nthese e ffects. Far-field Three-Dimensional X-Ray Di ffraction\n(ff-3DXRD) is an ideal technique for measuring in-situ defor-\nmation response at the mesoscale. Using ff-3DXRD, per-grain\ncentre-of-mass positions, orientations, and Type II strain states\ncan be evaluated for many hundreds of grains simultaneously in\nlarge samples [ 31], and recent advancements to the technique\ncan also measure Type III stresses and grain morphologies [ 32].\nIn the extreme case, with far-field 3DXRD alone, highly accu-\nrate characterisation of grain neighbourhoods can be performed\nin-situ for thousands of grains [ 33]. These measurement capa-\nbilities have enabled newfound microstructural insights into the\ndeformation response of simple steel alloys [ 34,35,7]. TheseTable 1: Nominal chemical composition of DX54 steel [9].\nElement Fe C P S Mn\nwt.% Balance≤0.06≤0.025≤0.025≤0.35\nσ\nσPilatus 2M CdTe 2D area detectorDiffracting grains\nIncident X-ray beam\nBeam stop\nFerritic Steel Specimen\nProbed X-ray volume\nFigure 1: Tensile specimen and 3DXRD configuration at the I12 beamline,\nDiamond Light Source.\ntechniques can also be utilised for the analysis of more compli-\ncated alloys such as shape-memory alloys [ 36], duplex stainless\nsteels [ 37] and transformation-induced plasticity (TRIP) steels\n[38]. Of particular note is recent work by El Hachi et al. [36],\nwho used in-situ 3DXRD to observe grain-neighbourhood ef-\nfects deformation-induced martensitic transformations (DIMT)\nin a Cu-Al-Be alloy.\nIn-situ 3DXRD experiments are currently limited to spe-\ncialised synchrotron beamlines around the world and are there-\nfore not commonplace. New instruments that enable reliable\ncollection and analysis of 3DXRD data are highly desired by\nthe materials science community. The aim of this study is two-\nfold: to establish and evaluate in-situ far-field 3DXRD at the I12\nbeamline at Diamond Light Source, and to use the technique to\nexplore the e ffect of residual stress and grain neighbourhood on\ngrain responses to in-situ deformation in 3D.\n2. Experimental Method\n2.1. Material\nA tensile dog-bone sample with a 1 mm wide gauge sec-\ntion and a 6 mm gauge length was produced from a 1 mm thick\nZn-galvanized sheet of DX54 steel, a single-phase ferritic steel\nwith a body-centered cubic crystal structure. The nominal com-\nposition is given in Table 1. Prior to any testing, the galva-\nnized surface was removed with abrasive media, then annealed\nat980 °C for1 hand slow-cooled at ∼1 °C min−1to achieve a\ncoarse equiaxed microstructure.\n2.2. 3DXRD Data Acquisition\n3DXRD data were collected in Experimental Hutch 1 of the\nI12 beamline at Diamond Light Source [ 39]. The experimental\n2geometry is shown in Figure 1; the axes represent the sample\nreference frame that will be hereon used. Prior to loading a sam-\nple, a multi-distance calibration [ 40], executed within DAWN\n[41,42] was performed with a NIST 674b CeO 2standard ref-\nerence sample [ 43]. The energy of the monochromatic X-ray\nbeam was determined as 60.2 keV and the sample to detector\ndistance was 550.3 mm . The tensile specimen was placed in a\nDeben CT5000 5 kN load frame designed for X-ray tomography\nscans, with the loading axis set to be axisymmetric with the sam-\nple stage rotation axis. 3DXRD data were collected at multiple\napplied vertical ( y-axis) loads to allow tracking of individual\ngrains under di fferent applied external loads.\nTen successive “letterbox” 3DXRD scans were taken along\nthe sample gauge ( ydirection) with a 1.5 mm×0.15 mm beam,\nand0.05 mm of overlap between each. The total illuminated\nsample volume at each load step was ∼1 mm×1 mm×1 mm .\nDuring each letterbox scan, the sample was rotated about the\nyaxis from−180° to180° . Diffraction patterns were acquired\nwith a Pilatus 2M CdTe area detector, recording data at 1°incre-\nments and a 1 sexposure time. This data collection procedure\nreplicated a previously established routine for the beamline [ 44].\n0.00 0 .01 0 .02 0 .03 0 .04 0 .05 0 .06\nStrain020406080100120140160Applied Stress (MPa)\n0 MPa100 MPa132 MPa142 MPa162 MPa\n35 MPaDiscrete Load Steps\nFigure 2: Stress-strain increments at which 3DXRD measurements were made.\nThe load frame was operated in displacement control at a\nconstant rate of 0.2 mm min−1until the desired force targets\nwere reached. At each deformation step, the applied load was\nheld constant. Six load steps, shown in Figure 2, were chosen to\nexplore the deformation response up to a small degree of plastic-\nity (maximum strain of ∼5 %). The final load step was recorded\nafter unloading the sample as far as possible, to investigate the\nremaining residual stresses in the sample. Two small strips of\nKapton tape were a ffixed at each end of the sample gauge area\nto act as fiducial markers. The macroscopic strain on the sample\nwas measured from large field-of-view radiographs, acquiredwith an end-of-hutch X-ray imaging camera, at each load step\nusing the fiducial marker separation.\n2.3. EBSD data collection\nFor EBSD data collection, the sample was polished to a\n0.04µmsurface finish using colloidal silica, then examined\nwith a Zeiss Merlin field emission gun scanning electron mi-\ncroscope (FEG-SEM). A Bruker e−FlashHREBSD detector was\nused to collect EBSD maps at a 5 nA probe current and a 20 keV\nbeam energy. Electron backscatter di ffraction patterns (EBSPs)\nwere recorded at high angular resolution (HR-EBSD), 800×600\npixels2and were saved for post-collection analysis; the scan and\nindexing was performed using Esprit 2.0 software. A low magni-\nfication map ( 5 mm×6.5 mm ,2µmstep size) of the sample was\nalso taken in the undeformed grip region. A higher spatial reso-\nlution map ( 1.0 mm×0.8 mm ,2µmstep size) of the deformed\nspecimen, within the gauge of the tensile specimen was also\nperformed.\nHR-EBSD was used to estimate intragranular residual elastic\nType III stresses from the deformed specimen using an in-house\nwritten method that measures subtle changes in the crystal geom-\netry, inferred from the EBSPs. This method extracts a reference\nEBSP within a grain, from which image shifts are measured via\na cross correlation function between the reference pattern and\ntest patterns within this grain. A deformation gradient tensor is\ndefined from the di ffraction pattern shifts, from which strain and\nrotation components can be separated using a finite decomposi-\ntion framework [ 45]. The method has a strain sensitivity of at\nleast 1×10−4[46]. The resulting strain tensor, for each pixel,\nis multiplied by the sti ffness tensor for ferritic steel, using the\nV oigt notation, to obtain the stress tensor. Comprehensive details\nof the HR-EBSD method are provided elsewhere [ 47,48], as\nwell as the mathematical descriptions [49, 50].\n2.4. 3DXRD analysis\nIndexing and analysis of collected 3DXRD data broadly\nfollowed routines established in [ 44]. However, a number of\nimprovements have since been made to the analysis procedure\nthat yield significant gains in data quality. A flowchart showing\nthe data processing steps utilised in this study are shown in Fig-\nure 3. A number of pre-processing stages have been introduced\nto reduce error in detector peak positions. Peaks close to gaps\nin the modules of the Pilatus di ffraction detector were deemed\nunreliable due to inaccurate intensity profile shape and were\nremoved. As the position and orientation for each module on the\ndetector is independent, small distortions in peak position may\nhave been introduced if these displacements were not corrected\nfor. Following an established routine [ 51] applied to a series of\nreference CeO 2calibration patterns, the module displacements\nwere determined—this allowed a correction file to be generated\nwhich specifies the sub-pixel adjustments required to correct\nthe peaks data. Di ffraction peak searching and determination\nof grain positions, orientations and strain states was performed\nwith ImageD11 [ 52], with each individual letterbox scan indexed\nseparately.\nTo enable higher-level analyses of the 3DXRD data, as a\ncomplete dataset, in-house pre- and post-processing software\n3Detector \nImagesPeaksearch at\nmultiple intensity\nthesholds\n(ImageD11)Peak /f_iles at\nmultiple\nthresholdsRemove \npeaks near \nmodule gapsFiltered \npeaks /f_ile\nMerge peaks\ntogether\n(ImageD11)Merged\npeaksCorrect detector\nmodule distortionCorrected\npeaks\nG-vectors Trimmed peaks\nGrain mapsRe/f_ined\ngrain mapsGrain maps\nwith errorsSplit peaks for\neach phaseCalculate\nscattering \nvectors\n(ImageD11)Grid indexing\n(ImageD11)\nIterative\nre/f_inement\n(ImageD11)Bootstrap \nerror\ndetermination\nFor each phaseFigure 3: Indexing and analysis procedure for 3DXRD datasets.\nwas developed using the Python programming language. This\nsoftware coordinated the parallel initial indexing of each Im-\nageD11 letterbox scan. An object-orientated model was devised\nto store and manipulate grain data at a number of di fferent levels:\nthe sample as a whole; individual load steps; individual scans\nwithin those load steps; single phases within those scans; and\nfinally individual grains within each phase. This data model\nenabled sophisticated post-processing and data analysis, such\nas duplicate grain detection, sample rigid body transform detec-\ntion, grain nearest neighbour identification, and positional grain\nfiltering.\nA new “bootstrap” approach was devised to determine errors\nin grain positions, orientations, and strains. More specifically,\nthe grain Biot strain and stress tensors were determined from\nthe reciprocal lattice lengths of the grain as described in Ap-\npendix A. After each grain was indexed with the grid indexing\nprocedure, the detector peaks associated with that grain were\nisolated. Next, 100 copies of these peaks were generated. For\neach copy, a random 50 % of the peaks were removed. Then,\nusing each copy containing only 50 % of the peaks, the grain\nposition, orientation and strain was refined using ImageD11.\nThis allowed the convergence of the grain parameter refinement\nto be probed—if a large variation in grain parameter outcomes\nwas observed, it would indicate a high degree of error in each\npeak, leading to a poor degree of convergence of the refinement\nroutine. Conversely, if only a small variation in grain parameter\nwas observed between refinements, the error in each peak would\nhave to be lower. The means of the grain positions, orientations\nand strain tensor elements of each copy were taken as the fi-\nnal parameters for that grain. The standard deviations of the\nparameter distributions were used as the errors for those grain\nparameters.\nAfter the bootstrap routine, a cleanup process removed any\nduplicated grains created by the grid index procedure. Using\nthese cleaned grain data, each individual “letterbox” scan was\nstitched together to form a single grain map for each load step,\nwith duplicate grains removed in the overlap regions. Grains\nwere then tracked across subsequent load steps to identify grains\ncommon to all load steps. For the cleanup, stitching and trackingstages, the same grain de-duplication algorithm, described in\ndetail in Appendix B, was used to identify and group together\nrepeated grain observations.\nOnce grains were tracked across multiple load steps, tracked\ngrain positional data was used as an input to a rigid body trans-\nformation solver using the coherent point drift algorithm as\nimplemented in the pycpd Python library [ 53]. The resultant\norientation matrix was then used to generate a modified orienta-\ntion matrix at a load-step level in which individual grains could\nbe subsequently assessed, with any sample rotations e ffectively\nremoved.\n(a)\n100µm\n(b)\n100µm0.00.51.0σVM(GPa)[0 0 1] [0 1 1][1 1 1]\nFigure 4: HR-EBSD map obtained from the sample gauge, post deformation\nwith∼5 %plastic strain; (a) IPF- Zcolouring and (b) the corresponding per-pixel\nType III von Mises stresses. The white regions are grains determined to have a\nlow quality and were eliminated from the analysis. The loading axis is left-right.\n43. Results\n(a)\n[0 0 1] [0 1 1][1 1 1]\nz (mm)−0.50\n−0.25\n0.00\n0.25\n0.50x (mm)−0.50\n−0.25\n0.00\n0.25\n0.50y (mm)\n−0.50−0.250.000.250.50(b)\n(c)\nZX\n09182736455463Count\nFigure 5: In the condition prior to loading, grains within the tensile specimen\ngauge region have crystal orientations shown on Z-axis inverse pole-figure (a),\nwith the corresponding colours shown in (b) grain position map, and (c) a texture\nevident on the Z-axis direct{1 1 0}contoured pole-figure.3.1. Microstructure\nThe grain mean spherical equivalent diameter was deter-\nmined to be 130µm, calculated from a low-magnification EBSD\nscan of 1797 grains with a 2°misorientation tolerance to deter-\nmine the grain boundaries. The reader is referred to Figure 1 in\nthe Supplementary Materials for an IPF- ZEBSD map of these\nresults. The sample examined with 3DXRD was also charac-\nterised post-deformation with a HR-EBSD scan acquired within\nthe sample gauge. An IPF- Zmap is shown in Figure 4a. Using\nthe stress (residual elastic Type III) tensor elements obtained\nfrom the cross-correlation method, the per-pixel von Mises stress,\nσVM, was calculated, shown in Figure 4b. The reader is referred\nto Figure 3 in the Supplementary Materials for maps of each\nstress tensor component of the HR-EBSD scan. The magnitude\nof these localised intraganular stresses is seen to far exceed the\nmacroscopic yield stress of the material ( ∼100 MPa ), with evi-\ndence of steep stress gradients in most grains. Stress banding is\nalso evident in several grains, featuring as yellow, approximately\nvertical streaks that traverse part or whole grains; these struc-\ntures are evidence of residual stresses developing from plasticity\nstructures.\n3.2. 3DXRD grain indexing\nIn total, 18 320 raw grains were indexed over the six load\nsteps. After the grain de-duplication and letterbox stitching\nroutines, 10 741 grains remained, with an average of 1790 grains\nper load step. Table 2 shows the number of stitched grains\nremaining at each load step.\nTable 2: Number of stitched grains remaining per load step.\nσApplied (MPa) Number of stitched grains\n0 2012\n100 2343\n132 1999\n142 1817\n162 1275\n35 (unload) 1295\nAn overview of the sample volume probed, given for an ex-\nample unloaded state (prior to deformation) is shown in Figure 5.\nThe distribution of the grain orientations are represented on an\ninverse pole figure, and the corresponding grain position map\nis shown in Figure 5a and Figure 5b, respectively. Here, the\ngrains are coloured by their orientation and the size of each point\nin the grain position map is scaled by the grain volume. The\ncentre-of-mass positions of indexed grains is seen to conform\nwell to the sample geometry, representing the probed volume\nof the tensile specimen within the gauge. A rolled texture is\nobserved in grain orientations as shown in the {1 1 0}pole figure;\nFigure 5c.\nAs the original DX54 sample sheet was hot-dip galvanized\nwith a Zncoating, a variation in lattice parameter was expected\nin grains located close to the original galvanised surfaces. This\narises from any remnant surface Zn di ffusing into the surface\nduring the heat treatment stage used to tailor the grain size. To\n5investigate this, grain unit cells were extracted, as shown in\nFigure 2 in the Supplementary Materials. For grains close to\nthe surfaces where a variation in lattice parameter is evident, a\ngeometric filter was applied to the 3DXRD dataset to remove\ngrains with centre-of-mass positions more than 0.3 mm from the\norigin along xorz. The remaining grains can then be safely\nevaluated for changes in strain.\n3.3. Stress development\nz (mm)−0.3\n0.0\n0.3 x (mm)−0.3\n0.0\n0.3y (mm)\n−0.30.00.3\nσApplied = 0 MPa(a)\nz (mm)−0.3\n0.0\n0.3 x (mm)−0.3\n0.0\n0.3y (mm)\n−0.30.00.3\nσApplied = 100 MPa(b)\nz (mm)−0.3\n0.0\n0.3 x (mm)−0.3\n0.0\n0.3y (mm)\n−0.30.00.3\nσApplied = 132 MPa(c)\nz (mm)−0.3\n0.0\n0.3 x (mm)−0.3\n0.0\n0.3y (mm)\n−0.30.00.3\nσApplied = 142 MPa(d)\nz (mm)−0.3\n0.0\n0.3 x (mm)−0.3\n0.0\n0.3y (mm)\n−0.30.00.3\nσApplied = 162 MPa(e)\nz (mm)−0.3\n0.0\n0.3 x (mm)−0.3\n0.0\n0.3y (mm)\n−0.30.00.3\nσApplied = 35 MPa(f)-50 0 50 100 150 200σyy(MPa)\nFigure 6: Grain maps at multiple applied loads, coloured by the vertical, σyy,\nstress in the sample reference frame (tensile direction).\nWith each indexed grain assigned a stress tensor, position,\norientation and size for every load step, general trends for the\ndeformation accumulation in the ferritic steel can be described.\nFigure 6 shows the filtered grain centre-of-mass map at multiple\nload steps, coloured by the vertical component of the stress\nvector in the lab frame (co-axial with the applied load from\nthe load frame). The changes in grain stress over the loadingsequence are clearly visible; the mean average stresses follow\nthe applied stress steps, with a wide distribution. Notably, a wide\nrange in initial grain stresses was observed in the first load step\n(σApplied =0 MPa ), although this becomes less pronounced after\nunloading ( σApplied =35 MPa ). To investigate the grain stress\ndistributions further, histograms are plotted for each component\nof the 3DXRD Type II stress tensor, σ, in the lab frame, as\nshown in Figure 7. The components of the HR-EBSD Type\nIII stress tensor from the post-deformation condition have also\nbeen plotted. At no applied load, a broad distribution in σyy,\ncentered roughly at the origin, is observed. This distribution\nnarrows significantly and shifts positively as strain increases in\nthe sample, indicating that the substantial initial residual stresses\nplay a less influential role as plasticity builds. The distributions\nin the transverse normal stresses ( σxxandσzz) are also initially\nbroad, but the degree of narrowing is reduced compared to σyy.\nThe shear components, σxy,σxz&σyzshow a stress distribution\nthat is characteristically narrower than the linear components. As\nstress is applied, these distributions broaden slightly, with bumps\ndeveloping in the leading and trailing tails. This demonstrates\nseveral grains that di ffer from the majority as plasticity builds.\nIn every direction, the Type III grain stress distributions from the\nHR-EBSD dataset show substantially broader distributions than\nthe 3DXRD Type II stresses, indicating that the localised grain\nstresses can deviate significantly, higher or lower, than the grain\naveraged Type II stress as measured by 3DXRD. Additionally,\ntheσxyshear component in the EBSD scan is narrower than the\naxial stress component ( σxx), which is narrower still than the\ntransverse stress component ( σyy), again mirroring the behaviour\nof the Type II grain stresses.\n3.4. Grain tracking\nAfter the individual letterbox scans were stitched together,\nthe stitched grain maps were tracked over all applied load steps,\nyielding 674fully-tracked grains. 178fully-tracked grains re-\nmained after geometric filtration. The evolution of grain pa-\nrameters can therefore be explored across multiple load steps.\nFigure 8a depicts the development of σyystress (tensile direc-\ntion) in the lab frame for each tracked grain. In general, good\nagreement is observed between the external applied load from\nthe load frame and the individual tracked grain data. Most grains\nfollow the trend of the macroscopic stress-strain curve. Grains\nwith high initial stresses (residual stress) tend to maintain a\nhigher stress state throughout the loading series. There are sev-\neral grains that experience a stress drop when passing the yield\npoint, a phenomenon also observed by Hedstr ¨om et al. [37] (in a\nduplex steel) and Abdolvand et al. [21] (in Zr and Ti).\nThe influence of grain initial (residual) stress on the change\nin grain stress between the first two load steps is examined in\nFigure 8b, where a clear negative correlation is observed between\nthe initial total stress of a grain and its ability to further increase\nin total stress. The relationship here indicates a grain with a low\ninitial stress may experience a significant increase in von Mises\nstress, whereas a grain with a high initial stress is more likely to\nsee only a small increase, or a stress drop.\nThe influence of the initial grain orientation on the grain\nstress development can be analysed, as per Figure 8c. Grains\n60.0000.0020.0040.0060.0080.0100.012(a)σxx\nσApplied (MPa)\n0\n100\n132\n142\n162\n35 (unload)\n0 (EBSD)(b)σyy\n(c)σzz\n−400 −200 0 200 4000.0000.0020.0040.0060.0080.0100.012(d)σxy\n−400 −200 0 200 400(e)σxz\n−400 −200 0 200 400(f)σyz\nGrain stress tensor component (MPa)DensityFigure 7: 3DXRD Type II grain stress distributions across multiple load steps, with added EBSD Type III grain stress distributions.\nwith a lower initial Schmid factor are much more likely to in-\ncrease rather than decrease their von Mises stress over the load-\ning series. Grains with a higher Schmid factor have a wider\nrange of permissible von Mises stress changes over the loading\nseries; the stress of these grains may increase or decrease. This\nprovides strong evidence that the orientation alone does not de-\ntermine whether a grain hardens or softens, at the grain average\nstress level.\nFigure 8d shows that the extent of the stress drop (after\nplastic deformation has commenced) appears to be negatively\ncorrelated with grain residual stress ( σApplied =0 MPa ). Grains\nwith a higher residual stress tended to experience a greater stress\ndrop. As proposed by Hedstr ¨om et al. [37], this may be evidence\nof the activation of multiple competing slip processes beyond a\ncertain threshold strain that act to lower the overall lattice strain\nof the grain.\n4. Discussion\nDuring this investigation, the deformation behaviour of a\nsingle phase ferritic steel has been studied to reveal the interplay\nbetween the size, orientation, position and elastic stress /strain\nstate, on a per-grain basis, within a volume of interest. Theinterplay between these parameters is considered key to under-\nstanding the macroscopic behaviour of engineering alloys, where\nthe method of far-field 3DXRD has been utilised for this purpose.\nFollowing an initial proof of concept on the I12 beamline [ 44],\nthe method implementation and its analysis have been developed\nsignificantly, as part of this work, enabling a first in-situ 3DXRD\ninvestigation at Diamond Light Source. This discussion includes\nfirstly a critique of the data processing and analysis method cre-\nated for Diamond, followed by the resulting phenomenological\nmicromechanical mechanisms that govern and control the tensile\nresponse of the ferritic steel during the onset and low levels of\nplasticity. The addition of complementary HR-EBSD orientation\nand Type III stress measurements to supplement the per-grain\n3DXRD observations are also discussed.\n4.1. Indexing quality\nAn average of 1790 grains per load step were indexed, after\ngrain de-duplication and letterbox stitching, with 2012 grains\nremaining in the first load step. The success of the indexing\nstrategy utilised here is evident in the grain map (Figure 5); the\nedges of the sample are well defined, matching the macroscopic\ndimensions of the sample. Any spurious “satellite” grains in\nunfeasible positions relative to the sample are absent, indicating\n70.00 0 .01 0 .02 0 .03 0 .04 0 .05 0 .06\nεApplied−1000100200300σyy(MPa)(a)\nTracked grain σyy\nσApplied\n100 200 300 400 500\nGrain initial von Mises stress, σ0 MPa\nVM (MPa)−400−300−200−1000100200300σ100 MPa\nVM −σ0 MPa\nVM (MPa)(b)\n0.350 0 .375 0 .400 0 .425 0 .450 0 .475 0 .500\nGrain initial Schmid factor−200−1000100200300σ162 MPa\nVM −σ0 MPa\nVM (MPa)(c)\n−100−50 0 50 100 150\nσ132 MPa\nyy −σ100 MPa\nyy (MPa)−100−50050100150200250Grain initial stress, σ0 MPa\nyy (MPa)(d) Best fitFigure 8: Tracked grain correlations: (a): following macroscopic yield, coloured by their initial σyyvalues; (b): initial von Mises stress vs change in von Mises\nstress between first two load steps ( σApplied =0 MPa and100 MPa ); (c): initial Schmid factor vs change in von Mises stress between no load and max applied load\n(σApplied =0 MPa and 162 MPa); (d): σyychange between σApplied =132 MPa and 100 MPa vs initial σyy.\nsatisfactory convergence in the grain centre-of-mass position re-\nfinement. The measured texture also matches other prior studies\nfrom DX54 steel of similar pedigree [9, 10, 44].\nTable 3: Indexing technique precision summary for 3DXRD datasets obtained\nfrom I12, Diamond.\nParameter Error\nOrientation (°) 0.03\nPosition, horizontal ( µm) 8\nPosition, vertical ( µm) 6\nεxx(×10−3) 0.2\nεyy(×10−3) 0.1\nεzz(×10−3) 0.2\nA summary of the grain parameter precision is given in\nTable 3. The quoted values are derived from the parameter\ndistributions, produced from the bootstrap data analysis; theyrepresent one standard deviation from each distribution. The\nmagnitudes are comparable to those obtained by other authors\n[54]. The orientation precision achieved here is attributed to the\ndetector module distortion corrections; without this inclusion\nthe orientation error is significantly higher ( 0.1°[44]). The po-\nsition and strain accuracy are determined by the indexing and\nrefinement procedure, as well as grain errors calculations. The\nhigh precision is evidence of a low degree of divergence between\nindividual refinements during the bootstrap error determination\nprocess—this is due to the high number of average peaks per\ngrain, and an accurate peak location a fforded by the detector\nmodule distortion corrections. The grain strain error of 3DXRD\nat I12 at Diamond is considered acceptable in the context of\nengineering alloys, and is approximately one decade from state-\nof-the art implementations of the method, such as 1×10−5at\nthe ID11 beamline of the ESRF [ 31]. Further improvements to\ngrain strain accuracy with the current detector are only likely\nto be possible by reducing the ωstep size in subsequent exper-\n8iments. Using the present data analysis method, the precision\nin orientation and grain centre-of-mass position are now limited\nby the experimental geometry, detector pixel size, and detector\ndynamic range.\n4.2. Errors\nUsing the bootstrap approach for grain error determination,\nFigure 9 shows the distributions of errors in grain centre-of-mass\nposition (a), orientation, (b) and stress tensor elements (c). Grain\nvolume-weighted averages of these distributions are provided in\nTable 4. Position errors follow a broadly bimodal distribution,\nwith most position errors around 5µm. The outlying grains with\nlarger position errors are likely caused by grains truncated by\nthe X-ray beam at the top and bottom of the illuminated volume.\nAs these values remain significantly below the average grain\nsize, the conclusions of this work reliant on grain position re-\nmain valid. Orientation and stress error histograms are normally\ndistributed, and increase slightly with increasing applied load,\nas expected from the increased di ffracted peak spread due to\nplastic mosaicity [55].\nTable 4: Mean grain parameter errors at each load step.\nσApplied\n(MPa)Position\n(µm)Orientation\n(°)Stress\n(MPa)\n0 3 0 .02 62\n100 3 0 .02 50\n132 4 0 .03 53\n142 4 0 .03 52\n162 5 0 .03 58\n35 5 0 .03 64\n4.3. Grain parameter correlations\nGrain volumes measured using far-field 3DXRD alone are\ndetermined using the mean intensity of the X-ray scattering\npeaks of that specific grain relative to the intensity of scattering\npeaks from all other grains [ 56]. From these values, the grain\nsize can be estimated by multiplying each 3DXRD grain volume\nby a constant scale factor. Here, this corresponded to a value\nto match the grain size distribution obtained from EBSD mea-\nsurements. Comparing grain diameter against grain stress in the\nlab frame, as shown in Figure 10, yields an inverse correlation\nbetween the grain diameter and both von Mises stress and error\nin von Mises stress. Stresses were smaller and closer to the\nmacroscopic applied stress for larger grains than smaller grains.\nThis behaviour is evidence of a Hall-Petch dependency at the\nintergranular level, which is heavily governed by intragranular\ngrain-size dependent backstresses [ 57]. To validate this as a\nreal phenomenon, and not an artefact from the grain parame-\nter refinement process, the grain-averaged misorientation from\nHR-EBSD is used as an analog for the 3DXRD intergranular\nstress. As is evident in Figure 10, both follow a matching in-\nverse relationship to the grain size. The misorientation-grain\nsize relationship is explained by the heterogeneous nature of\nplasticity; greater misorientation is correlated with increased\nresidual elastic strains /stresses [58].\n0 5 10 15 20 25 30\nPosition error ( µm)0.000.020.040.060.080.100.12Probability Density(a)σApplied (MPa)\n0\n100\n132\n142\n162\n35 (unload)\n0.00 0 .01 0 .02 0 .03 0 .04 0 .05 0 .06\nOrientation error ( °)020406080Probability Density(b)\n0 20 40 60 80 100 120\nStress error (mean of diagonal tensor elements) (MPa)0.0000.0050.0100.0150.0200.0250.0300.035Probability Density(c)Figure 9: Per-grain error parameter distributions, shown for each loading step.\nAgain using the grain-averaged misorientation from EBSD\ndata as an analogue for intergranular stress, the relationship\nbetween 3DXRD-measured grain stresses and the final Schmid\nfactor is explored in Figure 11. The relationship replicates\nthe observation of the intergranular von Mises stress change at\n9100200300400500600σ35 MPa\nVM (MPa)(a)\n0 50 100 150 200\nGrain diameter ( µm)406080100120Error in σ35 MPa\nVM (MPa)(b)0.350.400.450.500.550.600.650.700.75\nGrain-average misorientation ( °)3DXRD\nEBSDFigure 10: Grain diameter relationship in the deformed, unloaded condition\n(σApplied =35 MPa ) to the (a) 3DXRD grain von Mises stress & EBSD grain-\naveraged misorientation, and (b) the 3DXRD von Mises stress error.\nyield, as in Figure 8c, where grains with a high Schmid factor\nhave a larger allowed range of stresses, and grains with a lower\nSchmid factor are much more restricted in stress. This is an\nintriguing observation as it implies that even though a grain may\nbe favourably orientated for easy slip, it may experience elastic\nstresses that are (i) very high (behaving as a hard grain), (ii)\nmoderate stresses where plasticity is easy, (behaving as a soft\ngrain), or (iii) stresses below the critical resolved shear stress, so\nno slip at all. The range of conditions indicates that grains with\na high Schmid factor are more susceptible to grain-neighbour\ninteractions, in a way that grains with a lower Schmid factor are\nnot.\nWhilst intergranular stresses as measured via 3DXRD are\nclearly important, grain neighbour interactions from compatibil-\nity (amongst other mechanisms) can generate significant intra-\ngranular stress gradients [ 59]. Any stress localisation, governed\nby Type III stresses in particular for BCC steels, is predicted\nto ultimately determine locations of failure [ 60]. Given the\n0.36 0 .38 0 .40 0 .42 0 .44 0 .46 0 .48 0 .50\nSchmid factor50100150200250300350400σ35 MPa\nVM (MPa)\n0.350.400.450.500.550.600.650.700.75\nGrain-average misorientation ( °)3DXRD\nEBSDFigure 11: Tracked grain unloaded condition ( σApplied =35 MPa ) von Mises\nstress from 3DXRD data (blue dots) and EBSD grain-averaged misorientation\n(red crosses) vs grain Schmid factor.\nmagnitude of the Type III stresses present in this material, signif-\nicantly higher than the 3DXRD measured Type II stresses, their\nrole cannot be neglected. The enormous di fference between the\nType II and Type III stress distribution widths were similarly ob-\nserved by Hayashi et al. [61], which also proposes that locations\nwithin grains of high triaxial stresses have low plastic strains,\nwith adjacent regions compensating with large plastic strains.\nDeformation in the present study is highly heterogeneous (see\nFigure 4b), with the behaviour of a given grain influenced by\nboth Type II and Type III stress, its own and of the neighbours.\n4.4. Grain rotation during straining\nGiven there was strong evidence that the orientation (i.e.\nSchmid factor) will influence the stress development, the orien-\ntation changes of individual grains are considered. Figure 12a\nshows the evolution in tracked grain orientation, plotted on an\ninverse pole figure, over all load steps. Most grain rotations are\nseen to be small in magnitude. This may be expected give the\ntotal plastic strain is ∼5 %. Interestingly, grain transformations\nare markedly di fferent across load steps, with large changes in\norientation direction visible once the applied stress passes the\nmacroscopic yield point (at σApplied =100 MPa ). For certain\nsimilarly-orientated grains, such as those labelled G54 & G247,\nthe grain orientation changes are very similar in direction, indi-\ncating a strong correlation between grain initial orientation and\nsubsequent rotation transformation under load. Other similarly-\norientated grains, such as grains G137 and G219, have markedly\ndifferent orientation changes under load. One key di fference\nbetween these two pairs of grains is their Schmid factor, with\nvalues for G54 & G247 significantly lower than G137 & G219.\nThis may be the reason a narrower stress range was observed for\n10grains with a lower Schmid factor (see Figure 8c), and a wide\nrange of stress states for those with a high Schmid factor, a ffect-\ning their magnitude of rotation. Whilst the observations here\nare limited to a few grains, observations of grains with similar\ninitial orientations but dramatically di fferent stress development\nwere reported by Hedstr ¨om et al. [37] in an in-situ investigation.\nThis also reflects the findings of Juul et al. [7], who found that\nin-situ grain rotations under load were much more scattered than\npredicted by finite-element models. Figure 12b further explores\nthis discrepancy by plotting the overall grain orientation to the\nmaximum applied load—the majority of tracked grains are ob-\nserved to rotate as predicted for BCC crystals deforming by\n/angbracketleft1 1 1/angbracketrightpencil glide [ 62] as per Figure 12c, although some grains\ndo not rotate as expected from their starting orientation. This is\nprimarily because the rotation of a grain, at any single moment\nin time, is governed by its intergranular and intragranular stress\nstate. It is evident from this study that the global Schmid factor\nalone cannot be used to predict the stress state for a given grain;\nit must instead be controlled by grain interactions.\n4.5. Grain neighbourhood\nThere are several factors that are known to contribute to\nthe stress state of the grain in polycrystals, which are often\nwell correlated to significant orientation gradients. Firstly, this\ndepends on the initial orientation of a strained crystal and its\norientation path as plasticity develops [ 63,64,65]. The magni-\ntude of per-grain stresses may well develop from any remnant\nresidual elastic stress, accumulating from low strains, or present\nfrom prior processing [ 66,67]; this was unequivocally evident\nin the present study for grains with a measured stress initial\nstress inversely proportional to the subsequently accumulated\nstress change (see Figure 8b and 8d). Within a grain, there may\nbe domains or di ffering orientations that gives rise to di fferent\ndislocation glide systems, and thereby hardening [ 68,66,13].\nContinued subdivision of cell structures developing within a\ngrain, arising from plasticity structures, will themselves influ-\nence both the inter- and intragranular stresses [ 69,66]. Finally,\nthe grain stress will also depend on the stress and crystallogra-\nphy attributes of the neighbouring microstructure environment\n[12, 13, 14, 21].\nFor a given grain of interest, the influence of neighbour-\ning grains is explored here. For each grain, a Schmid factor\nwas devised by taking the highest Schmid factor for slip in\nany/angbracketleft1 1 1/angbracketrightdirection. Here, a softgrain is defined as a crystal\nexhibiting a high Schmid factor that is well aligned for easy\nslip, relieving stress, whereas a hard grain is orientated poorly\nfor easy glide and would be expected to build higher elastic\nstresses. The existence of such e ffects are well reported in al-\nloys with HCP crystal structures [ 70], but its significance is\nseldom reported for highly symmetric cubic systems. To ascer-\ntain the e ffect in ferritic steel, the nearest neighbours of each\ngrain using a Delaunay triangulation [ 71] were identified at (i)\nthe yield point, σApplied =100 MPa , and (ii) at∼5 %plastic\nstrain,σApplied =162 MPa . Neighbours were then subdivided\ninto series and parallel neighbours, depending on their position\nrelative to the central grain. From a vector that connects grain-to-\ngrain centroid coordinates, an inclination angle was calculatedbetween this and the tensile axis. Serial neighbours were classi-\nfied as those with an inclination angle less than 45°and parallel\nneighbours were those with an inclination angle greater than\n45°; such classification has been established by other authors\n[21]. Whether a grain is hard orsoftcould next be ascertained\nfor serial and parallel neighbours of a central grain of interest,\ncalculated for the tensile direction stress, σyy. This analysis is\nreported in Figure 13; stresses of a central grain are provided as\na function of the average Schmid factor of its serial and parallel\nneighbours, but also for its components cosλandcosφto ex-\nplore which of these components govern the response, as defined\nby the Schmid factor equation for the critically resolved shear\nstress,τ:\nτ=cos(φ) cos(λ)σ (1)\nwhereφis the angle between the slip plane normal and the ten-\nsile axis,λis the angle between the slip direction and the tensile\naxis, andσis the applied stress. While the scatter is large, a\nnegative correlation is evident between the mean Schmid factor\nof serial neighbours and the stress achieved by that grain, Fig-\nure 13c at the onset of yield ( σApplied =100 MPa ). If a grain\nis in series with hard grains (low Schmid factor), it tends to\nexperience a greater stress in the loading direction compared to\nsofter grains (high Schmid factor). This finding, reported in-situ\nfor a BCC system for the first time, replicates the trend for HCP\ncrystals [ 21], though critically, is a weaker e ffect. By plotting\nthe individual components of the Schmid factor, the influence\nof slip direction vs slip plane normal orientation relative to the\nloading axis can be separately observed. For serial neighbours,\nthe negative correlation between a central grain σyyand the\nSchmid factor is also observed with cosλ; interestingly, this\nis the converse to the slip plane normal component behaviour,\ncosφ(Figure 13b), showing a positive correlation with σyystress.\nThis indicates that the slip direction in an adjoining grain deter-\nmines the Schmid factor dependent neighbourhood e ffect. The\nanalysis is also shown for σApplied =162 MPa (Figure 13d–f).\nThe trends replicate the observations at the yield stress, but the\ngradients are shallower. This indicates that the neighbourhood\neffect diminishes as plastic strain increases. One can postulate\nthat such neighborhood e ffects may disappear entirely as plas-\nticity builds; grains must continue to rotate, which is likely to\neliminate any grain-to-grain load partitioning as their respective\nSchmid factors become more similar. Given there are 48 slip\nsystems for BCC, with a great degree of freedom for crystal dis-\ntortion /rotation, this operation will be easy. It is also plausible\nthat the elimination of grain neighbour stress partitioning, which\nwould otherwise be detrimental to fracture strain, is a key reason\nwhy these materials exhibit exceptional ductility.\nThe e ffect of parallel neighbours on central grain maximum\nstress is minimal in this case (see Figure 13g–l), matching ob-\nservations by Abdolvand et al. [21]. There is no strong re-\nlationship between the mean Schmid factor of neighbouring\ngrains, and the central grain σyystress; the trend lines when at\nσApplied =162 MPa (Figure 13j–l) are notably flat. In short, the\naxial stress for a central grain is correlated with the orientation of\nneighbouring grains located in series along the loading axis with\nit. Neighbouring serial grains with a lower Schmid factor lead to\n11(a)\n[1 0 0] [1 1 0][1 1 1]σApplied increments (MPa)\n0 - 100\n100 - 132\n132 - 142\n142 - 162\n162 - 35\n0.30 0 .35 0 .40 0 .45 0 .50Maximum Schmid Factor[1 0 0][1 1 1]\n[1 11][1 1 0](b)\n[1 0 0][1 1 1]\n[1 11][1 1 0](c)G54G247\nG137G219Figure 12: Z-axis inverse pole-figure, stereographic projection (shaded by Schmid factor), showing tracked grain rotations across multiple load steps (a), between no\nload (σApplied =0 MPa) and max load ( σApplied =162 MPa) (b), and predicted rotations of BCC grains deforming by /angbracketleft1 1 1/angbracketrightpencil glide [62] (c).\nhigher overall axial stresses in the central grain. This supports\nthe conclusions of prior modelling studies of cubic systems by\nBretin et al. [14], who found that the influence of neighbouring\ngrain orientations on the stress state depends on the relative\nposition of the central grain and the neighbouring grain with\nrespect to the loading axis. This observation has implications\nfor future modelling endeavours—for crystal plasticity finite\nelement method (CPFEM) simulations of a very small number\nof grains, for example, simply randomising all grain orientations\nmay not be su fficient to remove influences of nearest neighbour\norientation on grain stress states. Instead, a number of model\niterations with shu ffled grain orientations may be required, as\nobserved in face-centered cubic systems by Kocks et al. [12].\nNa¨ıve simulations (with a larger number of grains) of macro-\nscopic stress anisotropy due to sample texture may be similarly\naffected by “unlucky” shu ffles of grain orientation due to this\neffect. To establish the significance of the grain neighbour ef-\nfect observed in the present study, on a cubic system, future\ninvestigations at high plastic strains, for other cubic structurepolycrystals, and the role of Type III intragranular stresses are\nexciting future areas for exploration in this field.\n12−150−100−50050100150200250\n(a)\nσApplied = 100 MPa\nσyy=−164 cos λ+ 193(b)\nσApplied = 100 MPa\nσyy= 214 cos φ+−73(c)\nσApplied = 100 MPa\nσyy=−401S+ 278\n0.4 0 .5 0 .6 0 .7 0 .8\ncosλ−150−100−50050100150200250\n(d)\nσApplied = 162 MPa\nσyy=−101 cos λ+ 188\n0.6 0 .7 0 .8 0 .9\ncosφ(e)\nσApplied = 162 MPa\nσyy= 125 cos φ+ 29\n0.375 0 .400 0 .425 0 .450 0 .475 0 .500\nSchmid Factor(f)\nσApplied = 162 MPa\nσyy=−287S+ 259Serial Neighboursσyyof central grain (MPa)\n−150−100−50050100150200250\n(g)\nσApplied = 100 MPa\nσyy=−74 cos λ+ 136(h)\nσApplied = 100 MPa\nσyy= 114 cos φ+ 2(i)\nσApplied = 100 MPa\nσyy=−119S+ 146\n0.4 0 .5 0 .6 0 .7 0 .8\ncosλ−150−100−50050100150200250\n(j)\nσApplied = 162 MPa\nσyy=−2 cos λ+ 126\n0.6 0 .7 0 .8 0 .9\ncosφ(k)\nσApplied = 162 MPa\nσyy=−18 cos φ+ 139\n0.375 0 .400 0 .425 0 .450 0 .475 0 .500\nSchmid Factor(l)\nσApplied = 162 MPa\nσyy=−102S+ 172Parallel Neighboursσyyof central grain (MPa)Figure 13: Dependence of grain stress in the tensile direction, σyy, on the slip direction component, cosλ, the slip plane normal component, cosφ, and the Schmid\nfactor. This is shown for both serial and parallel neighbours, and for each at the onset of plasticity, σApplied =100 MPa and at∼5 %plastic strain, σApplied =162 MPa .\nThe size of the circles represent the relative volumes of the grains.\n135. Conclusion\nAn in-situ three-dimensional X-ray di ffraction experiment\nduring mechanical loading of a low-carbon ferritic steel has been\nperformed to reveal individual crystal behaviour. Significant de-\nvelopments on existing 3DXRD analysis methods have revealed\nseveral insights, which are summarised here:\n1.This experiment is the first example of an in-situ 3DXRD\nexperiment at Diamond Light Source, demonstrating the\nfeasibility to track the response of a polycrystalline mate-\nrial during deformation on a per-grain level.\n2.A rigorous approach to quantifying uncertainties related\nto per grain stress and orientation analysis provide con-\nfidence that reliable in-situ 3DXRD measurements can\nbe performed on I12 at Diamond. Confidence in orienta-\ntion is stable at around 0.03°, and the stress error varies\nbetween 50 MPa to 64 MPa\n3.The initial Schmid factor of a tracked grain was found to\ninfluence the increase in the per-grain von Mises stress.\nGrains with a low initial Schmid factor were found to\nalmost exclusively increase in stress, whereas grains with\na high Schmid factor had a wider range of allowed changes\nin stress.\n4.The change in per-grain von Mises stresses between the\nfirst two load steps was found to be strongly negatively\ncorrelated to the initial von Mises stress of a grain. This\ndemonstrates that tracked grain residual stresses are domi-\nnant over subsequent grain stress evolution.\n5.Grainσyy(axial) stresses were found to broadly follow the\nmacroscopic stress-strain curve, but a significant fraction\nof grains experienced a stress drop beyond the global yield\npoint. The magnitude of the stress drop was found to be\nproportional to the grain residual stress—grains with a\nhigher starting σyytended to experience a greater stress\ndrop.\n6.Most grains followed an orientation change that is ex-\npected from/angbracketleft1 1 1/angbracketrightpencil glide, however, examples that\ndo not follow this trend were evident; similarly orientated\ngrains may posses quite dissimilar orientation paths dur-\ning straining. This may be explained by their Schmid\nfactor, stress state, and grain neighbour interactions.\n7.A grain neighbourhood e ffect is evident at low plastic\nstrains. Grains with hard series neighbours were found\nto have a higher axial stress at the macroscopic yield\npoint, whilst parallel neighbours had limited e ffect on a\ngrain stress state. The strength of this e ffect diminishes at\nhigher macroscopic strains, as grains rotate and Schmid\nfactors between neighbours become more similar, load\npartitioning becomes less significant. The ability for fer-\nritic steels to eliminate these neighbour e ffects is believed\nto contribute to the exceptional ductility they possess.\n6. Acknowledgments\nThis work was supported by Diamond Light Source, instru-\nment I12 [NT26376] and the Engineering and Physical SciencesResearch Council [EP /R030537 /1]. James Ball would like to\nthank the Diamond Light Source and the University of Birming-\nham for jointly funding his PhD program, as well as Anastasia\nVrettou and Neal Parkes for their help with the sample prepa-\nration process, and Younes El-Hachi and Jon Wright for their\nassistance with the bootstrap method of grain parameter error\ndetermination.\n14References\n[1]P. Withers, H. Bhadeshia, Residual stress part 1 – measurement techniques,\nMater. Sci. Technol. 17 (2001) 355–365.\n[2]H. Mughrabi, Dislocation wall and cell structures and long-range internal\nstresses in deformed metal crystals, Acta Metall. 31 (1983) 1367–1379.\n[3]L. E. Levine, P. Geantil, B. C. 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Hagberg, P. Swart, D. S Chult, Exploring network structure, dynamics,\nand function using networkx, in: SCIPY 08, Pasadena, CA, United States,\n2008, p. 960616.\n16Appendix A. Strain and stress tensor determination\nThe Biot strain is used as the default representation of lattice\nstrain for grains defined by ImageD11. The derivation of the\nBiot strain tensor from grain lattice lengths is provided here.\nFirst, the deformation gradient tensor Fis determined:\nF=(U·B)−/intercal·B/intercal\n0(A.1)\nwhere Urotates a vector in the Cartesian grain system, /vectorGc, to\nthe sample reference frame, /vectorGs:\n/vectorGs=U/vectorGc (A.2)\nBtransforms a vector in reciprocal space, /vectorGhkl, to the Cartesian\ngrain system /vectorGc[72]:\n/vectorGc=B/vectorGhkl (A.3)\nand is defined by the deformed lattice parameters of the grain in\nreciprocal space (a∗,b∗,c∗,α∗,β∗,γ∗):\nB=a∗b∗cos(γ∗) c∗cos(β∗)\n0b∗sin(γ∗)−c∗sin(β∗)cos(α)\n0 0 c∗sin(β∗)sin(α)(A.4)\nand:\ncos(α)=cos(β∗)cos(γ∗)−cos(α∗)\nsin(β∗)sin(γ∗)(A.5)\nB0is calculated from the reference unit cell lattice lengths. Fis\nthen used to determine the Biot strain tensor E:\nE=C1/2−I=(F/intercal·F)1/2−I (A.6)\nwhere Cis the right Cauchy-Green deformation tensor and I\nis the identity matrix. ImageD11 uses polar composition to\ndetermine C1/2:\nF=V·R=R·C1/2(A.7)\nThe stress tensor σis then determined from the strain tensor as\ndefined by Oddershede et al. [31] using the sti ffness values in\nTable A.5.\nTable A.5: Sti ffness constants used for ferrite phase [73].\nStiffness Constant Value (GPa)\nc11 231.4\nc12 134.7\nc44 116.4\nIt is noted that the Biot strain tensor Edetermined by Im-\nageD11 is not equivalent to the linear Lagrangian strain tensor\nεused by Oddershede et al. [31]. Therefore, the Biot stress\ntensor equivalent does di ffer fromσ. However, the tensors must\nbe equivalent in the small strain limit, and the error between\nrepresentations is assumed to be negligible at the strain levels\nobserved in this study. The strain and stress tensors in the sample\ncoordinate system ( Esandσsrespectively) are also determined\nby rotating Eandσ[31]:\nEs=UEU/intercal(A.8)\nσs=UσU/intercal(A.9)Appendix B. Grain de-duplication algorithm\nA new de-duplication algorithm has been devised to identify\nrepeated observations of the same grain, given a list of grains, a\ndistance tolerance, and a misorientation tolerance. First, a list\nof all possible grain pairs within a specified centre-of-mass dis-\ntance (the distance tolerance) is produced. A nearest-neighbour\nsearch algorithm using K-dimensional trees (as originally de-\nfined by Maneewongvatana and Mount [74]) is performed using\nan implementation in the scipy Python library [ 71] to gener-\nate a list of candidate grain pairs. Once the candidate grain\npairs list is identified, the misorientation between the grains of\neach pair is calculated. The misorientation algorithm, originally\ndefined by Proudhon [75] in the pymicro Python library, has\nbeen vectorised with the help of the Numba Python library [ 76],\nthen parallelised. This is much faster than the single-threaded\napproach—with a modern 12-core AMD Ryzen processor, cal-\nculating the misorientation of 100 000 grain pairs takes 5 swith\nthe parallelised approach, vs 47 s with the original approach.\nOnce the misorientation for each grain pair is determined, grain\npairs with a misorientation that is greater than the misorientation\ntolerance are removed. The remaining grain pairs are therefore\nclose together in position and orientation.\nOnce the final grain pairs list is produced, a graph is con-\nstructed using the NetworkX Python library [ 77]. Each node in\nthe graph represents a grain, and nodes are connected with edges\nif the grains in the nodes are paired. Then, a list of all connected\ncomponent sub-graphs is generated. Each connected component\nsub-graph represents a group of nodes that is connected together.\nFor example, if grains AandBare paired, and grains Band\nCare paired, grains A,BandCform a connected component\nsubgraph. This connected component sub-graph ABC is then\nsaid to represent a single physical grain, observed more than\nonce.\nIn practice, the outputs of this algorithm are used slightly\ndifferently depending on the purpose. At the grain cleaning\nstage, all grains provided in the grain list are from the same\nindividual 3DXRD scan. Therefore, each grain group returned\nfrom the algorithm is associated to a single “clean grain”, which\nmay consist of multiple raw observations, or “raw grains”. The\nparameters of the “clean grain” (such as grain position, orienta-\ntion, and strain state) are determined from the parameters of the\nraw grains. A volume-weighted average of raw grain positions\nis used for the position of the clean grain. The UBI matrix of the\nclean grain (where UBI=(U·B)−1) is taken from the UBI of\nthe largest contributing raw grain. The volume of the clean grain\nis determined by the sum of the volumes of the raw grains. The\nsame procedure applies to the letterbox stitching stage. When\nthe algorithm is applied to grains across load steps, multiple\ngrains from each load step may be found in each grain group\nreturned by the algorithm. In this instance, only the largest grain\nin each load step is used.\n17Per-grain and neighbourhood stress interactions during\ndeformation of a ferritic steel obtained using\nthree-dimensional X-ray diffraction\nJames A. D. Balla, b, Anna Kareerc, Oxana V. Magdysyukb, Stefan Michalikb, Thomas Connolleyb,\nand David M. Collinsa,*\naSchool of Metallurgy and Materials, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom\nbDiamond Light Source Ltd., Harwell Science and Innovation Campus, Didcot, OX11 0DE, United Kingdom\ncDepartment of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, United Kingdom\n*Corresponding author: D.M.Collins, D.M.Collins@bham.ac.uk\nSupplementary Content\nFigure 1 is a low magnification EBSD map from the undeformed grip section of the DX54 ferritic steel sample. A\nlarge equiaxed single-phase ferritic microstructure is observed. These data were used to determine the average grain\nsize of the material.\n500µm[0 0 1] [0 1 1][1 1 1]\nFigure 1: Low magnification IPF-Z map from EBSD data on undeformed DX54 ferritic steel.\nFigure 2 is a map of 3DXRD grain centre-of-mass positions for the first load step ( σApplied = 0 MPa). Grain points\nin the figure are coloured by the mean crystal unit cell length of the grain, as determined from the Bmatrix by\nImageD11. A significant near-surface variation in grain lattice parameter is observed at the surfaces perpendicular to\nthez-axis. These were the original external galvanized surfaces of the steel sheet from which the samples were cut.\nNote the effect was not observed on sample surfaces perpendicular to the x-axis, which were originally in the bulk of\nthe sheet.\nFigure 3 is a map of each component of the σstress tensor obtained from the processed HR-EBSD data. Outlying\ngrains with non-physical stress distributions (due to poor quality data for these grains) have been masked in white.\nSignificantly wide stress distributions (on the order of ∼3 GPa) are observed in the in-plane axial and transverse\ndirections ( σxxandσyyrespectively). Narrower distributions are observed in the shear directions ( σxy,σxzandσyz).\n1arXiv:2302.10711v1  [cond-mat.mtrl-sci]  21 Feb 20232.868 2 .870 2 .872 2 .874 2 .876 2 .878 2 .880Lattice parameter ( /RingA)z (mm)−0.50\n−0.25\n0.00\n0.25\n0.50x (mm)\n−0.50−0.25 0.00 0 .25 0 .50Figure 2: Grain map at no applied load, observed as a cross section of the dog bone specimen. The points here\nrepresent grains, which are coloured by lattice parameter and their size is scaled by their volume.\nσxx\n100µmσxy\n100µmσxz\n100µm\nσyy\n100µmσyz\n100µm\nσzz\n100µm\n−1.5−1.0−0.50.00.51.01.5σ(GPa)\nFigure 3: HR-EBSD maps obtained from the sample gauge, post deformation with ∼5 % plastic strain and per-pixel\nType III stress tensor elements. The loading axis is left-right.\n2"    },    {        "title": "0801.4073v1.Magnetoelectric_Control_of_Domain_Walls_in_a_Ferrite_Garnet_Film.pdf",        "content": "Magnetoelectric Control of Domain Walls \nin a Ferrite Garnet Film  \n \nA.S. Logginov, G.A. Meshkov, V.A. Nikolaev, A.P. Pyatakov*  \n Physics Department, M.V. Lomonosov MSU, Moscow, Russia, 119992; \nand A.K. Zvezdin \nA.M. Prokhorov General Physics Institute, 38, Vavilova st, Moscow, 119991 \n* Corresponding author: pyatakov@phys.msu.ru \n \nThe effect of magnetic domain  boundaries displacement induced by electric field is observed in epitaxial ferrite \ngarnet films (on substrates with the (210) crystallographic or ientation). The effect is odd with respect to the electric \nfield (the direction of  wall displacement changes with the polarity of  the voltage) and even with respect to the \nmagnetization in domains.  The inhomogeneous magnetoelectri c interaction as a possible mechanism of the effect is \nproposed.  \nDOI: 10.1134/S0021364007140093 \n \nThe last few years marked the great progress in the field of magnetoelectric materials [1-\n3]. The newly discovered ferroelectricity induced  by spiral magnetic ordering [3-7] and early \ndiscovered inverse effects of el ectrically induced spin modulat ion [8-10] not only provide a \ndeeper insight into the mechanisms of magnetoelectric coupling (the so-called inhomogeneous \nmagnetoelectric interaction [9-11]) but also map out the route for electric control of \nmicromagnetic structure in solids, the possibility predicted in 1980-ies [11] and still not realized.  \n The best candidates for the specific char acter of micromagnetism in magnetoelectric \nmaterials are thin films of ferrite garnets. From  the one hand they are classical object to study \nmicromagnetism, [12-15] from the other hand they  exhibit a magnetoelectric effect that is an \norder of magnitude greater than the co rresponding effect in the classical Cr 2O3 magnetoelectric \n[16].  Electromagneto-optical effects on local areas of a ferrite-garnet film revealed that effect is \nvanishingly small in a homogeneously magnetized film  but it increases drastically in the vicinity \nof domain walls. This effect was attributed to  “breathing” of domain wall in electric field \nproviding indirect evidence for the influence of el ectric field on micromagnetic structure [17].  \nIn this Letter we report  on the direct magnetooptical obser vation of a new manifestation \nof the magnetoelectric effect, name ly, the electric-field-control di splacement of domain walls in \nferrite garnet films, that is of interest from  the fundamental point of view and opens up new \npossibilities for the development of multipurpos e spintronic and magnetophotonic devices on a \nsingle material platform. \nIn our experiments we used the 9.7- μm-thick epitaxial (BiLu) 3(FeGa) 5O12  ferrite garnet \nfilms grown on a Gd 3Ga5O12 substrate with the (210) crystall ographic orientation. The substrate \nthickness was about 0.5 mm. The period of  the strip domain structure was 34.5 μm, and the \nsaturation magnetization was 4 πMs= 53.5G. To produce a high-strengt h electric field in the \ndielectric ferrite garnet film, we used a 50 μm-diameter copper wire with a pointed tip, which touched the surface of the sample (Fig. 1). The diameter of the tip of the copper “needle” was \nabout 20 μm. This allowed us to obtain an electric field strength of  up to 1500 kV/cm near the tip \nby supplying a voltage of up to 1500 V to the need le. The field caused no dielectric breakdown, \nbecause it decreased rapidly with distance from the needle and, near th e grounding electrode (a \nmetal foil attached to the substrate), did not exceed 600 V/cm. The absence of the possible leakage currents between the need le and the grounding electrode (e.g., over the sample surface) \nwas verified by a milliampermeter. To observe the domain structure, the polarization method \nbased on the Faraday effect was used. For the observation in transmitted light, a hole ~ 0.3 mm \nin diameter was made in the grounding electr ode. The image of the magnetic structure was \nobtained using a CCD camera conn ected with a personal computer. \n1\n3 \n2 4 \n9 6 \n7 8\nEr\n z \ny mA \n5\n \n \n \nFig. 1  Schematic representation of the geometry of the experiment and the configurations of the electric field and \nmagnetization. The electric field (the field lines are shown by the dashed line s) is formed in the dielectric medium of \nthe sample between the needle ( 1) and the metal foil ( 2),which plays the role of the grounding electrode. The \nmaximum field strength (above 1 MV/cm) is reached in the magnetic film ( 3) near the tip; it decreases rapidly in the \nbulk of the substrate ( 4) and does not exceed 600 V/cm near the grounding electrode ( 2). The abscence of the \nleakage currents is controled with the milliampermeter (mA). The light (beams are denoted with wavy arrows) is \nincident along the normal to the surface.  The objective lens (5) is placed behind the pinhole in the foil (2). The inset \nshows the magnetization distribution in the film: the domain wall ( 6) separates two domains ( 7, 8)  with opposite \nmagnetization directions; the tip ( 9) touches the ferrite garnet surface near the domain wall.  \n \nIn the experiment, we measured the static di stribution of magnetizati on before and after \nthe electric field was switched on. As a result, we obtained pairs of images for different voltage \npolarities and different  needle positions. In al l of the series, when a dc voltage was applied \nbetween the needle and the substrate, we observed a local displacement of the domain wall near the tip (Fig. 2). The magnitude of the  displacement increased with voltage.  \nWe found three characteristic features of the phenomenon, which serve as the basis for \nour subsequent consideration.  \n(i) The direction of the domain wall displacem ent depends on the pola rity of the voltage \n(and, hence, on the direction of th e electric field): in the case of positive polarity, the wall was \nattracted to the needle, and, in the cas e of negative polarit y, it was repulsed.  \n(ii) The direction of the wall displacement did not depend on the direction of \nmagnetization in the domain (along the z axis or against it, see the inset in Fig. 1).  \n(iii) The magnitude of the effect increase d strongly with a decrease in the distance \nbetween the tip and the domain wall. The most  pronounced effect was observed at a positive \nvoltage, when the domain wall was pulled into the region of high electric field strength in the \nimmediate vicinity of the point of contact  of the needle and the surface (Fig. 2).  \n (0,0,1)\n(1,2,0)\nFig. 2 The effect of electric field in the vicinity of electrode (1) on magnetic domain wall (2) in the films \nof ferrite garnets: a) initial state b) at the voltage of +1500 V applied  \n \nThe characteristic features listed above  allow us to exclude the effects of non-\nmagnetoelectric nature that could lead to displacements of domain walls: the magnetic fields caused by possible leakage currents and the ma gnetostrictive phenomena caused by the pressure \nof the tip on the sample due to electrostatic at traction. Indeed, the dependence on the polarity of the voltage applied to the needle (feature (i)) allows  us to exclude the effect  of the tip pressure on \nthe sample, because the tip polarizes the sample surface and is attracted to  it irrespective of the \nsign of the potential at the needle; hence, th e effect caused by the ti p pressure should be \nindependent of polarity. Feature (ii) testifies that , even in the presence of leakage currents, the \neffect cannot be related to the magnetic field generated by the electric  currents, because, \notherwise, the domain wall woul d be displaced in opposite direc tions for domains with opposite \nmagnetization directions. Thus, features (i) and (ii) of the phenomenon under study allow us to \nconclude that the latter is  of magnetoelectric nature. \nConcerning feature (i ), it should be emphasized that be cause of the presence of an \ninversion center in the crystal symmetry group of bulk ferrite garnet samples only the effects that \nare proportional to even powers of electric field are possible, which manifests itself in quadratic \nmagnetoelectric [18] and electroma gnetooptic [19] effects. The de pendence of the direction of \nthe domain wall displacement on the electric polarity (the  oddness of the effect with respect to \nelectric field) testifies to the violation of the spatial inversion in films , unlike the case of bulk \nferrite garnet samples.  \nThe third feature (iii), namely, the sharp increase in the magnitude of the effect observed \nwhen the domain wall is pulled closer to the tip, indicates that the electric field selectively acts \non the regions with inhomogeneous magnetization, i. e., the domain walls. Indeed, in this case, \nthe domain wall approaches the point of the need le-surface contact and comes into the region of \nhigh electric field strength (Fig. 1). If we dealt with a homogene ous magnetoelectric effect, the \nelectric field would also act upon the regions of homogeneous magnetization, i.e., domains, by \ndecreasing (or increasing) the domain over which th e tip is positioned. In this case, one should \nexpect an opposite behavior of the magnitude of  the effect as a function of the tip–domain wall \ndistance, because a displacement of the wall to ward the needle would bring the neighboring \ndomain into the region of high elect ric field strength as well. The fact that the needle selectively \nacts on the domain walls suggests that the phe nomenon observed is a manifestation of the \ninhomogeneous magnetoelectric effect  related to micromagnetic inhomogeneities of the material \n[11,20]. \nl k j ijkl L M M F ∇⋅⋅⋅=iEγ ,  (1) \nwhere  M=M(r) is magnetization distribution, E is electric field,  is vector differential \noperator, ∇\nijklγ is the tensor of inhomogeneous magne toelectric that is determined by the \nsymmetry of the crystal. One can learn imediately from the equation (1) that the effect is odd in \nelectric field E, and doesn’t change th e sign with magnetization M reversal, that agrees with the \nmain features of the effect. It follows from (1) that effect is selective to the magnetic inhomogeneities, i.e. it influences on the domain walls rather than domain itselves. This property \nagrees with the enhancement of the effect observed when the wall was drawn in the region of the \ncontact where the electric fi eld strength was maximal.  \n   The influence of the inhomogeneous magneto electric effect on the domain walls is the \nstronger, the greater co mponents of magnetoelectri c interaction tensor γˆ and the smaller the \ncharacteristic size of magnetic inhomogeneity Δ are in the given material. The components of \ntensor γˆ can be estimated from Eq. (1) and from the condition of the equality of the \nmagnetoelectric and magnetostatic energies with the us e of the known magnetic field parameters \nand experimental data: th e saturation magnetization Ms= 53.5 G, the domain wall width Δ= 100 \nnm, the electric field E=1MV/cm = 3.3 103CGS, and the volume density  of magnetostatic energy \nFm–st=0.1 erg/cm3 (estimated from the deflection of the do main wall) [12]. The value obtained for \nthe inhomogeneous magnetoelectric in teraction tensor components are γ~10-9 CGS (for \ncomparison, a similar constant for th e effect that gives rise to spat ially modulated spin structures \nin bismuth ferrite [10] is γ=10-11 CGS). \nDespite the fact that all the features of the effect testif y that the observed phenomenon is \ncaused by the inhomogeneous magnetoelectric inter action in the ferrite ga rnet material [11,20], \nwe cannot completely exclude the mechanism de termined by the local an isotropy variation due \nto mechanical stress associated with the piezoelectric effect. To distinguish between these mechanisms with better accuracy, additional studie s are necessary, for example, the study of the \neffect as a function of the characteristic size of micromagnetic inhomogeneities and dependence on the direction of the electric field. \n \nWe are grateful to A.V. Khval’ kovskii for the interest  in our study and for valuable discussions. \nThis work was supported in part by the Russia n Foundation for Basic Research (project no. 05-\n02-16997) and the “Dynasty” Foundation. \n \n1. Manfred Fiebig, “Revival of  the magnetoelectric effect”, J. Phys. D: Appl. Phys . 38, R123–\nR152 (2005) \n2  W. Eerenstein, N. D. Mathur & J. F. Scott,   Multiferroic and magnetoelectric materials, \nNature, 442, 759 (2006) \n3. S.-W. Cheong, M. Mostovoy, Multiferroics: a ma gnetic twist for Ferroelectricity, Nature \nMaterials, 6, 13 (2007)  \n4. Maxim Mostovoy, Ferroelectricity  in Spiral Magnets, PRL, 96, 067601 (2006)  \n5. A. M. Kadomtseva, Yu.F. Popov, G.P. Vorob’ev, K. I. Kamilov, A. P. Pyatakov, V. Yu. \nIvanov, A. A. Mukhin, A. M. Balbashov, Specifity of magnetoelectric effects in new GdMnO 3, \nmagnetic ferroelectric, JETP Letters, 81, iss.1, pp.19-23 (2005) \n6. Yoshinori Tokura, Multiferroics as Quantum Electromagnets, Science,  312 , 1481 (2006) \n7. 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Smirnov, A. F. Popkov, et al., Sov. Phys. JETP 67, 2269 (1988) \n15.  A. S. Logginov, A. V. Nikolaev, V. N. Onishchuk, and P. A. Polyakov, JETP Lett. 66, 426 \n(1997). \n16. B. B. Krichevtsov, V. V. Pavl ov, and R. V. Pisarev, JETP Lett. 49, 535 (1989). \n17. V. E. Koronovskyy, S. M. Ryabchenko, and V. F. Kovalenko, Electromagneto-optical effects \non local areas of a ferrite -garnet film, Phys. Rev. B 71, 172402 (2005)  \n18. T. H. O’Dell, Philos. Mag. 16, 487 (1967). \n19. B. B. Krichevtsov, R. V. Pisarev, and A. G. Selitskii, Sov. Phys. JETP Lett.  41, 317 (1985) \n20. A.S. Logginov, G.A. Meshkov, A.V. Nikolaev, A. P. Pyatakov, V. A. Shust, A.G. Zhdanov, \nA.K. Zvezdin, Electric field control of micr omagnetic structure, Journal of Magnetism and \nMagnetic Materials , 310 ,  2569  (2007) \n "    },    {        "title": "1706.10148v1.Investigation_on_nickel_ferrite_nanowire_device_exhibiting_negative_differential_resistance_____a_first_principles_investigation.pdf",        "content": "CondensedMatterPhysics,2017,Vol.20,No2,23301:1 –12\nDOI:10.5488/CMP.20.23301\nhttp://www.icmp.lviv.ua/journal\nInvestigationonnickelferritenanowiredevice\nexhibitingnegativedifferentialresistance —a\n/uniFB01rst-principlesinvestigation\nV.Nagarajan,R.Chandiramouli∗\nSchoolofElectricalandElectronicsEngineering,ShanmughaArtsScienceTechnologyandResearchAcademy\n(SASTRA)University,Tirumalaisamudram,Thanjavur,Tamilnadu —613401,India\nReceivedSeptember23,2016,in /uniFB01nalformDecember13,2016\nTheelectronicpropertyofNiFe 2O4nanowiredeviceisinvestigatedthroughnonequilibriumGreen ’sfunctions\n(NEGF)incombinationwithdensityfunctionaltheory(DFT).TheelectronictransportpropertiesofNiFe 2O4\nnanowirearestudiedintermsofdensityofstates,transmissionspectrumand I–Vcharacteristics.Thedensity\nofstatesgetsmodi /uniFB01edwiththeappliedbiasvoltageacrossNiFe 2O4nanowiredevice,thedensityofchargeis\nobservedbothinthevalencebandandintheconductionbandonincreasingthebiasvoltage.Thetransmis-\nsionspectrumofNiFe 2O4nanowiredevicegivestheinsightsonthetransitionofelectronsatdifferentenergy\nintervals.The /uniFB01ndingsofthepresentworksuggestthatNiFe 2O4nanowiredevicecanbeusedasnegativedif-\nferentialresistance(NDR)deviceanditsNDRpropertycanbetunedwiththebiasvoltage,whichmaybeused\ninmicrowavedevice,memorydevicesandinfastswitchingdevices.\nKeywords:nickelferrite,nanowire,negativedifferentialresistance,densityofstates,electrondensity\nPACS:31.10.+z,31.25.-v,61.46.+w,61.66.Fn,73.63.Rt,85.30.-z\n1. Introduction\nThe spinel ferrite is one type of soft magnetic materials with the general formula of MFe 2O4,\nwhere “M” represents the divalent metal ions such as Mg, Zn, Mn, Cu, Co, Ni, etc., which are the\nmost attractive magnetic material owing to their significant magnetic, magnetoresistive and magneto-\noptical properties. The other fascinating characteristics of MFe 2O4are its low melting point, large\nexpansion coefficient, low magnetic transition temperature and low saturation magnetic moment [1].\nIn spite of these properties, the spinel ferrites have been utilized in many technical applications, such\nas in catalysis [2], photoelectric devices [3], nano-device [4], sensors [5], magnetic pigments [6] and\nmicrowave devices [7]. The remarkable magnetic and electronic property of ferrites mainly depends\nupon the cations, their charges and the distribution of cations along tetrahedral (A) and octahedral (B)\nsites[8].Nickelferrite(NiFe 2O4)isoneofthemostversatilematerialsduetoitssoftmagneticproperty,\nlow eddy current loss, low conductivity, catalytic behaviour, high electrochemical stability, abundance\ninnature,etc.,[7].NiFe 2O4isakindofferromagneticoxidewithinversespinelstructureinwhichFe3+\nions are equally distributed between both octahedral B-sites and tetrahedral A-sites, whereas Ni2+ions\noccupy only octahedral B-sites [9]. The inverse spinel ferrites are represented by the general formula\nof (Fe3+)A(Ni2+Fe3+)BO2\u0000\n4[10]. NiFe 2O4powders have been used as catalysts [11], ferrofluids [12],\nbiomedicine [13] and gas sensors [14, 15]. Various methods have been employed for the synthesis of\nnanoscale NiFe 2O4, which includes solid-state reaction [16], sol-gel [17], rheological phase reaction\nmethod[18],mechanochemical[19],pulsedwiredischarge[20],electrospinning[21],hydrothermal[22]\nand sonochemical methods [23].\n∗Corresponding author\nThisworkislicensedundera CreativeCommonsAttribution4.0InternationalLicense .Furtherdistribution\nofthisworkmustmaintainattributiontotheauthor(s)andthepublishedarticle ’stitle,journalcitation,andDOI.23301-1arXiv:1706.10148v1  [cond-mat.mes-hall]  22 Jun 2017V.Nagarajan,R.Chandiramouli\nThenanoscaledeviceshaveattractedresearchersandthesedevicesmayhavehighpackingdensityand\nare more efficient than microelectronic devices. Moreover, the junction properties of nanoscale devices\nplay a vital role in the charge transport across the semiconductor/metal interfaces [24]. Furthermore,\nthesemiconductor/metalinterfacemayalsoformSchottkyorohmiccontact.IfSchottkytypeofcontact\nis present, rectifying action takes place. The transport characteristics of nanoscale contacts must be\ninvestigated before the amalgamation of these structures in nanoscale electronic devices [25]. Transport\npropertiesofthesenanoscaledevicecontactsarealsoinfluencedbythechargecarriersandthegeometryof\nthesemiconductor/metalinterface.Negativedifferentialresistance(NDR)behaviourisamostsignificant\nelectronic transport property for various electronic components [26]. The NDR effect can be observed\nfrom low dimensional nanostructures like nanowire when connected between two electrodes [27]. In a\nnegative differential resistance device, the occupied states on one side may get aligned with the gap on\nthe other side, when the voltage across the device is increased. Moreover, the current reduction may\nalso occur due to the position of the resonant states of the molecule, which move within the gap of one\nof the contacts. In the case of carbon nanotube junctions, the reduction in the current for an increased\nbias voltage is due to the mismatch in the symmetry of incoming and outgoing wave functions of the\nsame energy. Besides, the NDR effect observed between gold electrodes and scattering region is due to\nthe lack of orbital matching between the contacts. The potential barriers in 2D graphene sheets are due\nto the linear dispersion of electrons, which shows a gap in their transmission across the barrier [28].\nThus,negative resistanceprovidesa physicalsignificancein nonlinearelectroniccomponents. NDRhas\nattracted scientific community due to its vast applications in electronics, such as in oscillators, memory\ndevicesandfastswitchingdevices[29].Nowadays,NDRhasbeendemonstratedinvarioussemiconductor\nsystems, including molecular nanowire junctions [30], organic semiconductor [31] and single electron\ndevices [32]. The NDR effect is associated with a variety of phenomena, including Coulomb blockade\n[33],tunnellingandchargestorage[34].Ling[35]reportedthenegativeresistancepropertyintriangular\ngraphene p–n junctions induced by vertex B–N mixture doping. Liu and An [36] investigated the\nnegative resistance property in metal/polythiophene/metal structure. Chen [37] investigated NDR in\noxide-based resistance-switching devices. Gupta and Jaiswal [38] reported NDR in nitrogen terminated\ndoped zigzag graphene nano-ribbon field effect transistor. Zhao et al. [39] studied NDR property and\nelectronic transportproperties of agated C60dimer molecule sandwichedbetween two goldelectrodes.\nThe inspiration behind the present work is to study the transport property of NiFe 2O4nanowire and\nto investigate its NDR property. In the present work, the transport characteristics of NiFe 2O4nanowire\ndevice and its NDR properties are explored at an atomistic level and the results are reported.\n2. Computationalmethods\nThe first-principles calculation on inverse spinel NiFe 2O4molecular device is investigated through\nnonequilibriumGreen’sfunctions(NEGF)incombinationwithdensityfunctionaltheory(DFT)method\nutilizing TranSIESTA module in SIESTA package [40]. NiFe 2O4nanowire is optimized by reducing\nthe atomic forces on the atoms in nickel ferrites to be less than 0.05 eV/Å. The Brillouin zones of\nNiFe 2O4aresampledwith 1\u00021\u00025k-points.Thegeneralizedapproximation(GGA)alongwithPerdew-\nBurke-Ernzerhof(PBE)exchangecorrelationfunctionalisusedtostudytheelectron-electroninteraction\n[41, 42]. The negative differential resistance property of NiFe 2O4is also studied through SIESTA\npackage, in which the core electrons are suitably replaced by Troullier-Martins pseudopotentials for\nnickel, iron and oxygen atoms. Moreover, the electronic wave functions of nickel, iron and oxygen\natoms are demonstrated in terms of a basis set, which are mainly related to the numerical orbitals. The\noptimizationofbandstructureandelectronicpropertiesofNiFe 2O4nanowireareimplementedusingthe\ndoublezetapolarization(DZP)basissetfortheright-hand,left-handelectrodesandthescatteringregion\nin the present study [43]. In order to investigate the electronic properties of NiFe 2O4and to exclude the\ninteraction of NiFe 2O4nanowire with its periodic images, 10 Å vacuum padding is modelled along x\nandyaxes. This makes the computation process easy while examining the density matrix Hamiltonian.\nTheatomsinNiFe 2O4nanowirefreelymovealongtheirrespectivepositionsuntiltheconvergenceforce\nsmaller than 0.05 eV/Å is achieved.\nSen et al. [44] studied the transport properties of trimer unit of cis-polyacetylene and fused furan\n23301-2Investigationonnickelferritenanowire\ntrimer using DFT in combination with NEGF ab initio method. They observed the NDR over a bias\nvoltageof( +2.1to +2.45V).Yuetal.[45]investigatedthetransportpropertiesofafewnmlongsingle-\nwalledcarbonnanotube(SWCNT)p–njunctionsusingthe ab initioquantummethod.Thefindingreveals\nthatnmlongSWCNTshowsnegativedifferentialresistance.Songetal.[46]reportedNDRbehaviourin\n(8,0)carbon/boronnitridenanotubeheterojunction.Theyreportthatunderpositiveandnegativebias,the\nvariation in the localization of corresponding molecular orbital under the applied bias voltage leads to\nNDRbehaviour.MahmoudandLugli[47]studiedmoleculardeviceswithnegativedifferentialresistance.\nThe molecular device is composed of diphenyl-dimethyl connected to the carbon chain linked to gold\nelectrodes.TheyobservedNDRbehaviouronlyforanoddnumberofcarbonatomsinthechainbetween\nthe gold electrodes. In the present work, NDR behaviour is observed along NiFe 2O4nanowire. The\nadoptedmethodinthepresentworkresemblesthemethodusedintheabovementionedliterature,which\nconfirmsthereliabilityoffirst-principlesstudyonNiFe 2O4nanowiremoleculardevice.Thenovelaspect\nofthepresentworkisNDRpropertiesofNiFe 2O4nanowiredevicewhichisdiscussedintermsofdensity\nof states spectrum, transmission and I–Vcharacteristics.\n3. Resultsanddiscussion\n3.1. StructureofNiFe 2O4nanowire\nThe NiFe 2O4nanowire device is built using International Centre for Diffraction Data (ICDD) Card\nnumber:03-0875,whichexhibitstheinversespinelstructure.ThedesignedNiFe 2O4nanowiremolecular\ndeviceisdividedintothreeregions,namelyleft-handelectrode,scatteringregionandright-handelectrode\nregions. The scattering region of NiFe 2O4nanowire device is placed in between two electrodes. The\ncorresponding width of the scattering region, left-hand electrode and right-hand electrode are 25.02 Å,\n8.34Åand8.34Å.TheNiFe 2O4nanowireisrepeatedfivetimesalong c-axis.Initially,inordertooptimize\nthedimensionofthemoleculardevice,theNiFe 2O4moleculardeviceisbuiltwithdifferentdimensions.\nMoreover,whenthedimensionofthescatteringregionissmall,itgivesrisetothetunnellingofelectrons\nacross the NiFe 2O4device. However, if the dimension is too long, the magnitude of the current flowing\nacross the NiFe 2O4device decreases. When NiFe 2O4device is of the order of the above mentioned\ndimensions, a significant current flows across the NiFe 2O4device. Along NiFe 2O4scattering region, a\nbiasvoltageismaintainedbetweentheleft-handelectrodeandright-handelectrodefortheflowofcurrent.\nThe scattering region of NiFe 2O4nanowire consists of twenty four nickel atoms, forty eight iron atoms\nandninetysixoxygenatoms.Theregionontheleft-handandright-handelectrodesincludeseightnickel\natoms,sixteenironatomsandthirtytwooxygenatomseach.Thepotentialdifferenceof \u0000V2and+V2\nis maintained across the right-hand and left-hand electrode in NiFe 2O4molecular device. Besides, the\n \nFigure1.(Color online) NiFe 2O4molecular device.\n23301-3V.Nagarajan,R.Chandiramouli\nvariation in the bias voltage leads to the change in the density of states and transmission along NiFe 2O4\nnanowire device. Figure 1 represents the schematic diagram of NiFe 2O4molecular device.\n3.2. BandstructureofNiFe 2O4nanowire\nThebandstructureofNiFe 2O4nanowireprovidestheinsightsonthematerialspropertiesofNiFe 2O4\nnanowire. The band structure of NiFe 2O4nanowire can be described in terms of conducting channels\nacross the Fermi energy level ( EF) between the conduction band and the valence band [48]. Figure 2\nrepresents the band structure of NiFe 2O4nanowire. From the observation, it is known that NiFe 2O4\nnanowire has the band gap of 2.65 eV for the whole nanostructure, which exactly matches with the\nreported theoretical work [49]. The experimental direct band gap value of NiFe 2O4is 2.5 eV, which is\nalmostequaltotheobtainedtheoreticalvalueasshowninfigure2.Thus,itcanbesuggestedthatSIESTA\nmay be used as a significant computational tool for studying electronic properties of nanostructured\nmaterialswithsuitablebasissets.Moreover,thebandgapof2.65eVforNiFe 2O4isoneofthefavorable\nconditions for the application in electronic devices.\n \nFigure2.(Color online) Band structure of NiFe 2O4nanowire.\n3.3. DensityofstatesandelectrondensityacrossNiFe 2O4nanowiredevice\nThe density of states (DOS) spectrum provides a clear picture regarding the density of charge in\nenergy intervals along NiFe 2O4nanowire [50–52]. Besides, the variation in bias voltage along NiFe 2O4\nnanowire leads to the change of the density of charge in the energy interval. In the present work, the\nvariationinDOSisobservedonlybeyondathresholdvoltageof2.5V;whichyieldsasignificantchange\nin the density of charge. On behalf of this reason, the bias voltage from 2.5 V to 7.5 V is carried out\nin the present study. In addition, the Fermi level ( EF) is kept at zero, since the bias window between\nright-hand and left-hand electrode is set as \u0000V2,V2in NiFe 2O4nanowire device. Figure 3 illustrates\nthe projected density of states (PDOS) of NiFe 2O4base material. The base material refers to the basic\nelementforbuildingthemoleculardevice.Inthepresentwork,NiFe 2O4isthebasematerialthatisused\nas electrodes and scattering region in the molecular device. Moreover, the major contribution in PDOS\nspectrum arises from dorbitals of Ni and Fe, whereas for O, it is due to porbitals as observed in total\nDOS. The peak maxima at different energy levels are governed by the orbital overlapping of dandp\norbital projected in NiFe 2O4base material. Furthermore, the peak maxima are observed near the Fermi\nlevel,whichuponapplyingthebiasvoltageresultsinthetransitionofelectronsfromthevalencebandto\nthe conduction band.\n23301-4Investigationonnickelferritenanowire\n \nFigure3.(Color online) PDOS spectrum of NiFe 2O4.\nFigure4refersthedevicedensityofstatesspectrumfor0.0V,2.5V,3.0V,3.5V,4.0V,4.5V,5.0V,\n5.5 V, 6.0 V, 6.5 V, 7.0 V and 7.5 V bias. For 0 V bias, the DOS spectrum across NiFe 2O4nanowire is\nobservedtobemoreintheconductionbandthaninthevalenceband.Thepeakmaximumisrecordedto\nbe around 0.85 eV in the conduction band. Interestingly, at zero bias voltage condition, the peaks arise\nduetothemismatchofelectronicchemicalpotentialbetweentheelectrodes,thuslocalizationofcharges\nis observed in the conduction band.\nThere is no significant peak maximum observed in the valence band of NiFe 2O4nanowire device\nat 0 V. Furthermore, on applying the bias voltage of 2.5 V across the electrodes, the localization of\ncharges is recorded near the Fermi level as shown in figure 3. In addition, increasing the bias voltage\nto 3.0 V across NiFe 2O4nanowire device, results in peak maximum at \u00002.5 eV in the valence band.\nWhen the bias voltage is set to 3.5 V, localization of charges is observed on both the valence band and\nthe conduction band within the energy interval of \u00002.4 and 1.75 eV, respectively. This infers that the\nbias voltage drives the charges from the valence band to the conduction band along NiFe 2O4scattering\nregion.Thesametrendisobservedatthebiasvoltageof4.0V.Theonlydifferenceisthatthelocalization\nof charges is shifted towards the conduction band on increasing the bias voltage. When the bias voltage\nis switched to 4.5 V, the localization of charges is noticed in the valence band at \u00002.1 eV. However, the\ncharge transition takes place for the bias voltage of 5.0 V and the peak is observed at 1.4 eV. In the case\nof bias voltage for 5.5 and 6.0 V, the peak maxima are observed on both the conduction band and the\nvalenceband.Bycontrast,thelocalizationofchargesisobservedonlyontheconductionbandatdifferent\n23301-5V.Nagarajan,R.Chandiramouli\n \nFigure4.(Color online) Device DOS of NiFe 2O4nanowire.\n \nFigure5.(Color online) Electron density of NiFe 2O4nanowire.\n23301-6Investigationonnickelferritenanowire\nenergy intervals in the case of 6.5 and 7.0 V bias voltages. Thus, it is inferred that the density of charge\nalong NiFe 2O4nanowire device can be finely tuned with the bias voltage. The electron density across\nNiFe 2O4nanowire is shown in figure 5. The density of electrons is observed to be more in oxygen sites\nthan in iron and nickel sites along NiFe 2O4nanostructure. Since the atomic number of the oxygen atom\nis eight and it is belongs to the group VIA element, due to the electronegative property of oxygen, it\nresultsintheaccumulationofmoreelectronsacrossoxygensitesinNiFe 2O4nanowire.Oneofthemost\nsignificant chemical properties of the oxygen atom is the electronegativity property, which is accredited\nas the tendency of oxygen to attract electrons towards it. Moreover, the electron density is larger along\ntheoxygensitesowingtotheelectronicconfigurationoftheoxygenatomwhenbondingwithnickeland\nironatomsinNiFe 2O4nanowire.Besides,theelectronegativityoftheoxygenatomisalsoinfluencedby\nthedistancebetweennucleusandvalenceelectronsinNiFe 2O4nanowire.Theelectrondensityprovides\nthe insight on the chemical and electronic properties of NiFe 2O4nanowire.\n3.4. TransportpropertiesofNiFe 2O4nanowiredevice\nThe electronic transport of NiFe 2O4molecular device can be ascribed in terms of transmission\nspectrum [53–55]. The transport characteristics of NiFe 2O4nanowire devices are investigated using\nTranSIESTA module in SIESTA package. The transmission function T¹E;Vºof NiFe 2O4molecular\ndevice can be expressed as the sum of the probabilities of transmission for all the channels at energy E\nbeneath external bias voltage Vas shown in equation (3.1)\nT¹E;Vº=Tr\u0002\n\u0000L¹VºGR¹E;Vº\u0000R¹VºGA¹E;Vº\u0003\n; (3.1)\nwhere \u0000R,Lis the coupling function of the right-hand and left-hand self-energies, respectively. GAand\nGRare the advanced and retarded Green’s function. Furthermore, the molecular orbitals nearer to the\nFermi energy level ( EF) facilitate the electronic transport across NiFe 2O4nanowire even for the low\nbiasvoltage.Thegeneralrelationbetweentheconductanceandtransmissionprobabilityunderzerobias\ncondition is given as\nG=G0T¹E;V=0º; (3.2)\nwhere G0isthequantumunitofconductanceanditisequalto 2e2h,hisPlanck’sconstantand eisthe\nelectronic charge. The potential of \u0000V2and+V2is maintained between the right-hand and left-hand\nelectrode across NiFe 2O4molecular device, respectively.\nThe current through the NiFe 2O4nanowire device can be calculated from the Landauer-Büttiker\nformula [56]\nI¹Vº=2e2\nh\u0016R¾\n\u0016LT¹E;VbºdE; (3.3)\nwhere eistheelementarycharge, 2e2histhequantumconductance, \u0016L,Ristheelectrochemicalpotential\nof left-hand and right-hand electrode, respectively.\nWhenzerobiasissetacrossNiFe 2O4nanowiredevice,theFermilevelofleft-handelectrodeandright-\nhand electrode gets aligned and the electronic transmission between right-hand and left-hand electrode\nis equal in both directions, hence Fermi level is considered as zero. Figure 6 depicts the transmission\nspectrum of NiFe 2O4nanowire for different bias voltages. (The transmission spectrum is drawn in a\nthreedimensionalmulti-curvefashion;themagnitudeistakenintoconsiderationalong yaxis.)Besides,\nthe transmission peaks recorded for the zero bias voltage are owing to the mismatch in the electronic\nchemical potential across right-hand electrode and left-hand electrode in NiFe 2O4nanowire device. By\ncontrast, low peak amplitude is recorded in the conduction band. On applying the bias voltage above\nzero, the molecular orbitals in NiFe 2O4nanowire get delocalized. In that case, the mobility is recorded\nto be more in these energy intervals in the transmission spectrum [57]. This gives rise to a certain peak\nmaximum in the transmission spectrum of NiFe 2O4nanowire device [58]. However, on increasing the\nbias voltage across NiFe 2O4scattering region, the transmission pathways increase along the NiFe 2O4\nnanowire;thisgivesrisetoashiftinthepeakmaximum[59].Besides,whenthebiasvoltageof2.5Vis\napplied between the electrodes, the peak maximum is observed around 2.6 eV.\n23301-7V.Nagarajan,R.Chandiramouli\n \nFigure6.(Color online) Transmission spectrum of NiFe 2O4nanowire.\nThe increase of the bias voltage leads to the flow of electrons along the scattering region and the\npeak maximum moves towards the conduction band for the potential difference of 2.5 V. In the case\nof 3.0 V, the peak maximum is observed at \u00002.5 eV on the valence band and the peak gets shifted to\nthe conduction band on applying the bias voltage of 3.5 V as shown in figure 5. Furthermore, due to\nthe transition of electrons across the scattering region along NiFe 2O4, the peak maximum shifts to a\ndifferentenergyintervalonvaryingthebiasvoltage.Theappliedbiasvoltagedrivestheelectronsacross\nthe NiFe 2O4molecular device, in which the peak maximum gets shifted. For the applied bias of 4.0 V,\nthepeakmaximumisobservedonboththevalencebandandtheconductionbandat \u00001.65and2.75eV,\nrespectively.Furtherincreasingthebiasvoltagefrom4.5to7.5V,thepeakmaximumgetsshiftedalong\nthe valence band and the conduction band. The transmission spectrum has a peak maximum along\ndifferent energy levels. The change in the current for different voltages should not be correlated directly\nwith transmission spectrum with that of I–Vcharacteristics curve. The transmission spectrum indicates\nthat the transmission of charges is larger for a particular energy interval to the applied bias voltage.\nHowever, the net current flowing through the molecular device depends on overall transmission for a\ndifferent energy interval. This clearly suggests that the bias voltage is adequate enough for the transition\nofelectronsalongNiFe 2O4nanowiredeviceandthetransmissionisgovernedbytheappliedbiasvoltage.\nThus,itcanbeconcludedthatthetransportpropertyofNiFe 2O4nanowiredevicecanbefinelytunedby\napplying the proper bias voltage and can be used as a chemical sensor in microwave devices.\n3.5.I–VcharacteristicsofNiFe 2O4nanowiredevice\nNegative differential resistance behaviour is the most significant electronic transport property for\nvarious electronic components [26]. In the present study, the NDR behaviour is observed in the I–V\ncharacteristics of NiFe 2O4nanowire as shown in figure 7. The behaviour of NiFe 2O4molecular device\nis similar to that of an n-type semiconductor. At the beginning, the current flowing through NiFe 2O4\nnanowire device shows almost a linear increase across NiFe 2O4scattering region on increasing the bias\nvoltage.Uptothethresholdlimitof5Vbias,thecurrentincreaseslinearlyfortheappliedbias.TheNDR\nisobservedforthebiasvoltageof5.0Vto6.0V.Moreover,whenNiFe 2O4nanowiredeviceisoperated\n23301-8Investigationonnickelferritenanowire\nFigure7.(Color online) I–VCharacteristics of NiFe 2O4nanowire.\nin this bias voltage, it exhibits NDR. Further increasing the bias voltage beyond 6.0 V along NiFe 2O4\nnanowire device, the NDR behaviour vanishes and the device obeys the ohm’s law.\nInthepresentwork,N-shapedNDRisobservedforNiFe 2O4moleculardevice.TheNDRbehaviour\nin NiFe 2O4nanowire device originates from the inhibition of the conduction channels at a certain\nbias condition [60]. Besides, the frontier orbitals localized in any part of the scattering region will not\ncontribute to the transmission spectra and the current conduction may be suppressed. By contrast, a\ncompletely delocalized molecular orbital may contribute more to the transmission probabilities than\nthat of the localized one in NiFe 2O4nanowire device. Figure 8 illustrates the schematic diagram of\n \nFigure8.(Color online) Schematic diagram of NiFe 2O4as NDR device.\n23301-9V.Nagarajan,R.Chandiramouli\nNiFe 2O4nanowiredevice,whichcanbeusedasNDRdevice.Lietal.[61]observedtheN-shapedNDR\nin GaAs-based modulation-doped FET along with InAs quantum dots. Xu et al. [62] reported a similar\nN-shaped negative differential resistance in GaAs-based modulation-doped FET with InAs quantum\ndots. The NDR effect observed in the device is not only related to a single physical mechanism. Many\nphenomenagiverisetotheNDRproperty,namelytunneling,Coulombblockade,Gunneffect[63],metal\nandsemiconductorcontact,chargestorageandgeometryofthenanodevice.Furthermore,thecylindrical\ngeometryandhighsurface-to-volumeratioofnanowireresultsindeeppenetrationofthesurfacecharge,\nwhichlargelyaffecttheconductionpropertyofnanowire.FromtheLandauer-Büttikerrelation,itiswell\nknown that the current through the device depends on T¹E;Vº. The current in the NiFe 2O4device is\nthe integral of the transmission coefficient in the bias window of [ \u0000V2,V2]. In the present work, the\nNDR effect is observed in the bias voltage of around 5 V to 6 V. Moreover, the device DOS (figure 4)\nindicates a peak in the conduction band for 5 V at the energy level of 1.4 eV, whereas for 5.5 V and\n6 V bias, the peaks are observed both in the conduction band and in the valence band. Thus, for the\napplied bias voltage of 5 V, the current increases drastically, and the further increase in the bias voltage\ngivesrisetoadecreaseinthecurrentduetotheCoulombblockadethatariseduetothegeometryofthe\ndevice. Furthermore, for the bias of 5 V to 6 V, the bias window makes transition of electrons between\nthe highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO)\ndecrease.Thedecreaseinthetransmission(figure6)takesplacebecausealargerwavefunctionoverlaps\nbetween the scattering region and electrodes, the degree of coupling between the molecular orbitals and\nelectrodes becomes weaker with an increase in the bias voltage beyond 5 V. Moreover, such a decrease\nmaynotbecompensatedbytheincreaseinthebiasvoltage,thustheintegralareagetssmaller.However,\non further increasing the bias voltage beyond 6 V, the degree of coupling between the electrodes and\nscattering region is overcome by the bias voltage and the current increases further more for the applied\nbias voltage.\nThe negative differential resistance properties are observed on various materials with different mor-\nphology such as ZnO nanorod, porous silicon devices and graphene nanoribbon FET [38, 64, 65]. The\nNDR property of NiFe 2O4nanowire device is similar to the reported works, which further strengthens\nthe present work. Thus, the negative differential resistance property of NiFe 2O4nanowire can be finely\ntuned by applying a proper bias voltage.\n4. Conclusions\nInthepresentstudy,NiFe 2O4nanowirebasedmoleculardeviceisstudiedusingDFTmethod.Under\nvarious bias voltages, the electronic transport properties of inverse spinel NiFe 2O4nanowire device is\ninvestigated. The density of charges among different energy intervals of NiFe 2O4nanowire is clearly\nstudied with the help of projected density of states spectrum. Moreover, the peak maximum is observed\non both the valence band and the conduction band, which is influenced by the applied bias voltage.\nThe electron density is observed to be more on oxygen sites along NiFe 2O4nanowire. The transmission\nspectrum of NiFe 2O4nanowire device shows a larger peak maximum in the valence band at the zero\nbias condition. However, on increasing the bias voltage, a larger peak maximum in the conduction band\nisobserved,whichclearlysuggeststhatthebiasvoltagedrivesthechargestowardstheconductionband.\nTheNDRpropertiesofNiFe 2O4nanowireareinvestigatedusing I–Vcharacteristics.TheNDRproperty\nof NiFe 2O4nanowire device depends on the applied bias voltage. Thus, the NDR property can be finely\ntunedwiththebiasvoltage.ThefindingsofthepresentworkinNiFe 2O4nanowiredevicecanbeusedas\nNDRdevice,whichmayfinditspotentialapplicationinmicrowavedevices,memorydevicesandinfast\nswitching devices.\nReferences\n1. Xu Q., Wei Y., Liu Y., Ji X., Yang L., Gu M., Solid State Sci., 2009, 11, 472,\ndoi:10.1016/j.solidstatesciences.2008.07.004.\n2. Słoczyński J., Janas J., Machej T., Rynkowski J., Stoch J., Appl. Catal. B, 2000, 24, No. 1, 45,\ndoi:10.1016/S0926-3373(99)00093-4.\n23301-10Investigationonnickelferritenanowire\n3. 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Kathalingam A., Kim H.-S., Kim S.-D., Park H.-M., Park H.-C., Physica E, 2015, 74, 241,\ndoi:10.1016/j.physe.2015.06.030.\nДослiдження з перших принцип iв нiкель-феритового\nнанодротового пристрою з негативним диференц iйним\nопором\nВ.Нагараджан,Р.Чандiрамулi\nШкола електротехн iки та електрон iки,Академiя мистецтв,науковихiтехнологiчних дослiджень Шанмуга\n(унiверситетSASTRA), Танджавур,Тамiл-Наду—613401,I ндiя\nЕлектроннiвластивостiNiFe 2O4нанодротового пристрою досл iджується з використанням методу нер iв-\nноважних функц iй Грiна в комбiнацiї з теорiєю функцiоналу густини .Властивостiелектронного переносу\nNiFe 2O4нанодроту вивчаються в терм iнах густини стан iв,спектру трансм iсiї таI–Vхарактеристик .Гу-\nстина станiв змiнюється при прикладанн iзмiщувальної напруги через NiFe 2O4нанодротовий пристр iй,\nгустина заряду спостер iгається як у валентн iй зонi,такiв зонiпровiдностiпри збiльшеннiнапруги змiще-\nння.Спектр трансм iсiїNiFe 2O4нанодротового пристрою дає уявлення про перех iд електронiв на рiзних\nенергетичних iнтервалах.Результати даної роботи наводять на думку ,щоNiFe 2O4нанодротовий при -\nстрiй може бути використаний як негативний диференц iйний опiр,iця його властив iсть може бути ре -\nгульована за допомогою напруги зм iщення,що може мати потенц iйне використання у м iкрохвильових\nпристроях,пристроях пам ’ятiiв перемикальних пристроях .\nКлючовiслова:нiкель ферит,нанодрiт,негативний диференц iйний опiр,густина станiв,електронна\nгустина\n23301-12"    },    {        "title": "1212.2538v1.Synthesis_and_Properties_of_Bismuth_Ferrite_Multiferroic_Nanoflowers.pdf",        "content": " 1 Synthesis and Properties of Bismuth Ferrite Multife rroic \nNanoflowers \nK. Chybczy ńska a, P. Ławniczak a, B. Hilczer a, B. Ł ęska b, R. Pankiewicz b, A. Pietraszko c, L. K ępi ński c, T. Kałuski d,  \nP. Cieluch d, F. Matelski e and B. Andrzejewski a* \na Institute of Molecular Physics \nPolish Academy of Sciences \nSmoluchowskiego 17, PL-60179 Pozna ń, Poland \n* corresponding author: bartlomiej.andrzejewski@ifm pan.poznan.pl \nb Faculty of Chemistry \nAdam Mickiewicz University \nUmultowska 89b, PL-61614 Pozna ń, Poland \nc Institute of Low Temperature and Structure Researc h \nPolish Academy of Sciences \nOkólna 2, PL-50422 Wrocław, Poland \nd Research Centre of Quarantine, Invasive and Genetic ally Modified Organisms \nInstitute of Plant Protection – National Research I nstitute \nWęgorka 20, PL-60318 Pozna ń, Poland \ne Faculty of Technical Physics \nPoznan University of Technology \nNieszawska 13a, PL-60965 Pozna ń, Poland \n \nAbstract — The method of microwave assisted hydrothermal \nsynthesis of bismuth ferrite multiferroic nanoflowe rs, their \nmechanism of growth, magnetic as well as dielectric  properties \nare presented. The nanoflowers are composed of nume rous petals \nformed by BiFeO 3 (BFO) nanocrystals and some amount of \namorphous phase. The growth of the nanoflowers begi ns from \nthe central part of calyx composed of only few peta ls towards \nwhich subsequent petals are successively attached. The \nnanoflowers exhibit enhanced magnetization due to s ize effect \nand lack of spin compensation in the spin cycloid. The dielectric \nproperties of the nanoflowers are influenced by wat er confined to \ncrystal nanovoids resulting in a broad dielectric p ermittivity \nmaximum at 200 K ÷ 300 K and also by Polomska trans ition \nabove the temperature of 450 K.  \n \nKeywords-component; bismuth ferrite, multiferroic, nanoflowers, \nmagnetic properties, dielectric properties  \nI.  INTRODUCTION \nRecent development of nanoscience and nanotechnolog y has \nbrought about the discovery of miscellaneous forms of \nnanoobjects. Among them one can mention nanopowders , \nnanotubes, nanowires, nanorods, nanosheets, nanoclu sters, \nnanocones and even such sophisticated nanoobject li ke \nnonobowlings, nanobottles and nanonails. One of the  most \nrarely met nanobjects are nanoclusters in the form of nanoflowers reported in a nice review by Kharisov [ 1]. \nNanoflowers can be composed of various elements and  \ncompounds like: metals (Au [2], Ni [3], Zn [4], Sn [5], Co[6]) \nand carbon [7, 8], metal oxides (ZnO [9, 10], MgO [ 11], CuO \n[12], α-MnO 2 [13], SnO 2 [14]), hydroxides and oxosalts \n(Mg(OH) 2 [15], Cu 2(OH) 3Cl [16]), sulphides, selenides and \ntellurides (CdS [17], ZnSe [18], PbTe [19]), nitrid es and \nphosphides (lnN [20], GaP [21]), organic and coordi nation \ncompounds (CdQ 2 complexes [22], alkali earth \nphenylphosphonates [23]).  \nThe nanoflowers exhibit different dimensions from a  few \ndozen of nanometers (about 40 nm for In 2O3 nanoflowers [24]) \nup to micrometers (about 50 µm for SnO 2 nanoroflowers [14]) \nand also a variety of morphological details. For ex ample, \nperforation of petals in NiO [25] nanoflowers, shee t-like petals \n[2] in Au nanoflowers, hollow cores [7] in carbon n anoflowers, \nplate-like and brush-like shapes composed of pyrami dal \nnanorods in ZnO nanoflowers [9] have been observed.  Other \nexamples are MgO nanoflowers consisting of nanofibe rs [11], \nCuO nanoflowers with petals branched into tips [12] , α-MnO 2 \nwith nanocrystalline petals [13], nanobelt-like pet als in MoO 3 \n[26], hexapetalous snowflake-like Cu 7S4 nanoflowers [27], \ndowny-velvet-flower-like nanostructures of PbS [28] , vaselike \nCuInS 2 nanostructures [29], hexangular shapes of InN  2 compound [30] and at last bundles of nanorods of Cd Q 2 \ncomplexes [22].  \nMethods of synthesis of nanoflowers include reducti on of \nmetal salts [2, 3], electrodeposition [31], catalyt ic pyrolysis \nmethod [32], zero-valent metal oxidation or decompo sition of \ncompounds to obtain metal oxides [10], chemical vap or \ntransport and condensation [14, 30, 33], calcinatio n [25] and \nmany others. However, it seems that the solvotherma l and \nhydrothermal techniques are here especially effecti ve [5, 13, \n15, 27, 28, 34-37]. In some cases the hydrothermal process was \nactivated by microwave heating [3, 17, 38].  \nThe nanoobjects in form of nanoflowers are importan t from \nthe point of view of future applications because th ey exhibit \nexcellent electrocatalytic activity [39], high-diel ectric \npermittivity, [11], photocatalitic activity [34] an d they can be \nused as excellent field emitters [40, 41], effectiv e solar cells \n[15] or amperometric biosensors [42].  \nIn this work, we report for the first time on the m icrowave \nassisted hydrothermal synthesis, structure, electri c and \nmagnetic properties of bismuth ferrite BiFeO 3 (BFO) \nnanoflowers. Bismuth ferrite, even in bulk is an un usual \nmaterial because it belongs to magnetoelectric (ME)  \nmultiferroics that exhibit simultaneously charge an d magnetic \nordering with some mutual coupling between them at room \ntemperature [43-46]. These materials have recently attracted \nworld-wide attention because of their interesting p hysical \nproperties and large technological potential to be applied as \nfour-state memory and in spintronic devices. Other important \napplications of BFO in piezoelectric devices, as TH z radiation \nemitters or catalysts are not related to ME effects  [47].  \nFrom the point of view of structure BFO oxide is  \na rhombohedrally distorted perovskite with space gr oup R3c at \nroom temperature. The ferroelectric (FE) properties  appear in \nBFO below ferroelectric Curie temperature TC=1100 K, \nwhereas the antimagnetic ordering together with wea k \nferromagnetic (FM) moment appear below the Néel \ntemperature TN=643 K. These FE and FM properties result \nfrom the charge ordering caused by lone electron pa irs of Bi 3+  \nions and from the complex ordering of Fe 3+  spins, respectively. \nBFO compound exhibits an antiferromagnetic (AFM) G- type \nordering and superimposed long range incommensurate  \ncycloidal modulation with the period λ=62 nm [45, 48].  \nThe spin cycloid can propagate along three equivale nt \ncrystallographic directions [1,-1,0], [1,0,-1] and [0,-1,1] \n(pseudocubic notation). The spins in the cycloid ro tate in the \nplane determined by the direction of the vector of cycloid \npropagation and the [1,1,1] direction of the vector  of \nspontaneous electric polarization. Weak FM moment i n this \nAFM compound originates from an interaction similar  to \nDzyaloshinskii-Moriya one which forces small cantin g of the \nspins out of the rotation plane. The weak FM moment  increases \ndue to the size effect apparent when the dimensions  of BFO \nparticle become comparable to the modulation period  λ. In this \ncase, the lack of magnetic moment compensation in t he spin \ncycloid is responsible for enhanced magnetic proper ties of \nBFO nanoobjects, which is much desired from the poi nt of \nview of ME coupling magnitude [49].  The aim of this work is to present the method of mi crowave \nassisted hydrothermal synthesis of BFO multiferroic  \nnanoflowers as well as to study their basic ferroel ectric and \nmagnetic properties with a special attention to the  size effect.  \nII.  EXPERIMENTAL \nA.  Sample Synthesis \nBFO ceramic powder-like samples were synthesized by  means \nof microwave assisted hydrothermal Pechini method [ 50, 51] \nusing bismuth and iron nitrates as precursors: Bi(N O 3)3·5H 2O, \nFe(NO 3)3·9H 2O. The nitrates together with Na 2CO 3 were \nadded into a KOH solution of a molar concentration of 6 M. \nThe mixture was next transferred into a Teflon reac tor (XP \n1500, CEM Corp.) and loaded into a microwave oven ( MARS \n5, CEM Corp.). The reaction was carried out at the same \ntemperature for all samples (200 0C) for a short time equal to \n20 min (samples labeled ST) or for one hour which w as the \nlong time synthesis (samples labeled LT). After pro cessing, \nBFO powders were first cooled to 20 0C, next collected by \nfiltration kit, rinsed with distilled water and pla ced in a dryer \nfor 2 h. The final products were brown powders of B FO \nagglomerates. In this way a set of ST and LT sample s \ncomposed of grains in the shape of nanoflowers was \nsynthesized. Another sample, labeled STC was obtain ed after \nair calcination of ST sample at 500 0C for 1h.  \nB.  Sample Characterization \nThe crystallographic structures and phase compositi ons of the \nBFO samples were studied by means of x-ray diffract ion \nmethod (XRD) using an ISO DEBYEYE FLEX 3000 \ndiffractometer equipped with a HZG4 goniometer in B ragg-\nBrentano geometry and with a Co lamp ( λ=0.17928 nm).  \nThe morphology and microstructure of the samples we re \nstudied by FEI NovaNanoSEM 230 scanning electron \nmicroscope (SEM) and also by Philips CM20 SuperTwin  \ntransmission electron microscope (TEM). Mmagnetomet ric \nmeasurements were performed using a Vibrating Sampl e \nMagnetometer (VSM) probe installed on the Quantum D esign \nPhysical Property Measurement System (PPMS) fitted with  \na superconducting 9T magnet. Dielectric properties and \nelectric conductivity of the BFO samples were studi ed by \nmeans of an Alpha-A High Performance Frequency Anal yzer \n(Novocontrol GmbH) combined with a Quatro Cryosyste m for \nthe low temperature control. Before the measurement , the \nBFO fine powders were pressed to a compact form of pellets \nof dimensions 5.15 mm in diameter and ca. 1 mm in t hickness, \nonto which silver paste electrodes were deposited.  \nThe measurements were recorded on heating at the ra te of 0.25 \nK/min while the frequency was varied from 1 Hz to 1  MHz at \nthe oscillation voltage of 1 V.  \nIII.  RESULTS AND DISCUSSION \nThe time of microwave reaction and the concentratio n of KOH \nwere found to be crucial factors determining the sh ape of the \nBFO agglomerates obtained. At the KOH concentration  of 6 \nM and after 20 min of the reaction, numerous agglom erates in \nthe shape of nanoflowers were formed, (ST sample).   3 The processing time of about 1 h allowed getting la rge but \nsingular BFO nanoflowers (LT sample). Besides the \nnanoflowers, other forms like large BFO microsphere s were \npresent. Higher molar concentration of KOH equal to  10 M \nand the synthesis time of 30 min promoted formation  of \nmicrocubes or spheres instead of the nanoflowers.  \nFigs. 1 and 2 present SEM micrographs of ST sample \nobtained at the KOH concentration of 6 M and 20 min  \nsynthesis. This sample contains only agglomerates i n the form \nof nanoflowers, which are indeed very similar to nu merous \nflower-like structures reported in ref. [1]. The me an size of the \nnanoflowers is of about 15 µm. The nanoflowers are not \nperfectly spherical and exhibit kind of hollows whe re the \npetals are packed less dense. The structure of the hollows and \ndetails of petals ordering in their vicinity is sho wn in Fig. 3 \nwhich presents a small nanoflower very similar to a  rose.  \nThe petals in this nanoflower have more irregular s hapes than \nin the large nanoflowers and form a kind of hole, o r using the \nbotanic terminology “bottom of calyx” near the cent er.  \nThe size of this hole is less than 1 µm. The bottom of calyx is \nvery pronounced in the structure shown in Fig. 4 wh ich \npresents the BFO nanoflower “in statu nascendi”.  \nThe structure of this nanoflower indicates the proc ess of its \nformation. One can assume that at early stages of g rowing  \na central part composed of only a few petals (or ev en four as \nin Fig. 4) is formed. The petals are connected at t he edges and \nform a kind of rectangular cage. Next, around this structure, \nsuccessive petals nucleate. For small flowers, like  in Fig. 4 the \npetals are mainly located in the plane perpendicula r to the \nflower axis and perpendicularly to the petals in th e center.  \nIn the bigger flowers, the petals can take various orientations. \nWe assume that the big, spherical nanoflowers in Fi gs. 1 and 2 \nalso contain cages inside but they are well masked by \nnumerous petals around them. This explanation is su pported \nby the observation that the holes in the center of the \nnanoflower are apparent mainly for small nanoflower s (about \n5 µm in diameter in Fig. 3 or about 2.5 in diameter µm in Fig. \n4) being in the initial stages of formation. The bi g nanoflowers \nare always almost spherical.  \n \nFigure 1.  The SEM micrograph of ST sample composed of nanoflo wers only.  \n \nFigure 2.  The SEM micrograph presenting the order of petals n ear the \nhollow part of one of the selected small nanoflower s in ST sample.  \n \nFigure 3.  SEM micrograph of the nanorose found in the ST samp le.  \n \nFigure 4.  SEM micrograph of the initial stage of formation of  a nanorose \nobserved in the ST sample. Four petals in the cente r of the nanorose form a cage.   4 Fig. 5 presents SEM micrograph of LT sample process ed \nfor 1 h. The image shows large, well formed spheric al \nnanoflowers of about 20 µm in diameter. The nanoflowers are \ncomposed of hundreds of closely packed petals. Besi des \nnanoflowers, in this sample there are numerous othe r \nstructures like microspheres and other irregular fo rms. It was \nobserved that the long time of synthesis and crysta llization \npromoted formation of large nanoflowers and also mo re dense \nstructures composed of massive BFO microcrystals. P robably, \nthe spherical nanoflowers transform to microspheres  because \nof the crystal growth of petals as the reaction pro ceeds.  \n \nFigure 5.  The SEM micrograph of a BFO nanoflower found in LT sample. \nThe thickness of petals of BFO nanoflowers varies f rom \nabout 30 nm to 200 nm as presented in the histogram  in Fig. 6. \nMost of the petals have thickness in the range 50÷1 00 nm.  \nThe distribution of the petals thickness can be fit ted using log-\nnormal function well describing granular random sys tems \n[52]:  \n \n \n(1) \n \nwhere 〈D〉 denotes the median thickness of the petal and σ is \nthe distribution width. The best fit of eq. (1) to the data \npresented in the histogram is obtained for 〈D〉=78(3) nm and \nσ=0.40(3). The maximum value of the distribution f(D) of \npetal thickness appears at 67 nm.  \nNanostructure of the petals of nanoflowers was anal yzed \nby means of TEM and the image is presented in Fig. 7. It turns \nout that the petals are composed of BFO blocks with  \ndimensions exceeding 100 nm. Some of them, like the  biggest \nblock in Fig. 7, exhibit well crystallized bismuth ferrite phase, \nwhich was confirmed using selected area diffraction  (SAD) \nmethod (see Fig. 8). An example of the agglomerate \ncontaining very small BFO nanocrystals probably emb edded \nin the amorphous matrix is shown in Fig. 9. Wide di ffraction \nrings in the SAD pattern of this agglomerate presen ted in Fig. \n10 confirm that the size of the nanocrystals does n ot exceeds  \na few nm. Agglomerates composed of bigger BFO cryst allites with dimensions about 10-20 nm, embedded in the amo rphous \nphase are also observed.  \n \nFigure 6.  The histogram of thickness of petals of BFO nanoflo wers. The \nsolid line is the best fit of the log-normal distri bution eq. (1) to the data.  \n \nFigure 7.  The TEM micrograph of the BFO crystallites from the  ST sample. \n \nFigure 8.  The SAD of the biggest crystallite in Fig. 7.  \n\n\n\n\n\n\n\n− =DD\nDDf2\n22ln \n21exp 1\n21)(\nσ πσ  5 \n \nFigure 9.  The TEM micrograph of the BFO agglomerate in the ST  sample. \n \nFigure 10.  The SAD pattern of the BFO agglomerate presented in  Fig. 9.  \nThe crystalographic structure of the BFO samples wa s \nstudied by means of x-ray diffraction at room tempe rature. \nThe XRD patterns for as-prepared powders containing  \nnanoflowers synthesized during 20 min (ST sample) a re \npresented in panel “a” of Fig. 11, for the sample s ynthesized \nduring 1h (LT sample) in panel “b�� and for the calc ined ST \nsample (STC sample) in panel “c”. The solid lines i n Fig. 11 \ncorrespond to the best fits by means of Rietveld me thod \nperformed using FULLPROF software to the experiment al data \nrepresented by open points. The lines below the XRD  data \nillustrate the difference between the data and the fit.  \nThe vertical sections indicate the positions of ind ividual Bragg \npeaks. Analysis of these XRD patterns reveals the p resence of \nthe BFO rhombohedral phase with R3c  space group in all \nsamples. The parameters of the hexagonal crystallog raphic cell \nin ST sample are: a=b=5.577(1) Å and c=13.862(2) Å.  \nThe unit cell parameters of STC sample obtained aft er air \ncalcination of ST sample at 500 0C for 1 h are unchanged, \nbecause: a=b=5,576(1) Å and c=13,858(2) Å. However, the \nLT nanoflowers obtained in the long-time synthesis exhibit \nsmaller crystallographic cell as compared to those in the ST and STC samples:  a =b=5.572(2) Å and c=13,847(5) Å. \nProbably, these parameters can be related to irregu lar BFO \nagglomerates present in this sample. The peak at ab out 34 deg \nmarked by asterisk in Fig. 11 originates from a sma ll content of \nBi 25 FeO 40  parasitic phase. The crystallographic parameters o f \nBFO cubes and spheres are close to those of the nan oflowers \npresent in ST and STC samples.  \n \n \nFigure 11.  The XRD-pattern of ST sample prepared during 20 min  processing \n(panel a), LT sample synthetized in 1 h (panel b) a nd calcined STC sample \n(panel c). The solid line is the best fit to the ex perimental data represented by \nopen points. The line below the fits represent the difference between the data \nand the fit. The vertical sections represent positi ons of the Bragg peaks.  \nThe asterisk indicates the trace of Bi 25 FeO 40  phase. \nMean size of crystallites in the nanoflowers was es timated \nusing the Scherrer’s equation [53]: D=K λ/βcos Θ, where D is \nthe crystallite size (in nm), β is the half-width of the \ndiffraction peak and Θ represents the position of the Bragg \npeak. The constant K in the Scherrer’s equation dep ends on \nthe morphology of the crystal [54] and here it was assumed \nthat K=0.93 as in the Scherrer report [53]. Mean si ze of the \ncrystallites in ST and LT samples was assessed to a mount to \nabout D=37 nm. It seems however, that the calcination slig htly \nincreased the mean size of crystallites in STC samp le to about \nD=39 nm. The growth of crystallites is probably due to \ncrystallization of the amorphous phase. The crystal lographic \nparameters of the nanoflowers studied are collected  in Table I.  \n \n  6 Table I. Hexagonal unit-cell parameters and the cry stallite size of BFO \nnanoflowers \nsample  a=b [Å] c [Å] α, β, γ  [deg]  D [nm] \nLT nanoflowers  5,572(2) 13,847(5) α=β, γ=120 0 37 \nST nanoflowers  5,577(1) 13,862(2) α=β, γ=120 0 37 \nSTC nanoflowers 5,576(1) 13,858(2) α=β, γ =120 0 39 \n \nThe mean size of the crystallites can be compared t o the \nthickness of the nanoflower petals whose thickness \ndistribution is presented in the histogram Fig. 6. It is evident \nthat the mean size of the crystallites well corresp onds to the \nthickness of the finest petals, equal to about 40 n m. Therefore, \none can assume that the thinnest petals are compose d of  \na single layer of BFO crystallites.  \nBesides the morphology and the structure of BFO \nnanoflowers we have also studied their magnetic and  dielectric \nproperties i.e. basic parameters characterizing the  \nmultiferroics. Fig. 12 presents the magnetic hyster esis \nmeasurements at 300 K for the samples composed of S T, STC \nand LT nanoflowers.  \n \nFigure 12.  Magnetic hysteresis curves M(H) recorded at 300 K for the BFO \nnanofloweres.  \nThe shape of the magnetization curves M(H) is similar for \nall samples however the differences in the value of  \nmagnetization are substantial. The highest magnetiz ation was \nmeasured for the samples containing LT and ST nanof lowers \nand it was almost 3 times higher than the BFO bulk \nmagnetization of about 0.3 Am 2/kg reported by Park [49] (all \nmeasurements performed at 2 T). The enhanced magnet ization \nof nanoflowers can be related to the size effect be cause of the \nuncompensated spins near the surface [49]. This eff ect appears \nwhen the size of the BFO particle is lower than the  period of \nthe spin cycloid λ=62 nm. Indeed, the thickness of the finest \npetals evaluated for 40 nm is smaller than λ length and the \nnanoflowers exhibit higher magnetization equal to 0 .94 \nAm 2/kg for ST and to 0.86 Am 2/kg for LT samples (both measured at 2 T) than that of the bulk BFO material . This \nresult well corresponds to the data published in ea rlier report \n[49] where the saturation magnetization of BFO nano particles \nwith the mean size of 41 nm was about 0.8 Am 2/kg.  \nThe decrease in magnetization observed in STC nanof lowers \nafter calcination can be caused by an increase in m ean size of \ncrystallites (see table I) or by the crystallizatio n of the \namorphous phase. The amorphous phase exhibits highe r \nmagnetization that that of crystalline BFO phase du e to spin \nglass behavior [55] and its reduction can lead to a  decrease in \nthe total magnetization of the STC nanoflowers.  \nTemperature variation of the real ε’ and imaginary ε” parts \nof dielectric permittivity dependence on temperatur e for ST \nand SCT samples are shown in Fig. 13.  \n \nFigure 13.  Real ε’ and imaginary ε” part of permittivity for as prepared ST \nsample (open symbols) and calcined STC sample (soli d symbols).  \nTwo main features can be found in this figure – a b road \nmaximum around 200-300 K apparent in ε” and also in ε’ \ndependences (for low frequency only) and an increas e in both \nreal and imaginary parts of the permittivity above about 450 \nK. The low temperature broad maximum is probably du e to \nwater or ice surface condensation [56] or due to th e water \nconfined to nanocavities of the samples because of the wet \nchemistry synthesis. To check this interpretation, STC \ncalcined sample with reduced water content was meas ured.  \nIt turns out that the broad maximum is suppressed i n STC \ncalcined sample, which supports our interpretation of the \norigin of low temperature maximum being due to the water \nconfined to nanovoids. The strong increase in both \ncomponents of permittivity above 450 K can be inter preted in \nterms of Polomska transition [57, 58] related to th e surface \nphase transition and anomalous magnon damping [56].   \nIV.  CONCLUSIONS \nIn summary, for the first time we report the method  of the \nmicrowave assisted wet chemistry synthesis of bismu th ferrite \nmultiferroic nanoflowers and the necessary conditio ns \npromoting the formation of these nanoobjects.  \nThe morphology of nanoflowers is sensitive to KOH c ontent  7 and the time of microwave processing, so that homog eneous \npowders composed of nanoflowers only are obtained f or KOH \nconcentration of 6 M and 20 min synthesis. Differen t KOH \ncontents lead to formation of other BFO structures,  whereas \nthe long-time synthesis leads to powders containing  large \nnanoflowers but also irregular BFO agglomerates.  \nAt early stages of the growth process the central p art of \nnanoflowers or bottom of calyx composed of only a f ew petals \nis formed. The next petals are attached successivel y to this \nstructure as the reaction continues. The finest pet als with \nthickness of about 40 nm are composed of single lay er of BFO \nnanocrystals and also some amount of the amorphous phase.  \nThe size of the nanocrystals forming the petals is smaller \nthat the period of spin cycloid in bismuth ferrite which leads to \nenhanced magnetization of nanoflowers when compared  to \nthat of bulk BFO materials. Dielectric properties o f the \nnanoflowers are dominated by a broad maximum at 200  K ÷ \n300 K and an increase in both real and imaginary co mponent \nof the dielectric permittivity above 450 K. 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Phys Rev B \n2002;65:064422-1-8 \n[53]  Scherrer P: Bestimmung der Grösse und der inn er Structur von \nKolloidteilchen mittels Röntgenstrahlen. Nachrichte n Gesell. Wiss. \nGöttingen 1918;26:98–100  \n[54]  Li Z: Characterization of different shaped na nocrystallites using X-ray \ndiffrction line profiles. Part Part Syst Charact 20 11;28:19-24 \n[55] Nakamura S, Soeya S, Ikeda N, Tanaka M: Spingl ass behavior in \namorphous BiFeO 3. J Appl. Phys 1993;74:5652-5657 \n[56]  Kumar A, Scott JF, Katiyar RS: Magnon Raman s pectroscopy and in-\nplane dielectric response in BiFeO 3: relation to the Polomska trnsition. \nPhys Rev B 2012;85:224410-1-4 \n[57]  Polomska M, Kaczmarek W, Pajak Z: Electric an d magnetic properties \nof (B 1-xLa x)FeO 3 solid solutions: Phys Stat Solidi A 1974;23:567-57 4 [58]  Fischer P, Polomska M, Sosnowska I, Szyma ński M: Temperature-\ndependence of the crystal and magnetic structures o f BiFeO 3. J Phys C \n1980;13:1931-1940 \n "    },    {        "title": "1506.04968v1.Parametric_excitation_of_surface_magnetostatic_modes_in_an_axially_magnetized_elliptic_cylinder_under_longitudinal_pumping.pdf",        "content": "arXiv:1506.04968v1  [cond-mat.mes-hall]  16 Jun 2015M.A. Popov\nM.A. POPOV\nTaras Shevchenko National University of Kyiv\n(64/13, Volodymyrs’ka Str., Kyiv 01601, Ukraine; e-mail: m axim_popov@univ.kiev.ua)\nPARAMETRIC EXCITATION\nOF SURFACE MAGNETOSTATIC MODES\nIN AN AXIALLY MAGNETIZED ELLIPTIC\nCYLINDER UNDER LONGITUDINAL PUMPING PACS 76.20.+q\nA rigorous analytical theory of parametric excitation unde r the longitudinal pumping has been\ndeveloped for the surface magnetostatic modes of a long elli ptic ferrite cylinder magnetized\nalong its axis with regard for the boundary conditions at the surface of the cylinder. It is shown\nthat a pair of frequency-degenerated counter-propagating surface modes at half the pumping\nfrequency can be parametrically excited, and the expressio ns for the corresponding paramet-\nric excitation threshold have been derived. The threshold d emonstrates a strong dependence\non the mode number and elliptic cylinder’s aspect ratio and t ends from above for the large\naspect ratio to the value deduced on the basis of the plane-wa ve analysis. The simple analytical\nrelation between the ratio of axes of the high-frequency mag netization polarization ellipse of\nexcited surface magnetostatic oscillations and the parame tric excitation threshold is obtained,\ndiscussed, and graphically illustrated.\nKey word s : parametric processes, surface magnetostatic oscillatio ns, elliptic cylinder,\nyttrium-iron garnet, film ferrite resonator.\n1. Introduction\nThe parametric excitation of spin waves due to the\nintrinsic nonlinear properties of ferromagnetic mate-\nrials plays a key role in practical applications. While\nthe magnetization oscillations with small amplitudes\ncan be safely analyzed in the linear approximation\n[1], the nonlinear properties of ferrite for relatively\nlarge amplitudes of the high-frequency magnetiza-\ntion lead to various nonlinear effects [2], including\nthe parametric excitation (PE) of spin waves [3]. On\nthe one hand, such phenomenon restricts the dynamic\nrange of the input RF power of magnetostatic res-\nonators. On the other hand, a number of nonlinear\ndevices, such as a power limiter and a signal-to-noise\nenhancer are based on this effect [4]. Therefore, the\ncareful examination of parametric excitation with re-\ngard for the specific features of a ferrite resonator and\nexcitation conditions is of importance for the applied\nresearch.\nSuhl [3] developed the basics of the PE theory,\nas applied to a transversely pumped isotropic ferro-\nmagnet. Subsequently, the theoretical model was im-\nc/circlecopyrtM.A. POPOV, 2015proved to account for the arbitrary orientation and\npolarization of a microwave pumping field [5]. Finally,\nit was generalized for the first and second bands un-\nder the dual pumping of ferrite materials with either\nuniaxial or cubic magnetocrystalline anisotropy, by\nusing arbitrary polarized and oriented RF fields [6].\nHowever, Suhl’s theory utilizes the expansion of the\nhigh-frequency magnetization in uniform plane spin\nwaves, which is justified only when the wave number\nkof excited spin waves is much larger than the inverse\ndimensions of a sample. But, for thin ferrimagnetic\nfilms with thickness of the order of a few to a few tens\nof microns, the typical wave numbers of resonator\neigen-excitations – surface magnetostatic oscillations\n(SMSO) – are much less than the inverse thickness. In\nthis case, one ought to expand the magnetization vec-\ntormand the RF magnetic field in problem’s normal\nmodesmn:m=/summationtext\nn(Anmn+c.c.)[7, 8], instead of\nplane spin waves.\nIn this paper, the parametric excitation of SMSO\nin a longitudinally magnetized yttrium-iron garnet\n(YIG) film resonator with elliptic cross-section un-\nder the longitudinal pumping will be considered, by\ntaking the actual boundary conditions at the res-\nonator surface into account. The ferrite anisotropy is\n452 ISSN 2071-0186. Ukr. J. Phys. 2015. Vol. 60, No. 5Parametric Excitation of the Surface Magnetostatic Modes\nneglected, by assuming the external static magnetic\nfield to be much larger than typical YIG cubic and\nuniaxial anisotropy fields ( ≈50 Oe).\n2. General Theory\nThe exact analytical theory of SMSO in infinitely long\nisotropic ferrite resonator magnetized along its axis\nwith elliptic cross-section (Fig. 1) in the nonexchange\nlimit is presented in [9].\nIt was shown [9,10], that the eigenfrequency of the\nSMSOn-th mode of such resonator is given by:\nω2\nn= (ωH+ωM/2)2−1/4ω2\nM((a−b)/(a+b))2|n|.(1)\nThe magnetostatic modes of the infinitely long el-\nliptic resonator can be characterized by three indices\n[7], namely, the number of nodes in the circumfer-\nential direction n, indexrof a solution of the char-\nacteristic equation, and wavenumber βcorrespond-\ning to the propagation along the cylinder axis. For\nthe axially uniform oscillations, β= 0,and the\nsurface modes are labeled by r= 0,according to\n[7]. Hereafter, we will designate each mode with the\nsingle subscript ninstead of all three indices (n,0,0),\nfor the sake of brevity.\nSinceωn=ω−n(1), the external pumping RF\nmagnetic field h(applied in parallel to the DC field\nH0) with the frequency ωp= 2ωncan parametri-\ncally excite two frequency-degenerated counterpropa-\ngating surface magnetostatic modes with the indices\nnand−n.\nThe magnetostatic potential Ψfor the SMSO n-th\nmode in an elliptic ferrite cylinder with semiaxes a\nandbcan be expressed in the modified elliptic coor-\ndinate system (ρ,φ,z)as [9]\nΨn(ρ,φ) =Bn(R+\nn(ρ)cos(nφ)−\n−isgn(n)µ+1/A\nµaR−\nn(ρ)sin(nφ)), (2)\nwhereBnis the mode amplitude, R+\nn(ρ) = (ρ|n|+\n+(c/2)2|n|ρ−|n|),R−\nn(ρ) = (ρ|n|−(c/2)2|n|ρ−|n|),\nA= (1−(c/(a+b))2|n|)/(1+(c/(a+b))2|n|,c=\n=√\na2−b2,µ= (ω2−ω2\n1)/(ω2−ω2\nH),µa=\n=ωωM/(ω2−ω2\nH),ω2\n1=ωH(ωH+ωM),ωM=\n=γ4πM0,ωH=γH0,γis the gyromagnetic ratio,\nandM0is the saturation magnetization. In (2), one\nshould treat ωas the eigenfrequency ωnof then-th\nmode at a given magnetic field H0(see (1)).\nFig. 1. Longitudinally magnetized ferrite elliptic cylinder un-\nder parallel pumping\nIn view of the standard expressions\nhn= gradΨ n,mn= ˆχhn,\nˆχ=ωM\nω2\nH−ω2/parenleftbiggωHiω\n−iω ωH/parenrightbigg\n,\nrelations (2) yield the explicit formulae for the com-\nponents of the high-frequency magnetization vector\nmnfor each SMSO mode. Then we verified,by\nstraightforward calculations, the orthogonality re-\nlation/integraltext\n(mρnmϕm−mϕnmρm)dV= 0and calcu-\nlated the eigenmode normalization constant Dn=\n=−i/integraltext\n(mρnm∗\nϕn−m∗\nρnmϕn)dV[7]. It was found, that\nDn=D−n=2πC2\nn\n|n|En, whereCnis some expression\ndepending on the frequency, magnetic field, satura-\ntion magnetization, and geometric parameters of a\nsample, and En= 2ωn(2ωH+ωM(1+((a−b)/(a+\n+b))|n|))−1. Note that, in modified elliptical cylindri-\ncal coordinates [9], we have dV≡ρh2\nρdρdϕdz ,hρ=\n=/radicalBig\n(1−c2/4ρ2)2+(c2/4ρ2)sin2ϕ.\nIn [7], the general expression for the experimentally\nobserved parametric excitation threshold RF field hc\nwas found to be\n(γhc)2=4ωrnωrm\nλn,mλ∗m,n, (3)\nwhereλn,m= (1/Dn)/integraltext\n(m∗\nnm∗\nm)dV, andωrnis the\nrelaxation frequency of the proper mode.\nUsing (2), we obtain\n/integraldisplay/parenleftbig\nm∗\nρnm∗\nρm+m∗\nϕnm∗\nϕm/parenrightbig\ndV=πC2\nn\n|n|(1−E2\nn),\nwhen|n|=|m|, and is equal to zero otherwise (here,\nCnandEnare exactly the same as in the expression\nforDn). Therefore,\nλn,m=(1−E2\nn)\n2Enδ|n|,|m|=λ∗\nm,n.\nISSN 2071-0186. Ukr. J. Phys. 2015. Vol. 60, No. 5 453M.A. Popov\nThis means that the PE process like ωp=ωn+ωm,\n|n| /negationslash=|m|, which could be allowed by the energy\nconservation law, would have, nevertheless, the in-\nfinite threshold due to the zero overlapping inte-\ngralλn,m. Thus, only the parametric excitation of\ntwo SMSO modes having opposite azimuthal indices\nn=−mis allowed and would be considered further.\nBy substituting all the previously calculated expres-\nsions into (3), we obtain\nhc=/parenleftbigg2ωrn\nγ/parenrightbigg2En\n(1−E2n).\nAfter some cumbersome calculations, the final ex-\npression takes the form\nhc=/parenleftbigg2ωrn\nγ/parenrightbigg2ωn\nωM/parenleftbigga−b\na+b/parenrightbigg|n|=ηωp\nωM/parenleftbigga−b\na+b/parenrightbigg|n|∆Hk,\n(4)\nwhereη=∂ωn/∂ωH= (ωH+ωM/2)/ωnis the el-\nlipticity factor [11], and ∆Hkis the ferromagnetic\nresonance linewidth.\nApparently, the excitation threshold for SMSO un-\nder the longitudinal pumping strongly depends on\nthe geometric parameters of a sample (e.g., the as-\npect ratio a/b). But otherwise, expression (4) is sim-\nilar to that deduced on the basis of the plane-wave\nanalysis [1].\nAs it was pointed out earlier [8], the PE process\nefficiency strongly correlates with the polarization of\nexcited spin waves. Next, we will elucidate this state-\nment for the problem under consideration and express\nit in the strict mathematical form.\nUsing the explicit expressions for mn, our calcula-\ntions show that the ratio of axes of the high-frequency\nmagnetization polarization ellipse is given by the for-\nmula\nmmin/mmax= tan(1/2arcsin(2 En/(1+E2\nn))).\nIt is worth noting that mmin/mmaxdoes not depend\non coordinates, i.e., it is spatially uniform.\nAfter some simplifications, we obtain a formula\nthat explicitly expresses the ratio of axes of the eigen-\nmode polarization ellipse in terms of the magnetic and\ngeometric parameters of the SMSO resonator:\nmmin\nmmax= tan/parenleftbigg\n1/2arcsinωn\nωH+ωM\n2/parenrightbigg\n. (5)Considering the expression for the ellipticity fac-\ntor and relation (1), one can see that magnetization’s\npolarization is defined by the ellipticity factor ηonly,\naccording to mmin/mmax= tan(1/2arcsin(1 /η)).\nSince the critical field hcand the polarization state\ndepend on the same coefficient En, we can express\none physical quantity directly via another one. Thus,\nwe have\nhc=/parenleftbigg2ωrn\nγ/parenrightbiggmmin/mmax\n1−(mmin/mmax)2. (6)\nThe physical origin of such correlation is clear:\nfor the circular precession of the magnetization\n(mmin/mmax= 1), the longitudinal component of\nthe magnetization mzis absent, and no coupling\nwith the pumping field is possible ( hc→ ∞). For\na more elliptic precession, mzbecomes correspon-\ndently larger. Hence, the interaction is stronger, and\nthe threshold is lower [8].\n3. Discussion\nLet us consider two limiting cases of (4): a circular\nferrite rod ( a=b) and a very elongated elliptic cylin-\nder (a≫b). In the first case, hc→ ∞, since the char-\nacteristic equation admits a solution only for n >0,\nand a pair of counterpropagating surface modes re-\nquired for the PE process is absent, as it was cor-\nrectly pointed out in [7]. As for the second case, let\nus use the previously published expression for the\nPE of traveling surface magnetostatic waves with the\nwavevectors ±kin a thin magnetic film with thick-\nnessd[12]. In that situation, the threshold is equal to\nhc= (2ωrn/γ)(ωp/ωM)exp(|k|d), which for kd≪1\nreduces to hc= (2ωrn/γ)(ωp/ωM)(1+|k|/d). On\nthe other hand, expression (4) for b/a≪1reduces to\nhc= (2ωrn/γ)(ωp/ωM)(1+2b|n|/a). Those two for-\nmulae would be identical, if we make the natural re-\nplacement 2b→dand assume that an “equivalent”\nwavevector |k|=|n|/acan be assigned to each eigen-\nmode with index n. Since1/a→0,the discrete set of\nmode indices smoothly transforms into a continuous\nmanifold of k. Thus, expression (4) gives the correct\nresults in both limiting cases.\nThe dependence of the threshold on the cylinder\nshape is illustrated in Fig. 2, where the normalized\nmicrowave threshold field hc/∆Hkfor a few lowest-\norder SMSO modes is depicted as a function of the\naspect ratio a/b. In calculations, we used H0= 1kOe\n454 ISSN 2071-0186. Ukr. J. Phys. 2015. Vol. 60, No. 5Parametric Excitation of the Surface Magnetostatic Modes\nand the value of 4πM0= 1.75kG typical of YIG. The\ndash-dotted line shows the normalized threshold for\ninfinite isotropic media (that is equal to ωp/ωM), by\nassuming that ∆Hkin both cases are identical. One\ncan see that the parametric excitation threshold dras-\ntically increases for cylinder’s shape close to the cir-\ncular one. But, for elliptic cylinders with a large as-\npect ratio (for example, thin-film resonators), it is\napproaching the value for plane spin waves. Speci-\nfically, for the n= 1mode, the difference from the\n“bulk” value is less than 15% for a/b >20. Moreover,\nthe threshold noticeably increases with the mode\nnumber.\nThus, the calculations presented here allow one to\nevaluate the PE threshold for any given mode of an\nelliptic resonator and give the more flexibility to an\nSMSO resonator designer in choosing the dynamic\npower range of a device. For example, if the opera-\ntion at a larger input power is required, the resonator,\naccording to (4) and Fig. 2, should work on higher\nmodes with large hcor must be manufactured as a\ncircular cylinder. On the other hand, for the appli-\ncations like a power limiter, one can precisely set the\ndesired resonator’s threshold power, by simply select-\ning the appropriate axis ratio (see Fig. 2).\nThe analysis of expression (5) demonstrates that,\nfor a circular cylinder ( a=b), all modes without\nexception have circular polarization. However, when\ncylinder’s aspect ratio a/bincreases, the mmin/mmax\nratio start decreasing, and the modes with larger in-\ndexnare always being more “circular”. In addition,\nthe polarization ellipse aspect ratio increases with H0,\ntending to 1 for the large bias (see Fig. 3).\nThe threshold vs. polarization dependence, as de-\nscribed by expression (6), is illustrated in Fig. 4. It\nis clearly seen that the more elliptic precession of\nthe magnetization (smaller mmin/mmax) facilitates,\nindeed, PE under the longitudinal pumping, as was\npointed out earlier. Note that the very elliptic (close\nto linear) precession is beyond the scope of the cur-\nrent theory, since the assumption mz≪mx,my(mi\nbeing the dynamical (high-frequency) components of\nthe magnetization) used when deriving the expres-\nsions for tensor magnetic permeability is no longer\nvalid in this case.\nIn order to define the limits of current theory’s ap-\nplicability, the investigation of the longitudinal and\ntransversal high-frequency components of the mag-\nnetization vector, assuming |M|=const, was done,\nFig. 2. Normalized microwave threshold field as a function of\nthe elliptic cylinder aspect ratio\nFig. 3. Modes magnetization ellipse axis ratio as a function\nof the bias magnetic field (cylinder’s aspect ratio a/b= 3)\nFig. 4. Parametric excitation threshold vs. the polarization\naxis ratio of excited magnetostatic oscillations\nISSN 2071-0186. Ukr. J. Phys. 2015. Vol. 60, No. 5 455M.A. Popov\nresulting in the following formula:\nmmin\nmmax=/radicalBigg\n4/parenleftbiggα\nmx0/parenrightbigg2\n+1−2α\nmx0. (7)\nExpression (7) defines the polarization ellipse as-\npect ratio mmin/mmax,for which mzreaches a value\nequal to αmy,0< α <1, for the given polarization\nellipse normalized major semiaxis mx0=mx/M0.\nThe parameter α=mz/mydetermines the small-\nness of the longitudinal component mzrelative to\nthe transversal component my. Ifα≪1, the stan-\ndard expressions for tensor magnetic permeability are\nentirely valid. Otherwise, those expressions are no\nlonger applicable, and all the theoretical results pre-\nsented here are doubtful. Formula (7) allows one to\nestimate the range of mmin/mmax, for which our the-\noretical model remains correct. For example, for fixed\nmx/M0= 0.1,αis equal0.05formmin/mmax= 0.41,\nandα= 0.1formmin/mmax= 0.24.Thus, the\nsafe interval is roughly 0.5< mmin/mmax<1. For\nsmallermx/M0we will always get lesser values of the\nlower boundary of mmin/mmax. Therefore, it would\nbe safe to assume that, for relatively small mx/M0\n(which is typical of the parametric excitation pro-\ncesses under the parallel pumping), the curve in\nFig. 4 is trustworthy for mmin/mmaxabove approx-\nimately 0.5.\n4. Conclusions\nAnalytical calculations by the theory of parametric\nexcitation of magnetostatic surface oscillations in lon-\ngitudinally magnetized elliptic cylinders under the\nlongitudinal pumping have been conducted. The fi-\nnal expressions are obtained in the simple convenient\nform suitable for the further analysis.\nThe parametric excitation threshold for various\nmode numbers and cylinder aspect ratios has been\nderived and analyzed. It is shown that the paramet-\nric excitation threshold for SMSO in a thin ferrite\nfilm is of the same order of magnitude with that\ncalculated within the classical theory for plane spin\nwaves (SW). The interpretation of the experimen-\ntal results and the thorough analysis of both pos-\nsible mechanisms of parametric excitation are car-\nried out. Indeed, we have the relation hSMSO\nc/hSW\nc=\n=/parenleftbig\n∆HSMSO\nk/∆HSW\nk/parenrightbig\n(a+b)|n|/(a−b)|n|. For exam-\nple, for the pumping field frequency ωp= 10 GHz,the parametric SMSO excited at ωp/2in an YIG res-\nonator biased with H0= 1000 Oe will have low k\nand∆HSMSO\nk→0≈0.25Oe [13]. At the same time, the\nplane spin waves with equal frequency would have\nk≈105cm−1and much larger ∆HSW\nk≈0.7Oe\ndue to the additional contribution from the dipolar\n3-magnon confluence process [14]. In this situation,\nthe SMSO main mode ( n= 1) in a resonator with\naspect ratio a/b >2will have a lower parametric\nexcitation threshold than plane spin waves.\nThe analytical expression for the ratio of axes of\nthe high-frequency magnetization polarization ellipse\nis obtained, and the correspondence between the\npolarization state and the PE threshold is investi-\ngated. The expression directly connecting the ratio\nof axes and the PE threshold is found and graphi-\ncally illustrated, and the bounds of its applicability\nare indicated.\nEarlier [9], it was shown that it the nonexchange\nlimit SMSO spectrum of a long YIG longitudinally\nbiased resonator with rectangular cross-section can\nbe calculated with the use of the geometric approx-\nimation of the resonator cross-section with inscribed\nellipse. Thus, the presented theory, though being de-\nrived for an elliptic resonator, can be potentially ap-\nplied to the widely used film ferrite resonators with\nrectangular shape.\n1. A.G. Gurevich and G.A. Melkov, Magnetization Oscilla-\ntions and Waves (CRC Press, Boca Raton, 1996).\n2. V.S. L’vov, Nonlinear Spin Waves (Nauka, Moscow, 1987)\n(in Russian).\n3. H. Suhl, J. Phys. Chem. Solid 1, 209 (1957).\n4. I.V. Zavislyak and M.A. Popov, in Yttrium: Compounds,\nProduction and Applications , edited by B.D. Volkerts\n(Nova Science Publ., New York, 2011).\n5. Yu.M. Yakovlev, Fiz. Tverd. Tela 10, 2431 (1968).\n6. I.V. Zavislyak, Ph. D. thesis (Kyiv Univ., 1980).\n7. E. Schlomann and R.I. Joseph, J. Appl. Phys. 32, 1006\n(1961).\n8. Ya.A. Monosov, Nonlinear Ferromagnetic Resonance\n(Nauka, Moscow, 1971) (in Russian).\n9. I.V. Zavislyak, G.P. Golovach, M.A. Popov, and V.F. Ro-\nmanyuk, J. of Commun. Technol. and Electron. 51, 203\n(2006).\n10. R.E. De Wames and T. Wolfram, Appl. Phys. Lett. 16,\n305 (1970).\n11. V. Kambersky and C.E. Patton, Phys. Rev. B 11, 2668\n(1975).\n12. G.A. Melkov and S.V. Sholom, Zh. Tekh. Fiz. 60, No. 8,\n118 (1990).\n456 ISSN 2071-0186. Ukr. J. Phys. 2015. Vol. 60, No. 5Parametric Excitation of the Surface Magnetostatic Modes\n13. M.A. Popov and I.V. Zavislyak, J. of Commun. Technol.\nand Electron. 57, 468 (2012).\n14. M. Sparks, R. Loudon and C. Kittel, Phys. Rev. 122, 791\n(1961).\nReceived 29.04.14\nМ.О.Попов\nПАРАМЕТРИЧНЕ ЗБУДЖЕННЯ\nПАРАЛЕЛЬНОЮ НАКАЧКОЮ ПОВЕРХНЕВИХ\nМАГНIТОСТАТИЧНИХ КОЛИВАНЬ\nВ ПОЗДОВЖНЬО НАМАГНIЧЕНОМУ\nЕЛIПТИЧНОМУ ЦИЛIНДРI\nР е з ю м е\nРозроблено аналiтичну теорiю параметричного збуджен-\nня паралельною накачкою поверхневих магнiтостатичнихколивань нескiнченно довгого елiптичного феромагнiтного\nцилiндра, намагнiченого вздовж осi, з урахуванням грани-\nчних умов на поверхнi феромагнетика. Показано можли-\nвiсть параметричного збудження пари вироджених мод з\nпротилежними напрямками поширення, з частотами, що\nдорiвнюють половинi частоти накачки, та отримано вирази\nдля порога цього процесу. Знайдено, що порогова амплiтуда\nполя накачки сильно залежить вiд номера моди та вiдно-\nшення великої та малої пiвосi елiптичного цилiндра i при\nвеликому значеннi цього вiдношення прямує зверху до вели-\nчини, що розрахована на основi моделi плоских хвиль. Було\nотримано, проаналiзовано та графiчно проiлюстровано про-\nсте аналiтичне спiввiдношення мiж елiптичнiстю поляриза-\nцiї високочастотної намагнiченостi збуджених поверхневи х\nмагнiтостатичних коливань та порогом їх параметричного\nзбудження.\nISSN 2071-0186. Ukr. J. Phys. 2015. Vol. 60, No. 5 457"    },    {        "title": "1803.06690v1.Non_reciprocal_Components_Based_on_Switched_Transmission_Lines.pdf",        "content": "1\nNon-reciprocal Components Based on Switched Transmission Lines\nAravind Nagulu, Tolga Dinc, Zhicheng Xiao, Mykhailo Tymchenko, Dimitrios Sounas, Andrea Al `u, and Harish\nKrishnaswamy\nAbstract —Non-reciprocal components, such as isolators\nand circulators, are critical to wireless communication and\nradar applications. Traditionally, non-reciprocal components\nhave been implemented using ferrite materials, which exhibit\nnon-reciprocity under the influence of an external magnetic\nfield. However, ferrite materials cannot be integrated into IC\nfabrication processes, and consequently are bulky and expensive.\nIn the recent past, there has been strong interest in achieving\nnon-reciprocity in a non-magnetic IC-compatible fashion using\nspatio-temporal modulation. In this paper, we present a general\napproach to non-reciprocity based on switched transmission\nlines. Switched transmission lines enable broadband, lossless and\ncompact non-reciprocity, and a wide range of non-reciprocal\nfunctionalities, including non-reciprocal phase shifters,\nultra-broadband gyrators and isolators, frequency-conversion\nisolators, and high-linearity/high-frequency/ultra-broadband\ncirculators. We present a detailed theoretical analysis of the\nvarious non-idealities that impact insertion loss and provide\ndesign guidelines. The theory is validated by experimental\nresults from discrete-component-based gyrators and isolators,\nand a 25 GHz circulator fabricated in 45 nm SOI CMOS\ntechnology.\nIndex Terms —Isolators, gyrators, circulators, non-reciprocity,\nfull-duplex, radars, CMOS, millimeter-wave passive components,\nlinear periodically time-varying (LPTV) circuits, UWB circuit\ntechniques, ultra-wide-band communication (UWB).\nI. I NTRODUCTION\nNon-reciprocal components, such as gyrators, isolators,\nand circulators, have numerous applications in the fields\nof wireless communication and radar. Frequency-modulated\ncontinuous-wave (FMCW) radars typically operate in\nsimultaneous-transmit-and-receive (STAR) mode, and a\ncirculator is critical to enable the transmitter and the receiver\nto share the same antenna and avoid saturation of the receiver.\nFull-duplex wireless is an emerging wireless communication\nparadigm which has drawn significant research interest in\nrecent years [1]–[9] due to its potential to double the\nspectral efficiency in the physical layer and offer numerous\nother benefits in higher layers. Unlike traditionally-used\ntime-division or frequency-division duplexing schemes, in\nfull-duplex wireless, the transmitter and the receiver operate\nat the same time and at the same frequency . Once again, the\ncirculator is critical in allowing the the transmitter and the\nreceiver to share the same antenna. Isolators are commonly\nused to protect power amplifiers from reflections at the\nA.N. and H.K. are with the Department of Electrical Engineering, Columbia\nUniversity, New York, NY 10027, USA. e-mail: hk2532@columbia.edu.\nZ.X., M.T., D.S., A.A. are with the Department of Electrical and Computer\nEngineering, University of Texas at Austin, TX 78712, USA.\nA.A. is also with the Advanced Science Research Center, City University of\nNew York, New York, NY 10031, USA.\nThis paper is an expanded version from the International Microwave and RF\nConference (IMaRC 2017), Ahmedabad, India from 11-13 December,2017.antenna interface. The gyrator was postulated by Tellegen as\nthe fifth linear circuit element after the resistor, capacitor,\ninductor and transformer [10], providing a non-reciprocal\nphase equal to π, and is a basic non-reciprocal element\nwhich can be used to realize arbitrary passive non-reciprocal\ncircuits [11], [12].\nLorentz Reciprocity states that any linear and time-invariant\nmedium with symmetric permittivity and permeability tensors\nis reciprocal. Historically, reciprocity has been broken\nby biasing magnetic (ferrite) materials using permanent\nmagnets [13], [14]. However, magnetic materials are\nincompatible with IC fabrication technology, and therefore\nmagnetic non-reciprocal components tend to be bulky\nand expensive. IC-compatible circulators have been\nproposed using the inherent non-reciprocal behavior of\nactive voltage-/current-biased transistors [15]–[18]. However,\nactive non-reciprocal components are severely limited by\nthe linearity and noise performance of the active transistors,\nand are therefore not suitable for wireless applications\nwhich demand stringent performance on those fronts [19].\nNonlinearity can be used to violate reciprocity, and has been\nextensively studied in the optical domain [20]–[23], but these\ntechniques have limited utility in wireless communication\napplications due to their stringent linearity constraints.\nRecently, exciting research efforts have been made to break\nreciprocity using spatio-temporal modulation [24]–[27], with\npermittivity being the material parameter that is modulated.\nIn [24], a transmission line is periodically loaded with\nvaractors, and a traveling-wave modulates the varactors along\nthe line. The unidirectional modulation signal imparts a\ndirection-dependent frequency conversion to the input signal\nas it travels along the line. Thus, forward and reverse traveling\nsignals are separated in frequency, and can be isolated from\neach other using a frequency diplexer. In this structure,\nthe length of the transmission line required is inversely\nproportional to the modulation contrast. In general, however,\npermittivity modulation is weak, with varactors exhibiting\na typicalCmax/Cmin ratio of 2-4 on chip. As a result, a\nlarge form-factor is required to achieve strong non-reciprocity.\nFurthermore, varactors exhibit very poor quality factor as\nthe operation frequency is increased to millimeter-wave\nfrequencies, resulting in higher insertion loss. In [25], [26],\nit was demonstrated that traveling-wave modulation in a\nresonant ring results in angular momentum biasing and strong\nnon-reciprocity without frequency conversion. The ring can\nbe miniaturized using lumped components, but this degrades\nthe operating bandwidth. Most importantly, as before, the\nlimitations of permittivity modulation using varactors, namely\narXiv:1803.06690v1  [eess.SP]  18 Mar 20182\nlimited modulation contrast and the degradation of the quality\nfactor as frequency increases, remain.\nOn the other hand, conductivity can be very efficiently\nand strongly modulated in semiconductor media, with CMOS\ntransistors exhibiting ON/OFF condutance ratios as high as\n103-105[28]. It has been shown that by modulating the\nconductivity, i.e. switching transistors, on either side of a\ndelay medium, extremely strong, low-loss, and broadband\nnon-reciprocity can be achieved within a small form factor\n[29]–[34]. The first demonstration involved staggered (i.e.,\nphase-shifted) switching of transistors across a capacitor bank\n(commonly called an N-path filter [35], [36]) [29]. The\nphase-shifted N-path filter realizes an electrically-infinitesimal\ngyrator, which was then embedded within a ring to realize\na compact circulator. The placement within the ring was\nfurther optimized to suppress the voltage swing at the gyrator\nfor transmitter-port excitations, enhancing the linearity and\npower handling. However, this approach suffers from low\nbandwidth because the capacitors do not provide a true time\ndelay. In addition to this, N-path filters are not realizable at\nmillimeter-wave frequencies due to their stringent clocking\nrequirements, as they require multiple phases of a clock at the\nfrequency of operation. When the delay medium is replaced\nby a transmission line of appropriate length, non-reciprocity\ncan be observed over much wider bandwidths. In addition\nto this, the modulation frequency can be greatly reduced\nrelative to the operating frequency [30], [31]. These concepts\nwere leveraged to realize the first millimeter-wave (25 GHz)\npassive non-magnetic circulator in CMOS technology in [30],\n[31]. Similar concepts were used in [32] to demonstrate\nan ultra-broadband circulator operating from 200 KHz to\n200 MHz using discrete switches and co-axial cable delays,\nand in [33] to implement a 0.2 µm GaN HEMT MMIC\ncirculator operating up to 1GHz.\nThis paper presents an exhaustive set of\nvarious non-reciprocal structures possible by using\nspatio-temporal conductivity-modulation across\ntransmission line delays, including an arbitrary\nphase-non-reciprocal element, an ultra-broadband gyrator, a\nfrequency-conversion isolator, an ultra-broadband isolator,\nand ultra-broadband/high-linearity/high-frequency circulators.\nA detailed analysis for estimating the transmission losses is\npresented, along with results from a 25 GHz 45 nm SOI\nCMOS circulator prototype as a case study. This analysis\ncan aid in choosing between different implementation\ntechnologies, fabrication processes, and non-reciprocal\nelement topologies, and in performance optimization during\nthe design phase. The rest of the paper is organized as\nfollows. Section II discusses various phase-non-reciprocal\nconfigurations. Section III discusses various isolator and\ncirculator topologies. Section IV contains a detailed analysis\nfor estimating losses. Section V considers case studies\nimplemented using discrete switches and co-axial cables,\nwhile Section VI details the 25 GHz 45 nm SOI CMOS\ncirculator case study. Section VII concludes the paper.\nFig. 1. Single-ended phase-non-reciprocal element: (a) circuit diagram and\noperation, and (b) fundamental-to-fundamental scattering parameters for fm\n= 8.33 GHz.\nII. P HASE NON-RECIPROCITY\nA. Single-ended Phase-Non-Reciprocal Element\nFig. 1(a) depicts the circuit diagram and the operation of\na single-ended phase-non-reciprocal element. It consists of\na transmission-line segment whose characteristic impedance\nis equal to the port impedance and is sandwiched between\ntwo switches which are modulated using signals LO 1(t)and\nLO 2(t). The switches toggle between zero resistance (ON\nstate) and infinite resistance (OFF state) when their modulation\nsignals change between high and low values (1 and 0),\nrespectively. The modulation signals LO 1(t)andLO 2(t)are\n50%duty cycle square-wave signals with a period of Tm.\nThe modulation signal of the right-hand-side switch, LO 2(t),\nis delayed with respect to that of the left-hand-side switch,\nLO 1(t), byTm/4, which is also the propagation delay of the\ntransmission line.\nThe operation of this structure can be explained using\ntime-domain analysis. For signals travelling from left to right,\nany arbitrary signal incident at port 1 when LO 1(t)is high\nwill be transmitted to port 2 in one pass, experiencing a delay\nofTm/4because the propagation delay of the transmission\nline is equal to the delay between the modulation signals. The\nsignal incident at port 1 gets reflected when LO 1(t)is low.3\nHowever, for signals travelling from right to left, the signal\nincident at port 2 when LO 2(t)is high will be transmitted\nto port 1 after three passes, experiencing a delay of 3Tm/4,\nbecause at the end of the first two passes, the signal sees an\nopen termination due to an OFF state switch and experiences\na total reflection. When LO 2(t)is low, the signal incident at\nport 2 gets reflected. As a result of the additional reflections for\nincident signals from port 2 compared to port 1, the structure\nexhibits a different phase response for the signals travelling\nfrom left to right and right to left, making the structure\nnon-reciprocal in phase. The behavior of the structure can be\nexpressed in the time domain as\nv−\n1(t) =v+\n1(t) (1−LO 1(t)) +v+\n2/parenleftbig\nt−3Tm\n4/parenrightbig\nLO 2/parenleftbig\nt−3Tm\n4/parenrightbig\n,\n(1)\nv−\n2(t) =v+\n1/parenleftbig\nt−Tm\n4/parenrightbig\nLO 1/parenleftbig\nt−Tm\n4/parenrightbig\n+v+\n2(t) (1−LO 2(t)).\n(2)\nIn general, LPTV circuits are represented using harmonic\ntransfer functions (HTFs) which capture frequency conversion\neffects [37], [38]. For simplicity, here we only show the\nfundamental-to-fundamental S-parameter matrix by taking\nFourier transform of the time-domain equations:\nS=/parenleftBigg\n1\n21\n2e−jω3Tm\n4\n1\n2e−jωTm\n41\n2/parenrightBigg\n. (3)\nOne-fourth of the power incident into the port is\ntransmitted to the other port, one-fourth of the power is\nreflected due to imperfect matching, and the remaining\nhalf is converted to other frequencies (i.e., intermodulation\nproducts) due to the switching action. Fig. 1(b) shows the\nfundamental-to-fundamental S parameters around 25 GHz for\nfm=8.33 GHz.\nB. Balanced Configuration: Arbitrary Phase-Non-Reciprocal\nElement\nHarmonic conversion and the effect of imperfect matching\nof the single-ended phase-non-reciprocal element can be easily\novercome by adding another parallel branch switched with\ncomplementary clocks LO 1(t)andLO 2(t), as shown in Fig.\n2(a). The incident signal at port 1 (port 2) travels through the\nfirst transmission line when LO 1(t) (LO 2(t))is high, and the\nsecond when LO 1(t) (LO 2(t))is low, making the structure\ncompletely matched with lossless transmission. The behavior\nof the structure can be expressed in the time domain as\nv−\n1(t) =v+\n2/parenleftbigg\nt−3Tm\n4/parenrightbigg\n, (4)\nv−\n2(t) =v+\n1/parenleftbigg\nt−Tm\n4/parenrightbigg\n. (5)\nBy taking Fourier transform of the time-domain equations, the\nS-parameter matrix can be calculated as\nS=/parenleftBigg\n0e−jω3Tm\n4\ne−jωTm\n4 0/parenrightBigg\n. (6)\nFig. 2. Balanced arbitrary phase-non-reciprocal element: (a) circuit diagram\nand operation, and (b) scattering parameters for fm= 8.33 GHz.\nInterestingly, in this balanced configuration, intermodulation\nproducts generated in the first path and the second path will\nhave opposite phase and cancel each other. Therefore, while\nthe structure is internally linear and periodically-time-varying\n(LPTV), at its ports, it features no frequency conversion and\nappears linear and time-invariant (LTI).\nAt the odd multiples of the modulation frequency, ω=\n(2n+1)ωm, whereωm= 2π/Tmandn=0,1,2,3..., the phases\nfrom left to right and right to left are −(2n+ 1)π/2and\n−3(2n+1)π/2respectively. In this case, the structure behaves\nas a gyrator, providing a non-reciprocal phase equal to π[10].\nFig. 2(b) shows the S parameters around a gyration frequency\nof 25 GHz for fm=8.33 GHz. In addition to this, at any\ngiven input frequency, a value of the modulation frequency ωm\ncan be chosen to realize a non-reciprocal phase difference of\narbitrary value, ∆Φ =π(ω/ωm). Of course, reconfiguring the\nvalue of this non-reciprocal phase shift after implementation\nrequires reconfiguring the transmission line so that its delay\nmatches the modulation frequency change.\nWhile ideally lossless, in practice, the transmission loss will\nbe limited by the quality factor of the transmission line and\nnon-zero switch resistance. The bandwidth of operation will\nbe limited by the dispersion characteristics of the transmission4\nline, switch parasitic capacitance, and rise/fall time of the\nmodulation clocks. In the presence of a finite quality factor\nin the transmission lines, signals incident at port 2 experience\nmore loss because they get transmitted after three passes when\ncompared to signals incident at port 1 which get transmitted\nafter one pass.\nC. Doubly-balanced Con���guration: Ultra-broadband Gyrator\nEven though the balanced structure provides non-reciprocal\nphase response over an infinite bandwidth, it behaves as a\ngyrator only at discrete frequencies. In addition to this, in\na practical implementation, the transmission loss in forward\nand reverse directions will be imbalanced. Another interesting\nvariant is a doubly-balanced non-reciprocal element, as shown\nin Fig. 3(a), which exploits differential signaling for both\nthe modulation signal as well as the input signal. It consists\nof a differential transmission line, with a characteristic\nimpedance equal to the differential port impedance, which\nis sandwiched between two doubly-balanced switch sets\n(often called Gilbert-quad mixers). A Gilbert-quad switch set\nconsists of four switches where two switches connect the\ninput and output directly when the modulation signal is high\nand the other two switches (crisscrossed) swap the signal\npolarity when the modulation signal is low. The switch sets\non the left and right of the transmission line are driven by\nmodulation signals LO 1(t)andLO 2(t)respectively. Similar\nto the balanced case, the modulation signal of the right-hand\nside switch, LO 2(t), is delayed with respect to that of the\nleft-hand side switch, LO 1(t), by a value Tm/4, which is\nalso the propagation delay of the transmission line.\nIn this configuration, signals travelling from left to right,\nincident at port 1, get transmitted to port 2 without any sign\ninversion in the first half cycle, and with two sign inversions\nthat cancel each other in the second half cycle. On the\nother hand, signals travelling from right to left, incident at\nport 2, get transmitted to port 1 with a sign inversion from the\nleft-hand-side mixer in the first half cycle and a sign inversion\nfrom the right-hand-side mixer in the second half cycle. The\nbehavior of the structure can be expressed in the time domain\nas shown in (7) and (8), where m(t)is the multiplication\nfactor due to the switching action of a quad-mixer, which is a\nsquare wave signal between +1 and -1 with 50 %duty cycle and\nperiod ofTm. By taking Fourier transform of the time-domain\nequations, the S-parameter matrix can be calculated as\nS=/parenleftBigg\n0−e−jωTm\n4\n+e−jωTm\n4 0/parenrightBigg\n. (9)\nAs expected, this doubly-balanced structure also appears\nLTI externally while being internally LPTV . Most importantly,\nthis structure describes an ideal gyrator over an infinite\nbandwidth. Fig. 3(b) shows the S parameters around 25 GHz\nforfm=8.33 GHz. Ideally, zero insertion loss and a perfect\nnon-reciprocal phase shift of πare possible over infinite\nbandwidth at arbitrarily small sizes through appropriate\nincrease of the modulation frequency. In practice, transmission\nlosses will be limited by the quality factor of the transmission\nFig. 3. Doubly-balanced ultra-broadband gyrator: (a) circuit diagram and\noperation, and (b) scattering parameters for fm= 8.33 GHz.\nlines and non-zero switch resistance, the bandwidth of\noperation by the dispersion characteristics of the transmission\nline, switch parasitic capacitance, and rise/fall times of\nthe modulation clock, and the size by the practicalities\nof switching solid-state switches at increasing modulation\nfrequency and the associated power consumption. It should\nalso be mentioned that in the presence of losses, this structure\ncontinues to behave as a perfect gyrator with symmetric losses,\nbecause the phase non-reciprocity is achieved due to sign\ninversion as opposed to path length differences.\nD. Frequency-Domain Analysis of the Ultra-broadband\nGyrator\nThe ultra-broadband gyrator can be interpreted as a\ntransmission line sandwiched between two mixers, and can be\nanalyzed in the frequency domain. The multiplication factors\nof the left-hand-side mixer and the right-hand-side mixer, m(t)\nandm(t−Tm\n4), can be expressed in the frequency-domain as\nshown in (10) and (11).\nAn input signal, ejωttraveling from left to right, after\npassing through the left-hand-side mixer, will be multiplied\nwith the multiplication factor m(t), resulting in (12). From this\nequation, we can see that the transmission line supports signals\nat infinitely-many intermodulation frequencies, and therefore,\ndisperson-free operation of the line is critical. As can be5\nv−\n1(t) =m(t)×v+\n2/parenleftbigg\nt−Tm\n4/parenrightbigg\n×m/parenleftbigg\nt−Tm\n2/parenrightbigg\n=−v+\n2/parenleftbigg\nt−Tm\n4/parenrightbigg\n(7)\nv−\n2(t) =m/parenleftbigg\nt−Tm\n4/parenrightbigg\n×v+\n1/parenleftbigg\nt−Tm\n4/parenrightbigg\n×m/parenleftbigg\nt−Tm\n4/parenrightbigg\n=v+\n1/parenleftbigg\nt−Tm\n4/parenrightbigg\n(8)\nm(t) =2\njπ∞/summationdisplay\nn=11\n2n−1(ej(2n−1)ωmt−e−j(2n−1)ωmt) (10)\nm(t−Tm\n4) =2\njπ∞/summationdisplay\nn=11\n2n−1(e−j(2n−1)π\n2ej(2n−1)ωmt−ej(2n−1)π\n2e−j(2n−1)ωmt) (11)\nv+\n1(t)×m(t) =2\njπ∞/summationdisplay\nn=11\n2n−1(ej(ω+(2n−1)ωm)t−ej(ω−(2n−1)ωm)t) (12)\nv+\n1(t−Tm\n4)×m(t−Tm\n4) =2\njπe−jωTm\n4∞/summationdisplay\nn=11\n2n−1(e−j(2n−1)π\n2ej(ω+(2n−1)ωm)t−ej(2n−1)π\n2ej(ω−(2n−1)ωm)t) (13)\nv−\n2(t) =/bracketleftBigg\n2\njπe−jωTm\n4∞/summationdisplay\nn=11\n2n−1(e−j(2n−1)π\n2ej(ω+(2n−1)ωm)t−ej(2n−1)π\n2ej(ω−(2n−1)ωm)t)/bracketrightBigg\n×\n2\njπ∞/summationdisplay\np=11\n2p−1(e−j(2p−1)π\n2ej(2p−1)ωmt−ej(2p−1)π\n2e−j(2p−1)ωmt)\n\n=−4\nπ2e−jωTm\n4∞/summationdisplay\nn=11\n(2n−1)2(−ej(ω+(2n−1)ωm−(2n−1)ωm)t−ej(ω−(2n−1)ωm+(2n−1)ωm)t)\n=8\nπ2e−jωTm\n4ejωt∞/summationdisplay\nn=11\n(2n−1)2\n=e−jωTm\n4ejωt\n=e−jωTm\n4v+\n1(t)(14)\nseen, the amplitude of higher-order intermodulation products\nbecomes progressively smaller, and therefore, dispersion at\nfrequencies further away from the operation frequency will\nhave a progressively smaller effect. A quantification of the\nimpact of transmission-line dispersion is given in Section\nIV. This signal experiences a delay of Tm/4through the\ntransmission line, at which point, it can be expressed as shown\nin (13). This gets multiplied with the multiplication factor of\nthe right-hand-side mixer, resulting in the output signal shown\nin (14). This agrees with the result from the time-domain\nanalysis in (8). At frequencies other than ω, due to cancellation\nof the intermodulation products, the amplitude of the signal\nbecomes zero, making the structure appear externally LTI,\nwhile being internally LPTV , as mentioned earlier. For a signal\ntravelling from right to left, the result in (7) can be obtained in\nthe frequency domain as well by following a similar approach.\nIII. A MPLITUDE NON-RECIPROCITY : ISOLATORS AND\nCIRCULATORS\nSo far, we have discussed various topologies of\nspatio-temporal conductivity-modulation that realize phase\nnon-reciprocity. In this section, we discuss topologies and\ncircuits exhibiting amplitude non-reciprocity, such as isolators\nand circulators.\nA. Frequency-conversion Isolator\nAn isolator is a two-port non-reciprocal component that\nsupports transmission in one direction but not in the reversedirection. A frequency-conversion isolator can be realized\nfrom the doubly-balanced phase-non-reciprocal element\nby modifying the delay between the modulation signals\nof the left-hand-side Gilbert-quad switch set, LO 1(t), and\nright-hand-side switch set, LO 2(t), to a value Tm/8, which\nis also the modified propagation delay of the differential\ntransmission line between the switch sets, as depicted in Fig. 4.\nA signal travelling from left to right, incident differentially\nat port 1, transmits to port 2 with no sign inversion in the\nfirst half of the modulation cycle, and two inversions in\nthe second half of the cycle, similar to the doubly-balanced\nphase-non-reciprocal element. A signal travelling from right\nto left, incident differentially at port 2, will be transmitted\nwithout sign inversion during the first and third quarter cycles.\nHowever, during the second and fourth quarter cycles, the\nsignal will be transmitted with a sign inversion from the left\nand right Gilbert-quad switch sets respectively, as depicted in\nFig. 4. Equivalently, the signal travelling from right to left will\nbe multiplied with a +1/−1square wave of 50% duty cycle\nand2ωmangular frequency, and gets up-/down-converted to\nother frequencies (i.e., inter-modulation products), leading to\nan isolation at the input frequency. The operation of this\nstructure can be expressed in the time domain as\nv−\n1(t) =m/parenleftbigg\nt−Tm\n4/parenrightbigg\n×v+\n2/parenleftbigg\nt−Tm\n8/parenrightbigg\n×m(t) (15)6\nFig. 4. Frequency-conversion isolator: (a) circuit diagram and operation, and\n(b) fundamental-to-fundamental scattering parameters for fm= 8.33 GHz.\nv−\n2(t) =m/parenleftbigg\nt−Tm\n8/parenrightbigg\n×v+\n1/parenleftbigg\nt−Tm\n8/parenrightbigg\n×m/parenleftbigg\nt−Tm\n8/parenrightbigg\n=v+\n1/parenleftbigg\nt−Tm\n8/parenrightbigg .\n(16)\nThe fundamental-to-fundamental S-parameter matrix is\nobtained by taking Fourier transform of the time-domain\nequations:\nS=/parenleftbigg0 0\n+e−jωTm\n80/parenrightbigg\n(17)\nFrom (17), it can be seen that the power incident at port 2\nat any input frequency gets translated to other frequencies,\nwhile the power incident at port 1 is transmitted losslessly,\nmaking it a frequency-conversion isolator that can operate\nover an infinitely-wide range of operating frequencies. It\nshould be emphasized, however, that this configuration has\nFig. 5. Ultra-broadband dissipative isolator: (a) circuit diagram and operation,\nand (b) scattering parameters for the case with fm= 8.33 GHz.\nlimitations when it comes to instantaneously wideband signals\nat port 2. For signals with bandwidth greater 2 ωm, the\nfrequency-translated signals will fall within the desired signal’s\nbandwidth, compromising the isolation from port 2 to port 1.\nFig. 4(b) shows the fundamental-to-fundamental S parameters\naround 25 GHz for fm=8.33 GHz.\nB. Ultra-broadband Dissipative Isolator\nIsolation through frequency conversion is often undesirable,\nas isolators are typically used to protect sources and amplifiers\nfrom back reflections. In such situations, it is desirable for\nsignals incident in the reverse direction to be dissipated\nwithin the isolator. Indeed, such perfect isolation and matching\ncannot be realized without the presence of loss in the\nsystem [39]. An ultra-broadband dissipative isolator with\nno frequency conversion can be realized by modifying the\nbalanced phase-non-reciprocal element by adding a pair of\nparallel switches with a terminating resistor, of value equal to\nthe port impedance, at the left-hand-side switches. The new\npair of switches is modulated using the complementary clocks\nof the left-hand-side switches, as shown in Fig. 5. For signals7\ntravelling from left to right, this structure behaves exactly like\nthe balanced phase-non-reciprocal element, imparting a delay\nofTm/4. However, for signals travelling from right to left,\nafter the first pass, instead of reflecting, the signals travel\nthrough the newly-added switches and get dissipated in the\nterminating resistor. The operation of the structure can be\nexpressed in the time domain as\nv−\n1(t) = 0, (18)\nv−\n2(t) =v+\n1/parenleftbigg\nt−Tm\n4/parenrightbigg\n. (19)\nBy taking Fourier transform of the time-domain equations,\nthe S-parameter matrix can be calculated as\nS=/parenleftbigg0 0\n+e−jωTm\n40/parenrightbigg\n. (20)\nTheoretically, this structure appears externally-LTI, exhibits\nlossless transmission in the forward direction, perfect isolation\nin the reverse direction, and has infinite bandwidth (Fig.\n5(b)). In practice, the loss and bandwidth are limited by\nquality factor of transmission line, non-zero switch ON\nresistance, dispersion of the transmission line, switch parasitic\ncapacitance, and non-idealities of the modulation clocks,\nsimilar to the phase-non-reciprocal elements.\nC. Ultra-broadband Circulator\nThe ultra-broadband dissipative isolator is actually a special\ncase of an ultra-broadband circulator whose third port has\nbeen terminated to the reference impedance. In other words, a\ncirculator with infinite bandwidth can be realized by replacing\nthe terminating resistor in the ultra-broadband isolator by\na third port as shown in Fig. 6. For such a circulator,\nit can be shown using time domain analysis that lossless\ntransmission happens from port 1 to port 2 , port 2 to port 3\nand port 3 to port 1, and there will be a perfect isolation in the\nreverse circulation direction. The S-parameter matrix for this\nultra-broadband circulator can be constructed as\nS=\n0 0 e−jω2Tm\n4\ne−jωTm\n4 0 0\n0e−jωTm\n4 0\n. (21)\nAs opposed to three switches in the signal path as in [32]\nthis structure has only two switches, enabling it to potentially\nhave superior insertion loss performance. In addition, this\nstructure needs only two transmission lines as opposed to six\ntransmission lines in [32], making it more compact and readily\nimplementable at RF frequencies. It is closely related to the\n4-port circulator recently described in [33].\nD. High-linearity Circulator\nThe ultra-broadband circulator features switches in the\nsignal path, which would limit the power handling of the\ncirculator, particularly in the TX-to-ANT path. Recently,\nwe demonstrated a circulator architecture with high power\nFig. 6. Ultra-broadband circulator: (a) circuit diagram, and (b) scattering\nparameters for fm= 8.33 GHz.\nhandling by wrapping a 3λ/4transmission line with a\ncharacteristic impedance equal to the port impedance around\nthe gyrator component with ±90◦non-reciprocal phase,\nand placing ports λ/4apart on the 3λ/4transmission\nline [29]–[31] (Fig. 7). For such a circulator, the S parameters\nat the center frequency can be computed as\nS=\n0 0−1\n−j0 0\n0−j0\n (22)\nThe power handling at port 1 of the circulator (TX port)\ncan be improved significantly by placing the third port at the\ngyrator element ( l=0 in Fig. 7), as the isolation from port 1\nto port 3 suppresses the voltage swing across the gyrator\n[29]. In [29], a 750MHz circulator was demonstrated in\n65nm CMOS using the N-path-filter-based gyrator discussed\nearlier. In [30], [31], a differential circulator was realized\nat 25GHz in 45nm SOI CMOS using the doubly-balanced\ntransmission-line based gyrator. In general, both single-ended\nand differential circulators can be realized using the balanced\nand doubly-balanced gyrator configurations (Figs. 7(a)\nand (b)). The balanced configuration achieves gyrator\nfunctionality at ω= (2n+ 1)ωm. The doubly-balanced8\nFig. 7. Highly-linear circulators: (a) single-ended configuration leveraging\nthe balanced gyrator, and (b) differential configuration leveraging the\ndoubly-balanced gyrator. (c) Internal node voltage swings of both\nconfigurations when l=0 for 1V TX port voltage at 25GHz.\nconfiguration achieves a non-reciprocal phase of πacross\ninfinite bandwidth, but the need for ±90◦phase once again\ndictatesω= (2n+ 1)ωm. In the 25GHz circulator in [30],\n[31], the modulation frequency was one-third of the operating\nfrequency, which critically enabled scaling of the circulator\nto millimeter-waves. An important point of difference is that\nthe internal gyrator nodes also experience suppressed voltage\nswing for TX-port excitations in the differential configuration,\ndue to the fact that a pair of switches is always on, shorting\nthe internal nodes to the external nodes (Fig. 7(c)). This,\nhowever, is not the case for the single-ended circulator\nconfiguration that uses the balanced gyrator, limiting the TX\nFig. 8. Port 1-Port 3 isolation of the differential circulator featuring the\ndoubly-balanced gyrator and the single-ended circulator featuring the balanced\ngyrator, depicted in Figs.7(a) and (b).\npower handling enhancement. This can be restored by adding\na pair of parallel switches to ground at both the left-hand side\nand right-hand side switches. The new pair of switches would\nbe modulated using the complementary clocks of the original\nswitches, making the nodes SBn2andSBn3quiet. However,\ndue to the additional switches, the power consumption of this\nsingle-ended circulator would become equal to that of the\ndifferential circulator, and so it is beneficial to implement the\ndifferential configuration owing to its benefits of lower clock\nfeedthrough, and +3dB higher power handling.\nIsolation from port 1 to port 3 is due to destructive\ninterference of signals from λ/2reciprocal path and the\nλ/4transmission line and gyrator path. Perfect isolation\noccurs at the center frequency where the signals from both\nthe paths are exactly out of phase. For frequencies off\ncenter, the condition of destructive interference starts to fail\nand isolation will be limited to a finite value. Hence, the\nisolation bandwidth is dependent on the phase response of\nthe3λ/4ring and the non-reciprocal phase response of the\ngyrator. The doubly-balanced gyrator exhibits a non-reciprocal\nphase response of 180◦over theoretically infinite bandwidth,\nwhile the balanced phase-non-reciprocal element behaves as\na gyrator only at discrete frequencies, as discussed earlier.\nHence, as shown in Fig. 8, the 20dB isolation bandwidth of the\ndifferential circulator featuring the doubly-balanced gyrator is\n2.34×higher than that of the single-ended circulator featuring\nthe balanced gyrator.\nIV. A NALYSIS OF INSERTION LOSSBASED ON VARIOUS\nNON-IDEALITIES\nIn this section of the paper, we derive formulae for\nestimating the transmission losses of the highly-linear\ncirculator architecture taking various non-idealities into\naccount using perturbation analysis. During implementation,\ntransmission lines of length comparable to the wavelength are\noften miniaturized using lumped L-C components to reduce9\nFig. 9. (a) Equivalent circuit for loss estimation from port 1 to port 2. (b)\nEquivalent circuit for loss estimation from port 2 to port 3.\nthe chip area and to absorb the parasitic capacitances of the\nswitches into the transmission lines. Hence, in our analysis,\nit is assumed that the λ/4transmission lines sections in the\n3λ/4ring were miniaturized using C-L-C sections. Various\nnon-idealities such as quality factor of the transmission\nlines/inductors, non-zero switch resistance of the switches in\nthe gyrator element, rise/fall time of the modulation signals,\nand the dispersion of the transmission line in the gyrator, limit\nthe insertion loss of the circulator. In this derivation, it has\nbeen assumed that the effect of the non-idealities are small,\nso that they do not degrade the matching of the gyrator element\nand the isolation of the circulator substantially. The effect of\neach non-ideality in the circulator is separately computed using\nperturbation analysis, and the total loss of the circulator is\nestimated by summing up the individual contributions.\nWhile estimating the loss from port 1 to port 2 in the\ncirculator, port 3 can be shorted to ground due to the\nport 1-to-port 3 isolation. As a result, the λ/4section between\nport 2 and port 3 will transform the short circuit to an open\ncircuit when seen from port 2 as will the the λ/4section\nbetween port 1 and gyrator when seen from port 1. Hence,\nthe circulator reduces to a λ/4transmission line connecting\nports 1 and 2 as shown in Fig. 9(a). A similar equivalent circuit\nfor port 2-to-port 3 transmission is shown in Fig. 9(b).\nA. Effect of Non-zero Switch Resistance, Rsw\nA switch with non-zero ON resistance, Rsw, can be modeled\nby an ideal switch with a series resistance, Rsw. Hence, a\ngyrator with non-zero switch resistance can be modeled in\nthe manner depicted in Fig. 10(a). As before, port 3 of the\ncirculator can be shorted to ground when estimating the loss\nFig. 10. (a) Equivalent circuit for a gyrator with non-zero switch resistance.\n(b) Equivalent circuit for loss estimation from port 1 to port 2 of the circulator\nwith non-zero switch resistance. (b)Equivalent circuit for loss estimation from\nport 2 to port 3 of the circulator with non-zero switch resistance.\nfrom port 1 to port 2, as depicted in Fig. 10(b). When one\nof its ports is shorted to ground, the input impedance of the\ngyrator shown in Fig. 10(a) at the other port is 2Rsw. This\n2Rswresistance will be transformed to Z2\n0/2Rswat port 1\nas depicted in Fig. 10(b). Consequently, the transmission loss\nfrom port 1 to port 2 can be written as\n|S21|≈Z0\nZ0+Rsw=Qsw\nQsw+ 1. (23)\nwhereQsw, the equivalent quality factor of the switch, is\ngiven byZ0/Rsw. Similarly, port 1 can be shorted to ground\nwhile estimating transmission loss from port 2 to port 3. Using\nsimilar arguments, the circulator can be reduced to the circuit\ndepicted in Fig. 10(c). Interestingly, the non-zero switch\nresistance will have no effect on the transmisison loss from\nport 2 to port 3 because the switch resistance is in series with\nan open termination.\n|S32|≈1. (24)\nB. Effect of the Quality Factor, Q1, of Inductor L1\nSimilar arguments can be made for the quality factor, Q1, of\ninductorL1. Figs. 11 (a) and (b) depict the equivalent circuits\nforS21andS32loss estimation. We have assumed that the\ninductor losses are purely in the form of series resistance in\nwriting the equivalent parallel resistance as ωL1Q1=Q1Z0.10\nFig. 11. Equivalent circuits for loss estimation of the circulator with finite\nquality factor, Q1, in inductor L1: (a) port 1 to port 2, and (b) port 2 to port 3.\nFig. 12. (a) Equivalent circuit for loss estimation from port 1 to port 2 of the\ncirculator with inductor L2of finite quality factor, Q2. (b)Equivalent circuit\nfor loss estimation from port 2 to port 3 of the circulator with inductor L2of\nfinite quality factor, Q2.\nFurther, we have assumed that the capacitors used to lump the\ntransmission line are lossless. By analyzing these circuits, we\ncan write the transmission losses as\n|S21|≈|(1\nQ1−j)\n(1 +1\nQ1) +j\nQ1|≈Q1\nQ1+ 1, (25)\n|S32|≈2Q1\n2Q1+ 1. (26)\nC. Effect of the Quality Factor, Q2, of Inductor L2\nMaking similar arguments, the equivalent circuits in Fig. 12\nenable loss estimation in the presence of finite quality factor\nQ2in inductorL2. By analyzing these circuits, we can write\nthe transmission losses as\n|S21|≈2Q2\n2Q2+ 1, (27)\n|S32|≈|(1\nQ2−j)\n(1 +1\nQ2) +j\nQ2|≈Q2\nQ2+ 1. (28)\nD. Effect of the Quality Factor, Q3, of Inductor L3\nSimilarly, the circulator with finite quality factor, Q3, in\ninductorL3can be reduced to a C-L-C section with a shunt\nresistor,Q3Z0, at port 1 while estimating loss from port 1 to\nport 2 and a C-L-C section with a shunt resistor, Q3Z0, at\nFig. 13. Filtering due to finite rise/fall time of the modulation signals.\nport 3 while estimating loss from port 2 to port 3. By analyzing\nthese circuits, we can write the transmission losses as\n|S21|≈2Q3\n2Q3+ 1, (29)\n|S32|≈2Q3\n2Q3+ 1. (30)\nE. Effect of the Rise/Fall Time of the Modulation Signal\nIn Section II-D, we have presented a frequency-domain of\nthe doubly-balanced ultra-broadband gyrator with the switch\nsets viewed as mixers. We considered the modulation signals\nto be perfect square waves with negligible rise/fall time, but\nin a practical implementation, the rise/fall time ( tr/tf) can\nbe a significant portion of the time period, particularly for\nhigh-frequency operation. For a trapezoidal modulation signal\nwith rise/fall time of tr=tf, the frequency content of\nmodulation signal will be limited to a cutoff frequency of\n(0.35/tr)Hz. This will correspondingly establish an upper\nlimit on the frequency content of the multiplication factor,\nm(t), as shown in Fig. 13. The value of this cutoff frequency\ncan be obtained from simulations. The effect of this cutoff\nfrequency in the multiplication factor is a limit to the\nintermodulation products that are generated and travel down\nthe line, as depicted in Fig. 13, resulting in loss of signal\npower. In our analysis, the filtering profile of this effect is\nassumed to be a brick wall, and mathematically, this can be\nexpressed as a truncation in the higher frequency terms in\n(10) - 14. For the example shown in Fig. 13, where the cutoff\nfrequency is between 3ωmand5ωm, 14 can be modified as\nv−\n2(t) =−4\nπ2e−jωTm\n4[(−ej(ω+ωm−ωm)t−ej(ω−ωm+ωm)t)\n+1\n32(−ej(ω+3ωm−3ωm)t−ej(ω−3ωm+3ωm)t)]\n=80\n9π2e−jωTm\n4ejωt\n=80\n9π2e−jωTm\n4v+\n1(t)\n≈0.9e−jωTm\n4v+\n1(t).\n(31)\nSimilarly, for the signal traveling from right to left, the\nsignal transmitted can be written as11\nFig. 14. Filtering of the inter-modulation products higher than the Bragg\nFrequency.\nv−\n1(t)≈−0.9e−jωTm\n4v+\n1(t). (32)\nAssuming that the matching of the gyrator has not degraded\ndue to the finite rise and fall time, the S-parameter matrix of\nthe gyrator at the center frequency can be written as\nS=/parenleftbigg0krise/fallejπ\n2\nkrise/falle−jπ\n2 0/parenrightbigg\n(33)\nwherekrise/fall is the attenuation factor due to the filtering\nof higher inter-modulation products. For the example shown\nin Fig. 13,krise/fall≈0.9. Using the perturbation analysis\ndiscussed before, the S-parameters of the circulator can be\nexpressed as\n|S21|���Qrise/fall\nQrise/fall + 1, (34)\n|S32|≈Qrise/fall\nQrise/fall + 1, (35)\nwhereQrise/fall is an effective quality factor associated with\nthe filtering due to finite rise/fall time of the modulation\nsignals, and is given by 2(1+k2\nrise/fall\n1−k2\nrise/fall).\nF . Effect of the Bragg Frequency of the Gyrator Transmission\nLine\nFrom the frequency-domain analysis, it is clear that the\ntransmission line(s) between the switches must be able\nto support all the inter-modulation frequencies with the\nsame group delay for a perfect lossless gyrator. If the\ntransmission lines are miniaturized through a quasi-distributed\nimplementation consisting of periodic lumped-LC sections,\nthen the lines will have a cutoff frequency, after which\nthe periodic structure no longer supports a traveling wave.\nThis cutoff frequency is known as the Bragg frequency.\nInter-modulation products above the Bragg frequency will be\nfiltered and the signal power associated with them will be\nlost as depicted in Fig. 14. Similar to effect of finite rise and\nfall time of the modulation signal, the filtering profile due\nto Bragg frequency is also assumed to be a brick wall. All\nthe inter-modulation terms with frequency above the Bragg\nfrequency should be truncated from (14), and an associated\nattenuation factor, kBragg can be calculated. Assuming thatthe matching of the gyrator has not degraded due to Bragg\nfrequency effects, the S-parameter matrix of the gyrator at the\ncenter frequency can be written as\nS=/parenleftbigg0kBraggejπ\n2\nkBragge−jπ\n2 0/parenrightbigg\n. (36)\nFrom a similar perturbation analysis, the S-parameters of\nthe circulator can be expressed as\n|S21|≈QBragg\nQBragg + 1(37)\n|S32|≈QBragg\nQBragg + 1(38)\nwhereQBragg is an effective quality factor associated with\nthe Bragg effect, and is given by 2(1+k2\nBragg\n1−k2\nBragg).\nBoth finite rise/fall time and finite Bragg frequency\nresult in suppressing the higher frequency content of the\ninter-modulation products. Hence, one has to be careful while\ntruncating the terms in (14) because the minimum of these\ncutoff frequencies will define the actual cutoff frequency for\nthe inter-modulation products. In the final loss estimation,\nto avoid repetition of the loss due to filtering of higher\nmodulation products, a general attenuation factor kfilter can\nbe calculated by truncating all the terms which are required to\nbe truncated from Bragg frequency effects and finite rise/fall\ntime effects. In general, for circulators operating at high\nfrequencies, finite rise/fall time effects will dominate, while\nfor circulators operating at lower frequencies, the need to\naggressively miniaturize the transmission lines will cause\nBragg frequency effects to dominate. For instance, in the\n25GHz 45nm SOI CMOS circulator case study presented later\nin this paper, truncation was limited by the rise/fall time of\nthe modulation signal, making kfilter =krise/fall .\nG. Effect of the Quality Factor, QNR, of the Gyrator\nTransmission Line\nDue to ohmic losses, the signal attenuates exponentially\nas it travels through the transmission line in the gyrator,\ncreating another loss mechanism. As mentioned earlier, all\ninter-modulation products travel through the transmission line,\nand the attenuation suffered by each inter-modulation product\nis different because attenuation constant depends on frequency.\nThe attenuation suffered by a signal at a frequency ωas\nit travels through the transmission line with a delay of\nTm/4ise−π\n4Qω\nωm, whereQis the quality factor of the\ntransmission line at ω. Due to the skin effect, ρ∝√ω,\nwhereρis the resistivity of the metal. A general attenuation\nfactor can be written by taking the skin effect of the metal\ninto consideration as e−(2n+1)π\n4QNR√ω\n(2n+1)ωm, whereQNR is\nthe quality factor of the transmission line at a frequency\nω= (2n+1)ωm. By modifying (13), the signal after traveling\nthrough the transmission line can be expressed as shown\nin (39). This signal gets multiplied with the multiplication\nfactor of the right-hand-side mixer. Similar to (14), the signal\nvoltage at the input frequency, ω, at the other port of the\ngyrator can be written as shown in (40), where kQNR21is12\nv+\n1(t−Tm\n4)×m(t−Tm\n4)\n=2\njπe−jωTm\n4∞/summationdisplay\np=11\n2p−1(e−j(2p−1)π\n2e−(2n+1)π\n4QNR/radicalbigg\nω+(2p−1)ωm\n(2n+1)ωmej(ω+(2p−1)ωm)t−ej(2p−1)π\n2e−(2n+1)π\n4QNR/radicalbigg\n|ω−(2p−1)ωm|\n(2n+1)ωmej(ω−(2p−1)ωm)t)\n(39)\nv−\n2(t) =e−jωTm\n4ejωt×4\nπ2∞/summationdisplay\np=11\n(2p−1)2(e−(2n+1)π\n4QNR/radicalbigg\nω+(2p−1)ωm\n(2n+1)ωm+e−(2n+1)π\n4QNR/radicalbigg\n|ω−(2p−1)ωm|\n(2n+1)ωm)\n=e−jωTm\n4ejωtkQNR21\n=e−jωTm\n4v+\n1(t)kQNR21(40)\nthe attenuation factor for signals traveling from left to right\ndue to the finite quality factor of the transmission line in\nthe gyrator. For a more accurate expression that includes\neffects due to the finite Bragg frequency of the line and\nfinite rise and fall times in the modulation signals, the higher\ninter-modulation terms in (40) should be truncated, and the\nobtained (kQNR21)truncated should be divided by kfilter ,\ni.e.,kQNR21=(kQNR21)truncated /kfilter , so thatkQNR21isolates the effect of the tranmission-line’s finite quality factor.\nSimilarly,kQNR12can be calculated for the signals traveling\nfrom right to left.\nFor the balanced gyrator, kQNR12andkQNR21will be\ndifferent because the signals follow a different path for forward\nand reverse directions, and the attenuation factor depends on\nthe length traversed. However, for the doubly-balanced gyrator,\nthese factors will be equal, since the phase non-reciprocity is\nachieved due to sign inversion from the quad-mixer, rather\nthan path length differences. Assuming that the matching of\nthe gyrator has not degraded due to the quality factor of the\ntransmission line, the S-parameter matrix of the gyrator at the\ncenter frequency can be written as\nS=/parenleftbigg0kQNR12ejπ\n2\nkQNR21e−jπ\n2 0/parenrightbigg\n. (41)\nFrom perturbation analysis, the S-parameters of the\ncirculator can be expressed as\n|S21|≈QNReffec\nQNReffec + 1(42)\n|S32|≈QNReffec\nQNReffec + 1(43)\nwhereQNReffec is the effective quality factor associated with\nthe loss of the transmission line in the gyrator, and is given\nby2(1+kQNR12kQNR21\n1−kQNR12kQNR21)).\nH. Estimating Transmission Losses of the Circulator\nUnder the assumption that all the non-idealities are small,\nthe losses due to the individual effects can be added\nwhen estimating the final loss of the circulator. Hence, the\ntransmission S-parameters of the circulator can be expressed\nas\n|S21|≈(Q1\nQ1+1)(2Q2\n2Q2+1)(2Q3\n2Q3+1)(Qfilter\nQfilter +1)(QNReffec\nQNReffec+1)(Qsw\nQsw+1)\n(44)|S32|≈(2Q1\n2Q1+1)(Q2\nQ2+1)(2Q3\n2Q3+1)(Qfilter\nQfilter +1)(QNReffec\nQNReffec+1).\n(45)\nV. D ISCRETE -COMPONENT -BASED CASE STUDIES\nA. Balanced Arbitrary-Phase Non-Reciprocal Element\nA balanced arbitrary-phase non-reciprocal element has been\nimplemented using coaxial cables and commercially-available\noff-the-shelf switches as shown in Fig. 15. It consists of\na two 10m long 50 Ωcoaxial cables with a time delay\nof 40.5ns and an insertion loss of 1dB. These cables are\nsandwiched between two single-pole double-throw (SPDT)\nswitches which are modulated at 6.17MHz. In practice,\nopen-reflective switches with fast switching time are not\ncommercially available. Alternatively, we used short-reflective\nswitches from Minicircuits (model number ZFSW-2-46) with\na fast switching time of 2ns, to minimize the insertion\nloss due to rise/fall time as discussed in Section IV. Using\na short-reflective switch does not change the principle of\noperation of the structure. Signals traveling in the forward\ndirection do not see the short termination. However, signals\ntraveling in the reverse direction experience two reflections\nfrom a short termination ( Γ=-1) instead of two reflections\nfrom an open termination ( Γ=1). The two additional negative\nsigns cancel each other, and result in an output signal which\nis equal to the open-reflective switch case. At 10MHz, the\nswitch has an insertion loss of 0.5dB in the on-state and\na reflection coefficient of 0.71 /negationslash178◦in the off-state. The\nmodulation signals are generated using arbitrary waveform\ngenerators (model: Agilent 33500B) and have a rise/fall time\nof 8.5ns.\nThe measured S-parameters between 0-50MHz are shown\nin Fig. 16. A non-reciprocal phase of 180◦has been observed\nat 6.043MHz, 18.11MHz, 30.2MHz, etc. This shift in the\ngyration frequency is due to the reflection co-efficient of\nthe off-state switch, which is 178◦instead of 180◦. Since\nthe reverse traveling wave is reflected twice, a total of -4◦\nphase shift is picked up by the signal. Due to this additional\nphase shift, the structure behaves as a gyrator at slightly\nlower frequencies, a similar effect to the observation in\n[34]. At 10MHz, the transmission losses in forward and\nreverse directions are -3.8dB and -17dB respectively. The\ntransmission losses in the reverse direction are higher due\nto the additional loss mechanisms, including 3 ×loss of13\nFig. 15. Photograph of the balanced arbitrary-phase non-reciprocal element\nassembly.\nthe transmission line due to reflections, non-unity reflection\nco-efficient of the reflective-short switch in its off-state, and\ndestructive interference between the leaked signal to port-1\nduring off-state (due to imperfect clocking) and the actual\nsignal.\nB. Ultra-broadband Dissipative Isolator\nAn ultra-broadband dissipative isolator has been\nimplemented by the replacing the left-side switch (SW-1)\nin the balanced configuration setup with a commercially\navailable absorptive switch from Minicircuits (model number:\nZFSWA-2-46). The measured S-parameters between 0-50MHz\nare shown in Fig. 17. At 10MHz, the transmission loss in the\nforward direction is -3.5dB, and the isolation in the reverse\ndirection is 40dB over the entire frequency range.\nVI. A 25GH Z45NMSOI CMOS C IRCULATOR CASE\nSTUDY\nA. Implementation Details\nA differential circulator based on the doubly-balanced\ngyrator was implemented at 25 GHz in 45nm SOI CMOS as\ndepicted in Fig. 18. Aside from achieving superior isolation\nbandwidth due to the use of the doubly-balanced gyrator,\nthe fully-differential architecture reduces LO feedthrough and\nenables 3 dB higher power handling at the expense of a\ndoubling in the power consumption. The quarter-wave sections\nbetween ports 1 and 2, and 2 and 3, were implemented\nusing differential conductor-backed coplanar waveguides with\na Q of 15 at 25 GHz. The gyrator element was placed\nsymmetrically between port 1 and port 3 so that the parasitic\ncapacitances from the mixer switches could be absorbed into\nthe quasi-distributed transmission line in the gyrator and the\nλ/8C-L-C sections on either side. The transmission line in\nthe gyrator is a combination of four π-type C-L-C sections,\nwith inductor Q of 20, and coplanar waveguides, with a Q\nof 15, connecting the C-L-C sections. The overall quality\nfactor and Bragg frequency of the transmission line are 17\nand 83.9 GHz respectively. The Bragg frequency was improved\nby a factor of two because only a part of the quarter-period\ndelay required was obtained from the four C-L-C sections,\nwith the rest obtained from the coplanar waveguides used for\nFig. 16. Measured S-parameters of the balanced arbitrary-phase\nnon-reciprocal element: (a) magnitude, and (b) phase.\ninterconnects. An extra quadrature phase path was added in\nthe gyrator element to counter the loss degradation due to\nduty-cycle impairments. The reader is directed to [40] for\nadditional details. The gilbert quad-mixers are designed using\n2×16µm/40nmfloating body transistors, achieving a Rswof\n8.66Ω. The placement of the gyrator symmetrically between\nport 1 and port 3 distributes the loss due to non-zero switch\nON resistance equally between S21ansS32, as opposed to\n(23) and (24). The new transmission losses due to non-zero\nON resistance can be expressed as\n|S21|≈2Qsw\n2Qsw+ 1, (46)\n|S32|≈2Qsw\n2Qsw+ 1, (47)14\nFig. 17. Measured S-parameters of the ultra-broadband dissipative isolator:\n(a) magnitude, and (b) phase.\nmodifying the circulator transmission losses in (44) and (45)\nto (48) and (49).\nFig. 19 shows the circuit diagram of the 8.33 GHz clock\npath. A two stage poly-phase filter is used to generate the\n8.33 GHz quadrature signals driving the mixer switches. After\nthe poly-phase filter, a three stage self-biased CMOS buffer\nchain with inductive peaking in the final stage generates the\nsquare wave clock signals with a rise/fall time of 7.5 ps for\nthe switches. Independently controlled NMOS varactors at the\ndifferential LO inputs provide I/Q calibration of range ±10◦.\nInterested readers can refer to [40] for more comprehensive\nimplementation details.\nB. Comparison between Theory and Simulations\nFor this 25 GHz circulator, as mentioned earlier, Q1andQ2\nat 25 GHz are 15 each as the λ/4sections were implemented\nusing coplanar waveguides, and Q3at 25 GHz is 20 as it\nFig. 18. Circuit diagram of the implemented 25 GHz doubly-balanced\ncirculator\nFig. 19. Circuit diagram of the quadrature modulation signal generation\nwas implemented using C-L-C sections. Qswis 5.77 as the\nsingle-ended port impedance is 50 ΩandRswis 8.66 Ω. The\nquality factor of the delay line in the gyrator, QNRat 25 GHz,\nis 17. The Bragg Frequency is 76 GHz and the bandwidth\nof the modulation signal, 0.35/tr, is 46.66 GHz. Hence, the\nmodulation signals are limited to their 5thharmonics, as\nare the inter-modulation terms formed. kfilter can therefore\nbe calculated as shown in (50). From (50), Qfilter can\nbe calculated to be 14.4. Similarly, (kQNR)truncated can\nbe calculated as shown in (51). From (51), kQNRcan be\ncalculated using (kQNR)truncated/kfilter . HenceQNReffec is\n7.18. From (44) and (45), S21andS32are -2.66 dB each.\nThe transmission losses, S21andS32, at the center frequency\nfrom post-layout simulations assuming ideal baluns are -2.8 dB\neach as shown in Fig. 21. Simulated transmission losses after\nde-embedding the on-chip baluns are ≈3 dB. Therefore, the\nsimulation results show a close agreement with our analysis.\nC. Measurements\nThe implemented circulator occupies an area of\n1.2 mm×1.8 mm and Fig. 22 shows the die micrograph.\nThe circulator was implemented with on-chip baluns for\nmeasurement purposes. The measurements were performed\nthrough RF probing in a chip-on-board configuration, and the\nbaluns are de-embedded to obtain the circulator performance.\nA 180◦hybrid (Krytar 4010265) is used to generate the\ndifferential (0◦/180◦) 8.33 GHz signals from a signal generator\nto drive the clock inputs of the circulator. A two-port Anritsu\n37397E Lightning VNA is used to measure the S-parameters15\n|S21|≈/parenleftbiggQ1\nQ1+ 1/parenrightbigg/parenleftbigg2Q2\n2Q2+ 1/parenrightbigg/parenleftbigg2Q3\n2Q3+ 1/parenrightbigg/parenleftbiggQfilter\nQfilter + 1/parenrightbigg/parenleftBigg\nQNReffec\nQNReffec+ 1/parenrightBigg/parenleftbigg2Qsw\n2Qsw+ 1/parenrightbigg\n. (48)\n|S32|≈/parenleftbigg2Q1\n2Q1+ 1/parenrightbigg/parenleftbiggQ2\nQ2+ 1/parenrightbigg/parenleftbigg2Q3\n2Q3+ 1/parenrightbigg/parenleftbiggQfilter\nQfilter + 1/parenrightbigg/parenleftBigg\nQNReffec\nQNReffec+ 1/parenrightBigg/parenleftbigg2Qsw\n2Qsw+ 1/parenrightbigg\n. (49)\nkfiltere−jωt=−4\nπ2[(−ej(ω+ωm−ωm)t−ej(ω−ωm+ωm)t) +1\n32(−ej(ω+3ωm−3ωm)t−ej(ω−3ωm+3ωm)t)]\n+1\n52(−ej(ω+5ωm−5ωm)t−ej(ω−5ωm+5ωm)t)]\n= 0.933.(50)\n(kQNR)truncatede−jωt=−4\nπ2[(−e−3π\n4QNR/radicalBigω+ωm\nωmej(ω+ωm−ωm)t−e−3π\n4QNR/radicalbigg\n|ω−ωm|\nωmej(ω−ωm+ωm)t)\n+1\n32(−e−3π\n4QNR/radicalBigω+3ωm\nωmej(ω+3ωm−3ωm)t−e−3π\n4QNR/radicalbigg\n|ω−3ωm|\nωmej(ω−3ωm+3ωm)t)]\n+1\n52(−e−3π\n4QNR/radicalBigω+5ωm\nωmej(ω+5ωm−5ωm)t−e−3π\n4QNR/radicalbigg\n|ω−5ωm|\nωmej(ω−5ωm+5ωm)t)]\n= 0.742.(51)\nFig. 20. Calculated transmission losses, S21andS32.\nFig. 21. Post-layout simulations results of transmission losses, S21andS32,\nand isolation S31of the 25 GHz 45 nm SOI CMOS doubly-balanced circulator\nFig. 22. Die micrograph of the 25 GHz doubly-balanced circulator\nimplemented in the GF 45 nm SOI CMOS process.\nby probing two ports at a time, while a millimeter-wave probe\nterminated with a broadband 50 Ωtermination is landed on\nthe third port (Fig. 23).\nWith the the circulator configured for clockwise circulation\nwith 8.33GHz modulation signals, the measured transmission\nin clockwise direction, S21,S32andS13, shown in Fig. 24,\nare -3.3 dB, -3.2 dB and -8.7 dB, respectively, without any\nport impedance tuning. The measured isolations in the reverse\ndirection,S12,S23andS31, shown in Fig. 24, are -10.9 dB,\n-9 dB and -18.9 dB, respectively, without any port impedance\ntuning. The design of the circulator was optimized to give best\nperformance for S21,S32andS31because these (TX-ANT\nloss, ANT-RX loss and TX-RX isolation) are the most critical\nparameters in wireless applications.\nImportantly, the measured insertion losses are close to\nthe post-layout simulations and the analytical calculations,\nvalidating the ability of the analysis presented in this paper to\nilluminate design trade-offs and provide accurate estimates of\nperformance. The measured isolation of -18.9 dB is limited by\nthe measurement setup. For all circulators, if the ANT port is\nterminated with an imperfect impedance, the TX-RX isolation\nwill be limited to the return loss at the ANT port. Here, the\n-18.9 dB isolation is limited by the reflection coefficient of16\nFig. 23. S-parameter measurement by probing two ports at a time, with a\nbroadband 50 Ωmillimeter-wave probe termination on the third port.\nFig. 24. (a) Measured S-parameters between port 1 and port 2 when port 3 is\nterminated with a 50 Ωtermination. (b) Measured S-parameters between port\n2 and port 3 when port 1 is terminated with a 50 Ωtermination. (c) Measured\nS-parameters between port 3 and port 1 when port 2 is terminated with a 50 Ω\ntermination.\nthe millimeter-wave probe and its broadband termination. In\ngeneral, and at millimeter-wave frequencies in particular, an\nantenna tuner must be tightly integrated with the circulator to\nobtain best TX-RX isolation.\nVII. C ONCLUSION\nThis paper presented various non-reciprocal structures\nsuch as a frequency-conversion isolator, a broadband\nisolator, gyrators, a broadband circulator, and highly-linear\nsingle-ended and differential circulators, enabled by using\nswitch-based spatio-temporal conductivity modulation around\ntranmission lines. The working principle of these structures\nhas been explained using time-domain analysis. In addition,\nwe have also provided a detailed analysis for estimating the\ntransmission losses by factoring in various non-idealities\nencountered during implementation. We have shown that\nthe theoretically-estimated loss matches closely with our\nsimulation and measurement results. Such an analysis can aid\nthe designer in choosing between different implementation\ntechnologies, fabrication processes, and non-reciprocalelement topologies, and in optimizing performance by\nmaking better trade-offs.\nTopics for future research include the discovery of new\ntopologies that further improve insertion loss, isolation\nbandwidth, and linearity, the incorporation of antenna tuning\nfunctionality into the circulator architecture, and investigation\nand mitigation of the impact of clock phase noise on circulator\nperformance.\nACKNOWLEDGMENT\nThis research was sponsored by the DARPA ACT and SPAR\nprograms. 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Krishnaswamy, “A Millimeter-wave\nNon-Magnetic Passive SOI CMOS Circulator Based on Spatio-Temporal\nConductivity Modulation,” IEEE Journal of Solid-State Circuits , vol. 52,\nno. 4, pp. 3276 – 3292, Dec 2017."    },    {        "title": "1401.4930v1.Determining_polarizability_tensors_for_an_arbitrary_small_electromagnetic_scatterer.pdf",        "content": "arXiv:1401.4930v1  [physics.optics]  20 Jan 2014Determining polarizability tensors for an arbitrary\nsmall electromagnetic scatterer\nViktar S. Asadchya,b, Igar A. Faniayeub, Younes Ra’dia, Sergei A. Tretyakova\naDepartment of Radio Science and Engineering, Aalto Univers ity, FI-00076 Aalto, Finland\nbDepartment of General Physics, Gomel State University, Sov yetskaya Str. 104, 246019,\nGomel, Belarus\nAbstract\nIn this paper, we present a method to retrieve tensor polarizabilitie s of general\nbi-anisotropic particles from their far-field responses to plane-wa ve illumina-\ntions. The necessary number of probing excitations and the direct ions where\nthe scattered fields need to be calculated or measured have been f ound. When\nimplemented numerically, the method does not require any spherical harmonic\nexpansion nor direct calculation of dipole moments, but only calculatio ns of co-\nand cross-polarized scattering cross sections for a number of pla ne-wave exci-\ntations. With this simple approach, the polarizabilities can be found als o from\nexperimentally measured cross sections. The method is exemplified c onsidering\ntwo bi-anisotropic particles, a reciprocal omega particle and a non- reciprocal\nparticle containing a ferrite inclusion coupled to metal strips.\nKeywords: Bi-anisotropic particle, Polarizability, Dipole moments, Scattered\nfields\n1. Introduction\nArtificial materials (metamaterials) made of small inclusions (meta-a toms)\npositioned beside each other have become very popular, because w ith this con-\ncept it is possible to realize exotic electromagnetic properties which a re not\nfound in natural materials. These inclusions, so-called meta-atoms , which are\nelectrically small (in comparison to the wavelength in the medium), in th e sense\nof their electromagnetic response play the same role as atoms do in n atural\nmaterials. The averaged electric and magnetic properties of a meta material\nsample are determined by electric and magnetic properties of individu al inclu-\nsions and by their mutual interactions. The meta-atoms can be cha racterizedby\ntheir polarizabilities. The polarizabilities show how a single meta-atom be haves\nin responding to external electromagnetic fields. Knowing the elect romagnetic\n∗Corresponding author\nEmail address: viktar.asadchy@aalto.fi (Viktar S. Asadchy)\nPreprint submitted to Photonics and Nanostructures Januar y 21, 2014properties of each building block of the metamaterial allows us to und erstand\nthe electromagnetic properties of metamaterials as composite med ia. In partic-\nular, proper engineering of meta-atoms allows us to design metamat erials with\nrequired effective properties. There are several approaches to determination of\neffective parameters of a medium knowing the electric and magnetic m oments\nof each building block of the medium, both for volumetric (bulk) sample s and\nthin layers (metasurfaces), e.g. [1, 2, 3, 4, 5, 6]. In this paper, we discuss how\nthe meta-atom polarizabilities can be retrieved from the knowledge o f single\nmeta-atom far-field scattering response to plane-wave excitatio ns.\nThe spherical harmonic expansion theory introduced by Mie for a ho moge-\nneous sphere of any size and arbitrary refractive index is known as one of the\nmain tools in deriving the polarizabilities of a single dielectric sphere [7, 8, 9].\nLater, this theorywas extended to particles ofan arbitraryshap e [10]. Recently,\nthe extended theory has been used by many researchers to stud y the multipolar\nbehavior of special inclusions, e.g. [11, 12]. Using the generalized Mie theory\nand writing the scattered fields in terms of vector spherical harmo nics, multipo-\nlar moments of an arbitraryscatterer can be calculated. However , this approach\nimplies computationally heavy integrations of scattered fields over t he sphere\nsurrounding the particle that complicates the implementation of the method in\nnumerical calculations. Furthermore, it appears problematic to us e such meth-\nods for extracting polarizabilities from experimentally measured res ponse of the\nparticle.\nIn most cases when the particle is electrically small, electric and magne tic\ndipolar moments are the only significant and important moments in the Mie\nexpansion. This assumption allows us to dramatically simplify the scatt ering-\nbased polarizability retrieval and propose a much simpler method for extracting\npolarizability tensors of an arbitrary small scatterer from its resp onse in the\nfar zone. To extract one specific polarizability component of the sc atterer, our\nmethod implies determination of the scattered fields only in two specia l direc-\ntions. This significantly simplifies the realization of the method in numer ical\ncalculations. Furthermore, the discrete and minimal number of dire ctions in\nwhich the scattered fields must be probed allows us to utilize the meth od also\nexperimentally. This method for the first time was proposed in [13] fo r helical\nparticles possessing bi-anisotropic electromagnetic coupling. In th e present pa-\nper, we generalize the polarizability retrieval method so that it can b e utilized\nfor arbitrary small particles with the most general bi-anisotropic p roperties.\nThe method can be considered as a generalization of the approach u sed in [14]\nfor determination of the polarizablities of small chiral particles from their co-\nand cross-polarized scattering cross sections.\nIn the most general case, assuming that the induced dipole moment s in the\nparticle depend linearly on the applied fields, the dipolar moments induc ed in\nthe particle relate to the incident fields (at the location of the partic le) by the\npolarizability tensors as:\np=αee·Einc+αem·Hinc,\nm=αme·Einc+αmm·Hinc.(1)\n2These relations hold for bi-anisotropic particles of all known classes : recipro-\ncal chiral and omega, non-reciprocal Tellegen and “moving” partic les, and any\ncombination of these [15], [16]. For a special case of anisotropic part icles with-\nout electromagnetic coupling (e.g., small dielectric spheres) the rela tions are\nsimplified taking into account that αem=αme= 0.\nThe structure of the paper is as follows. In Section 2, we formulate the\nbasic idea and derive the proper expressions for the polarizabilities o f a general\nbi-anisotropic particle (assuming the particle is electrically small). In S ection 3,\nwe implement the method for two different particles: a reciprocal om ega parti-\ncle and a non-reciprocal particle possessing moving and chiral elect romagnetic\ncouplings.\n2. Basic formulation\n2.1. Polarizabilities of a bi-anisotropic particle\nIn order to determine the polarizabilities of an arbitrary particle, we analyze\nthe far-field response of the particle to incident plane waves. We st art from\nwriting the relations for the polarizabilities of the particle in terms of t he dipole\nmomentsinduced by aset ofprobingfields. Let usfix the position oft he particle\nat the center of a Cartesian coordinate system (see Fig. 1). The p article is\nFigure 1: An arbitrary particle in the center of a Cartesian coordina te system.\nsituated in free space with the characteristicimpedance η0. As an example, here\nwewritetheformulasforthecasewhentheincidentplanewavesillumin atingthe\nparticle propagate along the z0-axis. It will be shown that the directions can be\nchosen arbitrarily. Obviously, choosing the z0-directed incident waves, one can\nfind only the components of the polarizability tensors in the x0y0-plane. The\nother components can be determined similarly using incident wavespr opagating\nalong the x0andy0axes. Taking into account that the incident plane waves\nare transverse and propagate along the z0-axis, equations (1) simplify to:\n/bracketleftbigg\np1\np2/bracketrightbigg\n=/bracketleftbigg\nα11\neeα12\nee\nα21\neeα22\nee/bracketrightbigg\n·/bracketleftbigg\nEinc1\nEinc2/bracketrightbigg\n+/bracketleftbigg\nα11\nemα12\nem\nα21\nemα22\nem/bracketrightbigg\n·/bracketleftbigg\nHinc1\nHinc2/bracketrightbigg\n,\n/bracketleftbiggm1\nm2/bracketrightbigg\n=/bracketleftbiggα11\nmeα12\nme\nα21\nmeα22\nme/bracketrightbigg\n·/bracketleftbiggEinc1\nEinc2/bracketrightbigg\n+/bracketleftbiggα11\nmmα12\nmm\nα21\nmmα22\nmm/bracketrightbigg\n·/bracketleftbiggHinc1\nHinc2/bracketrightbigg\n.(2)\n3Hereafter we use numerical indices 1 ,2,3,representing the x0,y0andz0projec-\ntions, respectively. It can be seen from (2) that to determine any polarizability\ncomponent, it is insufficient to know the response of the particle to o nly one\nincident wave. The simplest way to find the component is illumination of t he\nparticle by two incident plane waves with the following polarization stat es:\nEinc=η0H0x0,Hinc=±H0y0, (3)\nin which the upper and lower signs correspond to waves propagating in +z0and\n−z0directions, respectively, and H0is the magnitude of the incident magnetic\nfield. Here, for simplicity, we assume that the two incident waves hav e equal\namplitudes and phases at the location of the particle. Although in pra ctice it is\ndifficult to generate two incident waves with precisely equal phases, the problem\ncan be solved similarly with the assumption that the waves in (3) have n ot\nonly different propagation directions, but also different amplitudes a nd phases,\nmeaning that this assumption is not limiting. Substituting (3) in (2), we get 8\nequations:\np±\n1=α11\neeηH0±α12\nemH0, p±\n2=α21\neeηH0±α22\nemH0,\nm±\n1=α11\nmeηH0±α12\nmmH0, m±\n2=α21\nmeηH0±α22\nmmH0,(4)\nwhere the double signs correspond to the double signs in (3). Next, the simple\nsolution of the equations with regard to the polarizability component s reads:\nα11\nee=1\n2ηH0(p+\n1+p−\n1), α12\nem=1\n2H0(p+\n1−p−\n1),\nα21\nee=1\n2ηH0(p+\n2+p−\n2), α22\nem=1\n2H0(p+\n2−p−\n2),\nα11\nme=1\n2ηH0(m+\n1+m−\n1), α12\nmm=1\n2H0(m+\n1−m−\n1),\nα21\nme=1\n2ηH0(m+\n2+m−\n2), α22\nmm=1\n2H0(m+\n2−m−\n2).(5)\nIn order to derive the other 8 polarizability components in the x0y0-plane, we\nchoose the incidence in the form:\nEinc=η0H0y0,Hinc=±H0x0. (6)\nLikewise, for these two different incident waves, we can write 8 equa tions for\nthe polarizabilities according to (2):\n¯p±\n1=α12\neeηH0±α11\nemH0,¯p±\n2=α22\neeηH0±α21\nemH0,\n¯m±\n1=α12\nmeηH0±α11\nmmH0,¯m±\n2=α22\nmeηH0±α21\nmmH0,(7)\n4where we use notations with bars in order to distinguish the induced d ipole\nmoments for different polarization states in (3) and (6). The double sign in (7)\ncorresponds to the double sign in (6). Similarly, we can derive expres sions for\nthe polarizability components:\nα12\nee=1\n2ηH0(¯p+\n1+ ¯p−\n1), α11\nem=1\n2H0(¯p+\n1−¯p−\n1),\nα22\nee=1\n2ηH0(¯p+\n2+ ¯p−\n2), α21\nem=1\n2H0(¯p+\n2−¯p−\n2),\nα12\nme=1\n2ηH0(¯m+\n1+ ¯m−\n1), α11\nmm=1\n2H0(¯m+\n1−¯m−\n1),\nα22\nme=1\n2ηH0(¯m+\n2+ ¯m−\n2), α21\nmm=1\n2H0(¯m+\n2−¯m−\n2).(8)\nThus, we have determined 16 polarizability components of the partic le in terms\nof the induced dipole moments by probing plane waves. The other 20 c om-\nponents one can derive in the same way illuminating the particle by wave s\npropagating along the x0andy0axes. In the next section, we determine the\ninduced dipole moments in the particle from the far-zone scattered fields.\n2.2. Induced electric and magnetic dipole moments\nAscattererwithinducedoscillatingelectricandmagneticmultipolesra diates\nenergy in all directions. Here, we study the case of an electrically sm all particle\n(the size of the particle is small compared to the wavelength of the in cident\nwaves) that allows us to take into account only the lowest multipoles, i.e. the\nelectric and magnetic dipoles. The scattered far fields from an elect rically small\nparticle are defined by the induced dipole moments in the form [17]:\nEsc=k2\n4πǫ0re−jkr/bracketleftbigg\n(n×p)×n−1\ncµ0n×m/bracketrightbigg\n,\nHsc=1\nη0n×Esc,(9)\nwherenis the unit vector in the direction of observation, ris the distance\nbetween the particle and the observation point, k=ω/cis the wave number\nin surrounding space and the time-dependence ejωtis understood. Since it is\nrequired to find only the x0andy0projections of the electric and magnetic\ndipole moments (according to (5) and (8)), we choose the observa tion direction\nto be along + z0and−z0(however, this choice is not compulsory). Taking this\ninto account, we can rewrite the scattered electric field in (9) as:\nzEsc=γ/bracketleftbigg\n(p1+1\nη0m2)x0+(p2−1\nη0m1)y0/bracketrightbigg\n,\n−zEsc=γ/bracketleftbigg\n(p1−1\nη0m2)x0+(p2+1\nη0m1)y0/bracketrightbigg\n,(10)\n5whereγ=k2\n4πǫ0re−jkris a parameter introduced for convenience. Although\nthese formulas are for the case of the incidence (3), similar formula s can be\nwritten also (with notations in bars) for the incidence defined in (6). Combining\nequations (10), we find formulas for calculation of electric and magn etic dipole\nmoments:\np±\n1=1\n2γ(zE±\nsc1+−zE±\nsc1), p±\n2=1\n2γ(zE±\nsc2+−zE±\nsc2),\nm±\n1=η0\n2γ(−zE±\nsc2−zE±\nsc2), m±\n2=η0\n2γ(zE±\nsc1−−zE±\nsc1).(11)\nAt this step, we are ready to write general formulas for calculating all the\npolarizability components in the x0y0-plane. First, we consider the case when\nthe incident fields equal Einc=η0H0x0,Hinc=±H0y0. Then, substituting\n(11) in (5), we write the expressions for the polarizability componen ts:\nα11\nee=1\n4γηH0/bracketleftbig\nzE+\nsc1+−zE+\nsc1+zE−\nsc1+−zE−\nsc1/bracketrightbig\n,\nα12\nem=1\n4γH0/bracketleftbig\nzE+\nsc1+−zE+\nsc1−zE−\nsc1−−zE−\nsc1/bracketrightbig\n,\nα21\nee=1\n4γηH0/bracketleftbig\nzE+\nsc2+−zE+\nsc2+zE−\nsc2+−zE−\nsc2/bracketrightbig\n,\nα22\nem=1\n4γH0/bracketleftbig\nzE+\nsc2+−zE+\nsc2−zE−\nsc2−−zE−\nsc2/bracketrightbig\n,\nα11\nme=1\n4γH0/bracketleftbig\n−zE+\nsc2−zE+\nsc2+−zE−\nsc2−zE−\nsc2/bracketrightbig\n,\nα12\nmm=η0\n4γH0/bracketleftbig\n−zE+\nsc2−zE+\nsc2−−zE−\nsc2+zE−\nsc2/bracketrightbig\n,\nα21\nme=1\n4γH0/bracketleftbig\nzE+\nsc1−−zE+\nsc1+zE−\nsc1−−zE−\nsc1/bracketrightbig\n,\nα22\nmm=η0\n4γH0/bracketleftbig\nzE+\nsc1−−zE+\nsc1−zE−\nsc1+−zE−\nsc1/bracketrightbig\n.(12)\nTo clarify the notation here we can use an example. −zE+\nsc1denotes the x0\nprojection of the scattered electric field in the −z0direction if the scatterer is\nilluminated by the incident wave Einc=η0H0x0,Hinc= +H0y0.\nNext, we study the case when the incident fields are defined as Einc=\nη0H0y0,Hinc=±H0x0. Likewise, the expressions for the other 8 polarizability\n6components can be found:\nα12\nee=1\n4γηH0/bracketleftbig\nz¯E+\nsc1+−z¯E+\nsc1+z¯E−\nsc1+−z¯E−\nsc1/bracketrightbig\n,\nα11\nem=1\n4γH0/bracketleftbig\nz¯E+\nsc1+−z¯E+\nsc1−z¯E−\nsc1−−z¯E−\nsc1/bracketrightbig\n,\nα22\nee=1\n4γηH0/bracketleftbig\nz¯E+\nsc2+−z¯E+\nsc2+z¯E−\nsc2+−z¯E−\nsc2/bracketrightbig\n,\nα21\nem=1\n4γH0/bracketleftbig\nz¯E+\nsc2+−z¯E+\nsc2−z¯E−\nsc2−−z¯E−\nsc2/bracketrightbig\n,\nα12\nme=1\n4γH0/bracketleftbig\n−z¯E+\nsc2−z¯E+\nsc2+−z¯E−\nsc2−z¯E−\nsc2/bracketrightbig\n,\nα11\nmm=η0\n4γH0/bracketleftbig\n−z¯E+\nsc2−z¯E+\nsc2−−z¯E−\nsc2+z¯E−\nsc2/bracketrightbig\n,\nα22\nme=1\n4γH0/bracketleftbig\nz¯E+\nsc1−−z¯E+\nsc1+z¯E−\nsc1−−z¯E−\nsc1/bracketrightbig\n,\nα21\nmm=η0\n4γH0/bracketleftbig\nz¯E+\nsc1−−z¯E+\nsc1−z¯E−\nsc1+−z¯E−\nsc1/bracketrightbig\n.(13)\nFrom (12) and (13) one can see that to extractone specific polariz abilitycompo-\nnent of the particle by this method, we need to probe (or measure) the scattered\nfields only in two directions (at any arbitrary point in far-field). In or der to find\nall 16 polarizability components, it is sufficient to know the scattered fields in\ntwo directions and to use only four different plane-wave illuminations. In the\nnext section, we show an example of implementation of this method fo r two\ndifferent particles, one reciprocal and the other one non-recipro cal.\n3. Polarizability retrieval applied to reciprocal and non- reciprocal bi-\nanisotropic particles\nHere,weutilizethemethodforextractingpolarizabilitiesoftwobi-an isotropic\nparticlesthat havebeen previouslyused as building blocks for metas urfacespos-\nsessing novel and exotic electromagnetic properties [18, 19]. In t his paper we\ndescribe the method of extracting the polarizabilities in detail. The an alyzed\nparticles are electrically small, therefore, the present method can be applied.\nWe determine the scattered fields by full-wave simulations using Anso ft High\nFrequency Structure Simulator. However, one can calculate the s cattered fields\nusing other approaches, e.g., based on the method of moments (Mo M) or finite\nelement method (FEM), or measure the far-fields experimentally.\nThe first example, which we consider here, is a reciprocal omega par ticle [15]\nshown in Fig. 2a.\n7(a)\n (b)\nFigure 2: (a) The omega particle and the related coordinate system . (b) The\nnon-reciprocal particle.\nIt was previously shown that this kind of inclusions can be utilized in thin\ncompositemetamirrors(full-reflection layers)which allowfull contr oloverphase\nof reflection [18]. The dimensions of the particle under study are as f ollows:\nThe radius of the loop is r= 7.45 mm, the half-length of the electric dipole is\nd= 18.1 mm, the radius of the wire is r0= 0.5 mm, and the pitch is 1 .45 mm.\nThe material of the particle is PEC. In the defined coordinate syste m only\nfour polarizability components of the particle are significant [15]: α11\nee,α22\nmm,\nα12\nem, andα21\nme. As it is seen from (5), we can find all these components using\ntwo incident waves with the polarization states defined by (3). In or der to\ndetermine the electric and magnetic dipolar moments in (5), we probe the fields\nscattered by the particle in the + z0and−z0directions, as it is dictated by\n(11). Next, using the final formulas (12), we plot the polarizability c omponents\nof the particle versus frequency (see Fig. 3).\n1.38 1.41.42−2−101\nFrequency [GHz]η0α11\nee[10−12m2s]\n  Real\nImaginary\n1.38 1.41.42−2−101\nFrequency [GHz]1/η0α22\nmm[10−12m2s]\n1.38 1.41.42−2−101\nFrequency [GHz]α12\nem[10−12m2s]\n1.38 1.41.42−1012\nFrequency [GHz]α21\nme[10−12m2s]\nFigure 3: Polarizabilities of the omega particle, normalized to the free -space\nimpedance.\nAs it is seen, at the resonance frequency of the particle electric an d mag-\n8netic polarizabilities become purely imaginary while the electromagnetic polar-\nizabilities are real. It was expected, since all the reactances are co mpensated\nand only dissipative terms remain for the particle at resonance. The condi-\ntionα12\nem=−α21\nmeholds for the particle, as it must be in accordance with the\nOnsager-Casimir principle [20]. Also it can be seen from Fig. 3 that the e lectric\nand magnetic polarizabilities satisfy the balance condition η0α11\nee=α22\nmm/η0.\nThis corresponds to the case of extreme response of balanced bi- anisotropic\nparticles [21], and it was the design requirement in [18].\nAs another example, we analyze a non-reciprocal particle possess ing moving\nand chiral electromagnetic couplings (see Fig. 2b). A planar array o f these par-\nticles acts as a non-reciprocal one-way transparent ultimately th in layer [19].\nThe layer is transparent from one side while from the opposite side th e layer\nacts as a twist-polarizer in transmission. A ferrite sphere magnetiz ed by exter-\nnal bias field is the non-reciprocal element in the particle. The ferrit e sphere\nwith the radius a= 1.65 mm is coupled to metal elements with the dimensions\nl= 18 mm and l′= 3 mm. The radius of the copper wire is δ= 0.05 mm. The\nferrite material is yttrium iron garnet: The relative permittivity ǫr= 15, the di-\nelectric loss tangent tan δ= 10−4, the saturation magnetization MS= 1780 G,\nand the full resonance linewidth ∆ H= 0.2 Oe. The + z0internal bias field is\nHb= 9626 A/m, corresponding to the desired resonance frequency. In the de-\nfined coordinate system, significant polarizabilities of the particle ar e those in\nthex0y0-plane. In the same way as for the omega particle, we find the norma l-\nized polarizabilities for moving-chiral particle (see Fig. 4).\n1.962 1.964 1.966 1.968−202\nFrequency [GHz]η0α11\nee[10−13m2s]\n1.962 1.964 1.966 1.968−202\nFrequency [GHz]1\nη0α22mm[10−13m2s]\n  real\nImaginary\n1.962 1.964 1.966 1.968−202\nFrequency [GHz]η0α21\nee[10−13m2s]\n1.962 1.964 1.966 1.968−202\nFrequency [GHz]1\nη0α12mm[10−13m2s]\n1.962 1.964 1.966 1.968−202\nFrequency [GHz]α12\nem[10−13m2s]\n1.962 1.964 1.966 1.968−202\nFrequency [GHz]α21\nme[10−13m2s]\n1.962 1.964 1.966 1.968−202\nFrequency [GHz]α22em[10−13m2s]\n1.962 1.964 1.966 1.968−202\nFrequency [GHz]α11me[10−13m2s]\nFigure 4: Polarizabilities of the non-reciprocal particle.\n9One can see from Fig. 4 that the Onsager symmetry relations α22\nem=−α11\nme\nandα12\nem=α21\nmehold for the particle.\n4. Conclusions\nHere, we have presented a method which allows us to find all polarizab il-\nity tensor components for an electrically small arbitrary bi-anisotr opic particle\nwith any complex shape and internal structure. We have assumed t hat only\ndipolar moments are significant in the particle, that is, that the part icle is elec-\ntrically small. In comparison to other known methods, this method re quires\nless complicated calculations. To determine one specific polarizability c ompo-\nnent of the particle, the method requires probing of the particle re sponse only\nby plane waves in two directions. For determining all the 36 tensor co mponents\nit is sufficient to find the scattered fields only in 6 directions for 12 diffe rent\nincidences. The scattering response is measured only in the far zon e and only\nin a few directions (6 for the most general particle). Due to simplicity of the\nmethod, the method can be utilized also experimentally. In order to d emon-\nstrate and illustrate the concept, we have derived formulas for 16 polarizability\ncomponents. One can similarly derive the formulas for the other 20 c ompo-\nnents. In the paper, the polarizability retrieval method has been a pplied for\ntwo specific bi-anisotropic particles, to give particular examples. Th e method\ncan be used to determine and optimize single inclusions in metamaterials and\nmetasurfaces with the goal to achieve desired electromagnetic pr operties of the\nwhole structure.\nReferences\nReferences\n[1] L. Lewin, The electrical constants of a material loaded with sphe rical par-\nticles, Radio and Communication Engineering 94 (1947) 65–68.\n[2] A. Sihvola, Electromagnetic Mixing Formulas and Applications, 1st e d.,\nLondon, IEEE Publishing, 1999.\n[3] S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Nor-\nwood, Artech House Publishers, 2003.\n[4] E. F. Kuester, M. A. Mohamed, C. L. Holloway, Averaged transit ion con-\nditions for electromagnetic field at a metafilm, IEEE Transactions on An-\ntennas Propagation 51 (2003) 2641-2651.\n[5] Y. Zhao, N. Engheta, A. Al` u, Homogenization of plasmonic meta surfaces\nmodeled as transmission-line loads, Metamaterials 5(2011) 90-96.\n[6] T. Niemi, A. Karilainen, S. Tretyakov, Synthesis of polarization tr ansform-\ners, IEEE Transactions on Antennas Propagation 61 (2013) 3102 –3111.\n10[7] G.Mie, Beitr¨ agezuroptiktr¨ ubermedienspeziellkolloidalermeta ll¨ osungen,\nAnnalen der Physik 330 (1908) 377–445.\n[8] L. V. Lorenz, Sur la lumi` ere r´ efl´ echie et r´ efract´ ee par un e sph` ere transpar-\nente, Oeuvres Scientifiques, Librairie Lehman et Stage, Copenhag en (1898)\n405–529.\n[9] C. F. Bohren, D. R. Huffman, Absoption and Scattering of Light b y Small\nParticles, 1st ed., New York, Wiley, 1983.\n[10] P. C.Waterman, Symmetry, unitary, andgeometryin electrom agneticscat-\ntering, Physical Review D 3 (1971) 825–839.\n[11] S. M¨ uhlig, C.Menzel, C.Rockstuhl, F. Lederer,Multipole analys isofmeta-\natoms, Metamaterials 5 (2010) 64–73.\n[12] F. B. Arango, A. F. Koenderink, Polarizabilitytensorretrieval for magnetic\nand plasmonic antenna design, New Journal of Physics 15 (2013) 07 3023.\n[13] V. S. Asadchy, I. A. Faniayeu, Simulation of the electromagnet ic properties\nof one-turn and double-turn helices with optimal shape, which prov ides\nradiation of a circularly polarized wave, Journal of Advanced Resea rch in\nPhysics 2 (2011) 011107.\n[14] S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, J.- P. Heliot,\nAnalytical antenna model for chiral scatterers: Comparison with numerical\nand experimental data, IEEE Transactions on Antennas and Prop agation\n44 (1996) 1006–1014.\n[15] A.N.Serdyukov,I.V.Semchenko,S.A.Tretyakov,A.Sihvo la,Electromag-\nnetics of Bi-Anisotropic Materials: Theory and Applications, Amster dam,\nGordon and Breach Science Publishers, 2001.\n[16] S. A. Tretyakov, A. H. Sihvola, A. A. Sochava, C. R. Simovski, M agne-\ntoelectric interactions in bi-anisotropic media, Journal of Electrom agnetic\nWaves and Applications 12 (1998) 481–497.\n[17] J. D. Jackson, Radiating Systems, Multipole Fields and Radiation, 3rd ed.,\nNew York, Wiley (1999) 407–455.\n[18] Y. Ra’di, V. S. Asadchy, S. A. Tretyakov, Tailoring reflections f rom thin\ncomposite metamirrors (2013) arXiv:1401.1677.\n[19] Y. Ra’di, V. S. Asadchy, S. A. Tretyakov, One-way transpare nt sheets\n(2013) arXiv:1310.4586.\n[20] L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Me dia, 2nd\ned., Oxford, England, Pergamon Press, 1984.\n[21] Y. Ra’di, S. A. Tretyakov, Balanced and optimal bi-anisotropic p articles:\nMaximizing power extracted from electromagnetic fields, New Journ al of\nPhysics 15 (2013) 053008.\n11"    },    {        "title": "2105.08931v1.Magnetic_structure_and_multiferroicity_of_Sc_substituted_hexagonal_YbFeO__3_.pdf",        "content": "1 \n  \n \nMagnetic structure and multiferroicity of Sc-substituted hexagonal YbFeO 3 \n \nY . S. Tang,1 S. M. Wang,1, L. Lin,1,* V . Ovidiu  Garlea ,2 Tao Zou,3,† S. H. Zheng,1 H.-M. \nZhang,4 J. T. Zhou,5 Z. L. Luo,5 Z. B. Yan,1 S. Dong,4,‡ T. Charlton,2 and J. -M. Liu1 \n1Laboratory of Solid State Microstructures , Nanjing University, Nanjing 210093, China  \n2Neutron Science  Division, Oak Ridge National Laboratory, Tennessee 37831, USA  \n3Collaborative Innovation Center of Light Manipulations and Applications, Shangdong \nNormal University, Jinan 250358, China  \n4School of Physics, Southeast University, Nanjing 211189, China  \n5National Synchrotron Radiation Laboratory, University of Science and Technology of China,  \nHefei 230026, China  \n \n  \n                                                 \n* llin@nju.edu.cn  \n† taozoucn@gmail.com  \n‡ sdong@seu.edu.cn  2 \n Abstract  \nHexagonal rare -earth ferrite RFeO 3 family represents  a unique class of multiferroic s exhibiting \nweak ferromagneti sm, and a strong coupling between magnetism and structural trimerization  is \npredicted . However, the hexagonal structure for RFeO 3 remains metastable in conventional \ncondition. We have  succeeded in stabiliz ing the hexagonal structure of polycrystalline YbFeO 3 \nby partial Sc substitution  of Yb . Using bulk magnetometry and neutron diffraction , we find that  \nYb0.42Sc0.58FeO 3 orders into  a canted antiferromagnetic state with the N éel temperature TN ~ \n165 K, below which t he Fe3+ moments form the triangular configuration in th e ab-plane and \ntheir in-plane projection s are parallel to the [100] axis , consistent with magnetic space group \nP63c'm'. It is determined that t he spin-canting is align ed along the c-axis, giving rise to the weak \nferromagnetism.  Furthermore, the Fe3+ moments reorient toward a new direction below \nreorientation  temp erature TR ~ 40 K, satisfying magnetic subgroup P63, while the Yb3+ moments \norder  independently and ferrimagnetically alo ng the c-axis at the characteristic temperature TYb \n~ 15 K . Interestingly , reproducible modulation of electric polarization induced by magnetic \nfield at low temperature is achieved,  suggesting that the delicate  structural distortion  associated \nwith two-up/one -down buckling of the Yb/Sc -planes and tilting  of the FeO5 bipyramids  may \nmediate the coupling between ferroelectric and magnetic orders under magnetic field . The \npresent work  represents a substantial progress to search  for high-temperature multiferroics in \nhexagonal ferrites and related materials.   3 \n I. Introduction  \nMultiferroics, where the ferroelectric and magnetic order s are coupled  with each other , \nhave stimulated enormous attention not  only because of a plenty of manifested exotic physical \nphenomen a but also promising functionalities for novel applications [1 -5]. However, most \nknown multiferroics do not show ferroic order ing until extremely low temperature ( T), which \nbecomes a serious barrier for applications. As such, searching for high -T multiferr oics is one of \nthe most vital tasks  in the magnetoelectric (ME) community, but in a long period only rare \nsystems can be qualified [6,  7]. \nRecently, it has been shown that hexagonal  ferrite  family  h-RFeO 3 (R = rare earth ions, Y) , \nthe so -called improper ferroelectrics , provides another approach to achiev ing high-T \nmultiferroicity [8-15], because of the stronger Fe3+ spin exchanges in RFeO 3 compared to  the \nMn3+ spin exchanges in RMnO 3. In this family , h-RFeO 3 is ferroelectric at room tempera ture, \nand ferroelectric (FE) polarization ( P) is generally introduced by the structural distortion in \nassociation with transition from high -T non-polar P63/mmc  phase  to low-T polar P63cm phase , \ne.g. the FE transition occurs  around  1000 K for h-LuFeO 3 film [11-12]. The cooperative tilting \nof FeO5 bipyramids and buckling of R layers lead to net electric polarization  [13]. This finding \nseems to make the h-RFeO 3 family another set of multiferroics in addition to well -investigated \nRMnO 3 family. Nevertheless, different from RMnO 3 that favors orthorhombic structure as R \nion is relatively large and hexagonal structure as R ion is small, free-standing RFeO 3 family all \nfavors orthorhombic structure (o-RFeO 3), as shown in Fig.1(a).  \nAlternatively , next to the stable o-phase , hexagonal structure (h-phase) as the metastable \none, as shown in Fig.1(b), can be obtained in thin film form if proper clamping from hexagonal \nsubstrate can be realized  [11-15]. This allows an exploration of multiferroic behaviors of \nmetastable h-RFeO 3. In fact, earlier work on h-LuFeO 3 films did receive attention , and the film \nsample first exhibited  an antiferromagnetic (AFM) transition at T = TN1 ~ 440 K . Subsequently, \na spin reorientation at T = TR = 130 K appear ed [11], and later neutron diffraction experiments \nsuggested a ferromagnetically canted AFM  orderi ng at the Néel point  TN ~ 155 K regardless of  \nthe film thickness [ 12]. This discrepancy might originate from the limited neutron diffraction \ndata for determin ing the magnetic structure of film samples , appealing for more experiment on \nbulk systems . Fortunately, Masuno et al. found that the bulk h-phase could be stabilized upon \nSc doping into Lu site  for LuFeO 3, and consequently , TN was significantly enhanced, e.g. from \n120 K in h-LuFeO 3 thin film  [16] to ~172 K in h-Lu0.5Sc0.5FeO 3 bulk [17, 18], while the \nferroelectricity remained robustly unaffected according to the first -principles calculations [18]. 4 \n Most recently, a record ed TN ~185 K was achieved by structural distortion strategy on some \nhexagonal ferrites [19]. In addition , large ME coupling or linear ME vortex structure in \nhexagonal ferrites  was theoretically predicted, but not yet experimentally confirmed  [20]. Even \nthough the cloverleaf FE-vortex domains and large -loop weak ferromagnetic domains were \nobserved in single crystal (Lu, Sc)FeO 3, no coupled ME effect between the FE domains and \nweak ferromagnetic  domain s has been demonstrated  [21].  \nFrom the above statement , one can find that  the h-RFeO 3 demonstrates more transitions \nthan h-RMnO3 in the magnetism and polarization due to the stronger Dzyaloshinskii -Moriya \ninteraction between Fe3+ ions under the formation of long -range magnetic ordering. It is worth \nnoting  that the h-RFeO 3 has a trimer distortion lattice  at ferroelectric phase ( P63cm) similar to \nthe h-RMnO3 features , while the particular Fe -Fe interaction is expected to instrinsically \ninfluence on trimerization by the spin -lattice coupling. Such as, earlier experiments on h-\nYbFeO 3 thin films reveal  two-step FE transitions occurring at T ~ 470 K and ~ 225 K \nrespectively, accompanied by pronounced magneto -dielectric effect near T ~ 225 K , while these \nproperties are not available in h-RMnO3. In addition, so far the 4f-3d coupling in h-RFeO 3, if \nR3+ is strongly magnetic, has not been paid sufficient attention for understanding the \nmultiferroic behaviors, while this coupling is non-negligible or even very important for the \nmultiferroicity in RMnO 3. Furthermore, this coupling can be remarkable far above the \nindependent 4 f spin ordering point (usually < 10 K), making substantial contribution to the ME \ncoupling at relatively high temperature [22 -27]. More interestingly,  the accordant steplike \nanom aly in magnetization and polarization at critical field in YbMnO 3 indicates strong \nassociation  with a change in the nature of the field-induced spin reorientation  below the Yb3+ \nordering  [28]. All this inspires us  to fu rther study the magnetic structure and multiferroicity  in \nh-RFeO 3 (R3+ is magnetic ions, such as Yb3+), which  are expec ted as a nontrivial candidate for \nhigh-T ME coupling and emergent physics beyond h-LuFeO 3 [13].  \nEarlier works on  h-YbFeO 3 thin films did show quite a few inconsistencies regard ing the \nFE transitions and ME coupling , and we highlight  them  here. First, are there indeed FE phase \ntransitions occurring at T ~ 470 K and ~ 225 K ? Second, is there any direct evidence with \nmagnetic anomaly associated with the se FE transitions?  Third, how about the magnetism  and \nME effect in this system?  While quite large magnetization at low T was reported  in Ref. [13], \ne.g., residual magnetization Mr > 2 B/f.u. at T = 3 K and ~ 0.7 B/f.u. at T = 20 K, and similar \ndata were reported in Ref. [29 ], only very small value Mr ~ 0.06  B/f.u. at T = 18 K was probed \nin Ref. [30 ]. Indeed, it is challenging to detect the intrinsic magnetism of thin film samples  for 5 \n ferrites not only due to magnetic signals from Fe 2O3-like impurities , if any,  but also \ncontributions from substrates , and thus a n experimental study on bulk h-YbFeO 3 becomes \nurgently needed  not only for clarifying these issues .  \nIn this work, we start from synthesis of high-quality bulk h-YbFeO 3 polycrystalline \nsamples by adopting substitution of Sc onto the Yb site . Using magnetic susceptibility and \nneutron powder diffraction measurements, we unveil convincingly three successive magnetic \nphase transitions at T = TN ~ 165 K , T= TR ~ 40 K , and T = TYb ~ 15 K, respectively . At TN ~ \n165 K, the system enters a canted AFM  state with Fe3+ moments  lying in the ab-plane  and a \nsmall canting moment along the c-axis giv ing rise to the weak ferromagnetism. Subsequently, \nthe Fe3+ spin projection in the basal plane reorient s to a new direction at TR ~ 40 K, followed \nby the independent ferrimagnetic Yb3+ spin ordering along the c-axis below TYb ~ 15 K. The \nobserved  of electric polarization and dielectric anomaly at TN provides clear evidence for \nsimultaneous magnetic and ferroelectric transitions, which is believed to attribute to the spin -\nlattice coupling [31 ]. In addition, remarkable ME effect in the low -T range has been confirmed  \namong t he hexagonal ferrite family .  \n \nII. Experimental details  \nThe p olycrystalli ne h-Yb0.42Sc0.58FeO 3 (h-YSFO) sample s were  synthesized using \nconventional solid -state reaction [9]. Stoichiometric mixtures of high purity Yb 2O3, Sc 2O3, and \nFe2O3 were thoroughly ground, and sintered at 1200  oC for 24 hours. Then the powder was \npelletized and sintered at 1400  oC for 24 hours with intermittent grindings.  \nThe crystallinity of the as -prepared sample s was checked using the X -ray diffraction (XRD, \nD8 Advanced, Bruker) in th e  -2 mode with Cu K  source ( λ = 1.5406 Å) at room temperature. \nTo monitor the  T-dependent structural evolution, in situ  high-resolution X -ray diffraction was \nperformed at the 14B beamline of Shanghai synchrotron radiation facility (SSRF) with a \nLinkam cryo -stage. Neutron pow der diffraction (NPD) data were collected at HB -2A powder \ndiffractometer at Oak Ridge National Laboratory , USA . The wavelength of λ = 1.54 Å was used \nat room temperature for the refinement of nuclear structure, while λ = 2.41 Å was used at low \ntemperatures  for the refinement of both nuclear and magnetic orderings. The refinement was \ncarried out using the Rietveld a nalysis program package FUL LPROF [32 ]. The symmetry \nallowed magnetic models were analyzed using the software package SARAh [33 ]. To check the \nsamples’ stoichiometry, the electron dispersion spectroscopy (EDS) (Quanta 200, FEI) was used 6 \n to analyze the chemical composition , noting that the Fe3+ valence in ( Lu1-xScx)FeO 3 is highly \npreferred  [18].  \nTo measure dielectric and ferroelectric properties, a sandwich -type capacitor was made by \ndepositing Au on the top/bottom surfaces of the disk -like sample  with 3.0 mm  in diameter and \n~ 0.2 mm  in thickness . The T-dependence of dielectric constant ( ) was measured using the \nHP4294A impedance analyzer (Agilent Technologies, Inc.) integrated with Physical Property \nMeasurement System (PPMS) (Quantum Design, Inc.). The specific heat ( CP) was measured  \nusing the PPMS in the standard procedure. The magnetic susceptibility ( ) as a function of T \nunder the zero -field co oling (ZFC) and field cooling (FC) modes was measured by the \nSuperconducting Quantum Interference Device Magnetometer (SQUID) (Quantum Design, \nInc.), using the measuring/FC field of 1 .0 kOe. The isothermal magnetic hysteresis \n(magnetization M vs magnetic f ield H) was also probed in the standard mode .  \nThe conventional pyroe lectric current method was used  to probe the ferroelectric \npolarizatio n [34, 35] . In detail, the sample was cooled down from 180 K to 10 K under a poling \nelectric field ( Ep = 10 kV/cm). Subsequently, the electric field was removed, followed by \nsufficiently long short -circuit  to reduce the background of electrical current with minimum level \n(i.e., <  0.2 pA ). The extrinsic contributions, e.g. trapped charges during the electri c field poling , \nor charges contributed from thermal activation during warming process  were carefully  \nexamined.  The T dependent pyroelectric current Ip was collected with the heating  rate of 4 \nK/min.  In addition, the  magnetic field -driven polarization  (P) was obtained by integrating the \nmagnetoelectric current IM (H), which was measured using the Keithley 6514  electrometer upon \nincreasing H from H = 0 to H =  7 T at ramping magnetic field rate 100 Oe/s  after the poling \nand the short -circuit process .  \n \nIII. Results and discussion  \nA. Crystal structure   \nFor the sample synthesis, a series of YSFO  (0.3 ≤ x ≤ 1.0) samples were  prepared in order \nto obtain pure hexagonal phase . At low x (x ≤ 0.6), the samples can be clearly indexed as the \ncombinations  of o-Yb1-xScxFeO 3, h-Yb1-xScxFeO 3, and very tiny ( Yb1-xScx)3Fe5O12 [36]. A \nfurther increasing of x makes the intensity of Bragg peaks of o-Yb1-xScxFeO 3 phase to gradually  \ndrop down, and almost disappear  in sample x = 0.6 where the h-YbFeO 3 phase is dominated, \naccompanied with tiny amount of bixbyite phase ScFeO 3. For higher x (> 0.7), t he bixbyite \nphase (see Fig. 1(c)) gradually increase s.   7 \n It is more challenging to obtain pure  h-YSFO phase  in comparison with previously \nsynthesized h-Lu1-xScxFeO 3 [17, 18]. The larger Yb3+ ions are disadvantageous to stabilize the \nhexagonal structure than Lu3+ and the ionic size mismatch between Yb3+ and Sc3+ is more \nobvious. These are non -favored for obtaining pure h-phase  and the ionic occupation disorder ing \nmust be more remarkable. According to the XRD pattern shown in Fig. 1(d), sample x = 0.58 \nhas the purest h-phase, with only very tiny bixbyite impurity , as indicated  in the inset. The tiny \nScFeO 3 as impurity can be safely neglected in our discussion since it is paramagnetic above T \n~ 20 K [ 37] and non -ferroelectric due to the cubic space group Ia-3 [38]. The high-quality data \nrefinement show s that the lattice structure of sample x = 0.58  is in good agreement w ith \nhexagonal symmetry  and the lattice constants are a = b = 5.8523 Å and c = 11.7008 Å. \nThe chemical compositions were checked using the EDS  measurement (Fig. 1(e)) and the \ndata show that the atom ic ratio Yb:Sc:Fe is 1 .0:1.182:2.270. While the sum of Yb  and Sc is \nlower (by 4%) than the Fe content , it is enough safe to conclude that the samples are \nstoichiometric considering the uncertainties of EDS technique .  \n \nB. Magnetic phase transitions   \nWe first check the magneti c phase transitions . The dc magnetic susceptibility  (T) for \nsample x = 0.58 in the T-range from 2 K and 300 K under a measuring field H ~ 1 kOe is plotted \nin Fig. 2(a). It is seen that the sample first undergoes a  weak anomaly around T = TC ~ 225 K , \nsuggesting some magnetic ordering event . The (T) curves under the ZFC and FC modes have \na weaker anomaly at TC, while the splitting emerges below  TN. Upon further cooling, a cusp \nfeature emerg ing at T = TN ~ 165 K was clearly identified , indicating another magnetic phase \ntransition. At T = TR ~ 40 K, a small bump shows up in the (T) curve, suggesting third  magnetic \nordering  event . For checking the nature of these anomalies, the specific heat divided by \ntemperature CP/T was measured as shown in Fig. 2(b) . Apparently, the AFM transition at TN is \nconfirmed by an anomaly, while n o identifiable anomaly at TC can be detected , raising question \non the long -range ordering nature of the event at TC.  \nOne may also plot the (d/dT) ~ T curve  in Fig. 2(a) , and identifies the three anomalies  at \nTC, TN, and TR. However, the feature at TC is very dim, suggesting no long -range magnetic \nordering. Considering the fact of no anomaly  at TC in the  CP(T) curve and splitting of the (T) \ncurves under  the ZFC and FC modes below TC, it is argued that m ost likely there appears \nferromagnetic ( FM)-like cluster ing just below TC and these clusters don’t merge together until \nT  TN, below which a long -range AFM ordering develops. Due to this long -range AFM 8 \n ordering, one sees a significant splitting between the ZFC and FC modes below TN, which \nreveals a weaker ferromagnetic ordering.   \nIn proceeding, we can evaluate the magnetic interactions by fitting  the (T) data under the \nZFC mode , as shown in Fig. 2(c) where the -1(T) curve  is plotted  as well . The Curie -Weiss \nfitting of the paramagnetic data between 350 K  ~ 400 K  gives a negative Curie -Weiss \ntemperature θcw ~ -185 K  and an effective moment eff  ~ 4.8 B/f.u, indicative of strong AFM  \ninteraction . Furthermore, the magnetic transition at TR ~ 40 K  identified here was also reported \nfor the h-Lu1-xScxFeO 3 single crystal  [17]. It is noted that all the h-RMnO 3 and those  h-RFeO 3 \ncontaining Sc  samples  exhibit spin  reorientation at low temperature , e.g., T = 37 K for h-\nHoMnO 3 [39], 43.5 K for h-YbMnO 3 [40], 40 K for h-LuMnO 3 [41], 35 K for h-YMn 0.9Fe0.1O3 \n[42], and 45 K for h-Lu0.5Sc0.5FeO 3 [17]. The earlier neutron scattering provided clear evidence \nfor the Mn/Fe ionic shift from the 1/3 position [ 17, 43]. Such shift does not suppress the \nrotational invariance in the Mn plane but clearly lifts the inter -plane frustration, thus leading to \nremarkable variations of the in-plane and out -of-plane exchange and magnetocry stalline \nanisotropy. The spin reorientation is believed to be related with these variations at TSR. In \ncontrast, the reported  AFM ordering point  TN of h-YbFe O3 is ~ 120 K, and a  spontaneous \nmagnetization reversal at T ~ 83 K  was claimed due to the competition between two \nmagnetocrystalline anisotropy terms associated with Fe3+ and Yb3+ moments [13].  \nThe macroscopic magnetism can be further characterized by measuring the  M-H hysteresis, \nas shown Fig. 3(a)  for several selecte d temperatures . The FM -like hysteresis is identified below \nTN, while the magnetic clusters can give rise to the weak hysteresis loops above TN until T ~ TC. \nTo better quantify the magnitude of its effects, the  specific  M-H curves near TC with magnetic \nfield between -0.2 T  to 0.2 T  are plotted in Fig. 3(b) .  We can see significant hysteresis loops \nbelow TC point, and a weak residual moment ( Mr) ~ 0.002 B/f.u. at T ~ 200 K is clearly \ndemonstrated. The evaluated residual moment  as a function of T are summarized in Fig. 3( c). \nHere it is noted that there is minor garnet impurity phase at low doping content. The garnet \nphase is  ferrimagnetic with about 1.0 B/Fe [44]. Therefore, even 1% of this phase can add 0.01 \nB/Fe to the total magnetization. Although we did not observe any magnetic impurity within \nresolution of XRD instrument, the possibility of the garnet impurity phase as the source of \nmagnetic clusters can not be excluded.  Previous first -principles calculatio ns have shown that \nthere is a small spin canting out of the xy plane that leads to a net magnetization Mz=0.02 B/Fe \nfor h-LuFeO 3 along the z-axis, displaying weak ferromagnetism  [20]. Our M-H data supports 9 \n the theoretical prediction. The measured  moment (Mr) varies from 0.01 B/f.u. at T=100 K to \n0.05 B/f.u. at TR, below which the net magnetization increases dramatically due to the \nintroduction of Yb3+ ordering. Therefore, t hese behaviors qualitatively agree with  theoretical \nprediction of weak but intrinsic ferromagnetism generated from the canted Fe3+ spins that order \nantiferromagnetically . It should be mentioned here that the weak FM hysteresis in h-YbFeO 3 \nthin films at low T was already reported [13], and the m easured residual moment is as large as \n~ 2.0 B/f.u.. Such a large moment should not be intrinsically associated with the spin -canting, \nnoting that the AFM order is highly favored by the very negative θcw ~ -185 K . In the present \nwork, our samples show much smaller residual moment  which is ~ 0.18 B/f.u. at T ~ 5 K and \n~ 0.09 B/f.u. at T ~ 20 K , comparable with the reported value of ~ 0.06 μB/f.u. at T ~ 18 K [ 30]. \nIt is thus well confirmed that the system exhibits the AFM order below TN with canted spin \nmoment, and it is also proposed that the system contains some FM -like clusters in TN < T < TC. \nThese will receive further checking by the neutron scattering investi gation to be shown below.  \n \nC. Neutron diffraction  for m agnetic structure    \nFor determining the magnetic structure, we performed the NPD measurements on the \nsamples at several selected temperatures. The neutron wavelength for probing at T = 300 K is \nλ = 1.54 Å, and that at T = 200 K, 125 K, 25 K , and 3 K is λ = 2.41 Å , noting that the temperature \nchoice was made referring to the values of TC, TN, TR etc.  \nAs shown in Fig. 4(a), all the Bragg reflections at T = 300 K can be well indexed with the \nhexagonal space group P63cm. The refined lattice constants  are a = b = 5.86054(8) Å, c = \n11.70606(22) Å, respectively, consistent with the XRD data  shown in Fig. 1 (d). The most \nimportant crystallographic information extracted from the  refinement and the related \ndiscrepancy factors are shown in Table. I. The cubic bixbyite -type ScFeO 3 (space group  Ia-3) \nwas added as the second phase for the refinement and it shows that this tiny phase occupies ~ \n3.8(3) wt%  in amount  at most .  \nNow we can discuss the magnetic structures associated with the various phases below TC, \nTN, and TR, respectively , by combining the neutron scattering data and magnetometry results. \nFirst, no magnetic peaks were detected in addition to the nuclear ones  at T ~ 200 K  and above . \nWe note that a ferromagnetic component of less than 0.1  μB is nondetectable using unpolarized \nneutrons, and therefore our  neutron  measurements cannot probe the moment increase occuring  \naround  TC ~ 225 K , where the remnant moment is smaller  than 0. 01 μB. Second, we collected \nthe NPD data at T = 125 K, 25 K , and 3 K  respectively . The data  and their  refinement  results at 10 \n the three temperatures  are plotted in  Fig. 4( b), (c), and ( d). Several issues can be clarified here \nbefore we discuss the magnetic structures at various temperatures : \n(1) No satellite reflection was observed, revealing that the magnetic unit cell is identical to \nthe nuclear (chemical) one. The magnetic propagation vector k = (0, 0, 0)  was confirmed  using  \nthe k-search function of the FullPROF package , consistent with other hexagonal manganites \nand ferrites [12].  \n(2) Symmet ry allowed  magnetic models were analyzed based on  the irreducible \nrepresentation analysis using the SARA h software  [33]. Four corresponding magnetic structure \nmodels are derived from the four irreducible representations 1, 2, 3 and 4 of symmetry \nP63cm and k = (0 0 0).  \n(3) It is noted that the only irreducible representation which gives rise to FM component \nalong the c-axis is 2, equivalent to the magnetic space group P63c'm'.  \n(4) Furthermore and even more important, we collected the intensity evolution data of the \nmagnetic peaks in the low-Q region as a function of T and the T-dependent contour map is \nplotted in Fig. 4 (e). To clearly show the transition temperatures , the peak intensities of  (101), \n(100) and (102) are also plotted as function of T respectively,  as shown  in Fig. 4( f). These data \npresent the basis for us to evaluate the magnetic structures in various T-ranges . \nWe first address the magnetic structure between  TR < T < TN. As shown in Fig. 4( e) and (f), \nit is clearly identified that (101) peak which is structurally forbidden by the P63cm space group \nsymmetry for nuclear diffraction emerges at TN = 165 K, indicating its entire magnetic origin. \nIn addition, the M-H hysteresis does show minor FM -like loop below TN, due to the spin canting \nalong the c-axis, as evidenced in Fig. 3( a), and this effect is similar to the case of h-LuFeO 3 \n[12]. Since only 2 (P63c'm') allows the existence of a ferromagnetic component along c-axis, \nit has to be included to  refine the magnetic structure. On the other hand, the irreducible \nrepresentations 1 and 3 can be safely ruled out since they must contribute to the (100) peak \nwhich however remains absent until TR. The best refinement can be reached with a nuclear \nmodel in combination with a magnetic structure characterized by 2, as shown in the Fig. 4( b). \nThe evaluated magnetic structure for Fe3+ moment is schematically drawn in Fig. 5(a). The \nrefined magnetic moments are ma = mb = 2.511 μB, mc = 0.783 μB, mtot = 2.63  μB, and the \nmagnetic R-factor is 15.9.  \nTowards T < TR ~ 40 K, another magnetic peak (100) appears by remarkable  increas e in \nintensity with decreasing T, as shown in Fig. 4( e) and ( f). Meanwhile , the intensity of (101) \npeak experiences an abrupt  drop below TR, accompanied by an obvious intensity increase of  11 \n (102). The appearance of (100) and lowering of (101) intensity can be explained either by Fe \nrotation in the plane or from the ordering of Yb moment. The model based on 2 with ordered \nYb moments can be use d to fit the data, but larger Fe3+ moments than 3 K are acquired. This \nmodel is thus discarded.  It is known that the irreducible representations 1 and 3 usually \ncontribute significantly to the (100) peak. An emergence of the peak (100) at TR simply implies \na reorientation of the Fe3+ spins toward a new direction in the basal plane characterized by either \n1 or 3. The reorientation is also captured by the anomaly around 40 K in the M-T curve. \nConsidering the observed hysteresis loop below TR, 2 has to be included for the refinement. A \nmagnetic model composed of basis vectors from two representations, 1 and 2, has given us \nthe best refinement result, as shown in Fig. 4(c). This model is equivalent to the magnetic space \ngroup  P63 (#173.129) , and the magnetic structure for Fe3+ moment is schematically drawn in \nFig. 5(b). The refined moments are ma = 3.532  μB, mb = 2.22  μB,  mc = 0.796  μB, mtot = 3.193  μB, \nand the magnetic R -factor is 16.1.  \nUpon further cooling to 15 K, the integrated intensity of (101) and (102) shows a n obvious  \nincrease, accompanied with  a minor decrease of (100) peak , as shown in Fig. 4 (f ). This feature \nis very similar to isostructural HoMnO 3, where Ho3+ magnetic moments order below T = 25.4 \nK [45]. Since rare earth Yb3+ ions are also magnetic, it is reasonable to take the ordering of Yb3+ \nmoments into consideration for the refinement at 3 K. The refinement using the same magnetic \nsymmetry at 25 K with ordered Yb3+ moments give us the best refinement results. At T ~ 3 K, \nthe Fe3+ moment components are  ma = 3.344 B, mb = 1.4 B, mc = 1.105 B, and the total \nmoment is mtot = 3.111 B, while for Yb3+ ions on site (2 a), mc = 1.802 B, Yb3+ ions on site \n(4b), mc = -1.511 B. The magnetic R -factor is 18.8.  \nWhile for (102), the intensity remarkably increases at TN and below due to the magnetic \ncontribution. This is consistent with the specific heat peak around 165 K , as shown in Fig. 2(b), \nsuggesting the onset  of intrinsic magnetic phase transition at this temperature. In addi tion, we \nfind that the intensity curve of (102) combines the features of the (101) and (100) reflections,  \nindicating the involvement of magnetic contribution from both Fe3+ and Yb3+, similar to the \ncase of YbMnO 3 [28]. In RFeO 3 and RMnO 3 systems, it is well known that the R3+-Fe3+/R3+-\nMn3+ coupling is strong and the Fe3+/Mn3+ spin ordering may induce the R3+ spin ordering too \nas the concurrent sequence.  In this scenario, one may  intuit ively  argue that the magnetic order  \nis driven by the competing exchanges in this system, a very common phenomenon for magnetic \noxides.  12 \n Besides the low -T data, the NPD patterns were collected up to 550 K . The refinements at \nvarious temperatures reveal that the polar hexagonal structure remains unchanged in the entire \nmeasured temperature range, as indicated by  the variation of  lattice constants as functions of T \nas shown in Fig. 4(g). It thus confirms tha t ferroelectric phase persists  above T ~ 550 K.  \nPrevious density functional theory (DFT) calculations has explained the stability of the \nhexagonal phase in h-LuFeO 3 upon Sc substitution, while the multiferroic properties, including \nthe noncollinear magneti c order and ferroelectricity remains robustly unaffected [18]. Under \nthe ferroelectric state, h-Yb0.42Sc0.58FeO 3 belongs to non -centrosymmetric P63cm space group \nat room temperature, similar to the isomorphic h-RMnO 3.  Here, it is worth noting  that Disseler \net al have summarized the Néel transitio n temperatures as a function of  the c/a ratio lattice \nparameters for hexagonal RMO3 (R=Lu, Dy, Sc, Y and M=Mn or Fe)  [17]. They find a linear \ntrend that is independent of both the R species as well as the tr ansition metal (Fe or Mn). Instead, \nTN is highly dependent of the ratio of  c to a.  The c/a ratio obtained from our neutron diffraction  \nis 1.997 (300 K) and 1.998 (5 K), compared with that in h-Lu0.5Sc0.5FeO 3 (1.998 at 300 K ) [9]. \nTherefore, the similar c/a ratio between h-Lu0.5Sc0.5FeO 3 and h-Yb0.42Sc0.58FeO 3 confirms that \nthe AFM transition at TN~165 K is intrinsic  [36].  \nTo this end, the magnetic space groups used to refine the magnetic str uctures are \nsummarized  in Table II, and the refined magnetic structures are plotted in Fig. 5.  \n \nD. Ferroelectricity and Magnetoelectric Coupling    \nME coupling effect in the multiferroics is typically manifested as control ferroelectric \npolarization ( P) or magnetization ( M) by applied magnetic field ( H) or electric field ( E). In \nsome type -II multiferroics, e.g., TbMnO 3 and DyMnO 3, the magnetodielectric effect is also \naccompanied by a magnetoelastically induced lattice modulation, which results in the \nemergence of ferroelectric polarization. In f act, while most earlier works on hexagonal ferrites \nfocused on nonmagnetic Lu3+/Y3+ based systems, indeed no direct observation of ME effect, \ne.g., magnetic field control of polarization, have yet been confirmed . One may recall previous \ntheoretical work on  the trimer structural distortion that induces not only a spontaneous \npolarization but also bulk magnetization and linear ME effect [20].  As h-Yb0.42Sc0.58FeO 3 is \nalready ferroelectric at room temperature, we are very interested in the low -T antiferromagne tic \nphase. Based on the magnetic structures evaluated above, one can now discuss the possible \nferroelectricity associated with the magnetic structure , noting that  both the magnetic space \ngroups P63c'm' and P63 are polar and allow the ME effect . In our h-YSFO system, the strong 13 \n Yb magnetism and Fe3+-Fe3+/Fe3+-Yb3+ coupling may make some difference, allowing our \nattention on the ferroelectric and dielectric properties of our sample  subsequently .  \nIt is verified  that the polar hexagonal lattice structure remains unchanged in the whole \ncovered T-range up to 550 K . Thus , the sample is already in the ferroelectric state at room \ntemperature . Here, we focus our attention on the potential correlation between magnetism and \nferroelectricty.  Using the hig h-precision pyroelectric current method , we determine d the change \nof polarization P (not the total polarization) as a function of T below room temperature . To \nproceed, a poling electric field ( Ep = 10 kV/cm) was applied from T ~ 180 K to T ~ 10 K, and \nthe corresponding pyroelectric current ( Ip) and integrated polarization P are presented in Fig. \n6(a). It is found that pyroelectric current Ip begins to emerge just around TC (with a small \nshoulder) . Upon cooling from TN ~ 165 K, a broad and large peak appears with the peak -location \nat T ~ 120 K.  This broad peak does not end until T = TR ~ 40 K, and thus a polarization change \nP as large as ~ 2.0 C/cm2 is obtained . Similar to that in RMnO 3, the ferroelectric -transition \ntemperature ( TFE) is about 570 -990 K, which is a  very broad polarization region, w hile it has \nbeen reported that the structural -transition occurs at Ts ~1350 K  [46]. \nOne may argue that the Ip feature should be magnetically induced , considering the broad \nIp(T) peak cover ing the T-range from TN to TR. In particular, the dielectric constant as a function \nof T, as plotted in Fig. 6(b) for zero magnetic field and H = 5 T, evidences a clear bump around \nTN, indicating the ME effect via the spin -lattice coupling. In addition, the dielectric loss is as \nsmall as 10-3 on the order of magnitud e for the whole T range , indicating that the sample is \nhighly insulat ing. It is thus speculated  that the broad Ip peak below TN originates from the AFM \nFe3+ spin ordering via the spin -lattice coupling . Let us clarify that the magnetism undergoes \nsome change in this interval.  A weak ferromagnetism with a net mc=0.783 B at T=125 K along \nthe c axis is formed. As T reducing, the moment has a n obvious change in plane from ma = \n2.511  B to ma = 3.532 B at T=125 K and 25 K, respectively, and the total moment increases \nsignificantly, while the mc nearly unchanged.  It is reasonable to understand that the polarization \nappearing in the TN is due to the stronger interaction of Fe3+ result in lattice distortion, similar \nto the  trimer distortion  from the paraelectric to ferroelectric phase. The moment  of in -plane \nchange s in succession, reveal s that  the spin -lattice coupling may continue to the TR. This \nproposed scenario also explains why the Ip peak in Fig.  6(a) and thus the P transition region \nare so broad.  \nTo shed more light on the ME effect , we performed the iso-thermal ME measurements by \nprobing the magnetoelectric current IM in response to the magnetic field and magnetic field -14 \n driven ferroelectric polarization ( P) at T = 2 K, as shown in Fig. 7(a) where the arrows indicate \nthe direction of current change. The measured IM ~ H butterfly loop represents the typical ME \nresponse, demonstrating the detectable ME effect. The coercive field corresponding to the ME \nsignal reversal is roughly Hc ~ 0.5 T. Unfortunately, the measured ME current IM seems to be \nsmall and the evaluated polarization response P is on the order of magnitude of C/m2, much \nsmaller than the polarization itself ( P ~ 2 C/cm2) shown in Fig. 6(a). Such a small P is \nsurely due to the robustness of the Fe3+ AFM order against magnetic field up to 7.0 T. It is also \nreasonable since the spin -lattice mechanism generated ferroelectric polarization always shows \nweak ME resp onse, like the cases of multiferroic RMnO 3.  \nThis ME effect is significant between ± 4 T. In Figs. 7(b) and 7(c), the ME effect repeats \nwell with H oscillating. The maximum of P is about -0.9 C/m2, indicating the coupling is \nweak.  Considering the polycrystalline nature of the sample, the ME coupling is often week and \nmight be obscure if the signal is intrinsically low, or if the magnetic energy gain is not sufficient \nto overcome the energy barrier between multiple grains. Nevertheless , it should be  mentioned \nthat a varied P of 8 C/m2 was observed in single crystal h-YbMnO 3 [28], thus the value in \nour case is considered as intrinsic signal. Hence, high -quality of single crystal hexagonal rare -\nearth ferrite  is highly recommended to quantify  the ME coupling  phenomena by applying \nmagnetic field along the different crystallographic directions, and unveil the underlying \nphysical origin that is highly correlated to the Fe3+ ordering, and Fe3+-R3+ interactions.   \n \nE. Discussion  \nWe note that the ferrimagnetic  phase is characterized by a linear ME effect as shown in Fig. \n7(c). According to  the NPD analysis , the magnetic point group is P63c'm' below 165 K, and \nthen evolve s into P63 at 40 K. Both P63c'm' and P63 are polar, and allow c-axis spontaneous \npolarization , consistent with previous report [13]. Furthermore, both P63c'm' and P63 allow \nlinear magnetoelectric effect from the space group theory. Nevertheless , our ME coupling effect \nexhibited in polycrystalline sample still shed more light on the hexagonal rare -earth ferrite  \nRFeO 3, thus high quality of single crystal is highly required.  \nIn addition, the high-resolution T-dependent synchrotron XRD was conducted to monitor \nthe expected structure transition in our Yb 0.42Sc0.58FeO 3 around the  ferro magnetic  transition \npoint TC. As shown in the XRD patterns in Figs. 8(a) and 8(b), there is no phenomenon such as \npeak splitting, appearing or vanishing, indicating none obvious phase transition occur in the T \nrange of 200~250 K , consistent  with the neutro n data.  Even though, obvious changes of peak 15 \n amplitudes of (008) and (222) planes are evidenced around TC, as shown in Figs. 8(c) and 8(d). \nThe T-dependent peak positions of selected crystal planes are presented in Figs. 8(e) and 8(f). \nBesides the overall decreasing trend caused by thermal expansion, clear turning points could be \nnoticed in the curves belonging to (115), (008), and (222) of Yb 0.42Sc0.58FeO 3. These result \nshows that a mild structural transition does occur around TC, which is in good agreement with \nthe electrical properties of Yb 0.42Sc0.58FeO 3. \n \nIV . Conclusion  \nIn summary, we have achieved good description of the magnetic properties and \nmagnetoelectric coupling in bulk h-Yb1-xScxFeO 3 though neutron diffraction and electrical \nmeasurements . We find that the canted AFM state with Fe3+ moments lying in the ab-plane \nemerges at TN = 165 K .  Upon cooling to TR = 40 K, the Fe3+ moments reorient toward a new \ndirection, while the Yb3+ moments order ferrimagnetically along the c-axis below  the \ncharacteristic temperature TYb ~ 15 K. Direct experimental observation of ME coupling was \nfirst verified in this hexagonal ferrite , which provide a promising candidate for hunting for \nmultiferroics in other hexagonal RFeO 3 system and related mate rials. Moreover, high quality of \nsingle crystals is highly required to distinguish the magnetic configuration  of two sites of the \nYb3+ ions, in particular when does the two Yb3+ions order independently, and which is expected \nto play an imp ortant role in the ME coupling that is associated with any possible field -induced \nmetamagnetic transitions .  \n \nAcknowledgment  \nThis work was supported by the State key Research Program of China (Program No. \n2016YFA0300101, and 2016YFA0300102), the National N atural Science Foundation of China \n(Grant Nos. 11874031, 11834002, 11774106, 51721001, and 11974167). Z. L. Luo thanks the \nstaff at beamline BL14B of SSRF for their support. The research at Oak Ridge National \nLaboratory’s High Flux Isotope Reactor was spon sored by the Scientific User Facilities \nDivision, Office of Basic Energy Sciences, US Department of Energy.  \n \n \n  16 \n References   \n[1] S.-W. Cheong, M. 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Mater. 13, 1497  (2001).  \n [46] G. Nenert, M. Pollent, S. Marinel, G. R. Blake, A. Meetsma and T. T. M. Palstra , \nExperimental evidence for an intermddiate phase in the multiferroic YMnO 3, J. Phys. \nCondens. Matter. 19, 466212 (2007).  \n \n  21 \n  \nTABLE I. Refined structure parameters from powder neutron diffraction data measured at 300 \nK. \n \nAtom (Wyck.)  x y z B \nYb1 (2 a) 0.0000  0.0000  0.26847(63)  0.552(111)  \nSc1 (2 a) 0.0000  0.0000  0.26847(63)  0.552(111)  \nYb2 (4 b) 0.33333  0.66667  0.23632(54)  0.501(61)  \nSc2 (4 b) 0.33333  0.66667  0.23632(54)  0.501(61)  \nFe1 (6 c) 0.33333  0 0.00000  0.025(28)  \nO1 (6 c) 0.33061  0 0.17040  0.522(76)  \nO2 (6 c) 0.63231  0 0.33802  0.139(12)  \nO3 (2 a) 0.0000  0.0000  0.47629  0.201(20)  \nO4 (4 b) 0.33333  0.6667  0.01236  0.678(68)  \nSpace Group: P63cm, a = b = 5.86054(8)  Å, c = 11.70606(22)  Å, Chi2 = 10.6, RBragg = 3.61, \nRp = 3.73, Rwp = 4.84  \n \n  22 \n TABLE II. Magnetic space group and models used to refine the magnetic structures at 125 K, \n25 K and 3 K. The magnetic subgroup is P63c'm' #(185.201) at 125 K, and P63(#173.129) at 25 \nK and 3 K.  \nT = 125 K  T = 25 K, T = 3 K  \n(x,0,zmx,0,mz) \n(0,x,z0,mx,mz) \n(-x,-x,z -mx,-mx,mz) \n(-x,0,z+1/2 -mx,0,mz) \n(0,-x,z+1/2 0,-mx,mz) \n(x,x,z+1/2 mx,mx,mz) (x,0,zmx, my,mz) \n(0,x,z -my,mx-my,mz) \n(-x,-x,z -mx+my,-mx,mz) \n(-x,0,z+1/2 -mx,-my,mz) \n(0,-x,z+1/2  my,-mx+my,mz) \n(x,x,z+1/2 mx-my,mx,mz) \n \n \n  23 \n  \n \nFig. 1. The crystal structure of (a) orthorhombic, (b) hexagonal, and (c) bixbyite RFeO 3 ferrites.  \n(d) Rietveld profile fitting result for the XRD pattern of Yb 0.42Sc0.58FeO 3. The tiny impurity of \nScFeO 3 is indicated in the inset. ( e) The cation/a nion stoichiometry indicated by EDS.  \n  \n24 \n  \n \nFig. 2.  (a) The T dependence of magnetic susceptibility  () of Yb0.42Sc0.58FeO 3 under ZFC and \nFC modes  measured at 1 kOe, respectively. And its temperature derivative ( d/dT) is also \nplotted.  (b) T-dependent specific heat divided by T (Cp/T). (c) The Curie -Weiss fitting -1 \nmeasured under ZFC mode.  \n  \n25 \n  \n \nFig. 3.  (a) Magnetic field dependence of magnetization ( M) measured at various T’s. (b) The \nM(H) curves around TC with magnetic field between -0.2 T to  0.2 T . (c) The residual \nmagnetization ( Mr) as a function of T.  \n  \n26 \n  \n \nFig. 4 . The NPD  patterns collected  at (a) T = 300 K , (b) T = 125 K, (c) T = 25 K, and (d) T = 3 \nK. (e) Intensity map of the neutron -diffraction intensity about the magnetic (100), (101), and  \n(102) reflections.  (f) Integrated neutron -diffraction intensities of the magnetic (100), (101), and \n(102) reflections  as a function of temperature . (g) Temperature dependence of refined  lattice \nparameters a, b, and c in T-range up to 550 K.  \n  \n27 \n  \n \nFig. 5. The refined magnetic structures  at (a) 125 K, (b) 25 K and (c) 3 K, respectively.  The red \nand blue arrows stand for the moments of Fe3+ and Yb3+. The magnetic moment components  of \nFe3+ and Yb3+ are listed in the table.  \n  \n28 \n  \n \nFig. 6 (a) T-dependence of pyroelectric  current ( Ip) and corresponding change of polarization  \n(P). (b) The  T dependence of  real part of dielectric constant. Inset: the dielectric loss.  \n  \n29 \n  \n \nFig.7 (a) H-dependence of current ( IM) at 2 K, measured after field cooling from 180 K. (b)(c) \nThe ME effect repeats with H oscillating between ±  4 T. \n  \n30 \n  \n \nFig. 8. T-dependence of XRD patterns of Yb 0.42Sc0.58FeO 3 in 2θ range of (a) 39.5~43.5  and (b) \n50~52.5. The angle resolution is ~0.0036 , T interval is 1 K, incident X -ray photon energy is \n10 keV . The contour maps consist of stacking XRD patterns measured at varied T where the \ncolors indicate the diffraction intensity. (c -d) The magnified view of selected regions. The T-\ndependent diffractive p eak positions of (115)(106)(300)(008)(220)(222) were plotted in (e) and \n(f) respectively, where the color lines are guide of eyes and the green shadows indicate the T \nregions of structural distortion. For a more clear vision to the readers, the error bar o f 0.0018 \nwas not plotted along with the data.  \n \n"    },    {        "title": "1704.02495v2.Effect_of_annealing_temperatures_on_the_electrical_conductivity_and_dielectric_properties_of_Ni1_5Fe1_5O4_spinel_ferrite_prepared_by_chemical_reaction_at_different_pH_values.pdf",        "content": "1 \n Effect of annealing temperatures on the electrical conductivity and dielectric properties  of \nNi1.5Fe1.5O4 spinel ferrite  prepared by chemical reaction at different pH value s \nK.S. Aneesh Kumar  and R.N. Bhowmik*  \nDepartment of Physics, Pondicherry University, R. Venkataraman Nagar,  \nKalapet, Pondicherry -605014, India  \n*Corresponding author: Tel.: +91 -9944064547; Fax: +91 -413-2655734 \nE-mail: rnbhowmik.phy@pondiuni.edu.in  \nAbstract  \n The electrical conductivity and dielectric properties of Ni1.5Fe1.5O4 ferrite has been \ncontrolled by varying the annealing temperature of the chemical routed samples . The frequency \nactivated conductivity obeyed Jonscher’s power law and universal scaling  suggested \nsemiconductor nature. An unusual  metal like state has been revealed in the measurement \ntemperature scale  in between two semiconductor states  with different activation energy . The \nmetal like state has been affected by thermal annealing of the material. The analysis of electrical \nimpedance and modulus spectra has confirmed non-Debye dielectric relaxation  with \ncontributions from grains and grain boundar ies. The dielectric relaxati on process is thermal ly \nactivated in terms of measurement temperature  and annealing temperature of the sample s. The \nhole hopping process, due to presence of Ni3+ ions in the present Ni rich ferrite , played a \nsignificant role in determining the thermal activated conduction mechanism . This work has \nsuccessfully applied the technique of a  combined variation of annealing temperature and pH \nvalue during chemical reaction  for tuning electrical parameters  in a wide range ; for example dc \nlimit of conductivity  10-4 -10-12 S/cm, and unusually high activation energy  0.17-1.36 eV.  \nKey words: Ni rich ferrite; Heat treatment ; Metal like conductivity; Dielectric properties . 2 \n 1. Introduction  \n Spinel ferrites are promising magneto –electronic materials that can be used in the field of \nelectromagnetic devices, power electronics, sprintronics, biomedical applications , MRI scan  and \nhyperthermia treatment  [1-4]. The basic needs for applying ferrites in multi-functional devices \nare soft ferromagnetic properties , flexible electrical conductivity, low dielectric loss, and good \nchemical stability [5-10]. NiFe2O4 ferrite is one such material that exhibited tunable electrical \nproperties [11-13] and controlled by the variation of grain size and heat treatment [5, 6, 14-15].  \nThe conduction mechanism in nickel ferrite  is controlled by hopping of charge carriers via super -\nexchange paths Fe3+-O2--Fe2+ (electron hopping) and Ni2+-O2--Ni3+ (hole hopping)  and \nsignificantly affected by the grain and grain boundary structure [5, 16-18]. It is understood that \nhole hopping  (Ni2+-O2--Ni3+) mechanism dominates at high temperature regime and electron \nhopping (Fe3+-O2--Fe2+) mechanism dominates  at low temperatures  [19]. The increase of Ni \ncontent (x >1) in NixFe3-xO4 ferrite increases a considerable amount of Ni3+ ions in B sites of the \nspinel structure. This provides  a scope of studying electrical properties in nickel ferrite  where \nhole hopping plays a major role.  \nMost of the reports [20-22] studied the electric properties of Fe -rich regime (0 ≤ x ≤ 1) of \nNixFe3-xO4. There is hardly few report on electrical conductivity and dielectric properties o f Ni \nrich regime (1 ≤ x ≤ 2). Hence, we have selected the composition Ni1.5Fe1.5O4, a slightly Ni rich \nferrite in comparison to NiFe2O4, to study the effect of the variation of pH value during chemical \nreaction and post annealing temperature on different physical properties . The dielectric [23] and \nroom temperature magnetic [24] properties for the Ni1.5Fe1.5O4 samples prepared at different pH \nvalue and annealed only at 1000 0C have been published. The variation of annealing temperature 3 \n of the as-prepared (chemically routed)  sample showed a significant change in micro -structure \n[25] and it can also lead to a remarkable change in material properties.  \nIn this work, we show the effects  of the variation of annealing temperatures on electrical \nconductivity and dielectric properties of Ni 1.5Fe1.5O4 ferrite samples, which were prepared using \nchemical reaction at pH values in the range 6 -12. Secondly, we have analyzed the experimental \ndata using existing models to present a  generalized mechanism of the electrical conduction and \ndielectric properties, and the role of grains and grain boundar ies as the function of annealing \ntemperatures  of the samples . Finally, we anticipate that this work can be used as a reference to \ntune electrical prop erties in different materials by varying the parameters in chemical route.     \n2. Experimental  \n The material was prepared by chemical reaction of the required amounts of Ni(NO 3)2. \n6H2O and Fe(NO 3)3. 9H2O solutions at selected pH values  6, 8, 10, and 12 . The chemical \nreaction of the mixed solution was performed at 80  °C by maintaining nearly constant pH value. \nThe as prepared samples, after washing, were annealed in the temperature range 500 °C -1000 °C \nwith heating rate 5 °C/min. The prepared samples are labe led as NFpHX_Y, where X is the pH \nvalue during chemical reaction and Y is the annealing temperature  in degree centigrade . The \nreaders can refer earlier work [25] to get details of the material prepared at pH values 8, 10 and \n12, and structural phase stabiliz ation by annealing  the samples in air . The material prepared at \npH value 6 was annealed under high vacuum (10-5 mbar) to get the single phased cubic spinel \nstructure, because impurity hematite phase (α-Fe2O3) appeared after annealing  of the as-prepared \nsample in air. The X -ray diffraction pattern  of all the samples, used in this work,  matched to \nsingle phased cubic spinel structure with space group Fd3m . The structural phase evolution with \nvariation of annealing temperature has been illustrated for the samples prepared at pH 6 and pH 8 4 \n (supplementary Fig.1s) . The details of the samples are provided in Table 1. The lattice constant \nof the samples in cubic spinel phase was found in the range 8.2895 -8.3490 A°. Grain size of the \nsamples prepared at pH value 6, 8, 10, and 12 are in the range 15 -95 nm, 10 - 90 nm, 15 -80 nm, \nand 5-30 nm, respectively.  The disc shaped (  ~ 12 mm, t ~ 1 mm) samples were used for \ndielectric measurements in the frequency (ν) range 1 -107 Hz at ac field amplitude 1 V  and \nmeasurement temperature 173 K -573 K with interval 20 K . The disc shaped samples were \nsandwiched between two gold coated plates  and connected to broadband dielectric spectrometer \n(Novocontrol, Germany) using shielded cables . WINDETA software was used for acquisition  \ndata during dielectric measurement . The magnetic coercivity of the samples at room temperature \nwas calculated from magnetic loop, measured using PPMS (quantum Design, USA).  \n3. Results and discussion  \n3.1. Analysis of AC conductivity spectra  \n Fig. 1 (a-k) shows the frequency (ν) dependence of the re al part of ac conductivity (σ′) at  \nselected measurement temperatures  for the samples prepared at specific pH values and annealed \nat selected temperatures . The σ′(ν) curves at low measurement temperatures (typically below 233  \nK) followed a linear frequency response and showed small dc limit of conductivity (10-12 to 10-13 \nS/cm). This is the nearly constant loss (NCL) region of conductivity [26], where conductivity is \ncontrolled by localized hopping of ions in an asymmetric double well potential  [27]. The \nincrease of measurement temperature (233 K to 573 K) activates the hopping of charge carrie rs \nbetween two ions at lattice sites. In this temperature range, σ′(ν)  spectra showed two conductivity \nregimes (marked by dotted lines in Fig. 1(c)). In regime 1, σ′(ν) is nearly frequency independent \nand denote d as the dc limit of conductivity (σ dc). The σ′(ν) curves are frequency activated  in \nregime 2. In addition to th ese two regimes , the σ′(ν) curves  slowly decreas ed at high frequencies 5 \n (regime 3) for some of the low measurement temperatures . The σ′(ν,T) spectra in the entire \nmeasurement temperature scale obeyed Jonscher’s power law with two separate contributions , in \naddition to dc limit of conductivity  [20, 28]. \nσ\u0000(ν,T)=σ\u0000\u0000(T)+σ\u0000\u0000(T)ν\u0000\u0000\u0000+σ\u0000(T)ν\u0000\u0000            (1) \nThe first term σ\u0000\u0000(T) arises due to thermal activated transition of electrons from valence band to \nconduction band  in semiconductor ferrites . The second term arises due to short range hopping of \ncharge carriers  at the grain boundaries  (regime 2). The third term denotes the conductivity at \nhigh frequencies (regime3) due to localized hopping of charge carriers within grains. In spinel \noxides, the slowly moving electrons under the application of ac electric field distort the electrical \ncharge configuration in lattice structure. A strong electron -phonon coupling between a moving \nelectron and vibrating lattice ions at finite temperature forms a bound charge carrier, known as \npolaron [29]. The spatial dimension of a polaron  extends in the lattice structure  of the order of \nlattice constant  for small polaron s or beyond the lattice constant  for large polaron s. Table 1 \nsummarizes the range of ngb and ng values (within error bar  ± 0.01) obtained for measurement \ntemperature range 173 K -573 K for the samples prepared at  pH 6, 8, 10 and 12. The exponents \n\u0000\u0000\u0000 and \u0000\u0000 measure the degree of interaction of the charge carriers with surrounded ions while \nperforming hopping between Fe (Fe2+ ↔Fe3+) and Ni (Ni2+ ↔Ni3+) lattice sites.  The temperature \ndependence of ngb (contribution from grain boundaries)  and ng (contribution from grains) (plot is \nnot shown) indicated a local minimum in the temperature range 200 K -500 K, whose position \nshifted depending on annealing temperature  of the samples . The local minimum in ngb(T) and \nng(T) curves , as plotted for the samples annealed at 1000 0C [23], suggest that  overlapping large \npolaron tunnelling (OLPT)  mechanism  dominates in the thermal activated charge conduction \nprocess [29], irrespective of the annealing temperatures .  6 \n To verify the universal conductivity mechanism in the σ\u0000(ν,T) data [30-31] with respect \nto variation of measurement temperature, pH value during material synthesis and post annealing \ntemperature , we have adopted the following scaling approach .  \n                    σ\u0000(ν,\u0000)\nσ\u0000(\u0000)=f\u0000ν\nν\u0000\u0000      (2) \nHere,ν\u0000 is the critical frequency where the conductivity crosses from dc to disper sion of ac \nconductivity , σ\u0000is the intercept of σ\u0000(ν) curve on σ\u0000 axis at ν→0 and its value is close to σ\u0000\u0000. \nThe ν\u0000 andσ\u0000 were adjusted to obtain the best scaling of σ\u0000(ν) data. Fig. 2(a-h) shows the \noverlapped curves measured in the temperature  range 173 -573 K for the sample prepared at pH \nvalues 6, 8, 10 and 12 , and each sample was annealed at two different temperatures . Fig. 2(i-j) \nshows the scaling of σ\u0000(ν) curves at measurement temperatures  313 K and 513 K  for the sample  \nprepared at pH 8  and annealed at different temperatures in the range 500 -1000 0C. Fig.2(k-l) \nshows the scaled σ\u0000(ν) curves at measurement temperatures  513 K and 313 K for the samples \nprepared at different pH values and annealed at 800 0C. We observed that the conductivity data \nare scaled into a single master curve over a wide range of measurement temperatures for all the \nsamples. The conductivity curves  merged into a single master curve throughout the frequency \nrange almost at all measurement temperature s for the samples annealed  at higher temperatures . \nThe conductivity data for the sample s annealed at low temperature ( 500 0C) deviated from the \nmaster curve at lower measurement temperatures (where NCL region  dominates in conductivity \ncurves) and at frequencies lower than ν p and some cases, at higher frequenc ies also. It may be \nmentioned that micro-structural heterogeneity determine the character of electrical properties , \narising from  grain boundaries , for the samples annealed at lower temperatures [32]. Fig. 3(a-l) \nshows the temperature dependence o f σ\u0000, which was used for scaling of ac conductivity  of the \nsamples. The ν\u0000(T) curves (not plotted) reproduced the features similar to that of σ\u0000(T) curves. 7 \n The σ\u0000(T) curves in Fig. 3(f) indicated  three conductivity states S1M1  (high temperature \nsemiconductor state), M1M2  (intermediate metal like state) and M2S2  (low temperature \nsemiconductor state) for all the samples. The conductivity transition temperatures are marked as \nTSM (S1M1 ↔ M1M2) and T MS (M1M2 ↔ M2S2) while measurement temperature decreased \nfrom 573 K to 173 K. The metal like state (negative slope in σ\u0000(T) curves) is more prominent for \nthe samples annealed at low temperatures, irrespective of pH value during chemical reaction of \nthe material preparation. The metal like state  shifts to higher measurement temperature and the \nnegative slope of σ\u0000(T) curves is reduced  for samples with higher annealing temperature .  Such \nmetal like state has been understood  [23, 26] to be related to change of thermal activated \nconduction mechanism and reconfiguration of the electronic spins of transition metal ions like \nFe2+/Fe3+ and Ni2+/Ni3+ ions among the triply degenerate t 2g states and less stable e g states during \nexchange of Fe and Ni ions between A and B sites of spinel structure  [33]. The high magnetic \nstate and high resistance state are associated with the inverse spinel structure of NiFe 2O4, \nwhereas local disorder in B sites due to site exchange of Ni2+ and Fe3+ ions promote s metal like \nstate [34]. The magnetic spin interactions inside the grain and grain boundar y of magnetic \nparticles are affected by heat treatment of the material. We have analyzed the complex \nimpedance  (Z*(ν)) and modulus (M*(ν)) spectra to understand t he grain and grain boundary \ncontributions of the samples  as the function of annealing temperature . \n3.2. Analysis of impedance spectra   \n Fig. 4 (a-l) shows the Cole -Cole plots in complex impedance diagram ( -Z′′ vs. Z′) for the \nsamples prepared at pH values 6-12, and annealed at different temperatures . We noted that t he \nresistance (R el) contribution from electrode –sample interface is not significant for the present \nferrite system, which generally exhibit s a straight line parallel to Z ′′ axis in the impedance plane. 8 \n Considering heterogeneous electronic structure in the samples , a constant phase element (CPE) \nhas been included in the equivalent circuit  consisting of two parallel R -C elements  in series to \naccount for the distribution of relaxation process  and (Z∗(ω)) of the samples has been analyzed \nby following equa tion [23, 35].  \nZ∗(ω)= \u0000\u0000\n\u0000\u0000+A\u0000(jω)\u0000\u0000\u0000\u0000\u0000\n+ \u0000\u0000\n\u0000\u0000\u0000+A\u0000\u0000(jω)\u0000\u0000\u0000\u0000\u0000\u0000\n               (3) \nThe impedance of CPE is given by Z CPE(ω) = A-1(jω)-α where A and α are fit parameters, where \nA = C (capacitor) with α = 1 for an ideal capacitor and A-1 = R (resistance) with α = 0 for an \nideal resistor. The parameters R g, Rgb, Ag, Agb, αg, and α gb were obtained by fitting the impedance \nplot at each measurement temperature . The subscripts ‘g’ and ‘gb’ represent the contributions  \nfrom grain (high frequency regime) and grain boundary (low frequency regime), respectively.  As \nshown in Table 1, th e αgb values varied in the range 0.98 -0.61, where as αg values varied in the \nrange 1.2-0.83. We have calculated the resistivity  (ρ) of the sample s by incorporating the \ndimensions in the expression of resistance (R =ρL/A ; A and L are surface area and thickness, \nrespectively). Fig. 5(a-h) shows temperature dependence of resistivity from grains (ρ g) and grain \nboundaries (ρ gb). The values of ρgb and ρg are found in a wide range 105 -1012 Ω-cm and 104-1011 \nΩ-cm, respectively . Both ρg(T) and ρ gb(T) curves suggested a transformation of semiconductor \n↔ metal like conductivity states with negative temperature coefficient of resistance (NTCR) in \nsemiconductor state and positive temperature coefficien t of resistance (PTCR) in metal  like state. \nThe ρg(T) and ρ gb(T) curves  also showed a shift of the metal like state to higher temperature with \nthe increase of annealing temperature of the samples. But, impedance analysis cannot resolve the \ncontribution of ρgb and ρg for the whole measurement temperature scale. The modulus formalism \nis the best option to extract the contributions  from grains and grain boundaries. This is confirmed \nfrom a comparative plot (Fig. 6) of the imaginary part of impedance (Z ′′()) and electric 9 \n modulus (M ′′()) at measurement temperatures 213 K ( at low temperature semiconductor state) \nand 413 K ( near to high temperature semiconductor  state). Fig.6(a-l) suggests that Z′′() peak \nposition ( gb) of the samples lies well below of 1 Hz  at low measurement temperature (213 K)  \nand showed only one peak at lower frequency side for all the samples at 413 K. On the other \nhand, M′′() data showed two peaks or signatures of two peaks in the applied frequency range 1 -\n107 Hz. The peak at lower frequency side ( gb) is related to long range hopping of ions from one \nsite to another (slow relaxation) at grain boundaries. The peak at high frequency side ( g) is \nattributed to short range hopping of ions (fast relaxation) confined inside the grains. From \nphysics point of view, a peak like behavior in both M ′′ (ν) and ε′′(ν) suggests that long range \nhopping (conduction process) coexist with localized charge hopping (dielectric relaxation) in the \nmaterial [36]. We show a detailed analysis of modulus spectra  at all measurement temperatures . \n 3.3. Analysis of Electrical modulus spectra  \nThe complex electrical modulus ( \u0000∗) is related to the complex dielectric constant ( \u0000∗) \nand complex impedance ( Z*) by the following relations [2 6, 37].  \nM∗= \u0000\nε∗=M\u0000+ iM\u0000\u0000=iωC\u0000Z∗ \n\u0000′= \u0000\u0000\n\u0000\u0000\u0000(\u0000\u0000\u0000\u0000\u0000)\u0000\n\u0000\u0000(\u0000\u0000\u0000\u0000\u0000)\u0000\u0000+ \u0000\u0000\n\u0000\u0000\u0000\u0000(\u0000\u0000\u0000\u0000\u0000\u0000\u0000)\u0000\n\u0000\u0000(\u0000\u0000\u0000\u0000\u0000\u0000\u0000)\u0000\u0000  \n\u0000′′=\u0000\u0000\n\u0000\u0000\u0000(\u0000\u0000\u0000\u0000\u0000)\n\u0000\u0000(\u0000\u0000\u0000\u0000\u0000)\u0000\u0000+ \u0000\u0000\n\u0000\u0000\u0000\u0000(\u0000\u0000\u0000\u0000\u0000\u0000\u0000)\n\u0000\u0000(\u0000\u0000\u0000\u0000\u0000\u0000\u0000)\u0000\u0000                                                    (4) \nWhere ω=2πν is the angular frequency  \n C\u0000= \u0000\u0000\u0000\n\u0000 is the empty cell capacitance  \n′A′ is the sample area and ′ d′ is the thickness of the sample.  Both \u0000′(\u0000) \u0000\u0000\u0000 \u0000′(\u0000) approaches \nto zero as the applied frequenc y tend to zero (static limit of dielectric relaxation) , as clearly seen \nat 413 K. This suggests the suppression of interfacial polarization effects  in modulus formalism . 10 \n The dielectric relaxation information can b e obtained by fitting M′′() spectra with double peaks \nin the frequency domain using the modified Bergman proposed function [ 23]. \nM\u0000\u0000(ω)=\u0000\u0000\u0000\u0000\u0000\u0000\n\u0000\u0000β\u0000\u0000β\n\u0000\u0000β\u0000\u0000β\u0000ω\u0000\u0000\u0000\nω\u0000\u0000\u0000ω\nω\u0000\u0000\u0000\u0000\u0000β,ω=2πν       (5) \nThe fit of M′′(ν) spectra using equation ( 5) are shown in Fig. 7(a-l) at selected measurement \ntemperatures  of the samples prepared at specific pH and annealed at different temperatures . The \nstretched exponential factor, β, represents the strength of interaction in dielectric material [11, \n38-39]. As shown in Table 1, t he βgb (due to grain boundary contribution) and βg (due to grain \ncontribution) are found in the range (0.45 -0.96) and (0.2 -0.7), respectively. The β values smaller \nthan 1 (0 ≤ β ≤ 1) suggest non –Debye type relaxation. A relatively higher value of βgb than \ncorresponding value of βg suggests that electronic interaction s between charge carriers inside the \ngrains are  comparatively stronger than that in the grain boundar ies. We observed an interesting \nchange that βgb increases while β g decreases  with the increase of annealing temperature . It shows \na cross over in the conduction mechanism from grain boundar y dominated process to grain \ndominated process with the increase of annealing temperature of the chemical routed  samples.  \nWe defined relaxation time (τ) from the M′′(ν) peaks at low frequency (τ gb for grain \nboundary) and (τ g relaxation time for grains) high frequency , respectively . The temperature \ndependence of  τgb and τg are given in Fig. 8(a-d) and Fig. 8(e-h). The temperature variation of \nrelaxation time confirmed the existence of three conduction regimes (semiconductor ↔ metal \nlike ↔ semiconductor ) in the measurement temperature scale. The relaxation time decreases \nwith the increase of measurement temperature in semiconductor state s. In metal like state, the \nrelaxation time increases with temperature due to scattering of delocalized charge carriers. The \nrelaxation time, τ gb is nearly two orders of magnitude higher than τg. This suggests a long range d \ncharge hopping process inside the grain boundaries , whereas  charge hopping process inside the 11 \n grains is short ranged type. The temperature dependence relaxation time (τ) follows Arrhenius \nlaw: τ(T)= τ\u0000exp(\u0000\u0000\n\u0000\u0000\u0000) [39]. Here, Ea is the activation energy for charge hopping or relaxation \nprocess, k B is the Boltzmann constant, τo is the high temperature value s of τ. The activation \nenergy (Ea) was calculated from the slope of lnτ vs. 1000/T plot s (Fig. 9) in the semiconductor \nstates of the samples. Fig. 9(g-h) shows the variation of activation energy ( E\u0000\u0000\u0000,E\u0000\u0000) with \nannealing temperature  (TAN) of the samples calculated for high temperature semiconductor state . \nThe essential feature is that activation energy contributed by grain ( E\u0000\u0000) and grain boundary ( E\u0000\u0000\u0000) \nare found in the range 1.12 -0.17 eV and 1.36 -0.37 eV, respectively. In this Ni rich ferrite , the \ngeneral trend for the samples prepared at pH 6 and 8 is that activation energy ( E\u0000\u0000\u0000,E\u0000\u0000) initially \nincreased when TAN increased from 500 °C and after attaining a maximum value at a typical \nannealing temperature, which also depends on pH values, the activation energy decreases at \nhigher annealing temperatures. The activation energy of the sample prepared at pH 10 showed a \nmonotonic increase with TAN up to 1000 °C. The activation energ y of the sample prepared at pH \n12 monotonically decreased from 0.84 eV to 0.37 eV for grain boundary conduction and 0.62 to \n0.3 eV for grains when T AN increases from 800 °C to1000 °C. The decreased of activation \nenergy for higher annealing temperatures is consistent to that reported for NiFe2O4 and similar \nsystems with the increase of annealing temperature [5,16, 39]. Among the prepared samples, the  \nsample prepared at pH 8 showed the largest activation energy for both grain ( E\u0000\u0000: 0.52 eV- 1.1 \neV) and grain boundary ( E\u0000\u0000\u0000: 0.66 eV -1.36 eV) contributions . The samples prepared at pH 6 \nshowed the smallest activation energy from grains (E\u0000\u0000\u0000: 0.19 eV-0.60 eV). The activation energy \nobtained for low temperature semiconductor state (not for all the samples) is signific antly small \nin comparison to the value s in high temperature semiconductor state. E\u0000\u0000 is also found to be 12 \n smaller than the E\u0000\u0000\u0000 for low temperature semiconductor state. For example, E\u0000\u00000.32 eV and \nE\u0000\u0000\u00000.35 eV for NFpH10_800 sample; E\u0000\u00000.23 eV and E\u0000\u0000\u00000.42 eV for NFpH12_950 sample; \nE\u0000\u00000.30 eV and E\u0000\u0000\u00000.35 eV for NFpH12_1000 sample. The important point is that activation \nenergy from grain contribution for some of the Ni1.5Fe1.5O4 ferrite samples ( prepared at  higher \nannealing temperature) only at high temperature semiconductor state and at low temperature \nsemiconductor state (for all annealing temperatures) matched to the range of activation energy \n(0.22-0.53eV) reported for NiFe2O4 ferrite [5, 39]. At this point, we would like to mention a basic \ndifference in the charge hopping mechanism in the present material  and NiFe 2O4. The electrical \nconduction in NiFe 2O4 ferrite is controlled mainly by electron hopping process between Fe ions \n(Fe2+ -O- Fe3+) and hole hopping between Ni ions (Ni2+- O- Ni3+ ) can have minor role . On the \nother hand , a greater amount of Ni3+ ions is expected in the B sites of Ni rich ferrite (NixFe3-xO4; \nx > 1) [40]. In our earlier work [24], we have estimated the ionic states and site distribution of Ni \nand Fe ions in the cubic spinel structure of Ni1.5Fe1.5O4 ferrite using Mössbauer spectroscopy , \nmagnetic data and matching of lattice parameters. The essential information is that ionic states \ncould be assigned as Fe2+, Fe3+, Ni2+, and Ni3+ in Ni1.5Fe1.5O4 ferrite and distribution of these ions \namong A and B sites of the cubic spinel structure depends on the pH value at which the material \nwas chemically  prepared. The general tendency on increasing the pH value during chemical \nreaction is that A site occupancy of Ni2+ ions decreases by increasing the amount of Fe3+ ions. \nSubsequently, B site occupancy of Ni2+ ions increases by decreasing the equal amount of Fe3+ \nions, whereas B site occupancy of Ni3+ ions kept fixed (0.5) to maintain the overall charge \nvalence state of the structure. This indicates the probability of more hole hopping process  \n(Ni2+- O- Ni3+ ) at B sites in the present Ni rich samples . It is well established that activation \nenergy for hole hopping process is larger than electron hopping  process in ferrite system  [40-41]. 13 \n Based on a wide difference in the activation energy in the measurement temperature scale, w e \nsuggest that charge hopping process at low temperature semiconductor state (M2S2) is \ndominated by electron hopping process, whereas hole hopping process dominates at high \ntemperature semiconductor state  (S1M1). The difference in activation energy between the low \ntemperature and high temperature semiconductor states  are used for the reconfiguration of low \nand high spin states of Ni2+ and Ni3+ ions, and introduction of local disorder at B sites by thermal \nactivated site exchange of Ni2+ and Fe3+ ions in spinel structure  [33-34].  Subsequently, the \nsystem exhibits an intermediate metal like state  (M1M2) and electrical conductivity is affected .  \n4. Summary of the electric al parameters with annealing temperature of the samples  \nWe present a comparative plot (Fig. 1 0) for selective electrical parameters with annealing \ntemperature of the samples chemically prepared at specific pH values. Fig. 10(a-d) shows the \nvariation of transition temperatures  of conductivity states  (grain contribution (\u0000\u0000\u0000\u0000, \u0000\u0000\u0000\u0000) and \ngrain boundar y contribution  (\u0000\u0000\u0000\u0000\u0000, \u0000\u0000\u0000\u0000\u0000)). The low transition temperature (T MS) and high \ntransition temperature (T SM) showed an increasing trend with annealing temperature of the \nsamples to achieve a maximum (hump) in the annealing temperature range 800 0C-950 0C, \ndepending on the pH value at which the m aterial was chemically prepared. T hen, both TMS and \nTSM decreases at higher annealing temperature of the samples. For some of the samples  with low \nannealing temperature  (prepared at pH 6 and 8) , the conductivity transition temperatures initially \ndecreased with the increase of annealing temperature. Similar fea ture was noted in the variation \nof activation en ergy with annealing temperature . We attribute such behavior to the gr ain \nboundary disorder effect in  small grain -sized samples (see Table 1). Fig. 10(e-f) shows the \nconductivity (inverse of  resistivity  from impedance analysis ) variation at measurement \ntemperature 413 K  with annealing temperature of the samples . The conductivity  (gb and g) 14 \n curves initially increased with annealing temperature to achieve a maximum at annealing \ntemperature, typically above 950 0C for the samples prepared at pH values 8 -12, and above 800 \n0C for the samples prepared at pH value 6 . Then, conductivity again decreases at higher \nannealing temperatures . The samples prepared at pH 6 have exhibited remarkably high  \nconductivity  in comparison to the samples prepared at pH 8 -12. Another interesting fact is that \nconductivity of the samples at any specific annealing temperature showed a general decreasing \ntrend with the increase of pH value from 6 to 12, except some differences in th is trend for the \nsamples prepared at pH 8 and 10.  The variation of conductivity transition temperatures and \nconductivity from grain and grain boundary contributions suggest that there is some internal \ntransformation in t he electrical conduction process . Such internal transformation depends on \nannealing temperature of the samples. If we look at the comparative plot (Fig. 6), a wide \nseparation in the positions of Z ′′() and M′′() peaks at low frequency side  was indicated for \nlow measurement temperature (213 K) . It represents a typical localized dielectric relaxation. T he \ndifference () between the positions of Z ′′() and M′′() peaks narrowed at higher \nmeasurement temperature (413 K) , which suggests thermal activated charge conduction process \nin the material  [36]. The variation of gb (from low frequency peak at 413 K) in Fig. 10 (g) \nshowed an initial increase with annealing temperature of the samples up to 800 0C-950 0C, \nfollowed by a noticeable decrease at annealing temperature 1000 0C. This result suggests that the \nlocalized character of charge hopping process initially increases with the increase of annealing \ntemperatures . At higher annealing temperature  (> 950 0C), the thermal activated  charge hopping \nprocess increases. We propose that localized hopping of ions in an asymmetric double well \npotential controls the conductivity mechanism in the low measurement temperature scale (NCL \nregime), typically below 233 K, whereas thermal activated charge (electrons and holes) hopping 15 \n process dominates at higher measurement temperature s. The charge hopping due to electron s \n(Fe2+-O-Fe3+) is a fast relaxation process in comparison to hole hopping process  (Ni2+-O-Ni3+). \nThe hole hopping process is activated mainly at higher measurement temperature  [19]. The fast \nrelaxation contribution of the electrons (Fe2+-O-Fe3+) may increase significantly in the high \ntemperature semiconductor state of the samples for annealing temperature below 800 °C and \nabove 950 °C, and slow relaxation contribution of holes  (Ni2+-O-Ni3) increases for annealing \ntemperature in the range 800 °C-950 °C. This means increase of the annealing temperature, \nirrespective  of the variation of pH value during chemical reaction, brings a transformation in the \nconduction mechanism . This is associated with s ome other intrinsic phen omena, e.g., grain \nboundary refine ment and exchange of Ni and Fe ions among A and B sites of the cubic spinel \nstructure as an effect of thermal annealing of the material . First, we look at the grain boundary \nrefinement effect. It is interesting that t he hump like feature in the variation of different electrical \nparameters is well comparable to the feature of magnetic coercivity at room temperature (Fig.  \n10(h)). Such typical pattern of the variation of magnetic coercivity is attributed  to grain size \neffect, where magnetic domain structure transforms from single domain (SD) to pseudo -single \ndomain (PSD) to multi -domain (MD) with the increase of grain size  or annealing temperature of \nthe samples [8, 17, 24]. Although  a detailed Mössbauer study was not performed for the samples \nprepared at a specific pH value and post annealed at different temperatures, but the results from \nRaman spectra and FTIR spectra [25] indicated a possible site exchange of cations (Ni, Fe) upon \nincreasing t he annealing temperature, in addition to the fact that site distribution of cations in our \nsamples depends on pH value during coprecipitation.  For example, the suppression of shoulders \nbelow T 2g (2) and A 1g modes in Raman spectra suggested more B sites pop ulation of Fe3+ ions by \nmigrating equivalent amounts of Ni2+ ions to A sites for the samples  prepared at pH 8 and 10 16 \n with annealing temperature below 800 0C. Raman spectra also suggested that the samples \npreferred to be in inverse spinel structure  (more population of Fe3+ in A sites by exchanging \nequal amount of Ni2+ in B sites) for annealing temperatures ≥ 800° C. Such site exchange of Ni \nand Fe ions is expected to establish a correlation between magnetic and electrical properties in \nthe present ferrites . The present work suggests that the grain boundary refinement process along \nwith contribution from Ni2+ ↔ Ni3+ paths in B sites increases the slow charge conduction for the \nannealing temperature  in the range 800 °C -950 °C. The  increase of annealing temperature above \n> 900 °C can increase the population of Fe ions in B sites that increases probability of electron \nhopping (Fe2+-O-Fe3+) with fast relaxation and low activation energy  in the material  [41-42].  \n5. Conclusions  \nThe present work suggests that thermal activated charge conduction in the chemical \nrouted Ni1.5Fe1.5O4 ferrite samples, irrespective of the annealing temperatures, is controlled by \noverlapping large polaron  tunnelling mechanism . We find that a c conductivity data obeyed a \nuniversal scaling over a large range of frequency and measurement temperatures  for the samples \nannealed at higher temperatures. The conductivity data deviated from a master curve mainly at \nlower measurement temperatures and at frequencies lower than ν p for the samples annealed at \nlow temperature s. The material exhibited a transformation of c onductivity states  from low \ntemperature semiconductor state to high temperature semiconductor state  with an intermediate \nmetal like state  for all the samples at different annealing temperatures . The metal like \nconductivity state  shifts to higher measurement temperature and the negative slope of the σ\u0000(T) \ncurves reduced for the samples with higher annealing temperature . A peak behavior in both M ′′ \n(ν) and ε′′ (ν) suggested a long range hopping (conduction process) coexist with localized charge \nhopping (dielectric relaxation) in the material.  The metal like state  is understood as  the effect of 17 \n a crossover of the charge hopp ing dynamics from localized hopping at lower measurement \ntemperature  to thermal activated hopping at higher measurement temperature s. A notably large \nvalue of activation energy in the high temperature semiconductor state  indicates a significant \ncontribution of hole hopping process through Ni2+-O-Ni3+ paths. The most attractive result of this \nwork is the establishment of a hump like feature  in dc conductivity, activation energy, and \nconductivity transition temperatures with the variation of annealing temperature . Such feature is \nunderstood as a transformation in the conduction process  from grain boundary dominated \nmechanism to grain dominated mechanism  with the increase of annealing temperature.  The fast \nrelaxation contribution of the electrons (Fe2+-O-Fe3+) increases significantly in the high \ntemperature semiconductor state for the samples annealed below 800 °C and above 95 0 °C. The \nslow relaxation contribution of holes  (Ni2+-O-Ni3) increases for the samples with annealing \ntemperature in the range 800 °C-950 °C. \nAcknowledgment  \nThe authors thank CIF, Pondicherry University for dielectric measurements. RNB thanks to \nUGC for supporting research Grant (F.No. 42 -804/2013 (SR)) for the present work.  \nReferences  \n[1] K Inomata, N Ikeda, N Tezuka, R Goto, S Sugimoto, M Wojcik and E Jedryka  Sci. Technol. \nAdv. Mater.  9 014101 (2008)  \n[2] R Valenzuela Phys. Res. Int.  2012 1 (2012) \n[3] J D Adam, L  E Davis, G F Dionne, E  F Schloemann, S  N Stitzer IEEE Transactions on \nMicrowave Theory and Techniques  50 721 (2002) \n[4] U Lüders, A Barthélémy, M Bibes, K Bouzehouane, S Fusil, E Jacquet, J  P Contour, J  F \nBobo, J Fontcuberta, A Fert Adv. 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Res. 15 428 \n(2012) \n \nFigure captions  \nFig. 1(a-k) /() curves at selected me asurement temperatures for the Ni1.5Fe1.5O4 ferrite  \nsynthesized at pH values 6,8,10, and 12, and annealed at different temperatures.  \nFig. 2 Scaling of /( data at different measurement temperatures of the samples (a -h), scaled \ndata at 513 K (i) and 313 K (j) for the samples at pH 8 and annealed at different  temperatures, \nscaled data at 513 K (k) and 313 K (l) for the samples prepared at different pH values in the \nrange 6-12 and annealed at 800 0C.   \nFig. 3 Temperature variation of the dc conductivity value ( 0) used for scaling of the frequency \ndependence of ac conductivity curves for different samples.  \nFig.4 (a-l) Complex impedance plot for selected samples. Inset of (a) shows the equivalent  \ncircuit used for fitting of experimental data (symbol)and lines showed the fit data.  \nFig. 5 Temperature dependence of grain boundary (ρgb) and grain (ρg) resistivity contribution s \nfrom impedance analysis of the samples at different annealing temperatures.  21 \n Fig. 6 Frequency dependence of imaginary part of impedance (left -Y axis) and electrical \nmodulus (Right - Y axis) measured at 213 K (a-f) and 413K (g-l). \nFig. 7 Fit of the Imaginary part of modulus spectra using Bergman proposed function at selected \nmeasurement temperatures of the samples prepared at specific pH value and  annealed at different \ntemperatures. Lines guide to the fit data and symbol  represents experimental data.  \nFig. 8 Temperatur e dependence of relaxation time τ (τgb,τg) calculated from imaginary part of \nmodulus spectra for Ni 1.5Fe1.5O4 samples annealed at different  temperatures.  \nFig. 9 Fit of the τgb(T) and τg(T)data to obtain the activation energy. The activation energy for \ngrain boundary (Ea\ngb) and grain (Ea\ng) contributions are shown in (g -h). \nFig. 10 Variation of the conductivity transition temperatures (TMS, TSM) (a-d), conduc tivity  \n(g, gb) at 413 K contributed from grains (e) and grain boundaries (f), difference of the  positions \nin low frequency Z//() and M//() peaks (g), and magnetic coercivity at 300 K (h) with annealing \ntemperature of the samples prepared at specific p H samples.  \n \n \n \n \n \n \n \n \n \n \n 22 \n Table. 1. Power law fitted exponent (n gb,ng), equivalent circuit fitted CPE parameter (α gb, αg), \nand stretched exponential parameter (β gb, βg) for Ni 1.5Fe1.5O4 ferrite system.  \n pH \nvalue TAN \n(°C) Sample  \ncode Grain \nsize (nm) ngb(±0.01) ng(±0.01) αgb(±0.01) αg(±0.01) βgb(±0.01) βg(±0.01) \n \n \n6 500 NFpH6_500  16 1.07 – 0.84 1.63 – 1.12 0.81 – 0.91 0.87 – 0.98 0.52 – 0.94 0.37 – 0.83 \n600 NFpH6_600  25 1.05 – 0.73 1.55 – 1.08 0.81 – 0.97 0.54 – 0.98 0.52 – 0.96 0.46 – 0.78 \n800 NFpH6_800  67 1.61 – 0.73 1.55 – 1.08 0.9 – 0.98 0.68 – 0.83 0.57 – 0.96 0.31 – 0.67 \n1000 NFpH6_1000  95 1.02 – 0.72 2.12 – 1.07 0.86 – 0.93 0.89 – 1.01 0.5 – 0.91 0.22 – 0.38 \n \n \n8 500 NFpH8_500  10 1.04 – 0.36 1.99 – 0.79 0.58 – 0.93 0.88 – 0.97 0.89 – 0.59 0.28 – 0.44 \n600 NFpH8_600  12 1.07 – 0.83 2.08 – 0.96 0.86 – 0.92 0.73 – 0.9 0.64 – 0.84 0.35 – 0.51 \n800 NFpH8_800  33 0.99 – 0.85 1.82 – 1.12 0.84 – 0.93 0.7 – 0.98 0.48 – 0.89 0.33 – 0.52 \n950 NFpH8_950  76 0.83 – 0.67 - 0.83 – 0.93 0.7 – 0.98 0.44 – 0.68 0.38 – 0.55 \n1000 NFpH8_1000  92 0.98 – 0.76 - 0.89 – 0.94 0.94 – 1.12 0.55 – 0.8 - \n \n \n10 600 NFpH10_600  15 0.94 – 0.8 1.2 – 0.97 0.84 – 0.93 0.87 – 1.1 0.62 – 0.96 0.45 – 0.5 \n800 NFpH10_800  48 0.99 – 0.7 1.3 – 0.99 0.78 – 0.99 0.82 – 1.14 0.46 – 0.99 0.36 – 0.68 \n950 NFpH10_950  49 0.92 – 0.62 1.3 – 1.07 0.87 – 0.96 0.87 – 0.94 0.5 – 0.98 0.3 – 0.67 \n1000 NFpH10_1000  75 1.19 – 0.71 - 0.94 – 0.97 0.93 – 0.97 0.88 – 0.97 0.44 – 0.77 \n \n12 800 NFpH12_800  6 0.69 – 0.97 0.94 – 1.23 0.86 – 0.91 0.6 – 0.9 0.64 – 0.77 0.28 – 0.55 \n950 NFpH12_950  14 0.79 – 0.97 1.01 – 1.4 0.84 – 0.97 0.71 – 0.91 0.48 – 0.81 0.44 – 0.5 \n1000 NFpH12_1000  29 0.93 – 1.15 - 0.93 – 0.96 0.73 – 1.02 0.87 – 0.96 0.43 – 0.48 \u0000 \u0001 \u0000 \u0002 \u0003 \u0001 \u0003 \u0002 \u0004 \u0001\u0005 \u0006 \u0007 \u0006 \b \u0006 \t \u0006 \n \u0006 \u000b \u0006 \f \u0006\n\r\u000e \u000f\n\u0010\u0011 \u0012 \u0013 \u0014 \u0015 \u0016 \u0017 \u0017\u0018\u0019 \u001a \u001b\u001c \u001d \u001e \u001f \u001f\n ! \" \" # $ ! % % & ! 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College of Chemistry  and Molecular  Engineering  \nPeking  University  (PKU), Beijing  100871 , China  \n25/Nov/2021  2. Depa rtment  of Material  Science and Engineering  \nHarbin Engineering Uni versity (HEU)  \nHarbin 150001, CN  \n \nAbstract ： \n   Nanometer copper ferrite, as a kind of nanometer particle with catalytic activity, and its photothermal and magnetothermal \neffects as ferrite, can be widely used in different fields. It is a general  way to obtain the nano effect of the target by controllin g \nthe particle size. In this paper, the crystallization process of hydrothermal/solvothermal synthesis was analyzed, and the \nnucleation model was established to simulate the effects of solvent, reaction temperature and cooling time on the particle si ze of \ncopper ferrite nanoparticles. Through Monte Carlo method and energy function, the ratio of nano particle agglomeration was \nestablished, and the influence of different reaction conditions on it was discussed.  \n \n \nIntroduction   \nWith the development of nano science, more and more \nresearches have been done on nano effects. At the same \ntime, researchers are also looking for method s to \ncontrol particle nano effects. The nano  effect mainly \ncomes from the nano scale of particles, at which the \nphysical and chemical properties of substances can be \ngreatly changed. The size of nanoparticles is directly \nrelated to their size, which depends on their \ncrystallization process. By chang ing the experimental \nconditions, such as solvent, temperature, pressure, \nreaction time, etc., the crystallization process of \nsubstances can be affected  [1], and then different \nparticle sizes can be obtained.  \n \nCopper ferrite has catalytic activity for hydro gen \nperoxide in nanometer scale, and different particle sizes \nhave great influence on its catalytic activity  [2]. \nTherefore, controlling the particle size during the \nsynthesis process is a widely used method. Then, by simulating  and analysing  the influence  of factors on \nparticle size in  synthesis, guidance for its preparation \nprocess  can be provide d. The hydrothermal/  \nsolvothermal  synthesis is a general synthesis method \nused when preparing copper ferrite  [3][4][5] , as it is \nlow cost and easy to operate. In this article, the \nmodeling of nucleation process will base on \nhydrothermal/  solvothermal  synthesis.  \n1  Model Design  \n1.1 Energy Change during  Nucleus  Formation  \nThe modeling focus on the influence of solvents on the \nsize of nanoparticles produced.  The crystal nucleus  are \nformed to reduce the Gibbs freedom energy of the \nsystem  [6]. The change of Gibbs freedom energy from \nions in solvents to crystal cell structure per unit volume \nwas written a s ∆GV. There is following chemical \nequation to express the crystallization.  \n \n2Fe3++Cu2++8OH−→CuFe2O4+4H2O  2 The standard Gibbs energy of molar  formation  of \nCuFe2O4 can be calculated by the above equation, \nwhich was expressed as formulation (F.1).  \n \n∆𝑓𝐺𝑚⊝ CuFe2O4=2∆fGm⊝ Fe3++∆fGm⊝ Cu2+ \n+8∆fGm⊝ OH−−4∆fGm⊝ H2O(F.1) \n \n∆fGm⊝ Fe3+=−4.7kJ/mol \n∆fGm⊝ Cu2+=65.49kJ/mol \n8∆fGm⊝ OH−=−157.2kJ/mol \n∆fGm⊝ H2O=−237.1kJ/mol \n \nThe alpha lattice parameter  𝛼 of CuFe 2O4 is 8.389(2) \nÅ [7]. The unit cell  of the spinel  structure can be \nexpressed as Cu8Fe16O32, the 32 oxygen ions  are faced \ncenter cubic ( FCC ). Each cubic  cell is consisted by \neight CuFe 2O4 units  [8]. Thus,  \n \n∆GV=8\n𝛼3∆𝑓𝐺𝑚⊝ CuFe2O4 (F.2) \n \nDuring the process of crystal nucleus  uniform \nformation , when  a sphere embryonic crystal  with radius \n𝑟 exists  in solvent , the total change of Gibbs freedom \nenergy is  \n \n∆𝐺=4\n3𝜋𝑟3∆GV+4𝜋𝑟2𝜎 (F.3) \n \nwhere the 𝜎 is specific surface energy . The c ritical \nradius  𝑟∗ is obtained by applying 𝑑∆𝐺\n𝑑𝑟=0 on (F.3),  \n \n𝑟∗=−2𝜎\n∆GV(F.4) \n \nThus, by applying formulation (F.4) in (F.3), the work \nneeded to form a critical crystal nucleus  can be \nexpressed by  \n \n∆𝐺∗=16𝜋𝜎3\n3(∆GV)2(F.5) \n \nTo form the crystal nucleus, ∆𝐺∗ need to be filled with \nthe energy fluctuation  in solvents , which was \nconsidered as the summation of kinetic energy Ek of all solvents molecules the nucleus contacted during its \nformation.  Thus, the amount of solvent molecules \ncontacted is the amount of solvent molecules in volume \nsame as the volume of crystal nu cleus. Ek is: \n \n𝐸𝑘=3\n2𝑘𝐵𝑇∙4𝜋𝑟∗3∙𝜌𝑠\n3𝑀𝑠∙𝑁𝐴(F.6) \n \nThe section before the first multiplication sign  is the \naverage kinetic of solvent molecules. 𝑘𝐵 is the \nBoltzmann constant , which is 1.381× 10-23 m2 kg s-2 /K, \n𝑇 is the absolute temperature  of solvent , 𝜌𝑠 is the \ndensity of solvent, 𝑀𝑠 is the molar mass of solvent, \n𝑁𝐴 is Avogadro constant , which is 6.022 ×1023 mol-1. \n \nAt the critical situation, 𝜎 at this situation can be \nsolved by applying formulation (F.6) equal to (F.5).  \nThen ∆𝐺∗ can be solved by (F.5).  \n1.2 Nucle ation Rate \nThe nucleation rate 𝑁 (m-3 s-1) is the amount of crystal \nnucleus formed by a unit volume solvent in a unit time  \n[9], it was controlled by two coefficients . In \nformulation (F.7), the one before the multiplication \nsign is coefficient of work of nucleation, the other one  \nis coefficient of diffusion probability of atoms . \n \n𝑁=𝐾exp(−∆𝐺∗\n𝑘𝐵𝑇)∙exp(−𝑄\n𝑘𝐵𝑇) (F.7) \n \nwhere 𝐾 is proportionality constant , 𝑄 (J/mol) is \nenergy of diffusion activity of atoms , which is related \nto atomic binding force  as well as mechanism of \ndiffusion . It can be calculated by introducing P -\nparameters with effective energy of paired interaction \nof atoms , as it was shown in (F.8)  [10]. \n \n1\n𝐸𝑏=2[(𝑟𝑖𝑛\n𝑃0)\n1+(𝑟𝑖𝑛\n𝑃0)\n2] (F.8) \n \nwhere  𝐸𝑏 is same meaning as the 𝑄, but  under eV \nunit. 𝑟𝑖 is the orbital radius of i –orbital  of the atom, 𝑛 \nis the  number of  effective valence electrons , 𝑃0 was \ncalled a spatial -energy parameter . For CuFe 2O4, it was \nconsidered that Cu atoms take the position of crystal  3 cell first then the iron and oxygen atoms diffused in. \nThe parameters needed for calculation were listed in \nTable.1  [10]. \n \nTable.1  𝑃0-parameters of valence orbitals of neutral atoms in \nbasic state  [10] \nAtom Valence orbitals  𝑟𝑖(Å) 𝑃0 (eVÅ)  \nO 2p1 0.414 5.225 \n2p1 0.414 12.079  \nFe(Ⅲ) 3d1 0.365 10.564  \nCu(Ⅱ) 3d1 0.312 6.191 \n1.3 Temperature Range  \nIn most preparation, after keeping a higher reaction \ntemperature, there is a cooling process over the system \nthat contains product  [11][12]. During this process, \nthere is a maximum and a minimum temperature , Tmax, \nTmin, for the crystal nucleus  forming and growing.  \n \nIf T max is above the reaction temperature T R, then T max \nwas equaled to T R. If not, o ver the maximum \ntemperature, the Gibbs freedom energy of n ucleation \nreaction  is >0, which cannot process s pontaneously . \nThus, by applying ∆𝑓𝐺𝑚⊝ CuFe2O4 in (F.1) as a \nfunction o f temperature (F.9), then equal it to 0, T max \ncan be solved. Although the Gibbs freedom energy of \nformation was seen as a constant  previous , the model \nneed a border condition for calculation , which has to \nconsider that at here it is related to temperature . \n \n∆𝑓𝐺𝑚⊝ =∆𝑓𝐻𝑚⊝−𝑇𝑚𝑎𝑥∆𝑓𝑆𝑚⊝=0(F.9) \n \nThe crystal is not able to growth below T min, which \nmeans that the atoms in solvent cannot get through in \ncrystal. Thus, the T min can be solved by a critical \nsituation, when the kinetic energy of 1mol atoms in \nsolvent is equal to  the energy of diffusion activity  of the \n1mol atoms to diffuse into the crystal structure . The \nformulation can be written as  (F.10).  \n \n𝑇𝑚𝑖𝑛=2𝑄\n3𝑘𝐵∙𝑁𝐴(F.10) \n \nConsider the cooling as a linear process, 𝛽 is lapse rate o f temperature, 𝑡 is time (s) since the temperature \nreaches T max. As the experiments systems cool down \nfrom 280 ℃ to 60℃ in 60min, so 𝛽 was set as \n0.0583 K/s. There is  \n \n𝑇=𝑇𝑚𝑎𝑥−0.0583 𝑡 (F.11) \n1.4 Average Radius of Nanoparticles  \nThe total amount of crystal nucleus  formed at time 𝑡 \nsince the temperature reaches T max is 𝑛𝑐, it is the \nsummation of the n ucleation rate  𝑁, which is a \nfunction of 𝑇, from T max to T min in the whole volume \nof solvent  𝑉𝑠. Thus,  \n \n𝑛𝑐=∫𝑁[𝑇(𝑡)]∙𝑉𝑠𝑑𝑡𝑡\n0(F.12) \n \nThe 𝑉𝑠 was set as 30mL  in experiments  [13]. When it \ncools down to T min, the total time cost can be calculated \nby (F.11), which is (T max-Tmin)/0.0583.  \n \nAssume the copper is excess in reaction, there are \nformulation (F.13 a) based on iron conservation . The \namount of Fe atoms in crystal equal to that in reactants.  \n \n16×4𝜋𝑅3\n3𝛼3∙𝑛𝑐=𝑛𝐷∙𝑁𝐴𝑡=∞→  𝑛𝐹𝑒∙𝑁𝐴(F.13a) \n \nwhere 𝑅 is the average radius of product nanoparticles,  \n𝑛𝐷 is the mole of iron diffused in nucleus, when 𝑡=∞, \nbecause of the  copper excess, all iron was considered \nto form product,  𝑛𝐹𝑒 is the mole of iron in reactants , \nwhich was set as 1mol in experiments . 𝑛𝐷 is denoted \nby (F.13b) . J is diffusive mass transfer flux  of Fe ions.  \n \n𝑛𝐷=∫J∙𝑛𝑐∙4𝜋𝑅2𝑑𝑡𝑡\n0(F.13b) \n1.5 Morphology  of Nanoparticles  Aggregation  \nThe aggregation of nano particles influence the \nmorphology seriously. As the aggregation can reduce \nthe specific surface energy of nanoparticles. However, \nthe particles have to overcome repulsion between each \nother during the aggregation process. To simulate the  4 aggregation process, Monte Carlo method was applied.  \n \nThe hexagonal  center packing (HCP) has  the highest \ndensity of packing modes, which is 74%  [14]. It is the \nmost efficient way to reduce specific surface energy of \nthe particles aggregation. In HCP, e ach ball is \nsurrounded by 12 other balls  (Fig.1) . Choose one  part \nL0 with vector R⃗⃗  of its location , which is a single \nsphere particle initial at origin , the next tangent particle \nL1 aggregate s from one direction of the 12 positions \nrandomly. The way is to choose one  random number p1 \ngenerated in (0, 12], and then create a random \ndisplacement vector  Ξ⃗  in the sphere reference  frame . \n \nFig.1 Structure of HCP and vector representation  in sphere \nreference frame  \n \nIf p1 is in (0, 1] , 𝛽=𝜋/2, 𝛾=0. If p1 is in ( 1, 2], 𝛽=𝜋/2, \n𝛾=0. If p1 is in ( 2, 3], 𝛽=𝜋/2, 𝛾=2𝜋/3. If p1 is in ( 3, 4], \n𝛽=𝜋/2, 𝛾=𝜋. If p1 is in ( 4, 5], 𝛽=𝜋/2, 𝛾=4𝜋/3. If p1 is \nin (5, 6], 𝛽=𝜋/2, 𝛾=5𝜋/3. If p1 is in ( 6, 7], 𝛽=𝜋/6, \n𝛾=𝜋/6. If p1 is in ( 7, 8], 𝛽=𝜋/6, 𝛾=5𝜋/6. If p1 is in ( 8, \n9], 𝛽=𝜋/6, 𝛾=3𝜋/2. If p1 is in ( 9, 10], 𝛽=5𝜋/6, 𝛾=𝜋/6. \nIf p1 is in ( 10, 11], 𝛽=5𝜋/6, 𝛾=5𝜋/6. If p1 is in ( 11, \n12], 𝛽=5𝜋/6, 𝛾=3𝜋/2. With 𝑅 is the average radius \nof product nanoparticles calculated in section 1.4, t he \nmodule  of Ξ⃗  is \n \n‖Ξ⃗ ‖=2𝑅 (F.14) \nΞ⃗ =[𝛽,𝛾,2𝑅] \n \nThere is a r estriction  that the position of the new \nparticle L 1 must not coincide with the existing particles \nin L 0. For an example, if 4 of 12 positions of L 0 were \nalready placed, L 1 can only be random placed in the \nremainder  8 positions.  \n \nThe energy difference during this process is 𝛿𝜀, which \nis expressed by the difference between specific surface \nenergy change ∆𝐸𝑆 to Lennard -Jones potential  𝑉𝐿−𝐽.  𝛿𝜀=|∆𝐸𝑆|−𝑉𝐿−𝐽 (F.15) \n \nCheck whether  𝛿𝜀 is positive or negative , if e is \npositive, r eplace R⃗⃗  by R⃗⃗ +Ξ⃗  and return to next \nparticle. If e is negative, pick a new random number k \nin [0, 1], and compare it with Boltzmann factor , exp( -\n𝛿𝜀/𝑘𝐵𝑇). If k is less than the factor, set R⃗⃗  as R⃗⃗ +Ξ⃗  and \nreturn to next particle.  \n \nThe specific surface energy ∆𝐸𝑆 change is calculat ed \nby (F.16), where 𝜁 (J/kg m2) is the specific surface \nenergy of unit area , to form an interface of CuFe 2O4 \ncrystal, there are 8 Fe -O bonds (710.5eV) and 6 Cu -O \nbonds  (936.7eV)  [15] needed to be break per area of \nlattice cell . Δ𝑆 is the change of specific surface are a. \n𝑚 is the mass summation.  \n \n∆𝐸𝑆=𝜁∙𝑚Δ𝑆 (F.16) \n \nThe Δ𝑆 is calculated by an approximat ion of area, \nwhich  is the summation 𝑆0 of area of the two parts L0 \n& L 1 of aggregation and then minus a correction item \nS’, by considering the surface c omposed of triangular \nfaces . \n \nΔ𝑆=(𝑆0+4𝜋𝑅2−𝑆′)/𝑛𝑀 (F.17) \n \n𝑆′ is related with the structure formed by the joint of  \nL1. 𝑛 is the amount of particles, 𝑀 is the mass per \nparticle. In Fig.2, by forming each e quilateral triangle  \nwith tangent particles, the surface reduce S 3. By \nforming each regular tetrahedron  with tangent particles, \nthe surface reduce S 4.  \n \nFig.2 Approximation of surface area change  \n𝑆4=2𝑆3=(2𝜋−√3)𝑅2(F.18) \n \nThe number of formed e quilateral triangle  and regular \ntetrahedron  can be calculated by the known coordinates \nof particles in L0. \n \nConsider a single sphere particle in L 0, its Lennard -\nJones potential with L 1 is \n 5 𝑉(𝑟)=4𝜖[(𝜎\n𝑟)12\n−(𝜎\n𝑟)6\n] (F.19) \n \nWhere 𝑟 is the distance between the two particles. 𝜖 \nand 𝜎 are parameters. The total potential between L 0 \nand L 1 is  \n \n𝑉=∑𝑉(𝑟𝑖)𝑛−1\n𝑖=1(F.20) \n \nBased on the above formulation, t he potential energy of \na single nanoparticle can be denoted by relative \ncohesive energy  Ea/E0 [16]. Ea is the cohesive energy \nper particle. E 0 is the total cohesive energy of the \naggregation of particles. As the packing density of HCP \nand face center packi ng is same , the relation between \nEa/E0 and size of nanoparticles was calculated and \nshown in Fig.3.  \n \nFig.3 The particle size dependence of the cohesive  energy of \nface-centered  cubic nanoparticles  [14]. \n \nThus, the potential energy change during the jth particle \nwith radius as 𝑅 aggregate on the initial particle  can \nbe expressed as  (F.21), where 𝜀0 is the potential \nenergy of the first two aggregated particles.  \n \n𝑉𝑛=𝜀0(𝐸𝑎\n𝐸0|\n𝑛=𝑅)𝑗\n(F.21) \n \nThe m orphology of nano particles aggregation  was \nevaluated by ratio 𝜔 of the module of vector of final \nparticle to the summation of diameter of all particles.  \nThe ideal uniform aggregation should be a sphere, consider the volume conservation, there is followin g \nformulation  \n \n𝑅′=2𝑅√𝑛3(F.22) \n \nwhere 𝑅′ is the radius of  the sphere aggregation  \nassumed , 𝑛 is the amount of total particles calculated \nin the simulation.  The closer ‖R⃗⃗ +Ξ⃗ ‖ to 2𝑛𝑅, the \nmore ununiform  morphology is . The closer ‖R⃗⃗ +Ξ⃗ ‖ to \n𝑅′, the more  uniform  the morphology is . Thus, 𝜔 can \nbe denoted as  (F.23), which should in [0, 1].  \n \n𝜔=‖R⃗⃗ +Ξ⃗ ‖−𝑅′\n2𝑛𝑅−𝑅′(F.23) \n2  Result  and Discussion  \n2.1 Relation of Nucleation Rate with  Temperature  \nTable.2  Parameters of solvents used in simulation  \nSolvent  Mi g/mol  Density g/mL  Boiling point  °C \nH2O 18.02 0.997 100 \nC6H14O4 150.17  1.1 285 \nC8H18O5 194.23  1.1 327 \nC18H34O2 282.47  0.895 360 \n \nThe above four  solvents were chosen to be calculate d \nin this article , H 2O is a reference.  The curves of \nnucleation rate with temperature of the four solvents \nwere shown in Fig.4. The calculation shows that T min is \nbelow the lowest temperature in experiments, T max is \nover 350 °C, so t he temperature range is f rom 50°C to \nboiling point of each solvent  until 350°C. \n \nFig.4 Modeling curve of nucleation rate  of four solvents  \n 6 As it was shown in Fig.4, the nucleation rate in water is \npretty low when the temperature is below its boiling \npoint  at normal pressure , so that the h ydrothermal \nsynthesis  is always done in high pressure container s. \nThe nucleation rate o f C18H34O2 is low due to its high \nmolar mass and viscosity. By setting the reaction \ntemperature between 250 °C to 300 °C, the nucleation \nrate in C6H14O4 and C8H18O5 will pass the highest peak  \nduring the cooling process , it can form the most nucle us \nwithout boiling the solvent at normal pressure , which \ncan r educe energy consumption and simplify \nequipment  used in preparation.  \n2.2 Relation of Nucle us Radius with  Time  \nThe diffusion coefficients of iron and copper ions in \nwater under temperature which below its boiling point \nat normal pressure are too low to be applied in the \nmodel.  The total cooling time means the time caused of \nthe total process from reaction temperature to 50 °C, \nwhich affects the lapse rate of temperature, it is not the \ntime since cooling beginning.  \n \nFig.5 Modeling curve of nucleus radius \n \nIn the Fig.5 , the increasing trend of C6H14O4 and \nC8H18O5 shorter than 10min, C18H34O2 shorter than \n30min , which is  the fast cooling of the system,  at the \nbeginning if the system cool s down to 50°C in an \nextreme short time, the nucleus es have no time to grow \nas the diffusion coefficients of ions were already \nreduced quickly to values too low to  grow . When t he \ncooling time was slightly extended  but still short, there \nwill not form a quantity of nucleus es, but the growth \ncould only happen on these few nucleus es then make their diameter reach a high value. As the total cooling \ntime was extended  longer, there will be more and more \nnucleus formed during the time, then the diameter of \neach particle of different cooling time system decreases.   \n2.3 Morphology Uniformity  \nAssume t he reaction temperature is 280 °C, then was \ncooled to 50 °C, the curve of 𝜔 with total cooling time \nwas shown in Fig.6.  The m orphology uniformity of fast \ncooling calculated by the model is meaningless and was \ndeleted from the figure.  It can be seen from Fig.6, as \nthe total cooling time extends, the morphology \nuniformity  increased, which is because there are more \ntime for particles to change location s until the low \nenergy position was found  [17]. \n \nFig.6 Modeling curve of morphology uniformity  \n3  Conclusion  \nBy simulat ing the nucleation proc ess of copper ferrite \nnano particles and calculated the average radius of \nparticles  in preparation systems , solvents as C6H14O4, \nC8H18O5 are the appropriate  choice s. Temperature  \nbetween 250 -300°C and cooling time  over 1h is \nsuitable for particles below 50nm . By applying Monte \nCarlo method to evaluate the morphology uniformity , \ncooling time is important to  the aggregation of particles.  \n \nWith the models built in this article, it will provide \nreference and choice of solvents, temperatures and \ncooling times when considering about optimi zation of \nthe nano properties of copper ferrite during preparation.  \n 7 Acknowledg ment  \nThe support from Bachelor program in Institute of Inorganic  and \nNonmetal  Matter , HEU.  Quite appreciate the care and support of \nW.B. Z. and PhD students in his group , CCME, PKU . \nReference  \n[1] Li, G. Z., et al. (2000). \"Effect of crystallization conditions \non single crystals of ladderlike polyphenylsilsesquioxane \n(PPSQ).\" Polymer 41(8): 2827 -2830.  \n[2]  Zhao, W. K., et al. 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(2019). \"Preparation and catalytic \nperformance of CuFe2O4 nanoparticles supported on \nreduced graphene oxide (C uFe2O4/rGO) for phenol \ndegradation.\" Materials Chemistry and Physics 238.  \n[13]  Zi H. H., et al. (20 21). “Diameter and Aggregation \nControlled Preparation by Solvothermal Synthesis of Ultra -\nsmall Particles CuFe2O4 ”. \n[14]  Choi, Y., et al. (2022). \"Density Functional Theory \nDescription of Paramagnetic Hexagonal Close -Packed \nIron.\" Materials 15(4).  \n[15]  Zhu, M., et al. (2013). \"Facile Fabrication of Hierarchically \nPorous CuFe2O4 Nanospheres with Enhanced Capacitance \nProperty.\" ACS Applied Materials &amp; Interfac es 5(13): \n6030 -6037.  \n[16]  Qi, W. H., et al. (2004). \"Calculation of the cohesive energy \nof metallic nanoparticles by the Lennard -Jones potential.\" \nMaterials Letters 58(11): 1745 -1749.  \n[17]  Wei, Z. Y., et al. (2010). \"Crystallization and melting \nbehavior of isotactic polypropylene nucleated with \nindividual and compound nucleating agents.\" Journal of \nThermal Analysis and Calorimetry 102(2): 775 -783. \n \n \n     \n \n \n \n "    },    {        "title": "1801.08042v1.Quantization_of_magnetoelectric_fields.pdf",        "content": " Quantization of magnetoelectric fields  \n \nE. O. Kamenetskii  \n \nMicrowave Magnetic Laboratory,  \nDepartment of Electrical and Computer Engineering,  \nBen Gurion University of the Negev, Beer Sheva, Israel  \n \nJanuary 22, 2018 \n \nAbstract  \nThe effect of quantum coherence involving macroscopic degree of freedom, and occurring in \nsystems far larger than individual atoms are one of the topical fields in modern physics. \nBecause of material dispersion, a phenomenological approach to  macroscopic quant um \nelectrodynamics, where no canonical  formulation is attempted, is used. The problem becomes \nmore complicated when geometrical forms of a material structure have to be taken into \nconsideration.  Magnetic -dipolar -mode (MDM) oscillations in a magnetically sa turated quasi -\n2D ferrite  disk are macroscopically quantized states. In this ferrimagnetic structure, long -range \ndipole -dipole correlation in positions of electron spins can be treated in terms of collective \nexcitations of a system as a whole. The near fiel ds in the proximity of a MDM ferrite disk have \nspace and time symmetry breakings. Such M DM-originated fields – called  magnetoelectric \n(ME) fields  – carry both spin and orbital angular momentums. By virtue of unique  topology, \nME fields are different from fr ee-space electromagnetic (EM) fields.  The ME fields are \nquantum fluctuations in vacuum. We call these quantized states ME photons. There are not \nvirtual EM photons. We show that energy, spin and orbital angular momenta of MDM \noscillations constitute the ke y physical quantities that character ize the ME -field configurations. \nWe show that vacuum can induce a Casimir torque between a MDM  ferrite disk,  metal walls, \nand dielectric samples .  \n \nPACS number(s): 41.20.Jb, 71.36.+c , 76.50.+g  \n \nI. INTRODUCTION  \n  \nEnergy, lin ear momentum, and angular momentum are constants of motion , which constitute \nthe key physical quantities that characterize the EM field configuration.  Their conservation can \nbe cast as a continuity equation relating to a density or a flux density, or a current, associated to \nthe conserved quantity [1].  Rece ntly, a new constant of motion and a current  associated to this \nconserved quantity have  been introduced in electrodynamics.  For a circularly polarized EM \nplane wave, t here are the EM -field chirality and the flux density of the EM -field chirality  [2, \n3].   \n     Circularly polarized EM photons carry a spin angular momentum. EM photons can also \ncarry an additional angular momentum, called an orbital angular momentum. Azimuthal \ndependence of beam phase result s in a helical wavefront . The photons, carrying both spin and \norbital angular momentums are twist ed photons [4].  The t wisted EM photons are propagating -\nwave, “ actual ”, photons. In the near -field phenomena, which are characterized by \nsubwavelength effects a nd do not radiate though space with the same range properties as do \nEM wave photons, the energy is carried by virtual EM photons. Virtual particles should also \nconserve energy and momentum. The question whether virtual EM photons can behave as \ntwisting exc itations is a subject of a strong interest . Recently, a new concept of the spin -orbit \ninteractions in the evanescent -field region of optical EM waves  has been proposed .  2 Theoretically, it was shown that an evanescent wave possesses a spin component,  which i s \northogonal to the wave vector.  Furthermore, such a wave carries a momentum component, \nwhich is determined by  the circular polarization and is also orthogonal to the wave vector. The \ntransverse momentum and spin push and twist a probe Mie particle in an  evanescent field. This \nshould allow  the observation of ‘impossible’ properties of light, which was previously \nconsidered as ‘virtual’  [5, 6].  \n     The e vanescent modes have a purely imaginary wave number and represent the \nmathematical analogy of the tunnel ing solutions of the Schrödinger equation.  However, the \nquantization of evanescent waves is not a straightforward problem.  Many fundamental optical \nproperties  rely on virtual  photons to act as the  mediator.  A well -known example of the Casimir \neffect can be  understood as involving the creation of short  lived  virtual photons from  the \nvacuum . Together with the calculation of the Casimir force between arbitrary materials , in \nnumerous publications , the question of the angular -momentum coupling  with the quantum \nvacuum fields  is considered  as a topical subject . Since photons also carry the angular \nmomentum,  the vacuum torque will appear between macroscopic  bodies when their \ncharacteristic properties are anisotropic  [7 – 13]. In particular, i n Refs. [ 7, 8, 10, 13 ], it was  \ndiscussed that vacuum can induce a torque between two axial birefringent dielectric plates. In \nthis case, the fluctuating electromagnetic fields have boundary conditions that depend on the \nrelative orientation of optical axes of the materials. Hence , the zero -point energy arising from \nthese fields also has an angular dependence. This leads to a Casimir torque that tends to align \ntwo of the principal axes of the material in order to minimize the system’s energy. A torque \noccurs only if symmetry betwee n the right -handed and left -handed circularly polarized light is \nbroken (when the media are birefringent).  \n     The contributions of plasmonic  modes to the Casimir effect  was analyzed  in recent studies \n[14 – 17]. Plasmon polaritons are coupled states betwe en electromagnetic radiation and \nelectrostatic (ES) oscillations in a metal. Instantaneous Coulomb  interaction are supplemented  \nwith retardation effects associated with ‘real’  electromagnetic fields. In a non -classical \ndescription, there is the effect of i nteraction between real and virtual photons.  On the other \nhand, r ecently, it was shown that some special -geometric plasmonic structures can be used to \ncreate so called superchiral fields [ 18, 19]. Could one consider these near fields as chiral virtual \nphotons? Whether the contribution  of such chiral virtual photons to  the Casimir torque can be \nanalyzed ? To the best of our knowledge, no such effects  of are discussed in literature.  \n     Magnetostatic (MS) oscillations in ferrite samples [ 20, 21] can be consi dered, to a certain \nextent, as a microwave analog of plasmo n oscillations in optics. T he coupled states between \nelectromagnetic radiation and MS oscillations may be viewed as a microwave MS -magnon \npolaritons, analogously to the ES -plasmon polaritons in opt ics. Such an analogy, however, is \nquite inaccurate , since dynamics of charge motion is completely different in these cases. In \ncontrast to ES -plasmon oscillations with linear electric currents, MS -magnon oscillations \nappear  due to precessing magnetic dipol es [20 – 22]. The problem of MS [or, in other words, \nmagnetic -dipolar -mode (MDM)] oscillations demonstrates  fundamental ly new properties when \nspecific  geometrical forms of a material structure is taken into consideration. In a quasi -2D \nferrite disk, one be comes evident with a fact that MDM oscillations are macroscopically \n(mesoscopically) quantized state s.  \n      The effects of quantum coherence involving macroscopic degrees of freedom and occurring \nin systems far larger than individual atoms are one of the  topical fields in modern physics [23].    \nThere is evidence that macroscopic systems can under appropriate conditions be in quantum \nstates, which are linear superpositions of states with different macroscopic properties. The \nAharonov -Bohm effect shows that  the characteristically quantum effect is reflected in the \nbehavio r of macroscopic currents  [24]. Much progress has been made in demonstrating the \nmacroscopic quantum behavior of superconductor systems, where particles form highly  3 correlated electron syste ms. The concept of coherent mixture of electrons and holes, underlying \nthe quasiclassical approximation  based on the Bogoliubov -de Gennes (BdG) Hamiltonian  [25], \nwell describes the topological superconductors.  \n     Macroscopic quantum coherence  can also be  observed in some ferri magnetic structures. \nLong range dipole -dipole correlation in p osition of electron spins in a ferrimagnetic sample \nwith saturation magnetization can be  treated in terms of collective excitations of the system as \na whole. If the sample  is sufficiently small so that the dephasing length \nphL  of the magnetic \ndipole -dipole interaction exceeds the sample size, this interaction is non -local on the scale of  \nphL\n. This is a feature of a mesoscopic  ferrite sample, i. e., a sample with linear dimensions \nsmaller than \nphL   but still much larger than the exchange -interaction scales. In a case of a \nquasi -2D ferrite disk, the quantized forms of these collective matter oscillations – MS (or \nMDM)  magnons – were found to be quasiparticles with both wave -like and particle -like \nbehaviors, as expected for quantum excitations.  The MDM oscillations in a quas-2D ferrite \ndisk, analyzed as spectral solutions for the MS-potential scalar wave fu nction \n( , )rt\n , has \nevident quantum -like attributes. The  oscillating state in a lossless ferrite -disk particle can be \ndescribed in terms of a Hilbert space spanned by a complete orthonormal set of  eigenvectors of \nsome observable A with eigenvalues ai [26 – 28]. The discrete energy eigenstates of the MDM \noscillations are well observed in microwave experiments [ 29 – 32]. Experimental ly, it was \nshown also that  interaction of the MDM ferrite particles with its microwave environment give \nmultiresonance Fano -type resonances  [31, 32]. The observed interference between resonant \nand nonresonant processes is  typical  for quantum -dot structures  [33]. In this case,  the Fano \nregime emerges because of  resonant tunneling between the dot and the channel . \n     MDM oscillations in a quasi -2D ferrite disk can conserve energy and angular momentum. \nBecause of these properties, MDMs strongly confine energy in subwavelength scales of \nmicrowave radiation.  In a vacuum subwavelength region abutting to a MDM ferrite  disk, one \nobserves the quantized state of power -flow vortices. Moreover, in such a vacuum \nsubwavelength region, the time -varying electric and magnetic fields can be not mutually \nperpendicular.  Such a specific near field – so-called magnetoelectric (ME) fi eld – give \nevidence for spontaneous symmetry breakings at the resonance state s of MDM oscillations  [34, \n35]. The ME fields are quantum fluctuations in vacuum. We call them ME photons. The \nelectric - and magnetic -field components of the ME photons have spin and orbital angular \nmomenta. These twisted evanescent fields  are neither  virtual  nor “real” (free -space \npropagating) EM photons  in vacuum.  \n     The fact of the presence of the power flow circulation inside and in a subwavelength \nvacuum region outside  a ferrite disk arises a question of the angular momentum balance. It is \nevident that at the MDM resonance, such a balance can be realized only when the opposite \npower flow circulations can be created due to metal (and also dielectric) parts of the \nmicrowave st ructure, in which a ferrite disk is placed. It means that at the MDM resonance in a \nferrite disk, certain conductivity (or displacement) electric currents should be induced in the \nmetal (or dielectric) parts of the microwave structure. T he coupling between  an electrically \nneutral  MDM ferrite disk and an electrically neutral  metal (or dielectric) objects  in a \nmicrowave subwavelength region,  should  involve the electromagnetic -induction  or/and \nCoulomb  interaction s through virtual photon s mediation . In Ref. [36 ], it was shown that due to \nthe topological action of the azimuthally unidirectional transport of energy in a MDM -\nresonance  ferrite sample there exists the opposite topological reaction on a metal screen  placed \nnear this sample. We call this effect topolog ical Lenz ’s effect.  The coupling energy depends on \nthe angle between the  directions of the magnetization vectors in a ferrite and electric -current \nvectors on a metal  wall. A vacuum -induced Casimir torque [7 – 13] allows for torque \ntransmission between the ferrite  disk and metal wall avoiding any direct contact between them.   4      In this paper , we analyze quantum confinement of MDM oscilla tions in a ferrite disk and  \ncoupling between  MDM bound states and microwave -field continuum . The paper is organized \nas fo llows.  In section II, we consider MDM  oscillations and ME fields based on a classical \napproach . We analyze the MS description in connection with the  Faraday law  and “Sagnac \neffect” in a MDM ferrite disk . We show that the near fields originated from magneti zation \ndynamics at MDM resonances – the ME fields – appear as the fields of axion electrodynamics. \nSharp  multiresonance oscillations, observed experimentally in microwave structures with an \nembedded quasi -2D ferrite disk give evidence for quantized states of the microwave fields \noriginated from MDM  oscillations . This is a subject of our detailed discussion in section III. In \nsection IV, we consider MDM bound states in the waveguide continuum . The questions of PT \nsymmetry and PT-symmetry breaking of MDM osci llations are the subject of section V. In \nsection VI, we analyze the effects in non -Hermitian MDM structures with PT-symmetry \nbreaking. Section VII is a conclusion of the paper.  \n \nII. MDM OSCILLATION S AND MAGNETOELECTRIC FIELDS : CLASSICAL \nAPPROACH  \n \nA. Magnetostati c description and Faraday law  \n \nIn a frame of a classical description, MDM oscillations in small ferrite samples are considered \nas an approximation to Maxwell equations when a displacement electr ic current is negligibly \nsmall . The physical justification for  such an approximation arises from the fact that in a  small  \n(with sizes  much less than a free -space electromagnetic wavelength) sample  of a magnetic \nmaterial with strong temporal dispersion (due to the ferromagnetic resonance) , one neglect s a \ntime variatio n of electric energy in comparison with a time variation of magnetic energy [ 20 – \n22]. In this case, we have a system of three differential equations for t he electric and magnetic \nfields     \n \n                                                                  \n0B \n ,                                                                 (1)                                                    \n \n                                                                  \n0H \n ,                                                               (2) \n \n                                                                  \nBEt \n .                                                          (3)                                                 \n \nIn such a system, there is evident electromagnetic duality breaking.  The spectral solutions for  \nMDM oscillations  in a small ferrite sample can be  obtained based on  the first two differential \nequations  in the system , Eqs. (1) and (2) . With a formal use  of equation \nH\n  and a \nconstitutive relation for a ferrite, one gets the Walker equation [ 37]  \n \n                                                                   \n0  \n                                                      (4) \n \ninside and the Laplace equation \n20  outside a ferrite sample.  Here \n  is a magnetostatic \n(MS) potential  and \n0 ( , )H\n  is a tensor of ferrite permeability  at the ferromagneti c-resonance \nfrequency range .  \n     To obtain the  MDM spectral solutions , the boundary conditions for the MS-potential scalar \nwave function \n( , )rt\n  and its space derivatives  should be imposed . In th ese spectral  solutions,  5 we do not use t he third equation in the system  – the Faraday equation  (3). From literature,  one \ncan see that for the MDM spectral problem formulated exceptionally for the MS -potential \nwave function \n( , )rt\n , no use of the alternative electric fields  is presumed  [20 – 22, 3 7 – 39]. \nFollowing  a formal analysis [ 40, 41] it can be also shown that for MDM oscillations, the \nFaraday equation is incompatible with Eqs. (1) and (2).  Really, from Eq. (3), we obtain \n2\n2EB\ntt \n. Excluding completely  the electric displacement current \n0D\nt\n , in a \nsample which does not possess any dielectric anisotropy, we have \n2\n20B\nt\n . It follows that the \nmagnetic field in smal l resonant magnetic objects varies  linearly with ti me. This gives , \nhowever, arbitrary large fie lds at early and late times which  is excluded on physical grounds. \nAn evident conclusion suggests itself at once: at the MDM resonances, the magnetic field s are \nconstant quantities.  Since  such a conclusion contra dicts the fact of tempo rally dispersive media \nand any resonant conditions, one can state that  the Faraday equation is incompatible with the \nMDM spectral solutions . In Ref. [42 ] it was merely stated that in a case of the MDM spectral \nproblem, an electric fi eld is completely absent and thus \n0E \n .  \n \nB. “Sagnac effect” in a MDM ferrite disk  \n \nIt appears, h owever, that in а case of MDM oscillations in a quasi -2D ferrite disk, the Faraday \nequation plays an essential role . These  oscillations are  frequency steady states characterizing \nby power -flow azimuthal rotations. I n a frame of reference co -moving with the orbital rotation, \nthe fields are constant. In every steady state, the Faraday law appears as the Ampere -law \nanalog for magnetic currents a s sources of the electric fields. The following consideration \nmakes this question clearer.  \n     An analytical approach for  MDM resonances in a quasi -2D ferrite disk is based  on a \nformulation of a spectral problem for a macroscopic scalar  wave function – the MS -potential \nwave function  \n [26, 28, 34] . In this  approach, the description  rests on  two cornerstones: (i) \nPrecession of all  electrons in a  magnetically  ordered ferrite sample is determines  by \n \nfuncti on, and (ii) the phase of this wave function is well defined  over the whole ferrite -disk \nsystem, i.e., MDMs are macroscopic states maintaining the global phase coherence.  For a \nferrite magnetized along z axis, the permeability tensor has a form [20]:  \n \n                                                            \n00\n0\n0 0 1a\nai\ni\n   \n\n\n  .                                                (5)  \n \nIn a quasi -2D ferrite disk with the disk axis oriented along z, the Walker -equation solution for \nthe MS-potential wav e function is written in a cylindrical coordinate system as [26, 28, 34, 35 ]: \n \n                                                                    \n( ) ( , )C z r   \n ,                                                 (6) \n                                          \nwhere \n\n  is a dimensionless membrane function, r and \n  are in -plane coordinates, \n()z  is a \ndimensionless function of the MS -potential distribution along z axis, and \nC  is a dimensional \namplitude coefficient. Being the energy -eigenstate oscillations, the MDMs in a ferrite disk  are \nalso characterized by topologically distinct structures of the fields. This becomes evident from  6 the boundary condition on a lateral surface of a ferrite disk of radius \n , written for a \nmembrane wave function as [26, 28, 34, 35 ]: \n \n                                        \n0a\nr r rirr                             \n .                                  (7)  \n \nEvidently, i n the solutions, one can distinguish the time direction (given by the direction of the \nmagnetization precession and correlated with a sign of \na ) and the azimuth rotation direction \n(given by a sign of \n\n\n\n ). For a given sign of a parameter \na , there can be  different  MS -\npotential wave functions, \n()\n  and \n()\n , corresponding to the positive and negative directions \nof the phase variations with respect  to a given direction of azimuth coordinates, when \n02\n. So a function \n\n  is not a single -valued function. It changes a sign when angle \n  is \nturned on  \n2. \n      Inside a ferrite disk, t he boundary -value -problem solution for Eq. ( 6) is written as  \n \n                            \n, 1, , , cos  sin  i i t\nnrr z t C J z z e e \n   \n  \n       .                    ( 8) \n \nHere  \n is a wave number of a MS wave propagating in a  ferrite a long the z axis, \n   is a \npositive integer azimuth number,  and \nJ  is the Bessel function of order \n  for a real argument. \nThis equation  show s that the modes in a ferrite disk are MS  waves standing  along the z axis \nand propagating along an azimuth coordinate  in a certain (given by a direction  of a normal bias \nmagnetic field)  azimuth direction.  When the spectral problem for the MS -potential scalar wave \nfunction \n( , )rt\n  is solved, distribution of magnetization in a ferrite disk is found as  \nm \n, where \n\n  is the suscep tibility tensor of a ferrite [ 20, 21 ]. The magnetization \nhas both the spin a nd orbital rotation. T here is  the spin -orbit interaction  between these angular \nmomenta . The electric field  in any point inside or outside a ferrite disk  is defined as [ 35, 43] \n \n                                                 \n()\n31( )  4m\nVj r r rE r dV\nrr \n\n ,                              (9) \n \nwhere \n()\n0mj i m\n  is the  density of a magnetic current. In Eq. (9), t he frequency \n  is a \ndiscrete quantity of the MDM -resonance frequency. The magnetic field inside a ferrite disk is \neasily defined from the equ ation \nH\n . Based on the known magnetization \nm\n  inside a \nferrite, one can find also the magnetic field distribution at any point outside a ferrite disk [ 1, \n43]: \n \n                            \n \n331()4VSm r r r n m r r rH r dV dS\nr r r r           \n .            (10) \n  7 In Eqs. (9) and (10), V and S are a volume and a surface of a ferrite sample, respectively . \nVector \nn\n  is the outwardly directed normal to surface S. \n     The azimuthally unidirectional wave propagation  of MDMs are o bserved in mechanically \nnon-rotating quasi -2D ferrite disks . It is worth comparing these oscillations with Sagnac -effect  \nresonances in microcavities. The Sagnac effect is manifested when a sample body rotates \nmechanically. In the  case of the 2D  disk dielectric resonator rotating  mechanically  at angular \nvelocity \n , the resonances are obtained by solving the stationary wave equation  [44]: \n \n                                            \n22\n22\n2 2 21120ik n kr r r r c              ,                           (11) \n \nwhere \n  is an electric field component. The solutions for \n,r  is given as \n ,imr f r e\n, where m is an integer. When the disk cavity is rotating, the wave function \nis the rota ting wave \n()im\nmmJ K r e , where \n2 2 22 ( )mK n k k c m   . The frequency difference \nbetween the counter -propagating waves is equal to \n22( )mn   . For a given direction  of \nrotation, the clockwise ( CW) and counterclockwise ( CCW ) waves inside a dielectric cavity \nexperience different refraction index n. Their azimuthal numbers are \nm .  \n     The wave equation  for MDM oscillations in a quasi -2D ferrite disk , are s imilar , to a certain \nextent,  to the solutions de scribing Sagnac effect in mechanically rotating optical microcavities.  \nWhen we  compare Eq. ( 7) with Eq. (1 1) we see that i n both cases, there are the terms with the \nfirst-order derivative of the wave function with respect to the azimuth coordinate.  Similar  to \nthe Sagnac effect in optical microcavities, MDM oscillations in a ferrite disk are described by \nthe Bessel -function azimuthally rotating MS-potential waves  [26]. However, in our case of \nMDMs, the field rotation is due to topological chiral currents on a lateral surface of a \nmechanically stable ferrite disk. These edge magnetic currents appear because of the spin -orbit \ninteraction in magnetization dynamics.  \n      The above discussion and statement that  the Faraday equation is incompatible with the \nMDM sp ectral solutions we can clarify now  for the case of a quasi -2D ferrite disk . It is known \nthat i n macroscopic electrodynamics, one can define three types of currents: density of the \nelectric displacement current \n0E\nt\n\n , the electric cur rent density arising from polarization \np\nt\n\n , \nand the electric  current density arising from magnetization \nm\n  [1, 22] . For a magnetic \ninsulator [such as yttrium iron garnet (YIG)], we have the macroscopic Maxwe ll equation : \n \n                                            \n00EpBmtt    \n ,                                       ( 12)  \n \nAt the MDM resonances  in YIG , orbitally  rotating MS-potential waves  \n( , )rt\n  provide \nmagnetization (\nm \n ) vectors with spin and orbital rotation and, as a result, the \nE\n \nand \np\n  vectors with spin and orbital rotation .  W e have the situation when the lines of the \nelectric field \nE\n  as well the lines of the polarization \np\n  are “frozen” in the lines of \nmagnetization \nm\n . It means that there are no time variations of vectors \nE\n  and \np\n  with respect \nto vector \nm\n  (and, certainly, with respect to space derivatives of vector \nm\n ). In other words, all \nthe time variations of vectors \nE\n  and \np\n  are synchronized with the time variations of vector \nm\n .  8 In other words, t he lines of the electric field \nE\n  as well the lines of the polarization \np\n  in a \nsample are “frozen ” in the lines of magnetization \nm\n . At this assumption, with taking into \naccount the constitutive relation \n 0 B H m\n , Eq. (12) is definitely reduced to Eq. (2). In \nsuch a case, the Faraday law (3) is not in contra diction with the MS equations (1) and (2 ). We \ncan say  that the entire system of Eqs. (1) – (3) is correct in the coordinate frame of orbitally \ndriven  field patterns .  \n     At the MDM resonance, t he electric field originated from a ferrite disk can cause el ectric \npolarization \np\n  in a dielectric  sample situated outside a ferrite . In the reference frame co -\nrotating with the magnetization  in a ferrite disk , this electric polarization in a dielectric sample \nis not time varying. In the orbitally rotating field patterns, originated from the magnetization \ndynamics in a ferrite disk,  no displacement electric current should be taken into consideration \nand, thus, the MS description is still valid  in a dielectric sample . Fig. 1 illustrates the e lectric -\npolarization dynamics in a dielectri c sample  placed  very closely to  a ferrite disk . For an \nobserver in a laboratory frame, the magnetic and electric fields outside a MDM ferrite disk are \nsynchronically rotating fields. The entire field structure is  called a ME field [35].  \n     It is worth noting , however,  that the basic condition  \n \n                                                             \n0Epmtt \n ,                                                 (1 3) \n \nnecessary for the MS descr iption,  can be  violated when the axes of a ferrite disk and a \ndielectric disk , shown  in Fig. 1 , are shifted or, in general, a dielectric sample is not \ncylindrically symmetric. In these cases the time variations of vectors \np\n  are not synchronized \nwith the time variations of vector \nm\n  and the electric polarization becomes time varying  with \nrespect to the magnetization. As a result, the electric displacement current is not negligibly \nsmall and a component of a curl magnetic field appears. This, however, does not lead to  \n“restoration” of a basic Maxwellian field structure. We concern this problem, more in details, \nin the next section .   \n \nC. Pseudoscalar h elicity parameters  for ME fields   \n \nIn Ref. [36] it was shown t hat at MDM resonances, one has nonzero product \n*mm \n . This \nproduct, characterizing  the way in which the field lines of magnetization curl themselves, we \ncall magnetization helicity parameter . For magnetization  defined  as \nm \n , with an  \nassumption  that MDMs are the fields rotating with an azimuth number \n  \nie , we \nobtain   \n \n                 \n*2 ()2 ( )a\nazm m i C zz r r r r                     \n .                        (14) \n \nHere \n  and \na  are, respectively, diagonal and off -diagonal components of the magnetic \nsusceptibility  tensor \n\n  [20, 21 ]. We can see that the magnetization helicity parameter is purely \nan imaginary quantity:  \n   9                                                  \n*Immm  \n .                                                          (15) \n \nThis parameter, appearing since the magnetization \nm\n  in a ferrite disk has two parts: the  \npotential and curl ones  [35], can be also represented as  \n \n                                                     \n*( ) ( )\n01Remejj\n ,                                               (16)  \n \nwhere \n()\n0mj i m\n  and \n()ejm\n  are, respectively, the magnetic and electric current \ndensities in a ferrite medium  [1, 35].  The pseudoscalar parameter \n  gives evidence for the \npresence of two coupled and mutually parallel currents – the electric and magnetic one s – in a \nlocalized region of a microwave structure.  The magnetization helicity parameter can be \nconsidered as a certain source which defines the helicity properties  of ME fields.  \n      An analysis of helicity properties of ME fields we should start with c onsideration of so-\ncalled optical chirality density . Recently, significant  interest has been aroused by a \nrediscovered measure of helicity in optical radiation – commonly termed optical chirality \ndensity – based on the Lipkin's \"zilch\" for the fields [ 45]. The optical chirality density for \npropagating electromagnetic waves is defined as [ 2, 3]: \n \n                                                     \n0\n01\n22optC E E B B\n   \n .                                     ( 17) \n \nThis is a time -even, parity -odd pseudosc alar parameter. Lipkin showed [ 45], that the chirality \ndensity is zero for a linearly polarized plane wave. However, for a circularly polarized wave, \nEq. ( 17) gives a nonvanishing quantity. Moreover, for right - and left -circularly polarized waves \none has o pposite signs of parameter \noptC . It is evident that for monochromatic electromagnetic \nwaves we have  \n \n                       \n *** 0 0 0 0Im Im Re 04 4 2optC E E H H E H           \n .         (18)  \n \n     Following Ref. [ 46] we consider , formally, two separate  terms in the left -hand side of Eq. \n(18). For EM fields, there are  two pseudoscalar parameters  \n \n                                \n*() 0Im4EF E E  \n    and   \n*() 0Im4HF H H  \n .              ( 19) \n \nTo satisfy Eq. (18) we may have  two cases  for mon ochromatic EM waves:  \n \n                                                           \n()0EF   and  \n()0HF                                                   (20) \n \nor \n \n                                                                    \n( ) ( )EHFF .                                                         (21)                                                   \n \nEvidently, f or Maxwellian fields, both Eqs. (20) and  (21) are trivial .    10       For quasistatic ME fie lds, however, the situation appears  quite different. Based on Eqs. (2) \nand (3) , one obtains for the near fields outside  a MDM ferrite sample [35]:  \n \n                                       \n *( ) * 0 0 0Im Re 044EF E E E H       \n                         (22)  \n \nand      \n \n                                                                            \n()0HF .                                                     (23) \n \nWe call the pseudoscalar parameter \n()E\nMEF  the “electric” helicity of a ME  field [35, 46]. This \nparameter gives e vidence for the presence of mutually parallel components of the electric and \nmagnetic fields in the ME -field structure. Both these , electric and magnetic,  fields – the fields \nin the right -hand side of Eq. (22) – are p otential fields. The outside electric field , however,  has \ntwo components: the potential  [defined by Eq. (9 )] and the curl ones.  The curl electric field is \nfound from the Faraday equation   \n \n                                                                     \n0HEt \n ,                                                (24) \n \nwhere the magnetic field is described by Eq. (10). For a monochromatic field, the curl electric \nfield is perpendicular to the potential magnetic field. This fields ( the curl electric and potential \nmagnetic ) form power -flow vortices  outside a MDM ferrite disk.  \n      Let us suppose now that in a combined structure shown in Fig. 1 , the axes of a ferrite disk \nand a dielectric disk are shifted  or, in general,  a dielectri c sample is not cylindrically \nsymmetric . The electric polarization \np\n  in this dielectric sample outside a ferrite is still induced \nby the electric field originated from a ferrite disk at the MDM resonance.  The electric -\npolarization dynamics  in a dielectric is strongly determined by the magnetization dynamics in a \nferrite, however, t he time variations of vectors \np\n  in are not synchronized with the time \nvariations of vector \nm\n  (which has  the sp in and orbital rotations in a ferrite) . So, the condition \n(13) is not fulfilled .  \n     At the  MDM resonances, the entire  structure  (a ferrite disk and a dielectric sample) is \nmanifested as a strongly temporally dispersive  system . Assuming that a dielectric  sample has \nsizes much less than a free -space electromagnetic wavelength, we can neglect in a dielectric a \ntime variation of magnetic energy in comparison with a time variation of electric energy. The \nsituation appears to be dual with re spect to the above considered quasi -magnetostatic  approach \nin a ferrite sample :  we can neglect the magnetic displacement current and have a just only a \npotential electric field  in a dielectric sample  [35]. In such a quasi -electrostatic approach in a \ndielectric sample,  the e lectric displacement current is not negligibly small  and so  the \ncomponent of a curl magnetic field appears.  In dielectric  regions , where the condition (1 3) is \nnot fulfilled, we have  \n \n                                                                  \n0D \n ,                                                               (25)                                                    \n \n                                                                  \n0E \n ,                                                              (26) \n \n                                                                  \nDHt \n ,                                                          (27)                                                  11  \nwhere \n0 D E p\n . In these equations, a potential  electric field \nE\n  is defined by Eq. (9 ) and a \ncurl magnetic field is accompanied also with a potential magnetic field described by Eq. ( 10). \nSimilar to the considered  above a pseudoscalar parameter of the “electric” helicity, we can \nintroduce now pseudoscalar parameter of the “ magnetic” helicity , \n()HF . Inside a dielectric , we \nhave  \n \n \n                                     \n *( ) * 0 0 0Im Re 044HF H H H E       \n .                   (28) \n \nand \n \n                                                                           \n()0EF .                                                       (29) \n \nThe parameter \n()HF  also gives e vidence  for the presence of mutually parallel components of \nthe electric and magnetic fields. Similar to the pseudoscalar parameter \n()EF [defined by Eq. \n(22)] , in a case of the parameter \n()HF  both the fields in the rig ht-hand side of Eq. (28) are \npotential fields. It is worth noting, however, the electric and magnetic components of the ME \nfields  (and, consequently, the entire structures of the ME fields) , described by Eqs. (22), (23) \nfrom one side and Eqs. (2 8), (29) from the other side, are absolutely not the same.  There are \ntwo different solutions: the quasi -magnetostatic and quasi -electrostatic ones. Moreover, these \ntwo different solutions are related to two different space regions.  \n      Follow ing Refs. [35, 46], we can introduce the normalized helicity parameters. The \nnormalized parameter of the “electric” helicity is expressed as  \n    \n                                                           \n*Im\ncosEE\nEE \n\n  ,                                            (30) \n      \nwhile the normalized parameter of the “magnetic” helicity is expressed as  \n \n \n                                                           \n*Im\ncosHH\nHH \n\n .                                           (31) \n \nThese normalized parameters define angles between the electric - and magnetic -field \ncomponents for two different structures of ME fields: the quasi -magnetostatic and quasi -\nelectrostatic ones.  \n \nD. ME fields and axion electrodynamics  \n \n     The ME fields, being originated from magnetization dynami cs at MDM resonances, appear \nas the fields of axion electrodynamics [47]. In axion electrodynamics, the coupling between an \naxion field (which we, in general, denote \n ) and the electromagnetic field is expressed by an \nadditional t erm in the ordinary Maxwell Lagrangian   12                                  \n                                                                \nEB\n ,                                                         (32) \n \nwhere \n  is a coupling constant . This coupling results in modified electrodynamics  equations \nwith the electric charge and current densities  replaced by [ 47 – 49] \n \n                                                         \n( ) ( )EEB       \n ,                                                   (33) \n  \n                                                  \n( ) ( )EEj j B Et    \n .                                         (34)  \n \nThe axion field \n  transforms as a pseudo scalar s under space  reflection P and it is odd under \ntime reversal T. Importantly, if \n  is a space –time constant then its contribution  to the classical  \nequations of motion vanishes.  \n      The magnetization helicity  V, defined by Eqs. (15), (16), is a pseudoscalar field.  We \nreassign  the magnetization helicity  V as an axion field \n . In the absence of free charges and \nconduction currents we have two equations :  \n \n                 \n                                            \nDB   \n                                                          (35)  \n \nand \n \n                                              \n00EB B Ett        \n .                                   (36)   \n \nIn a bulk magnetic insu lator, the first term in the right -hand side of Eq. (36) can be identified as  \nthe electric -polarization current , while the second term as the magnetization current [49, 50 ]. \nThe magnetization helicity , defined by Eqs. (15), (16), is a time averaged quantit y. So, in Eq. \n(36) we have \n0t . Also, a displacement current \n00E\nt\n\n  is a negligibly small quantity i n \nour consideration. T aking into account Eq. (2) , the constitutive relation \n 0 B H m\n ,  and \nwriting \n\n , where \n  is a dimensional coefficient, we represent  Eqs. (35), (36) as  \n \n                                                                 \n()EB    \n                                                      (37) \n \nand \n \n                                                             \n0mE\n   \n .                                                (38) \n \nEqs. (37), (38) clearly show that a MDM ferrite disk is a particle with the pro perty of \nmagnetoelectric polarizability. Based on these equations one can see how the distributions of \nelectric charges and magnetization are related to the magnetization helicity  factor and \ndistributions of the electric and magnetic fields . For example, l et us consider the main (the 1st \nradial) MDM  in a ferrite disk [28]. For this mode, on a surface of a ferrite in the central region \nof the disk,  we have for the fields are characterized by the following properties. T he rotating  13 electric -field vectors have only in -plane components [35, 43]. Also, \nm\n  and \nm\n  are rotating \nin-plane vectors. \n\n  has mainly the z component. Since vectors \nm\n  and \nm\n  are mutually \n90\n-shifted in space and time [36], we have from Eq. (38):  \n \n                                                                         \nˆ m z E\n ,                                                      (39) \n \nwhere \nˆz  is unit vector along z axis. Eq. (39) shows  that at the MDM resonance there is mutual \ncorrelation of the distribution of vectors \nm\n  and \nE\n  in a ferrite sample. It is evident that the in -\nplane vectors \nm\n  and \nE\n  are reciprocally perpendicular.  \n     At the MDM resonances, we observe special relationships between the magnetization and \nelectric polarization, both inside a  YIG disk and in dielectrics closely abutting to the ferrite. \nWhile , in classical electrodynamics, Maxwell equations well describe the dynamic relations \nbetween electric and magnetic fields, relations between electric polarization and magnetization \nappear as a highly  nontrivial issue. One of the basic reason is that the electrons contribute to \nthe electric polarization and magnetic moments in completely different ways. If these two ways \nare specifically correlated and both the ways of contributions are long -range ordered, the ME \ncoupling (the coupling between electric polarization and magnetization) could be enhanced. \nWith such a ME coupling, the electric and magnetic properties should be understood and \ntreated by a peculiar unified way.           \n     The ME-field helicity densities , defined by Eq s. (22), (30), from the one side, and Eqs. (28), \n(31), from  the other side, transform as pseudo scalar s under space reflection P and are odd \nunder time reversal T. It means that  the ME fields  are pseudo scalar axionli ke field s. Whenever \npseudo scalar axionlike field s, is introduced in  the electromagnetic t heory, the dual symmetry is \nspontaneously and explicitly  broken. This results in non -trivial coupling between pseudoscalar \nquasistatic ME fields and the EM fields  in m icrowave structures with an embedded MDM \nferrite disk.  A special role in this c oupling  plays the effect of interaction of MDM oscillations \nwith metal surfaces.  \n \nE. MDM oscillations in ferrite -metal structures. The ME-EM field interaction in a \nmicrowave wavegu ide \n \nIn the problems, where one considers an interaction of MDM oscillations with a metal surface, \nthe electric displacement current is also neglected. On a metal surface, however, Eq. (2) is \nreplaced by the Ampere equation  \n \n                                                                         \nSSHj  \n .                                                   (40) \n \nIn the MS -wave problems, t he current \nSj\n  is considered as a surface electric current which just \nonly give s discontinuity  of the tangential component of the magnetic  field on  a metal [ 51 – 53]. \nEven so, it is evident , however,  that to define the induced electric current in the Ampere \nequation, the Faraday law should be used. Experimental results of magnetic‐induction probing \nof the fields near a ferrite sample [ 54, 55 ], show that the Faraday equation yields the electric \nfield associated with the magnetic fields of MDMs. In a view of these experiments, one can \nconclude that the near -field interaction of MDM reso nances with external metal elements \ncannot be analyzed without the Faraday law.  \n     Let us consider a flat metal screen placed  in vacuum above a ferrite disk in parallel to the \ndisk plane and at a distance much less than the disk diameter.  The MDM ferrit e disk is a \nparticle with the property of ME polarizability. On a surface of a MDM ferrite disk, there are  14 both the regions of the orbital ly driven normal magnetic and normal electric field s. Moreover, \nthe max imums (minimums) of the electric and magnetic  fields normal to the disk are situated at \nthe same places on the disk plane  [36]. When a metal wall is placed closely to a ferrite -disk \nplane, t his field structure is projected on  the metal . Together with the surface electric charges \ninduced on a metal wall  by the electric field, there are also the Faraday -law eddy currents \ninduced on a metal surface by a time derivative of a normal component of the rotating MDM \nmagnetic field. Evidently, the induced surface electric current \nSj\n  should  have two \ncomponents. There are the linear currents arising from the continuity equation for  surface \nelectric charge s, for which \n0SSj  \n , and the Faraday -law eddy currents, for \nwhich\n0SSj  \n . As a result, we have the sources of the linear -current and eddy -current \ncomponents situated at the same place on a metal wall. Thus, the form of a surface electric \ncurrent is not a closed line. It  is a flat  a spiral  [36].  \n     Unique properties of interaction of MDM s with a metal screen  become more evident when \none analyze s the angular -momentum  balance conditions for MDM oscillations in a ferrite disk  \nin a view of the ME-EM field coupling  in a microwa ve waveguide. MDMs  in a quasi -2D \nferrite disk are microwave energy -eigenstate oscillations with topologically distinct structures \nof rotating fields and unidirectional power -flow circulations. The active power flow  of the field \nboth inside and outside a fe rrite disk  \n* 1Re2P E H\n  has the vortex topology . In the  MDM \nresonances , the orbital angular -momentum density is expressed as  \n \n                                                          \n* 1Re2r E H  \nL ,                                               (41) \n \nwhere \nr\n  is a radius vector from the disk axis. For a given direction of a bias magnetic field, the \npower -flow circulations are the same inside a ferrite and in the vacuum near -field regions  \nabove and below the disk.  Schematically, this is shown in Fig. 2.  \n     Depending on a direction of a bias magnetic field, we can distinguish the clockwise and \ncounterclockwise topological -phase rotation of the fields. The direction of an orbital angular -\nmomentum of a fer rite disk  \n \n                                                   \n*\n01Re2r E H dr  \n                                               (42) \n \nis correlated with the direction of a bias magnetic field \n0H\n  (along +z axis or –z axis). When we \nconsider a ferrite disk in vacuum environment,  such a unidir ectional power -flow circulation  \nmight seem to violate the law of cons ervation of an angular momentum in a mechanically \nstationary system . However, an angular momentum is seen to be conserv ed if topological \nproperties  of electromagnetic fields in the entire microwave structure  are taken into account. In \nRef. [36]  it was shown that due to the topological action of the azimuthally unidirectional \ntransport of energy in a MDM -resonance ferrite s ample there exists the opposite topological \nreaction  (opposite azimuthally unidirectional transport of energy ) on a metal screen placed near \nthis sample. T his effect is called topological Lenz’s effect. In a microwave structure with an \nembedded ferrite dis k, an orbital angular momentum, related to the power -flow circulation, \nmust be conserved in the process. Thus, if power -flow circulation is pushed in one direction in \na ferrite disk, then the power -flow circulation on metal walls to be pushed in the other direction \nby the same torque at the same time. Fig. 3 illustrates the orbital angular -momentum balance  \nconditions at a given direction of a bias magnetic field.    15      In Ref. [36], an analysis of conservation of an angular momentum  was made in microwave \nwaveguiding structures with the metal walls situated very close to the ferrite -disk surfaces . A \nthin ferrite disk is placed inside a rectangular waveguide symmetrically to its walls so that the \ndisk axis is perpendicular to a wall , as it is shown in  Fig. 4.  Vacuum gaps between the metal \nand ferrite 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 this  \nstructure, a role of a linear EM -wave momentum is negligibly small . So, t he orbital  angular -\nmomentum balance does not depend, actually, on the direction of the electromagnetic wave \npropagation in a waveguide. With taking into account that MDMs  in a quasi -2D ferrite disk are \nmicrowave energy -eigenstate oscillations , we have the angular momentum  quantization as \nwell. The observed properties of interaction between a MDM ferrite disk and a metal screen \nrely on ME virtual photons to act as the mediator.  There are two near-field elements of this \ninteraction:  (a) the static electric force and ( b) the electromagnetic induction.  \n     However, the fields in a ferrite disk rotate at microwave frequencies and situation becomes \nmore complicated when the vacuum -region scale is about the disk diameter or more  and the \nfinite speed of wave propagation in vacuum – the retardation effects – should be taken into \nconsideration. This means that for a brief period , the total angular momentum of the two \ntopological charges (one in a ferrite, another on a metal) is not conserved, implying that the \ndifference should be accounted for by an angular momentum in the fields in the vacuum space  \nin a waveguide . It means that the magnetization dynamics have an impact on the phenomena \nconnected with fluctuation energy in vacuum. A s the rotational symmetry is broken in this \ncase, the Casimir torque [ 56 – 60] arises because the Casimir energy now depends on the angle \nbetween the directions of the magnetization vectors in a ferrite and electric -current vectors on a \nmetal wall. A vacuu m-induced Casimir torque allows for torque transmission between the \nferrite disk and metal wall avoiding any direct contact between them.  \n     An e xperimental proof of the predicted above Casimir -torque effect in microwaves is a n \nimportant problem. H oweve r, even in optics, such an  experiment appears as  an open question. \nWhile the Casimir force has  been measured extensively, the Casimir torque has not been \nobserved experimentally though it was  predicted over forty years ago.  Some ideas proposed \nrecently to detect the Casimir torque with an optics (see, e. g. [ 61] and references therein) can \nbe usef ul for the proposal of analogous experimental methods in microwaves.  \n \nIII. EVIDENCE FOR QUANTIZED STATES OF THE ME FIELDS ORIGINATED \nFROM MDM OSCILLATIONS  \n \nThe eigenval ues of MDM resonances  are defined by discrete quantities of a diagonal \ncomponent  of the permeability  tensor [26 – 28, 34] . For a ferrite sample, magnetized at \nsaturation magnetization \n0M , the diagonal component  of the tensor (5)  is found  as [20] \n \n                                                             \n22\n00\n2 2 2 2\n01i\niMH\nH   ,                                                 ( 43) \n \nwhere \n is the gyromagnetic ratio and \niH  is a DC internal magnetic field. In neglect of \nmaterial anisotropy , the internal magnetic field is calculated as  \n \n                                                                    \n0 idH H H\n  ,                                                    (44) \n  16  where \n0H  is a bias magnetic field  and \ndH  is a demagnetization field [20]. An analysis [26 – \n28, 34] shows that real e nergy -eigenstate solutions for MS -potential wave functions in a quasi -\n2D ferri te disk are obtained when \n0 . This corresponds to the frequency range  [20]: \n \n                                                                     \nH   ,                                                      (45) \n \nwhere \n0 Hi H  , \nH H M    . Here \n00 M M  . The magnetic -field range for \nnegative quantities \n  are defined as  [20] \n \n                                                                    \n0 iiHH    ,                                               (46) \n \nwhere \n2 2\n00\n0 22iMMH\n   .  \n     Sharp  multiresonance oscillations, observed experimentally in microwave structures with an \nembedded quasi -2D ferrite disk  [29 – 32], are related to magnetization dynamics  in the sample . \nThis dynamics  have  an impact on the phenomena connected with the quantized energy \nfluctuation . For given sizes of a disk and a given quantity  of saturation magnetization  of ferrite \nmaterial \n0M , there are two different mechanisms of the MDM energy quantization: ( i) by a \nsignal frequency\n  at a constant bias magnetic field \n0H  and ( ii) by a bias magnetic field \n0H  at \na constant signal frequency \n . Fig. 5 shows the FMR diagonal component of the permeability \ntensor \n  versus frequency \n  at a constant internal magnetic field \niH . The FMR diagonal \ncomponent of the permeability tensor versus an internal magnetic field at a constant frequency \nis shown in Fig. 6. For both these cases, the discrete quantities of \n , in the region where \n0\n, are shown schematically  for the first four MDM resonances.  \n    Fig. 7 illustrates  correlation between the two  mechanisms of the MDM energy  quantization.  \nFor a certain frequency \nf , the energy quantizatio n is observed at specific bias -field \nquantities:\n(1)\n0H , \n(1)\n0H , \n(1)\n0H , …. Evidently, one has to use a statistical description of the \nspectral response functions of the system with respect to t wo external parameters – a bias \nmagnetic field \n0H  and a signal frequency\n  – and analyze the correlations between the \nspectral response functions at different values of these external parameters. It means t hat, in \nneglect of losses, there should exist a certain uncertainty limit  stating that  \n \n                                                          \n0fH    uncertainty limit .                                          ( 47)     \n \nThis uncert ainty limit is a constant which depends  on the disk size parameters and the ferrite \nmaterial property ( such as saturation magnetization) . Beyond the frames of the uncertainty \nlimit ( 47) one has continuum of energy. The fact that there are different mechani sms of \nquantization allows to conclude that for MDM oscillations in a qu asi-2D ferrite disk both \ndiscrete energy eigenstate and a continuum of energy can exist. In quantum mechanics, the \nuncertainty principle says that the values of a pair of canonically c onjugate observables cannot \nboth be precisely determined in any quantum state. In a formal harmonic analysis in classical \nphysics, the uncertainty princi ple can be summed up as follows:  A nonzero function and its \nFourier transform cannot be sharply localiz ed. This principle states also that there exist \nlimitations in performing measurements on a system without disturbing it. Basically,  17 formulation of the main statement of the MDM -oscillation theory is impossible without using a \nclassical microwave structure . If a MDM particle is under interaction with a “classical \nelectrodynamics” object, the states of this classical object change. The character and value of \nthese changes depend on the MDM quantized  states and so can serve as its qualitative \ncharacteristics.  The microwave measurement reflects interaction between a microwave \nstructure  and a MDM particle. It is worth noting that for different types of subwavelength  \nparticles, the uncertainty principle may acquire different  forms. An interesting variant of \nHeise nberg’s uncertainty  principle was shown r ecently in subwavelength optics [62 ]. Being \napplied to the optical field, this principle says that we  can only measure the electric or the \nmagnetic field with accuracy when the volume in which they are contained is  significantly \nsmaller than the wavelength of light in all three  spatial dimensions. As volumes smaller than \nthe wavelength  are probed, measurements of optical energy become uncertain,  highlighting the \ndifficulty with performing measurements  in this regime.  From this statement, we can see, once \nagain, that a MDM particle, distinguished by strong ME properties in subwavelength \nmicrowaves, is beyond the regular EM-field description.   \n     The fact that magnetization dynamics in a quasi -2D ferrite disk have an  impact on the \nenergy quantization of the fields in a microwave cavity, was confirmed experimentally in Ref. \n[32]. In this work, the MDM -originated quantized states of the cavity fields were investigated  \nwith variation of  a bias magnetic  field at a constan t operating frequency, which is a resonant  \nfrequency of the cavity.  The observed discrete variation of  the cavity impedances are related to \ndiscrete states of the cavity  fields . Since the effect was obtained  at a certain resonant \nfrequency,  the shown reson ances are not the modes related  to quantization  of the photon wave \nvector in a cavity.  It is evident that these resonances should be  caused by the quantized \nvariation of energy of a ferrite  disk, appearing due to variation of energy  of an external source – \na bias magnetic field.  In Ref. [32], the observed effect of energy quantization of the fields in a \nmicrowave structure in relation to quantization of magnetic energy in a  ferrite disk is analyzed \nqualitatively as follows. At the regions of a bias magnetic field, designated in Fig. 8 as A, a, b, \nc, d,  …, we do not have MDM resonances. In these regions, a ferrite disk is “seen” by \nelectromagnetic waves, as a very small obstacle which, practically, does not perturb a \nmicrowave cavity. In this case, the cavity (with an embedded ferrite disk) has good impedance  \nmatching  with an external waveguiding structure and a microwave energy accumulated in a \ncavity is at a certain maximal level. At the MDM resonances (the states of a bias magnetic \nfields designated in Fig. 8 by numbers 1, 2, 3, …), the reflection coefficient sharply increases. \nThe input impedances are real, but the cavity is strongly mismatched with an external \nwaveguiding structure. It means that at the MDM resonances, the cavity accepts smaller energy \nfrom  an external microwave source . In these states of a bias magnetic field, the microwave \nenergy accumulated in a cavity sharply decreases, compared to its maximal level in the A, a, b, \nc, d, … Since the only external parameter, which varies in this experimen t, is a bias magnetic \nfield, such a sharp ejection of the microwave energy accumulated in a cavity to an external \nwaveguiding structure should be related to emission of discrete portions of energy from a \nferrite disk. It means that at the MDM resonances, w e should observe strong  and sharp  \nreduction of magnetic energy of a ferrite sample.  \n     The question on a proper explanation of the experimentally observed q uantized states of the \nfields in a microwave structure  with an embedded ferrite disk remains stil l open. In Refs. [29, \n30, 63], it was stated that the main reason of appearing the multiresonance oscillations  is non -\nhomogeneity of an internal DC magnetic field in a ferrite -disk sample. To answer the question \nwhether non -homogeneity of an internal magne tic field can really lead to quantization of \nmagnetic energy of a ferrite disk, we analyze briefly the model suggested in Ref. [30] and \nmodified in Ref. [64].   18      In a supposition that saturation magnetization \n0M  is uniform everyw here inside the sample , \nthe demagnetization magnetic field  in a quasi -2D ferrite disk varies  only in the radial direction \nand is defined as [30] \n \n                                                                \n0 dH r M I r\n ,                                                 (48) \n \nwhere \nIr  is a dimensionless  scalar quantity and \n0r\n .  In Ref. [30], the multiresonance \nabsorption peaks are interpreted to be  caused by MS waves propagating radially acro ss the disk \nwith the internal magnetic field dependent on a radial  coordinate  in the  plane of a YIG film and  \nthe mode numbers are determined  based on the well -known Bohr –Sommerfeld quantization  \nrule. This model is illustrated in Figs. 9 and 10. Fig. 9 show s schematically the Bohr –\nSommerfeld quantization  rule for a certain MDM. A qualitative picture of the levels of an \ninternal magnetic field for standing waves corresponding to some MDMs is shown in Fig. 10. \nWith increasing the mode number, the effective dia meter \n()2n\neff\n  of the disk increases as well. \nThe ‘‘in -plane’’ MS -potential -function distribution is  supposed to be azimuthally \nnondependent.  \n     However, the Bohr –Sommerfeld quantization  used in Ref. [30] does not clarify the problem \nof the observed quantization of magnetic energy of a ferrite -disk sample. This quantization \nshould be observed as a Zeeman splitting of the energy levels created by a DC internal \nmagnetic field. In an assumption that saturation magnetization \n0M  is uniform inside the \nsample and is the same for every mode, one cannot suggest any mechanism of Zeeman \nsplittings  at the MDM resonances . When a ferrite disk is placed in a homogeneous external \n(bias) magnetic field \n0H , magnetic en ergy of the entire  sample is varied monotonically with \nvariation of \n0H , even if the MS -wave numbers are quantized and the effective diameter \n()2n\neff\n  \nis quantized as well.      \n     Magnetic energy o f a sample in an external (bias)  magnetic field  is determined by the \ndemagnetization field. The  demagnetization field  is the  magnetic field  generated by \nthe magnetization  in a magnet  and t he demagnetization factor det ermines how a magnetic \nsample responds to a n external (bias)  magnetic field.  It is evident that for the MDM spectra, \nobtained at variation of a bias magnetic field \n0H  and at a constant signal frequency \n , a \ndiscrete reduction of magnetic energy of a ferrite disk at the MDM r esonance should occur \nbecause of quantization of the demagnetization field. Contrary to Ref, [30], will consider of \nquantization of the demagnetization field due to quantization of DC magnetization. Following \nthe technique described in Ref. [64], we can pu t aside the question on a role of \nnonhomogeniety of an internal magnetic field in a ferrite disk.  With averaging  of the parameter \nIr\n on the  region  of the actual diameter of the disk,\n2\n , one can write [64]  \n \n                                                           \n 0 idaverage averageH H H ,                                        (49)  \n \nwhere  \n \n                                                               \n0 d averageaverageH M I .                                           (50) \n \nHere \n01averageI . The quantity of \naverageI  depends on the disk geometry. With an \napproximation of a homogeneous (averaged on the disk -diameter region) internal magnetic  19 field, one can obtain the diagonal permeability -tensor component  \n from Eq. (43). The \nspectral problem is analyzed with \n -function distribution s in a form of Eq. (6). The \neigenvalues of MDM resonances  are defined by discrete quanti ties of the  diagonal \npermeability -tensor component  \n. In this case, one can classify MDMs as the thickness, \nradial, and azimuthal modes. The spectral problem for these modes gives evidence for energy \neigenstate oscillations  [26, 2 8]. The technique used in Ref. [64] gives a good agreement with \nexperimental results  of the resonance mode position in the spectra.  \n     For simplicity of a further analysis, we will suppose  that \n1averageI . Assuming that \nquantization of m agnetic energy of a ferrite -disk sample is due to quantization of the DC  \nmagnetization, the demagnetization magnetic field for a certain MDM with number n (n = 1, 2, \n3, …) , is found as  \n \n                                                                 \n( ) ( )\n0nn\ndeffHM ,                                                  (51)  \n \nwhere \n()\n00n\neffMM  is a quantized DC magnetization . We can write  \n \n                                                                 \n( ) ( )\n00nn\neffM M K ,                                                (52) \n \nwhere the mode coefficient is a dimensionless quantity: \n()01nK . The number \n1n  \ncorresponds to the main MDM [28].  \n     For mode n, the magnetic energy of a samp le of volume V, placed in a magnetic field \n0H , is \ncalculated as  \n \n                                                 \n( ) ( )\n0011\n22nn\neff\nVW H MdV H M V   \n .                             (53)  \n \nDiscreteness of magnetic energy in a ferrite disk is d ue to discrete reduction:  \n \n                                      \n ( ) ( ) ( )\n0011122n n n\neff\nVW H M dV H K M V    \n .               (54)  \n \nThe energy \n()nW  is the microwave energy extracted  from  the magnetic energy of a ferrite \ndisk at the n-th MDM resonance . The effect is illustrated in Fig. 11. Based on t his model, we \ncan explain qualitatively how the multiresonance states in a microwave cavity, experimentally \nobserved in Ref. [32], are related to quantized variation of energy of a ferrite  disk, appearing \ndue to an external source  – a bias magnetic field.  When accepting this model, we can say that \nwe have a quantum effect of electromagnetically generated demagnetization of a sample.  \n     What is the physics of quantization of a DC magnetization in a ferrite  disk? Unidirectionally \nrotating fields with the spin and orbital angular momenta , observed at MDM resonances,  are \nrelated to precessing magnetic dipoles in a ferrite and to a double -valued -function magnetic \ncurrent on a later al surface of a ferrite disk [ 34, 35]. Circulation of the  chiral surface magnetic \ncurrent results in appearance of a DC gauge electric field [34] and thus appearance of DC \nelectric charges on the ferrite -disk planes. When a dielectric -disk sample is situated in a \nvacuum near a ferrite disk, it is subjected with an induced electric gyrotropy and orbitally \ndriven electric polarization  [35]. Because of the electric field originated from a ferrite disk,  20 every separate electric dipole in a dielectric disk  precesses around its own axis and fo r all the \nprecessing  dipoles, there is an orbital phase running  (see Fig. 1) . Due to the orbital angular \nmomentum of MDM oscillations, a torque exerting on the electric polarization in a dielectric \nsample should be equal to  a reaction torque exerting  on th e magnetization in a ferrite disk. \nBecause of this  reaction torque, the precessing magnetic moment density of  the ferromagnet \nwill be under additional mechanical rotation  at a certain frequency . At dielectric loadings, the \nmagnetization motion in a ferrite  disk is characterized by an effective magnetic field  [35, 65]. \nThis is a DC gauge (topological) magnetic field caused by precessing and orbitally rotating \nelectric dipoles  in a dielectric sample . As a result, an external  effective magnetic field becomes \nless than a bias magnetic field \n0H . So, at a dielectric  loading  of a ferrite sample, the Larmor \nfrequency \nH  should be lower than such a frequency for an unloaded ferrite disk.  In othe r \nwords, one can say th at due to a loading by an external dielectric sample, the effective DC \nmagnetic  charges appear on the ferrite -disk planes.  \n    At the same time, it is worth noting that a  ferrite disk is made of a magnetic dielectric  – \nyttrium iron garnet (YIG) – which has  sufficiently high permittivity, \n15r . Inside the YIG \ndisk, we also have the torque exerting on the electric polarization due to the magnetization \ndynamics of MDM oscillations. Because of the effective magnetic charges on a ferrite -disk \nplanes (caused now by the induced electric gyrotropy and orbitally driven electric polarization \ninside a ferrite), the demagnetizing magnetic field is redu ced. It means that the DC \nmagnetization of a ferrite disk is reduced  as well . We have the frequen cy \n( ) ( )\n00nn\nMeff effM  \n, which is less than such a frequency \n00 M M   in an unbounded \nmagnetically saturated ferrite. In connection with the effect discussed above, it is relevant to  \nrefer here to some recent studies in optics.  I n Ref. [66], l ight-induced magnetization using \nnanodots and c hiral molecules  was experimentally studied. It was shown that a torque \ntransferred  through the chiral layer to a ferromagnetic layer, can create  local perpendicular \nmagnetization.  The experiment is based on the chiral -induce d spin -selectivity effect described \nin Refs. [67 – 70]. An important conclusion a rises also from the above consideration . When at \na constant frequency of a microwave signal we vary a bias magnetic field, at MDM resonances \nwe ca n observe a DC magnetoelectric effect. We have  quantized DC electric and magnetic \ncharges on the ferrite -disk planes.  \n     Experiments in Ref. [32] shows that for the quantized states of microwave energy in a \ncavity and magnetic energy in a ferrite disk , the cavity input impedance s on the complex -\nreflection -coefficient plane (the Smith chart) [71] are real numbers  (see Fig. 12) . We have a \ntwo-state system. As the energy swept through an individual resonance, one observes evolution \nof the phase – the phase l apses. From the Smith chart one can also see that t he phase jump of \n\n is observed each time a resonant condition is achieved.  In a 2D parametric space – the \nimpedance space on a Smith chart – there is  clockwise or counter clockwis e circulation for the \nMDM states. Direction of the circulation should be correlated with the direction of a bias \nmagnetic field \n0H . In Fig. 12 we, contingently , showed the clockwise circulation.   \n     At the first glance, the anal yzed above processes in a structure shown in Fig. 8 can be \ndescribed based on a simple scheme shown in Fig. 13. At a given frequency \n , determined by \na RF source, a microwave waveguide is presented by a characteristic impedance \n0Z . The \nwaveguide is loaded by an impedance \nLZ , which depends on an external parameter – a bias \nmagnetic field \n0H . Such a simple scheme, however, leads us to a n evident  contradiction.  With \nacceptance of the fact that the microwave energy extracted from the magnetic energy of a \nferrite disk at the MDM resonance  is a quantized quantity, we cannot assume , at the same time,  \nthat the frequency of a microwave photon  propagating in a waveguid e is a constant quantity.   21 IV. MDM RESONANCES AND BOUND STATES IN THE MICROWAVE \nCONTINUUM  \n \nWhen analyzing the scattering of EM waves by MDM  disks in microwave waveguid es and \nenergy quantization of the field in a microwave cavity, it is relevant also to dwell o n some \nbasic problems of magnon -photon interaction and bound states in the microwave continuum. \nWe are witnesses that l ong-standing research in coupling between electrodynamic s and \nmagnetization dynamics noticeably reappear  in recent  studies  of magnon -photon interaction . In \na series of works [ 72 – 76], it was shown that magnetostatic modes in a small YIG sphere  can \ncoherently interact with photons in a microwave cavity. In a small ferromagnetic particle,  the \nexchange interaction can cause a very large  numbe r of spins to lock together into one \nmacrospin with a  corresponding increase in oscillator strength.  This results in s trong \nenhancement of spin -photon coupling relative to  paramagnetic spin systems  [72]. The total \nHamiltonian of the system incorporates the  magnetic  \nH\n and electric  \nE\n fields of the cavity and \nthe \nM\n magnetization of the ferromagnetic particle [ 1] \n \n                                                \n223\n0 0 01\n2H E H M d r     \n .                           (5 5) \n \nThe spatially uniform mode  of the magnetization dynamics  is called the Kittel mode. The \nKittel and cavity modes form magnon -polariton  modes, i.e., hybridized modes between the \ncollective spin excitation and the cavity e xcitation. In the theory, t he interaction between the \nmicrowave photon and magnon is described by the Hamiltonian with a rotating -wave \napproximation (RWA) [7 2, 73]. In a structure of a  microwave cavity with a YIG sphere inside,  \nthe avoided crossing in the microwave reflection spectra  (obtained with respect to the bias \nmagnetic field ) verifies  the strong coupling between the microwave photon and the magnon.  \nThe Zeeman energy is defined by a coherent state of the macrospin/photon system when a \nmagnetic dipole  is in its antiparallel orientation to the cavity magnetic field.  The coherent \nenergy exchange occurs back and forth between photon and a macrospin states. It is pointed \nout that  because of non -uniformity of the RF magnetic field  in sufficiently big YIG sp heres , \nhigh-order magnon modes – the MDMs – can be excited . For such  MDMs , the avoided \ncrossing in the reflection spectra, obtained with respect to the bias magnetic field, is observed \nas well  [73 – 76]. Importantly, for these modes, characterizing by non -uniform magnetization \ndynamics, the model of coherent states of the macrospin/photon system discussed above, is not \napplicable.  \n     The MDMs in YIG spheres were observed long ago [77]. The re are so -called Walker modes \n[37]. While for  a ferrite sphere one sees a few broad MDM absorption peaks , the MDMs in a \nquasi -2D ferrite disk are presented with the spectra of multiresonance  sharp peaks.  There are \nvery rich spectra of both types, Fano and Lortenzian, of the peaks [29 – 32]. The ferrite  disk is \nan open hi-Q resonator embedded in a microwave waveguide or microwave cavity. In such a \nstructure, sharp MDM resonances can appear as bound states in a microwave -field continuum.  \n     Bound states in the continuum (BI Cs), also known as embedded trapped modes,  are \nlocalized solutions which correspond to discrete eigenvalues coexisting with  extended modes \nof a continuous spectrum . The BICs are solutions having an infinitely long lifetime. Recent \ndevelopments show that in a large variety of electromagnetic structures there can be  different \nmechanisms that lead to BICs [78]. One of the main reasons for appearance of the MDM BICs \nis a symmetry mismatch. Modes of different symmetry classes (such as reflection or rotation) \nare completely decoupled. MDM oscillations do not exh ibit a rotational symmetry, while a  22 regular waveguide structure is rotationally symmetric. With such a condition, t he MDM bound \nstates observed in microwave structures can be classified as  the symmetry -protected BICs.  \n     The MDM bound states are embedded  in the microwave continuum but not coupled to it.  \nCertainly, if a bound state of one symmetry class is embedded in the continuous spectrum of \nanother symmetry class their coupling is forbidden. Moreover, t he ME  fields, originated from a \nMDM ferrite disk,  and the EM fields in a microwave structure are described by different types \nof equations. At any stable state, MDM s cannot radiate because there is no way to assign a far -\nfield EM-wave polarization that is consistent with vortex ME fields near a MDM ferrit e disk. \nIn a short -range interaction, an important aspect concerns the topological nature of the MDM \nBICs. These topological properties can be understood through eigen power -flow vortices with \ncorresponding topological charges  [34 – 36]. Quantized topologi cal charges cannot suddenly \ndisappear. They are protected by special boundary conditions in a quasi -2D ferrite disk. The \nMDM BICs cannot be removed unless MDM topological charges are cancelled with another \nstructure carrying the opposite topological charge s. Such opposite topological charges appear \non metal wal ls of a microwave waveguide . The probing of these BICs is due to topological -\nphase properties of MDMs resulting in appearance of spiral electric currents induced on metal \nparts of a microwave structur e [36].  The coupling is possible via the continuum of decay \nchannels  at the condition that phases of the MDM bound states are strongly determined phases.  \n     The region where a MDM ferrite disk is situated in a microwave waveguide is a contact \nregion. In this region, the conditions of  tunneling and pairwise coalescence of waveguide \ncomplex modes can be fulfilled . A model of i nteraction of MDM resonances with the \nmicrowave -waveguide continuum  is shown schematically in Fig. 1 4. In the system, the whole \nwave  function can be divided into an internal, localized MDM  system Q and an external part of \nenvironmental microwave  states  P [79 – 81]. The regions in a space between a ferrite disk and \nwaveguide walls are contact regions. Surface electric currents on wavegu ide walls are edge \nstates. For a whole spectrum of waveguide modes , there should be  a continuum of such edge \nstates. A ferrite disk is considered as a defect interacting with these edge states. We suppose \nthat the region where the defect is localized, is s mall compared with the cha racteristic length \nscale of waveguide modes. Interaction of MDM resonances with the microwave -waveguide \ncontinuum is realized by two ways (channels): ( i) interaction of MDMs with metal walls and \n(ii) coupling of surface helical bo und states (induced on the walls at MDM resonances [36]) \nwith a continuum of waveguide edge states on metal walls. In a structure of a  thin rectangular \nwaveguide with an embedded quasi -2D ferrite disk [36], the contact regions can be considered \nas two vacu um cylinders with the same diameter as a ferrite disk. This model is inapplicable in \na case of a  “thick” microwave waveguide [8 2, 83]. However, use of two dielectric cylinders \nwith a high dielectric constant allows considering these cylinders as the contac t regions in a \n“thick” microwave waveguide . Such structures are shown in Ref.  [83]. In the contact regions, \nabove and below a ferrite disk, we have helical -mode tunneling. There is an evidence for a \ntorsion structure of the fields in the contact regions  [36, 83] . Rotations of the power -flow \nvortices along an axis of contact regions are at different directions. The model is shown in Fig. \n15. \n     The appearance of BICs is directly related to the phenomenon of an avoided level crossing \nof neighbored resonance  states.  BICs  can occur due to  the direct and via -the-continuum \ninteraction between  quasistationary states and can be viewed as resonances  with practically \ninfinite lifetimes.  In a system of one open channel and two discrete resonances, these \nresonances ca n be coupled via the continuum. There is a short -range interaction with a purely \nimaginary coupling term . If two resonances pass each other as a function of a certain \ncontinuous parameter, one of the resonance states can acquire zero resonance width. This \nresonance state becomes a BIC. Whether or not two resonances interfere is not directly related \nto whether or not they overlap. Such a mechanism of cutting down of a discrete eigenstate from  23 any connection with a continuum was  described initially in Ref. [ 84] in a framework of a two -\nlevel model . In electromagnetic systems, t he phenomenon of avoided crossing of narrow \nresonance states under the influence of their coupling to the continuum is considered in many \nrecent publications  (see Ref.  [78] and references  therein) . The BICs remain perfectly confined \nwithout any radiation. For an open resonance structure , we have non -Hermitian effective \nHamiltonian with complex eigenvalues. The BIC can be found by the condition that at least \none of the complex eigenvalues b ecomes real.  \n    From a number of characteristic features inherent in the BICs, one should  distinguish such a \nproperty  as a Fano resonance collapse. For the MDM resonances the effect of Fano resonance \ncollapse  was clear ly demonstrated in Ref. [ 85]. Thanks  to the tenability of the ferrite -disk \nresonator by an external parameter (the bias magnetic field, for example) the MDM oscillations \ncan become close interacting modes. Fig. 1 6 shows how the physical parameter – a bias \nmagnetic field – tunes the shape  of the MDM resonance. The structure used in Ref. [85] is a \nmicrowave cavity with an embedded thin -film ferrite disk. It is seen that as we approach the \ntop of the cavity  resonance  curve , the minimum of the microwave transmission approaches the \nmaximum of the transmission. At the top of the cavity  resonance  curve,  these levels (minimum \nand maximum  transmission) are in contact. The Fano line shape is completely damped and one \nobserves a single Lorentzian peak . The scattering cross section corresponds to a pure d ark \nmode.  What are the neighbored resonance states in this MDM oscillation? There  are the dipole \n(dark) and quadrupole (bright) resonances [ 86]. Such  resonances are shown in Fig. 17 . These \nresonances, being energy degenerated, appear due to orbital angular  momentums originated \nfrom edge magnetic currents on a lateral surface of a ferrite disk [ 34].    \n     When t he contact regions – the space above and below a ferrite disk  – are dielectric \ncylinders with a high dielectric constant [8 3], the coupling of two resonance states is via the \ncontinuum of decay channels in a dielectric region. With variation of a permittivity of \ndielectric cylinders \nr  loading a ferrite disk, discrete eigenstates can be cut down from any \nconnection with a con tinuum at a certain threshold parameter\nr . This effect of the Fano \nresonance collapse  in such a structure is shown in Fig. 1 8. One can see that at \n38r , the \nFano line shapes are completely damped f or the 1st and 2nd MDMs and  single Lorentzian peak \nappear. The scattering cross section corresponds to pure bright modes.  \n     One of the features attributed to the BICs is also strong resonance field enhancement. In the \ncase of MDM oscillations in a microwave -field continuum, such a strong field enhancement is \nshown in numerous numerical studies [32, 35, 8 2, 86, 87]. For example, Fig. 1 9 shows passing \nthe front of the electromagnet ic wave, when the frequency is (a) far from the frequency o f the \nMDM resonance and ( b) at the f requency of the MDM resonance in  the disk.  This is an \nevidence for strong resonance field enhancement at the MDM resonance.  \n     Recently, the MDM BIC  phenomena , found further development in a novel technique based \non the combination of the micro wave perturbation method  and the Fano resonance effects \nobserved in microwave structures with embedded  small ferrite disks [8 5, 88]. When the \nfrequency of the MDM resonance is not equal to the cavity resonance frequency, one gets Fano \ntransmission intensit y. If the MDM resonance frequency is tuned to the cavity resonance \nfrequency, by a bias  magnetic field, one observes a Lorentzian line shape. The effect of Fano \nresonance collapse  has no relations to the quality factor of a microwave cavity. Use of an \nextremely narrow Lorentzian  peak allows exact probing of the resonant frequenc y of a cavity \nloaded by a  high lossy  material sample. With variation of a bias magnetic field, one can see  \ndifferent frequencies of Lorentzian  peaks  for different kinds of material samples.  This gives  a \npicture of precise spectroscopic characterization of high absorption  matter in microwaves, \nincluding biological liquids.  Importantly, there is no influence of the dissipation effects in the  24 microwave cavity on the quality of the MDM re sonances. The poles in the transmission \namplitudes are connected with the bound states and their lifetimes.  \n  \nV. PT-SYMMETRY OF MDMS  IN A FERRITE DISK  \n \nA MDM ferrite disk is an open resonant system. In the description of such a system, a non -\nHermitian effecti ve Hamiltonian can be derived from the Hermitian Hamiltonian including the \nenvironment. Because of strong spin –orbit interaction in the MDM magnetization dynamics in \nferrite disks, the fields of these oscillations  have helical structure s. The right -left as ymmetry of \nMDM rotating fields is related to helical resonances: The phase variations for resonant \n  \nfunctions are both in azimuth \n  and in axial z directions. This shows  that proper solutions of a \nspectr al problem for MDMs should  be obtained in a helical -coordinate system. The helices are  \ntopologically nontrivial structures, and the phase relationships  for waves propagating in such \nstructures could be very special.  Unlike the Cartesian - or cylindrical -coordinate systems, the  \nhelical -coordinate system is not orthogonal, and separating  the right -handed and left -handed \nsolutions is admitted.  In a helical coordinate system, the solution for MS -potential wave \nfunction for MDM oscillations in an open ferrite -disk resonator have four components. In a \nsingle -column matrix, this components are presented as  \n \n                                                                   \n(1)\n(2)\n(3)\n(4)\n\n\n\n\n\n  ,                                                          (56) \n \nwhere the components of the matrix are distinguished as forward right -hand -helix  wave, \nbackward right -hand -helix wave, forward left-hand -helix  wave, and backward left-hand -helix \nwave  [89].  \n     For helical modes, there are no properties of parity ( P) and time-reversal ( T) invariance —\nthe PT invariance.  The PT-symmetry breaking does not  guarantee real -eigenvalue spectra.  It \nwas shown, however, that by virtue of the  quasi -2D of the problem  for a thin -film ferrite disk , \none can reduce solutions from  helical to  cylindrical coordinates with a proper separation  of \nvariables . As shown [ 89], for the double -helix resonance,  one can introduce the notion of an \neffective membrane  function \n\n . This allows reducing the problem of parity  transforma tion to a \none-dimensional reflection in space and describe  the boundary -value -problem solution for the  \nMS-potential wave function by Eq. (8). In such a case, t he integrable solutions for MDMs  in a \ncylindrical -coordin ate system can be considered as PT invar iant. \n     At MDM resonances, we have BICs with two degenerated real -eigenvalue resonances and \nan imaginary -term coupling. The Fano -resonance collapse, observed at variation of a quantity \nof a bias magnetic field, corresponds to the BIC state. BICs are con sidered as resonances with \nzero leakage and zero linewidth (in other words, a resonance with an infinite quality factor). In \nour case of MDM oscillations, BICs appear when a microwave structure is PT–symmetrical . A \nPT-symmetric Hamiltonian is in general no t Hermitian,  but if the corresponding eigenstates are \nalso PT-symmetric, then the eigenvalues are  real and eigenstates may be complete  [90, 91] . In \nPT-symmetry MDM structures, the time direction is defined by the direction of the spin and \norbital  rotation s of the fields  in a ferrite disk  [35, 43] . Depending on a direction of a bias \nmagnetic field, we can distinguish the clockwise and counterclockwise topological -phase \nrotation of the fields. The direction of an orbital angular -momentum expressed by Eq. (42 ) is \ncorrelated with the direction of a bias magnetic field \n0H\n  (along +z axis or –z axis),  determines  25 the time direction in our “topological clock”.  Both directions of rotation are equivalent and the \nbiorthogonality relations for MD Ms can be used. Importantly, the MDM resonance in a PT \nsymmetrical microwave structure can reveal the properties analogous to the effect of spectral \nsingularities of complex scattering potentials  in PT-symmetric gain/loss structures [ 90]. The \nwave incident  on a MDM ferrite disk induce s outgoing (transmitted and reflected) waves of  \nconsiderably enhanced amplitude. The disk then uses a part of the energy of the magnetized \nferrite sample  to produce  and emit a more intensive electromagnetic wave. This spectral \nsingularity -related resonance effect  relies on the existence of a  localized region with a PT-\nsymmetric complex scattering potential . Because this is a characteristic property of resonance \nstates, spectral singularities correspond to the resonance states ha ving a real energy .  \n     In Refs. [34, 35 ], we analyzed two spectral models for the MDM oscil lations in a quasi -2D \nferrite disk. There  are models based on the so -called \nˆG  and \nˆL  differential operators – the G- \nand L-mode spectral solutions , respectively . For G modes one has the Hermitian Hamiltonian \nfor MS -potential wave functions. These modes are related to the discrete energy states of \nMDMs. The G-mode orthogonality condition is expressed as [34]  \n \n                                                       \n*0p q p qSE E dS \n ,                                                   (57) \n \nwhere the sign * means complex conjugation, \n\n   is the G-mode MS -potential membrane \nfunction, E is normalized energy of the mode, and S is an area of the disk plane.  In a case of the \nL modes, we have  a complex Hamiltonian for MS -potential wave functions . For eigenfunctions \nassociated with such a complex Hamiltonian, there is  a nonzero Berry potential (meaning the \npresence of geometric phases)  [34, 35] . The main difference between the G- and L-mode \nsolutions becomes evident when one considers the boundary conditions on a lateral surface of a \nferrite disk. In solving the energy -eigenstate spectral proble m for the G-mode states, the \nboundary conditions on a lateral surface of a ferrite disk  of radius \n  is expressed as  \n \n                                                       \n0\nrrrr\n         \n .                                            (58) \n \nThis boundary condition, however, manifests itself in contradictions with the electromagnetic \nboundary condition for a radial component of magnetic flux density \nB\n  on a lateral surface of a \nferrite -disk resonato r. Such a boundary condition, used in solving the resonant spectral problem \nfor the L-mode states, is written as \n r r ar r rH H i H      \n , where \nrH  and \nH  \nare, respectively, the radial and azimuth compon ents of a magnetic field on a border circle . In \nthe MS description, for the L-mode membrane function \n\n , this boundary condition on a lateral \nsurface  is described by Eq. (7). While the G mode s are s ingle -valued -function solutions, the L \nmodes are double -valued -function solutions.  It follows from the fact that at  a given dir ection of \na bias magnetic field, the two (clockwise and counterclockwise) types of resonant solutions  for \nL-mode states  may exist .  \n     When a ferrite disk is pl aced in a rectangular wa veguide, so that the disk plane  is in parallel \nto the wide waveguide walls and the disk position is symmetrical to these walls, MDM \noscillations are PT symmetric. It follows from the property  of PT-invariance of  the L-mode \nmembrane function \n\n . For a ferrite -disk axis directed along a coordinate axis y, on a lateral \nboundary of a ferrite disk we have the following condition [86] \n \n                                              \n*( ) ( ) ( )r r rz z z      \n .                                        (59)   26  \nThe bi orthogonality relation  for the two L modes can be  written as     \n \n                                  \n *, ( ) ( )\n                         = ( ) ( ) ,p q p qr r r r\nl\npqrr\nlz z dl\nz z dl   \n   \n  \n    \n\n                              (60) \n \nwhere \n2l\n  is a contour surrounding a cylindrical ferrite core. This relation may presume \nthe absence of complex eigenvalues for L modes. The phase acquired by the MDM fields \norbitally rotating in a ferrite disk is \nk , where k is an integer  [86]. The relation (60) has \ndifferent meanings for even and odd quantities k. For even quantities k, the edge  waves show  \nreciprocal phase behavior for propagation  in both azimuthal directions. Contrarily, for odd \nquantities k, the edge waves propagate only in one direct ion of the azimuth  coordinate . In a \ngeneral form, the inner product ( 60) can be written as \n , ,1k\np q p qrr  \n\n , where \n,pq  \nis the Kronecker delta.  \n     While a  PT-symmetric Hamiltonian is , in general , not Hermitian,  in the  problem under \nconsideration we can introduce a certain  operator  \nˆ\n, that the action of \nˆ\n  together with  the PT \ntransformation will give the hermiticity condition and  real-quantity energy eigenstates  [90]. \nThe operator \nˆ\n  is found  from an assumption that it acts only on the boundary  conditions of the \nL-mode spectral problem. This special differential operator has a form \nˆ a\nri\n\n , \nwhere  \nr\n  is the spinning -coordinate gradient. It means that, for a given direction of a bias  \nfield, operator \nˆ\n  acts only for a one -directional azimuth  variation. The eigenfunctions of \noperator \nˆ\n  are double -valued  border functions [ 34]. This operator allows performing the  \ntransformation from the natural boundary conditions of the  L modes, expressed by Eq. (7) , to \nthe essential  boundary conditions of G modes, which take  the form of Eq. ( 58) [34, 92, 93]. \nUsing operator \nˆ\n , we construct a  new inner  product structure for the boundary functions,  \n \n                                  \n ˆ , ( ) ( ) ,m n m nr r rr\nlz z dl        \n                           (61) \n \nAs a result, one has the energy eigenstate spectrum of MDM oscillations with topological \nphases accumulated by the  double -valued border functions [34, 8 6]. The topological effects \nbecome apparent through the integral fluxes of the pseudoelectric  fields. There are positive and \nnegative fluxes corresponding  to the clock wise and counterclockwise  edge -function chiral  \nrotations. For an observer in a laboratory frame, we have two  oppositely directed anapole  \nmoments \nea\n . This anapole  moment  is determined by the term \na\nri\n \n\n in Eq. ( 7). For a \ngiven direction of a bias magnetic field, we have  two cases : \n00eaH\n  and \n00eaH\n .  \n      In addition to the above consideration, other evident aspect s prove the PT–symmetry in  a \ngeometrically symmetric struc ture with a MDM ferrite disk . It follows from  the PT–symmetry \nof a scalar parameter characterizing properties of a ME field. At the MDM resonance, e nergy \ndensity of the ME-field structure is defined by the  helicity factor \n()EF [94]. There is the energy \ndensity of a quantized state of the near field originated from magnetization dynamics in a  27 quasi -2D ferrite disk . In a geometrically symmetric structure, the distribution of a helicity \nfactor \n()EF , expressed by Eq.  (22), is symmetric with respect to in-plane x, y coordinates and \nantisymmetric with respect to z coordinate [35 , 43]. The helicity factor \n()EF  is a pseudoscalar. \nIt changes a sign under inversions (also known as parity transformatio ns). On the other hand, \nthe factor \n()EF  changes its sign when one changes the direction of a bias magnetic field, that \nis, the direction of time ( see Fig. 20). In a view of PT-symmetry of MDM oscillations (which is \nproven by an analy ses of the MS -wave operator [8 6] and the time -space symmetry properties \nof the helicity factor \n()EF ), the presence of an axial -vector orbital angular -momentum of a \nferrite disk \n  presumes also existence a ce rtain polar vector directed parallel/antiparallel to \nthe vector \n . Such a polar ve ctor exists. This is an  anapole moment \nea\n  originated from the \n\n- flux resonances  due to surface m agnetic currents on a lateral surface of a ferrite disk [34, \n95].  \n     There can be different types of the microwave structures where PT-symmetry of MDM s in a \nferrite disk  is broken.  The effect of the PT-symmetry breaking can be observed when , for \nexample , a ferrite disk is placed in a rectangular waveguide non -symmetrically with respect to \nits walls. In such a case , the distribution of a helicity factor becomes assymetric with respect to \nz coordinate [35]. This is due to the role of boundary conditions on  a metal surface. Inside a \nmetal a helicity factor is zero. Fig. 2 1 shows schematically four cases of the helicity -factor \ndistributions in a waveguide structure when two external parameters – a disk position on z axis \nand a direction of a bias magnetic fie ld – change. At such a nonsymmetric disk position, the PT \nsymmetry of MDM oscillations is broken. Other cases are related to microwave waveguiding \nstructures with geometric nonsymmetry  or the structures with inserted special chiral objects. \nAs an important  criterion for PT-symmetry breaking, there  is the symmetry breaking of the \nhelicity -factor distribution. Nonreciprocity effects, observed in the scattering matrix parameters \nof a microwave structure at MDM resonances , are also due to PT-symmetry breaking. \nNumerous studies of these properties  for MDM structures , both numerical and experimental, \nare shown in Refs. [ 35, 65, 96, 97 ].  \n     To a certain extent, the above questions on the PT symmetry breaking are akin to recent \nresearch in optics . For example, in  publications [98 – 100], related to nonreciprocity in optics, \nit was shown that this effect takes place due to the PT symmetry breaking .  \n \nVI. PREDICTION OF EXCEPTIONAL POINTS FOR A STRUCTURE OF MDM s IN \nTHE CONTINUUM  \n \nMDM oscillations exhibit eigenstate chira lity which is related to the Berry phase . The phase of \nthe MDM wave function is not restored after one encircling around a lateral surface of the disk, \nbut is changed by \n . Only the second encircling restores the MS wave function including its \nphase [34]. There is the degeneracy of Hermitian operators ( G modes), at which the \neigenvalues coalesce while the eigenvectors remain different. The L-mode analysis [34, 89] \nshows that the eigenvector basis is skewed and the eigenvalues thems elves are complex. \nDepending on a direction of a bias magnetic field, o nly one MDM state (clockwise or \ncounterclockwise) dominates. In an analysis of MDM resonant structures, there are not only \nparameters controlling internal properties of the separated L-mode ferrite disk. There are also \nparameters by means of which the coupling strength between the MDM disk and the \nenvironment can be controlled.  When we consider our microwave structures with an embedded \nMDM ferrite disk and inserted chiral objects, we als o should consider the chirality of the \neigenstates and the external -parameter chirality. In Ref. [9 6], the observations support the \nmodel which assigns a special role of  the MDM chiral edge states in the unidirectional ME \nfield multi -resonant tunneling.  In Ref. [97 ], we have experimental observation  of microwave  28 chirality discrimination in liquid samples  due to MDM eigenstate chirality . There is an evident \ncorrespondence between topological edge modes of MDM oscillations and “external” chirality \nof inserted  objects.   \n    For open resonant structures, an important aspect concerns a behavior of non -Hermitian \nsystems at the spontaneous breaking PT–symmetry, marked by the exceptional point (EP). EPs \nare exhibited as a distinct class of spectral degeneracies, whi ch are branch points in a 2D \nparameter space. The EP is the degeneracy intrinsic to non -Hermitian Hamiltonians at which \ntwo eigenvalues and corresponding eigenvectors coalesce. The wave function at an EP is a \nspecific superposition of two configurations. T he phase relation between the configurations is \nequivalent to a chirality [101 – 103]. The PT-symmetry breaking is related to chirality at the \nexternal -parameter loop. The sign of the chirality is defined via the direction of time.  In the \nexperiment in Ref . [102, 103 ], the positive direction of time is given by the decay of the \neigenstates.  \n     In our case, the time  direction  is given by the direction of a bias field. This is an external \nparameter  of a type that can be controlled.  In the prediction of exc eptional points for a structure \nof MDMs in the continuum , we will use two tunable external parameters: (i) a bias magnetic \nfield H0 and (ii) the disk shift d in a waveguide  along z axis, when the disk plane is parallel to \nwaveguide walls . Evidently, if a MDM ferrite disk is placed in a waveguide symmetrically to \nits wall  (z = 0) , the direction of z axis is arbitrary and either (positive or negative) time \ndirections are the same. When the disk is shifted along z axis, we can distinguish a definite \ndirection of time. We chose the positive direction of time by an orientation of a bias magnetic \nfield along \nz axis. Now, let a bias magnetic field is varied in a range \n(1) (2)\n0 0 0H H H . A \nMDM ferrite disk be shifted along \nz axis and its positions are defined as  \n12d d d . The two \nexternal parameters compose a certain interaction parameter \n0,Hd .  \n     We suppose that, similar to the structure shown in Fig. 8, the fiel ds of a microwave \ncontinuum are the fields of a microwave cavity. Also, similar to the resonances shown in Figs. \n15 and 16, we have two modes. Following the picture in Fig. 16, t he modes are related to ( a) \nthe single -rotating -magnetic -dipole  (SRMD) resonan ce; ( b) double -rotating -magnetic -dipole  \n(DRMD) resonance. By variat ion of the quantity of a bias magnetic field, we have an \nadjustable frequency difference f1 – f2 for the se two modes.  When a ferrite disk is shifted, with \nvariation of quantity d, one obser ves variation of the helicity distribution. For the above \nresonances, we have variation of the field amplitudes and  variation of the phases between the \nelectric and magnetic fields  [35, 43] .  In the case of f1 = f2, there should be the possibility  to \nadjus t the phase difference between the modes (by variation of the disk shift along z axis). One \nof the main evidence for existence of the EP is exhibition of chirality when one encircles this \npoint in the space of \n [101, 102]. When th e EP is encircled, the eigenvectors pick up \ngeometric phases. One of the evidence for this is the presence of the \n - flux resonances.  There \nshould exist a critical value  \n( ) ( ) ( )\n0,EP EP EPHd , where the two modes coalesce.  \n     The EP can be found by looking at the behavior of the real and imaginary parts of the \neigenvalues. Such a basic analysis is beyond the frames of the present paper. It is a purpose of \nour future research. However, a qualitative analysis allows to predi ct encircling exceptional \npoints along two paths on a parametric space created by  a bias magnetic field and a disk shift. \nWe predict four types of contours on such a 2D parametric space  (see Fig. 2 2). When a ferrite \ndisk is shifted along \nz  axis at the positions defined as  \n12d d d , two types of contours \nexist. For the range of positive magnetic field \n(1) (2)\n0 0 0H H H , we have an e xceptional point \nEP 1CW inside the clockwise contour: \nA \nA\nB\nB\nA. For the range of negative magnetic \nfield \n(1) (2)\n0 0 0H H H   , an e xceptional point EP 2CCW is inside the counterclockwise  29 contou r: \nC\nC\nD\nD\nC. From Fig. 2 1, one can see that a simultaneous change of a sign \nof the bias magnetic field H0 and a sign of the disk shift d in a waveguide  gives symmetrically \nthe same pictures of the helicity distribution. It means that there exist two other contours on our \n2D parametric space. When a ferrite disk is shifted along \nz  axis at the positions \n12d d d  \n and t he range of positive magnetic field is \n(1) (2)\n0 0 0H H H , we have an \nexceptional point EP 3CCW inside the counter clockwise contour: \nE \nE\nF\nF\nE. At the \npositions \n12d d d    and the range of negative magnetic field \n(1) (2)\n0 0 0H H H   , an \nexceptional point EP 4CW is inside the counterclockwise contour: \nG \nG\nH\nH\nG. It is \nevident that EP 1CW is the same as  EP 4CW and EP 2CCW is the same as EP 3CCW. We can see that in \na case of a waveguide with a non -symmetrically embedded ferrite disk , the ME near -field \ndistribution  becomes not PT symmetrical. However, for the entire microwave  structure we \nhave PT symmetry together with chiral symmetry  of contours on a 2D parametric space .  \n     To verify the above p rediction of the EPs in a microwave waveguide with a non -\nsymmetrically embedded ferrite disk, we will do the following qualitative analysis of the \nproblem.  Energy density of the ME near -field structure is defined by the  helicity -factor \ndensity\n()EF [94]. We have the regions with the positive and negative ME -field densities, which \nare characterized by the positive and negative helicity -factor densities, \n( )( )EF and \n( )( )EF , \nrespectively . We define t he positive  helicity (the positive ME energy) as an integral of the ME -\nfield density over the entire near -field vacuum region \n()\n with the helicity -factor density \n( )( )EF\n: \n \n                                                            \n()( ) ( )( ) EFd\n\n .                                                   (62) \n \nSimilarly, the negative helicity (the negative ME energy) is defined as an integral of the ME -\nfield density over the entire near -field vacuum region \n()\n with the helicity -factor \ndensity\n( )( )EF : \n \n                                                           \n()( ) ( )( ) EFd\n\n .                                                    (63) \n \nIn a case of PT symmetri cal ME near -field distribution, \n( ) ( )\n . So, the total helicity  (the \ntotal ME energy) of the entire near -field vacuum region surrounding a ferrite disk \n( ) ( )\n is equal to zero [94 ]: \n \n                                       \n ( ) ( ) ( )( ) ( )( )0EEF F d       \n .                                   (6 4) \n \nIf, however, the PT symmetry is broken (in a case, for example, when a ferrite disk is shifted in \na waveguide  along z axis), we have \n( ) ( )\n . It means that we m ay have predominance of \nthe positive or negative ME energy. In particular, in Fig. 21  (a) we have predominance of the \nnegative ME energy, while in Fig. 2 1 (b), there is predominance of the positive ME energy.  On \na metal wall, the helicity -factor density (a nd so, the ME -energy density) is zero. Where does \nthe excess of ME energy go?  \n     The ME fields are quantized state of the near fields originated from the magnetization \ndynamics in a quasi -2D ferrite disk. There are pseudoscalars, which  appear as the fiel ds of \naxion electrodynamics. The coupling between an axion field and the electromagnetic field  30 results in modifi cation of the electric charge and current densities in electrodynamics \nequations. Such modified sources of electromagnetic fields were observed on metal walls of a \nmicrowave waveguide with an embed ded MDM ferrite disk [36, 82, 86 ]. It is worth noting here \nthat our ME-field structure s are fundamentally different from the ME  fields considered  in Ref. \n[104]. In Ref. [104] it has been proposed , hypoth etically, that local ME fields can be realized \ndue to  the interference of several plane waves.  In such a  case, however, the question , how one \ncan create a pseudoscalar field from interaction of regular electromagnetic waves, remains \nopen.  \n     Absorption of ME energy takes place only at the PT-symmetry breaking . When, in \nparticular, the PT-symmetry -breaking -behavior is realized by shifting a ferrite disk in a \nwaveguide  along z axis, the excess of ME energy  (positive or negative) leads to appearance of \ntopological charge s and current s induced on the upper and lower walls of a waveguide.  In Refs. \n[36, 8 2], it was shown numerically that in a structure with an embedded ferrite disk, s urface \nelectric current s on a waveguide  wall are the right -handed  and left -handed flat spirals.  For the \nPT-symmetric structure, we have chiral symmetry of such currents on the upper and lower \nwaveguide walls. In a case of the PT-symmetry breaking, s pontaneous chiral symmetry \nbreaking occurs and the net chirality  of the surface curre nts can fall into either the left -handed \nor the  right -handed regime.  This results in absorption of ME energy for MDMs.  \n     Let in the MDM oscillation spectra, there be two  Hermitian -Hamiltonian  G modes with \nenergies \n1E  and \n2E , which are sufficiently close  one to another. We assume that coupling \nbetween these modes is reciprocal and is characterized by a real coefficient \n . For an open \nstructure with PT–symmetry breaking, the a non -Hermitian Hamiltonian of the two -state \nsystem is expressed as  \n     \n                                                      \n11\n22Ei\nEi\n \n .                                                  (65)  \n \nReal quantities \n1  and \n2  characterize the absorption rates. For the two modes, the absorption  \nis due to the radiation inside a waveguide continuum caused be the currents induced on the \nwaveguide walls  of the PT-symmetry -breaking structure . The \n  signs in Eq. (6 5) characterize \ndifferent kinds of the ME -energy  excess (positive or negative).  When the signs of imaginary \ncomponents in Eq. (6 5) are the same, an analytical treatment of encircling of EPs can be \nsimilar to that used in Re f. [10 3].  \n     As we discussed above , due to the topological action of the azimuthally unidirectional \ntransport of energy in a MDM -resonance ferrite sample there should exist  the opposite \ntopological reaction  (opposite azimuthally unidirectional transport  of energy ) on a metal screen \nplaced near this  sample. Conservation of the  orbital angular momentum, related to the power -\nflow circulation, has been proven in a PT-symmetrical  microwave structure with an embedded \nferrite disk  [36]. It was also discussed th at a vacuum -induced Casimir torque allows for torque \ntransmission between the ferrite disk and metal wall avoiding any direct contact between them. \nFor the PT-symmetrical microwave  structure, the Casimir energy  is the same above and below \na ferrite disk. W hen the PT is broken¸ the Casimir energy  above and below a ferrite disk is not \nthe same . The orbital angular momentum  is not conserved now. All this means that together \nwith chirality of the MDM ferrite disk we should observe chirality of the microwave \ncontinuum. At the MDM resonances, the eigenvector basis of continuum becomes skewed and \nthe eigenvalues of continuum become complex. This appears  clearer  from the fact that  in \nmicrowave structures with  MDM  ferrite particles  we can observe  path-dependent inter ference  \n[82]. In general, we have a case of non-integrable systems.  Following Ref. [10 5], there is the  31 possibility to introduce two topological numbers: ( i) related to the Berry -curvature chirality of \nthe eigenstates and ( ii) the exceptional -point chiralit y in the external -parameter space . \n     It is worth noting that  when a MDM ferrite disk is placed in a rectangular waveguide \nsymmetrically with respect to its walls, PT symmetry is valid when one can neglect asymmetry \nin the ME -field structure due to the w ave propagation direction in a waveguide. This is \npossible in a case of a quasi -2D microwave structure with a “thin” waveguide (see Fig. 4), \nanalyzed in Ref. [36]. In a microwave structure with a “thick” waveguide, there is an evident \nasymmetry in the dist ribution of a helicity factor with respect to x, y coordinates [35, 82, 83 ]. \nSuch a symmetry due to the wave propagation direction in a waveguide can lead to unique \ntopological effects. In Ref. [82 ], it was shown that in a microwave structure of two ferrite  disks \nplaced on a waveguide axis symmetrically to the waveveguide walls, the coupling of two \nidentical MDM resonances is nonreciprocal.  \n \nVII. DISCUSSION : ON THE COMPLEX -WAVE INTERACTION BETWEEN EM \nAND ME PHOTONS   \n \nIn Ref. [106], an optical system that combine s a t ransverse spin and an evanescent \nelectromagnetic wave in semi -space geometry, has been proposed. In a theoretical mode l, it is \nassumed that in such surface modes with strong spin -momentum locking the  real and \nimaginary EM wave  vectors are mutually per pendicular . Nevertheless, it becomes clear that a \nslight deviation of the spin vector from the  normal to the surface -wave propagation direction, \nwill lead to appearance of a leaky wave, which is treated, mathematically, as a guided complex \nwave .  \n    The q uestion of a complex -wave behavior is very important in our case of a microwave \nstructure with an embedded ferrite disk. In the near -field region, ME photons , originated from \na quasi -2D ferrite disk, are characterized by the fields orbitally rotating in a plane parallel to \nthe disk plane and decaying along the disk axis. Such a field structure can be described by the \nMS-wave functio n with mutually perpendicular a real wave vector  (in a plane parallel to the \ndisk plane) and an imaginary wave vector  (directed  along the disk axis). When analyzing in  \nRef. [36] the electric current  induced on a metal wall in a waveguide with an embedded MDM \nferrite disk, we found the vortex stru cture of this current , we can also assume  that the field near \nto this wall has mutuall y perpendicular  real (in a plane parallel to the metal wall) and \nimaginary (along a normal to the wall) wave vectors . However, when we take into account the \nwaveguide thickness, that is when we consider a 3D microwave structure, the real and \nimaginary wave  vectors of the ME -field structure will have co -parallel components . It means \nthat in the near -field region, the ME field is viewed as a complex -wave field. This complex -\nwave behavior is observed together with the spin -orbit coupling in the ME -field struct ure. It \nbecomes clear that together with localized and quantized ME -field complex waves, we have to \npresume the presence of the complex -wave continuum in a microwave waveguide.    \n     In the above discussions, regarding experimental results obtained in Re f. [32] (and shown in \nthe present paper in Fig. 8) , we stated that for the microwave photon propagating in a \nwaveguide  at a constant frequency, the  energy extracted from the magnetic energy of a ferrite \ndisk at the MDM resonance s cannot be a quantized quan tity. So, the question of which \nmicrowave continuum we have when observing discrete states of MDM resonance in a \nmicrowave cavity with a constant frequency , remains open.  To answer this question, we should \nconsider the spectrum of so -called complex modes i n lossless microwave waveguides. In a \nclosed lossless waveguide, t here can be  a finite number of real positive eigenvalues and an  \ninfinite number of complex conjugate pairs of eigenvalues  [107, 108]. Following the parameter \ncirculation in Fig. 12, we can s ee that  for a give n frequency \n , the input impedance of the \ncavity passes through complex quantities  for every separate MDM . Assuming the presence of  32 complex waves in the waveguide spectrum we should conclude that in Fig . 13 not o nly \nimpedance \nLZ  is a complex quantity but also a characteristic impedance of a lossless \nwaveguide  \n0Z is a complex quantity . Generally, we have to take into account  the possibility of \ninteraction between di screte -state complex -wave MDMs and the microwave complex -wave \ncontinuum.  \n       Let us consider general relations for mode orthogonality in a lossless regular waveguide  \nassuming the presence of complex waves in the waveguide spectrum . We write homogeneous \nMaxwell equations in an operator form:  \n \n                                                                     \n0 MU\n ,                                                             (66) \n \nwhere \nM  is the Maxwell ope rator:  \n \n                                                           \n0\n0iM\ni\n\n                                                       (67) \n \nand \nU\n  is a vector function of the fields:  \n \n                                                                 \nEU\nH\n\n .                                                               (68)  \n \nIn a regular waveguide, the solution of Eq. ( 66) for electromagnetic wave propagating along z \naxis is presented based on the mode expansion:  \n \n                                                             \nˆmz\nmm\nmUА U е\n ,                                                     ( 69)                      \nwhere \nm m m i    is a propagation constant of a normal  mode m and \nˆ\nmU\n  is a membrane \nfunctions on the waveguide cross section:  \n \n                                                        \n\nˆ, ˆ,ˆ,m\nm\nmE x yU x y\nH x y\n\n\n .                                                 (70) \n \nFor homogeneous Maxw ell equations, we can write : \n \n                                                                  \nˆˆ\nm m m M U QU\n ,                                                  (71) \n \nwhere \nM  is  differential -matrix operator similar t o the operator \nM but operating over cross \nsection coordinate of a waveguide and  \n   \n                                                             \n0\n0z\nzeQe\n .                                                     (72) \n  33 Here \nze\n  is the unit vector directed along z axis.   \n     We juxtapose now the differential problem described by Eq. ( 71) with the conjugated \nproblem. For the conjugated problem, we have : \n \n                                                                \nˆˆ\nm m mM V QV\n ,                                                      (73) \n \nwhere \nm m m i  \n  is a propagation constant of a normal mode \nm\n  and \nˆ\nmV\n  is a vect or \nfunction of the membrane -function fields of the conjugated problem:  \n \n                                                          \n\nˆ, ˆ,ˆ,m\nm\nmE x yV x y\nH x y\n\n\n .                                                 ( 74)  \n \nA form of the conjugate operator \nM\n  is defined from integration by parts [9 2]: \n \n                                 \n**\n* ˆ ˆ ˆ ˆ ˆ ˆ  m m m m m m\nSSM U V dS U M V dS U RV d    \n   ,                    (75) \n \nwhere S is the waveguide cross -section area, \n  is the contour surrounding S. The operator R \nhas a form  \n \n                                                               \n0\n0nRn\n ,                                                      (76) \n \nwhere \nn\n  is the unit vector directed along the extern al normal to contour \n . For a lossless \nwaveguide and homogeneous boundary conditions, operator \nM\n  is the same as operator \nM . \nAs a result, one has the orthonormality condition  \n \n                                                       \n** ˆˆ 0m m m m\nSQU V dS \n .                                           (77) \n \nEq. (68) is correct for a general case of complex waves in a lossless waveguide. In this case, \nwe have biothogonality for conju gate modes. The conjugate modes are m and \nm\n are orthogonal \nwhen \n*0mm\n . Otherwise, these modes constitute a norm:  \n \n                                      \n * * * ˆ ˆ ˆ ˆ ˆ ˆ mm m m m m m m z\nSSN QU V dS E H E H e dS    \n .                    ( 78)                \n \nIn the spectrum of  a lossless waveguide , there are four complex modes : \nm m m i   , \nm m m i    \n, \nm m m i     , and \nm m m i   \n , which exist in fours : \n*\nmm\n , \n*\nmm\n, \n*\nmm\n , \n*\nmm\n . Positions of these complex  wave numbers are shown in Fig. \n23. An interaction of  modes with \nm  and \nm\n  or \nm and \nm\n  cause appearance  of active power \nflows; meanwhile an interaction of modes  with \nm and \nm\n  or \nm  and \nm\n lead to appe arance  34 of reactive power flow s. The pairs of modes which realize the  carrying over of energy are \ncharacterised by the same  direction of phase velocity and different sign of amplitude  changing  \n[107, 108].  \n     In a particular case of a propagating -wave beh avior, we have \nm m m i  \n . For \npropagating modes \nˆ ˆ ˆ ˆ, m m m mE E H H\n , the electric and magnetic fields oscillate in phase. So, \nthe norm ( 78) is a real quantity. On the other hand, in a particular case of  an evanescent wave, \nthe ele ctric-field oscillations are \n90\n  phase shifted in time with respect to the magnetic -field \noscillations. So, the norm defined by Eq. ( 78) is an imaginary quantity. At such an evanescent -\nwave behavior no transmission of energy occurs.  However, one can have power transmission \n(tunneling) at certain boundary conditions of a section of a below -cutoff waveguide. \nEvanescent waves are non -local waves. So, normalization is non -local as well. To get \ntransmission in a below cut -off section we h ave to take into account the load conditions. Let \nthe incident field  be the positive decaying mode  and the reflected field be the negative \ndecaying mode  (see Fig. 24) . Such reflected and incident fields can be related through a certain \nload-dependent refle ction coefficient  [109]. Fig.  24 shows the complex -plane combination of \nincident and reflected field s that will exist at the end of a below -cutoff waveguide section. The \nport 1 is considered as a source. The section is terminated in a certain load  at port 2. The above \nexpressions involve both normal modes, with fields decaying in positive and ne gative , \nrespectively.   We consider \nˆˆ, m i m rE E E E\n  and \nˆˆ ,  m i m rH H H H\n , where \n, irEE\n  \n, irHH\n  \nare the  incident and reflected electric and magnetic fields, respectively. We also assume that \nboth incident and reflected electric and magnetic fields are linearly polarized in space and that  \nm m m   \n. When , due to a load in port 2,  the electr ic- and magnetic -field oscillations of \nmodes \nm  and \nm\n  become \n90\n  phase shifted in time, the total electric \nirEE\n  and total \nmagnetic\nirHH\n  fields are in phase and the norm ( 78) is a real quantity. So, a n interaction of  \nmodes with \nm  and \nm\n  cause s appearance  of active power flow s through a below -cutoff \nwaveguide section. This effect is similar to the tunneling effect in quantum mechanics \nstructures. The two orthogonal basis states \nˆ1mU\n  and \nˆ2mV\n  oscillate in time if they are \nsuperimposed according to relation \nˆˆ12mm i U iV  \n . This coalesce nt state is a chiral state. \nAs an interaction parameter of a system – a below -cutoff waveguide section  – there is a \nboundary condition at port 2. The pairwise coalescence  occurs only at a certain boundary \ncondition at port 2 that is at a certain value of a n interaction parameter .  \n     Now, we come back to the structure shown in Fig. 8. Following the above analysis, we \nsuppose that the port 1 is the cavity input . The localized region in a waveguide where a MDM \nferrite disk is situated , is the port 2  with a certain load . We also suppose that in a waveguide \nsection between ports 1 and 2 the complex waves can be observed. Fig. 25 shows a possible \nregion of existence of such complex waveguide modes on the \nk diagram, where k is a real \nwavenumber. The phases of the MDM bound states  are strongly determined phases. W e can \nassume that for every MDM there are certain quantities of a bias magnetic field that provide \n90\n phase shift in time for the waveguide fields at the region of a ferrite disk, that is the \nregion of the port 2. This exactly corresponds to points 1, 2, 3, … in Figs. 8 and 12.  \nDiscretization  of the microwave response is only due to this complex -wave behavior of \nwaveguide modes.  The conditions for normaliza tion of these complex -wave modes is \nillustrated in Fig. 23.  \n     How can we view  a general picture of scattering of these complex waveguide modes by the \nresonant MDM particle?  We suppose , initially,  that a magnonic -resonance in a small ferrite  35 particle is  deprived of an y orbital rotation s (in other words, there are no vortices in the resonant \nmodes) . There is a resonant magnetic particle dual to plasmonic -resonance subwavelength \nparticle. In this case,  the scattering of propagating EM waves is well  describ ed by Mie theory. \nThe particle near field is a decaying field  described by  Laplace equation . Now, we take into \naccount that the quasistatic fields of this magnonic -resonance particle are orbitally rotating \nfields. Because of such a behavior, we have strong ly determined localized phases. At the MDM \nresonances, t his gives a “proper ” scattering for evanescent EM waves. All this means that at \nMDM resonances, we observe scattering of  waveguide complex -wave modes. In other words, \nwe have a n interaction of MDM dis crete states with the  complex -wave microwave continuum.         \n     Till now, we do not have direct experimental and/or numerical proves  of the proposed \nmodel of  the complex -wave interaction between EM and ME photons. Nevertheless , some \nunique topological -phase properties shown in Ref. [82] should deserve our special attention \nregarding this aspect.  It was shown that in a microwave structure of two small ferrite disks \nplaced on a waveguide axis symmetrically to the waveguide walls  [82], the frequency split  of \nthe interacting MDM resonances is extremely narrow and (what is especially unique) is quite \nindependent on the distance between the disks.  Since this  distance varied from the near -field to \nfar-field regions of the waveguide mode, the coupling between MDM disks is not due to the \npropagating EM wave, but because of the presence of the evanescent part of the complex EM \nwave. One of specific properties of evanescent and tunneling modes  is that they are non -local . \nFor EM waves, the MDM disks appear as locali zed-phase singularities and interaction between \nthe disks is due to the microwave complex -wave continuum.  \n  \nVIII.   CONCLUSION  \n \nIn small ferromagnetic -resonance samples, macroscopic quantum coherence can be observed. \nLong range magnetic dipole -dipole correlation can be treated in terms of collective excitations \nof the system. In a case of a quasi -2D ferrite disk, the quantized forms of these collective \nmatter oscillations – the MDM  magnons – were found to be quasiparticles with both wave -like \nand particle -like beh aviors, as expected for quantum excitations.  \n     In an electromagnetically  subwavelength ferrite  sample one neglects a time variation of \nelectric energy in comparison with a time variation of magnetic energy. In this case, the \nFaraday -law equation is inc ompatible with the spectral solutions  for magnetic oscillations . It \nappears, h owever, that in а case of MDM oscillations in a quasi -2D ferrite disk, the Faraday \nequation plays an essential role . In a ferrite -disk sample, the magnetization has both the spin \nand orbital rotations. There is the spin -orbit interaction between these angular momenta. Thi s \nresults in unique properties when t he lines of the electric field as well the lines of the \npolarization  in a sample are “frozen” in the lines of magnetization . In such a case, the Faraday \nlaw is not in contra diction with the magnetostatic equations descr ibing the MDM -oscillation \nspectra.  \n     The MDMs in a ferrite disk are characterized by the pseudoscalar  magnetization helicity \nparameter  which gives evidence for the presence of two coupled and mutually parallel currents \n– the electric and magnetic ones – in a localized region of a microwave structure.  The near \nfields of a MDM sample are so -called ME fields. The magnetization helicity parameter can be \nconsidered as a certain source which defines the helicity properties of ME fields. The ME \nfields, being or iginated from magnetization dynamics at MDM resonances, appear as the \npseudo scalar axionlike field s. Whenever the pseudo scalar axionlike field s, is introduced in  the \nelectromagnetic t heory, the dual symmetry is spontaneously and explicitly  broken. This res ults \nin non -trivial coupling between pseudoscalar quasistatic ME fields and the EM fields  in \nmicrowave structures with an embedded MDM ferrite disk.   36      Unique properties of interaction of MDMs with a metal screen become more evident when \none analyzes the  angular -momentum balance conditions for MDM oscillations in a ferrite disk \nin a view of the ME-EM field coupling  in a microwave waveguide. MDMs  in a quasi -2D \nferrite disk are microwave energy -eigenstate oscillations with topologically distinct structures \nof rotating fields and unidirectional power -flow circulations.  Due to the topological action of \nthe azimuthally unidirectional transport of energy in a MDM -resonance ferrite sample there \nexists the opposite topological reaction  (opposite azimuthally unidir ectional transport of \nenergy ) on a metal screen placed near this  sample. T his effect is called topological Lenz’s \neffect. In a microwave structure with an embedded ferrite disk, an orbital angular momentum, \nrelated to the power -flow circulation, must be co nserved in the process. Thus, if power -flow \ncirculation is pushed in one direction in a ferrite disk, then the power -flow circulation on metal \nwalls to be pushed in the other direction by the same torque at the same time.  A vacuum -\ninduced Casimir torque al lows for torque transmission between the ferrite disk and metal wall \navoiding any direct contact between them.  \n     The fact that magnetization dynamics in a quasi -2D ferrite disk have an impact on the \nenergy quantization of the fields in a microwave cavi ty, was confirmed experimentally . Sharp  \nmultiresonance oscillations, observed experimentally in microwave structures with an \nembedded  quasi -2D ferrite disk , are related to magnetization dynamics in the sample. This \ndynamics have  an impact on the phenomena connected with the quantized energy fluctuation . \nInside the YIG disk, we have the torque exerting on the electric polarization due to the \nmagnetization dynamics . Because of the effective magnetic charges on a ferrite -disk planes , the \ndemagnetizing magnetic  field is reduced. It means that the DC magnetization of a ferrite disk is \nreduced as well.  At the MDM resonances, we observe quantization of the DC magnetization of \na ferrite disk . It is worth noting also that at the MDM resonance, a quasi -2D disk is mani fested \nas a ME particle with two DC moments directed along the disk axis: ( i) a DC magnetic \nmoment (due to saturation magnetization) and ( ii) a DC electric moment – the anapole \nmoment.    \n     In the paper, we proved PT-symmetry of MDM s in a ferrite disk  and analyzed the \nconditions of the PT-symmetry  breaking. When analyzing the scattering of EM waves by \nMDM  disks in microwave waveguid es and energy quantization of the field in a microwave \ncavity, we dwell ed on some basic problems of magnon -photon interactio n and bound states in \nthe microwave continuum.  For a PT-symmetrical structure, o ne of the features attributed to the \nBICs is  a strong resonance field enhancement. In the case of MDM oscillations in a \nmicrowave -field continuum, such a strong field enhanceme nt was confirmed in numerous \nnumerical studies . In microwave -cavity  structures with embedded  small ferrite disks, t he effect \nof Fano resonance collapse  was shown at variation of an external parameter – a bias magnetic \nfield.  When a PT symmetry of a microwa ve structure with an embedded MDM ferrite disk is \nbroken, one can predict existence of certain branch points – the exceptional points – in a 2D \nparameter space of the structure.  \n     To answer the question of which microwave continuum we have when observi ng discrete \nstates of MDM resonance in a microwave cavity with a constant frequency , we take into \nconsideration the spectrum of so -called complex modes in lossless microwave waveguides. \nThe near fields of a MDM -resonance particle are orbitally rotating fie lds. Because of such a \nbehavior, we have strongly determined localized phases. For incident EM waves, this particle \nappears as a localized phase singularity. As an important problem , we discussed the possibility \nof interaction between discrete -state comple x-wave MDMs and the microwave complex -wave \ncontinuum.  \n     Finally , we have to note that interdisciplinary studies of EM -field chirality and magnetism \nis a topical subject in optics.  The magnetic dipole precession has neither left -handed nor right -\nhanded q uality, that is to say, no chirality [20, 22]. The synthesis of chiral and magnetic  37 properties in molecular structures [1 10] allow the observation of strong magneto –chiral \ndichroism, where unpolarized light is absorbed differently for parallel and antipara llel \npropagation with respect to an applied magnetic field. If the chiral medium itself is \nferromagnetic, the large internal magnetic field should provide a significant boost to such \nphenomena compared with para - and diamagnetic media [1 11, 112]. 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Due to  \nmagnetization dynamics, a n electric field inside a ferrite disk has both spin and orbital angular \nmomentums. Electric dipoles  induced in  a dielectric sample precess and accomplish an orbital \nrotation. A bias magnetic field is directed normally to the disk plane. For opposite directions of \na bias magnetic field, one has opposite rotations , both orbital and spin, of the fields  inside a \nferrite and a dielectric . \n \n \n \n \nFig. 2. An orbital angular momentum of a ferrite disk at a MDM resonance . For a given \ndirection of a bias magnetic field, the power -flow circulations are the same inside a ferrite and \nin the vacuum near -field regions  above and below the ferrite disk.  \n  43 \n \n \nFig. 3. The ME -EM field interaction in a  microwave waveguide. Evidence for the angular \nmomentum balance conditions at a given direction of a bias magnetic field. In a view along z \naxis, there are opposite power flow circulations in a ferrite disk and on a metal surface.  \n \n \nFig. 4. A quasi -2D mo del of a thin rectangular waveguide with an embedded thin-film ferrite \ndisk. An insert shows the disk position inside a waveguide.  \n \n  44 \n \n \nFig. 5. FMR diagonal component of the permeability tensor \n  versus frequency at a constant \nbias magnetic field. The discrete quantities, shown for the first four MDM resonances, are in \nthe region where \n0  \n H   . \n \n  45 Fig. 6. FMR diagonal component of the permeability tensor versus a DC internal magnetic \nfield at a constant frequency. The discrete quantities, shown for the first four MDM \nresonances, are in the region where \n0  \n 0 iiHH    . \n \n \n \nFig. 7. Correlation between two mechanisms of the MDM energy quantization:  Quantization \nby signal frequency and quantization by a bias magnetic field.  \n \n \n \n \n  46 Fig. 8.  Relationship between quantized states of microwave energy in a cavity and magnetic \nenergy in a ferrite disk. ( a) A structure of a rectangular waveguide cavity with a  normally \nmagnetized ferrite -disk sample.  (b) A typical m ultiresonance spectrum of modulus of the \nreflection coefficient.  (c) Microwave energy accumulated in a cavity; \n()n\nRFw  are jumps of \nelectromagnetic energy  at MDM resonances.  \n \n \n \nFig. 9. A model showing a radial distribution of the MS -potential wave function based on the \nBohr –Sommerfeld quantization  rule. The region of an internal magnetic field where such a \nstanding wave takes place is defined as  \n0 iiHH    . The in ternal field \niH  increases with \nincrease of a bias magnetic field \n0H .   47 \n \n \nFig. 10. A qualitative picture of the Bohr –Sommerfeld -quantization  levels of an internal \nmagnetic field \niH  for standing waves corresponding to the first three MDMs.  With increasing \nthe mode number, the effective diameter \n()2n\neff\n  increases as well.   \n \n  48  \nFig. 1 1. A model illustrating discretization of magnetic energy in a ferrite disk at \nconst . \nThe slope of the straight lines is determined by  DC magnetization. \n0M  is saturation \nmagnetization of a homogeneous ferrite material. At MDM resonances in a ferrite disk, t he \nslope of the straight lines is de termined by  \n()\n0n\neffM . \n()nW  shows  the microwave energy \nextracted from a  ferrite disk at the n-th MDM resonances. D iscretization of the magnetic \nenergy is shown for the first four MDMs.  \n \n \n \n \nFig. 12. Schematic repre sentation of the cavity input impedance s at MDM resonances on the \ncomplex -reflection -coefficient plane (the Smith chart). The numbers and letters correspond to \nnumbers and letters in Fig. 8 ( b). As the energy swept through an individual resonance, one \nobserves evolution of the phase – the phase lapses. The phase jump of \n  is observed each time \na resonant condition is achieved.   \n \n  49 Fig. 13. An equivalent  scheme describing a structure shown in Fig. 8. At a given frequency \n , \ndetermined by a RF source, a microwave waveguide is presented by a characteristic impedance \n0Z\n. The waveguide is loaded by an impedance \nLZ , which depends on an external parameter – \na bias magnetic field \n0H .   \n \n \n \n \n \nFig. 1 4. An interaction of a MDM f errite disk with a microwave waveguide. The structure is \nviewed as the P + Q space . It consists of a  localized quantum system (the MDM ferrite disk), \ndenoted as the  regio n Q, which is embedded within an environment of scattering  states  (the \nmicrowave waveguide), denoted as the regions P. The coupling between the regions  Q and P is \nregulated by means of the two “contact regions” in the waveguide space.   \n \n   \n  \n \nFig. 1 5. Directions of rotations of the power -flow vortices along the axis of  a contact region . In \nthe contact regions, above and below a ferrite disk, we have helical -mode tunneling.  \n  50 \n \n \nFig 16. Modification of the Fano -resonance shape. At variation of a bias magneti c field, t he \nFano line shape of a MDM resonance can be completely damped. The scattering cross section \nof a single Lorentzian peak corresponds to a pure dark mode. Reference to [ 85]. \n \n \n    \n  \n                                  (a)                                                                           (b) \nFig. 17. An example of t he power -flow density distributions in  a near -field vacuum region \nabove a ferrite disk. A ferrite disk is placed in a rectangular waveguide symmetrically to \nwaveguide walls. (a) Single-rotating -magnetic -dipole  (SRMD) resonance;  (b) double -rotating -\nmagnetic -dipole  (DRMD) resonance . There is the dissimilar radiation rates of the SRMD and \nDRMD resonances. Radiation of the SRMD results in strong reflection while at the DRMD \nmode one has  electromagnetic -field transparency and cloaking for a ferrite  particle.  Reference \nto [86]. \n \n(a)  51 \n \n(b) \n \n \n(c)  \n \n(d) \n  52 Fig. 18. Fano resonance collapse observed in the microwave transmission. (a) The structure of \na rectangular waveguide with a MDM ferrite  disk and loading dielectric cylinders.  (b) \nTransmission characteristics at a dielectric parameter\n30r , (c) at \n38r , (d) at \n50r . For \nthe 1st and 2nd MDMs, the Fano line shape s are complet ely damped  at \n38r . In this case, \nsingle Lorentzian peak s appear . The scattering cross section corresponds to pure bright mode s. \nReference to [8 3]. \n \n \n \nFig. 19. Evidence for strong resonance field enhancement at the MDM resonance . Passing the  \nfront of the electromagnet ic wave, when the frequency is (a) far from the frequency o f the \nMDM resonance and ( b) at the f requency of the MDM resonance in  the disk.  Reference to [8 2]. \n \n  \n  \n                                        (a)                                                                   (b) \nFig. 20. The helicity factor \n()EF distribution near a ferrite disk in a  geometrically symmetrical \nstructure . In a red region, \n()0EF , in a blue region, \n()0EF , in a green region, \n()0EF . The \nfactor \n()EF  is characterized by antisymmetrical distribution  with respect to z axis. Along with \nthis, the helicity factor changes a sign at time reversal.  \n \n              \n  \n                                      (a)                                                                       (b) \n     53  \n          \n  \n                                        (c)                                                                    (d) \n      \nFig. 21. Four cases of the helicity -factor distributions when two external parameters – the disk \nposition on z axis and the direction of a bias magnetic field – change. When a MDM ferrite \ndisk is placed in a geometrically nonsymmetric s tructure, the distribution of a helicity factor \n()EF\n becomes nonsymmetric as well. R eflection with respect to the xy plane with simultaneous \nchange of a direction of a bias magnetic field completely restore the same picture of a heli city \nfactor \n()EF  [look at the distributions ( a) and ( d) and the distributions  (b) and ( c)]. At such a \nnonsymmetric disk position, the PT symmetry  (with respect to the symmetry plane of the disk)  \nof MDM oscillations is broken and the biorthogonality condition is not satisfied.  \n \n \n \nFig. 22. Four types of contours encircling exceptional points along the paths on a 2D \nparametric space, a bias magnetic field and a disk shift.  \nExceptional point EP 1CW is inside the clockwise contour: \nA \nA\nB\nB\nA. \nExceptional point EP 2CCW is inside the counterclockwise contour: \nC \nC\nD\nD\nC . \nExceptional point EP 3CCW is inside the counterclockwise contour: \nE \nE\nF\nF\nE. \nExceptional point EP 4CW is inside the clockwise contour: \nG \nG\nH\nH\nG. \n  54 \n \nFig. 23. Position s of the waveguide wave numbers on a complex plane . \n \n \n(a) \n \n(b) \n \nFig. 24. The  complex -plane incident and reflected field conditions for a below -cutoff \nwaveguide section. ( a) The incident  (at port 1)  field is the positive decaying mode  and the \nreflected (from  port 2) field is the negative decaying mode . (b) Due to the boundary conditions \nat port 2, the electric and magnetic fields of conjugate modes are \n90\n  phase shifted in time.     \n \n  55 \n \n \nFig. 25 . Waveguide dispersion characteristics and  MDM -oscillation BICs in a microwave \nwaveguide. The MDM  wavenumbers are beyond  the region of the waveguide modes.  The \nfigure shows a region of existence of complex waveguide modes .  \n \n "    },    {        "title": "1207.0899v1.Phenomenological_theory_of_spin_orbit_phase_transitions.pdf",        "content": " 1\n                                    Phenomenologica l theory of spin-orbit phase transitions \nKhisa Sh. Borlakov* and Albert Kh. Borlakov \nNorth Caucasian State Humanitarian and Technologica l Academy, \n36 Stavropolskaya str., Cherkessk, Russia, 369001 \nThis work is devoted to the logical proof of the Go odenough and Khomskii idea of the existence of spin 6\norbit transitions in transition magnetic crystals. In agreement with the basics of the Landau theory o f \nphase transitions the phenomenological theory of sp in6orbit transitions is constructed. The general \nscheme of the theory is illustrated by the applicat ion to the description of magnetic and structural \ntransformations in the copper ferrite CuFe 2O4. \nPACS number(s): 61.50.Ah, 64.70.Kb, 71.70.Ej, 75.30 .Gw \n \n1. Introduction \n \nThis paper is the development of the contents of th e papers [163]. In those works we have \nshown existence of the cooperative effect of relati vistic interactions in magnetic compounds of 3d \nelements. That idea has been expressed even earlier  by a number of authors [466] who suggested \nexistence of spin6orbit phase transitions. But in t hose earlier works (done in late 80s) the idea was \nnot properly formalized and there was neither a phe nomenological nor the statistical theory of spin6\norbit phase transitions. Now the situation is quite  different and the existence of spin6orbit phase \ntransitions has gained wide support. Leading expert s in the field (the authors of [769] and a number \nof their followers) offer a variety of microscopic and semi6microscopic models of phase transitions \nand their applications to specific crystals. Howeve r, our phenomenological approach has several big \nadvantages. It allows accounting for the symmetry o f all the occurring phases, and its methods are \nmore closely related to empirical studies. In this work, we suggest a phenomenological theory of \nspin6orbit (or, more generally, relativistic) phase  transitions in the spirit of the thermodynamic \nLandau theory [10]. As an illustration, we show tha t this theory agrees well with physical effects \nobserved in the copper ferrite  O CuFe4 2  in a wide temperature range from above the Curie p oint \n730=cT  K to room temperatures and even lower. \n2. Symmetry of the magnetic Hamiltonian \nFollowing Van Vleck [11], we write the magnetic Ham iltonian in the form \nˆ ˆ ˆ ˆ \nex dd anis H H H H = + +                                                            (1)  2\nwhere \nˆ\nex ik i k \ni k H A S S \n<=∑r r \n                                                                    (2) \nis the exchange Hamiltonian; the second term in (1) , written as \n2 ˆ [ 3 ( )( )] dd ik i k ik ik i ik k \ni k H D S S r r S r S −\n<= − ∑r r r r r r                                              (3) \ncorresponds to the dipole6dipole interaction; and t he third term \n4 2 2 ˆ ( ) ( ) anis ik ik i ik k ik H K r S r S r −=∑r r r r                                                     (4) \ntakes into account the magnetic anisotropy. In (2)6 (4), iSr\n, is the vector operator of the spin of the \nith  atom measured in h units; ikrr is the position vector connecting the ith  and kth atoms; ikA , the \nexchange integrals; ikD , the parameters of the dipole6dipole interaction; an d ikK , the parameters of \nquadrupole6quadrupole interaction. As was indicated  by Van Vleck [11], the simplest microscopic \nmodel that allows for magneto6crystalline anisotrop y suggests that the spin6spin interaction is of the  \nquadrupole character. \nLet us analyze the symmetry properties of the magne tic Hamiltonian. It is obvious that the first \nterm is invariant with respect to the simultaneous rotation of all spins through the same angle. \nMoreover, under the action of operations of the sym metry group of a paramagnetic crystal, the \natoms will move into crystallo6graphically equivale nt positions. The exchange integrals ikA  only \ndepend on the absolute value of the position vector  ikrr that connects the interacting atoms and do \nnot change upon transformations caused by the symme try elements of the space group G of the \nparamagnetic crystal lattice. Thus, the symmetry gr oup of the Hamiltonian (2) is the exchange \nparamagnetic group ) 3 (OG× , where O(3) is the three6dimensional group of spin r otations. The \n) 3 (OG×  group is a color symmetry group of the so6called P type [12]. The second and third terms \nare related to relativistic interactions and have a  lower symmetry as compared to that of the \nexchange Hamiltonian. This symmetry coincides with the symmetry of the crystal lattice of the \nparamagnetic crystal, which is described by the spa ce group G. As can be seen from (3) and (4), the \nrotating part of the symmetry element must act not only on the position vector ikrr but also on the \nspin vectors iSr\n, and kSr\n. As to pure translations, both (3) and (4) are obv iously invariant with \nrespect to them, since the parameters of the dipole 6dipole (D ik ) and quadrupole6quadrupole (K lk ) \ninteraction depend only on the spacing ikrr between the spins. Moreover, expressions (3) and (4 ) are \ninvariant with respect to the operation of inversio n of the spin direction i i S Srr\n−→ . This means that \nthe symmetry group of the total Hamiltonian is the Shubnikov paramagnetic group } 1, 1 { 1−×=′G G .  3\nThe energy spectrum of a macroscopic body is quasi6 continuous [10]; therefore, at the Curie \npoint Tc, states that are associated with zero magnetization vectors neighbor with states that are \nassociated with nonzero magnetization. The symmetry  of the magnetic Hamiltonian and the \nsymmetry of the statistical6equilibrium state above  the Curie point are the same, whereas below the \nCurie temperature, the symmetry of the statistical6 equilibrium state becomes lower than the \nsymmetry of the magnetic Hamiltonian. Such thermody namic systems are called systems with a \nspontaneous symmetry breakdown [13]. Above the Curi e point, the convenient Bogolyubov \naverages and quasi6averages coincide, whereas below  the Curie point, this is not the case [13]. \nIt is commonly accepted that the symmetry of the to tal magnetic Hamiltonian (1) breaks down \nspontaneously at the Curie point. However, the hypo thesis on the existence of spin6orbit phase \ntransitions [466] states that the spin6orbit intera ction occurs at a temperature c lsT T<. This \nhypothesis is, thus, equivalent to the following st atement: at the temperature Tc, there occurs a \nspontaneous breakdown of the exchange Hamiltonian ( 2), whereas at the temperature c lsT T<, there \noccurs a spontaneous breakdown of the total Hamilto nian, i.e., a spin6orbit (relativistic) phase \ntransition takes place. Below the Curie point, ther e arises a spontaneous magnetization vector Mr\n, \nwhose orientation relative to the crystallographic axes is arbitrary, i.e., the resultant ferromagneti c \nphase is isotropic. Below lsT , the relativistic phase transition leads to the appe arance of a magnetic \nanisotropy and of magnetostrictive distortions of t he crystal, i.e., the resultant ferromagnetic phase  \nis anisotropic. \nA question that arises is in which crystals we shou ld expect the existence of an isotropic phase. \nThe main carriers of magnetic properties in crystal s are the elements of the iron (3 d) and rare6earth \n(4 f) groups. In the former, the LS coupling arises, whereas in the latter, this is the  jj coupling. The \nelectron cloud of a 3d ion is spherically symmetric, and the cloud shape h as no significant \ncorrelation with the spin direction. In the 4 f ion, on the contrary, the electron cloud is strong ly \nanisotropic (\"rigid\"), and there exists a strong co rrelation between the distribution of the electric \ncharge and the direction of the magnetic moment of the ion. For this reason, the magnetic \nsymmetries of the exchange and nonexchange magnetic  materials should be described differently \neven in the paramagnetic phase. The exchange magnet ic materials that contain magnetically active \n3d ions are invariant in the paramagnetic phase with r espect to the group of three6dimensional spin \nrotations O(3), because for a spin that is independ ent of the distribution of the electric charge, no \npotential barriers exist that separate one spin dir ection from any other. If the electron cloud is \nanisotropic, then there are several states (orienta tions of the electron cloud) in the center of a \ncoordination polyhedron that correspond to the mini mum energy of the electrostatic interaction. A \nfixed position of the anisotropic electron cloud co rresponds to a fixed orientation of the total  4\nmagnetic moment of the ion jr\n. In fact, no such static picture is realized; rath er, there takes place a \ndynamic effect, i.e., continuous transitions occur between all available states of the ion; note that \nthe number of these states is finite. Exchange magn ets also exhibit transitions between possible \nstates, but these states form a two6dimensional con tinuum equivalent to the surface of the unit \nsphere. \nThe concept of spin6orbit phase transitions was int roduced in [466] and some other works to \nexplain some experimental situations; we believe, h owever, that these situations are unconvincing; \nmoreover, their discussion was purely qualitative. Therefore, the question of whether or not spin6\norbit phase transitions exist in reality and how th ey can be unambiguously identified remains open. \nAs a possible experimental support for the existenc e of spin6orbit phase transitions and of an \nisotropic magnetic phase, we consider the behavior of various physical characteristics of copper \nferrite ( 42O CuFe ) crystals. \n \n3. Experimental data and selection of a model \nThe copper ferrite 42O CuFe  has been studied starting from the beginning of th e 1960s (see [146\n20] and references to chapter 29 of [14]). The main  interest in this compound is the structural phase \ntransition existing in it, which is usually associa ted with the cooperative Jahn6Teller effect. The \ntransition occurs as the temperature decreases to 631 ≈JTT K [5] and is accompanied by the \nlowering of the cubic symmetry of its crystal latti ce to tetragonal and an anomalous behavior of the \nmagnetic and some other physical properties [15620] . An interesting feature of the Jahn6Teller \neffect in copper ferrite is the fact that it is obs erved in the magnetically ordered phase: at 730 =cT  \nK [14], there occurs a ferromagnetic phase transiti on in the crystal. \nCopper ferrite has the spinel structure, whose symm etry at sufficiently high temperatures is \ncharacterized by the space group 7\nhO. An analysis of the experimental data given in [1462 0] \nsuggests that in the temperature range of 6306730 K  (1) the crystal lattice has a cubic symmetry \ncharacteristic of the paramagnetic phase; (2) the m agnetic anisotropy of the crystal is anomalously \nsmall; and (3) the ferromagnetic structure is colli near, of the Neel type. With the transition to the \ntetragonal phase (at JTTT< ), the magnetic anisotropy increases sharply and the m agnetization \nsuffers a small jump in the vicinity of the transit ion point. All these properties  suggest, that copp er \nferrite  is a typical exchange magnet, and the \"tet ragonal\" phase transition is the transition from an  \nisotropic to an anisotropic ferromagnetic phase ind uced by the cooperative effect of relativistic \ninteractions.  5\nLet us now turn to the construction of a theoretica l model which would consistently describe the \nbehavior of the copper ferrite crystal in a suffici ently wide temperature range from slightly above \nthe Curie point to room temperature and below. We w ill neither discuss nor criticize theoretical \nworks devoted to the Jahn6Teller effect, because we  set a quite different task of including the \nstructural transition associated with this effect i nto the general picture of phase transitions occurr ing \nin the ferrite with the spinel structure ( 42O CuFe ) rather than of explaining and describing the Jahn 6\nTeller effect. In our opinion, such a picture is gi ven by the thermodynamic Landau theory [10], \nwhich has been developed in much detail, sufficient ly to allow for the magnetic and crystal6\nchemical symmetry of the lattice; this theory proce eds from the fundamental principles of statistical \nthermodynamics and permits one to sufficiently easi ly describe experimental results. \nTo describe the whole body of experimental data acc ording to the Landau theory, one should, \nfirst, select the symmetry group of the initial (hi gh6symmetry) phase and, second, choose a critical \norder parameter. By the critical representation, it  is usually meant an irreducible representation of \nthe symmetry group of the initial phase which deter mines the lowering of symmetry at the phase \ntransition point and according to which there occur s transformation of the critical order parameter. \nAfter this, all subsequent actions are performed by  the standard scheme which is described, e.g., in \n[21,22]. As we have already seen in Section 2, we s hould select the exchange paramagnetic group \nas the symmetry group of the paramagnetic phase for  exchange magnets. For the ferrite with spinel \nstructure, this will be the ) 3 (7O Oh×  group. The transition to the ferromagnetic phase i s \naccompanied by the development of a spontaneous mag netization and, consequently, the role of the \ncritical order parameter will be played by the magn etization vector Sr\n. We designated the \nmagnetization vector by the letter Sr\n in order to emphasize its purely spin origin in th e isotropic \nphase. If this vector is considered as a stationary  matrix (see below) in the spirit of [23], it is \nassociated with a stationary matrix with one column  and three rows \n\n\n\n\n=\n321\nSSS\nSr\n \nThe three components form a vector with respect to the group of three6dimensional spin \nrotations O(3), and each of these components is a s calar with respect to the transformations due to \nthe elements of the 7\nhO group. Therefore, the critical representation is th e V Ag′×1  irreducible \nrepresentation (IR) of the magnetic group ) 3 (7O Oh× , where gA1 is the unit IR of the cubic group \n7\nhO, and V′ is the pseudo6vector IR of the O(3) group. The the rmodynamics of the transition \naccording to the V Ag′×1  IR was described in detail in [24], so that we wil l not consider this  6\nquestion and proceed with the description of the tr ansition from the isotropic ferromagnetic phase \ninto possible anisotropic phases. \n3. Determining symmetry groups of anisotropic ferro magnetic phases \nUpon the transition from the paramagnetic phase int o the isotropic ferromagnetic phase, the \nsymmetry is lowered as follows: ) 1 ( ) 3 (7 7SO O O Oh h ×→×  (at cTT=), i.e., the cubic symmetry of \nthe crystal lattice is retained and the magnetic sy mmetry of the crystal is lowered. The three6\nparameter group of improper spin rotations transfor ms into a single6parameter group of proper \nrotations about the direction of the magnetization vector Sr\n, and the operation of inversion of the \nspin direction is impossible, since the phase is ma gnetically ordered. In order to describe the \ntransition from the isotropic into the anisotropic phase, we cannot construct a new theoretical \nscheme and conjecture what will be the new critical  irreducible representation. Following our main \npremise, we assume that the critical irreducible re presentation of the 7\nhO group, which induces the \ntransitions from the isotropic into possible anisot ropic phases, should unambiguously be deduced \nfrom the critical V Ag′×1  IR of the ) 3 (7O Oh×  group. Such a group6theoretical operation exists a nd \nis called the restriction of the IR of the direct p roduct of two groups to one of the multipliers [12] . \nLet us designate the restriction of the V Ag′×1  IR to the 7\nhO group by the same symbol but taken in \nbrackets, i.e., []V Ag′×1 . In the general case, such a restriction is reduci ble and is the direct sum of \nother IRs of the same space group belonging to the star of the same wave vector as the first \nmultiplier and having the same parity relative to t he spatial inversion [25]. This restriction is call ed \nthe exchange multiplet [12]. Calculations using the  general group6theoretical formulas lead to the \nfollowing result: []g g F V A1 1=′× ; that is, the exchange multiplet degenerates to a si nglet, and we \nhave no problem of selecting the critical IR. Thus,  the critical IR that induces phase transitions \nfrom the isotropic ferromagnetic phase to anisotrop ic ferromagnetic phases is the three6dimensional \npseudo6vector irreducible representation gF1 of the cubic space group 7\nhO. \nTable 1.  Low6symmetry phases induced by the IR gF1, of the Oh group.  \n              1 6\n43\n22\n3321\ni h h i D C C C C Gccc coo occ ccccr\n \nLet us now determine the magnetic groups of symmetr y of the anisotropic phases. This may be \ndone using a purely algebraic method [21,22], which  consists in the following. For the three6\ndimensional IR gF1, a set of matrices of the representation M(g) is specified, where g are the  7\nelements of the 7\nhO group and cr is assumed to be an order parameter that is transf ormed through \nthe IR gF1. We may believe that the order parameter cr is a vector of some space of a proper \ndimension. This space is called the image space [26 ]. The arrangements of the order parameter in \nthe image space that differ in symmetry are associa ted with different subgroups of the 7\nhO group. \nThese can be found by solving the algebraic equatio ns ccgMrr=)(  for the matrices of the \nrepresentation. The vector that remains unaltered u pon the action of the matrix on it is called the st a6\ntionary vector of this matrix. For each subgroup of  the initial group, there exists its own stationary  \nvector. In order to find this vector, it is not nec essary to use all the matrices of the representatio n, \nbut it is sufficient to use the generating matrix. The calculations for the  7\nhOgroup were performed \nin [21] and the results that are necessary for our work are given in Table 1. \nNow, we may turn to the calculation of the temperat ure dependences of the physical \ncharacteristics of ferrites. \n4. Temperature dependence of magnetic anisotropy co nstants \nIn order to obtain the temperature dependences of m agnetic anisotropy constants, the \nthermodynamic Landau potential of a cubic crystals should be constructed. It is well known that the \nLandau potential should depend on invariants that c onsist of the components of the order parameter \ncr. The IR gF1 in the space of the order parameter is associated w ith the point group O, which is \ncalled the image group [26]. All invariant polynomi als composed of the components of the order \nparameter are expressed through several polynomials  that form an integer rational basis of \ninvariants (IRBI). For the L group coinciding with the point group O, the IRBI consists of the fol6\nlowing three polynomials [26]: \n            2\n32\n22\n1 32\n12\n32\n32\n22\n22\n1 22\n32\n22\n1 1 ; ; cccIccccccIcccI =++=++=                  (5) \n \nApart from these three polynomials, the IRBI also c ontains a polynomial of the ninth order. If \nwe can restrict ourselves to the Landau potential t hat only takes into account changes in two \ncontrol parameters, e.g., the temperature and the e xternal magnetic field, when considering the \nthermodynamics of the transition to the anisotropic  magnetic phase, then we may restrict our6\nselves to a potential of the sixth degree (see belo w), and we will not need the ninth6degree \npolynomial. It is desirable that the Landau potenti al be reduced to the form that is usually \nemployed in an analysis of spin6reorientational tra nsitions. To this end, we express the order \nparameter cr through the unit vector mr by the formula mccrr= . In this case, the basis invariants \nacquire the following form:  8\n           2\n32\n22\n16\n34\n22\n1 ; ; mmmc Isc IcI ===  ,                                     (6) \nwhere 2\n12\n32\n32\n22\n22\n1 mm mm mms ++= . As to the Landau potential, we only know that it is  an invariant \nfunction of the order parameter. In addition, it is  known that it has a critical point, which is \ndegenerate in the general case. An arbitrary smooth  k6parameter function may be represented in the \nvicinity of a degenerate critical point in the so6c alled normal form, i.e., in the form of a polynomia l \nof a finite degree, and the procedure of reducing t o the normal form at a given number of control \nparameters is quite unambiguous [27]. If there are only two governing parameters, e.g., temperature \nT and pressure P, then the normal form of the Landau potential is a polynomial of the sixth degree \n                                      213 32 213\n132\n12 11 )( IIbIbIbIa IaIa cF +++++=r .          (7) \nSubstituting the explicit expressions for the invar iants into (7), we obtain \n                   2\n32\n22\n16\n26\n34\n16\n34\n22\n1 ) ( )( mmmcbscb cb ca ca ca cF +++++=r                  (8) \nLet us compare our potential with the classical pot ential that takes into account only the \nanisotropy energy of a cubic crystal \n                                    2\n32\n22\n1 2 1 )( )( mmmTKsTK Uanis +=   .                                 (9) \nwhere )(1TK  and )(2TK  are the first and second constants of the magnetocr ystalline anisotropy of \ncubic crystals. Comparing (9) with the two last ter ms of (8), we find expressions for these \nanisotropy constants through the absolute value of the order parameter \n                               6\n2 26\n34\n1 1 )(; )( cbcKcb cbcK = +=  ,                            (10) \nGiven the temperature dependence of the order param eter c = c(T), formulas (10) yield the \nsought6for temperature dependence of the anisotropy  constants. Since T T cls−≈  in the Landau \ntheory, we have \n                                 3\n22\n1 ) ( ;) ( T T KT T Kls ls −≈−≈                                    (11) \nNote that the above formulas for the temperature de pendences are valid near the isotropic6\nanisotropic phase transition temperature lsT. Formulas (11) remain valid also for nonexchange \nferromagnets, in which the isotropic phase does not  exist. In this case, the lsT  temperature should \nbe replaced by the Curie temperature Tc. Formulas (10)6(ll) give zero values of magnetic ani sotropy \nconstants at T > lsT  in agreement with experiment. Note that our formula s differ from the classical \nformulas of Akulov6Zener, which express the magneti c anisotropy constants through the \nmagnetization \n                                                    21\n210\n1 ; M K M K ≅≅   .                                               ( 12)  9\nWhen applied to nonexchange ferromagnets, formulas (11) yield \n                                                  6\n24\n1 ; M KM K ≅≅  \nLet us make a remark concerning the physical meanin g of the order parameter c that describes \nthe phase transition from the isotropic magnetic ph ase into the anisotropic phase. Since the \nmagnetizations of the sublattices in the isotropic phase differ from zero and have finite values, \nwhereas the order parameter should be zero in the i sotropic phase, the order parameter is not equal \nto the magnetization vector Sr\n. In fact, this is the orbital contribution to the vector of spontaneous \nmagnetization caused by the cooperative effect of r elativis6tic interactions, and the total \nmagnetization is equal to the sum of terms of the e xchange and relativistic origin cS Mrrv\n+= . The \nappearance of this contribution causes both the cha nges in the magnitudes of the magnetizations of \nthe sublattices and the rotation of the magnetizati on vector from the arbitrary direction along one of  \nthe easy axes. The absolute value of the order para meter c gives the value of that small jump of the \nmagnetization that is observed experimentally near the tetragonal phase transition. \n5. Physical effects that accompany the isotropic-an isotropic phase transition \n \nUpon the phase transition to an anisotropic phase, not only the magnetic state changes, but various \naccompanying effects take place as well. Let us con sider them from the general group6theoretical \nviewpoint, following [21,22]. As we have already sa id above, the gF1 irreducible representation is \n\"principal\" in some way, since it is this IR that d etermines the decrease in crystal symmetry at the \nphase transition. In the immediate vicinity of the transition point, effects that are due to the criti cal \nIR are most pronounced (small to the first order in  c). However, the phase transition is accompanied \nby a number of physical changes that are compatible  with the symmetries of the newly forming \nphases. To each of these changes, there corresponds  its own IR, which is called noncritical or \nassociated [21]. The critical and noncritical IRs a re determined in terms of linear algebra quite \nunambiguously and, taken jointly, form the so6calle d condensate. The information concerning the \ncondensate of the IR gF1which we need is contained in [28], and the results  are given in Table 2. \nNote that the IR gF1 itself also enters into the condensate as a second ary IR, although this is not \nshown in Table 2. The components of the critical an d secondary parameters may be used to form \nmixed invariants. In the mixed invariants, the comp onents of the secondary order parameters are \ncontained to the first degree, and the components o f the critical parameter to higher degrees. These \ndegrees are given in Table 3. The symbol \"+\" in the  table indicates the presence, and the symbol \"6\" \nmeans the absence of the corresponding invariant in  the Landau potential.  10 \n \nTable 2.  Total condensate of stationary vectors of the crit ical IR gF1, \nbelonging to the star of the wave vector 0=kr\n of space group 7\nhO \nabc ba a a C ccca a a C cooaab ao a C occaaa a a C cccF E A A G c\nihhig g g g D\n,3,\n1\n3216\n43\n22\n32 2 1\n−−−−−r\n \n \nTable 2.  Entering of the noncritical IRs of space group 7\nhOinto \nthe direct symmetric degrees of the critical IR gF1. \n \n                                                    \n++++++++−+++++++−+−+\n54322 1 2 1 g g g g g s F F E A A m\n \n \nLet us now clarify the physical meaning of the conc rete degrees of freedom associated with each \nsecondary order parameter. To the unit IR gA1, there corresponds isotropic relativistic magne6\ntostriction. The corresponding secondary order para meter is equal to the sum of the diagonal \nelements of the deformation tensor zz yy xx u u ua ++= . The components of the deformation tensor \niju may form two more secondary order parameters, namel y, a two6dimensional order parameter \nwith components \n                                 ) (\n21); 2 (\n61\n2 1 zz yy yy xx zz u u a u u u a −=−− =                                (13) \nwhich is transformed through the IR gE, and an order parameter \n                                  xy zx yz u au au a ===3 2 1 ; ;   .                                                 (14) \nwhich is transformed via the three6dimensional IR gF2. \nThe two6dimensional order parameter (13) describes compressive or tensile deformations, and the \nthree6dimensional order parameter (14) describes sh ear deformations. For describing the relativistic  11 \nmagnetostriction, the Landau potential (8) should b e augmented with mixed invariants composed of \nthe components of the secondary order parameters (1 3)—(14) and with quadratic invariants of the \nsame parameters. To facilitate comparison with the commonly used designations [29], we write the \nelastic and the magnetoelastic energies explicitly through the components of the deformation tensor \n) ( ) (2) (2122 2 2 44 2 2 2 11\nxx zz zz yy yy xx zx yz xy zz yy xx y uu uu uuC u u uCu u uCF ++ ++++++= ,          (15) \n        \nsu u uc umm umm ummcu u u um um umc F\nzz yy xx zxx z yzz y xyy xzz yy xx zzz yyy xxx му\n) ( ) ( 2)] (31[\n4\n32\n22 2 2 2\n1\n++ + + + ++++−++ =\nα αα\n                         (16) \n \nIn the above formulas, ijC are the elastic constants of the cubic crystal; iα, are the \nmagnetoelastic constants; and y xmm,, and zm are the components of the unit vector directed along \nthe antiferromagnetism vector. Note that, unlike the commonly  accepted approach [29], we \nseparated the temperature dependences from the magnetoelastic cons tants using the factors c2 and \nc4. By differentiating the sum of the elastic and magnetoelastic energy with respect to the \ncomponents of the deformation tensor and equating the deriv ative to zero, we obtain the \ndependence of the components of the tensor of relativistic magnet ostriction on the components of \nthe order parameter \n                                            ki ls ki ik mmTT mmc u ) (2−≅≅                                             (17) \nFor nonexchange ferromagnets, the temperature of the relativistic t ransition lsT in (17) should be \nreplaced by the Curie temperature  cT.  \nTable 4.  Vector basis for the positions 32( e) of the space group 7\nhO.. \n \n \nIR  \n1  \n2  \n3  \n4  \n5  \n6  \n7  \n8 \n \ngE  \n112\nеео   \n112\nеео  \n1 12\nеео  \n112\nеео   \n112 \nеео   \n1 12\nеео  \n112\nеео  \n112 \nеео  \n \ngF1 110\n101 \n01 1 1 10\n101 \n011 110 \n101\n011 110 \n101\n011 110 \n101\n011 110 \n101\n011  1 10\n101 \n011  110\n101 \n011 \n  12 \n6. Analysis of the structural changes in copper fer rite \nBelow the temperature of the Iahn6Teller (in fact, the relativi stic) transition in the copper ferrite, \ntetragonal distortions of the crystal lattice are observed, and  the tetragonal axis becomes the easy \naxis of magnetization. X6ray diffraction and other structur al studies (whose results are given in [156\n20]) suggest that the tetragonal phase has a symmetry described b y the space group 19\n4hD. However, \nit follows from our theory that the tetragonal phase shoul d have a symmetry that is characterized by \nthe space group 6\n4hC. Thus, the theory appears to contradict the experiment. We sho w that this \ncontradiction is only apparent. \nThe space group 19\n4hD appears if the phase transition occurs through the ir reducible representation \ngE of the  space group 7\nhO. We believe, however, that the critical irreducible representation i s the \npseudovector irreducible representation gF1 resulting in a tetragonal phase with symmetry 6\n4hC. \nThe nature of displacements for the IRs  gE and gF1 is quite different, which can be seen from the \ncalculations performed by the formula \n                                                  ∑=\nαααϕ )( )( r c rui irr rr\n ,                                              (18 ) \nwhere ) (ruirr the displacement of the ith atom in the unit cell of the spin el structure; and iαϕr are \nthe basis functions of the representation considered. For bo th IRs, only the 32(e) positions occupied \nby oxygens suffer displacements. This can be expressed in the g roup6theoretical language as \nfollows: the IRs gE and gF1, enter into the mechanical representation at positions 32( e). In the \ncubic unit cell (Fig. la), there are eight oxygen positions , which are rigorously enumerated (Fig. lb). \nThe vector basis functions, which were borrowed from [30], are given in Table 4. The symbol \"e\" \nin the table denotes the number 3 , and the bar over the symbol means that the number is \nnegative. The transition into the tetragonal phase through th e IR gE is associated with the \nstationary vector (C, 0), and the transition through the I R gF1 with the stationary vector (C, 0, 0). If \nwe substitute the basis functions from Table 4 and the stat ionary vectors (C, 0) and (C, 0, 0) into \n(18), we will see that the displacements of each of the eight ox ygen atoms are only determined by \nthe first line of the vector basis for both the gE and gF1 IRs.  \nFigure la shows a portion of the unit cell of the spinel st ructure which is equivalent to its \nprimitive unit cell. The A atoms, which occupy the tetrahedral p ositions 8( a) are shown as stroked \ncircles. The B atoms, which occupy the octahedral positions 16( d),  are shown by the solid circles, \nand the oxygen atoms are given by the empty circles. Figure lb displays the upper projection of this \nunit cell (only oxygen atoms are shown). Figure schematically s hows the anion displacements for  13 \nthe gE and gF1 IRs. These figures correspond to Fig. lb. which also show s the oxygen anion \nnumbers. \n \n \n \n \n \n \n \n \n \n \n \n \n \n           А             В \n \n            а) \n \n           5                           1            7                        3 \n \n \n \n               6                         2             8                         4 \n \n          А                                         В \n           б) \n  \nFig. 1. A portion of the unit cell of the spinel structure.  \n \n \n                                       Paramagnetic Phas e ) 3 ( O Oh×                                                                             \n \n                                     Isotropic Phase ) 3 (O Oh×  \n \n                                                                     iC3 \n                                     hC4 \n \n \n                                              iC               hC2 \n \n \nFig. 2. The diagram of possible magneto6crystalline states of a cubic fer romagnet  \nFigure 2 displays the diagram of possible magneto6crystalli ne states of a cubic ferromagnet, \nwhich contains paramagnetic, isotropic, and anisotropic phases ( we gave the symbols of point \ngroups instead of the total symbols of space groups). So lid lines correspond to first6order \ntransitions. The single6parameter phases hC4 and iC3 always neighbor with the paramagnetic phase, \nso that the transition of the crystal into the ferromagnetic s tate is accompanied by the appearance of \nan easy axis [100] or [111]. The transition into these st ates is a second6order phase transition. Under  14 \ncertain conditions, the paramagnetic phase may also neighbor wit h a single6parameter phase hC2, \nwhich has an easy axis of the [011] type. \nIn this case, the boundary between the hC4 and iC3 phases degenerates into a tetra6critical point, \nat which the paramagnetic and three single6parameter phases meet. Th e dashed lines between the \nregions of existence of the ferromagnetic phases correspond to the lines of spin6reorientation \ntransitions, which are first6order phase transitions. For ex ample, when the representative point \ncrosses the boundary between the tetragonal phase hC4 and the triclinic phase iC a spin6\nreorientation transition into a canted phase with an easy axis of  the [lmn]  type occurs. \nIf we take a solid solution of two magnetic compounds that h ave the same crystal6lattice type but \ndifferent easy6axis types below the relativistic phase transitio n temperature lsT , then a change in the \nconcentration of one of the components may result in a spin6  reorientation transition of the [111] \n↔ [100] type. Such a transition can occur at temperatures sufficien tly close to the lsT point. At \nlower temperatures, this transition occurs through an interm ediate phase: [111] ↔ [011]  ↔ [100]. \n \n \n \n \n \n \n \n \n \n \n      IR gF1 \n \n \n \n \n \n \n \nIR gE \n \nFig. 3.  The character of displacements for two IRs  \n  \n \nIt can be seen from Fig. 3 that the character of displacements for  these two IRs is quite different. \nHow can we explain that the pattern of displacements that are observ ed experimentally corresponds \nto the IR gE, whereas the critical irreducible representation is gF1. The IR gE enters into the total \ncondensate of the critical IR gF1 as a secondary IR. The relative contributions of the critical and  15 \nnoncritical displacements to the final pattern of displacements in  the vicinity of a phase transition \npoint are different; namely, the critical displacements are large compared to the noncritical ones. \nFar from the phase transition point, the noncritical displacemen ts can exceed the critical \ndisplacements and the X6ray diffraction will show the pattern  of displacements corresponding to the \nspace group 19\n4hD, although the true symmetry group of the crystal lattice is  6\n4hC  (see [21] for more \ndetail). \n6. Conclusion \nFrom the above, the following conclusion may be drawn: we indeed succeeded in theoretically \nand experimentally substantiating the existence of spin6orbit ( relativistic) phase transitions. The \nmeaning of the above6said consists in the following. We developed a phenomenological theory of \nrelativistic phase transitions in terms of the thermodynamic L andau theory. The main postulate of \nthe Landau theory, i.e., the statement that the symmetry grou ps of low6symmetry phases are \nsubgroups of the symmetry group of the high6symmetry ph ase, remained hard6and6fast. But the \nidea that the low6symmetry phases correspond to the absolute minima in the nonequilibrium \npotential which is invariant with respect to the symmetry gro up of the high6symmetry phase had to \nbe sacrificed. In this case, the exchange paramagnetic group ) 3 (OG×  should be selected as the \nsymmetry group of the high6symmetry (paramagnetic) phase. The absolute minima of the non6\nequilibrium thermodynamic potential that is invariant relati ve to this group correspond to different \nisotropic magnetically ordered phases. In order to consider tran sitions from the isotropic into \nanisotropic phases, we should construct a new potential which i s invariant relative to only the space \ngroup. The critical irreducible representation of group G that induces the transitions into different \nanisotropic phases enters into the exchange multiplet generated by  the critical irreducible \nrepresentation of the ) 3 (OG×  group that induces phase transitions at the Curie point. This \nideology was illustrated using the copper ferrite 42O CuFe , which is ferromagnetic. Our theory \npermits us to describe changes in the magnetic and crystallogra phic symmetry at the Curie point \nand at the point of the tetragonal phase transition, as well as  changes of some physical properties. \nNote that the attempt to retain the Jahn6Teller interpretation of  the tetragonal phase transition at \n631≈JTT  K leads to serious difficulties. Indeed, as was said in [24, p . 203], \"when magnetized \nalong the direction of the cubic unit6cell edge, a cubic crystal  becomes weakly tetragonal, while \nmagnetized along the spatial diagonal of the unit cell cube, it b ecomes rhombohedral.\" This means \nthat if we assume that the easy axes appear directly at the Curie p oint, the crystal of the copper \nferrite indeed becomes tetragonal already at the Curie point Tc = 730 K, while at 631 ≈JTT K  there \noccurs an isostructural phase transition with a sharp increase in the degree of tetragonality due to  16 \nthe cooperative Jahn6Teller effect. But, with this approach, it is difficult to find a simple and clear \nlink between the changes in the symmetry and the magnetic and oth er physical properties. The \nattempt to discard the general scheme of the Landau theory and to introduce the \"foreign\" idea of \nthe Jahn6Teller effect into the picture of magnetostructural chang es in the crystal with the only \npurpose of not recognizing the existence of the isotropic ma gnetic phase is rather eclectic, in our \nopinion. The situation resembles the events of the last quarter  of the last century, when various \narguments were given in order to retain the mechanical supports f or the Maxwell field equations \nand not to recognize the independent existence of the field. \nThus, the T JT  temperature ( 631≈ K) is most likely to be the temperature of the relativistic phase \ntransition ( lsT) in the copper ferrite, while in the temperature range of 630  to 730 K, the \nmagnetocrystalline constants appear to be exactly zero. \n__________________________________ \n* Electronic address: borlakov@mail.ru  \n1.  Borlakov, Kh. Sh., SpinLorbit coupling and symmetry of the magnetically ord ered phase of \nmagnetic materials. 1. The symmetry of the exchange phase.  Available from VINITI, 1988. \nMoscow, No. 39736В88. \n2.  Borlakov, Kh. Sh., SpinLorbit coupling and symmetry of the magnetically ord ered phase of \nmagnetic materials. 2. Thermodynamics of the spinLorbital  phase transition . Available from \nVINITI, 1989. Moscow, No. 1086В89. \n3.  Borlakov, Kh. Sh., On One Consequence of the Hypothesis of  the Existence of Spin6Orbit Phase \nTransitions , Phys. Met. Metallogr. 1999, vol. 86, no. 1, pp. 19627. \n4.  Goodenough, J.B., Magnetism and the Chemical Bond. New York: Wiley6Interscience, 1963. \n5.  Kugel', K.I. and Khomskii, D.I., Jahn6Teller Effect  and Magnetism: Transition6Metal Compounds, \nUsp. Fiz Nauk, 1982, vol. 136, no. 4, pp. 6216664. \n6.  Belov, K.P., Goryaga, A.N., and Lyamzin, A.N., Anom alies of Magnetic Properties of the Cobalt \nFerrite, Fiz Tverd. Tela (Leningrad), 1989, vol. 31, no. 2, pp. 28631. \n7.  Y. Tokura and N. Nagaosa, Orbital Physics in Transition6Met al Oxides// SCIENCE . 21 APRIL \n2000, VOL 288. Pp. 4626468.  \n8.  Andrzej M. Oles’, Spin6Orbital Physics in Transition Met al Oxides//  ACTA PHYSICA \nPOLONICA A.  2009. Vol. 115, no.1. pp 36646. \n9.  Tapan Chatterji, Orbital ice and its melting phenomenon//  Indian J. Phys . 80 (6), 6656675 \n(2006). \n10.Landau L.D. and Lifshits E.M.: Statistical Physics, Oxf ord:Pergamon, (1980) 1, 3rd edition,  17 \n11.  Van Vleck, D., On the Theory of Ferromagnetic Reson ance Absorption// Phys. Rev.61950.6 V.78.,6 p. \n266 \n12.  Izyumov, Yu.A., Naish, V.E., and Ozerov, R.P.,  Neutr on Diffraction of Magnetic Materials; \nConsultants Bureau: New York, NY, USA, 1991. \n13.  Akhiezer, A.I. and Peletminskii. S.V., Metody statisticheskoi fiziki (Statistical Physics Methods), \nMoscow: Nauka, 1977. \n14.  Tablitsy fizicheskikh velichin (Tables of Physical Quantities), Kikoin, I.K., Ed.,  Moscow: Atomizdat, \n1976. \n15.  Prince, E. and Treuting, R.G., The Structure of Tet ragonal Copper Ferrite, Acta Ciystallogr., 1956, \nvol. 9. no. 12, pp. 102561028. \n16.Ohnishi, H. and Teranishi, T, Crystal Distortion  in Copper Ferrite6Chromite Series, J. Phys. Soc. \nJpn., 1961. vol. 16, no. I, pp. 35643. \n17.  Nagata, N., Miyagata, T., and Miyahara, S., Magnetocrystallin e Anisotropy of Tetragonal Phase \nof Copper Fer6rite, IEEE Trans. Magn., 1972, vol. MAG68, no. 3/1, pp. 145161453. \n18.  Levinstein, H.J., Schinetter, F.J.. and Gyorgy, E.M., Mag netic and Crystallography Study of the \nPhase Transition in Copper Ferrite, /. Appl. Phys., 1965, vol. 36, no. 3. pp. 116361164. \n19.  Malafaev, N.T., Murakhovskii, A.A., Popkov, Yu.A., an d Vorob'ev, V.V., Jahn6Teller Effect and \nthe Magnetic Properties of CuFe 204, Ukr. Fiz. Zh. (Russ. Ed.), 1984, vol. 29, no. 2, pp. 2866290. \n20.  Malafaev, N.T., Murakhovskii, A.A., Popkov, Yu.A., Prok openko, V.K., and Shemyakov, A.A., \nMagnetic Anisotropy and Nuclear Magnetic Resonance Spectra of t he Tetragonal and Cubic \nCopper Ferrites, Ukr. Fiz. Zh. (Russ. Ed.), 1988, vol. 33, no. 1, pp. 1236127. \n21.  Sakhnenko, V.P., Talanov, V.M.. and Chechin, G.M., Group6T heoretical Analysis of the Full \nCondensate Arising upon Structural Phase Transitions, Fiz. Met. Metalloved., 1986, vol. 62, no. \n5. pp. 8476856. \n22.  Stokes, H. T. & Hatch, D. M. (1988) Isotropy Subgroups of the 230 Crystallographic Space \nGroups . Singapore: World Scientific. \n23.  Borlakov, Kh.Sh., Generalization of Equal6Modulus Exchange M agnetic Classes// Low Temp. \nPhys.6  1998.6Vol. 24, no. 9, pp. 6476651. \n24.  Landau, L.D. and Lifshitz, E.M., Electrodynamics of condens ed matter, 1, 3rd edition, \nOxford:Pergamon, (1980). \n25.  Borlakov, Kh.Sh., On the Physical Meaning of Exchang e Multiplets, Phys. Met. Metallogr.. \n1998, vol. 86, no. 2, pp. 1236128. \n26.  Izyumov, Yu.A. and Syromyatnikov, V.N., Phase transitio ns and crystal symmetry, Kluwer,  18 \nDordrecht, 1990. \n27.  Kut'in, E.I., Lorman, V.L., and Pavlov, S.V.. Methods o f the Theory of Singularities in the \nPhenomenology of Phase Transitions// Usp. Fiz. Nauk, 1991, vol. 161, no. 6, pp. 1096148. \n28.  Ivanova, T.I., Kesoretskikh, V.N., Sakhnenko, V.P., and Ch echin, G.M., Group6Theoretical \nAnalysis of the Structure of Low6Symmetry Phases Arising upon Phase Transitions in Crystals \nwith a Space Group   /G731/Gbdb/Gb39, Available from VINITI, 1986, Moscow, no. 52636186. \n29.  Vonsovskii, S.V., Magnetizm (Magnetism). Moscow: Nauka, 1971. \n30.  Sakhnenko, V.P., Talanov, V.M., Chechin. G.M., and Ul'yano va, S.I., Vozmozhnye fazovye  \nperekhody i atomLnye smeshcheniya v kristallakh s prostranst vennoi gruppoi  /G731/Gbdb/Gb3b 2. Analiz  \nmekhanicheskogo i perestanovochnogo predstavlenii (Possible Phase Transitions and Atomic  \nDisplacements in Crystals with a Space Group /G731/Gbdb/Gb3b: 2. Analysis of Mechanical and  \nPermutation Representations), Available from VINITI, 1983 . Moscow, no. 6379683. "    },    {        "title": "2203.16061v1.Origin_of_Perpendicular_Magnetic_Anisotropy_in_Co__x_Fe___3_x__O___4_δ___Thin_Films_Studied_by_X_ray_Magnetic_Circular_and_Linear_Dichroisms.pdf",        "content": "arXiv:2203.16061v1  [cond-mat.mtrl-sci]  30 Mar 2022Origin of Perpendicular Magnetic Anisotropy in Co xFe3−xO4+δThin Films\nStudied by X-ray Magnetic Circular and Linear Dichroisms\nJun Okabayashi1,∗, Masaaki A. Tanaka2, Masaya Morishita2, Hideto Yanagihara3, and Ko Mibu2\n1Research Center for Spectrochemistry, The University of To kyo, Bunkyo-ku, Tokyo 113-0033, Japan\n2Graduate School of Engineering, Nagoya Institute of Techno logy, Nagoya 466-8555, Japan\n3Institute of Applied Physics, University of Tsukuba, Tsuku ba 305-8573, Japan\n(Dated: March 31, 2022)\nWe investigate the element-specific spin and orbital states and their roles on magnetic anisotropy\nin the Co-ferrite (Co xFe3−xO4+δ(001)) thin films which exhibit perpendicular magnetic anis otropy\n(PMA). The origin of PMA in the low xregion (x <1) can be mainly explained by the large perpen-\ndicular orbital magnetic moments in the Co2+(3d7) states detected by X-ray magnetic circular and\nlinear dichroisms (XMCD/XMLD). The XMLD for a PMA film ( x= 0.2) with square hysteresis\ncurve shows the oblate charge distribution in the Co2+site, which is consistent with the change in\nlocal nearest neighbor distance in Co detected by extended X -ray absorption fine structure analysis.\nOur finding reveals that the microscopic origin of PMA in Co-f errite comes from the enhanced or-\nbital magnetic moments along out-of-plane [001] direction through in-plane charge distribution by\ntensile strain, which adds the material functionalities in spinel ferrite thin films from the viewpoint\nof strain and orbital magnetic moments.\nPerpendicular magnetic anisotropy (PMA) is one of\nthecrucialissuesinthespintronicsresearchfieldfromthe\nviewpoints of thermal stability enhancement and high-\ndensity storage technology with the low current magne-\ntization switching. Until now, tremendous efforts have\nbeen devoted to design the materials with large PMA to\naddthenovelfunctionalitiesandexternalfieldcontrolling\nof magnetic properties [1–5]. Among them, the magnetic\noxides possess a number of attractive advantages for the\nspintronics devices because of their semiconducting or\ninsulating behavior with high spin polarization. These\nmagnetic oxides can be utilized as spin-filtering effects,\nresulting in high tunnel magnetoresistancewith multifer-\nroic properties by controlling the tensile or compressive\nstrains in the thin films [6–8].\nMostfamousspinel-typecompoundsforspintronicsap-\nplication might be Co-ferrites (CoFe 2O4: CFO) epitax-\nial thin films because of their insulating advantages as\nspin-filtering, high-frequencyperformance, and magneto-\nstrictive properties [9–16]. Further, the interfacial prox-\nimity effect, emerging on films with non-magnetic heavy-\nmetalelementswhichpossessalargeatomicspin-orbitin-\nteraction[17–19], alsodevelopsasthe spin-orbitronicsre-\nsearches through the interfacial spin-orbit coupling. Re-\ncent development appends further functionality of large\nPMA in CFO in the order of 106J/m3by tuning the\nCo compositions [20–23]. Since CFO exhibits high mag-\nnetostriction coefficient ( λ100= –590×10−6), the strain\nfromthesubstratespropagatesintotheCFOlayer,which\nbrings the local distortion around the transition-metal\n(TM) cation sites. In fact, the lattice mismatching be-\ntween MgO (4.21 ˚A) and CFO (8.38 ˚A) forming almost\nthe double period brings the epitaxial crystalline growth\nwith a few percentages of tensile strain into CFO thin\nlayer. Depending on the Co compositions, the lattice\nconstants of CFO can be tuned systematically. Struc-\ntural and magnetic properties in the CFO thin films\ncan be controlled by the cation vacancies and anti-phaseboundaries which are generated during the crystalline\ngrowth and suppress the saturation magnetization val-\nues through the antiparallel spin coupling [24–27]. His-\ntorically, magnetic anisotropy in bulk CFO has been un-\nderstood as an effect that the average of <111>easy\naxes direction produces cubic anisotropy in Co2+[28–\n30]. Recent theoretical and experimental investigations\nfor CFO thin films suggest that the tensile strain into\nCFOlayerproducesthePMA[31–38]. Squarenessofout-\nof-plane hysteresis curves depends on the growth mode,\nsuch as molecular-beam epitaxy, pulsed-laser deposition\n(PLD), or sputter deposition. It is also reported that the\nPMA increases in off-stoichiometric CFO films with low\nCo compositions [22, 23]. Although the PMA in CFO is\nstrongly demanded for spinel-type oxide spintronics and\nspin-orbitronics by controlling the orbital states [39], the\nmicroscopic origin of the PMA appearing in the low Co\ncompositions of CFO has not been clarified yet.\nIn order to understand the PMA in CFO, the element-\nspecific orbital magnetic moments ( morb) and their or-\nbital anisotropy have to be examined explicitly. In par-\nticular,morbis expected to be sensitive to the strain\nand local symmetry from the ligand fields around the\nTM ions. X-ray magnetic circular and linear dichro-\nisms (XMCD / XMLD) with their magneto-optical sum\nrules are powerful tools to detect the element-specific\nspin and orbital states in each site with the charge as-\nphericity [40]. Especially, XMLD sum rules can probe\nthe orbital anisotropy ( Q= 3L2\nz−L2) in the notation of\norbital angular momentum ( L) and electric quadrupoles\n(Q). Furthermore, since the Fe L-edge spectra consist of\nthe signals of three kinds of Fe2+or Fe3+states with oc-\ntahedral Ohand tetrahedral Tdsymmetries, the XMCD\nand XMLD magneto-optical sum rules for the integrals\nof FeL-edge spectra cannot be adopted without sepa-\nrating each contribution. Site- and state-resolved anal-\nyses become a crucial role for understanding the origin\nof PMA. For the analysis of XMCD and XMLD spec-2\ntra, the ligand-field multiplet (LFM) cluster-model sim-\nulations including the configuration interaction (CI) ap-\nproach are used to determine the site-specific electronic\nstructure parameters accompanied by spin and orbital\nmagneticmomentsandtheiranisotropies[41]. Untilnow,\nthe XMCD and XMLD for CFO with in-plane anisotropy\nhave been extensively studied with angular dependence\nbetween incident polarized beam, magnetic field, and\nsample surface normal [42–45]. However, the X-ray mag-\nnetic spectroscopy investigations for the PMA films with\nsquare magnetization–magnetic field ( M-H) hysteresis\nshape in low Co compositions have not been pursued.\nWe haveproceededthe XMLD measurementsforprobing\nthe orbital anisotropy in several PMA systems by using\nremanent magnetization states in the polar Kerr geome-\ntry [46, 47]. On the other hand, recent operando-XMCD\ntechnique enables manipulation of out-of-plane and in-\nplanemorbby controlling strain in the magnetic multi-\nlayers, which is a microscopicoriginofmagneto-striction;\nthat leads to the observation of an effect proposed as\n”orbital-striction effect” [48]. The PMA in strained CFO\ncan be also categorized as a novel orbital-controlled ma-\nterial system by using the orbitaldegeneracyin Co2+3d7\nsystem in Ohligand field. Furthermore, extended x-ray\nabsorption fine structure (EXAFS) analysis through the\nmultiple-scattering process is alsoemployed to determine\ntheelement-specificstructuralcharacterization. Thepre-\ncise determinations of spin and orbital states become a\nkey solution to develop a strain-induced orbital physics\nand their applications using CFO.\nIn this study, considering above research motivations,\nwe aim to investigatethe origin of PMAin CFO from the\nviewpoint of element-specific spin and orbital magnetic\nmoments and their anisotropy using XMCD and XMLD\nwith LFM calculations and local structure detection by\nEXAFS analysis.\nThe Co xFe3−xO4+δ(001) samples ( x=0.2 and 0.6)\nwere prepared by PLDon the MgO (001) substrates with\nthe similar recipe of reference [23]. The PLD was per-\nformedundertheconditionsofthesubstratetemperature\nof 300◦C, the background oxygen pressure of 6.0 Pa, and\nthe deposition rate of 0.03 nm/s. A neodymium-doped\nyttrium-aluminum-garnet laser at the double frequency\n(532 nm) with the pulse width of 6 ns and the repeti-\ntion rate of 30 Hz was used. The energy density of the\nlaser beam was controlled to 1 J/cm2by an optical lens.\nThe 13-nm-thick CFO (001) layer with 1-nm-thick Cu\ncapping layer was deposited for various Co compositions\n(x). Excess oxygens accompanied by possible cation va-\ncancies are described as δ. The case of x=0.2 and 0.6\nexhibits the PMA and in-plane anisotropy, respectively.\nThein-planeanisotropyinCo 0.6Fe2.4O4+δisproducedby\nthe post annealingat 400◦C after the growthby releasing\nthe strain. The results of sample characterization by X-\nray diffraction (XRD) and magnetization measurements\nare shown in Supplemental Material (Fig. S1) [49].\nThe XMCD and XMLD were performed at BL-7A and\n16A in the Photon Factory at the High-Energy Acceler-ator Research Organization (KEK-PF). For the XMCD\nmeasurements, the photon helicity was fixed, and a mag-\nnetic field of ±1.2 T was applied parallel to the inci-\ndent polarized soft x-ray beam, to obtain signals defined\nasµ+ andµ−spectra. The total electron yield mode\nwas adopted, and all measurements were performed at\nroom temperature. The X-ray absorption spectroscopy\n(XAS) and XMCD measurement geometries were set to\nnormal incidence, so that the directions of photon helic-\nity axis and the magnetic field were parallel and normal\nto the surface, enabling measurement of the absorption\nprocesses involving the normal components of the spin\nand orbital magnetic moments. In the XMLD measure-\nments, the remanentstates magnetized out-of-planewere\nadopted. For grazing incident measurements in XMLD,\nthe tilting angle between incident beam and sample sur-\nface normal was kept at 60◦. The direction of the electric\nfield (E) of the incident linearly polarized synchrotron\nbeam was tuned horizontally and vertically by undula-\ntor. We define the sign of XMLD by the subtraction of\nthe (M/bardblE)−(M⊥E) spectra with respect to the magne-\ntization M. The EXAFS measurements at Co K-edge\nwere performed at BL-12C in KEK-PF using the flores-\ncence yield mode with a 19-element solid-state detector\nat room temperature.\nFigure 1 showsthe XAS and XMCD ofCo 0.2Fe2.8O4+δ\nfor Fe and Co L-edges at the normal incidence setup.\nSpectra are normalized at each absorption edge. Because\nof the composition ratio of Co:Fe=1:14, the XAS intensi-\nties of Co are suppressed. XMCD intensity ratio to XAS\nis estimated to be 5% and 47% for Fe and Co L3edge,\nrespectively. The raw data are displayed at the Supple-\nmental Material (Fig. S2). XAS and XMCD line shapes\nfor FeL-edges show distinctive features due to the three\nkinds of Fe states (Fe3+inOh, Fe3+inTd, and Fe2+\ninOh). For the Fe L-edges, although the difference in\nXAS is small, clear differential XMCD line shapes are\ndetected. The Fe3+state with Tdsymmetry exhibits op-\nposite sign, which is common for the spinel ferrite com-\npounds. On the other hand, in the case of Fe 3O4, the\nFe2+component is more enhanced [50]. Large XMCD\nsignals in Co L-edge correspond to the saturated magne-\ntized states. Within the orbital sum rule, the large morb\ngives rise to the asymmetric XMCD line shapes. Since\nthe Co site is almost identical as Co2+(Oh) symmetry,\nthe sum rules can be applicable for the Co XMCD spec-\ntra. The spin and orbital magnetic moments for Co2+\nsites are estimated as 1.32 ±0.20 and 0.63 ±0.09µB, re-\nspectively, using an electron number of 7.1. The error\nbars mainly originate from the estimation of background\nin XAS which is used for the sum rule analysis. Large\nmorboriginatesfromtheorbitaldegeneracyin d7electron\nsystem in the strained Co site. The multiplet structures\nare almost similar to the previous report by the LFM\ncalculations along the easy-axis direction [44, 45]. In or-\nder to confirm the PMA, the element-specific magnetic\nhysteresis curves are measured at each peak in Fe and Co\nL3edge. As shown in Fig. 1(c), clear square shapes are3\ndetected at each photon energies due to the out-of-plane\neasy axis, which is almost consistent with the magneti-\nzation measurement shown in Fig. S1 [49]. The square\nM-Hcurves for all energies suggest the strong exchange\ncoupling among each site. The opposite sign in the M-H\ncurves at Fe3+Tdsite and Fe2+/Co2+Ohsites origi-\nnates from the antiferromagnetic super-exchange inter-\naction among these sites. These suggest that the small\namounts of anisotropic Co sites with large morbgovern\nthe easy axis direction of Fe sites and stabilize the PMA.\nFigure 2 shows the Evector polarization dependent\nXAS, where the electric field Eis parallel and perpen-\ndicular to the out-of-plane magnetization direction. The\ndifferential line shapes were similar to those of previously\nreported spectra [44]. Because of small difference in XAS\nby the horizontal (parallel) and vertical (perpendicular)\nbeams as shown in the inset of Fig. 2, XMLD intensities\nare also suppressed and displayed in the different scales.\nFor FeL-edge XMLD, the overlapping of three kinds of\ncomponents brings complex differential line shape. We\nnote that the integrals of the XMLD line shapes are pro-\nportionaltothechargeasphericitywithintheXMLDsum\nrule [51]. For Co L-edge XMLD of the Co2+Ohsite, we\nconfirmed that the integral of XMLD converges to a neg-\native value, deducing that the sign of Qzz, where Qzz\nis the diagonal tensor component of electric quadrupole\nmoment describing the charge distribution asphericity as\nQxx+Qyy+Qzz= 0, is negative with the order of 10−1.\nThe result means the in-plane orbital states are strongly\ncoupled with E. This value can be also estimated from\nthe XMCDspinsumruleconsideringthe magneticdipole\nmTzterm. Since the measurement geometry of 60◦tilted\nfrom the sample surface normal cannot detect precise\northogonal direction, the oblique angle suppresses the\nlinear dichroism intensity to√\n3/2. The orbital polar-\nization of Co 3 dstates means the pancake-type oblate\ncharge distribution, which enhances out-of-plane compo-\nnent ofmorb. Therefore, with both XMCD and XMLD\ncombined, it can be concluded the in-plane tensile strain\ntriggers the changes of charge distribution along in-plane\ndirection, resultingin the largeout-of-plane morband the\nPMA.\nFor the analysis of XMCD and XMLD spectra, we\nemployed cluster-model calculations including the CI\nfor Co2+and Fe sites in Co 0.2Fe2.8O4+δas tetrahedral\n(Td) TMO 4and octahedral ( Oh) TMO 6clusters, mod-\neled as a fragment of the spinel-type structures. The\nHamiltonian included the electronic structure param-\neters of full on-site TM 3 d–3d(valence–valence) and\n2p–3d(core–valence) Coulomb interactions ( U) and the\nTdorOhcrystal fields (10 Dq) in the TMs, along with\nthe hybridization between the TM 3 dand O 2 pwave\nfunctions. The charge-transfer energy was defined as\n∆=E(3dn+1L)−E(3dn), where Ldenotes a hole in a\nligandporbital. The hybridization between the TM 3 d\nand O 2pstates was also parameterized in terms of the\nSlater–Koster parameters ( pdσ) and (pdπ), where the re-\nlation (pdσ) = –1/2(pdπ) was used [52]. The param-eters (ppσ) and (ppπ) were always set to zero. In all\ncases, a Gaussian broadening was used to simulate spec-\ntral broadening.\nAs shown in Fig. 3, spectral line shapes of XMCD\nand XMLD can be reproduced by the LFM calculations\nqualitatively, at least the peak positions, with three Fe\nstates and uniformed Co state. The fitting parameters\nare listed in Table I. We emphasize the same adjusted\nparameters of ∆, Uandpdσand intensity ratios of each\ncomponent for XMCD and XMLD are applicable for the\nfitting. In comparison with the previous first-principles\ncalculations of CFO, the values of U= 5 eV are also\nplausible from the viewpoints of LFM calculations [53–\n56]. The multiplet parameter values are almost similar\nto the previous report [57]. For the Co2+3d7case, the\nt2gstates are split into two levels by the tetragonal dis-\ntortion, and the lowest xystates are occupied by one\nof the down spin electrons. The other electron occupies\nthe degenerated yzorzxstates [35]. We determined the\ntetragonal distortion for the Co 3 dstates (Dtet) to be\n0.02 eV in order to reproduce the XMCD and XMLD\nspectral line shapes qualitatively. However, the best fit-\nted parameter sets shown in Table I are not sensitive to\nthe parameters caused by the strain. The intensity ratio\nof Fe3+Td, Fe3+Oh, and Fe2+Ohis deconvoluted to be\n1:1.3:0.25 in the fitting of Fe L3-edge XMCD and XMLD\naffecting the decrease of Fe2+states in the inverse spinel\nstructure. The Co L-edge line shapes are reproduced\nwith only a single Co2+site, except the peak at 779.5 eV\nin XMLD.\nIn order to confirm the local environment around the\nCo atoms, XAFS measurements with EXAFS analysis\nin the Co K-edge were performed to deduce the near-\nest neighbor distance through the Fourier transform. As\nshown in Fig. 4, the Co K-edge absorption spectra with\nEXAFS oscillatory behaviors are observed for x= 0.2\n(PMA) and x=0.6 (in-plane anisotropy). We note that\nthe FeK-edge XAFS cannot be separated in the Fourier\ntransform without some assumption because of the con-\ntributions of three kinds of states. Clear XAFS oscil-\nlation is detected and plotted in the wave number kin\nthe inset of Fig. 4(a). XAFS oscillation functon k3χ(k)\ncan be fitted by FEFF8 program [58] and the Fourier\ntransformed EXAFS profile is also displayed in Fig. 4(b)\nconsidering the information up to second nearest neigh-\nbor from the Co site. By using the fitting procedure,\nthe nearest Co-O bond length in the Co sites is 2.09 and\n2.08˚Afor PMA and in-plane cases, respectively, which is\nalmostidentical withthe previousstudy [59]. Theexpan-\nsion of nearest neighbor in the PMA film is qualitatively\nconsistent with the XRD shown in Fig. S1 [49]. Because\nof the effect of phase shift, the peak position in EXAFS\nis not related to the bond length directly. Coordination\nnumber also becomes large in the case of PMA because\nof the distorted local environment around Co site, which\ncorresponds to the intensity of EXAFS profile in Fig.\n4(b). These suggest the tensile local distortion triggers\nthe PMA in the Co site of strained CFO.4\nConsideringtheaboveresults, wediscusstheelectronic\nstructures of CFO. First, we discuss the suppression of\nFe2+with increasing the Co composition. The previous\nreport of M¨ ossbauer spectra for related CFO with PMA\nsuggests the suppression of Fe2+[23], where ∼100 nm\nfrom the interface was probed using incident γ-rays and\nemitted conversion electrons. Since the probing depth\nregion of XMCD is beneath 3 nm from the surface, finite\nFe2+states in XMCD is located at the surface region.\nThe previous report suggests that the Fe2+state is lo-\ncated at the surface region with inevitable lateral inho-\nmogeneity [60].\nSecond, we discuss the occupation of cation sites\nby the charge neutrality and the number of cation\nelements. Using the Co concentration and charge\nneutrality in the inverse spinel structure of the\nformation of spinel structure of AB2O4formula\nunit; Fe3+(Td)(Co2+(Oh)Fe2+(Oh)Fe3+(Oh))O4, the fol-\nlowing two equations are deduced for Co 0.2Fe2.8O4+δ;\n14nCo2+=nFe2++nFe3+, and 2nCo2++2nFe2++3nFe3+=8,\nwherenis the number of each element and the cation\nvacancy is implicitly incorporated as δ. Since the ra-\ntio ofnFe3+(Td),nFe2+(Oh), andnFe3+(Oh)is estimated\nfromtheXMCDandXMLDanalyses,thesefourparame-\nters are calculated to be nFe3+(Oh)=1.34,nFe3+(Td)=1.03,\nnFe2+(Oh)=0.26, and nCo2+(Oh)=0.19. These suggestthat\nthe existence of excess cation vacancies ( nv) can be esti-\nmated to be 0.185 from the equation for the cation site,\nnv= 3−nFe3+(Oh)−nFe3+(Td)−nFe2+(Oh)−nCo2+(Oh).\nWe emphasize that the cation vacancy can be estimated\nquantitatively from the analysis of XMCD.\nThird, we discuss the relationship between strain and\norbital magnetic moments. For the Co2+3d7system,\nthe ground state4Fmultiplet energy split by the cubic\ncrystal field to Γ 4state, and further effects by trigonal\nor tetragonal field and exchange coupling stabilize the\nlowest 3d7states. The splitting of t2gstate promotes\nthe degeneracy for spins as analogous to the 3 d2sys-\ntem of spinel-type V3+compounds [61, 62]. Large morb\nis recovered by the spin-orbit coupling ξCoof 70 meV.\nSince the uniaxial anisotropy in spinel-type Co ferrite\nis inevitably formed from the structural and electronic\nconfiguration, the out-of-plane easy axis can originate\nfrom the tensile strain effect. In the theoretical model\ncalculation, the PMA energy ( K) deduced from the en-\nergy difference along the axis direction can be expressed\nasK=E(100)−E(001) = −B1χ, where B1andχ\nis the magneto-elastic coefficient and strain, respectively\n[34]. Phenomenological B1of 1.4×108J/m3consists of\nthe elastic constant and magneto-striction coefficient at\nroom temperature for CoFe 2O4. In the case of x=0.2,\nB1xis applied to estimate K. Considering the lattice\nconstants of a/bardbl= 8.42˚A anda⊥= 8.29˚A estimated\nfrom the XRD, the strain of ( a/bardbl−a⊥)/a0∼1.5 % was de-\nduced using the bulk CFO value a0= 8.38˚A [32]. There-\nfore, the PMA energy is roughly estimated to 0 .51×106\nJ/m3. On the other hand, microscopic estimation of the\nPMA energy can be performed using anisotropic morbasK= 1/4ξ∆morb, where spin-orbit coupling constant is\nξand ∆morbis the difference between out-of-plane and\nin-plane morb. Although the in-plane morbcannot be de-\ntected unless applying high magnetic field to saturatethe\nmagnetization along the hard axis, we used the value in\nthe reference with extrapolation in low Co concentration\n[44]. Assumingthevalueof∆ morbtobe0.2 µB, thePMA\nenergy of Co can be estimated to be 1 .6×106J/m3with\ntheassumptionoftheCocomposition x=0.2andstrained\nCFO volume. The value of Kfrom XMCD is overesti-\nmated compared with that from magnetization measure-\nments, in general, by a pre-factor of 0.1 [63]. These esti-\nmations suggest that the PMA energy can be explained\nmainlybythecontributionofsmallamountofanisotropic\nCosites. Furthermore, the macroscopicmagnetostriction\neffect, which the relation between strain and K, can be\nrecognized including the morbfrom the viewpoint of elec-\ntron theory as orbital-elastic effect. The contributions\nfrom the Fe3+(3d5) sites are negligible because of half-\nfilled quenched orbital magneticmoments. The contribu-\ntion from the Fe2+(3d6) sites is also quenched since the\nstrained undegenerated level is occupied. Therefore, the\norigin of PMA in Co 0.2Fe2.8O4+δcan be predominantly\nexplained by the enhanced morbof 1/14 Co compositions\nthrough the strained charge distributions, which can be\nrecognized as diluted doped magnetic oxide systems.\nFinally, wediscusstheoriginofPMAinthespinel-type\noxidesfrom the viewpoint of tensile or compressivestrain\nandmorb. The strain in CFO is derived from the tensile\nstrain in the film, which is detected by the EXAFS anal-\nysis and confirmed by the sign of XMLD. Recent report\nshows that the PMA in spinel-type NiCo 2O4originates\nfrom the compressive strain in the film [64, 65] and that\nthe enhancement of morbin XMCD measurements is not\ndistinctive with the 2+ and 3+ mixed valence states of\nboth Co and Ni sites [66]. Therefore, the PMA in spinel-\ntype oxides can be categorized by the tensile or compres-\nsive strain. Because of tensile strain, oblate-type charge\ndistribution induces morbalong out-of-plane direction,\nwhich is similar to the case of enhanced morbin Co/Pd\nmultilayer [67]. On the other hand, in-plane compres-\nsion or out-of-plane elongation modulates the prorate-\ntype charge distributions, which induces the quadrupole\nmomentswithoutenhancing morbasdiscussedinstrained\nMn3−xGa system [47].\nInsummary, usingXMCDandXMLD, weinvestigated\nthe element-specific orbital magnetic moments and their\nanisotropyin Co-ferritethin films with lowCoconcentra-\ntion which exhibit the PMA. The origin of PMA in CFO\nis explained by the large morbin the Co2+(3d7) states.\nThe tensile lattice strain, which is deduced from EXAFS,\ninduces the out-of-plane morbthrough the anisotropicin-\nplane charge distribution. Furthermore, a novel method\nto estimate the number of cation vacancy from XMCD\nline shapes was also proposed. Our finding clearly re-\nveals that the controlling strain modulates morb, which\nopens up the material functionalities in the oxide spin-\norbitronicsby strain engineering, especially in the TM d75\nsystems.\nACKNOWLEDGMENTS\nThisworkwaspartiallysupportedbyJSPSKAKENHI\n(GrantNo. 16H06332),the KatoScience andTechnologyFoundation, the Izumi Science and Technology Founda-\ntion, and the Telecommunications Advancement Foun-\ndation. Parts of the synchrotron radiation experiments\nwereperformedundertheapprovalofthePhotonFactory\nProgram Advisory Committee, KEK (Nos. 2019G028\nand 2021G069).\n[1] S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey,\nB. D. 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Mater. 31, 1805260 (2019), Magnetotransport\nAnomaly in Room-Temperature Ferrimagnetic NiCo 2O4\nThin Films\n[65] Hiroki Koizumi, Ikumi Suzuki, Daisuke Kan, Jun-ichiro\nInoue, Yusuke Wakabayashi, Yuichi Shimakawa, and\nHideto Yanagihara, Phys. Rev. B 104, 014422 (2021),\nSpin reorientation in tetragonally distorted spinel oxide\nNiCo2O4epitaxial films\n[66] Daisuke Kan, Masaichiro Mizumaki, Miho Kitamura,\nYoshinori Kotani, Yufan Shen, Ikumi Suzuki, Koji\nHoriba, and Yuichi Shimakawa, Phys. Rev. B 101,\n224434 (2020), Spin and orbital magnetic moments in\nperpendicularly magnetized Ni 1−xCo2+yO4−zepitaxial\nthin films: Effects of site-dependent cation valence states\n[67] Jun Okabayashi, Yoshio Miura, and Hiro Munekata, Sci.\nRep.8, 8303 (2018). Anatomy of interfacial spin-orbit\ncoupling in Co/Pd multilayers using X-ray magnetic cir-\ncular dichroism and first-principles calculations\nTABLE I. The Electronic structure parameters used in the\nLFM cluster calculation. The units of each value is in eV.\n∆ U (pdσ) 10Dq\nFe2+(Oh)6.5 6.0 1.2 0.9\nFe3+(Oh)0.5 6.0 1.2 0.9\nFe3+(Td)4.5 6.0 2.0 -0.5\nCo2+(Oh)6.5 6.0 1.3 0.58\u0001 \u0002 \u0003 \u0004 \u0000 \u0005 \u0006 \u0007 \b \t \n \u000b \f \r \u000e \u000f \u0010\u0011 \u0012 \u0013 \u0014 \u0015 \u0016 \u0017 \u0018 \u0019 \u001a \u001b \u001c \u001d \u001e \u001f  !\n\" # $ % & ' ( ) * + , - . / 0 1 2/g894/g258/g895 /g894/g271/g895 /g894/g272/g895\n/g87/g346/g381/g410/g381/g374/g3/g28/g374/g286/g396/g336/g455/g3/g894/g286/g115/g895 /g87/g346/g381/g410/g381/g374/g3/g28/g374/g286/g396/g336/g455/g3/g894/g286/g115/g895\n3 4 56 7 8 9 : ; < = > ? @ A Bµ/g1085\nµ/g882/g121/g4/g94 /g121/g68/g18/g24/g38/g286/g3/g62/g882/g286/g282/g336/g286 /g18/g381/g3 /g62/g882/g286/g282/g336/g286C D E F GH I\nJ KL MN\nOP Q R S TU V\nW XY Z[\n\\] ^ _ ` ab c\nd ef gh\nij\nk\nlmno\np\nq\nrstu\nv\nw\nxyz{FIG. 1. XAS and XMCD of Co 0.2Fe2.8O4+δfilm.µ+ and\nµ−in XAS denote the magnetic field direction along the in-\ncident photon beams at normal incident geometry. Difference\nofµ+−µ−is defined as XMCD spectra for (a) Fe L-edge and\n(b) CoL-edge. Both XAS and XMCD intensities are normal-\nized by photon flux. (c) Magnetic-field dependence at each\nfixed photon energy; 708, 709, and 778 eV corresponding to\nFe2+(Oh), Fe3+(Td), and Co2+(Oh) peaks, respectively.9|}~\n\n\n/g894/g258/g895 /g894/g271/g895\n/g87/g346/g381/g410/g381/g374/g3/g28/g374/g286/g396/g336/g455/g3/g894/g286/g115/g895 /g87/g346/g381/g410/g381/g374/g3/g28/g374/g286/g396/g336/g455/g3/g894/g286/g115/g895/g121/g4/g94 /g121/g68/g62/g24/g38/g286/g3/g62/g882/g286/g282/g336/g286 /g18/g381/g3/g62/g882/g286/g282/g336/g286                \n  \nFIG. 2. XAS and XMLD of Co 0.2Fe2.8O4+δfilm of (a) Fe\nand (b) Co L-edges. Spectra were taken at the grazing in-\ncident setup where Eof the incident beam and direction of\nmagnetization Mwere parallel (horizontal) and perpendicu-\nlar (vertical), respectively. The inset displays an illust ration\nof the XMLD measurement geometry.10\nFIG. 3. Calculated XMCD and XMLD spectra for (a,b) Fe\nand (c,d) Co L-edges. For Fe L-edge, three components of\nFe2+(Oh), Fe3+(Oh), and Fe3+(Td) sites are also shown.\nTotal spectra are shown in black solid line. Open dotted\ncurves are experimental XMCD and XMLD shown in Figs.\n1 and 2.11\n/g87/g346/g381/g410/g381/g374/g3/g28/g374/g286/g396/g336/g455/g3/g894/g286/g115/g895\n/g24/g349/g400/g410/g258/g374/g272/g286/g3/g894 /g973/g895/g121/g4/g38/g94\n/g364\n/g454/g894/g364/g895\n/g116/g258/g448/g286/g3/g69/g437/g373/g271/g286/g396/g3/g894 /g973\n /g895/g454/g1089/g1004/g856/g1006/g3/g894/g87/g68/g4/g895\n/g454/g1089/g1004/g856/g1010/g3/g894/g349/g374/g882/g393/g367/g258/g374/g286/g895/g894/g258/g895\n/g894/g271/g895\n/g24/g349/g400/g410/g258/g374/g272/g286/g3/g894 /g973/g895/g454/g1089/g1004/g856/g1006/g3\n/g454/g1089/g1004/g856/g1010/g28/g121/g4/g38/g94\n¡¢£\n6 5 4 3 2 1 0\n¤ ¥ ¦§ ¨ ©ª « ¬\n8200 8100 8000 7900 7800 7700 7600\n ® ¯ °\n± ² ³ ´\nµ ¶ ·\n¸ ¹ º\n» ¼ ½¾ ¿ À Á  à ÄÅ Æ Ç È É\nFIG. 4. XAFS and EXAFS of Co 0.2Fe2.8O4+δ(PMA) and\nCo0.6Fe2.4O4+δ(in-plane anisotropy). (a) Co K-edge XAFS.\nInset shows corresponding koscillation in k3χ(k) forx= 0.2.\nDotcurveshowsthefittingresult. (b)EXAFSprofileafterthe\nFourier transform. Inset shows the fitting result in x= 0.2."    },    {        "title": "1510.05359v1.Ferroelectric_polarization_switching_with_a_remarkably_high_activation_energy_in_orthorhombic_GaFeO3_thin_films.pdf",        "content": "Ferroelectric polarization switching with a \nremarkably high activation -energy in orthorhombic \nGaFeO 3 thin films  \nRunning title : Polarization switching in o- GFO thin film at RT  \nSeungwoo Song,1 Hyun Myung Jang,1* Nam -Suk Lee,2 Jong Y. Son,3 Rajeev Gupta,4 \nAshish Garg,4 Jirawit  Ratanapreechachai,5 and James F. Scott5,6+ \n \n1Division of Advanced Materials Science,  and Department of Materials Science and Engineering, \nPohang University of Science and Technology (POSTECH), Pohang 790- 784, Republic of \nKorea.  2National Institute for Nanomaterials Technology , Pohang University of Science and \nTechnology (POSTECH), Pohang 790-784, Republic of Korea.   \n3Department of Applied Physics, College of Applied Science, Kyung Hee University, Giheung -Gu, \nYongin -City 446- 701, Republic of Korea.   \n4Department of Materials Science and Engineering, Indian Institute of Technology (IIT),  Kanpur, \nKanpur 208016, India.  \n5Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, \nUK. 6School  of Chemistry  and School of Physics, University of St. Andrews, St. Andrews  KY16 9ST , \nUK.   \n \n \n \n \n \n \nCorrespondence and requests for materials should be addressed to either H. M. J. \n(email: hmjang@postech.ac.kr  ) or J. F. S. (email: jfs32@cam.ac.uk  )  \n \n  \n1 \n  \nOrthorhombic GaFeO 3 (o-GFO) with the polar Pna 21 space group is a prominent \nferrite by virtue of its piezoelectricity and ferrimagnetism, coupled with magneto-\nelectric effects. Herein, we unequivocally demonstrate a large ferroelectric remanent polarization in undoped o -GFO thin films by adopting either a hexagonal strontium \ntitanate\n (STO) or a cubic yttrium -stabilized zirconia (YSZ) substrate.  The polarization -\nelectric -field hysteresis curves of the polar c- axis-grown o-GFO  film on a SrTiO 3/STO \nsubstrate show the net sw itching polarization of ~3 5 μC/cm2 with an unusually high \ncoercive field of ±1400 kV/cm at room temperature. The PUND measurement  also \ndemonstrates the switching polarization of ~26 μ C/cm2. The activation energy for the \npolarization switching, as obtained by density -functional theory calculations, is \nremarkably high, 1.05 eV per formula unit.  This high value accounts for the observed \nstability of the polar Pna2 1 phase over a wide range of temperature up to 1368 K.  \nKeywords : Oxide thin films, GaFeO 3, ferroe lectricity, ferrimagnetism, PLD, first- principles \ncalculations  \n \n \n      \n2 \n  \nINTRODUCTION  \nMultiferroics  are an interesting group of materials that simultaneously exhibit ferroelectricity \nand magnetic ordering with coupled electric, magnetic, and structural orders. Multiferroic \nmaterials with a pronounced degree of magnetoelectric (ME) coupling at room tempe rature \nare of great scientific and technological importance for their use in various types of electronic devices that include sensors, actuators, and electric- field controllable magnetic memories.\n1-3 \nAmong all the known multiferroics, BiFeO 3 is most extens ively studied owing to its large \nroom -temperature spontaneous polarization with improved magnetic properties in epitaxially \nstrained thin -film forms.4-7 \nPolar orthorhombic GaFeO 3 (o-GFO) is another prominent multiferroic  oxide by virtue \nof its room -temperature piezoelectricity (possibly ferroelectricity as well), near room -\ntemperature ferrimagnetism, and pronounced low -temperature ME effects. Since a linear ME \neffect was first reported in 1960s by Rado,8 magnetization -induced second harmonic \ngeneration,9 optical ME effect ,10 and other interesting studies11-22 keep renewing our research \nattention to this system. GFO crystallizes into the polar orthorhombic Pna 21 (equivalently, \nPc21n) space group with a ferrimagnetically  ordered spin structure. The four Fe3+ ions in a \nunit cell are antiferromagnetically coupled along a- axis in the Pna 21 setting. However, the \nintermixed Fe3+ ions occupying at the Ga sites (in other words, different Fe occupations at the \nFe1 and Fe2 sites) can lead to a ferromagnetic order with Fe3+ ions at the Fe sites, which \ngives rise to ferrimagnetic ordering at a temperature around 230 K .11,12  \nContrary to ferrimagnetic ordering and low -temperature ME effects,11 much less is \nknown on the ferroelectricit y of o- GFO. In principle, o -GFO should exhibit ferroelectricity \nup to high temperatures as the polar Pna 21 phase (with the corresponding point group of \n3 \n 2mm or C2v in Sch önflies notation) remains stable, at least, up to 1368 K.22 According to \nStoeffler,20,23 the computed ab initio polarization value of the undoped o-GFO is as high as 25 \nμC/cm2. In spite of these predictions, however, there has been no experimental demonstration \nof the room -temperature ferroelectric polarization switching with its polarizatio n value \ncompatible with the ab initio predictions. Several groups have studied the P -E (polarization -\nelectric -field) response of GFO ferroelectrics, mostly using thin -film forms.13-16 However, all \nthese studied revealed that the remanent polarization ( Pr) of GFO is less than 0.5 μC/cm2 with \nan unrealistically small value of the coercive field (E c ≤ 5 kV/cm) as well. More recently, Oh \net al.17 reported the epitaxial growth of o- GFO film on a SrRuO 3(111)/SrTiO 3(111) substrate \nwith the remarkably enhanced E c of ~150 kV/cm. However, their P r value still remains at  \n~0.05 μC/cm2. On the other hand, Mukherjee et al.18 recently reported room -temperature \nferroelectricity of the o -GFO thin film grown on an ITO(001)//YSZ(001) substrate  on the \nbasis of their observation of an 180o phase -shift of piezoresponse. However, the 180o phase -\nshift (or switching of the piezoelectric phase) does not necessarily indicate a ferroelectric \npolarization switching across the barrier of a double -well potential.  \nUp to now, the only unequivocal experimental demonstration of a reversible polarizat ion \nswitching in o- GFO thin films is made by Thomasson et al .19 According to their study, the 2  % \nMg-doped GFO film exhibits a well saturated P -E switching curve with a negligible tendency \nof the non- switching polarization.19 However, the measured P r value  is as small as 0.2 \nμC/cm2,19 which is less than 1 %  of the predicted P r value (25 μ C/cm2) of the undoped GFO. \nMoreover, this system is not a stoichiometric o- GFO (i.e., not an orthoferrite with Ga:Fe = \n0.6:1.4), in addition to 2% Mg- doping.19 Thus, all the reported P r values of o-GFO are in the \nrange of 0.05 and 0.5 μ C/cm2, which is unacceptably too small to be compatible with the ab \n4 \n initio  polarization of 25 μC/cm2.20,23  \nIn this article, we have clarified a puzzling discrepancy between the o bserved Pr value \n(≤ 0.5 μC/cm2) and the ab initio  prediction (25 μ C/cm2) and unequivocally demonstrated the \nferroelectric polarization switching with the net switching polarization value of ~30 μ C/cm2 \nby using the o-GFO thin films preferentially grown along the polar c -axis in Pna 21 setting ( b-\naxis in Pc 21n setting). In the present study, we adopt either a hexagonal or a cubic substrate \nto demonstrate the room -temperature polarization switching : (i) a SrRuO 3 (SRO) (1 11) \nbuffered hexagonal strontium titanate (STO) (111) substrate and (ii) an ITO (001) buffered \ncubic yttrium -stabilized zirconia ( YSZ ) (001) substrate.  \n \nMATERIALS AND METHODS   \nExperimental methods. Orthorhombic GaFeO 3 (o-GFO) film and SRO  bottom electrode \nwere grown on a Ti4+-single -terminated STO  (111) substrate by pulsed laser deposition with \nKrF excimer laser (λ  = 248nm) operated at 3 Hz and 10 Hz, respectively. GFO films were \ndeposited at 800 ºC in an oxygen ambient atmosphere (200 mTorr) with a fluence of 1  J/cm2 \nfocusing on a stoichiometrically sintered o- GFO target while 30 -nm-thin SRO  bottom -\nelectrode layers were grown at 680 ºC in a  100 mTorr oxygen atmosphere with a fluence of 2  \nJ/cm2. After the deposition, the SRO layer was  cooled down to room temperature under the \nsame oxygen pressure used in the o- GFO film deposition. We observed the thickness fringes \nof SRO(222) around the two-theta (2𝜃𝜃) value of STO  (222) , which strongly indicates an \nepitaxial growth of the SRO bottom electrode layer . The thickness of this SRO layer (30 nm)  \nwas also determined from the positions of the interference fringes (See Supplementary \nInformation for details).  \n5 \n Analysis of the domain orientation and phase formation in the films was done  by using a \nhigh- resolution x- ray diffractometer (D8 discover, Bruker) under Cu Kα radiation. Domain \nstructures  were investigated in detail by employing high- resolution transmission electron \nmicroscopy (JEM -2100F, JEOL with a probe Cs -corrector). Z-contras t high- angle  annular \ndark- field STEM (HAADF -STEM) image and selected -area electron diffraction (SAED) \nexperim ents were carried out under 200- kV acceleration voltage.  \nFor die lectric -ferroelectric  measurement s of the c -axis-grown o-GFO film , the Pt top \nelectrode with the diameter  of 100 μ m was deposited using a dc  sputtering system. Current \nand voltage (I -V) curves were recorded using a Keithley 2400 source meter. Polarization-\nelectric -field hysteresis loops (P -E curves) and positive -up and negative -down (PUND) pulse \nsequences were measured using a precision LC ferroelectric tester (Radiant technologies). A \ncommercial atomic force microscope (DC -EFM in XE- 100, Park Systems) was used for \nvertical piezoelectric force microscopy (vPFM) study to map piezoelectric properties of thin \nfilms. Pt/Ir coated tip was used for probing the signals. The input modulation voltage V ac \n(with the amplitude in the range of 0.1~2 V and the ac  frequency of 15 kHz ) was applied \nbetween the conductive tip and the bottom electrode using a function generator . The resulting \noscillations of the cantilever are read out with a lock in amplifier. Magnetic properties were measured by superconducting quantum interference device magnetometr y (SQUID, MPMS, \nQuantum Design) from 10 K to 300 K under various applied magnetic fields .  \n \nComputational methods. We have performed DFT calculations of o- GFO on the basis of the \ngeneralized gradient approximation (GGA)24 and GGA+U method25 implemented with \nprojector augmented -wave (PAW) pseudopotential26 using the Vienna ab initio Simulation \nPackage (V ASP) .27  All the DFT calculations were performed by adopting (i) a 11×6×6 \n6 \n Monkhorst -Pack k-point mesh28 centered at the Γ-point , (ii) a 650- eV plane -wave cutoff \nenergy, and (iii) the tetrahedron method with Blöchl corrections for the Brillouin -zone \nintegrations.29 The structural optimizations were performed for the 40 -atom -cell which \ncorresponds to an orthorhombic unit cell consisting of 8 form ula units. The ions were relaxed \nuntil the Hellmann -Feynmann forces on them were less than 0.01 eV/Å. The Hubbard U eff of \n4 eV and intra -atomic exchange parameter ( J) of 0.89 eV for the Fe 3d- orbital were chosen \non the basis of the previous work.20  \n \nRESULTS  \nIn-plane D omain O rientation of the Polar c -axis-grown F ilm \nTo fabricate the polar c -axis grown GFO film with a simpler domain configuration, we have \ncarefully chosen a hexagonal STO (111) substrate as an alternative to a cubic YSZ (001) \nsubstrate. F or im plementing this, we adopt a SRO (111) bottom electrode by considering its \ncompatibility with the STO substrate and fatigue and imprinting resistance .30,31 This scheme \nof the o- GFO film growth on a hexagonal substrate enables us to achieve a substantial  \nsimplification in the domain configuration, from twel ve orientations [on a cubic YSZ (001) \nsubstrate18] to six in -plane orientations. Figure 1a shows that the [001] -oriented GFO thin \nfilm is preferentially grown on a SRO(111) buffered STO (111) substrate. T he calculated c -\naxis parameter using this θ -2θ XRD pattern is 9.3996 Å which essentially coincides with the \nbulk c-axis parameter.13 This suggests that the present GFO film is fully relaxed along the \ngrowth direction. For the [001] -oriented film grown on a  STO(111), the most probable \ndomain orientation  that minimizes the in -plane lattice mismatch is represented by three in -\nplane domain configurations ( 𝐷𝐷1,𝐷𝐷2 and 𝐷𝐷3), as schematically depicted in Figure 1b.  \n7 \n To examine the validity of the proposed domain c onfiguration, we have measured in-\nplane XRD phi(𝜑𝜑)-scan and the result is presented in Figure 1c. These 𝜑𝜑-scan spectra were \nobtained by keeping the Bragg angle at (013) for o- GFO film (orange line) and at (110) and \n(100) for STO (blue and green lines, respectively). In the case of GFO (013), six peaks \nappear (orange color) and each peak is separated from the two neighboring GFO (013) peaks by 60\no. On the other hand, as shown in Figure 1c, these six (013) peaks are separated from \nSTO (110) and (100) peaks by 30o. These two observations clearly indicate that the GFO [013] \nhas six in -plane orientations with a successive tilting angle of 60o when projected on a STO \n(111) surface and each of these six in -plane orientations is rotated by 30o from the STO (111)  \nprojection of the STO [110] and [100] vectors. The domain configuration depicted in Figure 1d satisfies all these orientation relationships. The six projected normal vectors of the GFO (013) are shown using orange color in the lower corner of Figure 1d. T hus, the o- GFO film \ngrown on the STO (111) surface is characterized by three distinct crystallographic variants, \n𝐷𝐷\n1,𝐷𝐷2 and 𝐷𝐷3, and is represented by total six in -plane orientations, namely, 𝐷𝐷1,𝐷𝐷1′,𝐷𝐷2,𝐷𝐷2′ \nand 𝐷𝐷3,𝐷𝐷3′, where 𝐷𝐷 𝑛𝑛 and 𝐷𝐷𝑛𝑛′ are f acing to each other with the same crystallographic \nvariant. This domain configuration [Figure 1d ] agrees well with our previous proposition \ndeduced from minimizing the lattice mismatch [Figure 1b]. On the contrary, the o- GFO film \ngrown on an ITO/YSZ(001) s ubstrate is characterized by total twelve in -plane orientations \nwith a successive in -plane tilting angle of 30o (See Supplementary Information for details).  \nHowever, there exists a considerable degree of the lattice mismatch ( ∆) even though the \n(001) -orien ted GFO film possesses six in -plane orientations to reduce the lattice mismatch \nbetween the film and substrate. According to our estimate, ∆𝑎𝑎= 5.8654% �𝑎𝑎𝐺𝐺𝐺𝐺𝐺𝐺 =5.0806Å ,\n√2𝑎𝑎𝑆𝑆𝑆𝑆𝐺𝐺cos (30𝑜𝑜)� and ∆𝑏𝑏= 5.3388% ( 𝑏𝑏𝐺𝐺𝐺𝐺𝐺𝐺  = 8.7510Å,   3√2𝑎𝑎𝑆𝑆𝑆𝑆𝐺𝐺cos (60𝑜𝑜)).13,29 This \n8 \n indicates that the o- GFO film is not able to show a cube -on-cube type growth on the unit -cell \nbasis. However, if the o- GFO film growth is proceeded by the formation and deposition of a \nsupercell with the dimension of 16𝑎𝑎𝐺𝐺𝐺𝐺𝐺𝐺 ×18𝑏𝑏𝐺𝐺𝐺𝐺𝐺𝐺, the latt ice mismatch between the GFO \nlayer and the substrate surface can be effectively removed (with ∆𝑎𝑎= 0.01%, ∆𝑏𝑏= 0.08%). \nWe will examine this proposition by looking into scanning transmission electron microscopy \n(STEM) images.  \nA bright -field STEM image is shown in Figure 2a for the cross -section of the [001] -\noriented GFO film grown on a SRO(111) buffered STO (111) substrate. According to this \ncross- sectional mage, the thicknesses of the GFO layer and the SRO electrode are 200 nm \nand 30 nm, respectively. The selected area electron diffraction (SAED) pattern shown in \nFigure 2b is indexed by considering the superposition of the diffracted peaks along the zone \naxes [010 ] and [31�0] of GFO and the peaks along the STO [11�0] zone axis (red \nrectangle) used as the standard. The diffracted peaks along the GFO [010 ] zone axis (yellow \nrectangle) correspond to the 𝐷𝐷1 domain configuration (Figure 1). On the other hand, the \npeaks along the GFO [31�0] zone axis (white rhombus) correspond to the 𝐷𝐷 2 domain  \nconfiguration. The SAED patterns also indicate an epitaxial growth of the GFO film with the \npolar axis along the growth direction. One can notice two separated peaks by carefully \nexamining the area encircled by green dots. This demonstrates that the GFO film is fully \nrelaxed, rather tha n epitaxially strained.  \nHigh -resolution high- angle annular dark- field (HAADF) STEM images clearly show \nstructural domain boundaries that are formed by distinct in- plane orientations. A HAADF \nSTEM image of the area surrounded by a yellow rectangle is magni fied and presented in the \nright -hand side of Figure 2c to clearly visualize the structural domain boundary. The three \n9 \n fast Fourier transformed images (upper regions of the magnified STEM image) indicate that \nthe in -plane orientation of the central domain i s different from those of the two neighboring \ndomains. Figure 2c indicates that the width of the central domain is ~ 8 nm which nearly \ncoincides with the a -axis dimension of the supercell proposed previously ( 16𝑎𝑎𝐺𝐺𝐺𝐺𝐺𝐺 ×\n18𝑏𝑏𝐺𝐺𝐺𝐺𝐺𝐺,). Thus, the central doma in in Figure 2c represents the 𝐷𝐷1 domain configuration, as \ndepicted in Figure 1.  \n \nPFM Images of the Polar c -axis-grown Film  \nVertical -mode piezoresponse force microscopy (vPFM ) is a suitable method of measuring \nferroelectricity or piezoelectricity for a small local region. In Figure 3a and b, we respectively present vPFM amplitude and phase images acquired over the area of 1 μm×1μm. The phase \ncontrast image shown in Figure 3b ca n be interpreted as the existence of ferroelectric \ndomains with two antiparallel polarizations. However, this interpretation would be false if \nthese domains were not remained at a particular polarization state (i.e., remain either up or down state) after t urning off the bias E -field which had been used for the polarization \nswitching. To clearly resolve this critical issue, we have chosen a particular region of the o-GFO film and applied an alternative dc voltage from +10 V to \n-10 V . Thus, the corresponding \nE-field is ±500 kV/cm. The two applied voltages and the corresponding regions are marked \nin Figure 3c and d for the amplitude and phase contrast images, respectively. The two lower \nresolution vPFM images (10 μm×10μm) shown in Figure 3c and d were taken imme diately \nafter turning off these dc  voltages. Figure 3d clearly indicates that the 180o phase -shifted \ndomains by the bias E -field return to the initial unshifted state as soon as the bias E -field is \nturned off. Thus, the present vPFM results do not show any evidence of ferroelectricity with a \nnon-zero remanent polarization  up to a bias E -field of ±500 kV/cm.  \n10 \n  \nFerroelectric Polarization Switching of the Polar c -axis-grown Film s \nHaving failed in obtaining a clear evidence of the ferroelectricity up to ±500 kV/cm, we \nhave then carried out P -E hysteresis measurement  by applying much stronger E -fields, Emax \nbetween 2500 and 4500 kV/cm. This has been done  as our optimally processed [001] -\noriented GFO/SRO/STO(111) film capacitors are characterized by the I -V current density as \nlow as ~10-6 A/cm2 even at ±500 kV/cm (i.e., at ±10V; See Supplementary  Information for \ndetails). This low leakage current is a remarkable improvement over those of other reported \nGFO films . For instance, the current density of the previously reported GFO/SRO/STO(111) \nfilm capacitor (with 0  % of MgO doping) is in the order of a few A/cm2 for the same \nthickness of 200 nm (i.e., at ±500 kV/cm).32 Thus, our film capacitor processed by the \noptimized layer -by-layer growth of SRO shows a remarkable 106-times improvement in the \nleakage -current density (10-6 vs. 100). Similarly, the reported current density of the \nGFO/Pt/YSZ(111) film capacitor at 300 K is as high as 1.4x10-2 A/cm2 (as estimated using \n300 μm as the diameter of the top electrode) for the same thickness of 200 nm (i.e., at ±500 \nkV/cm).33 Again, our film capacitor shows a remarkable ~104-times improvement in the \nleakage -current density.   \nThe Pt/ o-GFO(001)/SRO(111 ) film capacitor grown on a STO (111) shows ferroelectric \nswitching behavior with a remarka bly high coercive field (E c) of ±1400 kV/cm at room \ntemperature [Figure 4a]. Thus, Emax value used in the PFM measurements is only 1/3 of the \nminimum electric -field needed for the polarization switching ( Ec). Thus, the ferroelectric \npolarization switching cannot be attained by PFM techniques (with E max of ±10 V) though a \n180o phase -shift of the piezoresponse is readily observed. As sh own in the inset of Figure 4a, \nthe Pr tends to reach its saturated value with increasing value of E max. Thus, the net switching \n11 \n polarization (2 Pr) estimated from Figure 4a is ~3 5 μC/cm2 which is much bigger than the \npreviously reported values of 2P r (between 0.1 and 1.0 μ C/cm2).14-16  \nA more reliable value of the switching polarization can be obtained by employing \nPUND (positive -up & negative -down) pulse test. Figure 4b presents the PUND result of the \n200-nm-thin GFO film capacitor obtained using the  pulse delay time of 100 msec. We have \nfound that the switching polarization decreases with increasing delay time and reaches a \nsaturated value for the pulse delay time longer than ~80 msec. The net switching polarization \n(2Pr) is evaluated using the following relation: 2𝑃𝑃𝑟𝑟=(±𝑃𝑃∗)−(±𝑃𝑃Λ). The 2 Pr value \nobtained from Figure 4b is ~26 μ C/cm2. Comparing this value with the 2P r value obtained \nfrom Figure 4a, one can conclude that the P -E curves overestimate the 2 Pr value substantially. \nThis suggests that the ac  frequency of 1 kHz used in the P -E measurement is not high enough \nto completely eliminate the responses of mobile space charges which tend to be significant at \nlower ac  frequencies.  \nTo examine universality of the present finding of a large 2P r, we have als o examined the \npolarization switching characteristics of the polar c -axis-oriented GFO film grown on a \nconducting ITO  buffered cubic YSZ (001) substrate. As shown in Figure 5a, the GFO film \n(the same 200 -nm thickness) grown on an ITO(001) /YSZ(001) substrate is characterized by \nthe net switching polarization of ~30 μ C/cm2 at 298 K. As compared with the Pt/GFO/SRO/  \nSTO(111) capacitor (Figure 4a), the P -E curves of the Pt/GFO/ITO/YSZ(001) capacitor \n(Figure 5a) tend to have  a mor e noticeable electrical -leakage  problem . The I -V current \ndensity data supports this tendency of the electrical leakage in the Pt/GFO/ITO/YSZ(001) \ncapacitor (See Figure S 3b of Supplementary  Information ). The PUND result shown in Figure \n5b also demonstrates that the Pt/GFO/ITO/YSZ(001) ca pacitor (with a 200 -nm-thin GFO \n12 \n layer) is ferroelectric with the net switching polarization of ~ 20 μC/cm2 at 298 K.  \nCompared with the 2P r of the Pt/GFO/SRO/STO(111) capacitor, the 2Pr of the Pt/GFO/ \nITO/YSZ(001) capacitor is thus substantially reduce d: ~3 5 μC/cm2 vs. ~30 μC/cm2 according \nto their P -E results. As stated previously , the c-axis-grown o-GFO film on a n ITO/ YSZ(001) \nsubstrate is characterized by the six crystallographic variants having total twelve in -plane \norientations with a successive in-plane tilting angle of 30o, in spite of little lattice mismatch \nbetween the GFO film and the YSZ substrate  (See Supp lementary  Information for details).  \nThe reduced 2Pr value can possibly be correlated with this increase in the crystallographic \nvariants  (i.e., the degree of complexity in the domain configuration) upon replacing the \nSRO/STO(111) substrate with the  ITO/YSZ(001) substrate . However, further in -depth studies \nshould be made before clearly resolving  the origin of the substrate -dependent 2P r. \n \nDISCUSSION  \nFor in -depth understanding of the atomic -scale origin of the observed polarization switching, \nwe have calculated the DFT polarization of the polar Pna 21 phase of o- GFO using the Berry-\nphase method.34,35 In order to obtain a correct evaluation of  the ferroelectric polarization, a \ncentrosymmetric prototypic phase of GFO should be first identified. For doing this, we have \nutilized the PSEUDO code of the Bilbao crystallographic server,36 which lattice dynamically \nallows one to determine the nearest supergroup structure for an input arbitrary structure. In \nthe present case, four centrosymmetric supergroup structures are examined. They are Pnna, Pccn, Pbcn, and Pnma . Among these four, the nearest reference structure, in terms of the total \ndisplacement of all atoms, is the Pnna phase, which is consistent with the previously reported \nresult.\n20 The optimized lattice parameters of the ferroelectric Pna 21 and prototypic Pnna \n13 \n phases were subsequently obtained by calculating the Kohn- Sham (K -S) energy as a fun ction \nof the unit -cell volume and finding its minimum which corresponds to the ground- state K -S \nenergy in the absence of any external pressure. The optimized lattice parameters are: (i) a = \n5.1647 Å, b  = 8.8197 Å, and c  = 9.5079 Å for the Pna 21 structure, and (ii) a = 5.2672 Å, b = \n8.9193 Å, and c  = 9.5684 Å for the Pnna structure. The optimized structures of the Pnna and \nPna21 phases of o- GFO are depicted in Figure 6a.  \nAccording to group theoretical analysis,37 there exist s only one conceivable transition \npath that connects the prototypic Pnna phase to the ferroelectric Pna2 1 phase. We have \ndecomposed the atomic displacements that relate the nonpolar Pnna phase to the polar Pna2 1 \nphase into the symmetry -adapted mode of the prototypic phase. The resulting symmetry-\nadapted mode is exclusively given by 𝛤𝛤4−. Thus, the Pnna–to –Pna2 1 phase transition should \nbe mediated by the freezing -in of the zone -center 𝛤𝛤4− polar phonon. Let us defi ne the \ndisplacement amplitude of the polar 𝛤𝛤4− phonon as 𝑄𝑄Γ4−. In Figure 6b, the K -S energy and \nthe Berry -phase polarization are plotted as a function of the mode amplitude 𝑄𝑄Γ4−.38 Here, the \npolarization [lower panel of Figure 6b] is given by the produc t of 𝑄𝑄Γ4− and the Born effective \ncharge tensor. As shown in Figure 6b, the computed K -S energy exhibits a double -well-type \npotential, which demonstrates the relative stability of the ferroelectric Pna21 phase over the \nprototypic nonpolar Pnna  phase with t he energy difference of 1.05 eV per formula unit (f.u.). \nHerein the equilibrium ferroelectric polarization of the Pna 21 phase is given by the computed \npolarization values (lower panel) at the two K -S energy minima, namely, ±25.67 μC/cm2, \nwhich corresponds to 𝑄𝑄Γ4− of ±1, respectively. Since the polarization switching is expected \nto occur along the Pna2 1–to–Pnna phase -transition path, i.e., 𝛤𝛤4−, the activation free -energy \nof the polarization switching between the double wells can be obta ined from Figure 6b, \n14 \n which is ~1.05 eV per f.u. This value lies within the two extreme ab initio values (0.52 and \n1.30 eV per f.u.) previously obtained by Stoeffler20 using ABINIT & FLAPW/ FLEUR codes. \nThe activation barrier of 1.05 eV per f.u. is about 2.5 times bigger than that of BiFeO 3, the \nmost extensively studied multiferroic, and 20 times bigger than that of Pb(Zr,Ti)O 3, the most \nwidely used displacive ferroelectrics.39 This unusually high activation barrier indicates that \nthe polar Pna 21 phase in o- GFO is very stable against thermally activated random dipole \nswitching across the centrosymmetric Pnna barrier. The observed high E c (±1400 kV/cm; \nFigure 4a) can also possibly be correlated with this high activation barrier .  \nFinally, we will correlate the reported  high ferroelectric transition temperature (> 1368 \nK; ref. 2 2) with this remarkably high activation free -energy. According to the transition -state \ntheory of rate processes,40 the frequency ( 𝜈𝜈) of the polarization swi tching across the Pnna  \npotential barrier can be written as  \n                           𝜈𝜈 =𝑘𝑘𝐵𝐵𝑆𝑆\nℎ𝑒𝑒−𝛷𝛷𝑜𝑜𝑘𝑘𝐵𝐵𝑆𝑆⁄                          (1)  \nwhere 𝛷𝛷𝑜𝑜 is the barrier height of the polarization switching which can be treated as the \ndifference in the Kohn- Sham energy between Pnna and Pna21 phases (1.05 eV per f.u.). \nStrictly speaking, Equation (1) is valid for describing the dipole -switching rate of order -\ndisorder ferroelectrics. For sufficiently high temperatures near the phase -transition point ( 𝑇𝑇𝑐𝑐), \nhowever, Equation (1) is also applicable to the dipole -switching rate of displacive \nferroelectrics41 that include the present Pna2 1–to–Pnna transition. As temperature increases \nand approaches 𝑇𝑇𝑐𝑐, the dipole -switching rate becomes so rapid that the mean residence time \n(𝜏𝜏𝑜𝑜) of the bound polarization in one of the two ferroelectric double wells becomes shorter \nthan a certain critical time for the experimental observation. Under this condition, the net \nferroelectric polarization effectively disappears because of the switching average of two \n15 \n opposite polarizations across the centrosymmetric barrier [ Pnna state in the present case; \nFigure 6b]. Let 𝜈𝜈𝑜𝑜 be the f requency at which the net bound polarization just begins to \ndisappear. In the vicinity of 𝑇𝑇𝑐𝑐, 𝛷𝛷𝑜𝑜<𝑘𝑘𝐵𝐵𝑇𝑇 for a fixed value of 𝛷𝛷𝑜𝑜. Under this condition, 𝜈𝜈𝑜𝑜 \ncan be approximated by the following expression:  \n                           𝜈𝜈 𝑜𝑜=1\nτ𝑜𝑜≈𝑘𝑘𝐵𝐵𝑆𝑆𝑜𝑜\nℎ�1−Φ𝑜𝑜\n𝑘𝑘𝐵𝐵𝑆𝑆𝑜𝑜�                        (2)  \nwhere 𝑇𝑇𝑜𝑜 denotes the temperature that corresponds to the critical switching frequency 𝜈𝜈𝑜𝑜. \nSince 𝑇𝑇𝑐𝑐 is expressed by 𝑇𝑇𝑐𝑐=𝑇𝑇𝑜𝑜+𝜖𝜖, where ϵ is a small positive number (including zero), \n𝑇𝑇𝑐𝑐 can be correlated wi th Φ𝑜𝑜 by the following relation for a fixed value of Φ𝑜𝑜:  \n                        𝑇𝑇 𝑐𝑐=(ℎ𝜈𝜈𝑜𝑜+Φ𝑜𝑜)\n𝑘𝑘𝐵𝐵+𝜖𝜖≈(ℎ𝜈𝜈𝑜𝑜+Φ𝑜𝑜)\n𝑘𝑘𝐵𝐵                        (3)  \nEquation (3) clearly indicates that 𝑇𝑇 𝑐𝑐 is linearly proportional to 𝛷𝛷𝑜𝑜. This explains the \nobserved high 𝑇𝑇𝑐𝑐 of o-GFO in terms of the unusually high activation free -energy barrier ( 𝛷𝛷𝑜𝑜) \nas predicted by ab initio calculations.  \n \nCONCLUSIONS  \nWe have clarified a puzzling discrepancy between the observed P r value s (≤ 0.5 μC/cm2) \nand the ab initio  prediction and have unequivocally demonstrated the ferroelectric \npolarization switching with the net switching polarization of ~30 μ C/cm2 by using the two \ndistinct types of o- GFO thin films preferentially grown along the polar c -axis in Pna21 \nsetting ( b-axis in Pc 21n setting). The estimated activation energy for the polarization \nswitching is ~1.05 eV per f.u. This high value accounts for the observed stability of the polar \nPna2 1 phase over the centrosymmetric Pnna phase for a wide range of temperature up to \n1368 K.  \n \n16 \n  \nCONFLICT OF INTEREST  \nThe authors declare no conflict of interest.  \n \nACKNOWLEDGEMENTS  \nThis work was supported by the National Research Foundation (NRF) Grant s funded by the \nKorea Government (MSIP) (Grant No.  2012R  1A1A2041628 & 2013R  1A2A2A01068274).  \nThe work at Cambridge was supported by the Engineering and Physical Sciences Research \nCouncil (EPSRC).   \n \nREFERENCES  \n1. Scott, J.F. Multiferroic memories. Nat. Mater. 6 , 256 (2007) . \n2. Chu, Y ., Martin, L. W., Holcomb, M. B. & Ramesh, R. Controlling magnetism with multiferroics. \nMater. Today  10, 16 (2007).  \n3. Spaldin, N. A. & Fiebig, M. The renaissance of magnetoelectric multiferroics. Science 309, 391 \n(2005).  \n4. Wang, J., Neaton, J. B., Zheng, H., Nagarajan, V ., Ogale, S. 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B 80, 132105 (2009).  \n \n \n \n \n  \n20 \n * Figure Captions  \n \nFigure 1  X-ray diffraction (XRD) data and in -plane domain orientations. (a)  θ-2θ XRD \npattern of the [001] -oriented 200 -nm-thin GFO thin film grown on a SRO(111) buffered \nhexagonal STO (111) substrate. (b) Schematic representation of the most probable domain \nconfiguration for the polar c-axis-oriented GFO film grown on a STO (111) substrate. Blue \nbi-directional arrows refer to in -plane orientation of the STO (111) substrate. (c)  In-plane \nXRD 𝜑𝜑-scan spectra of the polar c-axis grown GFO film, as obtained by keeping the Bragg \nangle at (013) for o- GFO film (orange line) and at (110) and (100) for STO (blue and green \nlines, respectively). A schematic diagram presented below the 𝜑𝜑-scan spectra denotes the \nprojec ted normal vectors of (013) for GFO on STO (111) and projected normal vector s of (110) \nand (100) STO on STO (111). (d) A schematic domain configuration of the polar c-axis-\noriented GFO film grown on a STO (111) substrate showing six in- plane domain orientatio ns \nwith three distinct crystallographic variants. Three different kinds of the projected normal \nvectors are shown in the lower corner.  \n \nFigure 2  STEM images of the polar c-axis-oriented GFO film. (a)  A cross- sectional BF -\nSTEM image of the polar c-axis-oriented GFO film grown on a SRO (111) buffered \nhexagonal STO (111) substrate . (b) A SAED pattern confirming the in -plane orientation \nbetween the GFO film and the STO substrate. The diffraction pattern is indexed by \nconsidering the superposition of the diffra cted peaks along the zone axes [31�0] and [010] of \nGFO and the zone axis [ 11�0] of STO. (c)  A HAADF  STEM image for the interfacial region \nbetween the polar c -axis-oriented GFO film and the SRO electrode layer. The area \n21 \n surrounded by a yellow rectangle is magnified in the right -hand side to clearly visualize the \nstructural domain boundary. Three fast Fourier transformed images are also shown in the \nupper region of the magnified STEM image.  \n Figure 3  Vertical -mode PFM images of the polar c -axis-oriented GFO film. The GFO film \nwas grown on a SRO (111) buffered STO (111) substrate. PFM (a) amplitude and (b)  phase \ncontrast images of the GFO film acquired over the area of 1 μm×1μm. PFM (c)  amplitude and \n(d) phase contrast images of the two concentric square regions obtained after applying an \nalternative dc  voltage from +10 V to \n-10 V and subsequently turning off these dc  voltages.  \n Figure 4 Ferroelectric responses of  the GFO film grown on a hexagonal STO substrate. (a) \nP-E curves of the Pt/GFO(001)/SRO(111 ) film capacitor grown on a STO (111) showing \nferroelectric switching behavior with a high coercive field (E\nc) of ±1400 kV/cm  at 298 K . (b) \nPUND test result of the polar c-axis-oriented 200- nm-thin GFO film capacitor grown on a \nSRO(111) /STO(111) obtained using the pulse delay time of 100 msec.  \n Figure 5  Ferroelectric responses of  the GFO film grown on a cubic YSZ substrate. (a) P-E \ncurves of the Pt/GFO(001)/ ITO(001) film capacitor grown on a YSZ (001). The coercive field \n(E\nc) deduced from these curves is ±1100 kV/cm  at 298 K. (b)  PUND test result of the polar \nc-axis-oriented 200- nm-thin GFO film capacitor grown on a n ITO (001) /YSZ (001).  \n Figure 6 Crystal structures and the associated ferroelectric double- well potential. (a)  \nOptimized unit -cell crystal structures of prototypic Pnna and polar Pna 2\n1 phases of o-GFO \n22 \n obtained after structurally relaxing the computed Kohn- Sham energy as a function of the  \nunit-cell volume. Eigenvectors of the 𝛤𝛤4− phonons at Ga and Fe ions are shown in the right -\nhand side using blue arrows. (b)  The computed Kohn- Sham energy plotted as a function of \nthe fractional amplitude 𝑄𝑄𝛤𝛤4−. The reference state of 𝑄𝑄𝛤𝛤4−=0 denotes the prototypic Pnna  \nphase. In the lower panel, the computed polarization values are plotted as a function of 𝑄𝑄𝛤𝛤4−. \n \n23 \n * Figure s \n \nFigure 1  \n \n \n \n \n \n \nFigure 2 \n \n \n \n \n \n \n \n \nFigure 3 \n \n \n \n  \n  \nFigure 4 \n \n \nFigure 5 \n \n \nFigure 6  \n \nSUPPLEMENTARY  INFORMATION   \n \n \nFerroelectric polarization switching with a remarkably high \nactivation -energy  in orthorhombic GaFeO 3 thin films  \n \nSeungwoo Song,1 Hyun Myung Jang,1* Nam -Suk Lee,2 Jong Y . Son,3 Rajeev Gupta,4 Ashish \nGarg,4 Jirawit Ratanapreechachai,5 and James F. Scott5,6+  \n \n \n1Division of Advanced Materials Science (AMS) and Department of Materials Science and \nEngineering, Pohang University of Science and Technology (POSTECH), Pohang 790- 784, \nRepublic of Korea.   \n2National Institute for Nanomaterials Technology (NINT), Pohang University of Science and \nTechnology (POSTECH), Pohang 790- 784, Republic of Korea.   \n3Department of Applied Physics, College of Applied Science, Kyung Hee  University, \nGiheung- Gu, Yongin- City 446- 701, Republic of Korea.   \n4Department of Materials Science and Engineering, Indian Institute of Technology (IIT), \nKanpur, Kanpur 208016, India.   \n5Cavendish Laboratory, Department of Physics, University of Cambridge , Cambridge CB3 \n0HE, united Kingdom . \n6School of Chemistry and School of Physics, University of St. Andrews, St. Andrews KY16 \n9ST, United Kingdom.  \n \n    *Correspondence and requests for materials should be addressed to either H. M. J. (email: hmjang@postech.ac.kr  ) or J. F. S. (email: jfs 32@cam.ac.uk\n )  \n \n \n \n   \n1. Magnetic properties  \n \n \nFigure S1. (a) Temperature -dependent f ield cooled (FC)  and zero -field cooled  (ZFC)  \nmagnetization curves of the polar c -axis-oriented o-GFO film (200 -nm-thin) grown on a SRO \nbuffered STO (111) substrate . Notice that a large discrepancy between the in -plane FC and \nZFC curves for temperatures below ~230 K. This indicates that the magnetic e asy axis lies \nalong the in- plane direction (i.e., perpendicular to c-axis) and a ( ferrimagnetic ) ordering \noccurs at ~230 K. (b)  Magnetization versus  magnetic -field (M-H) hysteresis curve s at three \nselected temperatures, indicating that the magnetic -ordering temperature is lower than 240 K. \n  \n \n \n2. In-plane domain orientation of GFO film grown on ITO/YSZ (001)  \n \nAs presented in Figure S2( a), the x -ray 𝜙𝜙-scan spectra of both GFO {122} and {013} \nplanes are characterized by 12 distinct peaks, and each peak is separated from the two \nneighboring peaks by 30o. Here, we have elucidated  the detailed in -plane orientations of the \nGFO domains by performing x- ray 𝜙𝜙-scan experiments for both GFO {013} and {122} \nplane s. From the three 𝜙𝜙-scan spectra shown in Figure S 2(a), one can establish the following \norientation relations hips among GFO, ITO, and YSZ : The projection of the normal vector of \nYSZ (111) on YSZ  (001) is nearly parallel to the projection of the normal vector \ncorresponding to the peak A of GFO ( 122). On the contrary , the projection of the normal \nvector of YSZ (111) on YSZ (001) is rotated by 45o from that of GFO (013). These \nrelations hips can be readily identified by examining our schematic drawing of GFO ( 122) and \n(013) and Y SZ (111) on YSZ (001) [Figure S 2(b)]. The three projected normal vectors are \nsummarized in the inset of Fig ure S 2(b).  \nThe domain orientation corresponding to the peak A of the c-axis grown GFO film [on \nYSZ (001) ] is denoted by D 1 in Fig ure S 2(b). On the other hand, the two additional domain \norientations correspond ing to the peaks B and C are denoted by D 2 and D3, respectively. In \nFigure S 2(b), 𝐷𝐷𝑥𝑥′ denote s the GFO domain which is rotated from the 𝐷𝐷𝑥𝑥-domain by 90 \ndegrees. Thus, the observed 12 peaks can be unequivocally explained by the three distinct \ndomains, D1, D2, and D 3, with a successive tilting angle of 30o, showing total twelve in -plane \norientation s.  \n \n \n \n \n  \n \n \nFigure S2.  (a) X-ray 𝜙𝜙-scan spectra of {111} planes of YSZ  (black line), { 122} planes of \nGFO (blue line) , and {0 13} planes of GFO  (red line). (b) Schematic drawing of the domain \norientations of the GFO film on YSZ  (001). Here in, (111) plane of YSZ is shaded as grey, \nwhile (122) and ( 013) plane s of the GFO film on YSZ  (001)  are shaded as pal e blue and \ndarker blue , respectively.  \n \n \n \n3. Current versus  voltage  (I-V) curve  \n \n \n \nFigure S3. Temperature- dependent I-V curve s of the polar c -axis-oriented o-GFO film (200 -\nnm-thin) grown on (a)  a SRO(111) buffered STO (111) substrate  and (b) an ITO(001) \nbuffered YSZ (001)  substrate.  \n \n \n \n4. Atomic force microscopy (AFM) image of the substrate.  \n \nOne side -polished STO (111) substrate was dipped into a hot, ultrasonically agitated \nbuffered hydrogen fluoride (BHF) solution (NH 4F:HF = 7 :1) for 30 seconds  at 63oC. Then , \nthe substrate was annealed  in a tube furnace at 1050oC under oxygen ambient for 1 hour.  \n \n \n \nFigure S4. A topographic  AFM image of the chemically and thermally treated S TO(111)  \nsubstrate .  \n \n \n   \n \n \n \n \n \n \n \n \n5. AFM and XRD data of the SRO bottom electrode  \n \nFigure S5 (a) shows  XRD 𝜃𝜃-2𝜃𝜃 scan of SRO/STO (111). One can notice the thickness \nfringes of the SRO(222) around the 2𝜃𝜃 of the STO (222) , which is a strong evidence of the \nepitaxial growth. The positions of interference fringes displayed in the inset of Figure S5 (a) \nwere used to determine the film thickness which is estimated be 30 nm .  \n \n \nFigure S5. (a) XRD 𝜃𝜃-2𝜃𝜃 scan of the SRO electrode layer on S TO(111) with the fringe \naround 2𝜃𝜃 of the STO(222) peak. The i nset shows  x-ray reflectivity (XRR) data for the 30-\nnm-thick S RO bottom electrode  layer . (b) An a tomically flat AFM image of the SRO layer \ngrown on a S TO(111) substrate.  \n \n  \n \n6. AFM image of the 200-nm-thick polar c-axis grown G FO film \n \nFigure S6 displays topographic image s of the polar c -axis-oriented GFO (200- nm-thick ) \nfilm grown on a SRO(30nm)/STO(111) substrate during the PFM measurement .  \n \n \n \nFigure S6. Atomically flat AFM image s of the o-GFO film grown on a S RO(111) buffered \nSTO(111) substrate.  \n \n \n               \n \n7. Computational procedures for lattice dynamics study of the Pnna- to-\nPna2 1 structural  phase transition  \n \n \n \n \nFigure S7. (a) Looking for a pseudo- symmetry among the supergroup structure s for the input \npolar Pna2 1 structure u sing the PSEUDO code  of the Bilbao crystallographic server. Amomg \nthese super group structures , Pnna is the most rele vant paraelectric centrosymmetr ic structure \nthat has the least atomic -displacement for each constituting atom . (b) AMPLIMODES \ncalculations were carrie d out for a symmetry -adapted mode analysis of a displacive phase \ntransition. Starting from the experimental structures of the high - and low -symmetry phases, \nthe program determines the global structural distortion that relates to the two phases. The \nsymmetry modes co mpatible with the symmetry break are then calculated. Their \northogonality permits the decomposition of the global distortion, obtaining the amplitudes of \nthe different symmetry -adapted distortions present in the structure, as well as their \ncorresponding polarization vectors. As indicated, the Pnna- to-Pna2 1 structural phase \ntransition in o- GFO is mediated solely by the freezing- in of the  zone -center 𝛤𝛤4− polar \nphonon.  \n \n"    },    {        "title": "1610.02130v1.Performance_Evaluation_of_Klystron_Beam_Focusing_System_with_Anisotropic_Ferrite_Magnet.pdf",        "content": "arXiv:1610.02130v1  [physics.acc-ph]  7 Oct 2016Performance Evaluation of Klystron Beam Focusing System wi th\nAnisotropic Ferrite Magnet\nYasuhiro Fuwa\nDepartment of Physics, Kyoto University, Kyoto, Kyoto 606- 8502, Japan\nYoshihisa Iwashita\nInstitute for Chemical Research, Kyoto University, Uji, Ky oto 611-0011, Japan\nAbstract\nA klystron beam focusing system using permanent magnets, wh ich increases reliability in com-\nparison with electromagnet focusingsystem, is reported. A prototype model has been designed and\nfabricated for a 1.3 GHz, 800 kW klystron for evaluation of th e feasibility of the focusing system\nwith permanent magnets. In order to decrease the production cost and to mitigate complex tuning\nprocesses of the magnetic field, anisotropic ferrite magnet is adopted as the magnetic material. As\nthe result of a power test, 798 ±8 kW peak output power was successfully achieved with the pro -\ntotype focusing system. Considering a power consumption of the electromagnet focusing system,\nthe required wall-plug power to produce nominal 800 kW outpu t power with the permanent mag-\nnet system is less than that with electromagnet. However, th e power conversion efficiency of the\nklystron with the permanent magnet system was found to be lim ited by transverse multipole mag-\nnetic fields. By decreasing transverse multipole magnetic fi eld components, especially the dipole\nand the quadrupole, the power conversion efficiency would app roach to that with electromagnets.\n1I. INTRODUCTION\nThe International Linear Collider (ILC) is an electron-positron collid er for high-energy\nphysics with a center-of-mass energy up to 500 GeV. In order to a chieve the final energy,\nthe ILC should be equipped with more than sixteen thousands of acc elerating cavities to\naccelerate electrons and positrons [1]. Order of one thousand klys trons are also needed to\nfeed the Radio Frequency (RF) power to the cavities. For example, the Distributed RF\nScheme (DRFS), one of the RF power feed scheme formerly propos ed for ILC, utilizes eight\nthousand midium-power klystrons [2]. Due to the large number of klys tron units, even a\nlow failure rate of each component could increase the load of mainten ance work and limit\navailability of the facility.\nFor DRFS, the reliability of klystron itself is estimated to be fairly high b ecause of its\nrelatively-low output power (800 kW per klystron). In addition to th e high reliability of\nklystron itself, that of the whole system is also important. The klyst ron beam focusing\nsystem is one of the key components in the whole system. In fact, a ccording to KEK\ninjector operating statistics during 10 years in 2000s, the failures in focusing solenoids were\nthe most frequent reasons to exchange klystron assemblies amon g the other reasons [3].\nFocusing systems with permanent magnets for klystron had been d eveloped in some\naccelerator laboratories. In the SLAC 2-mile linear accelerator [4] a nd the Photon Factory\nlinac in KEK [5], the focusing systems with Alnico magnets were develope d and utilized\nin accelerator operation. However, both laboratories have discon tinued them and have\nemployed solenoid coils as the klystron beam focusing system. The ma in reason for the\ndiscontinuances was the complexity in the field tuning. The magnetic fi elds generated by the\nmagnets in these laboratories were tuned by partial magnetization and/or demagnetization\nof the Alnico magnets or attaching magnetic material shims such as ir on bars. Because the\ntuning methods had poor reproducibilities due to irreversibility cause d by the low coercivity\nof Alnico, the tuning process had some difficulties which took a long time .\nBecause the number of magnet systems in ILC klystron is quite large , they must be\ncost effective and have ease of tuning. This paper reports on deve lopments of a prototype\nfocusing magnet with ferrite magnets for ILC DRFS klystron and th e result of performance\nevaluation for the focusing magnet by way of both experimental an d numerical studies. It\nshould be noted that the Technical Design Report (TDR) for the IL C adopts the Distributed\n2KlystronScheme (DKS)using theMulti-BeamKlystron(MBK) instead oftheDRFSmainly\nbecause of the higher efficiency after discussions during the GDE (G lobal Design Effort) for\nILC.\nThemainpurposeofthisstudy istoprove thatfocusing systems wit hpermanent magnets\ncan replace electromagnets with a similar performances (power con version efficiency and\npeak power). This paper consists of six sections as follows. In Sec. 2, the conceptual design\nof the magnet is briefly presented. Sec. 3 describes design proces s of prototype magnet and\nthe result of performance estimation using simulation. In Sec. 4, th e result of power test\nwith prototype magnet is presented. In Sect. 5, the effect of asy mmetric fields and error\nfields on output power is discussed. In Sect. 6, a brief summery is pr esented.\nII. CONCEPTUAL DESIGN OF FOCUSING SYSTEM\nIn this section, the conceptual design of our focusing system with permanent magnets\nis briefly described. A detailed description can be found in Ref. 6. The anisotropic ferrite\nis chosen as magnetic material because of its inexpensive cost, eno ugh remanent field ( ∼4\nkG), and coercivity. Because the required magnetic field in DRFS klys tron is up to 1 kG,\nthe ferrite magnets can generate magnetic field with the sufficient m agnitude.\nThe focusing field is uni-directional along the beam axis in contrast to the Periodic\nPermanent Magnets (PPM) configuration [7]. Then the field distribut ion can be the same\nas the field generated by a solenoid coil except for the reverse field s in the exterior regions.\nThe uni-directional field distribution can avoid the damage on the klys tron tube wall due to\na beam loss [8] caused by the energy stop-band on the electron bea m transmission. In order\nto generate the required magnetic field with less amount of ferrite m aterial, magnets are\nconfigured in a quasi-Halbach dipole configuration [9]. The configurat ion also contributes\nto reduce the stray magnetic field outside the system and to preve nt field distortion due\nto some disturbance caused by magnetic materials in vicinity. The imag e of the magnet\nconfigurationandthemagneticfieldfluxdistributioncalculatedbyPA NDIRA[10]areshown\nin Fig. 1. The magnets are divided into a number of segments and each segment is movable\nindependently. By virtue of this segmentation and movability, magne ts can be set near\nthe klystron tube after the insertion of the tube to the focusing a lcove. The placement of\nthe magnets near the tube can reduce the required volume of magn ets to generate focusing\n3field. The local tuning of the magnetic field can be also realized by fine p ositionings of the\nsegments. This tuning scheme has a good reproducibility because of the high coercivity of\nferrite magnets and is a much easier process compared with the ear ly studies where the field\ndistributions were tuned by the cumbersome tuning methods as sta ted in the earlier section\n[5].\nFIG. 1. Configuration of magnets (left side of the figure). The calculated field distribution is shown\nin the right half part.\nIII. DESIGN OF PROTOTYPE MODEL\nForevaluationoftheperformance ofthe focusing system describ ed inSect. 2, a prototype\nwas designed for DRFS klystrons. The klystron assumed to be used in DRFS was a E37501\n[11] manufactured by Toshiba Electron Tube & Devices Co., Ltd. A mo dulating anode\nimbedded in the klystron makes the radius of the high-voltage insulat or around the cathode\npart large, which requires the large outer radius of the oil tank, an d hence, inflates the\nmagnet volume for that part . RADIA 4.29 [12] was used for the magn etic field design.\nAfter the fabrication cost estimation, the basic dimension of the fe rrite piece was chosen\nas 150 mm ×100 mm cross section and 25.4 mm thickness. The optimized result of t he\nmagnet and the yoke configuration is shown in Fig. 2. The properties of each magnet\nare described in Ref. 6. The magnets are designed so as to avoid inte rference with the\n4cooling pipes and the RF input port on the klystron outer wall. The field s in cathode region\nwere shielded by the oil tank whose cylinder is made of ferromagnetic iron. By moving the\npositions of the magnets, the field distribution in the beam drift regio n can be adjusted.\nAll magnets can evacuate from the central region to make the spa ce to insert the klystron\nfrom the top. The tuned field distribution for efficient beam focusing is shown in Fig. 3. In\nthis figure, the normalized value of the magnetic field distribution is sh own instead of the\nabsolute value due to a non-disclosure agreement between author s and the tube supplier.\nIn the discussion afterward, the positions of cathode, input cavit y, and output cavity are\nindicated by zcathode,zinput, andzoutput, respectively. Compared with the field generated by\nelectromagnet (EM) in Fig. 3, the field by permanent magnet (PM) ha s ripples in the beam\ndrift region. These ripples were caused by longitudinal segmentatio n of the magnets. For\nestimation of the effect on beam focusing, beam envelope was calcula ted as cylindrically\nsymmetric problems by DGUN code [13]. Comparing the beam envelopes for PM and EM\n(Fig. 4), the difference between two envelopes is not significant, wh ich can conclude that\nsuch ripples can be neglected for a klystron design.\nMultipole components of magnetic fields in the transverse planes are also generated be-\ncause the distribution of magnets is not azimuthally symmetric. The d ominant component\nis quadrupole and the strength of the quadrupole component reac hes 20 G/cm (see Fig. 12).\nHigher order components such as an octupole component are also e xisting, but their magni-\ntude are less than one tenth of the quadrupole component in the be am region. Quadrupole\nfields deform a circular beam into an elliptical beam [14]. In the case of o ur magnet system,\nelectron beams are magnetically confined and rotate around the ce nter of beam (Brillouin\nflow). Therefore, the effect of the quadrupole field is not simple. Fo r evaluation of the\nelectron distribution in the beam propagation from the cathode to t he collector, 3D beam\ntracking simulation was performed with CST Particle studio [15]. Fig. 5 s hows calculated\nbeam profiles at several points along beam axis. From the results, w hile the beam profile\nbecame elliptical during beam propagation in the quadrupole compone nt field, all the par-\nticles reached to the collector without hitting the tube wall. Beam simu lations with RF\nmodulation were also performed with PIC (Particle-in-Cell) simulation a nd no beam loss\nwas observed in any beam slices.\n5FIG. 2. Designed configuration of magnets and iron yoke. (Lef t) Magnet position for beam\noperation. (Right) Magnet position for klystron installat ion. The objects in (b) and (d) are seen\nfrom upper side.\nFIG. 3. Magnetic field distribution along the beam axis. The e lectron beam from the cathode is\naccelerated and matched to the Brillouin flow optics in the ma tching region and is transported\nthrough beam drift region. It finally goes into the collector region to be damped.\n6FIG. 4. Beam envelopes calculated by 2.5 dimensional simula tion with DGUN.\nFIG. 5. Beam profiles at cavity positions calculated by 3 dime nsional simulation with CST particle\nstudio.\nIV. TEST WITH PROTOTYPE MAGNET\nA. Fabrication of Prototype Magnet\nBased on the design described in Sect. 3, the prototype magnet wa s fabricated. The\nmagnet pieces shaped from 150mm ×100mm×25.4mm ferrite ingots to designed shapes\nwere glued together by acrylic cured resin. Then, each glued segme nt was also glued on\nan iron plate or an aluminum block. All the magnets were supported by these metallic\n7structures. The pictures of the fabricated prototype are show n in Fig. 6. The magnets\ncould be moved as designed for the klystron insertion and the field tu ning. The typical\naccuracy of the magnet position alignment is less than 1 mm.\nIn field tuning process, the magnetic field distributions {Bx(z),By(z),Bz(z)}on the\nbeam axis were measured with 3-axis Hall probe (Senis 3MH3). After some iteration pro-\ncesses of tuning by adjusting magnet positions, the longitudinal fie ld distribution could be\nadjusted to the designed one within 2 % error in the beam drift region . The field in the\nmatching region (see Fig. 3) was more precisely adjusted to suppre ss ripples of the beam\nenvelope. The typical transverse field distribution is shown in Fig. 7. The transverse fields\nwere caused by residual positioning errors of magnets in the tuning process, the mechanical\nerrors of magnet shapes, and magnetization errors in the fabrica tion process. Careful field\ntuning can make this transverse field less than 10 G (corresponding to 10 % magnitude\ncompared with the longitudinal magnetic field). This magnitude (frac tion) of transverse\nmagnetic field on the beam axis was comparable to that of the focusin g system developed\nat SLAC [16].\nThe displacement of the beam center can be estimated with an assum ption that electrons\nfollow the magnetic flux lines. The displacement of the flux line ∆ rat the position zcan be\nestimated by integrating the flux line angle ( Btransverse/Bz) from the beam axis. Using these\nfacts, the beam displacements in x,ydirections canbe derived in a first order approximation\nas follows.\nThex-component of the transverse field can be described as\nBtransverse,x (x,y,z) =Bx(z)+x\nrBr(r,z), (1)\nwherer=√x2+y2is displacement from the beam axis, Bx(z) is the x-component of\nthe dipole field on the axis, and Br(r,z) is azimuthally symmetric field component at the\ndistance rfrom the axis. Using the condition div B= 0, the Br(r,z) can be derived from\nthe longitudinal field component on the axis as\nBr(r,z) =−1\n2∂Bz(z)\n∂zr. (2)\nHereby, the displacement of beam center ∆ rcan be calculated as\n∆x(z) =/integraldisplayz\nzcathodeBx(ξ)+∆x(ξ)\n∆r(ξ)Br(∆r(ξ),ξ)\nBz(ξ)dξ, (3)\n8∆y(z) =/integraldisplayz\nzcathodeBy(ξ)+∆y(ξ)\n∆r(ξ)Br(∆r(ξ),ξ)\nBz(ξ)dξ, (4)\n∆r(z) =/radicalBig\n∆x(z)2+∆y(z)2, (5)\nwhere ∆xand ∆yare the displacement of beam center in xandydirection, respectively.\nThe calculated result ofthe displacement of beamcenter due to the transverse magnetic field\nestimated by equations (2) - (5) is shown in Fig. 8 together with a res ult from CST particle\nstudio. Although the space charge effects are neglected in the firs t order estimation, the\nestimated beam displacement gives a rough guess close to the result with CST, which reduce\na lot of efforts and computation time. The reason of the discrepanc y should be non-linear\neffect due to space charges and beam-wall interactions.\nB. Klystron Power Test\nKlystron power test was performed in KEK STF. A 1 MW dummy load was connected.\nA modulator at STF building in KEK supplied the powers for the klystron , which includes\na heater power, a cathode high voltage and a modulating anode pulse . A preliminary result\nof the power test was presented in Ref. 11. In the initial stage of e xperiment, a diode test\nwas carried out. In this test, no RF power was fed to the input cavit y and the beam was\njust transported to the collector. The cathode voltage was set b etween 45 kV and 67.5 kV,\nand the modulating anode voltage at beam extraction was 0.26 times c athode voltage. The\nmeasured perveance of extracted beam was 1.29 ±0.03 [µA/V3/2]. This value is the same as\nthe result using the solenoid (1.29 ±0.04 [µA/V3/2]). During the beam extraction test, the\ntemperature rise of the cooling water for klystron body was monito red, and no significant\ntemperature rise was detected. These results suggested that t he amount of the electron loss\non the klystron tube body was ignorable and our prototype magnet s had sufficient beam\ntransport efficiency.\nThen, the test with RF excitation was carried out. RF input-output characterization was\nmeasured at several cathode voltages. The result is shown in Fig. 9 . At 65.8 kV cathode\nvoltage, the saturated output power reached 798 ±8 kW. Because the required output power\nper klystron for DRFS is 800 kW, the system with permanent magnet focusing had enough\n9FIG. 6. Fabricated prototype magnet. (Top) Klystron insert ion mode: magnets are retracted\ntowards the outer sides and making the space to let the cathod e assembly part go through, which\nis the most thick part of the klystron. (Bottom) klystron ope rating mode; magnets are moved to\ndesigned position to generate the proper focusing field.\ncapability to produce the RF power.\nV. DISCUSSION\nA. Output power compared with solenoid focusing\nFor comparison, the saturated output power measurement was a lso performed with elec-\ntromagnet (EM). The results for both PM and EM are shown in Fig. 11 . The output power\nwith the EM is slightly larger than that with PM by about 10 %. For produ ction of 5.6\nkW RF average power (nominal parameter for DRFS; 800 kW peak RF power with 5 Hz\n10FIG. 7. Typical measured transverse (dipole) magnetic field s along the beam axis.\nFIG. 8. The estimated beam-center displacement due to the tr ansverse magnetic field (see Fig. 7).\nBoth the calculated result of the 1st order approximation an d the result from a massive simulation\nwith CST particle studio are presented.\nrepetition rate and 1.5 ms pulse duration), the required average be am power is about 9.6\nkW (59 % efficiency for beam-to-RF energy conversion) with the sole noid magnet and 10.5\nkW (54 % efficiency) with the prototype permanent magnet. Becaus e the solenoid magnet\ncontinuously consumes 1.2 kW power, the total power consumption of the prototype PM\ncase exceeds the same level of the EM case. This is a good result con sidering the limited\nexperiment time duration of a few weeks. The power conversion effic iency of a klystron with\nan improved PM focusing system would be enhanced at least as the sa me as the EM case\nand required power from wall plug would be reduced. In DRFS, the nu mber of klystrons is\nabout 8,000, and the total eliminated power in the total facility would be 9.6 MW.\n11FIG. 9. The output RF power of the DRFS klystron with permanen t magnet focusing as function\nof input RF power at four cathode voltages.\nFIG. 10. The results of the power test. Output RF power is show n as functions of cathode voltage\nwith permanent magnet focusing and solenoid focusing.\nThe main reason of the discrepancy of output power between the P M case and the EM\ncase is considered to be the beam deformation and displacement due to the small multipole\nfieldcomponents. Thebeamdeformationanddisplacement leadthea symmetric distribution\noftheparticlesandmaycauseexcitationofhigherordermodesinth ecavities suchasTM110\nand TM210 modes. The kick due to the higher order modes and dipole m agnetic field might\ncause beam loss and decrease output power. However, a significan t temperature rise of the\ntube outer wall was not observed, indicating that there were no be am losses in the klystron\ntube during the power test. The deformation of the beam decreas es clearance between the\nbeam envelope and the tube wall (including cavity iris), and pushes up the global coupling\n12factors between the beam and the cavities, which represents a co upling factor averaged over\nall the electrons in the beam (see Appendix).\nThe calculated value of the global coupling factor Mfocused by the PM without dipole\nerror was about 1 %larger than the EM case. As the beams propaga tethroughthe tube, the\nbeams with the larger energy modulation would result the larger dens ity modulationand the\nlarger RF current of the beam. Since the cavities in the klystron are designed and tuned for\nthe beam focused with the EM focusing, the electron beam with large r beam-cavity coupling\nfactor tends to be longitudinally over-focused at the output cavit y. According to the 1D\nklystron simulation, while a small increase less than 1.7 % relative to the global coupling\nfactor value for the EM case lead to increase of the output power, larger increase of the\nglobal coupling factor causes decrease of the output power. The refore, with perturbation\non the global coupling factor larger than 1.7 %, the maximum bunching efficiency and the\ninduced RF current in the output cavity decrease compared with th e EM focusing. With\nthe PM case with dipole error, the perturbation of the global couplin g factor was larger than\n3 % and expected output power is less than the EM case.\nFor the evaluation of the effects of the beam deformations, PIC (P article in Cell) simula-\ntions with CST particle studio were also performed. The simulations wit h the PM focusing\nwere performed with two conditions; 1) without transverse dipole fi eld (without imperfec-\ntions of the magnets), 2) with the transverse error field. In the s imulation, the excited\nvoltages of higher order modes were less than those of the fundam ental mode by two orders\nof magnitude. In addition, any beam loss in the beam drift region was o bserved in the\nsimulations. The simulation results showed the tendency of changing the output power with\nthe PM focusing compared with the EM focusing as supposed by 1D sim ulation discussed\nabove. The calculated output power with the PM focusing (without d ipole field) was 874\n±6 kW, while that with the EM focusing was 865 ±5 kW. Therefore, the results suggest\nthat the quadrupole fields reduce the output power. With the dipole field shown in Fig. 7,\nthe peak output power was calculated as 809 ±7 kW (the output power was decreased to\n95±2 % of that with the EM focusing). The reduction of the output powe r is supposed to\nbe caused by the enhancement of the beam-cavity coupling factor due to the beam center\ndisplacement. In Fig. 11, the input-output characteristics both m easured in the power test\nand calculated by the PIC simulations are presented. In the PIC simu lations, the induced\nvoltages of idler cavities ( Vcavity) were also monitored (see Table 1). The calculated results\n13shows that all the induced voltages with the PM focusing are larger t han those with the EM\nfocusing, which indicates that the former case has a larger global c oupling factors than the\nlatter.\nFIG. 11. The input-output characteristic both measured in t he power test and calculated with\nPIC (Particle-in-Cell) simulation.\nTABLE I. Calculated voltages induced in the idler cavities w ith 20 W input power.\nIdler cavity # Vcavity(PM) [kV] Vcavity(EM) [kV] Ratio[Vcavity(PM) /Vcavity(EM)] [%]\n1 19.56 19.27 101.3\n2 6.86 6.84 100.3\n3 20.04 19.75 101.4\n4 34.93 33.83 103.2\nB. Suppression of the transverse field\nAs discussed in the previous subsection, the beam-to-RF power co nversion efficiency was\nsupposed to be deteriorated by the transverse multipole magnetic field on the klystron de-\nsigned for the axi-symmetric field focusing. After the completion of an allowed experiment\ntime in the busy STF schedule, improvements on the focusing system was studied. For the\nrecovery of the output power, the multipole field components have to be suppressed. The\ntransversal segmentation of the magnets with a higher symmetry would result less multipole\ncomponents. The magnet configuration with the two-fold axial sym metry about beam axis\n14generates the quadrupole components, which was supposed to ha ve not much effect on the\nefficiency at the design stage. It should be noted that the constru ction of a more highly\nsymmetric structure leads to more difficulty in the magnet position alig nment and the com-\nplexity of the mechanical support of magnets. Considering the abo ve mentioned situation,\nand the fabrication and the assembling cost, a transverse field sup pressor is favorable rather\nthan the modification of the magnet structure in a fancy way.\nAlthough the multipole field suppressor can be realized either by activ e methods, such as\ncorrection coils, or by passive methods such as additional magnetic materials, the passive\nmethods have the advantage of higher reliability. Passive field suppr ession can be achieved\nby installing a transverse field filter with anisotropic macroscopic per meability, where the\nrelative permeability in the longitudinal direction is close to unity and th at in transverse\ndirections aresufficiently larger than1. A filter with such ananisotro pic permeability can be\nrealized by sparsely stacked rings made of soft-magnetic material such as silicon steel sheets.\nFor our permanent magnet system, the inner radii of the rings can touch the klystron outer\nwall (60 mm) and the outer radii are 70 mm, while the thickness of the rings is 0.5 mm.\nFor evaluation of the suppression capability of the transverse field , magnetic fields were\ncalculated with CST EM studio. From the calculated results with variou s ring intervals, the\nring train with 12 mm spacing could suppress the multipole field compone nt most efficiently\nwithout a significant reduction of longitudinal magnetic field compone nt. With 12 mm\ninterval rings, the quadrupole field is suppressed down to less than 1/5 (see Fig. 12). The\ncalculated transverse dipole field was also reduced to less than 1/2 o f the field without the\nfilter. While the transverse multipole field components were suppres sed efficiently, the effect\nof installing the rings on the longitudinal field components around the axis was less than 1\n% (<0.5 mm / 12 mm).\nPIC simulation results with the filter showed the improvement of the o utput power.\nThe calculated output power with the filter-installed PM focusing was 875±5 kW. The\noutput power become almost the same as that with the EM focusing c ase. A PIC simulation\nincludingthemagnetimperfectionswithfieldfiltershowednooutputp owerreduction. These\nresults indicate that such a filter effectively reduces the transver se multipole components.\n15FIG. 12. Transverse quadrupole magnetic field calculated wi th CST EM studio. The transverse\nfield filter reduce the quadrupole field to less than 20 % of the fi eld without the filter.\nVI. CONCLUSION\nKlystron beam focusing systems with permanent magnets are well s uited to the large\naccelerator facilities such as future linear colliders to enhance their reliability and reduce\ncosts. Even the prototype system with the ferrite magnets for 1 .3 GHz 800 kW klystrons\nexhibits performance almost the same as with EM focusing. This PM sy stem could not be\noptimized because of the limited trial and the experiment time. For re duction of the fab-\nrication costs, anisotropic ferrite was adopted as magnetic mater ial and the magnets were\ndesigned to be pushed-in after the klystron tube insertion to the f ocusing alcove. The mov-\nable magnet feature could also be usable for field tuning process. Th e technologies adopted\nin the prototype system can be applied for other klystrons. Espec ially, the application of\nthe technology to the Multi Beam Klystron would be important for IL C. Furthermore, by\nmitigating the effects of the transverse magnetic field, the focusin g system with permanent\nmagnets can save not only the power consumption in the accelerato r facilities, but also push\nup the reliability of the whole system by eliminating the possible water lea kages from the\nsolenoid coils and the possible failures of power supplies.\nACKNOWLEDGEMENTS\nThis work was supported by the Collaborative Research Program of Institute for Chem-\nical Research, Kyoto University (grant # 2016-10). The authors thank Mr. Y. Okubo at\n16Toshiba Electron Tubes & Devices Co., Ltd for giving us the klystron d ata and for the\nfruitful discussions on the klystron, and thank Dr. H. Hayano at K EK for his continuous\nencouragement and fruitful comments. The authors also thank D r. S. Fukuda, Dr. S.\nMichizono, and Dr. T. Matsumoto at KEK for their support on execu ting the klystron\npower test. The experimental results presented in this paper cou ld not be aquired without\ntheir help.\n[1] The International Linear Collider Technical Design Rep ort, ISBN 978-3-935702-74-4, 2013.\n[2] S. Fukuda, Distributed RF Scheme (DRFS)Newly proposed H LRF scheme for ILC, in Proc.\nLINAC, Tsukuba, Japan, 2010, pp. 112-114.\n[3] M. Baba et al., ”Maintenance Activity of High-Power RF Sy stem in KEK Electron-Positron\nLinac”, Proceeding of the 7th Annual Meeting of particle Acc elerator Society of Japan, Hyogo,\nJapan, 2010, pp. 918-920 (in Japanese).\n[4] G. K. Merdinian, J. H. Jasberg and J. V. Lebacqz, ”High pow er, permanent magnet focused,\nS-band klystron for linear accelerator use”, 1064, Electro n Device Meeting, Washington, D.\nC., Oct. 29-31 (1964) / SLAC-PUB-56.\n[5] S. Fukuda, T. Shidara, Y. Saito, H. Hanaki, K. Nakao, H. Ho mma, S. Anami, and J. Tanaka,\nPerformance of high Power S band klystrons focused with perm anent magnet, KEK, Tsukuba,\nJapan, 198624009 KEK-86-9, Feb. 1987, KEK Report.\n[6] Y. Fuwa et al., ”Focusing System With Permanent Magnets f or Klystrons”, IEEE Transaction\non Applied superconductivity, vol. 24, no. 3, 2014.\n[7] J.T.Mendel, C.F.Quate, andW.H.Yocom, Electronbeamfo cusingwithperiodicpermanent\nmagnet fields, Proc. IRE, vol. 42, no. 5, pp. 800-811, May 1954 .\n[8] S. Matsumoto, M. Akemoto, S. Fukuda, T. Higo, H. Honma, S. Kazakov, N. Kudo, H. Naka-\njima, T. Shidara, and M. Yoshida, Study of PPM- focused X-ban d pluse klystron, in Proc.\nLINAC, Knoxville, TN, USA, 2006, pp. 628-630.\n[9] K. Halbach, Design of permanent multipole magnets with o riented rare earth cobalt material,\nNucl. Instrum. Methods, vol. 169, no. 1, pp. 1-10, Feb. 1980.\n[10] K. Halbach and R. F. Holsinger, ”SUPERFISH - A Computer P rogram for Evaluation of RF\nCavities with Cylindrical Symmetry”, Particle Accelerato rs 7 (1976) 213-222.\n17[11] http://www.toshiba-tetd.co.jp\n[12] A. Steprans, E. W. McCune, and J. A. Ruetz, High-power li near-beam tubes, Proc. IEEE,\nvol. 61, no. 3, pp. 299-330, Mar. 1973., P. Elleaume, O. Chuba r, and J. Chavanne, Computing\n3D magnetic field from inserted devices, in Proc. PAC, May 199 7, pp. 3509-3511., O. Chubar,\nP. Elleaume, and J. Chavanne, A 3D magnetostatics computer c ode for insertion devices, J.\nSynchrotron Rad., vol. 5, pp. 481-484, 1988.\n[13] A. Larionov and K. Ouglekov, DGUN-code for simulation o f inten- sive axial-symmetric elec-\ntron beams, in Proc. 6th ICAP, Darmstadt, Germany, 2000, p. 1 7.\n[14] M. A. Basten, J. H. Booske, and J. Anderson, ”Magnetic Qu adrupole Formation of Elliptical\nSheet Electron Beams for High-Power Microwave Devices”, IE EE Transaction on Plasma\nScience, vol. 22, no. 5, pp. 960-966 (1994).\n[15] https://www.cst.com\n[16] W. Horrald and W. Reid, Permanent magnets for microwave devices, IEEE Trans. Magn.,\nvol. MAG-4, no. 3, pp. 229-239, Sep. 1968.\nAppendix A: Beam-Cavity Coupling Factor\nThe coupling factor between a single particle and a cavity has the rad ius dependency\nthrough the transit time factor\nT(r) =/integraltextg/2\n−g/2Ez(r,z)cos(2πz\nβλ)dz\n/integraltextg/2\n−g/2Ez(r,z)dz, (A1)\nwhereg,βandλare the length of the cavity gap, the electron velocity relative to th e speed\nof light and the wave length at the klystron operating frequency, r espectively. In our study\n(electron energy of about 65 kV at the klystron operating freque ncy 1.3 GHz), βλis 0.11\nm.Ez(r,z) is thezcomponent of the electric field for the fundamental cavity mode. S ince\nthe radial component of electron velocity can be neglected compar ed with the longitudinal\ncomponent, rcan be treated as a constant in the integration. The global beam-c avity\ncoupling factor Mcan be evaluated as the average of T(r) for all electrons in the beam.\nUsing the beam-cavity coupling factor M, the relation between the amplitude of the RF\ncurrent in the cavity ( icavity) and that of the beam ( ibeam) can be written as\nicavity=M ibeam. (A2)\n18The voltage induced in the cavity ( Vcavity) is denoted as\nVcavity=R icavity, (A3)\nwhereRis the shunt impedance of the cavity. The energy modulation ∆ Uby the induced\nvoltage is described as\n∆U=T(r)Vcavitycosφ, (A4)\nwhererandφare the position from beam axis and RF phase at the time when the elec tron\narrives at the cavity gap. Since the T(r) is a monotonically increasing function of rfor the\nre-entrant cavity, large rleads increasing of Mand the modulation gain enhancement.\n19"    },    {        "title": "1910.13851v2.Coherent_x_ray_radiation_induced_by_high_current_breakdown_on_a_ferrite_surface.pdf",        "content": "1 \n Coherent x -ray radiation induced by high -current breakdown on a ferrite surface  \nIvan N. Tilikin, Sergey Yu. Savinov, Nikolai V . Pestovskii, Sergey A. Pikuz, Sergey N. Tskhai, \nand Tatiana A. Shelkovenko  \nP.N. Lebedev Physical Institute of RAS, 119991 Moscow, 53 Leninskii prosp.  \nWe for the first time observe that at the initial stage of a high -current discharge, a low -\ndivergence short (< 2 ns) electromagnetic pulse is formed over a ferrite surface. The 50% part of \nthis pulse lies in the region of fairly  hard x -ray radiation (hν >1 keV)  with the energy ∼0.6 mJ and \nthe average power  0.3 MW. The radiation propagates parallel to the surface in the anode direction \nwith the angle divergence < 2ȯ. The high directionality of the radiation in absence of the apert ure-\nlimiting devices for the radiation beam and the quadratic dependence of the spatial radiation \nenergy flux density on the active  part of the ferrite prism points to the coherent nature of the \nobserved radiation. A possible generation mechanism of the ra diation is proposed. It is based on \nthe short -lived magnetization of the unit areas on the ferrite surface by a high -power \nelectromagnetic pulse and subsequent coherent interference of the unit waves irradiated by these \nareas.  \n \nIn the studies of vacuum ult raviolet (VUV) plasma radiation in a high -current discharge on \na ferrite surface [1,2], a short ( τ < 2 ns) directional pulse of electromagnetic radiation was detected \nat the initial stage of the discharge. The 50% energy part of this radiation is fairly hard - the \ncorresponding energies of photons are higher than 1 keV . In this paper, we report the fir st results \non the study of this radiation and provide their possible interpretation. We show in the present \nwork that the radiation is has the low angle divergence ( < 2ȯ) and its high radiant intensity can be \ncaused by the coherent processes induced by an electromagnetic wave passage through the \ndischarge gap.  \nThe experiments were performed on a BIN generator with an output current amplitude of \nup to 270 kA and a rise time of 80 ns [3]. The impedance of the generator forming line was ~1 Ω, \nand the voltage a t the generator output reached 240 kV , with the charging voltage of the forming \nline being of about 350 kV . The generator load was a rectangular ferrite ((Ni -Zn) Fe 2O4 prism of \ngrade M1000NN with transverse dimensions 10 x 20 mm2. The prism was mounted per pendicular \nto the diode axis, see Fig. 1a. By changing the electrode length on the cathode side, the length of \nthe active  part of the ferrite prism was varied from 1.5 cm to 7 cm. The current flow path on the \nferrite surface was set by a pattern drawn with  a graphite pencil. This path was generally the same \nduring consecutive discharges in the experiment [1,2]. The generator load was unmatched, and its \nimpedance varied greatly during the pulse. The pressure in the discharge chamber did not exceed \n10-4 Torr.  \nThe radiation from the discharge was studied using calibrated diamond photoconductive \ndetectors (PCDs) with flat spectral response C=5 ·10 -4 A/W in the energy range from 10 eV to 4 \nkeV . In the high -energy region, the sensitivity smoothly decreases in acc ordance with the \nabsorption of carbon [4]. The transverse size of the detector crystals was ~3x1 mm, whereas in the \ndetection direction it was 0.5~mm, which provided sufficient sensitivity of the detector up to \nenergies of 10 keV . Detector response time wa s less than 0.3 ns. The total time resolution of the \nrecording channel was 2 ns, with the bandwidth of the Tektronix TDS 3104B oscilloscope and the \ncable lines taken into account. The detectors were typically placed at a distance of 20 cm from the \nend face  of the ferrite prism at various angles with respect to the discharge direction (x axis in Fig. \n1a).  2 \n \n \nFig. 1. (a) Experimental layout. (b) Angular intensity distribution for the radiation from ferrite.  \n \nThe arrangement of the detectors is shown in Fig. 1a. The angular range of radiation \ndetection in the ferrite surface plane xy was -8o < θ < +22o (azimuthal angle), and in the xz plane, \northogonal to the ferrite surface, it was -1o < α < +5 o (polar angle). To estimate the width of energy \ndistribution over the cross section of the generated beam, we used Fuji TR imaging  plates sensitive \nto both x -ray and UV radiation; the plates were placed at a distance of 26 cm from the end face of \nthe ferrit e prism. In order to filter the bright UV -emission of the discharge, the films were covered \nby an aluminum foil. The time dependence of the load current was calculated by numerical \nintegration of the signal from the Rogowski coil (with a bandwidth of >500 MHz). The output \nvoltage on the discharge gap was measured by a resistive -capacitance voltage divider.  \n \nFig. 2. The temporal dependencies of the applied voltage (1), discharge current (2), first  derivative \nof the applied voltage (3), the radiation intens ity observed along (4) and perpendicular (5) to the \nferrite active  surface (the positions 2 and 7 of the detectors in the Fig. 1a). The first detector was \nsituated at L = 26 cm  from the end of the prism, the second one was mounted at the L⊥ = 26  cm \nfrom th e ferrite surface.  \n \n3 \n The time dependences of the discharge current, applied voltage and the radiation intensities \nrecorded along and perpendicular to the ferrite surface (PCDs in positions 2 and 7) are shown in \nFig. 2. Also, a first deviation of the voltage  is presented in this figure. The experimental conditions \nwere the following: the length of the active  part of the ferrite prism was l = 6.5 cm; one detector \nwas placed parallel to the active  surface of the prism at a distance L = 26 cm from its end face, and \nthe other was placed perpendicular to the active  surface at the same distance.  \nIt is seen in Fig. 2a that at the initial (prebreakdown) stage of the discharge, when there is \nalmost no discharge current yet, a short ( τ < 2 ns) radiation pulse is observe d along the discharge \naxis, with the intensity of this pulse exceeding the intensities of radiation detected in the same time \npoints in the perpendicular direction by an order of magnitude. Note that the actual signal duration \nis probably shorter since the  measured value coincides with the time resolution of the recording \nchannel.  \nAlso, it should be noted, that the initial stage of the studied pulse formation coincides in \ntime with a maximum of the first derivative of the applied voltage . In this point the speed of the \nvoltage growth is maximal and at that time an inflection point in the dependence of the applied \nvoltage on the time is reached. However, it is seen from the fig. 2 that the discharge curre nt in this \npoint is zero. After this time, the voltage is still increased and only when it reaches the  maximum, \nthe non -zero discharge current appears and a breakdown of the gap occurs.  \nWe studied the angular distribution of radiation intensities. The de tectors were placed at a \ndistance L = 15 cm from the prism end face. The ferrite prism had a length l = 2 cm. The \nmeasurement results are presented in Fig. 1b. It can be seen that the radiation is concentrated in a \nregion with angular sizes of ~4o and ~ 5 o (± 2.5o) in the planes perpendicular and parallel to the \nactive surface of the ferrite prism. Regarding the fact that in our measurement geometry, the \nangular resolution is ~ 2o, the presented results should be thought of as an evaluation.  \n In vacuum dis charges at such a high voltage (100…300 keV) high -energy electron beams \ncan be formed [5,6]. Such a beam can cause a response on the registration equipment similar to \nthe response caused by high -energy photons. For this reason, some experiments providing a  proof \nof the electromagnetic origin of the studied radiation were carried out. First of all, a transmission \nof an aluminum filter with the thickness 50 μm and a beryllium filter with the thickness 10 μm \n(see below). Corresponding measurements allow to est imate an average energy of the electron \nbeam [7]. In the case of the present work, for the Al filter ( IAl = 0,09 ·Io, where  Io is an intensity \nwithout the filter) the energy of the beam should be Еe=95 keV when for the beryllium filter \n(IBe=0,56 ·Io) the bea m energy should be Еe = 55 keV. It is evident that these results contradict each \nother.  \n Another series of experiments was carried out when at the end of the ferrite prism was \nmounted a permanent magnet with the size h=1 cm providing a constant magnetic field with the \nintensity H = 700 G. An estimations show that a shift of the trajectory for electrons with the kinetic \nenergy Еe=100 keV is Δу ≈ 20 cm. At the electron energy Еe= 50 keV this shift is Δу ≈ 32 cm. Our \nexperiments  show (fig. 1) that the deviation of the peak amplitude of the studied radiation does \nnot exceed the 20% part of the peak average intensity. Magnetic field with the induction H = 700 \nG does not lead to measurable change in the amplitude of the radiation pe ak higher than this \ndeviation. Consequently, it is proved that the nature of studied radiation is electromagnetic one.  \n A spectral composition of the radiation was investigated. For this purpose, integral \nintensities IAl, IBe and IPP behind the aluminum, beryllium and polypropylene filters were measured. \nThe filters had the following transmissions: Al: d = 50 μm, hν > 4 keV , (λ ≤2,5 Å), IAl = 0,09 ·Io; 4 \n Be: d = 10 μm, hν > 0,5 keV ,, IBe = 0,56 ·Io; PP(С3Н6 - polypropylene) d = 4  μm, 0,1 ≤ hν ≤ 0,293 \nkeV , (124Å ≤ λ ≤ 42,4Å)  IPP = 0,66 ·Io]. Here the Io is an integral intensity measured in absence of \nthe filters, i. e. in the region of quanta of the energies 10-2 keV ≤ hν ≤ 10 keV (1,24Å ≤ λ ≤ 1241Å).  \n The results of this measurement allow to estimate a spectral composition of the studied \nradiation using the database on the x -ray transmission spectra of different materials [8]. The results \nof this estimation are presented in Table 1. A dependencies of the energy spectral densities of the \nradiation on the photon energies in the region 1 0-2 keV≤hν≤10 keV evaluated on the basis of the \nTable 1 data are presented in Fig 3.  \nTable 1. Energy fractions of the generated beam  \nEnergies of photons hν Radiation \nintensity  \nh\nν\nkeV, (λ≤1240 Å) 100%  \n1\n0\n \nk\ne\nV\n \n≤\nh\nν\n≤\n0\n,\n1\n \nk\ne\nV\n≤λ≤1241 Å) 10% \nkeV ≤hν≤0,293 keV, (42Å ≤ λ≤ 124Å)   \nkeV ≤hν≤0,5 keV, (42Å ≤ λ≤25Å)  \nkeV≤hν≤1 keV, (25Å ≤ λ≤12,4 Å)  \nkeV≤hν≤4 keV, (12,4Å ≤ λ≤3,1Å)  \nkeV ≤hν≤10 keV, (3,1Å ≤ λ≤1,2Å)  \n \nThe measurement error (electrical noise and possible background radiation) is the same for all  \nangles and does not exceed 10%  \n \n \nFig. 3. The Intensity spectral density dependence on the energy of electromagnetic quanta for the \nstudied radiation.  \n \n The Table 1 shows that the 50% part of the radiant energy corresponds to the x -ray radiation \n(>1 keV) . Also, the most of the UV radiation has the energy higher than 1 eV. Let us note, that the \n5 \n discharge emission in the perpendicular direction to the ferrit e surface lies in the region 10 -800 eV \n[1,2].  \nIndependent estimation of the angular distribution of the investigated radiation was done \nusing the energy distribution over the cross section of the generated beam. Fuji TR imaging plates \nwere used, placed at  a distance L = 26 cm from the end face of the ferrite prism, with the length \nof its active  part l = 4.5 cm.  \n \nFig. 4. (a) Image of the cross section of the generated x -ray beam obtained by Fuji BAS \nTR imaging plate coated with an aluminum filter (d= 15 μ m) and positioned at the distance L = \n26 cm from the end of the ferrite prism with the active  length l = 4.5 cm; (b) The angular \ndependence of the plates darkening coated by the aluminum filter (d= 15 μm) along the image; \n(c) The angular dependence of the plates darkening coated by the aluminum filter (d= 15 μm) \nacross the image; (e) The spectral densities of the studied radiation behind the aluminum filter: d \n= 4μm - solid curve and d = 15 μm – dashed curve.  \n \nThe result is shown in Fig. 4. The darkening of  the imaging  plate corresponds to the time -\nintegrated radiation energy emitted in the given direction. It is seen from the Fig 4, that the beam \ncross -section behind the aluminum filter (d=15 μm) is characterized by the angle size Δθ~2,2ȯ (the \ncorresponding linear size is b~ 0.5 cm ). The cross -section longitudinal length of the beam behind \nthe aluminum filter with the thickness d = 4  μm is  nearly not changed while the transverse length \nis grown up to Δα~1,4 ȯ (Fig. 4 d). The corresponding l inear length is b~ 0,75 cm. A comparison \nof energy spectral densities for radiation behind the aluminum filters with the widths d=4 μm and \n15 μm (Fig. 4 e) shows that the radiation behind the filters at d=4 μm a part of low -energy quanta \nis much more than at d=15 μm. \n6 \n The energy characteristics of the radiation were studied. The total energy registered by \nthe detector is determined as follows  \n𝜀=1\nС𝑅∫𝑉(𝑡)𝑑𝑡∞\n−∞.      (1). \nHere, R = 75Ω is the detector load resistance, V (t) is the instantaneous value of the signal, \nmeasured in V olts, at time t, C=5 ·10-4А/Вт is the detector sensitivity. The duration of the radiation \npulse may turn out to be significantly less than the time resolution of the recording channel (~2 \nns), and the pulse duration and shape cannot be obtained by electrical measurements only. \nNevertheless, since the signal spectral width (see Table I) is scarcely beyond the region of the \ndetector spectral sensitivity (10 eV - 4 keV), relation (1) can be used to estimate the total energy \nincident on the detector. The value of ∫𝑉(𝑡)𝑑𝑡∞\n−∞ was determined as the area under the curve V \n(t) (see Fig. 2a).  \nDuring measurements, the length of the ferrite active  part l was varied from 1.5 cm to 7 cm \nby changing the length of the negative electrode. In several series of experiments, the length of the \nactive  part was first increased (2, 3, 5, 6, and 7 cm), and next decreased (6.5, 5.5, 4.5, 3.5, 2.5, and \n1 cm). At each le ngth in each series, 2 or 3 consecutive shots were made, with the total length of \nthe ferrite sample being constant. For estimates, the beam cross section was considered a rectangle \nwith dimensions s = 1 cm (width) and h = 0,2 cm (height). The value of h i s smaller than the \ntransverse dimensions on the detector crystals. The crystal sizes of the diamond detectors were rD \n~ 0,3 cm  in the lateral direction. This sizes were almost half as much than the smallest cross -\nsectional dimension of the beam measured be hind the aluminum filter with the thickness d=15  μm. \nThereby, the signal measured by the detector is proportional to the spatial energy flux P of the \nstudied radiation.  \n𝑃=𝜀\n𝑆𝐷       (2). \n \nFig. 5. The energy flux spatial density P dependence on the dis charge gap length. The diamond \ndetector was placed at the position 2 in Fig. 1a)  \nThe energy flux dependence on the active  length of the ferrite surface is depicted in Fig. 5 \nby squares. This figure shows that the energy flux P in nonlinearly -growing functi on of the ferrite \nsurface active  length. A solid line in the Fig. 5 shows a fit of the experimental data by a quadratic \n7 \n dependence. For the case under consideration, the pure quadratic dependence is statistically \nsignificant at the probability level W > 0. 87. It was founded, that the maximal energy flux P at the \ndischarge gap 7 cm was ~ 1 mJ·cm-2. In order to estimate the total energy ℇ of a single pulse of the \nstudied radiation, we calculated a relation  \nℇ=𝑃·𝑆,     (3), \nwhere S is the beam cross -section, which was regarded as a rectangle with the widths a = 1.2 cm \nand the height b = 0.5 cm (this size was founded to be the smallest behind the aluminum filter with \nthe thickness d = 15μm). The maximal radiant energy of the radiation  was ~0.6 mJ at the discharge \ngap 7 cm and the average radiant flux was 0.3 MW.  \nWhat is the physical nature of the observed radiation? In view of the sharp asymmetry in \nthe angular distribution of the radiation intensity in the absence of focusing and lim iting devices, \none can conclude that the radiation is coherent, and the asymmetry in the spatial distribution is due \nto interference phenomena. We encounter a similar phenomenon when Cherenkov radiation arises, \nthat is electromagnetic radiation of opticall y transparent media occurs, caused by a charged particle \nmoving in a medium at a speed exceeding the speed of light in this medium [7 --9]. The Cherenkov \nradiation condition can be derived considering the interference phenomena and Huygens -Fresnel \nprinciple . \nWhat happens in our case? We assume that at the prebreakdown stage of the discharge, \nwhen a high voltage is applied to the cathode, a longitudinal electric field 𝐸 appears in the \ninterelectrode gap (a bias current with the spatial density 𝐽𝑜𝑓𝑓=1\n4𝜋𝜕𝐸\n𝜕𝑡 arises), which causes the \nmagnetic field in the direction perpendicular to both Joff and to the normal n to the ferrite surface. \nThus, a high -power magnetic field pulse passes through the discharge gap. This pulse induces \nshort -lived magnetization of the ferrite surface. All the induced magnetic dipoles are oriented in \nthe same direction with the forming magnetic field. In the geometry of the present experiment the \nsymmetry of the system leads to the emission of interfering cylindrical elementary wa ves forming \nthe resultant radiation.  \n \nFig. 6. Diagram explaining the formation of the coherent radiation region  \nFig. 6 shows the diagram explaining the formation of the radiation as a result of coherent \naddition of elementa ry electromagnetic waves. The excitation pulse and the radiation it generates \nmove in the same direction at the same speed. The envelope of the wavefronts of the elementary \nwaves exists only in a small region near the ferrite surface where the phase matchi ng of the \nradiation from elementary sources is ensured.  \nAccording to the Huygens -Fresnel principle,  the elementary waves are mutually canceled \nexcept for their common envelope. Thereby, the emitted radiation has low angular divergence and 8 \n propagates parall el to the ferrite surface towards the anode. The coherence of the radiation is, \nanalogous to the Vavilov -Cherenkov radiation, due to the equivalent excitation conditions for all \nemitters. In a first approximation, one can be assumed that the longitudinal e lectric field strength \nE is proportional to the applied voltage U. Thereby, a highly -directional electromagnetic pulse is \nformed when the bias current 𝐽𝑜𝑓𝑓=1\n4𝜋𝜕𝐸\n𝜕𝑡 achieves its maximal value (see Fig. 2).  \nIt can be shown that the energy flux P of the studied radiation in a case of the coherent \ninterference of cylindrical unit waves can be calculated by a following expression:  \n𝑃(𝑦,𝑙,𝐿)(𝑙)2\n𝐿(𝑆𝑖𝑛𝐶[𝜋𝑦2𝑙\n2𝜆𝐿2])2\n                                 (4). \nIn this formula 𝑦 is the spatial coordinate in t he detector plane, 𝑙 is the length of the active  \npart of the ferrite surface, 𝐿 is the distance between the end of the ferrite prism and the plane where \nthe detectors are installed. The expression (4) was obtained using the approximation based on the \nrelation L >> l,y. The conclusion from (4) is that the energy flux 𝑃 is proportional to the second \norder of the active length of the ferrite surface if the radiation is coherent. This dependence was \nexperimentally observed (Fig. 5).  \n \nIn conclusion, we emphas ize the important feature and novelty of the considered \nphenomenon: no optically transparent medium with a refractive index n and no charge moving at \na speed v > c/n  are involved. The studied radiation is formed as the electromagnetic excitation \npulse pass es over the surface of the ferrite prism. This pulse and the radiation it generates move in \nthe same direction at the same speed, and the radiation region is formed as a result of coherent \naddition of elementary electromagnetic waves. Consequently, the rad iation with high radiant \nintensity is emitted.  \n \nThis study was supported by the Russian Science Foundation, project no. 19 -79-30086.  \n \nLiterature  \n \n1. I.N. Tilikin, S.N. Tskhai, T.A. Shelkovenko, S.Yu. Savinov, S.A. Pikuz. Generation of \nIntense UV Radiation  during High -Current Breakdown over a Ferrite Surface. Plasma Physics \nreports 2018, v. 44, no. 6,  pp. 600 –604. DOI: 10.1134/S1063780X18060107  \n2. I.N. Tilikin, S.N. Tzhai, T.A. Shelkove nko, S.Yu. Savinov, S.A. Pikuz, and \nA.R.  Mingaleev. A Pulsed, High -Intensity Source of XUV Radiation Based on Ferrite Surface \nBreakdown at High Current. IEEE Transactions on Plasma Science 2018, v. 46, no. 11, pp. 3982 –\n3985. DOI: 10.1109/TPS.2018.2873065   \n3. S. A. Pikuz, T. A. Shelkovenko, and D. A. Hammer, “X -pinch. Part I,”  \nPlasma Phys. Rep., vol. 41, no. 4, pp. 291 –342, 2015.  \n4. Rick B. Spielman, Lawrence E. Ruggles, Richard E. Pepping, Stephen P. Breeze, John \nS. McGurn, and Kenneth W. Struve, Fielding a nd calibration issues for diamond photoconducting \ndetectors, Rev. Sci. Instrum.  V ol. 68, p.762, 1997.  9 \n 5. G. A. Mesyats, S. D. Korovin, K. A. Sharypov, V. G. Shpak, S. A. Shunailov, and M. I. \nYalandin, \"Dynamics of Subnanosecond Electron Beam Formation in G as-Filled and Vacuum \nDiodes\", Technical Physics Letters, 2006, Vol. 32, No. 1, pp. 18 –22. \n6. G. A. Mesyats and M. I. Yalandin, \"High -power picosecond electronics\", Physics -\nUspekhi, Volume 48, Number 3, p. 211 (2005).  \n7. L. Katz, A. S. Penfold, \"Range -energ y relations for electrons and the determination of \nBeta-ray end -point energies by absorption\", Review of Modern Physics, Vol. 24, N. 1, pp. 28 -44, \n1952.  \n8. https://henke.lbl.gov/optical_constants/filter2.html  \n9. P. A. Cerenkov, C.R. Ac. Sci. U.S.S.R. 1934, v. 8, p. 451.  \n10. P. A. Cerenkov, Phys. Rev. 1937, v.52, p. 378.  \n11. I. M, Frank and I. E. Tamm, C.R. Ac. Sd. U.S.S.R. 1937, v. 14, p. 109.  \n "    },    {        "title": "2202.08115v1.Unusual_ferrimagnetic_ground_state_in_rhenium_ferrite.pdf",        "content": "Unusual Ferrimagnetic Ground State in Rhenium Ferrite\nM. Hussein N. Assadi\u0003\nSchool of Materials Science and Engineering, University of New South Wales, Sydney NSW 2052, Australia.\nMarco Fronzi\nCollege of Engineering, Shibaura Institute of Technology, Toyosu, Koto City, Tokyo 135{8548, Japan.\nDorian A. H. Hanaor\nFachgebiet Keramische Werksto\u000be, Technische Universit at Berlin, 10623 Berlin, Germany.\n(Dated: 2022)\nThrough comprehensive density functional calculations, we predict the stability of a rhenium-\nbased ferrite, ReFe 2O4, in a distorted spinel-based structure. In ReFe 2O4, all Re and half of the Fe\nions occupy the octahedral sites while the remaining Fe ions occupy the tetrahedral sites. All Re\nions are predicted to be at a +4 oxidation state with a low spin con\fguration ( S= 3=2), while all\nFe ions are predicted to be at a +2 oxidation state with a high spin state con\fguration ( S= 2).\nMagnetically, ReFe 2O4adopts an unconventional ferrimagnetic state in which the magnetic moment\nof Re opposes the magnetic moments of both tetrahedral and octahedral Fe ions. The spin-orbit\ncoupling is found to cause a slight spin canting of \u00181:5\u000e. The predicted magnetic ground state\nis unlike the magnetic alignment usually observed in ferrites, where the tetrahedral cations oppose\nthe spin of the octahedral cations. Given that the density of states analysis predicts a half-metallic\ncharacter driven by the presence of Re t2gstates at the Fermi level, this compound shows promise\ntowards potential spintronics applications.\nKeywords: Rhenium ferrite, unconventional magnetism, ferrimagnetism, spin canting, density functional\ntheory, spin-orbit coupling\nINTRODUCTION\nSearching for and investigating exotic magnetic phases\ndeepens our fundamental knowledge of complex func-\ntional materials and opens new horizons for novel ap-\nplications [1]. Ferrites are among the most studied mag-\nnetic materials, with broad applications in spintronics\n[2], magnetic data storage [3], magnetically recoverable\ncatalysts [4-7], and microwave guides [8]. The utility\nof ferrites stems from their high magnetic saturation,\nhigh Curie temperatures and controllable coercivity. Fer-\nrites, most commonly synthesised by reactions of Fe 2O3\nwith a smaller proportion of other metal oxides, encom-\npass a wide range of chemical compositions, stoichiome-\ntries, and crystal structures. However, they mostly crys-\ntallise into cubic spinel structures, distorted spinels with\nlower symmetry, [9] or hexagonal structures [10]. Fer-\nrite cations are situated between octahedral and tetra-\nhedral voids, created through the oxygens' closed packed\narrangement [11]. Alongside Fe, cation sites may be oc-\ncupied by Sr, Ba, Pb, or transition metals (TMs). Gen-\nerally, ferrites are ferrimagnetic where the spin of the\ntetrahedral cations opposes but does not cancel the spin\nof the octahedral cations [12]. The second cation's type\nand its abundance determine the magnetic hardness and\nsaturation of the resultant ferrite, enabling the design of\nmaterials with desirable magnetic phase transition tem-\nperatures and magnetic coercivity [13].\nTypically, in complex materials containing multiple\nmagnetic cations, the magnetic ground state is stabilisedthrough competing ferromagnetic and antiferromagnetic\nsuperexchange interactions, resulting in frustrated sys-\ntems with multiple magnetic phase transitions. In fer-\nrites with only fourth row 3d TM ions, these competi-\ntions simply stabilise the ferrimagnetic state where the\ncations on the octahedral site align antiparallel to the\ncations on the tetrahedral sites [12]. However, in other\nclasses of complex oxides, such as double perovskites,\nmagnetic interactions between 3d and heavier 4d and\n5d elements has demonstrated substantially more com-\nplex magnetic behaviour [14,15], often deviating from the\nrules of thumb established by Goodenough and Kanamori\n[16,17], where exotic magnetic behaviours are often the\nresult of the strong spin-orbit coupling, structural distor-\ntions and higher bond covalency between heavier TM ions\nand oxygen. A detailed review of Goodenough-Kanamori\nrules can be found in the literature [18,19].\nWith all these exciting developments in perovskite\nmagnetism, one wonders if there are any similar coun-\nterparts in ferrites. The idea of harnessing heavier 4d\nand 5d TM ion ferrites to control anisotropy and magne-\ntostriction in ferrite was proposed by Hansen and Krish-\nnan in 1977 [20]. However, since then, this idea has not\nattracted the attention it deserves. In the present work,\nwe demonstrate the stability of a rhenium-based ferrite\nReFe 2O4and discusses its unconventional ferrimagnetic\nground state through comprehensive density functional\ncalculations.arXiv:2202.08115v1  [cond-mat.mtrl-sci]  16 Feb 20222\nCOMPUTATIONAL SETTINGS\nSpin-polarised collinear and noncollinear density func-\ntional calculations were performed with VASP code\n[21,22], using the projector augmented wave method\n(PAW) [23] and the Perdew{Burke{Ernzerhof (PBE)\nexchange-correlation functional [24,25]. To improve the\nelectronic band description, adequate intra-atomic inter-\naction terms ( Ue\u000b), based on the Liechtenstein et al. ap-\nproach [26], were added to the Fe 3d electrons. The Uand\nJparameters were 3.5 eV and 0.5 eV, respectively, result-\ning in an e\u000bective U(Ue\u000b) of 3 eV. Comparable values\nwere reported to improve the band description accuracy\nof ferrites [27,28]. More speci\fcally a Ue\u000bvalue of 3 eV\nis necessary to adequately describe the Fe 3d electrons in\noxides [29]. Furthermore, our comprehensive test demon-\nstrated the adequacy of these values (Fig. S1). Accu-\nrate electronic localisation through the GGA+ Uformal-\nism is essential for obtaining reliable structures as the\natomic forces are sensitive to magnetic moments borne\non cations [30]. The energy cut-o\u000b was set at 650 eV.\nThe precision key for the rest of the parameters was set\nACCURATE . The noncollinear calculations were initi-\nated with the WAVECAR \fles calculated with the spin-\npolarised collinear method to facilitate convergence.\nTo simulate the ReFe 2O4structure, as shown in Fig.\n1, two Fe ions in the primitive magnetite cell, with the\nchemical formula Fe 6O8[31], were substituted with Re\nions. As shown in Fig. 1a{j, we considered all possible\nrhenium placement scenarios, and for each scenario, we\nexamined various spin alignments, searching for the most\nstable structure. Substitution at the octahedral sites con-\nsistently resulted in lower total energy, so we further in-\nvestigated all plausible spin alignments in ReFe 2O4with\nonly octahedral Re. A dense 7 \u00027\u00027 k-point mesh, gen-\nerated with the Monkhorst-Pack scheme of \u00180:015\u0017A\u00001\nspacing, consisting of 172 irreducible sampling points in\nthe Brillouin zone, was used for geometry optimisation.\nFor geometry optimisation, the internal coordinates and\nthe lattice parameters were relaxed to energy and force\nthresholds smaller than 10\u00006eV and 0.02 eV \u0017A\u00001, re-\nspectively. No symmetry restriction was applied in ge-\nometry optimisation to allow relaxation to lower symme-\ntry, should it be more stable. It is well-known that even\nthe simplest of the spinel ferrites, Fe 3O4, although cu-\nbic (Fd\u00163m) at room temperature, transforms to a lower\nsymmetry monoclinic structure ( P21=c) below\u0018125 K,\nthrough a Verwey phase transition [32].\nRESULTS AND DISCUSSION\nTwo cation types based on coordination exist in the\nspinel ferrite structure: one that is tetrahedrally coordi-\nnated and another that is octahedrally coordinated withoxygen ions. The \frst type represents one-third, while\nthe second represents two-thirds of the total cations in\nthe crystal. We examined three possible rhenium place-\nments to identify the most stable position of the Re ions\nin ReFe 2O4. First, both Re ions were placed at the tetra-\nhedral sites while all Fe ions were left at the octahedral\nsite, which is the typical cationic distribution in spinels.\nSecondly, one Re ion was placed at the tetrahedral site,\nand the other Re ion was placed at the octahedral site,\na con\fguration usually referred to as an intermediate\nspinel. Lastly, both Fe ions were placed at tetrahedral\nsites, while the octahedral sites were equally occupied\nwith Re and Fe, which is referred to as an inverse spinel.\nWe investigated all these cationic distributions by cal-\nculating the total energy for two possible ferrimagnetic\nand ferromagnetic spin alignments. In the ferromagnetic\nstructure, all cations' spin was set parallel, while in the\nferrimagnetic structure, the tetrahedral cations' spin was\nset antiparallel to the spin of the octahedral cations. As\nshown in Fig. 1a and b, when Re ions are located at\ntetrahedral sites, the total energy is relatively high re-\ngardless of the direction of the Re spin. However, the\nferrimagnetic state is still more stable than the ferro-\nmagnetic one. For intermediate rhenium occupancy, as\nin Fig. 1c, d, the total energies for both spin alignments\nwere slightly higher than the previous case of complete\ntetrahedral Re occupation.\nWhen Re ions are located at octahedral sites (Fig. 1e{\nj), the compound's total energy generally decreased sig-\nni\fcantly, indicating greater stability. For instance, the\naforementioned ferrimagnetic spin alignment of Fig. 1e\nwas more stable than the counterparts with tetrahedral\nRe (Fig. 1a) and the mixed Re (Fig. 1c) con\fgurations\nby approximately 2 eV per unitcell. The stability of the\noctahedral Re warranted further investigations of other\npossible spin alignments of ReFe 2O4with octahedrally\ncoordinated rhenium. Accordingly, we calculated all dif-\nferent possible ferrimagnetic spin alignments for this fer-\nrite compound with octahedral Re [see con\fgurations (g)\nand (h)]. In particular, con\fguration (h) was the most\nstable among all. In this ferrimagnetic con\fguration,\nthe Re ions' spin direction is antiparallel to the spins of\nboth octahedral Fe and tetrahedral Fe ions. In this case,\nthe spins of the octahedral Fe and tetrahedral Fe ions\nwere parallel. Con\fguration (e), with similar spin align-\nment to a conventional ferrimagnetic inverse spinel such\nas magnetite, was higher in total energy relative to con-\n\fguration (h) by 0.4446 eV/u.c. (u.c. is unitcell). Con-\n\fguration (g), representing the other possible realisation\nof the ferrimagnetic alignment, also had higher total en-\nergy than con\fguration (h) by 2.1606 eV/u.c. Likewise,\nthe ferromagnetic state in con\fguration (f) had higher\ntotal energy of 1.1251 eV/u.c.\nTo unambiguously con\frm the stability of con\fgura-\ntion (h), we further calculated the total energy of con-\n\fguration (i), which is quite similar to con\fguration (h)3\nexcept that the spins of the two Re ions were set an-\ntiparallel to examine the strength of the magnetic cou-\npling among Re ions. We also examined con\fguration\n(j), which is antiferromagnetic; that is, every cation has\nan antiparallel spin to the one adjacent. Both con\fgu-\nrations (i) and (j) had total energies higher than that of\ncon\fguration (h) by more than 1 eV/u.c. The stability\nof con\fguration (h) relative to con\fgurations (i) and (j)\ndemonstrates that pairs of Re Oct, Fe Octand Fe Tetions\nhave a strong tendency towards parallel spin alignment\namong themselves.\nFor the most stable placement of Re which can be\nexpressed as Fe Tet(ReFe) OctO4, con\fguration (h), which\nrepresents the magnetic ground state, is remarkably sta-\nble as \ripping to even the second most stable spin con-\n\fguration (e) has an energy cost of 0.4446 eV/u.c. This\nlevel of stability is likely to correspond to an ambient\nCurie temperature in ReFe 2O4. This prediction is based\non a comparison with CoFe 2O4's stability margin. A\nspin-\rip from ferrimagnetic to ferromagnetic order in\nCoFe 2O4costs 0.536 eV/u.c. [33], resulting in a Curie\ntemperature ( TC) of 793 K in bulk cobalt ferrite [34].\nFurthermore, the mean-\feld approximation can estimate\nthe Curie temperature based on the magnetic exchange\nbetween Fe and Re sublattice systems as follows [35]:\n3\n2kBTC=X\ni6=jJij; (1)\nin whichkBis the Boltzmann constant and Jijis the\npair exchange coupling parameter between sites iand\nj. Assuming the nearest neighbour interaction between\nRe and Fe to be the most signi\fcant, Equation 1 yields\nTC= 734:9 K, which is close to the comparison made\nearlier.\nGiven that Re is a relatively heavier sixth-row tran-\nsition element, we anticipate that the role of spin-orbit\ncoupling (SOC) is potentially signi\fcant in ReFe 2O4[36].\nTo examine the signi\fcance of SOC, we re-optimised the\nmost stable con\fguration (h) with spin-orbit coupling\ntaken into consideration. The structural relaxation was\nminor as none of the internal coordinates and the lattice\nparameters changed by more than \u00181%. However, the\ntotal energy was lowered to \u0000107:7975 eV/u.c., indicat-\ning that SOC accounts for \u00180:67% of the total energy.\nSimilarly, SOC is not anticipated to change the magnetic\nalignment in ReFe 2O4as other competing con\fgurations\nare also estimated to have their total energy lowered by\napproximately the same amount when SOC is considered\n(Table S1). SOC brings about magnetic noncollinearity\nby coupling the spin to the orbital degrees of freedom, of\nwhich the latter depends on the lattice environment. The\nnoncollinear magnetic alignment of ReFe 2O4is shown in\nFig. 2a. More precisely, the spins of the Re ions formed a\ntight angle of 1 :329\u000ebetween each other, i.e., nearly par-\nallel; and the net spin of the Re ions formed an angle of177:593\u000ewith the net spin of the Fe ions, i.e., nearly an-\ntiparallel. It can be seen that the degree of noncollinear-\nity is relatively small as the spin directions are to a great\nextent similar to that of the collinear alignment of con-\n\fguration (h). The net magnetisation of the whole com-\npound was calculated to be 6.090 \u0016B/f.u. (f.u. stands for\nformula unit), which is approximately 1.5 times larger\nthan that of the saturation magnetic moment of mag-\nnetite [37]. A more qualitative description of the e\u000bect\nof theUe\u000bchoice on the total magnetisation and the local\nmagnetisation of the Re and Fe ions in this con\fguration\nis given in Table S2.\nThe density of states (DOS) in ReFe 2O4, calculated\nconsidering SOC, are presented in Fig. 3. Here, DOS is\nprojected onto the axes of an orthogonal frame of which\nthezaxis is parallel to the cpdirection of the ReFe 2O4\nprimitive cell. First, we notice that the DOS magnitude\nalong thexandyaxes is\u00183% of the DOS magnitude\nalong thezaxis, indicating a slight deviation from linear-\nity and corroborating the magnetic moment orientations\nobtained in Fig. 2a. Along the zaxis, (Fig. 3c), we can\nsee that both Fe Tetand Fe Octare in high-spin and +2\noxidisation states as one spin channel, comprising of \fve\nelectrons per Fe, for both Fe Tetand Fe Oct, is fully occu-\npied while the other spin channel is only partly occupied.\nTherefore, the electronic con\fguration for Fe2+\nTetise2\"t3\n2\n\"e1#, while for Fe2+\nOct, the electronic con\fguration is t3\n2g\n\"e2\ng\"t1\n2g#. Moreover, Re is in low-spin state with +4\noxidation. According to its partial DOS, Re has a fully\noccupiedt2g\"states, which are immediately followed by\nits emptyt2g#states. The Fermi level crosses the tail\noft2g\"states, giving rise to half-metallic conduction, as\nno states are available at the Fermi level with opposing\nnet spin direction. Moreover, because of the larger crys-\ntal \feld acting on Re's 5d states, Re's empty egstates\nare located at\u00184 eV above the Fermi level (Fig. S2).\nThe ReFe 2O4half-metallicity can be utilised for magneto-\nresistive response [38,39] or near-perfect spin-polarised\ncurrent injection [40,41].\nAs shown in Fig. 3, the Fe 3d states are spread through\nthe conduction band, hybridising extensively with O,\nwhile Re 5d states are mainly concentrated within 2\neV below the Fermi level. Nonetheless, in the region of\n\u00002< E\u0000EFermi<0, the spin-down Re and Fe states\nand O 2p states hybridise together, facilitating the mag-\nnetic superexchange interactions that stabilise the mag-\nnetic ground state. Furthermore, the net magnetic mo-\nments borne on all cations, as shown in Fig. 2, are smaller\nthan ideal ions. For high spin Fe2+(3d6), either in tetra-\nhedral or octahedral coordination, the magnetic moment\nshould have been 4 \u0016B. For octahedral Re4+(5d3), the\nmagnetic moment should have been 3 \u0016B. However, since\nFe{O or Re{O bonds are not purely ionic but possess a\ndegree of covalency, the magnetisation of transition metal\nions is expected to be lower than the purely ionic values\n[42]. The reduction in magnetisation is more profound4\n(b)\nEt = –103.7775 eV (FM)\n(f)\nEt = –105.9444 eV (FM)\n(j)\nEt = –104.2201 eV (AFM)ap\nbpcp\nEt = –105.9039 eV\n(i)\nFe\nRe\nO\n(e)\nEt = –106.6249 eV (FiM-a) Et = –107.0695 eV (FiM-c)\n(h)\nOctTet\n(g)\nEt = –104.9089 eV (FiM-b)\nOctTetOctTet(d)\nEt = –103.5364 eV (FM)\nOctTet(c)\nEt = –103.7854 eV (FiM) Et = –104.6162 eV (FiM)\n(a)\nTet\nOct\nAll starting at\nap = bp = cp = 5.961 Å\nαp = βp = γp =  60º\n(             )Fd¯3m\nFig. 1. The spin con\fgurations used for determining the magnetic ground state of ReFe 2O4. In con\fgurations a\nandb, Re ions are at the tetrahedral sites. In candd, one Re ion is located at the tetrahedral site while the other\nis located at the octahedral site. In e,f,g,h,i, and j, both Re ions are located at the octahedral sites. The density\nfunctional total energy ( Et) of each con\fguration is also shown. FM, FiM, and AFM refer to ferromagnetic,\nferrimagnetic and antiferromagnetic, respectively.\n∠ReOct–O–ReOct: 86.626º, 86.634º∠FeOct–O–FeOct: 93.52º, 97.13º∠FeOct–O–ReOct: 91.29º, 91.32º, 98.81º, 98.84º∠ReOct–O–FeTet: 123.55º, 124.98º, 128.61º, 134.50º∠FeOct–O–FeTet: 111.94º, 112.10º, 114.02º, 114.05º, 127.39º, 127.42º\nccacbc(b)ReOct4+\nFeTet2+\nap = 6.243, bp = 6.198, cp = 6.243 Åαp = 60.24º, βp = 54.43º, γp = 60.24º\nac = 5.710 , bc = 6.198, cc = 6.243 Åαc = 119.76º, βc = 117.21º, γc = 90.00ºFeOct2+\nReOct4+FeOct2+FeTet2+\nap\nbpcp3.549 μB3.549 μB\n3.614 μB3.614 μB1.280 μB\n1.092 μB(a)\nFeTet2+\nFeOct2+\nFe\nRe\nO\nFig. 2. aThe optimised primitive cell of ReFe 2O4with SOC taken into account, along with the magnetic moments\nborne on all cations. bThe conventional representation of the optimised ReFe 2O4, which has a triclinic symmetry.\nInb, all possible bond angles between cations are also presented. The corresponding yellow marks show a\nrepresentative of each given angle. The lattice parameters shown are either indexed with porc, indicating the\nprimitive and conventional cell dimensions, respectively.5\nin heavier transition metal ions are their bonds are more\ncovalent [33]. A more quantitative description of the elec-\ntronic localisation function is provided in Fig. S3.\nAs shown in Fig. 2a, the primitive cell of ReFe 2O4\nis substantially transformed by geometry optimisation.\nThe lattice parameters of the optimised primitive cell do\nnot conform to the high symmetry of the initial structure\nas its lattice parameters asymmetrically changed and its\nvolume expanded. The initial structure volume, which\nwas based on magnetite, was 155.225 \u0017A3, while the opti-\nmised structure had a volume of 163.025 \u0017A3. The ionic\nradius of Fe can explain the expansion of the volume\nupon Re substitution at the tetrahedral site. The radius\nof high-spin Fe2+\nTetin ReFe 2O4is 0.63 \u0017A. In magnetite,\nthe tetrahedral site is occupied by Fe3+with a smaller\nradius of 0.49 \u0017A. The Re4+radius in octahedral coordi-\nnation is 0.63 \u0017A which is quite close to the replaced Fe3+\nradius (0.65 \u0017A) and is not expected to be a substantial\ndrive in the structural transformation.\nWe examined the optimised ReFe 2O4primitive cell's\nsymmetry to investigate the magnetic exchange among\nall cations. A triclinic symmetry was detected through\nthe FINDSYM symmetry detection algorithm [43] with a\ntight tolerance of 0.00001 \u0017A for lattice parameters (CIF\nprovided in the supplementary information). The con-\nventional cell with triclinic symmetry is shown in Fig. 2b.\nDetecting all possible magnetic exchanges that stabilise\nthe predicted ground state magnetism|con\fguration\n(h) re-optimised with SOC considered{is easier in the\nsymmetry-imposed structure. The TM{O{TM bond an-\ngles for this structure are all listed in Fig. 2. The TM{\nO{TM bond angles between cations on tetrahedral and\noctahedral sites are obtuse and thus dominate the mag-\nnetic exchange interactions, as the superexchange inter-\naction magnitude is proportional to cos2(6TM{O{TM).\nThe higher total energy of con\fguration (g) indicates the\nsuperexchange between Re4+\nOctand Fe2+\nTetis antiferromag-\nnetic, while the higher total energy of con\fguration (e)\nindicates that the superexchange between Fe2+\nOctand Fe2+\nTet\nis ferromagnetic. The TM{O{TM bonds among tetrahe-\ndral sites are all nearly right angles, indicating a minimal\norbital overlap favouring weaker ferromagnetic superex-\nchange. The strength of this ferromagnetic exchange can\nbe estimated from the total energy of con\fgurations (g)\nand (h) of Fig. 1, showing that setting adjacent cations\nto antiferromagnetic coupling raises the total energy.\nFinally, we examine the stability of ReFe 2O4in com-\npeting metallic and oxide phases. The compound's for-\nmation enthalpy (\u0001 Hmetallic ) was calculated relative to\nthe metallic Re (hexagonal paramagnetic) and Fe (body-\ncentred ferromagnetic) phases, and gaseous O 2was cal-\nculated as\n\u0001Hmetallic =Et(ReFe 2O4)\u0000Et(Re)\u00002Et(Fe)\u00002Et(O2):\n(2)\nHere,Etis the density functional total energy. \u0001 Hmetallic\n-0.40.00.4 \nO 2p \nFeTet 3d \nFeOct 3d \nReOct 4dE\n − EFermi (eV)(a) ReFe2O4 (x)-\n0.30.00.3(\nc) ReFe2O4 (z)(b) ReFe2O4 (y)-\n8- 6- 4- 20 2 -303DOS (eV−1)D OS (eV−1)t\n2gt\n2gt2g & egt\n2ge  & t2et2g & ege & t2DOS (eV−1)Fig. 3. Partial density of states of ReFe 2O4at its most\nstable magnetic state [con\fguration ( h) of Fig. 1],\ncalculated with spin-orbital coupling considered. The\ndensity of states is projected along an orthogonal frame\nhaving the x,y, andzaxes. Thezaxis of this frame\ncoincides along the lattice parameter cpof the primitive\nlattice parameter, shown in the lower row of Fig. 1.\nwas found to be\u00005:1924 eV/f.u. The formation enthalpy\nrelative to the competing oxide phase ( \u0001 Hoxide) was\ncalculated as\n\u0001Hoxide =Et(ReFe 2O4)\u0000Et(ReO 2){2Et(FeO):(3)\nHere, ReO 2was the most stable Re4+oxide in orthorhom-\nbic structure (materials project identi\fer mp-7228 [44]),\nand FeO was the most stable Fe2+oxide in monoclinic\nstructure (materials project identi\fer mp-1279742 [44]).\n\u0001Hoxide was found to be\u00001:1903 eV/f.u. Given the neg-\native \u0001Hvalues, we can conclude that ReFe 2O4is stable\nagainst decomposition to oxides of its constituent ele-\nments and Re4+and Fe2+. For the future synthesis of\nReFe 2O4, one can draw inspiration from the recently de-\nveloped green fabrication methods for ferrites [45].\nCONCLUSIONS\nUsing density functional calculations, considering spin-\norbit coupling, we predict that a Re-based ferrite\nReFe 2O4is stable in a distorted spinel structure with\nreduced triclinic symmetry ( P\u00161), and adopts an uncon-\nventional magnetic ordering in where Re spin opposes\nthe spin of both Fe Tetand Fe Oct, while Fe Tetand Fe Oct6\nhave parallel spin alignment among themselves. The net\nmagnetic moment of this compound is evaluated at 6.090\n\u0016B=f:u:which is about 1.5 times greater than that of\nmagnetite. The magnetic ground state is remarkably sta-\nble as \ripping any spin incurs an energetic cost of at least\n0.2223 eV/f.u., which is likely to correspond to an am-\nbient Curie temperature. The compound is predicted to\nbe half-metallic, which implies that this compound may\nbe useful towards applications in spintronics where the\nspin polarisation of conduction electrons is desired.\nCONFLICTS OF INTEREST\nThe authors declare that there is no con\rict of interest.\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge the funding of this\nproject by computing time provided by the Paderborn\nCenter for Parallel Computing (PC2).\nVERSION OF RECORD\nM. Hussein N. Assadi, Marco\nFronzi and Dorian A. H. Hanaor\nUnusual ferrimagnetic ground state in rhenium ferrite\nEur. Phys. J. Plus (2022) 137, 21. https:\n//doi.org/10.1140/epjp/s13360-021-02277-z\nREFERENCES\n[1] A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Pre-\njbeanu, B. Dieny, P. Pirro, and B. Hillebrands, J. Magn.\nMagn. Mater. 509, 166711 (2020). https://doi.org/\n10.1016/j.jmmm.2020.166711\n[2] R. K. Kotnala and J. Shah, in Handbook of Mag-\nnetic Materials, edited by K. H. J. 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Li, Nanoscale\nHoriz. 4 (2), 434 (2019). http://dx.doi.org/10.1039/\nC8NH00278A"    },    {        "title": "1102.4115v2.The_energetic_and_structural_properties_of_bcc_NiCu__FeCu_alloys__a_first_principles_study.pdf",        "content": "arXiv:1102.4115v2  [cond-mat.mtrl-sci]  25 Apr 2011Manuscript\nThe energetic and structural properties of bcc NiCu, FeCu al loys:\na first-principles study\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: November 2, 2018)\nAbstract\nUsing special quasirandom structures (SQS’s), we perform fi rst-principles calculations studying\nthe metastable bcc NiCu and FeCu alloys which occur in Fe-Cu- Ni alloy steels as precipitated\nsecond phase. The mixing enthalpies, density of state, and e quilibrium lattice parameters of these\nalloys are reported. The results show that quasi-chemical a pproach and vegard rule can well\npredict the energetic and structural properties of FeCu all oys but fail to yield that of NiCu. The\nreason rests with the difference of bond energy variation with composition between NiCu and FeCu\nalloys induced by competition between ferromagnetic and pa ramagnetic state. Furthermore, the\ncalculated results show that the energetic and structural p roperties of these alloys can well explain\nthe local composition of the corresponding precipitates in ferrite steels.\nPACS numbers: 64.60.My, 61.43.Bn, 64.75.Op, 71.20.Be\n1I. INTRODUCTION\nThe transitional metals and their alloys related to the magnetism att ract much scientific\ninterests. Enormous experimental and theoretical investigation s were dedicated to deeper\nunderstanding ofthenatureoftheirpropertieswhichbecamecom plicatedduetomagnetism.\nExamples of these systems can be provided by artificial crystalline s tructures fabricated by\nfilm growth techniques and precipitated second phase in matrix. Spe cially, many impor-\ntant properties of materials, such as their mechanical strength, toughness, creep, corrosion\nresistance, and magnetic properties are essentially controlled by p recipitated particles of a\nsecond phase. Therefore, understanding of these alloys is also de sirable from the views of\napplications.\nThe Cu-rich precipitates, which are commonly found in alloy steels, ha ve good strength-\nening effect on steels. The high-strength low-alloy steels strength ened by copper rich pre-\ncipitates also retain the impact toughness, corrosion resistance, and welding properties[1–6].\nHowever, it is also confirmed that the presence of Cu-rich precipita tes in reactor pressure\nvessel(RPV) steels is the origin of embrittlement [7], which limit the reac tor operating life.\nThe embrittlement effect can be enhanced by the content of Cu or N i of alloyed steels con-\ntaining of both Cu and Ni[8–12]. Thus, the mechanism of strengthenin g and embrittlement\nof Cu precipitates attract much interest[13–22]. The structure a nd composition of precip-\nitated phase also became important, and it had been studied by many experiments, such\nas atom probe field ion microscopy(APFIM) [23–25], small-angle neutr on scattering(SANS)\n[26], high-resolution and conventional electron microscopy(HREM a nd CTEM) [27, 28], etc.\nIt was observed that small Cu-rich precipitates with diameters less than about 5 nm have\na meta-stable body-centered cubic (bcc) structure and are coh erent with the α-Fe matrix\n[26–28] in the initial stage of segregation. In addition, it was also fou nd that Ni also occurs\nin the Cu-rich precipitates beside Cu and Fe. For HSLA steels, Ni was observed at the co-\nherent matrix/Cu-precipitate heterophase interfaces [29, 30, 3 6, 37]. For RPV steels, it was\nalso observed Ni appeared in the Cu-rich precipitates at very initial stage and was rejected\nfrom the core with the precipitates growth[21, 22, 33, 34].\nThe bcc FeCu alloy has been studied intensively by theoretical invest igations for well\nunderstanding of precipitated phase. First-principles calculations based on the cluster ex-\npansion framework gave the composition range of mechanical stab ility of FeCu random\n2alloy[35]. A thermodynamic equilibrium analysis was used to investigate t he composition\ndependence of Gibbs energy of FeCu alloy as well as the interfacial e nergy between precipi-\ntates and matrix[36]. In the meanwhile, the vibrational energies of d ilute FeCu alloys were\ninvestigated by first-principles calculations in combination with therm odynamical model-\ning, which show that the vibrational energies can stabilize the alloys [3 7]. The dependence\nof magnetism on the structural characteristics of Cu nucleation a nd the electronic struc-\nture for bcc Fe xCu1−x(x >0.75) systems were also well investigated[38]. However few\nsystematic investigation of bcc NiCu alloys was reported. The bcc NiC u, like bcc FeCu, is\nnot only a typical system related to magnetism in solids, but also beca me interesting be-\ncause of its important contribution to strengthening and embrittle ment of the ferrite steels.\nTherefore, systematic investigations of their properties at elect ronic level have both founda-\ntional and engineering significance. In this paper, the composition- dependent bcc Ni xCu1−x\nand Fe xCu1−xrandom alloys are investigated by employing special quasi-random st ructures\n(SQS’s) in the frame of first-principles calculations.\nII. MODELING AND THEORETICAL METHODS\nThe SQS method was proposed by Zunger and Wei et al to overcome the limitations\nof mean-field theories [41, 42], without the prohibitive computationa l cost associated with\ndirectly constructing large supper cells with random occupy of atom s. The SQS method\nwas extensively used to study the properties of semiconductor allo ys which are all fcc-based\nsystem. Recently, this method was also used into both fcc and bcc t ransition metal systems\nand was proved to be useful [43, 44]. For a binary substitutional allo y, many properties are\ndependent on its configuration. A binary AB substitutional alloy with a lattice of Nsites\nhas2Npossible atomicarrangements, denotedasconfigurations σ. Themeasurable property\n< E >represents an ensemble average over all 2Nconfiguration σ,< E >=/summationtextρ(σ)E(σ).\nThestructure σcanbediscretizedintoitscomponent figure f, whichischaracterizedbyaset\nofcorrelationfunctions Πk,m. Therefore, < E >canberewrite as < E >=/summationtextΠk,mε(f). The\nSQS’s are special designed Natoms periodic structures whose distinct correlation functions\nΠk,mbest match ensemble averaged /angbracketleftΠk,m/angbracketrightof random alloys.\nThe choice of SQS’s are critical for the calculations[45]. In this paper , SQS’s containing\n32 atoms are constructed for bcc alloys, which are shown in Fig. 1. T he vectors of lattice are\n3− →a1= (1.0,−2.0,0.0)a0,− →a2= (0.0,−4.0,2.0)a0,− →a3= (−2.0,0.0,−2.0)a0, respectively, and a0\nis the lattice parameter of the bcc unit cell. The occupations of thes e sites for the SQS’s at\nx= 0.25 and x= 0.5 are given in Table 1, and their structural correlations function Πk,m\ncompared with ideal random alloy correlation functions are given in Ta ble 2. As can be\nseen, the quality of the SQS’s used in these calculation is reasonably g ood.\nFirst-principles calculations are performed using the density funct ional theory[46–48] as\nimplemented in the Vienna ab initio simulation package(VASP)[49]. The ge neralized gra-\ndient approximation(GGA)[50, 51] is used for the exchange correla tion functional. The\ninteraction between core and valence electrons is described with th e projector augmented\nwave (PAW) potential [52, 53]. The equilibrium structures were det ermined, up to a preci-\nsion of 10−4eV in total-energy difference and with a criterion that required the f orce on each\natom to be less than 0.01 eV/ ˚A in atomic forces. The convergence tests with k-point and\nenergy cutoff are presented in table 3. The k-point meshes 6 ×2×4 for the Brillouin zone\nintegration are used for SQS’s, which is equivalent k-point sampling wit h Monkhorst-Pack\nmesh 11×11×11 for bcc primitive cell. An energy cutoff of 280 eV is applied in all cases .\nIII. RESULTS AND DISCUSSIONS\nFirstly, basic properties of bcc Fe, Ni, Cu are presented. The lattic e constant and the\nmagnetic moment for bcc Fe at the equilibrium volume are 2.83 ˚A and 2.16 µB/atom,\nrespectively. The lattice constant and bulk modulus for bcc Cu at th e equilibrium volume\nare 2.89 ˚A and 130 GPa. These results are in agreement with earlier studies[35 , 39]. The\nmagnetic moment for bcc Ni at the equilibrium structure with lattice o f 2.81˚A is 0.55\nµB/atom. The calculated result is well consistent with previous works u sing GGA, which\npredicted bcc Ni is ferromagnetic[54, 55]. Previous theoretical wo rks using local density\napproximation (LDA) predicted bcc Ni is paramagnetic[39]. Howeve r, it is confirmed that\nbcc Ni do possesses a magnetic moment of 0.52 µB/atom by experiments more recently[56].\nIn addition, it was proved that the GGA calculations can describe sat isfactorily lattice and\nelastic constants of Ni in their observed structures[55].\nThe mixing enthalpies( △H) of bcc NiCu and FeCu random alloys obtained by first-\nprinciples SQS method are shown in Fig .2. The mixing enthalpy is heat qua ntity ab-\nsorbed(or evolved) during mixing two elements to make homogeneou s solid solution, which\n4is defined as:\n△H=EAB−xEA−(1−x)EB (1)\nwhereEA,EB, andEABare the total energy of A, B, and AB alloy, and xis the compo-\nsition of alloys. In the frame of quasi-chemical approach[57], the mix ing enthalpy is only\ndetermined by the composition-independent bond energy between adjacent atoms. Hence,\nmixing enthalpy can be written as:\n△H= Ωx(1−x), (2)\nwhere Ω = Naz(εA−B−1\n2(εA−A+εB−B)) is interaction parameter, N ais Avogadro’s number,\nεisbondenergy, and zisthenumber ofbondsperatom. Thus, thecomposition-independe nt\nbond energy between atoms would result a parabolic variation for mix ing enthalpies with\ncompositions. It can been seen from Fig. 2, the variation of mixing en thalpies of FeCu\nwith composition is parabolic, while that of mixing enthalpies of NiCu exhib it a strong\nasymmetry and change sign as function of concentration. These e ffect were also found for\nrandom fcc NiCu alloys in previous calculations[40].\nTo study the effect of degree of disorder on the mixing enthalpies of NiCu, we calculate\nthe ordered structures comparing disordered structures. Thr ee 4-atoms-super-cells are used\nto simulate order alloys withcomposition x=0.25, 0.50, 0.75. The mixing enthalpies ofthese\norder alloys are represented by solid squares in Fig. 2(a). It shows the mixing enthalpies of\nthis ordered structures of NiCu are all lower than those of disorde red alloy with the same\nextent, and the trend of mixing enthalpies variations with compositio ns are the same as that\nof disorder alloys. The mixing enthalpies of order alloy also exhibit a str ong asymmetry.\nTherefore, the degree of disorder is not the reason of the asymm etry of mixing enthalpies\nfor NiCu alloys.\nWe turn to investigate bonding property of NiCu and FeCu from elect ronic structures\nfor understanding strong asymmetry variation of mixing enthalpies with composition. The\nbond energy is critical factor that determines the mixing enthalpies . The variation of the\nbond energy can be reveal from comparison of the density of stat es(DOS) of Fe xCu1−xand\nNixCu1−x. Since the bond energies of transitional metal are determined by t he coupling of\nd-band, the d-bands of Cu, Fe, Ni in Fe xCu1−xand Ni xCu1−xwith different compositions are\nplotted in Fig 3. One can easily understand the difference of mixing ent halpies magnitude\nbetween FeCu and NiCu from electronic structure. In comparison w ith the occupation of\n5Ni, the occupation of Fe in the alloy is changed significantly from that o f pure Fe(see DOS\naround 2 eV in Fig. 3). Therefore, the energy of Fe-Cu bond is much larger than that\nof Ni-Cu bond, and the mixing enthalpies of CuFe are much larger than those of CuNi.\nFurthermore, bond energies in alloys vary with compositions can be a lso reflected by the\nDOS. The DOS peaks of Ni vary with composition, while that of Fe in alloy s with different\ncompositions are the same. These findings indicate that bond energ yεof Ni-Cu varies\nwith composition. It can be known from equation (2), the variation o f bond energy induce\ncomposition-dependent interaction parameter Ω, which result tha t the variation of mixing\nenthalpy with composition is not parabolic. These are consistent with calculated results in\nFig 2(a). Hence, the complex variation trend of mixing enthalpies of N iCu derive from the\ncomposition-dependent bond energy.\nIn addition, one can infer that the magnetic moment of Ni atom varie s with the com-\nposition and the magnetic moment of Ni 0.25Cu0.75disappears. A more clear presentation of\nmagnetic moment of alloys with different composition is given in Fig. 4. Th ese phenomena\ncan be well understand by using stoner criterion which succeeds in e xplaining the magnetic\nproperty of transitional metal alloys dictated by the filling of the d- band[39]. The Stoner\ncriterion states that ferromagnetism appears when the gain in exc hange energy is larger\nthan the loss in kinetic energy. Therefore, there is always a compet ition between ferro-\nmagnetic(FM) and paramagnetic(PM) solutions, and magnetic prop erties are determined\nby the state which has lowest energy. The DOS at the Fermi level is a key factor to affect\nthe competition between FM and PM state. A larger peak of DOS at Fe rmi level induces a\nlarger exchange energy and prefers to split into FM state. Fig. 5 illus trates the unpolarized\nDOS of NiCu and FeCu alloys. It is shown in Fig. 4 and Fig. 5, the trend of magnetic\nmoment of the system well consistent with the variation of DOS at Fe rmi level.\nFor these alloys, the d electrons increase with the Cu concentratio n, which result the\nFermi level of Ni(Fe) be promoted. The key point is, as shown in Fig. 5 , since the DOS\nat Fermi level of NiCu is much smaller than that of FeCu, the Fermi lev el of NiCu change\nmuch rapidly with Cu concentration than that of FeCu which induce th at the peak of DOS\nof NiCu is much sensitive to the alloy composition than that FeCu. Henc e, the magnetic\nmoment of Ni change with the variation of DOS at Fermi level when th e Cu concentration\nis changed, and it disappears when the DOS at the Fermi level is 0. No w, we can conclude\nthat the strong asymmetry of mixing enthalpies of NiCu with composit ion is induced by\n6band shift because of the competition between FM and PM state.\nIt is found that the composition-dependent electronic interaction s of NiCu alloys also\naffect the structural property. As shown in Fig. 6, the calculated volume per atom of FeCu\nalloy well obeys the vegard law, while that of NiCu deviates from what t he vagard law\npredicts. According to the vegard law[58], the lattice constant ha s a linear relationship with\nits composition, i.e. a(AxB1−x) =xa(A)+(1−x)a(B). This implies that the atom can be\nconsidered as rigid body whose volume is unchanged with its chemical e nvironment. How-\never, the composition-dependent electronic interaction in NiCu res ult the atomic volumes\ndepend its composition, which induces bowing effect.\nAs shown in Fig. 7, the partial atomic pair correlation functions give m ore detailed\ninformation of the alloy structures. It can show that how large is th e degree of distortion of\nequilibrium alloy structure from the perfect bcc structure. For bc c structure, the numbers\nof 1st, 2nd, 3rd, 4th, 5th neighbors are 8, 6, 12, 24, 8. As is show n in Fig. 7, both NiCu and\nFeCu alloys are still bcc structure and the lattice just has a minor ch ange, but the NiCu\nalloy is more close to a perfect bcc structure. Since the nominal num ber of minority d-band\nholes of Fe is more than that of Ni by 2 electrons, the partial charg e effect of Fe from Cu\nis stronger than that of Ni. The bond length can be affected by part ial charge. Hence, the\ndistance between atomic neighbors of FeCu more sensitive to their c hemical environment\nthan NiCu, which induce a larger distortion of FeCu from bcc.\nThe strain effect is also very important for the precipitated phase in ferrite matrix. We\nhave calculated the strain energy with lattice constant from 2.80 ˚A to 2.90 ˚A as shown in\nFig. 8, which covers the lattice constant range of bcc Fe and Cu. It is shown that NiCu\nalloys have smaller strain energy at lattice of bcc Fe, while FeCu alloys h ave smaller strain\nenergyat latticeofbccCu. Ata latticeconstant ofFematrixwith2.8 3˚A,thestrainenergies\nare 6.52, 0.78, 0.51 meV/atom for Ni xCu1−x(x=0.25, 0.5, 0.75) alloys, and 20.71, 13.51, 7.13\nmeV/atom for Fe xCu1−x(x=0.25, 0.5, 0.75). The strain energy of bcc Cu in ferrite matrix is\n21.6 meV/atom. These indicate that Ni and Fe can lower the bcc Cu st rain energy in ferrite\nmatrix, while the Ni is better that Fe in role of reducing strain energy .\nFrom the energetic and structural properties of alloys, the prec ipitated phase can be well\nunderstood. For HSLA steels, it was observed that NiCu alloys at th e matrix/precipitates\nheterophase interfaces area[29]. The calculated results show tha t the mixing enthalpies of\nFeCu are positive, which indicate that the FeCu alloys are unstable an d Cu atoms prefer\n7to segregate to form precipitates. The mixing enthalpies of NiCu alloy s are negative when\nNi concentrations below 30 at. %, indicating that NiCu alloys with lower Ni concentration\nare energetic favorable. The element of Ni appears in the matrix/p recipitates heterophase\ninterfaces area, which is mainly induced the factors relate to strain : i) The volume per\natom of NiCu is similar to that of Fe matrix rather than pure Cu, even r ather than that\nof FeCu alloy, and the stain energy can be much lower by Ni. ii) The stru cture of NiCu\nis more close to the bcc structure, which induce little mismatch of latt ice. These indicate\nthat NiCu alloys in the interface of matrix/precipitate can contribut e lower interface energy\nfrom reducing shear stress. However, the experimental peak co ncentration of Ni is about 3\nat. % in the Cu-rich precipitates which is lower than the Ni concentra tion of most favorable\nNiCu alloy with concentration of 20 at. %. This indicates that the distr ibution of elements\ndistribution may be affected by the size effects and vibrational entr opy in addition. In RPV\nsteels, the Ni in Cu-precipitates be rejected from the core after thermal-aging or neutron-\nirradiated[33, 34], and the experimental peak concentration of Ni is among 15-20 at. %,\nwhich is well consistent the composition of energetic favorable alloys more closed. These\nindicate that the element distribution of bcc precipitated phase at v ery initial of segregation\nstage can be well understood from the energetic and structural properties.\nIV. CONCLUSION\nIn summary, we propose three 32-atom SQS supercells to mimic the p air and multisite\ncorrelation functions of random CuFe and NiCu bcc substitutional a lloys which occur in\nFe-Cu-Ni alloy steels as precipitated second phase. Those SQS’s ar e used to calculate the\nmixing enthalpies, density of state, and lattice parameters of the m etastable random alloys.\nThe results show that quasi-chemical approach and vegard rule ca n satisfactorily predict the\nmixing enthalpies andstructure parameters of FeCualloys but failt o accurately yield that of\nNiCu. As can be obtained from the analysis of electronic structure, the magnetism induced\nbond energy variation with composition is the reason that quasi-che mical approach and\nvegard rule fail to predict the properties of NiCu alloys. Furthermo re, the calculated results\ncan well explain the previous experimental observation of local com position of coherent\nCopper-rich precipitates containing Nickel and confirm that segre gation of Ni is drove by\nthermaldynamic and chemical factor. These suggest that the pro perties of random alloys\n8have important implications to better understand the multi-compon ent precipitates.\nIn this work, we present the intrinsic bulk properties of metastable bcc FeCu and NiCu\nrandomalloysto understand thestructure ofCu-rich precipitate s. Since Cu-rich precipitates\nhave important roles on properties of alloy steels, understanding t he formation mechanism\nis highly desirable for altering the properties of steels by controlling C u-rich precipitates.\nFurther investigation would simulate realistic interface to calculate t he interfacial energy\nof precipitates/matrix and the diffusion properties of Cu and Fe ato m affected by Ni shell,\nwhich will clarify the role of Nickel on the evolution of the Cu-rich prec ipitates.\nThe authors thank G. Xu and Prof. B. X. Zhou for critical discussio ns. Y. P. Xie also\nthanks J. H. Yang and Prof. X. G. Gong for insightful discussion. T his work is finan-\ncially supported by National Science Foundation of China (Grant No. 50931003, 51001067),\nShanghai Committee of Science and Technology (Grant No. 095205 00100) , Shu Guang\nproject (Grant No. 09SG36) supported by Shanghai Municipal Ed ucation Commission and\nShanghai Education Development Foundation, and Shanghai Lead ing Academic Discipline\nProject (S30107). The computations were performed at Ziqiang S upercomputer Center of\nShanghai University and Shanghai Supercomputer Center.\n[1] M. E. Fine, R. Ramanthan, S. Vaynman and S. P. Bha Mechanic al properties and microstruc-\nture of weldable high performance low carbon steel containi ng copper. In International Sym-\nposium on Low-Carbon Steels for the 90’s, R. I. Asfahani and G . Tither (Eds.), Pittsburgh,\nPennsylvania: TMS. 1993, pp. 511-514\n[2] S. Vaynman, M. E. Fine, G. Ghosh, and S. P. Bhat, Copper pre cipitation hardened, high-\nstrength, weldablesteel. 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Phys. 5 (1921) 17\n12TABLE I: Atomic coordinates and occupation of the 32 atoms bc c SQS.\nAtomic coordinates\nAB\nA: 0.0 -4.0 0.0; 0.0 -2.0 0.0; -0.5 -0.5 -0.5; -1.0 -4.0 -1.0\nA: -0.5 -1.5 -0.5; 0.0 -1.0 0.0; -1.0 -3.0 -1.0; 0.5 -4.5 1.5\nA: 0.5 -5.5 1.5; -1.5 -2.5 -0.5; -1.5 -3.5 -0.5; -0.5 -2.5 0.5\nA: -1.0 -6.0 0.0; -0.5 -3.5 0.5; 0.0 -3.0 1.0; -1.0 -5.0 0.0\nB: -0.5 -4.5 0.5; -0.5 -5.5 0.5; 0.0 -5.0 1.0; 0.5 -2.5 0.5\nB: 0.5 -3.5 0.5; -1.5 -0.5 -1.5; -1.5 -1.5 -1.5; -1.0 -1.0 -1.0\nB: -1.0 -2.0 -1.0; -0.5 -2.5 -0.5; -0.5 -3.5 -0.5; 0.0 -3.0 0.0\nB: 0.0 -6.0 1.0; 0.0 -4.0 1.0; -1.0 -3.0 0.0; -1.0 -4.0 0.0\nAB3\nA: 0.0 -2.0 0.0; 0.5 -2.5 0.5; 0.5 -3.5 0.5; 0.0 -1.0 0.0\nA: -0.5 -3.5 -0.5; 0.0 -3.0 0.0; 0.0 -4.0 1.0; -1.5 -2.5 -0.5\nB: 0.0 -4.0 0.0; -1.5 -0.5 -1.5; -1.5 -1.5 -1.5; -1.0 -1.0 -1.0\nB: -0.5 -0.5 -0.5; -1.0 -4.0 -1.0; -1.0 -2.0 -1.0; -0.5 -1.5 -0 .5\nB: -1.0 -3.0 -1.0; -0.5 -2.5 -0.5; 0.0 -6.0 1.0; 0.5 -4.5 1.5\nB: 0.5 -5.5 1.5; -1.5 -3.5 -0.5; -1.0 -3.0 0.0; -0.5 -2.5 0.5\nB: -1.0 -6.0 0.0; -1.0 -4.0 0.0; -0.5 -3.5 0.5; 0.0 -3.0 1.0\nB: -1.0 -5.0 0.0; -0.5 -4.5 0.5; -0.5 -5.5 0.5; 0.0 -5.0 1.0\n13TABLE II: Comparisonof correlation function Πk,mof 32 atoms bccSQSandideal value of random\nalloy.\nAB AB 3\nrandom SQS random SQS\nΠ2,1 0 0 0.25 0.25\nΠ2,2 0 0 0.25 0.25\nΠ2,3 0 0 0.25 0.25\nΠ2,4 0 0 0.25 0.25\nΠ2,5 0 0.17 0.25 0.04\nΠ2,6 0 0 0.25 0.25\nΠ3,2 0 0.03 0.13 0.13\nΠ4,2 0 0 0.06 0.08\nTABLE III: Total energies of bcc primitive cell of Fe, Ni and C u as a function of the number of k\npoints and the cutoff energy.\nMonkhorst-Pack mesh energy cutoff(eV) bcc Fe bcc Cu bcc Ni\n3×3×3 280 -8.15 -4.13 -5.27\n5×5×5 280 -8.12 -3.71 -5.33\n7×7×7 280 -8.17 -3.70 -5.39\n9×9×9 280 -8.17 -3.70 -5.37\n11×11×11 280 -8.17 -3.70 -5.37\n13×13×13 280 -8.17 -3.70 -5.37\n11×11×11 180 -6.99 -1.97 -4.05\n11×11×11 260 -8.15 -3.68 -5.36\n11×11×11 300 -8.17 -3.70 -5.37\n14FIG. 1: Crystal structures of the SQS-32 supercell in their u nrelaxed forms. Blue and red spheres\nrepresent A and B atoms, respectively. (a) AB (b) AB 3\nFIG. 2: Mixing enthalpy of alloys: (a) disorderedand ordere d NixCu1−x, (b) disordered Fe xCu1−x.\nFIG. 3: Comparison of the d-band between pure metals and allo ys. The d-band of Ni, Fe, Cu in\npure Ni, Fe, Cu, NiCu alloy, FeCu alloy are plot in each panel.\n15/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s32/s32/s70/s101\n/s120/s67/s117\n/s49/s45/s120\n/s32/s32/s78/s105\n/s120/s67/s117\n/s49/s45/s120/s77/s101/s103/s110/s101/s116/s105/s99/s32/s77/s111/s109/s101/s110/s116/s32 /s77 /s40\n/s98/s47/s97/s116/s111/s109/s41\n/s65/s116/s111/s109/s105/s99/s32/s70/s114/s97/s99/s116/s105/s111/s110/s32\n/s32\nFIG. 4: Magnetic moments per atom in NiCu and FeCu alloys.\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50\n/s69/s110/s101/s114/s103/s121 /s40/s101/s86/s41/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s115/s116/s97/s116/s101/s115\n/s69/s110/s101/s114/s103/s121 /s40/s101/s86/s41/s45/s54 /s45/s52 /s45/s50 /s48 /s50/s69\n/s70/s69\n/s70\nFIG. 5: The unpolarized calculated density of state(DOS) of NiCu and FeCu alloys. The solid,\ndash, dot line denote for Ni(Fe) xCu1−x,x= 0.25,0.50,0.75, respectively.\n16FIG. 6: Atomic volume versus composition in bcc Ni xCu1−xand Fe xCu1−xalloys. The circles and\nsquares denote the calculated results of SQS method. The sol id lines denote the results predicted\nby vegard rule.\nFIG. 7: Partial atomic pair correlation function gij(r) in NiCu and FeCu alloys.\n17/s50/s46/s56/s48 /s50/s46/s56/s50 /s50/s46/s56/s52 /s50/s46/s56/s54 /s50/s46/s56/s56 /s50/s46/s57/s48/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s48/s46/s48/s49/s53/s48/s46/s48/s50/s48/s48/s46/s48/s50/s53/s48/s46/s48/s51/s48/s48/s46/s48/s51/s53/s48/s46/s48/s52/s48\n/s32/s78/s105\n/s48/s46/s50/s53/s67/s117\n/s48/s46/s55/s53\n/s32/s78/s105\n/s48/s46/s53/s67/s117\n/s48/s46/s53\n/s32/s78/s105\n/s48/s46/s55/s53/s67/s117\n/s48/s46/s50/s53/s83/s116/s114/s97/s105/s110/s32/s101/s110/s101/s114/s103/s121/s40/s101/s86/s47/s97/s116/s111/s109/s41\n/s76/s97/s116/s116/s105/s99/s101/s32/s99/s111/s110/s115/s116/s97/s110/s116/s40 /s197 /s41/s50/s46/s56/s48 /s50/s46/s56/s50 /s50/s46/s56/s52 /s50/s46/s56/s54 /s50/s46/s56/s56 /s50/s46/s57/s48\n/s76/s97/s116/s116/s105/s99/s101/s32/s99/s111/s110/s115/s116/s97/s110/s116/s40 /s197 /s41/s32/s70/s101\n/s48/s46/s50/s53/s67/s117\n/s48/s46/s55/s53\n/s32/s70/s101\n/s48/s46/s53/s67/s117\n/s48/s46/s53\n/s32/s70/s101\n/s48/s46/s55/s53/s67/s117\n/s48/s46/s50/s53\nFIG. 8: The strain energy of NiCu and FeCu as a function of latt ice constant.\n18"    },    {        "title": "0712.2305v1.Electric_self_inductance_of_quasi_2D_magnetic_dipolar_mode_ferrite_disks.pdf",        "content": "Electric self inductance of quasi-2D magnetic-dipolar-mode ferrite disks \n \nM. Sigalov, E.O. Kamenetskii, and R. Shavit \n \nBen-Gurion University of the Negev, Beer Sheva 84105, Israel \n \nDecember 12, 2007 \n \n   Abstract \n \nAn electric current flowing ar ound a loop produces a magnetic field and hence a magnetic flux \nthrough the loop. The ratio of the magnetic flux to the electric current is called the (magnetic) \nself inductance. Can there be a dual situation with a magnetic current flowing around a loop and \nproducing an electric field and hence an electric  flux through the loop? Following the classical \nelectrodynamics laws an answer to this quest ion should be negative . Nevertheless special \nspectral properties of magnetic dipolar modes in  a quasi-2D ferrite di sk show there are the \ndouble-valued-function loop magnetic  currents which may produce eigen electric fields and \nhence eigen electric fluxes through the loop. In th is case one can definitely introduce the notion \nof the electric self i nductance as the ratio of the electric flux to the magnetic current. In this \npaper we show experimentally th at in the magnetic-dipolar-mode ferrite disks there exist eigen \nelectric fluxes. These fluxes are very sensitive to  permittivity parameters of materials abutting to \nthe ferrite disk. Dielectric samples above a ferrite disk with a higher permittivity than air confine \nthe electric field clos ely outside the ferrite, thereby cha nging the loop magnetic currents and \nthus transforming the magnetic-dip olar-mode oscillating spectrum. \n PACS numbers: 76.50.+g, 68.65.-k, 03.65.Vf, 41.20.Jb   \n                      21. Introduction \n \nMagnetostatic (MS) ferromagnetism has a charac ter essentially different from exchange \nferromagnetism [1, 2]. This statement finds str ong confirmation in confinement phenomena of \nmagnetic-dipolar-mode (MDM) oscillations. The dipole interaction provides us with a long-\nrange mechanism of interaction, where a magnetic medium is considered as a continuum. In a \ngeneral case, both short-range exchange an d long-range dipole-di pole (magnetostatic) \ninteractions contribute to eigenf requencies of the collective sp in excitation. The importance of \nMS energy increases gradually as the particle  size increases. MDMs ar e not eigen modes of \nmagnetization. Contrary to an exchange spin wave, in magne tic-dipolar waves the local \nfluctuation of magnetization does not propagate  due to interaction between the neighboring \nspins. When field differences across the sample become comparable to the bulk demagnetizing fields the local-oscillator approximation is no longer valid and ther e should be certain \npropagating fields – the MS fi elds – which cause and govern  propagation of magnetization \nfluctuations. In other words, space-time ma gnetization fluctuations are corollary of the \npropagating MS fields, but th ere are no magnetization waves.  The boundary conditions should \nbe imposed on the MS field and not on the RF magnetization.     Ferrite MDM samples are well localized in space, their extension is assumed to be much \nsmaller than the variation length of  the electromagnetic field. It is well known that in a general \ncase of small (compared to the free-space electromagnetic-wave wavelength) samples made of media with strong temporal disp ersion, the role of di splacement currents in Maxwell equations \ncan be negligibly small, so oscillating fields are the quasistationary fields. In small samples with \nstrong temporal dispersion of the permeability tensor: \n)( ωµµtt= , variation of the electric energy \nis negligibly small compared to variation of  the magnetic energy and so one can neglect the \nelectric displacement current in Maxwell equati ons [3]. These magnetic samples can exhibit the \nMS resonance behavior in microwaves [4 – 7]. For such resonance MDMs, the \nquasimagnetostatic fields are describe d by a scalar magneto static potential ψ. This potential \ndoes not have the same physical meaning as in a situation of pure magnetost atics. In fact, there \nis the MS-potential wave  function ) ,(trrψ  describing the resonant beha vior in a small magnetic \nobject. \n    Can the MS-potential wave function be  used for comprehensiv e description of MDM \noscillations in small magnetic samples? In attempts to use the notion of scalar wave function \n),(trrψ  for the spectral analysis one becomes faced  with evident contradictions with the \ndynamical Maxwell equations. This fact can be per ceived, in particular, from the remarks made \nby McDonalds [8]. For MS resonances in sma ll magnetic objects one ne glects the electric \ndisplacement current: 0 =∂∂\ntDr\n. From Maxwell equati on (the Faraday law), tB\ncE1\n∂∂−=×∇rr\n, one \nobtains 22\n 1\n tB\nc tE\n∂∂−=∂∂×∇r r\n. If a sample does not posses any dielectric anisot ropy, we have \n0 22\n=∂∂\ntBr\n. It follows that the magnetic field in small resonant magnetic objects vary linearly with \ntime. This leads, however, to arbitrary large fiel ds at early and late ti mes, and is excluded on \nphysical grounds. An evident conclusion suggests  itself at once: the magnetic (for MS \nresonances) fields are constant qua ntities. This contradicts to the fact of temporally dispersive \nmedia and any resonant conditions . Another conclusion is more unexpected: for a case of MS \nresonances the Faraday law is not valid. Con cerning the MS-wave propagation effects, it was \ndisputed also in [9 – 11] that from a classica l electrodynamics point of view one does not have a \nphysical mechanism describing the effect of tran sformation of the curl electric field to the \npotential magnetic field. Also th e gauge transformation in this de rivation does not fall under the  3known gauge transformations, neither the Lore ntz gauge nor the Coulomb gauge, and cannot \nformally lead to the wave equation.      MS waves can propagate only due to ferr ite-medium confinement phenomena [12, 13]. So the \npower flow density of MS waves should be cons idered via an analysis of the mode spectral \nproblem in a certain waveguide st ructure, but not based on an an alysis of the wave propagation \nin a boundless magnetic medium. It means that the MS-wave power flow density should have \nthe only physical meaning as a norm of a cer tain propagating mode in a magnetic waveguide \nstructure. As a necessary consequence, this leads to the question of orthogonality and \ncompleteness of MDMs in such a waveguide . In this problem MS-potential function \nψ acquires \na special physical meaning as a scalar wave functi on in a Hilbert functional space. In an analysis \nof the MDM oscillating spect ra, a ferrite-disk particle is consid ered as a section of an axially \nmagnetized ferrite rod. For a flat ferrite disk , having a diameter much bigger than a disk \nthickness, one can successfully use separation of variables for the MS-potential wave function \n[14, 15]. A similar way of separation variab les is used, for example, in solving the \nelectromagnetic-wave spectral proble m in dielectric disks [16].  \n    It was shown [10, 11, 14, 15] that for MD Ms in a ferrite disk one has evident quantum-like \nattributes. The spectrum is characterized by energy eigenstate  oscillations. The energy \neigenvalue problem is defined by the differential equation: \n \n                                                                    pp pE F ϕϕ~ ~=⊥),                                                          (1)                \n \nwhere pϕ~ is a dimensionless membrane MS-potential wave function and pE is the normalized \naverage (on the RF period) density of  accumulated magnetic energy of mode p. A two-\ndimensional (“in-plane”)  differential operator ⊥Fˆ in Eq. (1) is defined as: \n \n                                                                    2 16ˆ\n⊥ ⊥ ∇=µπpgF ,                                                       (2)                 \n \nwhere 2\n⊥∇ is the two-dimensional (with respect to cr oss-sectional coordinate s) Laplace operator, \nµ is a diagonal component of the permeability tensor, and pg is a dimensional normalization \ncoefficient for mode p. Operator ⊥Fˆ is a positive definite differential operator for negative \nquantity µ. The energy density of mode p is determined as \n \n                                                                    ()2\n16pzp\npgE βπ= ,                                                     (3)                \n \nwhere \npzβ is the propagation constant of mode p along disk axis z. For MDMs in a ferrite disk \nat a constant frequency one has the energy orthonormality condition: \n  \n                                                               0~~) ( = −∗\n′ ′∫dS E Epp\nSp p ϕϕ ,                                             (4)       \n \nwhere S is a cylindrical cross section of an open di sk. For constant frequency, different mode \nenergies one has at different quantities of a bias magnetic field. From the principle of \nsuperposition of states, it follows that wave functions pϕ~ ( ,... 2,1=p ), describing our \"quantum\" \nsystem, are \"vectors\" in an abstract space of an  infinite number of dimensions – the Hilbert  4space. In quantum mechanics, this is the case of  so-called energetic representation, when the \nsystem energy runs through a discrete sequence of values [17, 18].      It was shown, however, that because of th e boundary condition on a late ral surface of a ferrite \ndisk, membrane functions \npϕ~ cannot be considered as single- valued functions [10, 11]. This \nfact raises a question about validity of the energy orthogonality relation for the MDMs. The \nmost basic implication of the ex istence of a phase factor in ϕ~ is operative in the case on the \nborder ring region. In order to  cancel the \"edge anomaly\", th e boundary excitation must be \ndescribed by chiral states. Because of such chiral states, the double-valued-function loop \nmagnetic currents occur. These currents may pr oduce eigen electric fields and hence eigen \nelectric fluxes through the loop [ 10, 11]. In this case one can de finitely introdu ce the notion of \nthe electric self inductance as  the ratio of the electric flux to the magnetic current.  \n    In this paper we show experimentally that  in a quasi-2D ferrite disk  with MDM oscillations \nthere exist eigen electric fluxes. One has an ev idence for strong sensitivity of the MDM spectral \ncharacteristics to the dielectric-load permitti vity quantity. The oscillating MDMs can be \ncharacterized by certain electric self-inductance parameters. We star t with a brief description of \nsuch non trivial notions as eigen electric fl uxes, persistent magnetic  currents and anapole \nmoments in MDM ferrite disks . These notions are partly known from the literature. \nNevertheless, an initial classification of uniqu e topological effects in the MDM ferrite disks \ncould be very useful for proper characte rization of our experimental results.  \n  \n2. Eigen electric fluxes, persistent magnetic currents and anapole moments in MDM ferrite \ndisks \n \nThe topological effects in the MDM ferrite di sk are manifested through the generation of \nrelative phases whic h accumulate on the boundary wave functions ±δ [10, 11]:  \n \n                                                                       θδ±−\n±±≡iqef ,                                                        (5) \n \nwhere θ is an azimuth coordinate in a cylindr ical coordinate system. The quantities ±q are \nequal to 21l± , ... ,5 ,3 ,1=l  For amplitudes  f we have − +−=f f  with normalization ±f = 1. To \npreserve the single-valued nature  of the membrane functions of  the MDM oscillations, functions \n±δ must change its sign when a disk angle coordinate θ is rotated by π2 so that 12−=−πmiqe . A \nsign of a full chiral rotation, πθ=+q  or πθ−=−q , should be correlated with a sign of the \nparameter aiµ – the off-diagonal component of the permeability tensor µt. This becomes \nevident from the fact that a sign of aiµ is related to a precession di rection of a magnetic moment \nmr. In a ferromagnetic resonance, the bias field sets up a preferential pr ecession direction. It \nmeans that for a normally magnetized ferrite disk with a given direction of a DC bias magnetic \nfield, there are two types of re sonant oscillations, which we c onventionally designate as the (+) \nresonance and the (–) resonance. For the (+) resonance, a directi on of an edge chiral rotation \ncoincides with the precession magnetization directi on, while for the (–) resonance, a direction of \nan edge chiral rotation is opposite to the precession magnetization direction.  \n    For a ferrite disk with r and θ in-plane coordinates, the total MS-potential membrane \nfunction ϕ~ is represented as a product of two functions [10, 11]: \n \n                                                                 ± =δθηϕ  ),(~~r ,                                                           (6)                  \n \nwhere ),(~θηr is a single-valued membrane function, and ±δ is a double-valued edge (spin-\ncoordinate-like) function. We may introduce a \"spin variable\" σ, representing the orientation of  5the \"spin moment\" and two double-valued wave functions, )(σδ+  and )(σδ− . The two wave \nfunctions are normalized and mutually or thogonal, so that they satisfy the equations                \n1  )(2=∫+σσδ d ,1  )(2=∫−σσδ d , and 0  )( )( =− +∫σσδσδ d . A membrane wave function ϕ~ is \nthen a function of three coordinates,  two positional coordinates such as , ,θr and the \"spin \ncoordinate\" σ. For the positional wave function ) ,(~θηr, which is a solution of the Walker \nequation for a ferrite disk with the so-called essential boundary conditions [15], there could be \ntwo equiprobable solutions for the membrane wave functions: )( ),(~ ~σδθηϕ+ +=r  and \n)( ),(~ ~σδθηϕ− −=r . \n    The geometrical phase factor is not singl e-valued under continuati on around a circuit and can \nbe correlated with the vector potential. The vector  potential is considered  to be nonobservable in \nMaxwellian electromagnetism. At the same time, the vector potential can  be observable in the \nAharonov-Bohm [19] or Aharonov-Casher [20] e ffects, but only via its  line integral, not \npointwise. From an analysis in [10, 11] it fo llows that restoration of singlevaluedness (and, \ntherefore, Hermicity) of the MDM spectral problem in a ferrite disk is due to a line integral \n() dC i\nC )(*∫±±∇or\nδδθ , where ℜ=  2π C  is a contour surrounding a cylindrical ferrite core, ℜ is a \nferrite disk radius, and o\n±δ are the boundary wave functions for the conjugate problem. Such a \nline integral is an observable quant ity which can be represented as an integral of a certain vector \npotential:   \n                                       \n()± ± ℜ=±± =⋅≡ ∇ℜ∫∫q Cd A d i\nCm\nr π θδδπ\nθ θ 2 ]))( [(2\n0*rr ro                                   (7) \n \nOne can see that in the problem under considerati on the Berry's phase [21] is generated from the \nbroken dynamical symmetry. To compensate for sign ambiguities and thus to make wave functions single valued we adde d a vector-potential-type term to the MS-potential Hamiltonian. \nA circulation of vector \nmAθr\n should enclose a certain flux. The corresponding flux of pseudo-\nelectric field ∈r (the gauge field) thr ough a circle of radius ℜ is obtained as: \n \n                                                       ()()()± ± ± Ξ=⋅=⋅∈∫∫e\nSCmCd A Sdrrrr\nθ ,                                         (8) \n \nwhere ()±Ξe is the flux of pseudo-electric field. There should be the positive and negative \nfluxes. These different-sign fluxes should be ine quivalent to avoid the cancellation. Superscript \nm in the vector-potential term means that there is the physical quantity associated with the \nmagnetization motion. A magnetic  moment moving along contour C in the gauge field feels no \nforce and undergoes the Aharonov-Bo hm-type interference effect.  \n    For every MDM, the total solution of the MS-potential wave function in a cylindrical ferrite \ndisk is represented as   \n                                                               ) ,(~)( ),,( θϕξθψ rz zr= ,                                               (9) \n \nwhile a single-valued membrane function is represented as  \n \n                                                                     ()()()θφθη rR r=,~.                                               (10)  \n         6In these equations ) (zξ  describes an axial distribution of the MS-potential wave function, ) (rR  \nis described by the Bessel functions and θνθφ ~)(ie−, .... 3,2,1±±±=ν   \n    Based on Eqs. (7) – (10) and taking into acco unt an analysis of self-adjointness of the MDM \ndifferential operators [10, 11], we can represent the total flux of pseudo-e lectric field for every \nMDM in a ferrite disk as   \n                                                          \n  )(  2\n02∫ℜ=±=Ξd\nre\ntotal dzz Rqξ π ,                                           (11) \n \nwhere d is a disk thickness. \n    In the topological effects of the generation of relative phases which accumulate on the \nboundary wave function ±δ, the quantity ±∇δθ can be considered as the velocity of an \nirrotational \"border\" flow:  \n \n                                                                 ()± ±∇≡δθ θrrv .                                                           (12)                 \n \nIn such a sense, functions ±δ are the velocity potentials. Circulation of ()±θvr along a contour C  \nis equal to ()± ± ±−=∇ℜ=⋅∫∫f d Cd\nC2   v2\n0π\nθ θ θδrr. Taking into account the \"orbital\" function ) (θφ , \nwe may consider the quantity ()[]ℜ=±∇r  δφθr\n as the total (\"orbital\" and \"spin\") velocity of an \nirrotational \"border\" flow:  \n \n                                                         ()()[]ℜ=± ±∇≡r V   δφθ θrr\n.                                                      (13)        \nIt is evident that  \n \n                                                     ()()()\nθθ\nθ e efqvi Vqvir r\n±+−±±\n±ℜ+−= .                                           (14) \n \nFor a given membrane function η~and given z-distribution of the MS potential, )(zξ , we can \ndefine now the strength of a vortex of a whole disk as  \n \n      () () ∫ ∫∫∫∫ℜ=± ± ℜ= ± ℜ=± −=⋅ ℜ=⋅ ≡d\nrd\nr\nCd\nredzz Rf de Vdzz R Cd Vdzz R s\n02\n0 0 0)( 2  )(  )( ξ θ ξ ξπ\nθθ θrr rr\n.         (15) \n \n    The quantity ()±θVr\n has a clear physical meaning. In th e spectral problem for MDM ferrite \ndisks, non-singlevaluedness of th e MS-potential wave function a ppears due to a border term \nwhich arises from the demand of conservation of the magnetic flux density on a lateral surface \nof a disk [10, 11]. This border term is defined as ()ℜ=−r aHiθµ , where aµ is the off-diagonal \ncomponent of the permeability tensor and ()ℜ=rHθ  is an annual magnetic field on the border \ncircle. It is evident that  \n \n                                                      ()()±ℜ=±−=θ θ ξ Vz zH\nrr r\n)( )() ( .                                                (16) \n \n    We define now an angular moment ea±r:  7 \n                                           ()[]e\na\nCr ad\nesi dzCde Hi a± ℜ= ± =⋅ −≡∫∫    \n0µ µθ θrr.                                   (17)       \n \nThis angular moment can be formally represented as a result of a circulation of a quantity, which \nwe call a density of an eff ective boundary magnetic current mir\n: \n \n                                                               )( 4\n0∫∫⋅ =± ±d\nCm eCdidzz arr\nξπ ,                                          (18) \n \nwhere   \n                                                                      \n()± ±≡θρV im mr r\n                                                       (19) \n \nand  \n \n                                                                   ℜ=≡ra mR i 4πµρ .                                                      (20) \n                                                                          \n    In our continuous-medium model, a characte r of the magnetization motion becomes apparent \nvia the gyration parameter aµ in the boundary term for the spectral problem. There is \nmagnetization motion through a non-simply-connected  region. On the edge region, the chiral \nsymmetry of the magnetization prec ession is broken to form a flux-closure structure. The edge \nmagnetic currents can be observable only via its circ ulation integrals, not pointwise. This results \nin the moment oriented along a disk normal. It  was shown experimentally [22] that such a \nmoment has a response in an exte rnal RF electric field. This clarifies a physical  meaning of a \nsuperscript \"e\" in designations of es± and ea±. In a ferrite disk particle, the vector ear is an \nelectric moment characterizing by special symmetry properties.  \n    An electric moment ea± is characterized by the anapole-mome nt properties. This is a certain-\ntype toroidal moment. Some important notes shoul d be given here to ch aracterize properties of \nmoment ear. From classical consideration it follo ws that for a given electric current eir\n, a \nmagnetic dipole moment is described as  ∫×= dvircme 21rrr, while the toroidal dipole moment is \ndescribed as  ()∫××= dvirrcte 31 rrrr (see e.g. [23]). When we  introduce the notion of an \nelementary magnet: eir Mrrr\n×≡ , we can represent the toroidal dipole moment as a linear integral \naround a loop: ∫×= dlMrct  31rrr. It is considered as a ring of elementary magnets Mr\n. In this \nformulation, it is clear that a toroidal moment is parity odd and time reversal odd. In a case \nwhen Mr\nis time varying, one has a magnetic current tMim\n∂∂≡rr\n and a linear integral of this \ncurrent around a loop defines a mo ment which is parity odd and tim e reversal even. This is the \ncase of an anapole moment ear which has the symmetry of an electric dipole. From classical \npoint of view such a definition presumes no azimuth variations of loop magnetic current mi. In \nour case, however, for oscillating MDMs one has the azimuth varying ring magnetic current.  8The magnetic current mi is described by the double valued f unctions. This results in appearance \nof an anapole moment ear.   \n \n3. Characterization of interaction of MDMs with the external electromagnetic fields \n \nMicrowave experiments [4, 6, 7]  give evidence for the multiresonance MDM oscillations in \nferrite disks. The absorption peak positions corresponding to these MDMs depend on the ferrite material parameters and the disk geometry but not on the type of a cavity. This statement was \nconfirmed more in details in  experimental paper [22].  \n    The mechanism of excitation of the multiresonance MDM spectra by the cavity electromagnetic fields is, however, not so obvious . Following the theory in [9 – 11] one sees \nthat the main factor of interaction of MDMs w ith the cavity electromagnetic fields is caused by \nthe presence of topological singularities on th e ferrite disk surfac es. These topological \nsingularities are described by the double valued functions and the main aspects concern the \nquestion how one can describe effective reso nance interactions be tween the double-valued-\nfunction edge states and single- valued-function cavity electromagne tic fields. The mechanism of \nsuch an interaction we illustrate he re by the following qualitative models. \n    Suppose that we have the (+) resonan ce which is characterized by azimuth number \n21+=q . \nFig. 1 (a) shows the dou ble-valued functions )(θδ′+  and ) (θδθ′∇+′ . Here we use designation θ′ \nto distinguish the \"spin\" angular coordi nate from a regular angle coordinate θ. Because of the \nedge-function chiral rotation [10,  11], for the (+) resonance one has to select only positive \nderivatives: 0>′∂∂+\nθδ. The corresponding parts of the graphs are distinguished in Fig. 1 (a) by \nbold lines. Figs. 1 (b) and (c) give two cases  of the single-valued -function cavity field )(θF  \nwhich may lead to resonance interactions with  the double-valued-function edge state. It is \nevident that in a case of Fig. 1 (b), a positive half of function ) (θδθ′∇+′  is phased for a \nresonance interaction with th e positive halves of function )(θF , while in a case of Fig. 1 (c) a \npositive half of function ) (θδθ′∇+′  is phased for a resonance interaction with the negative halves \nof function )(θF . Interaction with the cavity field shown in Fig. 1 (b) can be characterized as \nthe \"resonance absorption\" while interaction with the cavity field shown in  Fig. 1 (c) – as the \n\"resonance repulsion\". Both type s of interactions are equipr obable and can be exhibited \nseparately. One may also expect that in a ce rtain situation transitions between these two \nresonance behaviors can be demonstrated. A similar model can be used for illustration of a \npossible mechanism of interactions in  a case of the (–) resonance. For 21−=q , the double-\nvalued functions )(θδ′−  and ) (θδθ′∇−′  are shown in Fig. 2 (a). Here we have to select only \nnegative derivatives: 0<′∂∂−\nθδ. In this case, interaction with the cavity field shown in Fig. 2 (b) \ncan be characterized as the \"resonance absorption \" and interaction with the cavity field shown in \nFig. 2 (c) – as the \"resonance repulsion\".     An interaction of the ana pole moment of a ferrite disk with the cavity RF electric field was \nclearly demonstrated in [22]. Fo llowing the model shown in Figs. 1 and 2 one can describe this \ninteraction by the average quantity \n()∫⋅π\nθπ2\n0 21dEaerr, where Er\n is the cavity electric field. The \npositive quantity of this integral corresponds to the \"resonance absorption\" while for a negative \nintegral one has the \"resonan ce repulsion\". These two behavi ors we demonstrate in the \nfollowing experiments.     9   In experiments, we used a disk sample of a diameter mm 3 2=ℜ made of the YIG film on the \nGGG substrate (the YIG film thickness mkm 6.49=d , saturation magnetization \nG 1880  40=Mπ , linewidth Oe 8.0=∆H ; the GGG substrate thickness is 0.5 mm). A normally \nmagnetized ferrite-disk sample was placed in  a rectangular waveguide cavity with the 102TE  \nresonant mode. The disk axis was oriented along the waveguide E-field (see Fig. 3). We \nanalyzed the MDM spectra for the DC magnetic fiel d variation at certain constant frequencies. \nTo investigate behaviors of the \"resonance repul sion\" and \"resonance absorption\" and transition \nbetween these behaviors we analyzed the ferrite di sk spectra measured at different frequencies. \nThe multiresonance spectral pictures are shown in Fig. 4 (a). These frequencies, 3 ,2 1  and , f ff , \ncorrespond to different positions on the resonan ce curve of the cavity [see Fig. 4 (b)]. The \nspectral peaks of the MDM oscillations one obtains at certain permeability tensor parameters [11, 15]. Since the permeability tensor paramete rs are dependent both on frequency and a bias \nmagnetic field, we are able to match the peak position by small variations of a bias field. The bias magnetic fields corresponding to the first peaks are adduced in Fig. 4 (a). The digits characterize the MDM numbers. \n    In Fig. 4 (a), the spectrum corresponding to \n1f represents the \"resona nce repulsion\"  behavior. \nAt the same time, the spectrum corresponding to3f, clearly demonstrates the \"resonance \nabsorption\" behavior. It becomes evid ent that the spectrum corresponding to2f, shows the \ntransitions between the \"res onance repulsion\" and \"resonan ce absorption\". A qualitative \nexplanation of the observed three cases could be the fo llowing. Since at frequency 1f the cavity \nis \"viewed\" by the incoming signal as an activ e load, one can clearly observe the \"resonance \nrepulsion\" due to a ferrite disk.   Contrary, at frequency 3f the cavity is characterized mainly as \na reactive load. In this case one observes the \"resonance absorption\" behavior.  At frequency 2f \nboth cases are mixed and a transitional behavior takes place.  \n    The main conclusion following from the a bove consideration is that there exists a real \nmechanism of the effective re sonance interactions between the double-valued-function edge \nstates and single-valued-function electromagneti c fields. The character of the MDM resonance \nresponses may depend on the EM field structure but the resonance peak positions are fixed for a \ngiven ferrite disk. These results wi ll allow us to analyze the elect ric self-inductance properties of \nMDM ferrite disks.   \n4. Experimental evidence for th e electric self-inductance prop erties of MDM ferrite disks \n \nIn the above experiments we analyzed the MDM ex citation by the cavity fields as a result of \ninteraction of the eigen electric  (anapole) moments with the ex ternal RF electric fields. An \nanapole moment is created by a loop magnetic cu rrent. It can be supposed  that an equivalent \nalternative mechanism of excitati on which describes an induction of the loop magnetic current \nby the time varying electric- field flux can also  be used. A dual effect of induction of an electric \ndisplacement current in a dielectric-disk resonator due to the Faraday law was analyzed in [24]. \nThe physics of this effect, however, is completely different from our case. \n    First of all, let us define  the notion of the electric self inductance for MDMs. For every MDM \nin a ferrite disk, we define the electric self induc tance as the ratio of the electric flux to the loop \nmagnetic current. Based on Eq. ( 11) we express the eigen elec tric flux for a given mode p as  \n \n                                                           () ( )    2\n02∫ℜ=±=Ξd\np p r pedz Rq ξ π .                                       (21) \n \nFollowing Fig. 1 or Fig. 2 and based on Eqs. ( 19) and (20), the time average magnetic current \nfor mode p is expressed as  10 \n                                     ()()()()()p rpa\np rpa\npm\naverage R i d R i iℜ= ′ ℜ= =′∇ =∫2\n02 1\n4 πµ\nθδπ πµπ\nθ .                   (22) \n \nWe define the electric self  inductance for a given mode p as  \n \n                                               ()()\n()()()∫±ℜ= =Ξ=d\np p r\npapm\naveragepe\npedz q RiL\n03\n 4ξµπ.                               (23) \n \nOne may presuppose that this is a constant quantity for every MDM in a ferrite disk.     For experimental evidence of the electric self-inductance properties of  MDM ferrite disks we \nsuggest the following measurements. Suppose that we put a dielectric sa mple above a ferrite \ndisk. Because of the electric flux \neΞoriginated from a ferrite disk, the electric charges will be \ninduced on surfaces of a dielectri c sample. These surface charges cr eate their own electric flux \nwhich will pierce a ferrite disk in an opposite  direction with respect to the electric flux eΞ. As a \nresult of the reduction of the total electric flux, the loop magnetic current in a ferrite disk will be \nreduced as well, preservi ng a constant quantity of eL for a given mode p:                                                     \n \n                                                                 ()()\n()′′Ξ=\npm\naveragepe\npe\niL .                                                     (24) \n \nHere ()′Ξpeand ()′\npm\naveragei  are, respectively, the total electric  flux piercing a ferrite disk and the \nloop magnetic current co rresponding to mode p when a dielectric sample is put on. For a given \nmode structure, the requirement of constancy of quantity eL will lead to a shift of the MDM \nspectra to lower quantities of a modulus of aµ [see Eq. (22)]. In experiments, this should give \nan evidence for shift of the MDM position in the spectrum to a lower bias field (at a constant \nfrequency) or to a higher fre quency (at a constant bias field) . Obviously, this shift will be \ndependent on the permittivity parameter of a dielectric sample. For larger permittivity parameter \nrε of a dielectric sample, the shift should be larger as well . Moreover, since for a given \ngeometry of a dielectric sample, the flux ()′Ξpe will abate linearly as rε increases and the \ncurrent ()′\npm\naveragei  decreases linearly as a modulus of aµ decreases, there should be a linear \ndependence between rε and a modulus of aµ for preserving constant quantity ()peL. In our \nexperiments, we clearly observe th is effect for enough small quantities rε.      \n    Contrary to the cavity structure shown in Fi g. 3, in these experiments we will use a short-wall \nrectangular waveguide without an  entering iris. In th is case we can anal yze the MDM spectra \nwith respect to the frequency at constant DC ma gnetic field. We work at  a bias magnetic field \n=0H 4900 Oe. We start with an analys is of the spectra of a ferrite  disk without any dielectric \nloading at different disk positions in a wavegui de. The forms of the MDM resonance peaks may \nbe dependent on the EM field stru cture and, certainly,  the effectiveness of excitation of the \nmodes (the MDM amplitudes) can be different for different disk positions in a waveguide. At \nthe same time, as we discussed above, the resonance peak positions should be fixed for a given ferrite disk. Fig. 5 (a) shows the MDM spectra for different disk positions in a waveguide. These \npositions are clarified in Fig. 5 (b). In Figs. 6 (a) and 6 (b) one can see more detailed spectral pictures at the positions numbered 1, 6 and 3, 4, respectively. A shift of the disk may slightly  11perturb the cavity field structure. This pert urbation, however, doe s not influence on the \nresonance peak positions. In fact, one can see th at the peak positions in the MDM spectrum does \nnot depend on the EM field structure in a waveguide.     A ferrite disk is placed on a GGG substrate which has the dielectric permittivity parameter of \n15=rε . Now we put dielectric samples above a ferrit e disk (see Fig. 7). There are dielectric \ndisks of a diameter 3 mm and thickness 2 mm. We used a set of  disks of commercial microwave \ndielectric (non magnetic) materials with the dielectric permittivity parameters of 4.3=rε  \n(RO4003; Rogers Corporation), 15.6=rε  (RO3006; Rogers Corporation), 15=rε  (K-15; TCI \nCeramics Inc), 30=rε  (K-30; TCI Ceramics Inc), 50=rε  (K-50; TCI Ceramics Inc), and \n100=rε  (K-100; TCI Ceramics Inc). It can be suppos ed that an insertion of dielectric samples \nchanges the cavity field structure but, following the results shown in  Fig. 5, this field variation \nshould not influence on the absorption peak positi on. We, however, observed strong variations \nof the spectral pictures when dielectric disk s were placed above a ferrite disk. These \ntransformations of the spectra become most evid ent when we match (by proper small shifts of \nthe bias magnetic fields) positions of the first pe aks in the spectra. From Fig. 8 one can see that \nas the dielectric permittivity parameter of a di electric sample increases, the frequency shift of \nthe mode peak position increases as well.     A more detailed picture is shown in Figs. 9, 10 and 11 for the modes 3, 5, and 7, respectively. \nIt is evident that for a given mode (with the mo de number more than one) the peak shifts are \nsmall compared to the frequency distances between the peak of this mode and the first mode in \nthe oscillating spectra. In a pres upposition that the peak shifts are very small one can use the \nperturbation method assuming that the mode spectral portrait does not change under the \nperturbation. In this case we can use Eq. (23) for the self inductance characterization.  \n    To analyze the electric self-inductance prope rties we have to inves tigate relations between \nparameters \nrε and aµ for given modes. Fig. 12 shows dependences between rε and a modulus \nof aµ for modes 3, 5 and 7. The quantities aµ we calculated for given parameters of a ferrite \nmaterial, a given bias magnetic field, and for frequencies co rresponding to th e mode peak \npositions in Figs. 9, 10, and 11.  It is worth noting that for enough small quantities rε of a \ndielectric sample (for the permittivity less than 30 in our studies) we have almost linear \ncharacter of dependence between rε and aµ. For such small permittivity parameters, the Eq. \n(24) can be applicable.        \n     \n5. Conclusion  \n \nOur results show that external RF electric fi elds can induce loop ma gnetic currents in MDM \nferrite disks. To a certain extent, one can consider  this as a dual case with  respect to an induction \nof loop electric currents in diel ectric-disk resonators. The physics  of these two effects, however, \nis completely different. In this paper we analyzed the mechanism of interaction of MDMs in a quasi-2D ferrite disk with the external RF fiel ds. This is a special m echanism of interaction \nbetween the double-valued and single-valued f unctions. The double-valued functions describe \nthe MDM geometrical phase factor on the latera l boundary of a ferrite di sk, while the single-\nvalued functions describe the MDM membrane oscillations. We showed that due to this \nmechanism one has an evidence for the electric self-inductance properties of MDM ferrite disks. \n    It becomes evident that in the MDM oscillating problem, the permittivity parameters of a dielectric layer abutting to the ferrite are the material parameters that influence on the spectral \ncharacteristics of MDM oscillations. At the same time, an additional dielectric loading is a small \nperturbation parameter for the original spectral characteristics.   \n \n  12References \n \n[1] J. M. Luttinger and L. Tisza, Phys. Rev. 70, 954 (1946). \n[2] H. Puszkarski, M. Krawczyk, an d  J.-C. S. Levy, Phys. Rev. B 71, 014421 (2005). \n[3] L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media , 2nd ed. (Pergamon, \nOxford, 1984). \n[4] R.L. White and I.H. Solt, Jr., Phys. Rev. 104, 56 (1956). \n[5] L.R. Walker, Phys. Rev. 105, 390 (1957). \n[6] J.F. Dillon Jr., J. Appl. Phys. 31, 1605 (1960).  \n[7] T. Yukawa and K. Abe, J. Appl. Phys. 45, 3146 (1974). \n[8] K.T. McDonald, e-print: physics/0312026 (2003). [9] E.O. Kamenetskii, J. Magn. Magn. Mater. \n302, 137 (2006); arXiv:cond-mat/0505717. \n[10] E.O. Kamenetskii, Phys. Rev. E 73, 016602 (2006). \n[11] E.O. Kamenetskii, J. Phys. A: Math. Theor. 40, 6539 (2007).  \n[12] A.G. Gurevich and G.A. Melkov, Magnetic Oscillations and Waves  (CRC Press, New \nYork, 1996). \n[13] D.D. Stancil, Theory of Magnetostatic Waves  (Springer-Verlag, New York, 1992).  \n[14] E.O. Kamenetskii, Phys. Rev. E 63, 066612 (2001). \n[15] E.O. Kamenetskii, M. Sigalov, a nd R. Shavit, J. Phys.: Condens. Matter 17, 2211 (2005).  \n[16] M.E. Tobar and A.G. Mann, I EEE Trans. Microw. Theory Techniq. 39, 2077 (1991).  \n[17] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory , 3rd ed. \n(Pergamon, Oxford, 1977). \n[18] A.S. Davydov, Quantum Mechanics , 2nd ed. (Pergamon, Oxford, 1976). \n[19] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). \n[20] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 (1984). \n[21] M.V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984). \n[22] E.O. Kamenetskii, A.K. Sa ha, and I. Awai, Phys. Lett. A 332, 303 (2004). \n[23] A. Ceulemans, L.F. Chibotaru,  and P.W. Fowler, Phys. Rev. Lett. 80, 1861 (1998). \n[24] S.B. Cohn, IEEE Trans. Microw. Theory Techn. MTT-16 , 218 (1968).  \n                        13Figure captions \n \nFig. 1. Interaction of the double-valued-functi on edge state with the single-valued-function \ncavity electromagnetic fields in a case of the (+) resonance. (a) Double-valued edge function; (b) \nthe cavity field at the \"resonance absorption\" inte raction; (c) the cavity field at the \"resonance \nrepulsion\" interaction.   Fig. 2. Interaction of the double-valued-functi on edge state with the single-valued-function \ncavity electromagnetic fields in a case of the (–) resona nce. (a) Double-valued  edge function; (b) \nthe cavity field at the \"resonance absorption\" inte raction; (c) the cavity field at the \"resonance \nrepulsion\" interaction.  Fig. 3.  Rectangular cavity w ith an inserted  ferrite disk \n Fig. 4. Experimental evidence for the \"resonance repulsion\", \"resonance absorption\", and transitional behaviors. (a) The multiresonance sp ectral pictures for the \"resonance repulsion\" (at \nfrequency \n1f), \"resonance absorption\" (at frequency 3f), and transitional behavior (at frequency \n2f); (b) frequencies 3 ,2 1  and , f ff  on the resonance curve of the cavity.  \n \nFig. 5. The spectra of a ferrite disk without diel ectric loading. (a) The MDM spectra for different \ndisk positions; (b) disk positions in a waveguide.   Fig. 6. The spectra of a ferrite disk without diel ectric loading (detailed spec tral pictures). (a) The \nMDM spectra for positions numbered 1 and 6; (b ) the MDM spectra for positions numbered 3 \nand 4.  Fig. 7. A ferrite disk on a GGG substrate with a dielectric sample.  Fig. 8. The frequency shift of the mode peak positio ns for a ferrite disk with dielectric loading. \nThe first peaks in the spectra are matched by pr oper correlations of bias magnetic fields. \n Fig. 9. The frequency shift of the peak positions  of mode 3 when a ferr ite disk is loaded by \ndielectric samples.  Fig. 10. The frequency shift of the peak positions  of mode 5 when a ferrite disk is loaded by \ndielectric samples.  Fig. 11. The frequency shift of the peak positions  of mode 7 when a ferrite disk is loaded by \ndielectric samples.  \nFig. 12. Dependences between \nrε and a modulus of aµ for modes 3, 5 and 7. \n \n                                                                                     14\n \n \nFig. 1. Interaction of the double-valued-functi on edge state with the single-valued-function \ncavity electromagnetic fields in a case of the (+) resonance. (a) Double-valued edge function; (b) \nthe cavity field at the \"resonance absorption\" inte raction; (c) the cavity field at the \"resonance \nrepulsion\" interaction.                          \n ()θF  \n()θF   15\n \n \nFig. 2. Interaction of the double-valued-functi on edge state with the single-valued-function \ncavity electromagnetic fields in a case of the (–) resona nce. (a) Double-valued  edge function; (b) \nthe cavity field at the \"resonance absorption\" inte raction; (c) the cavity field at the \"resonance \nrepulsion\" interaction.    \n \nFig. 3.  Rectangular cavity w ith an inserted  ferrite disk \n       \n ()θF  \n()θF   163800 4250-10-8-6-4-20\nHo [Oersted]Cavity  reflection coefficient [dB]\n  \n1234597610 8 11\n1\n12\n23\n34\n45\n56\n67\n78\n89\n910\n1011\n11f1=7.085 GHzf2=7.088 GHzf3=7.09 GHz\n \n                                                                               ( a) \n  \n \n \n                                                                                  ( b) \n Fig. 4. Experimental evidence for the \"resonance repulsion\", \"resonance absorption\", and transitional behaviors. (a) The multiresonance sp ectral pictures for the \"resonance repulsion\" (at \nfrequency \n1f), \"resonance absorption\" (at frequency 3f), and transitional behavior (at frequency \n2f); (b) frequencies 3 ,2 1  and , f ff  on the resonance curve of the cavity.                                                        ()Oe H 41501\n0=()Oe H 41531\n0=()Oe H 41551\n0= 178.7 8.75 8.8 8.85 8.9 8.95 99.05 9.1 9.15 9.2-12-10-8-6-4-202\nFrequency [GHz]Reflection coefficient [dB]\n  \nPosition 1\nPosition 2\nPosition 3\nPosition 4\nPosition 5\nPosition 6\nPosition 7mode1mode 2\nmode 3mode 4\nmode 5mode 7mode 6\n \n                           \n                                                                           ( a) \n \n \n \n                                                                      (b) \n \n  Fig. 5. The spectra of a ferrite disk without diel ectric loading. (a) The MDM spectra for different \ndisk positions; (b) disk positions in a waveguide.   188.7 8.8 8.9 9 9.1 9.2-12-10-8-6-4-20\nFrequency [GHz]Reflection coefficient [dB]\n  \nPosition 1\nPosition 6\nmode 1mode 3 mode 5mode 7\n \n \n                                                                           (a) \n8.7 8.8 8.9 9 9.1 9.2-5-4-3-2-10\nFrequency [GHz]  Reflection coefficient [dB]Position 3\nPosition 4mode 1mode 2 mode 3\nmode 4\nmode 5\n \n (b) \n \nFig. 6. The spectra of a ferrite disk without diel ectric loading (detailed spec tral pictures). (a) The \nMDM spectra for positions numbered 1 and 6; (b ) the MDM spectra for positions numbered 3 \nand 4.  19 \n        Fig. 7. A ferrite disk on a GGG substrate with a dielectric sample.    \n8.7 8.75 8.8 8.85 8.9 8.95 99.05 9.1 9.15 9.2-25-22-19-16-13-10-7-4-12\nFrequency [GHz]Reflection coefficient [dB]\n  \nFerrite disk only\ndielectric load with εr=3\ndielectric load with εr=6\ndielectric load with εr=15\ndielectric load with εr=30\ndielectric load with εr=50\ndielectric load with εr=100mode 1mode 3mode 5 mode 7\n \n \n  Fig. 8. The frequency shift of the mode peak positio ns for a ferrite disk with dielectric loading. \nThe first peaks in the spectra are matched by pr oper correlations of bias magnetic fields. Dielectric sample\nGGG substrate Ferrite dis k  20    8.9-4-1\nFrequency [GHz]Reflection coefficient [dB]\n  \nFerrite disk only\ndielectric load with εr=3\ndielectric load with εr=6\ndielectric load with εr=15\ndielectric load with εr=30\ndielectric load with εr=50\ndielectric load with εr=100mode 3\n \nFig. 9. The frequency shift of the peak positions  of mode 3 when a ferr ite disk is loaded by \ndielectric samples. \n9.05-1\nFrequency [GHz]Reflection coefficient [dB]\n  \nFerrite disk only\ndielectric load with εr=3\ndielectric load with εr=6\ndielectric load with εr=15\ndielectric load with εr=30\ndielectric load with εr=50\ndielectric load with εr=100mode 5\n \n \nFig. 10. The frequency shift of the peak positions  of mode 5 when a ferrite disk is loaded by \ndielectric samples.  21 \n  \n9.15-1\nFrequency [GHz]Reflection coefficient [dB]\n  \nFerrite disk only\ndielectric load with εr=3\ndielectric load with εr=6\ndielectric load with εr=15\ndielectric load with εr=30\ndielectric load with εr=50\ndielectric load with εr=100mode 7\n \n \nFig. 11. The frequency shift of the peak positions  of mode 7 when a ferrite disk is loaded by \ndielectric samples. \n                 224.3 4.35 4.4 4.45 4.5 4.55102030405060708090100\nabs(µa) εr\n  \nmode 3\n3.55 3.6 3.65 3.7 3.75 3.8102030405060708090100\nabs(µa) εr\n  \nmode 5\n \n                                   ( a)                                                                  ( b) \n \n3.05 3.1 3.15 3.2 3.25 3.3102030405060708090100\nabs(µa) εr\n  \nmode7\n \n                                                                         ( c) \n \nFig. 12. Dependences between rε and a modulus of aµ for modes 3, 5 and 7. "    },    {        "title": "2402.09706v1.Site_selective_cobalt_substitution_in_La_Co_co_substituted_magnetoplumbite_type_ferrites_____59__Co_NMR_and_DFT_calculation_study.pdf",        "content": "Site-selective cobalt substitution in La–Co\nco-substituted magnetoplumbite-type ferrites:\n59Co-NMR and DFT calculation study\nHiroyuki Nakamura1, Hiroto Ohta2, Ryuya Kobayashi1, Takeshi\nWaki1, Yoshikazu Tabata1, Hidekazu Ikeno3and Christian\nM´ eny4\n1Department of Materials Science and Engineering, Kyoto University, Kyoto\n606-8501, Japan\n2Department of Molecular Chemistry and Biochemistry, Faculty of Science and\nEngineering, Doshisha University, Kyotanabe 610-0321, Japan\n3Department of Materials Science, Graduate School of Engineering, Osaka\nMetropolitan University, Sakai, Osaka, 599-8570, Japan\n4Institut de Physique et Chimie des Mat´ eriaux de Strasbourg (IPCMS), UMR7504,\nCNRS-Universit´ e de Strasbourg, 23, rue du Loess, F-67034 Strasbourg Cedex 02,\nFrance\nE-mail: nakamura.hiroyuki.2w@kyoto-u.ac.jp\nAbstract. The La–Co co-substituted magnetoplumbite-type (M-type) ferrites\nAFe12O19(A= Ca, Sr and Ba, ion sizes Ca2+<Sr2+<Ba2+) with Co compositions\naround 0.2 have been subjected to59Co-NMR. The results show that Co occupies the\n4f1, 2a and 12k sites, and that the smaller the Aion, the more Co tends to occupy\nthe 4f 1minority spin site, which is effective in enhancing both uniaxial anisotropy and\nmagnetisation. First-principles total energy calculations based on density functional\ntheory (DFT) of undoped AFe12O19and a supercell (2 ×2×1 of the unit cell) in which\n1/96 of Fe3+is replaced by Co2+were performed to predict the stable structure and\nCo occupancy sites. The results show that regardless of A, Co is most stable when it\noccupies the 4f 1site, followed by the 2a and 12k sites with energy differences on the\norder of 100 meV, and Co practically does not occupy the 2b and 4f 2sites. As the\nAion becomes smaller, the energy difference when Co occupies each Fe site tends to\nincrease, and the Co occupancy of the 4f 1site also increases. The site selectivity of\nCo can be roughly explained as a result of the difference in uniaxial strain along the\nc-axis associated with the difference in A. However, the influence of the Aion differs\nbetween the R and S blocks and the local strain also has a secondary effect on the\nCo distribution. Based on these results, the guidelines for improving the performance\n(anisotropy and magnetisation) of La–Co co-substituted M-type ferrite magnets with\na limited amount of Co can be summarised as follows: It is effective to select as small\nAions as possible and to post-anneal at low temperature or cool slowly to concentrate\nCo at the 4f 1site in tetrahedral coordination.\nKeywords : ferrite magnet, anisotropy, NMR, DFT calculation, magnetoplumbite-type\nferritearXiv:2402.09706v1  [cond-mat.mtrl-sci]  15 Feb 2024Site-selective Co substitution in M-type ferrites 2\nFigure 1. Crystal structure of the magnetoplumbite-type (M-type) AFe12O19drawn\nby VESTA [4]. The space group is P63/mmc . The unit cell is equal to two formula\nunits. The Aion (black circle) occupies the 2d site surrounded by 12 oxygen ions. Iron\n(red circles) has five crystallographically inequivalent sites (2a, 2b, 4f 1, 4f2and 12k) and\nits oxygen coordination polyhedra are represented by different colours. The 2a, 4f 2and\n12k sites are 6-coordinated (octahedral), the 4f 1site is 4-coordinated (tetrahedral) and\nthe 2b site is 5-coordinated (bipyramidal). The 2a, 2b, 4f 1and 4f 2sites have trigonal\nrotational symmetry around the c-axis, but the 12k site is not trigonally symmetric,\nwith the local principal axis tilted from the c-axis resulting in three types of 12k-Fe.\nOxygen (blue circles) has five crystallographically inequivalent sites (4e, 4f, 6h, 12k 1,\n12k2). All Fe is trivalent and is expected to have a spin moment of 5 µB/Fe. AFe12O19\nis ferrimagnetic; the Fe at the 2a, 2b and 12k sites has the majority spin and the Fe\nat the 4f 1and 4f 2sites has the minority spin. S and R denote blocks and the M-type\nstructure has SRS∗R∗stacking, where * denotes a 180◦rotation around the c-axis.\n1. Introduction\nFerrite magnets are used in a wide range of equipment because they are extremely cost\neffective compared with rare earth magnets [1]. Higher performance and lower cost of\nferrite magnets are the demands of society. Many attempts have been made to improve\nthe performance of ferrite magnets [2]. Many of these are elemental substitutions,\nand the substitution of Co for Fe is a successful example [3]. However, much of the\ndevelopment is trial and error and not necessarily based on clear guiding principles.\nAgainst this background, the aim of this study was to propose guidelines beyond trial\nand error for the development of the next generation of high performance magnets\nand the effective use of resources. To improve the performance of magnetoplumbite-\ntype (M-type) ferrites, improvements in magnetic anisotropy (coercivity), magnetisation\n(remanence), or both, are required. This paper focuses on methods to improve magneticSite-selective Co substitution in M-type ferrites 3\nanisotropy without degrading or even improving magnetisation.\nThe base material of ferrite magnets is the hexagonal M-type ferrite AFe12O19(A\n= Ba, Sr, Pb...) (figure 1). The space group of the M-type structure is P63/mmc , and\nthe unit cell contains two formula units. All Fe atoms are located in the gap between O\nandA. There are five crystallographically inequivalent Fe sites in the Wyckoff symbols:\n2a (octahedral coordination), 2b (bipyramidal), 4f 1(tetrahedral), 4f 2(octahedral) and\n12k (octahedral). The M-type structure can be divided into the S block (Fe 6O8)2+\nincluding 2a and 4f 1sites and the R block ( AFe6O11)2−including 2b and 4f 2sites, with\nthe 12k site in the plane perpendicular to the c-axis at the boundary between the S\nand R blocks. The stacking is described as SRS∗R∗, where * represents a 180◦rotation\naround the c-axis. All Fe ions in AFe12O19are trivalent (3d5) and have a spin magnetic\nmoment of 5 µB(S=5\n2). The M-type ferrites are ferrimagnets in which the magnetic\nmoment of Fe3+is ordered by superexchange interactions through O2−. The 2a, 2b and\n12k sites are the majority spin sites, and the 4f 1and 4f 2sites are the minority spin\nsites, resulting in a net magnetic moment of 20 µBper formula unit. Isolated Fe3+\nwith no orbital component in the magnetic moment do not normally cause magnetic\nanisotropy. The presence of the asymmetric 2b site, which exhibits strong single-ion\nanisotropy [5] via interaction with the coordinating oxygen ions, appears to be essential\nforuniaxial anisotropy in M-type ferrites. However, contributions other than the 2b site\nare also important for a quantitative explanation of the anisotropy, as has been better\nunderstood by recent theoretical work [6, 7, 8].\nVarious element substitutions have been attempted to improve the performance of\nM-type ferrite magnets. Among these, La–Co co-substituted M-type SrFe 12O19, where\nsome of the Fe3+(approximately 2–3%) in SrFe 12O19is replaced by Co2+and some of\nthe Sr2+by La3+as charge compensation, has been reported to improve the coercivity\nby approximately 30% and the residual magnetisation by a few percent compared to\nthe undoped material [3]. The residual magnetisation is slightly improved because\nCo2+, which has a smaller magnetic moment than Fe3+, mainly occupies the minority\nspin sites, thereby increasing the net magnetisation. It is easy to imagine that the\nincrease in coercivity is due to the enhanced uniaxial magnetic anisotropy caused by\nthe residual (or induced) orbital magnetic moment of Co2+. However, the mechanism\nof enhancement has long been unclear due to the ambiguity of the Co occupancy sites.\nHowever, recent studies have shown that Co2+mainly occupies the 4f 1site and partially\noccupies the 2a and 12k sites. The fact that Co occupies three sites at 2a, 4f 1and 12k\nwas first proposed by a combined Rietveld analysis of neutron diffraction, extended X-\nray absorption fine structure (EXAFS) and X-ray magnetic circular dichroism (XMCD)\n[9]. This result was supported by the atom location determined using the channeling-\nenhanced microanalysis (ALCHEMI) method [10]. Recent NMR [11] and M¨ ossbauer\nexperiments [12, 13] are consistent with the site occupancy of Co.\nLa–Co co-substituted M-type ferrite magnets with Aions other than Sr have also\nbeen developed. It is known that the case of A= Ca has a higher coercivity than A\n= Sr, even with the same amount of Co substitution [14]. In contrast, the coercivitySite-selective Co substitution in M-type ferrites 4\nforA= Ba is lower than for A= Sr [15]. This suggests that even with the same\namount of Co, the Co distribution is not the same and depends on the Aion. The Co\ndistribution in La–Co co-substituted CaFe 12O19has also been studied by a combined\nanalysis of Rietveld refinement of neutron diffraction, EXAFS and XMCD [16, 17, 18].\nThe occupancy sites of Co are the same as in SrFe 12O19, but the 4f 1-site occupancy of\nCo has been reported to be greater in the Ca–La ferrite.\nIt should also be noted that physical properties such as coercivity can vary greatly\ndepending on the synthesis process and heat treatment conditions, even for materials\nof the same composition. Such cases arise when dopants occupy several non-equivalent\nsites and their stability is antagonistic at relatively low energy scales. This is because the\ncation distribution at room temperature is not necessarily the ideal thermal equilibrium\nstate at room temperature, but is more or less a frozen state of the high temperature\ndistribution. In other words, the final physical properties depend on the temperature at\nwhich the cation distribution is frozen, that is, the heat treatment applied to the sample.\nSpinel compounds have long been the focus of studies of cation distributions at finite\ntemperatures [19]. Some simple models commonly discuss the temperature dependence\nof the cation distribution from the probability of occupancy via the Gibbs free energy\n[19]. In recent years, many examples have been reported where density functional theory\n(DFT) has been used to calculate the energy change of a state in which a particular\nsite is replaced by a particular element, which is then used to calculate the probability\nof occupancy at a finite temperature [20, 21, 22]. The application of this method to\nM-type ferrites is growing rapidly [21, 22], and similar discussions can be made in this\nstudy.\n59Co ferromagnetic nuclear resonance experiments are a convenient way to obtain\ninformation on the site-to-site distribution of Co in ferromagnets and microscopic\nmagnetism, provided that the occupancy sites can be assigned to the resonance signals.\nWe performed59Co-NMR experiments on samples of La–Co co-substituted SrFe 12O19\nwith widely varying Co compositions (from 0.15 to 0.93 Co) to investigate the Co\ndistribution over multiple Fe sites and the role of Co occupying each site [23]. The\nresults showed that Co was commonly distributed over three Fe sites (4f 1, 2a and 12k),\nwith the 4f 1site being the most preferentially occupied. We evaluated the Co content\ndependence of the local anisotropy at each site by measuring the signal enhancement\nfactor, and found that the macroscopic anisotropy was positively correlated with the\nlocal anisotropy of Co at the 4f 1site and inversely correlated with those at the 2a and\n12k sites: see the second paragraph of section 3.2 of [23] for details, and [24, 25] for\nthis methodology. Based on these results, we proposed for the first time that only Co\nat the 4f 1site contributes to the enhancement of uniaxial anisotropy. Although no\norbital degeneracy of Co2+at the tetrahedrally coordinated site would be expected if\nthe coordination polyhedron were a regular tetrahedron, we concluded that the Co at\nthe 4f 1site, which is trigonally symmetric, is the main cause of the enhancement of\nuniaxial anisotropy. From these results, we proposed that the magnetic anisotropy and\nsaturation magnetisation of ferrite magnets can be improved by concentrating Co at theSite-selective Co substitution in M-type ferrites 5\n4f1site, and the production cost can also be reduced by the effective use of Co. However,\nthe actual method for concentrating Co at the 4f 1site remains debatable. In La–Co co-\nsubstituted SrFe 12O19, the 4f 1-site occupation ratio of Co was unexpectedly increased\nwith increasing La–Co substitution. It was suggested that the 4f 1-site selectivity of\nCo could be enhanced by a decrease in c/a, that is, compressive strain along the c-axis,\nassociated with the substitution of La3+, which is smaller in size than Sr2+. However, as\na series of samples with very different Co concentrations were used for the measurements,\nwe could not rule out the possibility that the Co distribution changes due to changes\nin the Co concentration itself and could not clearly conclude that uniaxial strain affects\nCo site selectivity.\nIn this study, we focused on the change in Co distribution in La–Co co-substituted\nAFe12O19due to the difference in A-ion size. In other words, similar to the previous\nstudy [23],59Co-NMR experiments were performed on La–Co co-substituted AFe12O19\n(A= Ca, Sr and Ba) with a common Co composition to compare the Co site selectivity\nand investigate the effect of A-ion size. We also evaluated the stability of Co occupying\neach Fe site and the crystal structure parameters of structure-optimised AFe12O19and\nCo substituted AFe12O19by first-principles total energy calculations based on DFT and\ndiscussed the agreement with experiments, as well as the factors controlling the Co site\nselectivity. As mentioned above, to further improve the coercivity and magnetisation\nof La–Co co-substituted M-type ferrite magnets and to make effective use of Co,\nit is necessary to concentrate Co2+more efficiently at the 4f 1site. Establishing a\nsimple guiding principle for this will accelerate the development of next-generation high\nperformance ferrite magnets. In this study, we show that the selection of Aions that\nreduce the c-axis length is effective for this purpose. We also discuss the secondary effect\nof the local strain on Co site selectivity. From the results of DFT calculations, we show\nthat the Co site-to-site distribution is temperature sensitive in the temperature range\nwhere the M phase is stable, and that post-annealing at low temperature and/or slow\ncooling is effective in concentrating Co at the 4f 1site.\n2. Methods\n2.1. Our strategy\nSince the development of La–Co co-substituted high-performance ferrite magnets [3],\nthe mechanism of coercivity (uniaxial magnetic anisotropy) enhancement has been\nextensively studied, but the debate has long remained confused. The main reason\nfor this confusion was that the Co occupancy sites were experimentally ambiguous\n[26]. The amount of Co substituted for Fe is at most a few percent of Fe, and Co is\ndistributed over several sites in five crystallographically inequivalent Fe sites. In other\nwords, the identification of Co occupancy sites in La–Co co-substituted M-type ferrite\nis an ultra-trace element analysis in a multi-site compound and requires extremely high\nexperimental precision. Despite these difficulties, early studies paid insufficient attentionSite-selective Co substitution in M-type ferrites 6\nto sample homogeneity, compositional reliability, experimental precision and analytical\nvalidity.\n2.1.1. Sample In many previous studies of M-type ferrites, the reagents were mixed\nto a certain composition and then calcined, and the initial composition was taken as\nthe nominal composition. Compositional analysis after synthesis was rarely performed.\nThis is because the main objective of these studies was to achieve high performance. In\nsuch samples, the actual composition rarely matches the initial composition due to the\nformation of secondary phases, precipitation of different phases at grain boundaries and\nsegregation of dopants on grain surfaces. Permanent magnets are composite materials\nthat skilfully exploit the different functions of crystal grains and grain boundaries. The\ndirect use of such materials is inappropriate for evaluating the intrinsic properties of\nthe matrix phase. In fact,59Co-NMR signals of various secondary phases have been\nobserved in NMR experiments on La–Co co-substituted M-type ferrite magnets, making\nidentification of the essential signal difficult. It is also known that Co segregates on the\ngrain surface depending on the heat treatment conditions and plays a different role\nfrom that in the grain [27]. More importantly, the valence of transition metals in M-\ntype ferrite is generally unstable. La–Co co-substituted M-type ferrites were originally\ndesigned to have equal amounts of La and Co, with all Co being divalent, but in reality\nthe divalent (3d6) and trivalent (3d5) states of Fe are energetically antagonistic and\ngenerally coexist [28, 29]. The abundance of Fe2+and Fe3+is dependent on the oxygen\npotential during synthesis and requires a sufficiently high oxygen potential to maintain\nall Fe in the trivalent state [30, 31]. In other words, if a sample is synthesised at a\nlow oxygen potential, the formation of Fe2+is inevitable. This results in a discrepancy\nbetween the design (initial) composition and the actual composition.\nThe study of the intrinsic properties of materials requires the use of quality\ncontrolled samples, whose phases and compositions have been identified from a solid\nstate chemistry point of view. For this purpose, it is essential to determine the actual\ncomposition of the sample after synthesis by compositional analysis, even if the sample\nis single phase. As mentioned in section 1 and discussed in more detail in section 4.2,\nthe dopant distribution in M-type ferrite can be strongly dependent on the cooling\nconditions or post-annealing of the sample, even if the compositions of the samples are\nthe same. This means that the heat treatment conditions should be carefully controlled\nso that they are the same when comparing the properties of different samples. The\nnecessary conditions for the sample used in this study were as follows: (i) a state close\nto equilibrium was achieved by properly controlling the heat treatment conditions, (ii)\nit was single phase (no secondary phases), (iii) the composition was uniform throughout\nthe sample, and (iv) the actual composition was determined. To satisfy these conditions,\nwe used single crystal samples grown by the flux method using a platinum crucible, which\nis a popular growth method for M-phase single crystals [32]. To satisfy (i), the sample\nwas slowly cooled at a predetermined cooling rate. With natural cooling, it was difficult\nto standardise the cooling conditions for each sample. The samples were grown in airSite-selective Co substitution in M-type ferrites 7\n(oxygen partial pressure 0.2 atm) or in oxygen gas flow (1 atm); these atmospheres\nare easy to handle and allow stable crystal growth, facilitating (i)–(iii). However, Fe2+\ncontamination is inevitable and care must be taken to ensure that the effect of Fe2+\ndoes not vary significantly from sample to sample. In the following NMR experiments,\npowdered crystals were used; the use of single crystals as raw material guarantees (ii).\nAlthough other crystal growth methods, such as the travelling solvent floating zone (TS-\nFZ) method, are known to grow large single crystals [33], crystals grown in a crucible\nwere used in this study. This is because, while the FZ method facilitates compositional\ncontrol by growing crystals from the melt in a closed space isolated in air, it limits the\nfreedom of atomic flow in the melt, and the sample is generally cooled rapidly, which may\ncause the synthesised crystals to deviate from their equilibrium state. In the crucible flux\nmethod, the crystals grow gently in the melt without significant perturbation, resulting\nin the growth of crystals in a state close to equilibrium (note that this is not necessarily\nthe ideal equilibrium phase, as described in section 1), and (i) and (iii) are satisfied.\nIndeed,59Co-NMR spectra of La–Co co-substituted SrFe 12O19show slightly different\ncharacteristics for crystals synthesised by the TS-FZ method and crystals synthesised\nby the flux method using a crucible [11].\nWavelength dispersive X-ray spectrometry (WDX) with a scanning electron\nmicroscope (SEM) was used to satisfy condition (iv). Unlike energy dispersive X-\nray analysis (EDX), WDX can completely separate the Fe and Co spectra, providing\na highly accurate analysis of Co composition. X-ray analysis, which only observes\nthe sample surface, has been criticised for its low accuracy. However, comparison\nwith inductively coupled plasma atomic emission spectrometry (ICP-AES), which gives\nthe average value for the bulk sample, shows that the two analyses are in good\nagreement [33, 34]. The high reliability of WDX has been demonstrated in many studies\n[28, 31, 34, 35, 36, 37, 38, 39, 40, 41]. In addition, WDX, which analyses a narrow spot\nwhere X-rays are irradiated, guarantees (iii) by sampling different parts of the crystal.\nWe have applied this sample strategy to systematic studies of La substituted\nand La–Co co-substituted SrFe 12O19and CaFe 12O19. Their structural and magnetic\nproperties have been reported in detail in [28, 34, 35]. In this study, among the single\ncrystals synthesised in these series of studies, we selected those with a Co composition\nclose to 0.2 and in sufficient quantity (see table 1 and section 2.2 for details). Not all\nsamples had a Co composition of exactly 0.2. This is because the final composition\nof flux-grown M-phase crystals is difficult to deliberately control; the M-phase melts\nincongruently [42], and the elemental fraction of the reagent before melting and the\ncomposition of the grown crystal are generally different. Nevertheless, the compositions\ndetermined by WDX were sufficiently accurate and the samples used in this study satisfy\nconditions (i)–(iv) above. It should be noted that the information used in this study is\nthe ratio of Co occupancy among sites, not the absolute amount, so that differences in\nCo composition within reasonable limits will not affect the outcome of the discussion.\nAs the samples used in this study were synthesised at relatively low oxygen potentials,\nthe formation of Fe2+was inevitable. However, since the design concept of La–Co co-Site-selective Co substitution in M-type ferrites 8\nsubstitution is to prevent the formation of Fe2+, and the purpose of this study was\nto investigate the site-to-site distribution of divalent Co, we assumed that the effect\nof Fe2+in the samples used in this study could be neglected. In fact, the effect of\nFe2+on the magnetic properties is not significant in the present samples; the presence\nof Fe2+reduces the uniaxial magnetic anisotropy in regions of low Fe2+concentration\n[33, 43]. The physics of Fe2+in M-type ferrite is certainly important [44, 45], but is\nnot considered in this study; the discussion is limited to outside its influence. In La\nsubstituted CaFe 12O19, the Fe/(Ca + La) content ratio is known to be less than 12\n[46]. Some Ca may occupy Fe sites, but the origin of this anomaly has not been fully\nunderstood [18, 35, 47, 48]. We will not address this point in detail in this study.\n2.1.2. Evaluation method for site-to-site distribution of Co What we need in this\nstudy is a rapid method for assessing the site-to-site distribution of Co. The\ncurrent understanding of the Co occupancy sites in La–Co co-substituted SrFe 12O19\nand CaFe 12O19was first achieved by combining synchrotron radiation with neutron\nexperiments [9, 16, 17, 18]. Experiments requiring large-scale facilities are certainly\nuseful, but lack immediacy and availability. Rapid on-site evaluation of synthesised\nsamples is needed to accelerate research.\nNuclear methods, such as M¨ ossbauer spectroscopy and NMR, have been widely\nused to assess cation distribution with high immediacy and availability. However,\ntheir application has historically caused confusion in the identification of Co sites in\nLa–Co co-substituted M-type SrFe 12O19[26]. Recently, a series of studies have been\ncarried out to investigate the validity and interpretation of57Fe M¨ ossbauer spectroscopy\nand57Fe,59Co-NMR in this research area [11, 12, 13, 49]. As a result, we have\nconfirmed that although the57Fe probe methods are useful for discussing microscopic\nmagnetism, for example through experiments under a magnetic field, they are not ideal\nfor quantifying the Co distribution. In the case of M-type ferrite, quantitative analysis\nof the Co distribution requires the detection of trivial intensity changes in the57Fe\nspectrum, which is composed of overlapping multi-components. Information on Co is\nobtained from changes in the number of Fe atoms, but the effect of Co substitution\nis attenuated because Co is distributed over multiple sites. The intensity of the57Fe\nsignal is more sensitive to the substitution modified environment than to the number\nof Fe atoms occupying the site [11, 12]. In addition, the lattice anharmonicity of M-\ntype ferrites [50, 51, 52, 53] makes it difficult to evaluate the true signal intensity and\nchallenges the assumption that the signal intensity is proportional to the abundance of\nthe corresponding nuclei. Thus, the shortcomings of the57Fe probe methods are more\npronounced in the quantitative evaluation of the Co distribution. This is in contrast to\nthe fact that the57Fe probe experiment works well for discussing more dramatic changes\nsuch as the crystal structure transition in M-type ferrites [43, 45].\nIn59Co-NMR, on the other hand, the integrated intensity of the signal corresponds\ndirectly to the number of Co atoms occupying a particular site. In principle, therefore,\n59Co-NMR provides simple and direct information on the Co occupancy sites andSite-selective Co substitution in M-type ferrites 9\nsubstitution amounts, provided that the site assignment of the signal is complete.\nThe NMR signal intensity of59Co nuclei is generally extremely strong, and especially\nin ferromagnetic materials the signal intensity is greatly enhanced; thus, significant\ninformation can be extracted, even with a small amount of Co. The problem in previous\nstudies of59Co-NMR on La–Co co-substituted M-type ferrites [49, 54, 55] was the site\nassignment of the signal and possibly the sample used. The site assignment of the59Co-\nNMR signal of La–Co co-substituted SrFe 12O19has already been completed with respect\nto the distinction between tetrahedrally and octahedrally coordinated sites (see section\n2.1 in [23]). We believe that this conclusion is also valid for other La–Co co-substituted\nM-type ferrites.\nIn this study, the NMR spectra were recorded on a home-built state-of-the-art\nbroadband spectrometer at the Institut de Physique et Chimie des Mat´ eriaux de\nStrasbourg (IPCMS) using an untuned (high-pass design) probe [24, 25], which ensures\ngood reproducibility of the frequency spectrum. This system is particularly suitable\nfor the measurement of ferromagnetic (and ferrimagnetic) materials, whose resonance\nsignals are distributed over a wide frequency range. When there are multiple magnetic\nsites in different local environments, as in M-type ferrites, the local anisotropy of each\nsite may be different. In such cases, the signal enhancement factor will be different at\neach site, and it is necessary to correct for the frequency dependence of the enhancement\nfactor in order to evaluate the distribution of nuclei corresponding to the resonance\nfrequency. The enhancement factor at each frequency point was evaluated by measuring\nthe signal intensity at each frequency using multiple radiofrequency (RF) field powers.\nThe RF power dependence of the signal intensity was fitted with a suitable function,\nthe width of which was also taken into account. From the above series of procedures,\nthe enhancement factor was used to correct the signal intensity. Monitoring the\nenhancement factor also facilitates the differentiation of NMR signals from the magnetic\ndomains and walls. Intensities were corrected by dividing by the square of the frequency,\nwhich is standard in NMR methods. The NMR spectrum corrected by the above process\ncorresponds to the distribution of nuclei corresponding to the resonance frequencies and\nis independent of the value and distribution of the enhancement factor. Furthermore, it\nis easy to calculate the local magnetic anisotropy of each environment from the signal\nenhancement factor, providing a unique insight into the magnetic properties of the\nsample, which has been described in detail in [24, 25].\nNote that the signal intensity is corrected by dividing it by the square of the\nfrequency and that the resonance frequency for our materials is significantly spread\nover the frequency domain (the lower and upper frequency limits differ by an order of\nmagnitude, see section 3.2). This means that the correction factor differs by a factor of\n102between the lowest and highest frequencies. Thus, if the correction factors contain\npotential errors, however small, these errors will be amplified in the final intensities after\ncorrection. Therefore, the final quantitative value of the Co distribution may contain\nan error of the order of 10%. Nevertheless, the quantitative nature of the system at\nIPCMS is exceptional for this type of experiment. Moreover, the errors contained in theSite-selective Co substitution in M-type ferrites 10\ncorrection coefficients are expected to vary systematically with frequency, so that the\nqualitative trend of the intensity distribution cannot be seriously affected. Furthermore,\nit should be noted that even sophisticated EXAFS and XMCD measurements show\noverlapping spectra of several components derived from Co occupying several sites.\nIn order to quantitatively assess the amount of Co at each site, the intensity of each\ncomponent must be decomposed by fitting. In contrast, in the59Co-NMR experiment,\nthe signal derived from Co occupying each site was observed without overlap (see figure\n2). In short, the direct quantification, speed and simplicity of the59Co-NMR method\nare outstanding among experimental techniques for the same purpose. It is therefore\nundeniable that the59Co-NMR is the most appropriate method for this study. This\nstudy follows a series of studies in which59Co-NMR experiments have been applied to\nLa–Co co-substituted M-type ferrites from the above perspective [23, 49].\nIn this study single crystals were powdered for NMR measurements. As explained\nin section 2.1.1, the reason for using single crystals is to ensure that the sample is\nsingle phase. As the main objective of this study was to assess the Co occupancy\nbetween sites, single crystals were powdered. Sakai et al [11] reported a difference in\nNMR results between powdered and single crystal samples for the same sample type;\nthere is anisotropy in the excitation of Co nuclei in this system, and this anisotropy\nis site dependent [23]. This complicates the process of evaluating Co occupancy ratios\nfrom single crystal NMR spectra. In particular, the fact that the 12k site has low\nlocal symmetry and multiple local symmetry axes further complicates the analysis. In\nthis study, we have avoided this problem by powdering the crystals and measuring the\naverage intensity.\n2.1.3. Prediction of Co site preference First-principles total energy calculations based\non DFT have been used to help interpret the experimental results. The subject of\ndiscussion is the site-to-site distribution of Co, that is, atomic-level issues. One of the\nobjectives is also to perform structural optimisation calculations to discuss the stable\nstructure of the crystal. Therefore, the Vienna Ab initio Simulation Package (VASP)\n[56, 57, 58, 59] with a plane-wave basis and the projector-augmented wave (PAW)\nmethod [60, 61], which is suitable for total energy calculations with fast structure\noptimisation, was chosen as the calculation code. The use of the PAW method\nreduces the computational load. However, the PAW method does not explicitly treat\nthe core orbitals. Therefore, VASP is less effective in discussing phenomena where\nthe contribution of the core electrons is dominant, such as hyperfine interactions at\ntransition metal sites. Therefore, this DFT calculation is only used to predict the Co\nsite preference. The physics of the anharmonic lattice vibrations of Fe at the 2b site\n[50, 51, 52, 53] was also not investigated in depth in this study. Anharmonic fluctuations\nat the 2b site are likely to be intrinsically important for the multiferroic nature of\nhexagonal ferrites [53, 62]. However, we assume that the effect of Co substitution on\nthe anharmonicity of the 2b site is not pronounced, and the possibility of Co substitution\nat the 2b site has been experimentally ruled out [9, 16, 17, 18].Site-selective Co substitution in M-type ferrites 11\nPrevious DFT studies have attempted to predict Co occupancy sites in La–Co co-\nsubstituted SrFe 12O19[10, 63]. Hui et al [63] predicted that the priority occupancy site of\nCo is the 2a site in their calculations using generalised gradient approximations (GGA)\n[64, 65]. This differs from the current view. Ohtsuka et al [10] performed calculations\nusing GGA with Hubbard-type orbital interaction (GGA+ U), but failed to reproduce\nthe experimental results of the Co occupancy site. Calculations using GGA+ Uare\nrelatively expensive, but require the provision of appropriate Hubbard Uvalues for\nthe transition metal atoms. However, in addition to the fact that the appropriate\nUis not clear a priori, it has already been pointed out that DFT calculation results\nfor M-type ferrites are sensitive to U[45, 66, 67]. Our preliminary calculations using\nGGA+ Ushowed that even if a particular Ucould explain the properties of a particular\nsubstance, the same Ucould not uniformly explain the overall trend among different\nsubstances. Therefore, we will attempt a more accurate calculation that is feasible\non realistic time scales and does not include arbitrary parameters, such as U. For\nthis purpose, we have chosen the Heid-Scuseria-Ernzerhof hybrid exchange-correlation\nfunctional (HSE06) [68]. To facilitate the convergence of the calculations, we will first\nperform a structure optimisation with the GGA+ Ucalculation using the appropriate U\nand use the resulting structure parameters as initial values for the HSE06 calculation.\nCalculations using HSE06 are relatively computationally intensive, and in order\nto perform the calculations in a realistic time, the structural model must be suitably\nsimplified. The model for the Co substituent of AFe12O19is described below. First,\nthe interaction between different Co atoms was neglected, assuming that the Co atoms\nwere sufficiently dilute. This is because considering the interactions between Co atoms\nrequires considering many cases with different arrangements of neighbouring Co atoms.\nWe consider a supercell of size 2 ×2×1 times the unit cell as the smallest supercell that\nsatisfies the condition that Co is sufficiently dilute and replaces one of the 24 ×4 = 96\nFe atoms with a Co atom. This corresponds to a substitution of 0.125 Co for a Fe+Co\ncomposition of 12. This Co composition is lower than that of the samples used in\nthe NMR experiment ( ∼0.2). However, it was assumed that the condition where Co\nis sufficiently diluted is also fulfilled in the experiment, and a comparison was made\nbetween the calculations and the experiment. Supercells of the five types, with Co\nsubstituted at each Fe site, were studied.\nSecond, the effect of La substitution was assumed to be replaced by the effect of one\nadditional electron per supercell. In real materials, La3+is replaced by A2+at the same\ntime as the Co substitution for charge compensation. This means that Fe3+inAFe12O19\nis replaced by Co2+instead of Co3+. However, in models with the co-substitution of\nLa and Co, there are multiple relative positions of Co and La, and the number of such\ncases is large. This assumption is equivalent to neglecting the interaction between La\nand Co, the model seems reasonable as a first approximation. The following calculations\nconfirm that the value of the Co bond valence sum is in the range of +1.9 to +2.3 in all\ncases (except for 4f 2-Co for A= Ca, see section 3.3.3). The magnitude of the magnetic\nmoment at this time was approximately 2.6 µB/Co. In other words, Co was convergingSite-selective Co substitution in M-type ferrites 12\nTable 1. List of samples used for59Co-NMR measurements, their lattice constants at\nroom temperature and their anisotropy fields at 5 K.\nLabel Chemical formula r(A2+)†a c c/a H ARef.\n(˚A) ( ˚A) ( ˚A) (T)\nCa17 Ca 0.77La0.37Fe11.70Co0.17O19 1.34 5.8890 22.919 3.8918 2.689 [35]\nSr28 Sr 0.68La0.32Fe11.72Co0.28O19 1.44 5.8839 23.005 3.9098 2.540 [23]\nBa20 Ba 0.70La0.30Fe11.80Co0.20O19 1.61 5.8910 23.141 3.9282 2.083 –\n†12-coordinated ionic radius [69].\nto a high spin state, close to the divalent state. This suggests that the structural model\nis working to some extent. In fact, it is quite possible that the partial substitution of the\nAsite by La affects the electronic state of the system and the priority occupancy site of\nCo. To confirm this, it is necessary to perform structural optimisation calculations for\nsystems with both La and Co substitutions; this will be reserved for future research.\n2.2. Sample preparation and characterization\nThe samples used for the59Co-NMR experiments were selected from our stock of\ncrystals, one each for A= Ca, Sr and Ba in La–Co co-substituted AFe12O19, following\nthe strategy described in section 2.1.1. Samples with Co compositions around 0.2 were\nselected (see table 1). These three samples are referred to as Ca17, Sr28 and Ba20,\nrespectively. For comparison of the A-ion size, the ionic radius of the 12-coordination\nby Shannon [69] is added to table 1.\nCrystal growth techniques for Ca17 and Sr28 have been reported in [35, 23].\nSystematic studies of flux-grown single crystals of La–Co co-substituted SrFe 12O19and\nCaFe 12O19have been reported in detail [28, 35]. The Ba20 crystals were grown under\nthe same conditions as Ca17. All crystals were cooled at −5◦C/h above 1000◦C and\n−200◦C/h below 1000◦C to ensure the same cooling conditions. All compositions\nshown in table 1 are actual compositions determined for the grown crystals using SEM-\nWDX (INCA wave 500, Oxford Instruments attached to SEM S-3500H, Hitachi). For\nSr28 and Ba20, the (Fe+Co)/( A+La) composition ratio was 12, whereas for Ca17, the\n(Fe+Co)/(Ca+La) ratio was greater than 12, which is characteristic of CaFe 12O19as\nmentioned in section 2.1.1. The room temperature lattice constants given in table 1\n[23, 35] were estimated by X-ray diffraction (XRD) using a PANalytical X’Pert PRO\nAlpha-1 for partially powdered single crystals.\nThe anisotropy field HAat 5 K shown in table 1 [23, 35] was obtained from the\nanalysis of single crystal magnetisation curves, which were measured with a SQUID\nmagnetometer (Quantum Design, MPMS). The anisotropy energy EAwas evaluated\nfrom the region bounded by the easy and hard axis magnetisation curves. Using the\nvalues of EAand the saturation magnetisation Ms,HAwas obtained from the relationSite-selective Co substitution in M-type ferrites 13\nHA= 2EA/Ms. The results of our systematic magnetic measurements of La–Co co-\nsubstituted CaFe 12O19and SrFe 12O19have been reported previously [28, 35]. The\nmagnetisation curve of single crystalline BaFe 12O19[70] is qualitatively the same as\nthat of SrFe 12O19, except that the magnetic field at which the hard axis magnetisation\nreaches saturation (i.e., HA) is slightly lower. In the B20 material used in this study,\nthe effect of La–Co co-substitution on BaFe 12O19is qualitatively the same as that on\nSrFe 12O19. Although the Msvalues of the three samples were almost the same, the\nabsolute value of Msis not discussed in this paper for the following reasons. Taking into\naccount the Co distribution revealed in section 3.2, the change in magnetisation with Co\nsubstitution in this system is expected to be at most 0.5%. Co substitution makes the\nmagnetisation curve convex, and suppresses the saturation of magnetisation even under\na magnetic field of 7 T [28, 35]. In addition, when measuring the hard axis magnetisation\nof single crystals, the sample may tilt slightly during the measurement due to the large\ntorque exerted on the sample by the magnetic field. Therefore, it is technically difficult\nto experimentally detect the expected change in Msdue to Co substitution, and even if\nsuch a change is observed, it is not appropriate to simply discuss its origin.\n2.3. NMR\nThe59Co-NMR experiments were performed at IPCMS, following the strategy described\nin section 2.1.2. The nuclear quantities for59Co are the nuclear spin I= 7/2, the\ngyromagnetic ratio γ/2π= 10.05 MHz T−1, the quadrupole moment Q= 42.0×10−30\nm2and a natural abundance of 100%. For the reasons described in section 2.1.2, the\nsingle crystals were powdered for NMR measurements. Measurements were performed at\n2 K under zero external field in the frequency range 40–600 MHz using 1.28 MHz steps.\nA classical spin-echo sequence of RF pulses was used. To observe magnetic domain\nexcitations, we used wide pulses (12.8 µs) separated by a 5 µs delay. The maximum\nRF power range of our setup was used ( ≤32–40 dBm). Each spectrum results in 5,000\nto 10,000 frequency scans, depending on the mass of the sample (100 to 200 mg). To\nevaluate the signal enhancement factor, the signal intensity was measured as a function\nof RF power (four or more points) at each frequency. The NMR data for Sr28 have\nalready been reported in [23] and some of the results of this study are quoted from\nthem.\n2.4. DFT calculations\nThe stable structures and total energies of the undoped materials and Co substituents\nofAFe12O19(A= Ca, Sr and Ba) were evaluated by structure-optimisation calculations\nusing VASP. The convergence condition was set to 1 ×10−5eV (1×10−4eV for GGA+ U\npreliminary calculations) for the electronic self-consistent loop and 0.01 eV/ ˚A for the\nstructure optimisation. The k-mesh was 2 ×2×1 and the energy cutoff was 500 eV. The\nexperimental values reported in [52, 71] were used as the initial values of the structural\nparameters in the GGA+ Ucalculations. However, since CaFe 12O19does not exist inSite-selective Co substitution in M-type ferrites 14\nthe equilibrium phase, the value of SrFe 12O19was used for the calculations when A=\nCa.\nCalculations for undoped AFe12O19were carried out using the following procedure,\nassuming that the Fe sublattice has a conventional ferrimagnetic structure collinear\nalong the c-axis. (i) Preliminary GGA+ Ucalculations using arbitrary values in the\nrange 4 to 6 eV as Ufor Fe. (ii) HSE06 calculations with the structural parameters\noptimised in (i) as initial values (the results are treated as final). (iii) Optimal GGA+ U\ncalculations using the Uvalue ( U(Fe) = 4.1 eV) obtained by the method described at\nthe end of this section (the results are treated as reference data).\nFor the Co substituents, as described in section 2.1.3, the 2 ×2×1 supercell of the\nAFe12O19unit cell was considered, in which one of the Fe ions was replaced by Co2+\n(valence initially set to 2), and then an electron was added. Supercells of the five types,\nwith Co substituted at each Fe site, were studied. The magnetic moments mof the host\nFe and the doped Co were assumed to be collinear along the c-axis. In this case, m(Co)\ncan be parallel or antiparallel to m(Fe). We calculated both cases for A= Sr, but only\nthe parallel case for A= Ca and Ba, considering the results for A= Sr (section 3.3.2).\nThe calculations were carried out as follows: (i) Preliminary GGA+ Ucalculations with\nU(Fe) = 6 eV and U(Co) = 5 eV. (ii) HSE06 calculations with the structural parameters\nobtained in (i) as initial values (the results are treated as final).\nThe optimal Uvalues in the GGA+ Ucalculations were sought to reproduce\nthe total energy difference given by the HSE06 calculations for the Co substituents\nof SrFe 12O19. Under the constraint U(Fe)/ U(Co) = 1.2, the HSE06 results are\nbest reproduced when U(Fe) = 4.1 eV and U(Co) = 3.4 eV. Based on this result,\nthe structural parameters of undoped AFe12O19were again optimised by GGA+ U\ncalculations assuming U(Fe) = 4.1 eV (reference data in tables 3 and 5). However,\nthe GGA+ Ucalculations for the Co substituents of CaFe 12O19using the same Uvalues\nfailed to reproduce the experimental tendency of the Co distribution (section 3.2); the\ncalculations predicted that 12k-Co and 2a-Co are more stable than 4f 1-Co in CaFe 12O19.\n3. Results\n3.1.A-ion dependence of structural and magnetic parameters\nThe lattice constants of the three samples measured at room temperature and the\nanisotropy field, HA, at 5 K are listed in table 1. The values for Ca17 and Sr28 are\ntaken from [28, 35]. The parameter achanges little and the difference in ais at most\n0.1%. In contrast, cis highest for A= Ba and lowest for A= Ca with a difference of\napproximately 1%. This means that the effect of the A-ion size is anisotropic. This may\nbe because ais determined by the skeleton of the oxygen ions, whereas c(particularly\nthe R-block length) directly reflects the effect of the A-ion size. Thus, the change in A\nions acts as a uniaxial strain. In the following, c/ais used as a parameter to characterise\nthe structure of the three samples.Site-selective Co substitution in M-type ferrites 15\nAs shown in table 1, the anisotropic field HA(at 5 K) is negatively correlated with\ntheA-ion size and the value of c/a.\n3.2. NMR\nFigure 2(a) shows the59Co-NMR spectra of the three samples measured at zero field\nand 2 K. Three types of signals, S1, S2 and S3, were observed in all samples as in the\nprevious studies [23, 49]. An additional signal (S4) was observed at ∼530 MHz in Sr28\n[23]. The origin of S4 has been attributed to high spin Co3+, which may be present in\nvery small amounts due to an unexpected charge imbalance. However, its intensity is\nless than 0.2% of the total Co signal intensity and is therefore ignored in this study. No\nS4 signals of significant intensity were observed for Ca17 and Ba20. Although the signal\nintensity ratios vary from sample to sample, it is noteworthy that three similarly shaped\nsignals were observed in almost all common frequency ranges. This indirectly suggests\nthat various parameters other than the differences in Aions remain nearly constant\nduring crystal growth. The intensity of S1 was the highest for all samples. In figure\n2(a) the signal intensity is normalised to the intensity of S1. The centre frequencies of\nS1, S2 and S3, evaluated by fitting to a Gaussian function, are listed in table 2. They\ndiffered slightly from sample to sample. This frequency shift is discussed in the last part\nof section 4.3.\nFigure 2(a) shows that the intensities of S2 and S3 tend to decrease with decreasing\nA-ion size ( A= Ba→Sr→Ca). The proportions P(j) (j= S1, S2, S3) of the integrated\nintensities of S1, S2 and S3 for each sample are shown in figure 2(b). As the A-ion size\ndecreased, P(S1) increased. P(S3) decreased slightly and the change was small. In\ncontrast, P(S2) decreased significantly and at A= Ca it decreased to approximately\n1.5% of the total. That is, although S2 and S3 were observed in similar frequency ranges,\ntheA-ion dependence of their intensities was large for P(S2) and small for P(S3).\nThe site assignments of the S1, S2 and S3 resonances when A= Sr have been\ndescribed in detail in section 3.1 of [23]. In this study, we apply this conclusion to\nthe other A-ion cases as well; the S1 resonance is assigned to 4f 1-Co and the remaining\nS2/S3 are assigned to 2a/12k-Co. Kobayashi et al [16, 17, 18] reported that Co occupies\nthree sites: 2a, 4f 1and 12k, with 4f 1occupying the largest fraction, also for La–Co co-\nsubstituted CaFe 12O19, according to studies using synchrotron radiation and neutrons.\nThis result supports our site assignment. In [23] we could not conclude whether S2 or S3\nwas assigned to 2a-Co or 12k-Co. However, considering the cation diffusion processes\nduring the synthesis procedure, the occupancy of 12k-Co is expected to be sensitive\nto various conditions (including A-ion differences), whereas that of 2a-Co is relatively\ninvariant (see section 4.2 for details). The A-ion dependence of the signal intensity was\nlarge in S2 and small in S3 (figure 2(b)), qualitatively suggesting an S2 →12k and S3\n→2a assignment.\nBased on the site assignment above, the Co 4f 1site occupancy P(4f1) was highest\natA= Ca and lowest at A= Ba. Figure 3(a) shows the anisotropy field HAof eachSite-selective Co substitution in M-type ferrites 16\nSpin-echo intensity (arb. units)6005004003002001000Frequency (MHz)(a)La-Co co-substituted AFe12O19A = CaSrBaS1S2S3\nCa17 Sr28 Ba2000.20.40.60.81\nS1S2S3P(j )(b)\nFigure 2. (a)59Co-NMR spectra for La–Co co-substituted AFe12O19measured at\nzero applied field and at 2 K. The intensity was normalised so that the maximum\nintensity of S1 is the same. (b) Intensity ratios of S1, S2 and S3 for the three samples.\n0.70.80.91\n3.88 3.89 3.9 3.91 3.92 3.93 3.94P(4f1)\nc/aA = CaBaSrLa–Co co-substituted AFe12O19(b)\n123\n0.6 0.7 0.8 0.9 1HA (T)\nP(4f1)CaSrA = BaLa–Co co-substituted AFe12O19(a)\nFigure 3. (a) Dependence of the anisotropy field on the 4f 1site selectivity of Co. (b)\nc/adependence of the 4f 1site selectivity of Co in La–Co co-substituted AFe12O19(A\n= Ca, Sr and Ba) evaluated from the59Co-NMR signal intensity. The straight lines\nin (a) and (b) are visual aids.\nsample at 5 K plotted against P(4f1) (= P(S1)). These values are almost proportional\nto each other, indicating that the 4f 1site selectivity of Co does indeed dominate the\nmagnetic anisotropy.\nAs mentioned in section 3.1, the difference in Aions appears as a uniaxial strain.\nFigure 3(b) shows the c/adependence of P(4f1) for each sample. P(4f1) increased with\ndecreasing c/a. It is therefore hypothesised that the uniaxial compressive strain of the\ncrystal improves the 4f 1site selectivity of Co and consequently the uniaxial magnetic\nanisotropy. As the 4f 1site is a minority spin site, the magnetisation is simultaneously\nincreased. To quantitatively evaluate the effect of different Co compositions of theSite-selective Co substitution in M-type ferrites 17\nTable 2.59Co ferromagnetic nuclear resonance frequencies of La–Co co-substituted\nAFe12O19at zero field and at 2 K, corresponding internal fields and their site\nassignments. Each59Co-NMR line was fitted with a Gaussian function and the median\nwas taken as the resonance frequency. The direction of the local magnetic moment at\neach site and the direction of the internal field are defined as positive if they are parallel\nand negative if they are antiparallel.\nSignal νres(MHz) Hint(T) Site\nCa17 Sr28 Ba20 Ca17 Sr28 Ba20\nS1 82.4(4) 83.6(4) 89.4(2) −8.20(4) −8.32(4) −8.90(2) 4f 1\nS2 327.0(29) 308.3(6) 301.3(4) 32.54(29) 30.68(6) 29.98(4) 12k\nS3 384.5(15) 389.1(5) 391.8(2) 38.26(15) 38.72(5) 38.99(2) 2a\nTable 3. Lattice parameters, c-axis lengths of S and R blocks, and unit cell volumes\nofAFe12O19(A= Ca, Sr and Ba) obtained by structure-optimisation DFT calculations\nwith HSE06 and GGA+ U(U(Fe) = 4.1 eV). Available experimental values at room\ntemperature are also given. No experimental values are available for A= Ca because\nthe undoped CaFe 12O19is not stable.\nA Method a(˚A) c(˚A) c/a c (S) (˚A) c(R) ( ˚A) V(˚A3) Ref.\nCa HSE06 5.880 22.892 3.893 5.031 6.415 685.4\nGGA+ U 5.917 22.963 3.881 5.041 6.441 696.2\nExperiment – – – – –\nSr HSE06 5.885 23.017 3.912 5.025 6.483 690.4\nGGA+ U 5.931 23.158 3.905 5.052 6.527 705.5\nExperiment 5.884 23.050 3.917 5.039 6.486 691.1 [52]\nBa HSE06 5.897 23.162 3.928 5.013 6.568 697.5\nGGA+ U 5.943 23.307 3.921 5.038 6.615 713.0\nExperiment 5.892 23.183 3.935 5.019 6.573 697.0 [71]\nsamples used in the experiments, one can refer to a previous study on La–Co co-\nsubstituted SrFe 12O19[23]. In this system, the change in c/awas approximately 0.08%\nwhen the Co concentration was changed by approximately 0.1. Assuming that the\nCo occupancy ratio depends only on the value of c/a, the expected change in P(4f1)\ndoes not exceed 0.02 (see figure 3(b)), which is sufficiently small compared with the\nchange when the Aion is replaced. When A= Ca, the Ca composition is greater\nthan its stoichiometric composition. If excess Ca occupies the 4f 1site, as proposed by\nKobayashi et al [18], the number of 4f 1sites available for Co to occupy would be reduced,\nand hence P(4f1) in Ca17 would be expected to be suppressed compared to Sr28 and\nBa20. However, this was actually increased. This suggests that even if Ca partially\noccupies the 4f 1site, the effect is insignificant and does not affect the discussion here.Site-selective Co substitution in M-type ferrites 18\nTable 4. Atomic coordinates of AFe12O19(A= Ca, Sr and Ba) obtained by structure-\noptimisation DFT calculations with HSE06. Experimental data for SrFe 12O19and\nBaFe 12O19were taken from [52] and [71], respectively.\nElement/Site HSE06 GGA+ U Experiment\nx y z x y z x y z\nCaFe 12O19\nCa/2d 0.6667 0.3333 0.2500 0.6667 0.3333 0.2500 – – –\nFe/2a 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 – – –\nFe/2b 0.0000 0.0000 0.2500 0.0000 0.0000 0.2500 – – –\nFe/4f 1 0.3333 0.6667 0.0270 0.3333 0.6667 0.0275 – – –\nFe/4f 2 0.3333 0.6667 0.1913 0.3333 0.6667 0.1915 – – –\nFe/12k 0.1691 0 .3382 0.8902 0.1688 0.3375 0.8902 – – –\nO/4e 0.0000 0.0000 0.1521 0.0000 0.0000 0.1525 – – –\nO/4f 0.6667 0.3333 0.0556 0.6667 0.3333 0.0557 – – –\nO/6h 0.1817 0 .3633 0.2500 0.1821 0.3643 0.2500 – – –\nO/12k 1 0.1572 0 .3144 0.0527 0.1563 0.3125 0.0530 – – –\nO/12k 2 0.5056 0 .0112 0.1525 0.5061 0.0123 0.1523 – – ��\nSrFe 12O19\nSr/2d 0.6667 0.3333 0.2500 0.6667 0.3333 0.25001\n32\n31\n4\nFe/2a 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0 0\nFe/2b 0.0000 0.0000 0.2500 0.0000 0.0000 0.2500 0 0 0.2542\nFe/4f 1 0.3333 0.6667 0.0270 0.3333 0.6667 0.02711\n32\n30.0272\nFe/4f 2 0.3333 0.6667 0.1909 0.3333 0.6667 0.19101\n32\n30.1909\nFe/12k 0.1689 0 .3378 0.8908 0.1688 0.3376 0.8909 0.1689 2 x0.8907\nO/4e 0.0000 0.0000 0.1516 0.0000 0.0000 0.1520 0 0 0.1516\nO/4f 0.6667 0.3333 0.0552 0.6667 0.3333 0.05542\n31\n30.0552\nO/6h 0.1817 0 .3634 0.2500 0.1819 0.3638 0.2500 0.1817 2 x1\n4\nO/12k 1 0.1570 0 .3141 0.0524 0.1568 0.3136 0.0525 0.1565 2 x0.0527\nO/12k 2 0.5041 0 .0083 0.1511 0.5044 0.0088 0.1512 0.5047 2 x0.1508\nBaFe 12O19\nBa/2d 0.6667 0.3333 0.2500 0.6667 0.3333 0.25002\n31\n31\n4\nFe/2a 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0 0\nFe/2b 0.0000 0.0000 0.2500 0.0000 0.0000 0.2500 0 01\n4\nFe/4f 1 0.3333 0.6667 0.0272 0.3333 0.6667 0.02721\n32\n30.0273\nFe/4f 2 0.3333 0.6667 0.1903 0.3333 0.6667 0.19031\n32\n30.1904\nFe/12k 0.1687 0 .3374 0.8918 0.1686 0.3372 0.8919 0.1687 2 x0.8917\nO/4e 0.0000 0.0000 0.1508 0.0000 0.0000 0.1511 0 0 0.1504\nO/4f 0.6667 0.3333 0.0555 0.6667 0.3333 0.05492\n31\n30.0548\nO/6h 0.1818 0 .3635 0.2500 0.1820 0.3640 0.2500 0.1821 2 x1\n4\nO/12k 1 0.1568 0 .3136 0.0519 0.1565 0.3130 0.0520 0.1563 2 x0.0521\nO/12k 2 0.5021 0 .0042 0.1491 0.5023 0.0046 0.1492 0.5024 2 x0.1493\n3.3. DFT calculations\n3.3.1. Undoped AFe12O19We have carried out structure-optimisation DFT\ncalculations on AFe12O19to obtain information on the stable lattice constants and local\nstructure of AFe12O19in the undoped form. The lattice constants obtained from theSite-selective Co substitution in M-type ferrites 19\nTable 5. Volume of the Fe–O coordination polyhedron of each Fe site in undoped\nAFe12O19(A= Ca, Sr and Ba) calculated from atomic coordinates obtained by\nstructure-optimisation DFT calculations with HSE06 and GGA+ U(U(Fe) = 4.1 eV)\n(table 4). Experimental values calculated from the available atomic coordinates at\nroom temperature are also given. No experimental values are available for A= Ca, as\nthe undoped material CaFe 12O19is not stable.\nA Method V(˚A3) Ref.\n2a 2b 4f 1 4f2 12k\nCa HSE06 10.714 6.650 3.454 10.527 10.926\nGGA+ U 10.805 6.758 3.557 10.711 11.039\nExperiment – – – – –\nSr HSE06 10.693 6.730 3.461 10.646 10.946\nGGA+ U 10.922 6.863 3.559 10.871 11.187\nExperiment 10.706 6.737 3.498 10.730 10.893 [52]\nBa HSE06 10.671 6.860 3.475 10.829 10.983\nGGA+ U 10.897 7.002 3.572 11.062 11.223\nExperiment 10.671 6.871 3.482 10.783 10.999 [71]\ncalculations are given in table 3, the atomic coordinates in table 4 and the volume Vof\nthe Fe–O coordination polyhedron of each Fe site calculated from the atomic coordinates\nin table 5. The values obtained using GGA+ Uare also listed for reference, together\nwith calculations using HSE06. The experimental values at room temperature [52, 71]\nare also given for Sr and Ba ferrites. The S-block length c(S) and R-block length c(R)\ncalculated for AFe12O19are also given in table 3, together with the c-axis length. The\nsum of c(S) and c(R) is equal to c/2. As shown in figure 1, the 2b and 4f 2coordinated\npolyhedra belong to the R block, while the 2a and 4f 1coordinated polyhedra belong to\nthe S block. The 12k site is located at the boundary between the R and S blocks. As\na general trend, the experimental values in tables 3, 4 and 5 are closer to the values\ncalculated using HSE06 than those calculated using GGA+ U. This indicates that the\ncalculations using HSE06 are more accurate than those using GGA+ U. In the following\ndiscussion only the values calculated using HSE06 are used.\n3.3.2. Co substituted SrFe 12O19The total energy E(i) of the supercell, in which some\nFe3+in SrFe 12O19was replaced by Co2+, was evaluated by DFT calculations with HSE06,\nwhere idenotes the Fe sites (2a, 2b, 4f 1, 4f2and 12k) occupied by Co. In all cases, Fe\nand Co converged to trivalent and divalent high spin states, respectively. Calculations\nwere carried out for both cases where the moment of substituted Co is in the same\ndirection ( ↑↑) and opposite direction ( ↑↓) to the magnetic moment of Fe in the host\n(Fe site where Co is substituted). When Co occupies the majority spin sites (12k, 2a,\n2b),↑↑indicates that the Co moment is parallel to the net magnetisation, ↑↓indicates\nthat the Co moment is antiparallel to the net magnetisation, and when Co occupies the\nminority spin sites (4f 1, 4f2), note that ↑↑indicates that the Co moment is antiparallel\nto the net magnetisation and ↑↓indicates that the Co moment is parallel to the netSite-selective Co substitution in M-type ferrites 20\nTable 6. Energy change evaluated by DFT calculations with HSE06 when one of\nthe Fe atoms in the AFe12O19supercell is replaced by a Co atom, based on the energy\nwhen Co occupies the 4f 1site and has a moment in the same direction as the host Fe.\nA Momenta∆E(meV)\n2a 2b 4f 1 4f212k\nCab↑↑ 43 1628 0 – 174\nSr ↑↑ 30 840 0 818 139\nBa ↑↑ 66 796 0 797 109\nSr ↑↓ 334 954 260 1222 362\naRelative spin direction of substituted Co and host Fe. ↑↑and↑↓denote cases where\nthe moment of the substituted Co is in the same or opposite direction to the host Fe,\nrespectively.\nbNo stable solution with Co2+occupying the 4f 2site was obtained in the case of A=\nCa (see text).\n0200400600800100012001400ΔE (meV)4f2\n4f12a12k2bCo in SrFe12O19\n↑↑↑↓\nFigure 4. Energy change evaluated by DFT calculations with HSE06 when each\nFe site in SrFe 12O19is replaced by Co. Both cases where the Co moment is in the\nsame direction as and opposite to the magnetic moment of the host Fe are shown. The\nenergy of the case where Co occupies the 4f 1site and the moment of Co is in the same\ndirection as the magnetic moment of the host Fe is taken as the origin.\nmagnetisation.\nThe total energy was minimised when Co occupied the 4f 1site with ↑↑. The energy\ndifference ∆ E(i) was then defined with respect to E(4f1) for↑↑. These values are shown\nin table 6 and figure 4. First, regardless of the site occupied by Co, the ↑↑case has\nan energy of 100–400 meV lower than the ↑↓case. Therefore, the magnetic moment of\nCo in the ground state is always stable in the same direction as that of the host Fe.\nAlthough there is no experimental evidence that AFe12O19is↑↑in La–Co co-substitutedSite-selective Co substitution in M-type ferrites 21\n050100150200\nCaSrBa4f12a12k\n050010001500\nCaSrBaΔE (meV)Co in AFe12O194f22b\nFigure 5. Energy change of AFe12O19(A= Ca, Sr and Ba) evaluated by DFT\ncalculations with HSE06 when each Fe site is replaced by Co. The origin is the energy\nwhen Co occupies the 4f 1site. In the case of A= Ca, a state where Co2+occupies the\n4f2site was not obtained as a stable solution (see text). The right figure is an enlarged\nview of the low energy part.\nAFe12O19, the above results provide the first evidence for the assumption of ↑↑in this\nsystem. In the following, only the case of ↑↑is considered.\nThe total energy E(i) of Co substituted SrFe 12O19isE(4f1)< E(2a)< E(12k)≪\nE(4f2)< E(2b). Thus, Co occupies the 4f 1site in the ground state. However, as\ndiscussed in section 4.2, the cation distribution in a real material freezes at a certain\nfinite temperature. ∆ E(2b) and ∆ E(4f2) are large of ∼1 eV, whereas the synthesis\ntemperature of the M phase never exceeds 2000 K; therefore, Co practically does not\noccupy 2b or 4f 2sites. On the other hand, ∆ E(2a) and ∆ E(12k) are ∼100 meV;\ntherefore, if we assume that the cation distribution freezes at approximately 1000 K, Co\nis distributed in three sites, 4f 1, 2a and 12k. In principle, it is impossible to selectively\nreplace only one specific site with Co that is more or less distributed among the three\nsites. The above calculations are in agreement with the experimental results (figure 2),\nin which Co was distributed over three sites and mainly occupied the 4f 1site.\n3.3.3. Co-substituted CaFe 12O19and BaFe 12O19To test the hypothesis that the Co\ndistribution correlates with the uniaxial strain of the crystal, we performed the same\ncalculations as in the previous section for CaFe 12O19and BaFe 12O19with different A-ion\nradii, and compared the results. We assumed that ↑↑is generally more stable than ↑↓\nfor La–Co co-substituted AFe12O19, and for A= Ca and Ba we only calculated in the\n↑↑case.\nSince E(4f1) was minimal for all A= Ca, Sr and Ba, we defined ∆ E(i) with\nE(4f1) as the origin, as in the previous section. The ∆ E(i) values for each system\nare shown in figure 5 and table 6. In the case of A= Ca, calculations usingSite-selective Co substitution in M-type ferrites 22\nHSE06 for the supercell with Co substituted for Fe at the 4f 2site show that the\nCo2+state is unstable. In other words, the Co valence converged to a significantly\nhigher state ( ∼+2.5) even when the Co valence value was initially set to +2, and the\nFe valence at the 2a site decreased accordingly. As a result, the magnetisation per\nsupercell did not change from the undoped state. Therefore the total energy of the\n4f2-Co case for A= Ca cannot be compared with the other cases. The relationship\nE(4f1)< E(2a) < E(12k) ≪E(4f2)< E(2b) holds in all cases (except the Ca 4f 2-\nCo case), but the overall trend is that the distribution of ∆ E(i) is largest for A=\nCa and smallest for A= Ba. In all cases, ∆ E(2a) and ∆ E(12k) are of the order of\n100 meV, whereas ∆ E(4f2) and ∆ E(2b) are of the order of 1 eV. This implies that\nat finite temperatures of ∼1000 K, Co tends to aggregate at the most stable 4f 1site\nwhen A= Ca, whereas it tends to be evenly distributed among the three sites when A\n= Ba. This trend is in qualitative agreement with the experimental results (figure 2),\nconfirming that the Co distribution is correlated with the uniaxial strain of the crystal.\nThe quantitative trends are discussed in section 4.2.\n4. Discussion\n4.1. Orbital contribution to hyperfine fields\nThe internal magnetic field at the Co nucleus is given by Hint=Aspinmspin+\nAorbmorb, where AspinandAorbare the spin and orbital components of the hyperfine\ncoupling constants, respectively, and mspinandmorbare the spin and orbital moments,\nrespectively. Since |Aspin| ≪ |Aorb|in general, when the magnetic moment of Co has an\norbital component morb, the resonance frequency depends significantly on the magnitude\nofmorband cannot be predicted in advance. Conversely, the frequencies of S1, S2 and S3\ndiffer significantly due to the different magnitudes of morb. The S1 resonance shifts to\nthe high frequency side when an external magnetic field is applied [11]. In other words,\nan external magnetic field is added to the internal field at the 4f 1site. Since the 4f 1site\nis a minority spin site, the magnetic moment of Fe at the 4f 1site is antiparallel to the\nexternal field when the sample particles are oriented in the magnetic field. Furthermore,\nwe assumed that the magnetic moment of the substituted Co is in the same direction\nas the magnetic moment of the host Fe, considering the results of the DFT calculations\n(section 3.3.2). In this case, the above observation implies that the sign of the internal\nfield at Co occupying the 4f 1site is negative. However, since the 2a and 12k sites\nare in octahedral coordination, there may be degeneracy in the ground state orbitals\nand residual orbital moments. In general, since Aorbhas a large positive value, the\norbital moment produces a relatively large field. Therefore, the internal fields of S2\nand S3 were assumed to be positive, although their sign was experimentally unknown.\nThe magnitude of the internal field at each site, evaluated on the basis of the above\nassumptions, is shown in table 2.\nAssuming Aspin=−12 T/ µB[72] and Aorb= +65 T/ µB[73], and fixing mspinatSite-selective Co substitution in M-type ferrites 23\nCa Sr Ba1000 K12k2a4f1Ca Sr Ba800 K(a)(b)\n00.20.40.60.81\nCa Sr Ba400 KP(i )00.51\n0 500 1000 1500Co in AFe12O19P(i )\nT (K)12k2a4f1BaA = CaSr\nFigure 6. (a) Temperature dependence of the probability of Co occupancy at\neach Fe site evaluated based on the energy difference obtained from DFT calculations\nwith HSE06 for AFe12O19(A= Ca, Sr and Ba). (b) Schematic of Co distribution of\nAFe12O19expected at specific temperatures (400, 800 and 1000 K).\n3µB/Co for convenience, the Co orbital moments morbat 4f 1(S1), 12k (S2) and 2a\n(S3) for the sample with A= Sr are estimated to be approximately 0.4, 1.0 and 1.1\nµB/Co. Although these may be overestimated (if |Aspin|andmspinare overestimated),\nthe relative magnitude appears to be reliable. These results suggest that relatively large\nmorb’s remain in Co occupying the 2a and 12k sites in octahedral coordination, while a\nsmall morbis induced in Co at the 4f 1site in tetrahedral coordination through spin-orbit\ninteractions and interactions with ligands. The orbital field at the 4f 1site is not large\nenough to cancel the spin component of the internal field, resulting in a relatively small\nnegative internal field. At the 2a and 12k sites, the orbital field was approximately twice\nas large as the spin field, resulting in a large positive magnetic field.\nAccording to [23], only Co at the 4f 1site was effective in enhancing the macroscopic\nanisotropy. At first glance, it may seem strange that despite the large orbital moments\nof Co at the 2a and 12k sites, they do not contribute to the anisotropy enhancement.\nHowever, there is no contradiction if only the Co at the 4f 1site is uniaxially anisotropic\nand the Co at the 2a and 12k sites is in-plane anisotropic. Cluster calculations taking\ninto account interactions with ligands show that the sign of the magnetic anisotropy\nof each site is sensitive to the local strain and can be positive or negative [6, 7].\nTherefore, it is not unnatural for the above situation to be realised in M-type ferrites.\nIt is concluded that the presence of Co in octahedral coordination is undesirable for\nenhancing macroscopic uniaxial anisotropy, and that Co at the 2a and 12k sites should\nbe reduced as much as possible to improve performance as a hard magnet.\n4.2. Temperature dependence of Co distribution\nThe results of the DFT calculations provided information on the ground state. In\nreal materials, the cation distribution freezes at a finite temperature. Therefore,Site-selective Co substitution in M-type ferrites 24\nwe considered the cation distributions at finite temperatures using the Gibbs free\nenergy. The Gibbs free energy change when Co occupies site iis given by ∆ G(i) =\n∆E(i)+p∆V(i)−T∆S(i), where ∆ E(i) is the energy difference in the ground state when\nCo occupies site i, as defined in section 3.3.2. ∆ V(i) and ∆ S(i) are the corresponding\nvolume and entropy changes, respectively, pis the pressure, and kBis the Boltzmann\nconstant. The pVterm can affect the results, mainly through the thermal expansion\nof the material. However, the volume change of SrFe 12O19between 0 K and 1000 K\nis approximately 3% [74], and the change in pVat this time can be roughly estimated\nto be on the order of 1 neV. This is sufficiently small compared with the ∆ Ewe are\ndiscussing (on the order of 10–100 meV) to be negligible. The entropy change ∆ S(i)\nconsists of the contributions of the configuration and the vibrations (phonons, magnons,\netc). If the number of sites to be exchanged does not change with temperature, the\ncontribution from the configuration entropy does not affect the statistical average of the\nphysical quantity. The vibrational contribution is estimated to be approximately 0.1–0.2\nkB/atom for binary substitutional alloys [75]. The corresponding T∆S(i) is of the order\nof 10 meV at T= 1000 K, and the possibility of affecting the total Gibbs energy cannot\nbe ruled out. However, the site dependence of T∆S(i) due to vibrations is non-trivial\nand is ignored for simplicity in this analysis. In this case, the probability P(i) of Co\noccupying site iat temperature Tis determined by ∆ E(i). Using classical Boltzmann\nstatistics, P(i) is given by P(i) = giexp(−∆E(i)/kBT)/P\njgjexp(−∆E(j)/kBT),\nwhere giis the multiplicity in the unit cell of site i.\nFigure 6(a) shows the temperature dependence of P(i) for each sample. In the\nequilibrium state at 0 K, Co should only occupy the 4f 1site. However, as the freezing\ntemperature of the cation increases, the probability of occupancy of the 2a site in the\nfirst excited state P(2a) increases. Subsequently, the probability of occupancy of the 12k\nsite in the second excited state P(12k) increases. At relatively high temperatures, P(2a)\nandP(12k) are reversed . At temperatures above 1000 K, P(2a) saturates or slowly\ndecreases. This is because P(2a) approaches the upper limit with a small multiplicity\nof sites, whereas the 12k sites have a sufficient number of sites. Close to the upper\ntemperature limit where the M phase is stable, P(4f1) and P(12k) are comparable.\nAs mentioned in section 2.1.1, the samples used in this study may have contained\nFe2+. From the charge balance, the amount of Fe2+is approximately 0.04–0.1 relative\nto 12 Fe. It is known that Fe2+prefers to occupy the 2a site [44]. Therefore, in\nthe equilibrium state at 0 K, Fe2+is expected to occupy at most 10% of the 2a site.\nHowever, the extra electron associated with Fe2+is expected to hop from site to site as\nthe temperature increases [45]. The effect is averaged over all Fe sites and should not\nsignificantly affect the discussion here.\nTo observe the A-ion dependence in detail, figure 6(b) shows the Co distribution\nfor each system at specific temperatures (400, 800 and 1000 K). As a qualitative trend,\nP(4f1) is largest for A= Ca and smallest for A= Ba at relatively high temperatures,\nbut this relationship does not necessarily hold at temperatures below 500 K. That is,\nP(4f1) of BaFe 12O19increases relatively rapidly at low temperatures.Site-selective Co substitution in M-type ferrites 25\nIn actual material synthesis, the cation distribution at a given temperature is not\nnecessarily maintained at room temperature, even if heat treatment is carried out at\nthat temperature. For example, in the spinel compound MgAl 2O4, the time required to\nequilibrate the ion distribution is a few seconds above 1400 K, but more than 10,000\nmin below 900 K [19]. Assuming that our compounds require a similar duration to\nequilibrate, at an experimental cooling rate, e.g. −200◦C/h, the cation distribution\nabove 1200 K is transformed to a low temperature equilibrium state, whereas at a\nlower temperature below 900 K the temperature drops before the cation distribution\nreaches equilibrium. Therefore, under normal sample preparation conditions, a cation\ndistribution at ∼1000 K is maintained at room temperature. The Co distribution at 1000\nK in figure 6(b) shows that P(4f1) is the highest for A= Ca and the lowest for A= Ba.\nThis trend is qualitatively consistent with the trends observed experimentally (figure\n2). Quantitatively, P(4f1) tends to be overestimated in the59-Co NMR experiments.\nAs mentioned in section 3.2, the experimental error in P(j) seems to be too large to\nmake quantitative comparisons. Nevertheless, the calculations confirmed that uniaxial\ncompressive strain improves the 4f 1site selectivity of Co.\nAs mentioned above, P(4f1) increases as a lower temperature cation distribution\nis achieved. Since the time to reach equilibrium is longer at lower temperatures, it is\nexpected that P(4f1) will increase by post-annealing at lower temperatures or by slowing\nthe cooling rate. In other words, it is concluded that post-annealing and slow cooling at\nlow temperatures are effective in increasing the uniaxial anisotropy and improving the\ncoercivity and magnetisation of Co-substituted ferrite magnets. When the temperature\nvariation of P(4f1) is steeper in BaFe 12O19than in SrFe 12O19and CaFe 12O19, as shown\nin figure 6(a), the post-annealing effect is most pronounced in the BaFe 12O19system.\nThe site assignment of the S2 and S3 Co-NMR signals is mentioned at the end\nof this section. As mentioned above, when a sample is prepared by the usual solid\nstate reaction method, an equilibrium cation distribution of approximately 800–1500\nK is frozen, regardless of the cooling rate or post-annealing, taking into account the\ndiffusion rate of the atoms. Therefore, if we look more closely at the variation of P(i)\nin the 800–1500 K range in figure 6(a), the following general trends can be expected. (i)\nThe temperature variation of P(2a) is relatively small and almost independent of the\nfreezing temperature. (ii) Although the freezing temperature dependence of P(4f1) and\nP(12k) is large, the signs of their temperature coefficients are opposite, and the sum of\nP(4f1) and P(12k) tends to be constant. (iii) The A-ion dependence is largest at P(12k).\nThese facts suggest that P(2a) is relatively insensitive to perturbations such as A-ion\nexchange, freezing temperature variation, and concentration changes, whereas P(12k) is\nexpected to be sensitive to these perturbations. This may be mainly due to the higher\nmultiplicity of sites at the 12k site compared with the 2a site. Figure 2(b) shows that\nthe experimental A-ion dependence was large for S2 and small for S3. Furthermore,\naccording to the59Co-NMR spectrum of La–Co co-substituted SrFe 12O19reported in\n[23], the intensity of S3 hardly changes with the Co concentration, whereas that of S2\ndecreases significantly with increasing Co concentration (with compression along theSite-selective Co substitution in M-type ferrites 26\n-101234\nCaSrBaΔV(i ), Δ(c/a) (%)\n2a2b4f24f112kc/aAFe12O19(b)\n-101234\nCaSrBaΔc(S), Δc(R), Δc (%)\nc(S)c(R)cAFe12O19(a)\nFigure 7. A-ion dependence of the relative change in structural parameters of\nundoped AFe12O19with respect to CaFe 12O19obtained by DFT calculations with\nHSE06. (a) R-block length, S-block length and c. (b) Volume of the coordination\npolyhedron of each Fe site and c/a\nc-axis). In the light of the above expected trends and experimental results, it seems\nreasonable to assign S2 →12k and S3 →2a. It should be noted that this argument is\nonly an inference based on qualitative trends. However, this site identification, whether\ncorrect or incorrect, does not affect the discussion of the 4f 1site selectivity of Co in this\nstudy.\n4.3. Effect of local structural distortion on Co distribution\nIn section 3.3.3, we noted that Co distribution correlates with the uniaxial strain of the\ncrystal. If the energy change associated with the Co substitution (figure 5) is determined\nonly by the uniform uniaxial lattice strain, then ∆ E(i) is expected to vary monotonically\nwith the A-ion size, and indeed ∆ E(i) decreases monotonically with increasing the A-\nion size ( c/adecreases) at most sites. However, the 2a site does not follow this trend\nand ∆ E(2a) varies non-monotonically. That is, ∆ E(2a) decreases for A= Ca→Sr, but\nconversely increases for A= Sr→Ba. To investigate the cause, we discuss the effect of\nlocal deformation in the lattice based on the atomic coordinates of undoped AFe12O19\nevaluated by DFT calculations (table 4).\nThe relative changes in c(S) and c(R) (with respect to CaFe 12O19) are shown in\nfigure 7(a). As the A-ion size increased, c(R) increased in response to c, but the amount\nof change was approximately three times that of c. In contrast, c(S) showed a negative\ndependence on the A-ion size, but the change was small. This means that the effect of\nA-ion expansion is not uniform within the crystal and the S block absorbs some of the\nexpansion of the R block. This may be because the R block to which the Aion belongs\nis directly affected by the Aion, whereas the S block is not.\nSimilar non-uniform changes can also be observed in the volume V(i) (table 5)\nof the coordination polyhedron at each Fe site. Figure 7(b) shows the relative changeSite-selective Co substitution in M-type ferrites 27\nTable 7. Coordinates of oxygen ions forming 2a and 4f 1coordination polyhedra, and\naspect ratios of 2a and 4f 1coordinated polyhedra in undoped AFe12O19(A= Ca, Sr\nand Ba) obtained by DFT calculations with HSE06.\nA Oxygen coordinateaAspect ratiob\nx z z′r(2a) r(4f1)r(12k)\nCa 0.1572 0.0527 0.0556 1.0445 0.9848 0.9707\nSr 0.1571 0.0523 0.0553 1.0429 0.9823 0.9677\nBa 0.1568 0.0519 0.0547 1.0407 0.9790 0.9625\naCoordinate in O-12k 1(x,2x, z) and O-4f (2\n3,1\n3, z′).\nbRatio of the oblique (side length of side triangle) hto the base (side length of\nbase triangle) b,r=h/b.r(12k) is the average of the different rvalues in the 12k\noctahedron.\n∆V(i) ofV(i) (with respect to CaFe 12O19). The changes in c/aare included in the\nsame figure for comparison. The ∆ V(2b) and ∆ V(4f2) at the R-block sites increased\nsignificantly with the expansion of the Aion (increase in c/a). The ∆ V(12k) at the\nR/S block boundary increased only slightly with the expansion of the Aion, which\nmay be due to the combined effect of the large expansion of the R block and the small\ncontraction of the S block, as well as the greater degree of freedom in the position of the\noxygen ions forming the 12k coordination polyhedron than at other sites. In contrast,\n∆V(2a) decreased slightly with increasing A-ion size. This is because the height of\nthe 2a coordination polyhedron along the c-axis is strongly correlated with the S-block\nlength, as can be seen from the similarity with the relative change in the S-block length\nc(S) in figure 7(a).\nIt is interesting to note the A-ion dependence of the volume of the 4f 1coordination\npolyhedron belonging to the same S block. The A-ion dependence of ∆ V(4f1) (figure\n7(b)) shows a trend that contradicts the change in the length of the S block (figure\n7(a)) to which it belongs. This is due to the fact that ∆ V(2a) and ∆ V(4f1) are not\nindependent. Figure 8(a) schematically shows the projection of the S block onto the c-\nplane. The 4f 1tetrahedron shares the vertices of its base triangle with the 2a octahedron,\nand the sum of the lengths of the base of the 4f 1tetrahedron and the base of the 2a\noctahedron is equal to the lattice constant a. The Wyckoff positions of the coordinating\noxygens of the 2a octahedron are all 12k 1(x,2x, z), the length of the base is 3 x×a\nand the height along the c-axis is 2 z×c. The 4f 1tetrahedron shares the vertices of\nits base triangle with a 2a octahedron and has a base length of (1 −3x)×a. The\nremaining vertex of the 4f 1tetrahedron is 4f (2\n3,1\n3, z′) and has c-axis degrees of freedom\nindependent of the 2a octahedron. In this case, the height of the 4f 1tetrahedron along\nthec-axis is ( z+z′)×c. Thus, if the c-axis displacements of the oxygen ions at the\n12k 1and 4f positions are comparable, V(4f1) and V(2a) are determined by the in-plane\ndisplacement of the oxygen ion at the 12k 1position. If ais invariant, ∆ V(4f1) and\n∆V(2a) show a trade-off tendency.\nThe coordinates of the oxygen positions 12k 1(x,2x, z) and 4f (2\n3,1\n3, z′) in the 2aSite-selective Co substitution in M-type ferrites 28\n2a2a2a2a4f14f1a\nO-12kO-4f(a)\n-2-10\nCaSr BaΔz, Δz', Δc(S) (%)Δc(S)ΔzΔz'(c)-0.4-0.200.20.4\nCaSr BaΔx, Δa (%)(b)ΔaΔx\nFigure 8. (a) 2a and 4f 1coordination polyhedra viewed along the c-axis. Only\nthe oxygen ions are shown. The atomic coordinates of the ligand oxygen ions (closed\ncircles, O-12k 1site) in the 2a octahedron (yellow) are given by ( x,2x, z). In this case,\nthe length of the base is 3 x×aand the height along the c-axis is 2 z×c. The vertices of\nthe 2a octahedron are also those of the base triangle of the 4f 1tetrahedron (red). The\n4f1tetrahedron has a base length (1 −3x)×a. The sum of the base length of the 4f 1\ntetrahedron and the base length of the 2a octahedron is equal to the lattice constant\na. The atomic coordinates of the remaining vertex of the 4f 1tetrahedron (open circle,\nO-4f site) are given by (2\n3,1\n3, z′) and have c-axis degrees of freedom independent of the\n2a octahedron. The c-axis height of the 4f 1tetrahedron is ( z+z′)×c. If the c-axis\ndisplacements of O-12k 1and O-4f are comparable, the volumes of the 4f 1tetrahedron\nand the 2a octahedron are determined by the c-plane displacement of O-12k 1and\nshow a trade-off tendency if ais invariant. (b), (c) Relative changes in the coordinates\nof the oxygen atoms constituting the 2a and 4f 1coordination polyhedra ( xandzin\nO-12k ( x,2x, z) and z′in O-4f (2\n3,1\n3, z′)) and lattice constants ( aand S-block length\nc(S)) in undoped AFe12O19(A= Ca, Sr and Ba) (CaFe 12O19basis) obtained by DFT\ncalculations with HSE06.\n-10-50\n303540\nCaSrBaHint (T)\nHint (T)12k4f12aCo in AFe12O19(b)\n0.951.001.05\nCaSr Ba Aspect ratioAFe12O19(a)2a\n4f112k\nFigure 9. (a) The A-ion dependence of the ratio of the oblique side to the base of\nthe 2a, 4f 1and 12k coordinated polyhedra in AFe12O19(A= Ca, Sr and Ba) obtained\nby DFT calculations with HSE06. (b) Internal magnetic fields at nuclear sites of Co\noccupying 2a, 4f 1and 12k sites in La–Co co-substituted AFe12O19.\nand 4f 1coordination polyhedra in undoped AFe12O19(A= Ca, Sr and Ba) calculated by\nDFT calculations are given in table 7. The relative changes with respect to CaFe 12O19\nare shown in figures 8(b) and (c). The zandz′, which correspond to the heightsSite-selective Co substitution in M-type ferrites 29\nof the coordination polyhedra along the c-axis, vary almost similarly to A, and the\ndisplacement of the oxygen ions along the c-axis in the S block is almost uniform.\nHowever, x, which corresponds to the length of the sides of the triangle in the c-plane,\ndecreases with increasing A-ion size (slightly increasing for a). This means that the\nbase of the 4f 1tetrahedron is enlarged and the base of the 2a octahedron is reduced.\nThis explains why the A-ion dependence of ∆ V(2a) and ∆ V(4f1) show opposite trends,\nbut it is not clear whether the 2a octahedron or the 4f 1tetrahedron is expanding or\ncontracting.\nNext we look at the changes in the 2a-octahedron and the 4f 1-tetrahedron from a\nsymmetry point of view. Table 7 and figure 9(a) show the ratio r=h/bof the length\nbof the base (side of a triangle parallel to the c-plane) and hof the hypotenuse (side\nof a side triangle) of each polyhedron, calculated using DFT calculations, where r= 1\ncorresponds to a regular polyhedron. In the 2a octahedron, r > 1, the octahedron\nis elongated along the c-axis, but the stresses due to the A-ion expansion cause rto\ndecrease and the 2a octahedron approaches the regular octahedron. On the other hand,\nin the 4f 1tetrahedron, r <1, the tetrahedron contracts along the c-axis, but the stresses\ndue to A-ion expansion reduce rand further increase the distortion. That is, the 2a\noctahedron (containing isotropic Fe3+) has room to stabilise against c-axis stress by\nincreasing symmetry, whereas the 4f 1tetrahedron has no such room. As a result, the\n4f1tetrahedron expands, and the 2a octahedron, which has a high symmetry and spatial\nmargin, absorbs the distortion, thereby stabilising the whole system. It can be said that\nthe 2a octahedron plays a buffering role in absorbing strain in the crystal.\nComparing figure 5 and figure 7(b), the change in ∆ E(i) is approximately inversely\ncorrelated with the change in ∆ V(i). In other words, a large positive volume change\nin the coordination polyhedron tends to increase the probability of Co occupancy. This\nmay be due to the decrease in elastic energy loss when Co2+, which has a larger ionic\nradius than Fe3+, occupies the Fe sites. Finally, the non-monotonic A-ion dependence\nof ∆E(2a) shown in figure 5 can be interpreted as the 2a coordination polyhedron\ncontracting with the expansion of A, unlike the other sites. The fact that there is a trade-\noff between V(4f1) and V(2a) may become apparent during the final optimisation process\nfor the performance of the M-type ferrite magnet. In conclusion, lattice distortion is\nnot necessarily uniform and differences in local distortion also have a secondary effect\non the distribution of Co sites.\nWe mentioned in section 3.2 that the resonance frequencies of the S1, S2 and S3\nresonances (table 2) shift slightly depending on the Aion. The resonance frequencies\nwere converted to internal fields based on the site assignment described above, and are\nshown in figure 9(b). The internal field shifted positively at the 2a site and negatively\nat the 4f 1and 12k sites with increasing A-ion size. The aspect ratio rin figure 9(a)\ncorresponds to the deviation from the cubic symmetry. The rof the 12k coordination\noctahedron, which is the average of the base and oblique sides at the 12k site, has\nalso been added to the figure. Comparing (a) and (b) in figure 9 we can see that as r\napproaches 1, that is, as the coordination polyhedron approaches cubic symmetry, theSite-selective Co substitution in M-type ferrites 30\ninternal field shifts in the positive direction. Assuming that mspinis invariant, a positive\nshift in the internal field corresponds to an increase in morb. Thus, it can be seen that\nmorbtends to increase as the symmetry approaches cubic symmetry.\n5. Summary\nIn the La–Co co-substituted M-type ferrite AFe12O19, Co mainly occupies the 4f 1site\n(minority spin site) in tetrahedral coordination, and some Co occupies the 2a and 12k\nsites (majority spin sites) in octahedral coordination. However, as proposed in [23], only\nCo occupying the 4f 1site was found to be effective in enhancing uniaxial anisotropy. To\nimprove the performance (anisotropy and magnetisation) of M-type ferrite magnets with\na limited amount of Co, it is important to concentrate Co at the 4f 1site. However, their\nmagnetic properties depend on the type and concentration of the Aions, even if the\nCo content is the same. Previous studies have shown that the decrease in c-axis length\ndue to contraction of the Aion, that is, uniaxial compressive strain, increases the 4f 1\nsite selectivity of Co. In this study, we performed59Co-NMR on La–Co co-substituted\nM-type ferrites with different Aions, that is, A= Ca, Sr and Ba (ion sizes are Ca2+<\nSr2+<Ba2+) with Co compositions close to 0.2, and experimentally confirmed the site\ndistribution of Co. At the same time, DFT calculations were performed for undoped\nAFe12O19and ( AFe12O19)8with Co2+replacing one Fe atom, [( AFe11.875Co0.125O19)8]−,\nand the stable structure and preferential site occupancy of Co were predicted using\nfirst-principles calculations. The results were as follows.\n(i) In La–Co co-substituted AFe12O19, it was experimentally confirmed that Co\nconcentrates at the 4f 1site as Aions become smaller.\n(ii) The resonance frequency (resonance field) of Co occupying each Fe site was\ndistributed over a wide frequency (field) range, which was mainly attributed to\nthe difference in the orbital components of the magnetic moment. The increase in\nlocal symmetry at each site correlates with an increase in the orbital field.\n(iii) DFT calculations confirmed that when Co occupies the Fe site, its magnetic moment\nis stable in the same direction as the moment of the host Fe.\n(iv) According to DFT calculations, the most stable case for any Aion is when Co\noccupies the 4f 1site, followed by the 2a and 12k sites with energy differences of the\norder of 100 meV. Co at the 2b and 4f 2sites is unstable (the energy difference is\nof the order of 1 eV) and Co practically does not occupy these sites. Thus, Co is\nexpected to occupy mainly the 4f 1site, with some Co occupying the 2a and 12k\nsites, which is in agreement with the experimental results.\n(v) DFT calculations predicted that the energy difference of Co occupying different\nFe sites tends to be larger for smaller Aions. This means that the smaller the\nAion, the higher the 4f 1site selectivity for Co, which is in agreement with the\nexperimental results.Site-selective Co substitution in M-type ferrites 31\n(vi) According to the DFT calculations, the strain due to the contraction of the Aions\nis not uniform throughout the crystal, but has a positive effect (contraction) in\nthe R block and a negative effect (expansion) in the S block. Therefore, the Co\noccupancy of each Fe site is not necessarily determined by uniform strain, but is\nalso influenced by differences in the local strain. In particular, the order of Co\noccupancy at the metastable 2a and 12k sites may depend on the type of Aion.\n(vii) Because the ionic size of Co2+is larger than that of Fe3+, the probability\nof Co2+occupancy generally tends to increase with increasing volume of the\noxygen coordination polyhedron at each Fe site. However, the crystal structure,\nparticularly the S block, is characterised by a trade-off between the volume of the\n4f1coordination polyhedron and that of the 2a coordination polyhedron, which may\nbe important in the final stage of performance optimisation.\n(viii) As the cation distribution is frozen at a finite temperature in the actual sample\npreparation, the 4f 1site occupancy of Co increases as the cation distribution is\nrealised at a lower temperature, taking into account the effect of thermal excitation.\nTherefore, post-annealing at low temperature and slow cooling is expected to be\neffective in improving performance.\nThe above results are briefly summarised in terms of guidelines for improving\nperformance (anisotropy and magnetisation) of La–Co co-substituted M-type ferrite\nmagnets with a limited amount of Co. It is effective to select Aions with the smallest\npossible size and to post-anneal at a low temperature or cool slowly to concentrate Co\nat the 4f 1site in tetrahedral coordination.\nAcknowledgments\nThis work was supported by the Elements Strategy Initiative Center for Magnetic\nMaterials (ESICMM), Grant Number JPMXP0112101004, through the Ministry of\nEducation, Culture, Sports, Science and Technology (MEXT), Japan, and JSPS\nKAKENHI 17K06793, 19K05002 and 22H00263. The authors would like to thank H.\nNishida for his help in preparing Ba-based crystals.\nConflict of interest\nThe authors declare no competing interests.\nORCID iDs\nHiroyuki Nakamura https://orcid.org/0000-0001-7085-4800\nHiroto Ohta https://orcid.org/0000-0002-0268-9468\nTakeshi Waki https://orcid.org/0000-0002-3294-9814\nYoshikazu Tabata https://orcid.org/0000-0002-4447-2160Site-selective Co substitution in M-type ferrites 32\nHidekazu Ikeno https://orcid.org/0000-0002-3840-4049\nChristian M´ eny https://orcid.org/0000-0003-0718-161X\nReferences\n[1] For example, Pullar R C 2012 Prog. Mater. Sci. 571191\n[2] For example, de Juli´ an Fern´ andez C, Sangregorio C, de la Figuera J, Belec B, Makovec D and\nQuesada A 2021 J. Phys. D 5415300\n[3] Iida K, Minachi Y, Masuzawa K, Kawakami M, Nishio H and Taguchi H 1999 J. Magn. Soc. Jpn.\n231093\n[4] Momma K and Izumi F 2011 J. Appl. Crystallogr. 441272\n[5] Fuchikami N 1965 J. Phys. Soc. 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Phys. 7411"    },    {        "title": "1505.06438v2.Structural_Properties__Impedance_Spectroscopy_and_Dielectric_Spin_Relaxation_of_Ni_Zn_Ferrite_Synthesized_by_Double_Sintering_Technique.pdf",        "content": " \n*Corresponding author: mohi@cuet.ac.bd  Structural Properties, Impedance Spectroscopy and Dielectric Spin Relaxation  of Ni -Zn Ferrite \nSynthesized by  Double Sintering Technique  \nM. A. Ali1, M. N. I. Khan2, F.-U.-Z. Chowdhury1,S. Akhter2,and M. M. Uddin1* \n1Department of Physics, Chittagong University of Engineering and Technology (CUET), Chittagong -4349, Bangladesh.  \n2Materials Science Division, Atomic Energy Center, Dhaka -1000, Bangladesh.  \nAbstract  \nStructural properties, i mpedance,  dielectric and electric modulus spectra have been used to investigate the sintering \ntemperature ( Ts) effect on the single phase cubic spinel Ni 0.6Zn0.4Fe2O4 (NZFO) ceramics synthesize d by standard ceramic \ntechnique . Enhancement of dielectric constants is observed with increasing Ts. The collective contribution of n -type and \np-type carriers yields a clear peak in notable unusual dielectric behavior is successfully explained by the Rezlescu model. The  \nnon-Debye type lon g range dielectric relaxation phenomena is explained by electric modulus formalism.  Fast response of the \ngrain boundaries of the sample sintered  at lower Ts sample leading to small dielectric spin relaxation time,  (several \nnanoseconds) have been determined using electric modulus spectra  for the samples sintered  at different Ts. Two clear \nsemicircles in impedance Cole -Cole plot have also been successfully explained by employing two  parallel  RC equivalent \ncircuits in  series  configuration  taking into account no electrode contribution.  Such a long relaxation time in NZFO ceramics  \ncould suitably be used in nanoscale spintronic devices.  \n \nKeywords:  Ferrites ; spin relaxation time ; Dielectric relaxation ; complex impedance spectroscopy ; electric modulus.  \n1. Introduction  \nThe spinel ceramics of MFe 2O4 have attracted much attention in recent years for microwave applications owing to their \nremarkable magnetic properties  and high resistivity values and low eddy current loss in high frequency operations . \nThese properties make them  promising candidates for the applications in radio frequency coils, transformer cores, rod \nantennas, magnetic cores of read write heads , high-speed digital tape or disk recording [1 -5]. These magnetic insula tors \nwith high Neel temperature ( TN) can be used in spintronics and spin caloritronics as they naturally provide a non -zero \nmagnetic moment along with spin dependen t band gaps [6,7]. The spinel ferrites belong to AB 2O4 structure such as \nNi-Zn ferrites and having tetrahedral A site and octahedral B site in a unit cell are such potential candidates. The ( Ni, \nZn) Fe 2O4 ferrite with a face-centered Fd\n3 m structure  is one of the MFe 2O4 phase family and could be described as \n[Fe 1-x3+ Znx2+]A [Ni 1-x2+ Fe1+x3+]BO4, where “A” represents the tetrahedral site, “B”  represents the octahedral site, and “ x” \ndenotes the degree of inversion.  \nNickel ions have preference for B -sites while zinc ions retain at A -sites and iron ionsare distributed over both of \nthe sites [5].The Zn ions decrease during the calcination and sintering processes which creates  cation vacancies and \nunsaturated oxygen ions . The  excessive electrons on oxygen will bond with the neighboring Fe3+ions, which give rise \nto Fe2+ions. As a result, there is a possibility of electron hopping existing in B-sites within the ferrite leading to \nreduction of resistivity [8]. Additionally, the excellent relaxation behavior of Ni -Zn spinel ferrite makes it a strong \ncandidate  in the field of relaxors.  The parameters which can sensibly alter the properties of ferrites are method of \npreparation, composition, sintering temperature, sintering time and rate of sintering [9,10]. Strategies will now need to \nbe identified to probe and develop their properties in suitable conditions for technological applications.  The structural \nand electrical properties of spinel ferrites are closely related. In some pr evious report s the structural, electrical  and \nmagnetic properties of Ni-Zn ferrites , prepared by different techniques, have  been investigated [11-20]. \nThe effect s of sintering temperature  on the structural, dielectric, resistivity and magnetic properties of \n(Ni,Zn)Fe 2O4ferrite have been investigated by Costa et al.  [11], Kothawale  et al [19], Rao et al.  [20] and Hossain et al. \n[21]. Some authors have reported the impedance spectroscopy and dielectric relaxation as well as transport properties \nof (Ni,Zn)Fe 2O4ceramics [22,23] . But t o the best of our knowledge , there are no reporting on the effect of sintering \ntemperature  on the impedance spectroscopy, non -Debye type dielectric relaxation, electron spin life time and electric \nmodulus in conventional double sin tering derived  Ni0.6Zn0.4Fe2O4 ceramics. In order to take the advantages of this \ncompound for eventual technological applications, an overall investigation is necessary.  \nIn this work , we have reported the  structural  properties , impedance spectroscopy, dielectric spin relaxation and \nelectric modulus of Ni 0.6Zn0.4Fe2O4 ceramics over a wide frequency range, prepared by the conventional double \nsintering method. The contribution of grain and grain boundary resistance in impedance of  the NZFO has been studied \nby impedance spectroscopy. The  correlation between the dielectric spin relaxation time and sintering temperature,  Ts \nhave been elucidated  by analyzing the electric modulus  with the established formalism .  \n2. Materials and Methods  \nThe polycrystalline samples with basic chemical composition of Ni 0.6Zn0.4Fe2O4 were synthesized by solid state \nreaction route using high purity (99.5%) (US Research Nanomaterials, Inc.) oxide precursors of nickel oxide (NiO), \nzinc oxide (ZnO) and iron oxid e (Fe 2O3) nano powders with particle sizes  20-40, 15 -35 and 35 -45 nm, respectively  in \nrequired stoichiometry. The analyt ical grade of powders was weigh ed according to the corresponding composition, and were mixed and ground for 3 hrs using an agate morta r and pestle. The slurry was drie d and loosely pressed into cake \nusing a hydraulic press. The cake  is pre-sintered in air for 3 hrs at 9 00°C . The pre -sintered cake was removed from the \nfurnace and it was crushed  and ground again for 1 h r. The powder thus obtained was then pressed using a suitable die in \nthe form of pellets of dimensions 8.4mm diameter and 1.1 mm thickness with a hydraulic press with a pressure of 10 \nkN for pellets using 5% polyvinyl alcohol solution as a binder. The samples  were finally sintered at 1200, 1250 and \n1300°C for 4 hrs in air at atmospheric pressure. The crystalline phases of the prepared samples were studied by X -ray \ndiffraction (XRD) using Philips X’pert Pro X -ray diffractometer (PW3040) with Cu -Kα radiation (λ = 1.5405 Å). The \nXRD data were collected at slow scan (2 /min) with a step size of 0.02  in a wide range of Bragg angles 15 -70 at room \ntemperature. Dielectric parameters such as capacitance, loss factor, impedance, and phase angles were measured at \nroom t emperature by a Wayne Kerr precision impedance analyzer ( 6500B) in the frequency range of 10 Hz to 100 MHz  \nwith drive voltage 0.5V. All the experimental data presented in this study were obtained at room temperature.  \n3. Results and Discussion  \n \n3.1. Structural properties  \nThe XRD patterns of the samples  sintered  at different Ts are shown in Fig. 1. Very sharp and well -defined peaks have \nbeen observed from the XRD pattern correspond to cubic spinel phase of the NZFO with Fd\n3 m space group \nsymmetry. No secondary phase peaks have observed, indicating the high quality and single phase of synthesized \nsample. It is also observed that the diffraction peaks become narrower and sharper with the augment  of Ts, representing \nan increase of the c rystallite size, which has also been confirmed by SEM  micrographs of the NZFO  as shown in Fig. \n1(b-d).The corresponding positions of all the sharp peaks are used to obtain the lattice parameters such as the lattice \nconstant, a0 (to determine accurate a0, the extrapolation technique has been applied between the lattice parameter and \nthe Nelson -Riley function F(), and the average value of a0 is found to be 8.39187A ). The X-ray density, x=8M/NAa03 \n(NA is Avogadro's number, M is the molecular weight), bul k density, b=M/V (V is the volume of the samples and V = \nπr2h, r = radius of the samples,  h = height of the samples), and the porosity P(%)=  (((x- b)/x)  100) have also been \ncalculated . The average values of a0, x, b, and P(%) are found to be 8.39187 Å, 5.33 gm/cm3, 3.98 gm/cm3 and \n25.34%, respectively . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nA room temperature surface morphology or microstructure of the samples  sintered at different Ts has been studied \nusing SEM micrograph as shown in Fig. 1 (b -d). The images exhibit clear grains and grain boundaries. The average \ngrain size (grain diameter) has been determined from the micrograph s by linear intercept technique . The values are Fig. 1. (a) The XRD patterns of the Ni0.6Zn0.4Fe2O4(NZFO) ferrites at different sintering temperature ( Ts). \n(b-d) show r oom temperature scanning electron microscopy  (SEM)  images of the NZFO sintered  at Ts \n1200, 1250 and 1300 C, respectively.   \n \n20 30 40 50 60 700100200300400\n1300oC\n1250oC\n1200oC\n  Intensity (a. u.)\n(deg.)(220)(311)\n(222)(400)(422)(511) (440)\n(111)(a)(a)(a)\n(a)(a)\n(d)\n (c) (b)found to be  1.4 - 7.8m, which are almost homogenously distributed throughout the sample surface. However, the grain \nsize is obtained in the range of 6.2 - 7.8 m for the sample sintered at 1300  C. \n3.2 Dielectric relaxation properties  \nThe frequency dependence  of the dielectric permittivity ( ) [ =CL/0A where, C is the capacitance of the pellet, L is \nthe thickness of the pellet, A is the cross -sectional area of the flat surface of the pellet and ε0 is the constant of \npermittivity for free space] and tan of the NZFO ceramics  have been measured at room temperature as  depicted in Fig. \n2. It is seen that the value of  decreases with increase in frequency, which is typical characteristics of the polar \ndielectrics [24]. A remarkable increase in the value of  has been observed for the sample sintered at higher  \ntemperature . Commonly, four types of polarizations (i.e., interfacial, dipolar, atomic and electronic) contribute in \ndielectric permittivity in ferr ites [8, 22] , among them dipolar and interfacial polarizations are strongly temperature \ndependent. Above mentioned polarization dynamics occur at the low frequency region, so permittivity increase s faster \nleadin g to a high dielectric constant.  The phonon a nd electron hopping enhance at measuring room temperature results \nthe permittivity increases for the samples sintered at higher Ts (Fig. 2).  It is evident that Zn is evaporated at higher \nsintering temperature and creates vacancies in the crystal.  The Fe elements have two valence states of Fe2+ and Fe3+. \nSome of the ions Fe3+ from the B site convert to Fe2+ ions due to compensate the Zn loss which accelerates  the hopping \nprocess between the Fe ions in +2 and +3 states.  More and more charges reach t he grain boundary due to increase in \npolarization process results accumulation of charges at the grain boundary of the NZFO ceramics  upturns.  As a \nconsequence, the permittivity increases for the samples sintered at higher Ts.  Moreover , at the lower frequency region \nincrease of permittivity is more prominent. The  decrease of  permittivity with the increase of frequency could be \nunderst ood by Maxwell -Wagner type dielectric structure. The development of bigger grains and boundaries as well  as \nboundary reaction become more remarkable that  the dipoles and boundary ions cannot be able to follow changes of the \nfield causing faster polarizationin high frequencies region  leading to decrease even zero value of . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFrequency dependent loss factor  (tan, is defined as /) for the samples sintered at different Ts for the NZFO is \nshown in Fig. 2.  The curve shows the dielectric relaxation processes (peaks) at the particular frequency which  are \nshifted at the higher frequency value. The dielectric relaxation peaks appear when the externally applied AC electric \nfield becomes equal to that of the jumping frequency of localized electric charge carrier [25]. It is reported that a visible  \npeak in  tan with increasing angular frequency could be observed if the value of  decreases and imaginary dielectric \nconstant,, slowly decreases with frequency [26, 27]. The obtained unusual behavior of dielectric  properties  in the \nNZFO  could be explained by the Rezlescu model as shown in Fig. 2(b) [28]. It states  that the contributions of n -type \nand p -type carriers decrease very slowly and rapidly with angular frequency increases, respectively. However, the \ncollective contribution of bo th types of carriers to the dielectric polarization yields a clear peak with the frequency. The \nobserved dielectric peak of the NZFO is closely compared with the Rezlescu model  (red curve in Fig. 2 (b)).  The \nelectric conduction in ferrites arises due to th e electron exchange between Fe2+ and Fe3+ and hole transfer between Ni3+ \nand Ni2+ [29] at the octahedral ( B) sites which is similar to that of dielectric polarization in ferrites [28]. The Fe and Ni \nions are produced by the following mechanism;  Ni2+ + Fe3+ ↔ Ni3+ + Fe2+, Fe3+ ↔ Fe2+ + e−.  \n Fig. 2.Variation of the dielectric permittivity (ε´) and loss factor (tanδ) of the Ni 0.6Zn0.4Fe2O4 ferrites sintered at \ndifferent T s as a function of frequency. (b) shows the Rezlescu model for the dielectric relaxation peaks.  \n1021031041051061071080246\n  ´ (106)\nFrequency, f (Hz) 1200  C\n 1250  C\n 1300  C\n0246810\n 1200  C\n 1250  C\n 1300  C\ntan(a)  \n3.3 Electric modulus  \nComplex electric modulus is a well -known and powerful tool which has been effectively used to analyze the electrical \nresponse of the materials such as nature of polycrystalline in samples (homogenous or inhomogeneous ) and the \nelectrical relaxation in electronically and ionically conducting materials as a microscopic property of the materials [30]. \nThe real part of electric modulus ( M) and impedance ( Z) as a function of frequency  for the samples  sintered  at \ndifferent Ts has been depicted in Fig.  3. The value of M reaches a maximum at high frequency region with reaching to \nzero at low frequency indicating the electrode polarization contribution is negligible [31]. The value of Z decreases \nmonotonically w ith increases in frequency as seen in Fig. 3. It is also observed that the value of Z´ shows dispersion \npattern at different  sintering temperature,  Ts in low frequency region followed by a plateau . Finally  all the curves \ncoalesce with approaching zero value  representing  Z´ is independent of frequency. The space charge has lesser time to \nrelax at higher frequency region and the recombination is faster which reduces the space charge polarization leading to \na merge of the curves of all the samples sintered  at different  Ts [32].  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nA combined plot of imaginary electric modulus M  [the complex electric modulus M*= M + jM, M=/(2 + 2) \nand M=/(2 + 2)] and impedance Zas a function of frequency is useful to detect the effect of the smallest \ncapacitance, the largest resistance and assists to distinguish whether a relaxation process is due to short or long -range \nmovement of charge carriers as proposed by Sinclair and West [33]. They also reported that if the peak in M versus \nfrequency and Z versus frequency will occur at the same frequency then the process is long range and if these peaks \nwill occur at different frequencies then the process is localized. The plot M versus Z as a function of frequency for the \nsamples sintered  at different Ts is shown in Fig. 4  (a). The presence of localized movement of charge carriers and \ndeparture from ideal  \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig.3.  Frequency dependent impedance  real part ( Z) and electric modulus ( M) of the \nNi0.6Zn0.4Fe2O4ferrites  sintered  at different Ts. \n1011021031041051061071080510152025\n Frequency, f (Hz)(a)\n  Z'' (  105) \n  1200  C\n  1250  C\n  1300  C\n-5051015  1200  C\n  1250  C\n  1300  C\nM'' (  10-3)\n \n1200 1250 13000204060\n  \n (ns)\nTs (oC)(b)\n10110210310410510610710804812\n  Z' ( 104)\nFrequency, f (Hz) 1200  C\n 1250  C\n 1300  C\n024\n 1200  C\n 1250  C\n 1300  C\nM' ( 10-2)\nFig. 4. (a)Frequency dependence of imaginary part of impedance ( Z) and electric modulus ( M) of the \nNi0.6Zn0.4Fe2O4ferrites  sintered at various Ts. (b) The dielectric spin relaxation time  for the samples sintered  at different  \nTs. Debye -like behavior has been proved by the separation between these peaks as shown in Fig. 4  (a) [34]. Moreover, the \nrelaxation of the electric field in the materials is denoted by the electric modulus since the electric displacement \nremains constant. Thus, the peak value in Mrepresents the real dielectric spin relaxation process of the materials where \nthe peak  frequency is indicative  of transition from long to short range mobility with increase in frequency . The \nrelaxation time is determined by the relation M= 1/(2fM), fMis the frequency at which M(ω) is maximum ( Mmax). \nThe dielectric spin relaxation time  () has been calculated from M vs. frequency plot as shown in Fig. 4  (b). Particular \nnoteworthy is that the value of  is in the range of several nano  seconds  and decreases with increasing sintering \ntemperarure , Ts. At higher sintering temperarure , the grain boundary movement faster than that of low sintering \ntemperarure  results the dielectric  spin easily relax. However, t he spin -relaxation times of semiconductor materials are \ntypically ∼100 ns, 150 ns in cobalt nan oparticle and they are on the order of picoseconds in bulk metals due to the high \ndensity of scattering centres  [35, 36].  Such a long dielectric spin-relaxation time in the NZFO ceramics can  be very \nuseful in nanoscale spintronic devices. The imaginary par t of Z shows monotonous decrease with increasing  frequency \nof the samples sintered  at various Ts as illustrated in Fig. 4  (a).   \n3.4 Impedance spectroscopy  \nThe complex impedance spectrum (Nyquist plots/ Cole -Cole plot) technique is a well -known and powerful tool which \nhas been effectively used for complete understands the electrical properties of the electro -ceramic materials such as \nimpedance of electrodes , grain  and grain boundaries. It also provides the information about the resistive (real part) and \nreactive (imaginary part) components of a material.  A Cole -Cole plot  typically comprises of two successive \nsemicircles: first semicircle is due to the contribution of the grain boundary at low frequency and second is due to the \ngrain or bulk properties  at high frequency of the materials.  \n \n \n \n \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 shows the room temperature complex impedance or Cole –Cole plot for the samples sintered at different  Ts as \na function of frequency. Two partially overlapping semi -circles are found whose centers are lying below the real axis, \nindicating a non -Debye type of  relaxation process as discussed in previous section.  A sample is assumed that a \nmicrostructure made up of parallel conducting plates (grains) separated by resistive plates (grain boundaries). \nAccording to the Maxwell –Wagner two layer model, the phenomena is typically related to the existence of a \ndistribution of relaxation time, in which the resulting complex impedance is composed of two overlapping semicircles \n[37, 38].These two semicircles could be modeled  by two parallel  RC equivalent circuits  in series  configuration as \nshown in Fig. 5  (d).The complex impedance of a system at an applied frequency can be written as sum of the real and \nimaginary parts:  Z*() =Z(ω) + i Z(ω), where Z and Z of the impedance can be represented as,  \n   \n 2 21 1gb gb gbgb\ngggg\nRCR\nRCR\nZ\n  \n    (1) Fig. 5. (a-c) Cole–Cole plot of complex impedance  of the Ni0.6Zn0.4Fe2O4ferrites  sintered  \nat different  Ts. (d) shows the equivalent circuit.  \n0 1 2 3 4 5 6 7012\n  Z(\nZ(103) Expt.\n Fitted(c) 1300 C\n0 5 10 15 20024\n  Z ( 103)\nZ ( 103)\n  \n Expt.\n Fitted(b) 1250 oC\n0 2 4 6 8 10 12012\nZ ( 104) Z ( 104)\n  \n Expt.\n Fitted(a) 1200 C\nRg\nCgRgb\nCgb(d) Equivalent circuit   \n22\n22\n1 1gb gb gbgb\ngggg\nRCR\nRCR\nZ\n  \n    (2) \nWhere Rg and Cg represent the resistance and capacitance of the grain and Rgb and Cgb represent the corresponding terms \nfor grain boundary, while ωg and ωgb are the frequencies at the peaks of the semicircles for grain and grain boundary, \nrespectively. The resistances are calculated from the circular arc intercepts on the Z axis, while the capacitances are \nderived from the maximum height of the circular arcs.  The maximum height in each semicircle is Z = −Z. We fit our \ndata using the above equation s and obtained parameters are listed in Table -1. It is noticed that t he circular arcs  in the \nlow frequency regions are not completed which may be due to the fact that grain boundary resistance is out of \nmeasurement scale.  \nTable 1: Impedance parameters of the Ni0.6Zn0.4Fe2O4 ferrites at room temperature.  \nTS (°C) \nParameters  1200  1250  1300  \nτM(ns)  50 8 8 \nRg (kΩ)  29.9 2.66 0.93 \nRgb (kΩ)  101.2  17.7 5.3 \nCg (nF) 0.18 0.03 0.017  \nCgb(nF) 1.57 29.9 7.5 \nτg (g=1/g=RgCg)  (ns)  538 80 16 \nτgb (gb=1/gb=RgbCgb) (μs)  158 529 39.8 \n \n4. Conclusions  \nIn summary, we have prepared and studied the structural, dielectric and electrical properties of single phase cubic \nspinel Ni 0.6Zn0.4Fe2O4 ferrites a t various sintering temperature . An increase in dielectric constants is explained by the \nelectron hopping mechanism between Fe2+ and Fe3+ states. Remarkable unusual dielectric behavior is due to the \ncollective contribution of both n and p -type carriers which is fruitfully explained by Rezlescu model. Irregular long \nrange and non -Debye type dielectric relaxation are attributed by the electric m odulus. The complex impedance spectra \nshow contribution of both grain and grain boundary effects in the electrical properties, endorsed by the classical \nCole-Cole plot. Long dielectric spin relaxation times in nanosecond range have also been determined fro m the electric \nmodulus spectra that make the NZFO as a promising candidate for future nanoscale spintronic devices.  \nAcknowledgements: The authors are grateful to the authority of CUET (DRE grant) for providing financial support \nfor th is work .  \nReferences  \n1. B. D. Cullity, Elements of X -ray Diffraction, 2nd ed. (Addison -Wesley, Reading, MA, 1978).  \n2. M. Sugimoto, J. Am. Ceram. Soc. 82, 269 (1999).  \nDOI:  10.1111/j.1551 -2916.1999.tb20058.x.  \n3. Q. Chen, P. Du, W. Huang, L. Jin, W. Weng, and G. Han, Appl. Phys. Lett. 90, 132907 (2007).  \nhttp://dx.doi.org/10.1063/1.2718482  \n4. W. Eerenstein, N. D. 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J¨ onsson,1, 4Olle Eriksson,1, 5and Patrik Thunstr¨ om1,∗\n1Department of Physics and Astronomy, Uppsala University, Box-516,Uppsala SE-751 20 Sweden\n2Universit´ e de Strasbourg, IPCMS, CNRS-UNISTRA UMR 7504, 67034 Strasbourg, France\n3Orexplore AB, Torshamngatan 30B, 164 40 Kista, Stockholm, Sweden\n4School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, South Korea\n5Wallenberg Initiative Materials Science, Uppsala University, Box-516,Uppsala SE-751 20 Sweden\n(Dated: January 9, 2024)\nThe theoretical calculation of the interaction of electromagnetic radiation with matter remains a\nchallenging problem for contemporary ab initio electronic structure methods, in particular for x-ray\nspectroscopies. This is not only due to the strong interaction between the core-hole and the photo-\nexcited electron, but also due to the elusive multiplet effects that arise from the Coulomb interaction\namong the valence electrons. In this work we report a method based on density-functional theory\nin conjunction with multiplet ligand-field theory to investigate various core-level spectroscopies, in\nparticular x-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD).\nThe developed computational scheme is applied to the L2,3XAS edges of magnetite (Fe 3O4) as\nwell as cobalt ferrite (CoFe 2O4) and nickel ferrite (NiFe 2O4) and the corresponding XMCD spectra.\nThe results are in overall good agreement with experimental observations, both regarding the XAS\nL2/L3branching ratio, the peak positions as well as the relative intensities. The agreement between\ntheory and experiment is equally good for XAS and the XMCD spectra, for all studied systems. The\nresults are analyzed in terms of egandt2gorbitals contributions and the importance of optimizing\nthe Slater parameters. The analysis also highlights the strong effect of the 2 p-3dinteraction in x-ray\nspectroscopy.\nI. INTRODUCTION\nCore-level spectroscopy is one of the most powerful\ntechniques for probing the electronic structure of strongly\ncorrelated electron systems, as the interaction between x-\nrays and matter gives element-specific insights into the\nmagnetic and electronic properties of materials, as well\nas their structural and chemical properties. In partic-\nular, the development of synchrotron radiation facilities\nand recently, x-ray free electron lasers, have made the\nanalysis efficient and the resolution very high in energy\nand in specific cases, in time [1]. This increase in the\nexperimental resolution has made it possible to detect\nfine features and structures in the x-ray spectra [2, 3].\nThis development of experimental tools and techniques\ncalls for a novel theoretical approach to compute x-rays\nabsorption spectra, most notably, for systems with com-\nplex or correlated electronic structures, like magnetites\nas well as cobalt- and nickel ferrites. In this work, we\nfocus specifically on these materials and on new insights\nfor the theoretical description of x-ray absorption spec-\ntroscopy (XAS) and x-ray magnetic circular dichroism\n(XMCD).\nFor the theoretical modelling of core-level and x-ray\nabsorption spectroscopy, there have been two main dis-\ntinct methodologies that are primarily used today. The\nfirst one is based on density functional theory (DFT) and\nthe Kubo formula [4] while the second focuses on multi\nconfiguration effects [5]. Both methodologies are well\n∗patrik.thunstrom@physics.uu.sereviewed in the literature (see for example Ref. [6, 7]).\nThe DFT-based methods are essentially parameter-free\nand have, as a main advantage, the description of mag-\nnetism, structural properties and chemical bonding in\ngeneral. These methods are designed for the ground-\nstate properties of materials, and hence their description\nof the excited states probed with x-ray spectroscopy is\nvery limited. This is due to an incomplete treatment\nof electronic correlations, in particular the multiconfig-\nurational effects. Overall, these methods work best for\nwidely dispersive electron systems. In contrast, atomic\nmultiplet theory is essentially intended for a better de-\nscription of the multiconfigurational effects originating\nfrom on-site electron-electron interaction [8]. For practi-\ncal computational efficiency, the multiplet theory is often\nimplemented for an isolated ion and thus neglects hy-\nbridization effects. It is therefore not surprising that this\nimplementation of the theory has been found to work best\nfor localized electronic states, e.g., the 4 forbitals of rare-\nearth elements. However, effects like crystal-field split-\nting and orbital hybridization can be treated by adding\nsuitable model terms to the Hamiltonian of the system.\nThe resulting theory is often referred to as the multi-\nplet ligand-field theory (MLFT) [8]. This theory is very\npowerful in reproducing experimental spectra when the\nelectrons have a tendency to be localized, but has the\nobvious drawback in the ambiguity of choosing and jus-\ntifying the various parameters of the model Hamiltonian\n(see e.g. Ref. [9]). One way to reduce the ambiguity is\nto obtain the hybridization strength parameters and the\ncrystal-field splitting from a tight-binding parameteriza-\ntion of a DFT calculation [10]. For the 4 f-orbitals ofarXiv:2401.03858v1  [cond-mat.str-el]  8 Jan 20242\nrare-earth systems it is also possible to neglect the hy-\nbridization and extract the remaining MLFT parameters\ncompletely from DFT [11].\nTo describe both on-site electron-electron interaction\nand hybridization effects, many methods have been pro-\nposed, for example DFT+ U[12] and DFT in conjunction\nwith dynamical mean-field theory (DFT+DMFT) [13].\nSince DFT+DMFT takes into account the multiconfig-\nurational effects, it was shown to provide a better de-\nscription of valence-band spectral function of, e.g., the\n3dtransition-metal oxides (TMOs) [14, 15].\nIn the x-ray absorption process, the excitation involves\nthe promotion of an electron in a core orbital to an\nempty orbital, according to the selection rules of the elec-\ntric dipole operator [16]. The resulting core-hole inter-\nacts with the valence electrons and the photo-electron.\nFor example, for the L2,3-edges of transition metal com-\nplexes the presence of a 2 pcore-hole in the final state\ntends to further localize the valence states. In the final-\nstate approximation, the core hole is treated as an at-\ntractive static potential, and the valence electrons and\nthe photo-electron are allowed to relax in the presence\nof this potential. However, it was shown that applying\nthe final state approximation in DFT+DMFT does not\nimprove the calculated XAS spectra of Fe and Co, and\nin the case of Ni dramatically worsens the comparison\nwith the experiment [17]. Several propositions to include\nthe dynamics of the core-hole within a single compu-\ntational framework for calculating L-edges of transition\nmetal (TM) systems are available in the literature, such\nas the multiple scattering approach [18] with dynamical\nscreening from time-dependent DFT [19, 20], the appli-\ncation of configuration interaction to a set of molecular\norbitals [21] and the Bethe-Salpeter equation [22, 23].\nIn spite of a high level of theoretical complexity, most of\nthe methods mentioned above do not deliver a sufficiently\naccurate description of XAS of a wide range of materi-\nals, exemplified by the transition metal oxides. In recent\nyears, approaches based on extending the impurity model\nin DFT+DMFT with core states have also been used to\nsimulate core level x-ray photoemission spectroscopy [24]\nand resonant x-ray emission spectroscopy [25]. A good\noverview of most methods for calculating x-ray absorp-\ntion spectra is presented in [26].\nIn the present study we use MLFT in conjunction with\nDFT+ Uand DFT+DMFT to describe on a equal foot-\ning the multiconfigurational effects as well as the dynam-\nics of the core-hole photo-electron interaction and its ef-\nfect on XAS and XMCD. The method described here has\nkey components that are identical for MLFT combined\nwith DFT+ Uand DFT+DMFT, and so for simplicity\nwe have chosen to give most numerical examples from\nthe MLFT+DFT+ Uapproach. The aim of the present\ntheory is to rely on as few free parameters as possible. For\nexample, the hybridization of the orbitals is directly ex-\ntracted from the electronic structure calculation and the\nscreening of the local electron-electron interaction (Hub-\nbard U) is estimated using constrained DFT (cDFT).However, there currently is no reliable way to calculate\nthe screening of the zeroth order valence-core interaction\n(F0\npd), so it is treated as a free parameter. We also in-\nvestigate how sensitive the spectra are to the description\nof the screening of the higher-order Slater parameters.\nIn this work transition selection rules have been included\nwithin the electric-dipole approximation. We present the\ndetails of the implementation and the results from an ap-\nplication of the developed method to the description of\ntheL2,3-edges and XMCD spectra of magnetite, Fe 3O4,\nas well as cobalt- (CoFe 2O4) and nickel ferrite (NiFe 2O4)\nat room temperature in the inverse spinel structure. We\nshow that our theoretical results compare favorably well\nwith experimental observations. In particular, we show\nthat both the L3/L2experimental XAS branching ratio,\nas well as fine structures in the XAS and XMCD spec-\ntra, are well reproduced by our calculations, contrary to\nprevious DFT calculations.\nII. THEORY\nA. DFT+MLFT\nIn the following, we describe the DFT+MLFT ap-\nproach, and show how to use this theory to compute x-ray\nabsorption spectra and x-ray magnetic circular dichroism\nof the magnetite Fe 3O4as well as the cobalt and nickel\nferrites. The theoretical model can in a simplified way\nbe described as a transfer of parameters obtained from\nDFT (or DFT+ U) based methods, to a MLFT level of\ntheory. In this way, hybridization effects and crystal-field\nsplittings do not enter the theory as fitting parameters\nbut rather evaluated from ab initio theory. We describe\nthe essential aspects of this mapping below.\n1. Projection\nThe starting point of the theory involves the one-\nparticle DFT+ UGreen’s function of the lattice, which\nencodes the contributions from hybridization and crystal-\nfield splittings of the transition metal 3 dorbitals. It is de-\nfined as the resolvent of the lattice-momentum dependent\nHamiltonian ˜HDFT\nk and the static DFT+ Uself-energy\n˜Σk;\n˜Gk(ω) = (( ω+µ)˜1−˜HDFT\nk−˜Σk)−1, (1)\nwhere µis the chemical potential and ˜1 is the identity\noperator. The self-energy ˜Σkis obtained from the local\nself-energy ˜Σlocand a double-counting correction ˜Σloc\nDC\nthrough a projection from a set of impurity orbitals to\nthe lattice orbitals:\n˜Σk=˜Pk(˜Σloc−˜Σloc\nDC)˜P†\nk, (2)\nwhere ˜Pkis the projection operator from the orthonor-\nmal orbitals on the impurity site to the lattice orbitals at3\nlattice momentum k. The impurity orbitals are defined\nthrough the construction of the projection as detailed in\nRefs. [15, 27, 28]. In the construction of the projection\noperator we only use the DFT bands within an energy\nwindow [ −16,10] eV around the Fermi level, in order to\npreserve the TM 3 dcharacter of the impurity orbitals.\nThe lower bound of the energy window prevents the im-\npurity orbitals to obtain any oxygen 2 scharacter, which\nadversely affects the crystal-field splitting in the impurity\norbitals. The upper bound prevents the impurity orbitals\nto include high-energy TM 4 dand TM 5 dcharacters.\nThe local Green’s function is obtained from ˜Gk(ω)\nthrough the reverse projection:\n˜Gloc(ω) =1\nNkNkX\nk˜P†\nk˜Gk(ω)˜Pk, (3)\nwhere Nkis the number of k-points in the first Brillouin\nzone. The projection of the lattice Green’s function to\nthe local Green’s function is identical to that in dynam-\nical mean-field theory (DMFT). The hybridization func-\ntion\n˜∆(ω) = (ω+µ)˜1−˜Hloc\n0−˜Σloc+˜Σloc\nDC−˜Gloc(ω)−1(4)\ndescribes the hybridization of the 3 d-TM orbitals with\nthe orbitals of the rest of the material. The imaginary\npart of the hybridization function is similar to the density\nof states but has peaks at the energies of the hybridis-\ning orbitals. The intensity of these peaks corresponds to\nthe strength of the hybridization. The role of the ligand\norbitals is to mimic this ab−initio hybridization i.e. to\nallow ligand electrons to enter (and leave) the 3 d-TM or-\nbitals with the appropriate transition energy. In Eq. (4)\nthe local Hamiltonian ˜Hlocis calculated from ˜HDFT\nk by\nusing the same projection procedure as in Eq. (3).\n2. Discretization of the hybridization function ∆(ω)\nThe hybridization function operator in Eq. (4) can be\napproximated as\n˜∆ED(ω) =˜Vh\nω˜1−˜Hbathi−1˜V†, (5)\nwhere ˜Hbathis an effective Hamiltonian for a finite (usu-\nally small) set of auxiliary bath orbitals which hybridize\nwith the impurity orbitals according to the impurity-\nbath hopping operator ˜V. This approximation is rou-\ntinely done in the exact diagonalization (ED) solver in\nDMFT [29]. The matrix elements of the operators ˜V\nand ˜Hbathare parameterized and the parameters are set\nto reproduce the main peaks of the imaginary part of\nthe exact hybridization function ˜∆(ω). In our imple-\nmentation several energy windows are selected after an\ninspection of ∆( ω), and within each window the param-\neters for one bath orbital is fitted to minimize the differ-\nence between ∆EDand ∆( ω). The matrix representation∆(ω) of˜∆(ω) depends on the choice of impurity orbitals.\n∆(ω) becomes block diagonal if the impurity orbitals are\nsymmetrized such that they transform according to the\nirreducible representations of the system. Even in the\nabsence of global symmetries it is often advantageous to\nsymmetrize the impurity orbitals according to their local\nenvironment to minimize the off-diagonal elements in the\nfitting of ∆( ω).\n3. Impurity Hamiltonian\nThe total impurity Hamiltonian for the TM 3 dand 2 p\nshells is summarized as\nˆH=X\nijH3d\nijˆc†\niˆcj+X\nbb′Hbath\nbb′ˆc†\nbˆcb′+X\ni,b(Vi,bˆc†\niˆcb+ h.c.)\n+X\nijH2p\nijˆc†\niˆcj+ˆHU, (6)\nwith\nH3d=Hloc−Σloc\nDC+HSOC\n3d, (7)\nH2p=ϵp˜1 +HSOC\n2p, (8)\nwhere H3dand H2pcontains the 3 dand 2 pon-site\nenergies including the spin-orbit coupling (SOC) (see\nEq. (A8)), respectively. HbathandVdescribe the bath\nand its coupling to the impurity, and ˆHUis the Coulomb\ninteraction (see Eq. (A1) of the appendix).\nThe diagonalization of Eq. (6) results in the set of\nmany-body eigenstates |n⟩, each expressed by a sum of\nproduct states, and the corresponding eigenenergies En:\nˆH|n⟩=En|n⟩. (9)\nOnce the eigenstates with eigenenergies at most a few\nkBTabove the lowest eigenenergy are found using a\nLanczos algorithm, we use these states to calculate spec-\ntra and other observables such as occupation numbers\nand spin moments.\nB. Spectra\nUsing the eigenstates |n⟩and eigenenergies Enfrom\nthe Hamiltonian ˆHin Eq. (6), we calculate the XAS and\nXMCD spectra at finite temperature (300 K) according\nto\nI(ω) =1\nZX\nn−Im(Gn(ω)) exp( −βEn), (10)\nwith\nGn(ω) =⟨n|ˆT† 1\n(ω+En+iΓc)ˆ1−ˆHˆT|n⟩,(11)\nand where ωis the incoming photon energy, ˆ1 is the iden-\ntity operator, Zthe partition function and βthe inverse4\ntemperature. Here Γ cmodels the core-hole decay rate\nand defines the energy resolution in terms of the half-\nwidth at half-maximum (HWHM). The transition op-\nerator ˆTis, within the dipole approximation, equal to\nˆD=ϵ·ˆp, where ϵis the light polarization vector and\nˆpis the momentum operator. We have implemented\nEqns. (10) and (11) using a Lanczos algorithm in an\nopen-source software [30, 31].\nIII. DETAILS OF THE CALCULATIONS\nThe Hamiltonian in Eq. (6) contains several parame-\nters that either can be calculated or have been estimated.\nFrom the electronic structure calculation, we determine\nthe local impurity Hamiltonian from a direct projection\nand also construct the hybridization function. The hy-\nbridization strength ( V) and the bath state energies ( eb)\nare set to reproduce the hybridization function. The av-\nerage Coulomb repulsion between the electrons within\nthe 3 dand 2 porbitals of the TM atom can be expressed\nin terms of the Slater-Condon parameters Fk\nlili,Gk\nliljand\nFk\nliljand the Wigner 3 jsymbols as [5]\nUlili=F0\nlili−2li+ 1\n4li+ 1X\nk\u0012\nlik li\n0 0 0\u00132\nFk\nlili, (12)\nUlilj=F0\nlilj−1\n2X\nk\u0012\nlik lj\n0 0 0\u00132\nGk\nlilj, (13)\nwhere ldenotes the angular momentum of the orbitals.\nNotice that in these expressions kis not a crystal momen-\ntum (as it is in Section II A 1) but an angular momen-\ntum. The bare higher-order Slater parameters ( k≥1)\nare calculated by solving the Slater-Condon integrals. To\naccount for screening effects the parameters are then in-\ndividually reduced to between 70% and 90% of their bare\nvalues. The values within this interval are chosen to best\nfit the experiment. For comparison, we also performed\ncalculations where the higher-order Slater parameters are\nkept at a fixed ratio of 80% of their bare values, to show\nthe impact of the screening. The screening of the zeroth-\norder parameters, F0\nddandF0\npd, are typically so high that\nthey cannot be estimated by this approach. Therefore,\nwe have calculated F0\nddusing cDFT for NiFe 2O4using\nthe electronic structure code Wien2K [32]. There is cur-\nrently no accurate method for calculating the screening of\nF0\npd, so it was varied in the empirically motivated interval\nF0\ndd+ 1 eV ≤F0\npd<1.4F0\ndd.\nThe MLFT double counting of the d-orbitals is given\nby,\nΣDC=ndUdd−npUdp−δDC, (14)\nwhere ndis the occupation of the dorbitals, npis the oc-\ncupation of the pcore states and δDCis the charge trans-\nfer potential. The value of δDCwas taken from Ref. [33].\nFIG. 1: Conventional unit cell of XFe 2O4, where X is\nFe, Co or Ni. The silver-colored octahedral sites\nrepresent the Fe2+sites that are occupied by Co or Ni\nin the other ferrites . The blue octahedral sites and the\npurple tetragonal are occupied by Fe3+.\nCompared to the previous work the p-orbital potential\n∆ϵpwas introduced to reproduce the relative difference\nin the core-level binding energies of the different Fe sites\nin the DFT+ Ucalculation. The values of ∆ ϵpare within\n0.15 eV of the DFT calculated values. The 2 pand 3 d\nspin-orbit splittings were calculated using a relativistic\nDFT+ Ucalculation. All parameters used here are com-\npiled in Table I.\nIV. CRYSTAL AND MAGNETIC STRUCTURE\nMagnetite (Fe 3O4) above the well known Verwey tran-\nsition, has an inverse spinel structure, with an fcc Bra-\nvais lattice. For this structure, DFT+ Ucalculations, as\npresented here (for details, see Appendix A), result in\na half-metallic electronic structure. The Fe cations are\neither octahedrally or tetrahedrally coordinated to the\noxygen ions. Above the Verwey temperature, the six Fe\nions in the unit cell are separated into two tetrahedral\nand four octahedral sites. The two tetrahedral sites and\ntwo of the octahedral sites are occupied by nominally\nFe3+ions, while the remaining two octahedral sites are\noccupied by nominally Fe2+ions. When we go from mag-\nnetite to CoFe 2O4or NiFe 2O4, almost all Co/Ni atoms\noccupy the Fe atom at the octahedral +2 sites. Therefore\nthe structures can be expressed as (Fe3+)[X2+Fe3+]O4,\nwhere the X is Fe/Co/Ni and the parentheses indicate\ntetrahedral sites and the square brackets denote the oc-\ntahedral sites. A schematic overview of the inverse spinel\nstructure and the difference between the compounds is\nshown in Fig. 1. The Fe 3O4and the other ferrites are fer-\nrimagnetic spinels. The magnetic ordering results from\nthe exchange interactions between the cations. The Fe\n3dorbitals of the tetrahedral and octahedral sites overlap\nwith the intermediate 2 poxygen at an angle of ≈125◦,\nwhich according to the Goodenough-Kanamori-Anderson\nrules leads to antiferromagnetic exchange between the\ntwo sublattices. The octahedral sites couple ferromag-\nnetically to each other due to a 90◦superexchange. The5\nTABLE I: Summary of the double-counting correction, Slater-Condon integrals, and spin-orbit coupling parameters\nused in the MLFT calculations (unit in eV). F0\npdis treated as a free parameter. An additional potential ∆ ϵpis\nintroduced to shift the calculated 2 pcore energies to coincide with experimental values.\nδDC F0\nddF2\nddF4\nddF0\npdF2\npdG1\npdG3\npd ζpζd∆ϵp\nCoFe 2O4Co 1.5 6.6 9.23 5.73 7.60 5.18 4.22 2.49 9.85 0.078 0\nCoFe 2O4Feoct1.5 6.4 9.35 5.85 8.12 4.89 3.21 2.05 8.3 0.064 0\nCoFe 2O4Fetet1.5 6.3 9.35 5.85 8.12 5.19 3.51 2.05 8.3 0.074 0.25\nNiFe 2O4Ni 1.5 6.9 10.53 6.94 7.90 5.79 4.28 2.42 11.6 0.096 0\nNiFe 2O4Feoct1.5 6.4 9.35 5.85 7.87 5.80 3.28 2.40 8.3 0.063 0\nNiFe 2O4Fetet1.5 6.3 9.35 5.85 7.87 5.80 3.28 2.40 8.3 0.072 0.35\nFe3O4Fe2+\noct 1.5 6.4 8.63 5.36 8.45 4.69 3.18 1.92 8.3 0.062 4.02\nFe3O4Fe3+\noct 1.5 6.4 8.88 5.52 8.55 4.88 3.34 2.00 8.3 0.064 0\nFe3O4Fe3+\ntet 1.5 6.3 9.28 6.26 7.82 5.21 3.57 2.14 8.3 0.076 1.35\nexchange interaction between the tetrahedral sites is an-\ntiferromagnetic, however it does not result in an antifer-\nromagnetic ordering in the tetrahedral sites as the other\nexchange interactions are stronger [34]. In Table II we\ncompare the calculated spin and orbital moments with a\nprevious calculation from Ref. [35]. It can be seen that\nthe two sets of calculations compare rather well with each\nother. In Table III the calculated sizes of the total mag-\nnetic moments of the tetrahedral sites and the average\nmoment of the octahedral sites are compared to neu-\ntron scattering experiments [36] for Fe 3O4, CoFe 2O4and\nNiFe 2O4[37] taken at a temperature of 300 K. It is clear\nfrom the table that the theoretical results are in good\nagreement with experiment. From Table III it is also\nclear that the results of magnetic moments from DFT\nand MLFT are in very good agreement with each other.\nNote that the values obtained from MLFT are obtained\nas expectation values of the spin and orbital angular mo-\nmentum operators, using the ground-state wavefunction\n(without core hole excitation). The fact that the two\nsets of calculations give similar results is rewarding since\nit reflects on the accuracy on the mapping to the local\nHamiltonian.\nV. RESULTS AND DISCUSSION\nIn this section, the theoretical calculations of XAS\nand XMCD are compared with the experimental results\n(black curves), for all compounds of this investigation.\nThe theoretical calculation is obtained by varying the\nhigher-order Slater parameters ( k≥1) in Eq. (A7)\nindividually between 70% and 90% of their calculated\nvalues while making the best fit to the experimental\ndata. These parameters are shown in Table I and the\ntheoretical spectra based on them are labelled ”theory”\nin Figs. 2- 8, and are shown as a red line. In Figs. 2- 6\nwe also show results from an alternative theoretical\ncalculation, where all the higher-order Slater parameters\nare screened to 80% of their bare values (blue dashed\nline in Figs. 2- 6). Note also that in each subsection,we start with an analysis of XAS which is followed by\nthe results of the XMCD. In the case of CoFe 2O4and\nNiFe 2O4the spectral features from the Fe sites will be\npresented first.\nA. Spectra of Fe 3O4\nXAS . The dipole allowed Fe L2/3XAS edges of Fe 3O4\nare displayed in Fig. 2. The spectrum is composed of\ntwo main peaks, the L3- and L2-edges, due to the SOC\n∆SO= (3/2)ζpin the TM 2 pstate. This results in a split\nof the core-levels of 2 p J=3/2 and 2 p J=1/2 by ∆ SO.\nWhen photons excite the core electrons in the higher J\n=3/2 level and lower J=1/2 level they respectively pro-\nduce electronic transitions to the L3-edge and the L2-\nedge. The SOC of the core states is strong and almost\nindependent of the atomic environment.\nThe experimentally observed spectrum has a broad L3-\nedge (between 705 and 715 eV), which is composed of\ncontributions from all three different Fe sites. The L2-\nedge (between 718 and 728 eV) has a significantly lower\nintensity and shows a double-peak feature. Both edges\nare well reproduced by the theory with the L3-edge being\na bit broader on the low energy side in the experiment.\nThe theoretical results from the L2-edge do not show\nthe same double-peak feature as in the experiment with\nthe second peak missing, however, the relative intensity\nto the L3-edge is more or less the same in theory and\nin the experiment. In a pure one-electron picture, and\nwith uniform matrix elements for the excitation, the in-\ntensities of the L3- and the L2-edges would reflect the\nnumber of electrons available of the corresponding core\nlevels, i.e., 4 to 2. The data in Fig. 2 show that this is\nnot the case, and the so called branching ratio deviates\nsignificantly from the 4/2 ratio. It is gratifying that the\ntheory used here reproduces this aspect of the measured\nspectrum. Furthermore, we can see that the calculation\nis robust towards the choice of Slater parameters as the\ncurves match nearly perfectly.6\nTABLE II: DFT+ Ucalculated absolute value of the spin and orbital (in parenthesis) magnetic moments, in units of\nBohr magneton ( µB) per atom compared with DFT+ Ucalculation of Ref. [35].\nCoFe 2O4 NiFe 2O4 Fe3O4\nCalculation Co Fe oct Fetet Ni Fe oct Fetet Fe2+\noct Fe3+\noct Fe3+\ntet\nPresent 2.53(0.00) 4.09(0.00) 3.93(0.00) 1.66 (0.34) 4.07 (0.03) 3.92 (0.03) 3.59 (0.02) 4.08 (0.02) 3.82 (0.02)\nRef. [35] 2.57 (0.01) 4.04 (0.04) 3.90 (0.03) 1.54 (0.27) 4.09 (0.03) 3.99 (0.02) 3.54 (0.02) 4.00 (0.02) 3.84 (0.02)\nTABLE III: Calculated DFT+ Utotal magnetic moments, in unit of Bohr magneton ( µB) per atom, of the different\nsublattices compared with the experimental results for CoFe 2O4and NiFe 2O4of Ref. [36] and for Fe 3O4of Ref.\n[37]. In the table µoctis the average moment on the octahedral sites.\nFe3O4CoFe 2O4NiFe 2O4\nµtetµoctµtetµoctµtetµoct\nDFT+ U 4.23 3.85 3.93 3.31 3.95 3.05\nMLFT ground state 4.25 4.03 4.23 3.11 4.03 3.43\nExpt. 4.0 3.6 3.5 3.24 3.72 2.90\nXMCD . The measured XMCD spectrum (also shown\nin Fig. 2) has a characteristic down-up-down peak\nstructure at the L3-edge. Here, the first peak has the\nlargest intensity and the second peak the lowest. This\nstructure is caused by the ferrimagnetic ordering and\nthe spinel structure. In a perfect antiferromagnet no\nXMCD can be observed as the contributions vanish.\nThe antiferromangnetically aligned Fe cations occupy\ninequivalent lattice sites and have therefore slightly\ndifferent excitation energies. This results in a dichroic\nresponse from the +3 cations, which here adds to the\nresponse from the Fe2+cations. At energies just above\nthese peaks one can observe a small shoulder. The\nL2-edge shows two different peaks with the first feature\nhaving a larger intensity and a distinct two-peak feature.\nThe theoretical calculation reproduces the behaviour\nat the L3-edge very well. Even the small shoulder at\nenergies above the three-peak feature is reproduced\nand the oscillations right below of the L2-edge are\nreproduced. The only major difference is that a negative\npeak appears in the calculation between the L3- and\nL2-edge which is not in the experiment. Also the\nintensities of the L2-edge is overestimated on all peaks\nexcept the first peak of the double-peak feature. This\ncould be due to an overestimation of the 3 dSOC of the\ntetrahedral sites. Here, we can see that our choice of the\nhigher order Slater parameters enhances the signal at\nthe middle peak and suppresses that of the third peak\nand the multiplet effects.\nB. Spectra of CoFe 2O4\nFe-projected results\nXAS . The measured XAS of the Fe sites (Fig.3, upperpanel) shows a main peak with a low energy shoulder at\ntheL3-edge and a double peak at the L2-edge. The XAS\nof the Fe sites shows the same features in experiment\nand theory. However, the shoulder of the L3-edge is\nsmaller in intensity for the calculated spectrum than\nin the experiment. This shoulder is caused by the\noctahedral sites in the calculation, therefore a higher\nratio of octahedral to tetrahedral Fe sites than the equal\namount that the calculation used could explain this\neffect. A reason for this could be Co cations occupying\none or both tetrahedral sites in some cells, causing the\nFe sites to occupy three or four octahedral sites in that\ncell, which would lead to a higher contribution of the\noctahedral Fe sites in the signal. Other point defects in\nthe experimental sample could cause different oxidation\nstates, that also would influence smaller details in\nthe observed spectra, that would not be picked up by\ncalculations of a defect-free system. For the L2-edge\nthe theory clearly puts more intensity on the higher\nenergy peak, while they have the same intensity in the\nexperiment. It is also notable that the branching ratio\nis very similar for theory and experiment. Furthermore,\none can see that the calculation is robust to the choice\nof Slater parameters as the two curves match.\nXMCD . In the XMCD spectrum (Fig.3, lower panel),\none can observe the same characteristic down-up-down\nstructure as for Fe 3O4in the L3channel, however due to\nthe absence of Fe2+the first peak is no longer the largest\nin intensity, rather it is the smallest while the third peak\nin the L3channel has the largest intensity, with the\nsecond peak having similar intensity to the third. The\ndetails of these peaks are quite similar if one compares\ntheory and experiment. The measured L2-edge shows\ntwo low intensity peaks with the second one being flat\nand lower in intensity. In the calculation the general\nstructure of the L3-edge is well reproduced. However,7\nthe intensity for the first peak is underestimated and the\nsecond one is overestimated, this could also be linked\nto some Co occupying tetrahedral sites, because the\nsecond peak is caused by tetrahedral Fe and the first by\noctahedral. Here, the fine structure is not reproduced\nby the calculation as it points in the opposite direction\ncompared to the experiment. The two distinct L2peaks\nare well reproduced by the theory. However, the second\npeak in the experiment is broader than in the theory.\nFurthermore, one can see that the calculation is robust\nto the choice of Slater parameters as the two curves\nmatch.\nCo-projected results\nXAS . The XAS of Co (Fig.4, upper panel) shows a\nlot of features, both in theory and experiment. The\nL3-edge has a double peak, where the peak on the left\nis slightly higher in intensity. The peak is accompanied\nby a small shoulder on the left, a high intensity shoulder\nand a small shoulder to the right. The L2-edge is\ncharacterized by a single peak. The calculated XAS\nof the Co site agrees very well with the experiment.\nThe double-peak feature of the L3is well reproduced,\nhowever in the experiment the first is larger. The left\nshoulder is located at lower energies in theory compared\nto the experiment and it has also a lower intensity. The\nright shoulder matches the experiment perfectly, even\nthe small shoulder at the end is reproduced. At the\nL2-edge, experiment and theory mostly lie on top of\none another, however the theory shows a slightly higher\nintensity at the maximum of the peak. Here, one can\nsee that the two theoretical spectra look very similar\nand that the main difference is a shift of energy. This is\ncaused by the fact that we are aligning the peaks of the\nlargest intensity in the calculation to the largest peak in\nthe experiment and the intensities of the two peaks in\nthe double peak feature changes due to the choice of the\nSlater parameters.\nXMCD . The measured L3-edge XMCD of the Co site\n(Fig.4, lower panel) shows a small down-pointing peak\nfollowed by a thin up-pointing peak in front of the main\npeak, which has one shoulder on the left and two on the\nright. After the main peak, there is a small fine structure\npeak. At the L2-edge, there is a single flat peak. The\ntheory reproduces the shape of the L3-edge main peak\nand its shoulders well, even though the shoulders on the\nright have less intensity than in the experiment. The\npeaks before the main peak are not well reproduced,\nwith the first peak missing and the second peak being\nbroader and less intense than in the experiment. This\ncould be because these peaks are caused by Co occupying\nthe tetrahedral site, where it would be closer to a +3\nstate. Alternatively, other impurities in the sample\ncould play a role. The small fine structure after the\nmain peak is very well resolved by the calculation. The\nL2-edge in the theory shows a very clear peak with muchhigher intensity than the flat peak in the experiment.\nThis could again be caused by Co in tetrahedral sites\nas those would have the opposite spin and therefore\nwould reduce the signal from the peak. Here, we can see\nthat the choice of Slater parameters only affects the two\nshoulders closest to the main peak, where the intensity\nis shifted slightly from the left shoulder to the one on\nthe right.\nC. Spectra of NiFe 2O4\nHere, the experiment uses a superposition of 77%\ncircular polarized and 23% linear polarized light [38].\nThis was explicitly taken into account in the transition\ndipole operator.\nFe-projected results\nXAS . The measured L3-edge of the Fe sites (Fig.5,\nupper panel) has a main peak with a shoulder to the\nleft and a smaller shoulder to the right. The L2-edge\nhas a clear main peak with a distinct shoulder to the\nleft. In the calculations of the XAS of the Fe sites the\nleft shoulder of the L3-edge is not present. However,\nthe right shoulder is nearly perfectly reproduced by the\ntheory. The observed relative intensity of the L2-edge\ncompared to the L3-edge is slightly overestimated by our\ntheory and the calculated shoulder of the L2-edge could\nbe more distinct from the main peak. Furthermore,\nthe calculated energy difference between the L3- and\nL2-edges, due to the SOC of the p-electrons, is slightly\noverestimated. Here, the blue and red lines perfectly\nmatch as we have used the same parameters.\nXMCD . The measured XMCD (Fig.5, lower panel)\nshows the same down-up-down structure as the other\nFe sites. However, here, the up-pointing peak has the\nhighest intensity and the first peak has the lowest due to\nthe missing Fe2+sites. After the three peaks, one can see\ntwo fine structure features. The L2-edge is characterized\nby an up-down-up structure with the down-pointing\npeak having the highest intensity and the last peak the\nlowest and most elongated. The down-up-down peak\nstructure of the L3-edge is reproduced very well by the\ntheory with a slight shift towards higher energies for the\nfirst two peaks and an overestimation of the middle peak.\nThe two fine-structure peaks after the third peak are\nalso reproduced, however the intensity of the first peak\nis overestimated and of the second underestimated. In\ntheL2channel, one can notice the same structure in the\nfirst two peaks from the calculations, but for the third\nwe see a double-peak structure with higher intensities.\nJust like for the XAS, the SOC is overestimated and\ntherefore, the energy difference between the L3- and\nL2-edges are slightly larger than in the experiment.8\nNi-projected results\nXAS . The observed L3-edge of Ni (Fig.6, upper panel)\nis characterized by a main peak with a large shoulder\non the high energy side. The L2-edge has a peak with\na shoulder on the high energy side. Overall, the XAS is\nwell reproduced by the theory. The main peak and shoul-\nder of the L3-edge are located at the correct positions,\nhowever the width and intensity are somewhat underes-\ntimated. The shape and relative intensity of the L2-edge\ncompared to the L3-edge are very close to the experi-\nmental results. However, the intensity is slightly lower\nthan in the experiment. Additionally the energy differ-\nence between the L3andL2peaks, which is determined\nby the SOC of the p-states, is slightly overestimated by\nthe theory. From Fig.6 one can note that the calculated\nresult is robust with respect to an uncertainty in Slater\nparameters, as the intensity of the shoulders increases\nonly slightly when the paramaters are varied.\nTo further check the sensitivity of the theoretical\nresults, we performed additional calculations that\ndemonstrate how the theoretical spectra depend on the\nCoulomb core-valence (CV) interaction. This is shown\nin Figs. 7 and 8, for the Fe respectively Ni sites in\nNiFe 2O4. As the figure shows, it is crucial to include the\nCV interaction in order to get a good XAS and XMCD\nsignal and the effect is most clearly demonstrated for\nthe Fe atom.\nXMCD . The XMCD (Fig.6, lower panel) has a large,\nup-pointing shoulder at the L3-edge. At the L2-edge,\nwe observe a large peak with a shoulder to the left. The\ncalculation nearly perfectly matches the experimental re-\nsults. The shoulder of the L3-edge is slightly too low in\nintensity, whereas the energy difference between the L3-\nandL2-edges is slightly overestimated like in the XAS.\nHere, we can see that the calculation is just as robust to\nthe choice of Slater parameters as the XAS.\nTo summarize the previous discussion, we observe that\nall XAS excitations can be explained in terms of electric-\ndipole-allowed electronic transitions and are composed\nof two strong peaks in the L3andL2edges, due to the\nlarge SOC of the 2 pstates into p3/2andp1/2relativistic\nstates. The photon excitations of the core electrons in the\nhigher J=3/2 and low J= 1/2 levels produce electronic\ntransitions essentially towards the unoccupied 3 dstates,\nproducing respectively the L3- and L2-edges. The fea-\ntures within each of those edges are mainly due to the 3 d\nelectron-electron interactions as well as the Coulomb in-\nteraction between the 2 pcore hole and the photo-electron\nin the 3 dunoccupied states. It is therefore a good idea\nto decompose such excitations in terms of the 3 d t2gand\negstates to determine the various contributions to the\nXAS and XMCD spectra. For this we have inspected\nthe partial density of states (PDOS) to investigate which\norbitals resemble egandt2gthe most. Then we con-\nstructed operators that only excite the 2 pcore electrons\ninto either the egort2gorbitals. The difference betweenthe total spectra and the sum of the signals from egand\nt2g, corresponds to the interference between the excita-\ntions into t2gandegthat produce the same final state.\nWe refer to this as the off-diagonal contribution in the\nfollowing.\nFigure 9 shows for Fe 3O4the contributions from exci-\ntations between 2 p1/2as well as the 2 p3/2core levels to\nthet2gandegstates, resulting in symmetry projections\nof the final states reached by the light matter interaction.\nNote from Fig. 9 that we show both data for XAS as well\nas XMCD. Similar figures for the analysis of the XAS\nand XMCD spectra of CoFe 2O4and NiFe 2O4are shown\nin the supplementary information. It is clear from Fig. 9\nthat the intensity of the two symmetry components is\nnot quite a 3/2 ratio that would be expected from the\noccupancy of the t2g/eglevels, but only about 1.4. One\nmay also see that the t2gandegenergy splitting is not\nnoticeably conspicuous in these spectra. It is interesting\nto note that both states contribute almost equally to the\nL3andL2main peaks. For the XMCD spectrum, the dif-\nference between t2gandegprojected levels is much more\npronounced compared to the XAS signal. In particular\nwe note that for the L2-edge the t2gandeghave opposite\ncontributions at most of the energies where this peak has\nlarge intensity. In general, however, Fig. 9 shows that an\ninterpretation of the XAS and XMCD signals in terms of\ntransitions to t2goregprojections is not trivial, since the\noff-diagonal contributions are significant. Similar conclu-\nsions can be drawn for symmetry decomposed spectra of\nall the here investigated materials (see Appendix B 1).\nAs a final comment to this section we note that the\nXAS and XMCD spectra of NiFe 2O4are very similar for\ncalculations where the parameters of the MLFT Hamilto-\nnian are evaluated from LDA+ Uand from LDA+DMFT.\nThis is shown in Fig.14 for the Fe L-ege spectra and in\nFig.15 for the Ni spectra. Note that the two levels of the-\nory give almost identical results for the Fe XAS signal,\nbut that the XMCD signal agrees sightly better with ex-\nperiment for the DMFT based theory. For the Ni signal\nthe two theory curves are almost identical both for the\nXAS and XMCD signal.\nVI. CONCLUSION\nWe have used multiplet ligand-field theory in conjunc-\ntion with DFT+ Uor DFT+DMFT electronic structure\ntheory to describe on an equal footing the multiconfig-\nurational effects as well as the presence of the core-hole\nphoto-electron Coulomb interaction and its effect on XAS\nand XMCD spectra. The electronic transition selection\nrules have been included within the electric-dipole ap-\nproximation. We have presented the details of the im-\nplementation and its application to the L2,3-edges and\nXMCD spectra of the ferrites, Fe 3O4(magnetite), cobalt\nferrite, CoFe 2O4, and nickel ferrite, NiFe 2O4. We find\nthat the results are in overall agreement with available\nexperimental spectra. In particular, we find that the ex-9\nTABLE IV: Site electron occupation N3dof the tetrahedral and the octahedral transition metal ion in different\nferrites.\nFe3O4 CoFe 2O4 NiFe 2O4\nFe2+\noctFe3+\noctFe3+\ntet Co Fe octFetet Ni Fe octFetet\nN3d6.267 5.901 5.768 7.129 5.148 5.110 8.291 5.177 5.123\n700 705 710 715 720 725 730\nEnergy (eV)0510152025Intensity (arb. units)experiment\ntheory\n80% Slater\n700 705 710 715 720 725 730\nEnergy (eV)1.5\n1.0\n0.5\n0.00.5Intensity (arb. units)experiment\ntheory\n80% Slater\nFIG. 2: Top: Calculated XAS of the Fe L2,3edges for\nFe3O4compared to experimental data (black curve).\nBottom: Calculated XMCD of Fe 3O4compared to\nexperimental data[39] (black curve). The theory was\ncarried out with the higher order Slater parameters\nscreened between 70%-90% (red curve) and to 80%\n(blue dashed curve) of the calculated values, for details\nsee text.\nperimental L3/L2branching ratio of the XAS signal is\nwell reproduced by the here described calculations, in\ncontrast to an approximate 2:1 ratio produced by DFT\ncalculations. In addition, most of the details of the spec-\ntral features of the measured XAS and XMCD signals\nare reproduced by the theory. In addition, and as de-\ntailed in Appendix B 2, our analysis suggests that cal-\nculations with parameters obtained from DFT+ Uand\n700 705 710 715 720 725 730\nEnergy (eV)051015202530Intensity (arb. units)theory\n80% Slater\nexperiment\n700 705 710 715 720 725 730\nEnergy (eV)4\n2\n024Intensity (arb. units)theory\n80% Slater\nexperimentFIG. 3: Same as Fig. 2 but for Fe sites in CoFe 2O4.\nThe experimental spectra are reproduced from [40].\nDFT+DMFT give very similar results both for the XAS\nand XMCD spectra. If this can be demonstrated to\nhold for a wider range of compounds, it would in general\nsimplify the theoretical description of XAS and XMCD\nspectra of materials with complex electronic structure,\nsince the computational effort of DFT+ Uis significantly\nsmaller than DFT+DMFT.\nThe aim of the present theory is to rely on as few free\nparameters as possible as detailed in the method sec-\ntion III. We explored the renormalization of the higher-\norder Slater parameters due to screening by either renor-\nmalizing their values to a fixed 80% of their bare value\nor varying them between 70% to 90%. We found that\nthe spectra were only marginally affected by different10\n770 775 780 785 790 795 800\nEnergy (eV)02468Intensity (arb. units)theory\n80% Slater\nexperiment\n770 775 780 785 790 795 800\nEnergy (eV)2.0\n1.5\n1.0\n0.5\n0.00.5Intensity (arb. units)theory\n80% Slater\nexperiment\nFIG. 4: Same as Fig. 2 but for Co site in CoFe 2O4. The\nexperimental spectra is reproduced from [40].\ndescriptions of the screening within this range. In this\nwork, only the zeroth-order Slater parameter F0\npdand\nthe double-counting correction δDCwere chosen as free\nparameters. We kept the value of δDCfixed to 1.5 eV for\nall investigated compounds. There is currently no accu-\nrate method for calculating the screening of F0\npd, so we\nvaried it in the interval F0\ndd< F0\npd<1.4F0\ndd. We are cur-\nrently investigating how to obtain screening of F0\npdfrom\ncDFT in a reliable way.\nWe have attempted to make an interpretation in terms\nof excitations from 2 p1/2and 2 p3/2to the t2gandegirre-\nducible representations of the 3d valence electron states.\nHowever, due to large inteference terms between the eg\nandt2gexcitations, such an analysis was shown to have\na convoluted physical interpretation.\nACKNOWLEDGMENTS\nWe acknowledge support from the Swedish Research\nCouncil, Wallenberg Initiative Materials Science (WISE)\nfunded by the Knut and Alice Wallenberg Founda-\n700 705 710 715 720 725 730\nEnergy (eV)05101520Intensity (arb. units)experiment\ntheory\n80% Slater\n700 705 710 715 720 725 730\nEnergy (eV)2\n1\n0123Intensity (arb. units)experiment\ntheory\n80% SlaterFIG. 5: Same as Fig. 2 but for Fe sites in NiFe 2O4. The\nexperimental spectra is reproduced from [41].\ntion, the ERC (FASTCORR project), eSSENCE and\nSTandUPP. Calculations provided by SNIC/NAISS. We\nare grateful to Dario Arena, University of South Florida,\nfor allowing us to show his experimental data of NiFe 2O4,\nprior to publication. Valuable discussions with Dr. Chin\nShen Ong, Dr. Yaroslav Kvashnin and Prof. Mauritz\nHaverkort are acknowledged.\nAppendix A: Computational details\nThe XAS and XMCD spectra were calculated for\nNiFe 2O4, CoFe 2O4and Fe 3O4, which all share the same\ninverse-spinel structure with Ni and Co occupying the\noctahedral sites.\n1. DFT\nSelf-consistent spin-polarized DFT+ Ucalculations,\nwhere the spin polarization was introduced via the local\nHartree-Fock potential (+ U), have been performed with11\n840 845 850 855 860 865 870 875 880\nEnergy (eV)01234567Intensity (arb. units)experiment\ntheory\n80% Slater\n840 845 850 855 860 865 870 875 880\nEnergy (eV)3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.00.5Intensity (arb. units)experiment\ntheory\n80% Slater\nFIG. 6: Same as Fig. 2 but for Ni site in NiFe 2O4. The\nexperimental spectra is reproduced from [41].\n8×8×8k-point mesh sampling in the Brillouin zone\nfor NiFe 2O4and CoFe 2O4and 10 ×10×10 for Fe 3O4.\nWe used a full-potential linear muffin-tin orbital method\n(LMTO) as implemented in the “RSPt” code[42, 43] to\nsolve the DFT problem. The set of localized impurity\norbitals is constructed by projecting the total electron\ndensity on a set of L¨ owdin orthogonalized LMTOs for the\nTM 3 dorbitals, denoted as “ORT” in Ref. [15, 27, 28].\n2. Coulomb U\nThe interacting part of the impurity Hamiltonian for\na multi-orbital system reads\nˆHint.=1\n2X\nabcd,σσ′Uabcdˆc†\nb,σˆc†\na,σ′ˆcc,σ′ˆcd,σ, (A1)\nwhere each term describes a process and Uabcdis given\nby\nUijkl=Z Z\nd3rd3r′ψ∗\ni(r′)ψ∗\nj(r)1\n|r−r′|ψk(r′)ψl(r).\n(A2)By expanding the Coulomb interaction 1 /|r−r′|in\nterms of spherical harmonics and with basic functions of\nthe form\nψi(r) =fni,li(r)Yli,mi(θ, ϕ), (A3)\nthe Coulomb interaction tensor becomes [44]\nUabcd=δma+mb,mc+mdkmaxX\nk=0ck(lb, mb;ld, md)ck(lc, mc;la, ma)\n×Rk(nala, nblb, nclc, ndld).\n(A4)\nThe Gaunt coefficients,\nck(l, m;l′, m′) =r\n4π\n2k+ 1Z2π\n0dϕZπ\n0dθsinθY∗\nl,m(θ, ϕ)\n×Yk,m−m′(θ, ϕ)Yl′,m′(θ, ϕ),\n(A5)\ntake care of the angular integrals in Eq. (A2) and are\neasily evaluated. By considering the parity of the spher-\nical harmonics in Eq. (A5), only Gaunt coefficients with\nl+l′+kbeing an even number can be non-zero. The two\nGaunt coefficients in Eq. (A4) constrain the k-expansion\nto a maximum of kmax= min( |lb+ld|,|lc+la|). The last\nfactor to discuss in Eq. (A4) is the parameter,\nRk(nala, nblb, nclc, ndld) =Z∞\n0Z∞\n0drdr′r2r′2fna,la(r′)\n×fnb,lb(r)rk\n<\nrk+1\n>fnc,lc(r′)fnd,ld(r),\n(A6)\nwhere r<(r>) indicates min( r, r′) (max( r, r′)). It is cus-\ntomary to define the Slater-Condon parameters,\nFk(nl, n′l′) =Rk(nl, n′l′, nl, n′l′)\nGk(nl, n′l′) =Rk(nl, n′l′, n′l′, nl), (A7)\nwhere FandGdescribe the Coulomb and exchange\nintegrals, respectively. For the Coulomb interaction\nbetween d-orbitals, for any given principal quantum\nnumber, Fk=Gk, and only the three parameters F0, F2\nandF4are relevant, due to the constrains mentioned\nabove. The bare Slater-Condon integrals are calculated\nusing the projected 3 dand 2 pwave functions within the\nmuffin-tin sphere. However, the screened value of F0\npdis\ndifficult to calculate due to the strong screening effects\nfrom uncorrelated electrons and is treated as a tunable\nparameter.\nThe core-valence interaction ( Fk\npdandGk\npd) gives cru-\ncial contributions to the spectra. In Fig. 7 and Fig. 8 we\ncompare the calculated spectra of NiFe 2O4using the full\nHamiltonian and neglecting the core-valence interaction.\nThe L 2,3-edges in the spectra without core-valence in-\nteraction are characterized by having a single Lorenzian\npeak, except the XMCD of Fe where we have a single\npeak per site.12\n690 700 710 720 730 740 750\nEnergy (eV)05101520Intensity (arb. units)experiment\ntheory\nno CV\n690 700 710 720 730 740 750\nEnergy (eV)2\n1\n0123Intensity (arb. units)experiment\ntheory\nno CV\nFIG. 7: Calculated Fe L2,3XAS (top panel) and\nXMCD (bottom panel) edges in NiFe 2O4using the full\nCoulomb interaction (red solid) and neglecting the p−d\ninteraction (no CV, blue dashed) compared to\nexperiments (black solid).\n3. Spin-orbit coupling\nThe spin-orbit coupling (SOC) Hamiltonian is first\nquantized form for a ( n, l)-shell with Nelectrons and\nSOC parameter ζis\n˜HSOC=ζNX\ni=1˜li·˜si=ζNX\ni=1\u0012\n˜lz\ni˜sz\ni+1\n2(˜l+\ni˜s−\ni+˜l−\ni˜s+\ni)\u0013\n,\n(A8)\nwhere for particle i,˜li(˜si) is the orbital (spin) angu-\nlar momentum vector operator, ˜lz\ni(˜sz\ni) the z-projected\norbital (spin) angular momentum operator and ˜l±\ni(˜s±\ni)\nthe raising and lowering orbital (spin) angular momen-\ntum operators. In the last expression in Eq. (A8), the\nfirst term is diagonal in the ( l, m, σ ) basis and in second\nquantized form becomes\nζlX\nm=−lX\nσ∈{−1\n2,1\n2}σmˆc†\nl,m,σˆcl,m,σ. (A9)\n840 845 850 855 860 865 870 875 880\nEnergy (eV)01234567Intensity (arb. units)experiment\ntheory\nno CV\n840 845 850 855 860 865 870 875 880\nEnergy (eV)3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.00.51.0Intensity (arb. units)experiment\ntheory\nno CVFIG. 8: Calculated Ni L2,3XAS (top panel) and\nXMCD (bottom panel) edges in NiFe 2O4using the full\nCoulomb interaction (red solid) and neglecting the p−d\ninteraction (no CV, blue dashed) compared to\nexperiments (black, solid).\nThe other terms flip the spin and can be written as\nζ1\n2l−1X\nm=−lp\n(l−m)(l+m+ 1)(ˆ c†\nl,m+1,↓ˆcl,m,↑+ˆc†\nl,m,↑ˆcl,m+1,↓).\n(A10)\nFor 3 dorbitals of the 3 delements, ζis rather small (less\nthan∼100 meV) in comparison to other relevant ener-\ngies, e.g., the bandwidth. But for core 2 porbitals of the\n3dtransition metals ζis of the order of several eV and\nthus absolutely necessary to include in the calculation.\n4. Double counting\nAnother important aspect is the double counting (DC)\ncorrection δDC, which has to be subtracted from the\nDFT-derived Hamiltonian. This is done in order to re-\nmove the contribution of the Coulomb repulsion that is\nalready taken into account at the DFT level.13\nIn this work, we apply a DC that is often used in MLFT\nby considering the relative energy for different configura-\ntions [33].\nAppendix B: Supporting Results\n1. Symmetry-decomposed spectra\nIn this section, we describe the symmetry-decomposed\nspectra of all compounds investigated here. The sym-\nmetry decomposition is made onto egandt2gorbitals of\nthe 3d states of the transition metal atom, as described\nin the main part of this paper. The results of the eg\nandt2gprojections of the XAS and XMCD for Fe 3O4\nare shown in Fig. 9. In the XAS, we can see that the\nt2ghas a pronounced shoulder on the right of the main\npeak at the L3-edge, which in the total spectrum is en-\nhanced due to the off-diagonal elements, while the in-\ntensity of the main peak is suppressed. The egsignal\nshows a broad peak that begins at the same energy as\nthet2g, but peaks at its shoulder with approximately\ntwice the intensity. At the L2-edge, the contributions\nfrom egandt2gboth show a single peak, where the t2g\npeak is located at higher energies and keeps the intensity\nratio, while the off-diagonal suppresses the signal at the\negpeak and enhances the signal at the t2gpeak. The t2g\npeak appears broader because they are the lower-lying\norbitals in the lower energy octahedral Fe sites and the\nhigher in the higher energy tetragonal Fe sites. The in-\ntensity ratio can be explained by the fact that we can\nexcite into three t2gand two egper site. In the XMCD\nwe see that the first down-pointing peak of the L3-edge\nis caused by the lower lying t2gorbitals from the octa-\nhedral Fe sites that have lower binding energy than the\ntetragonal site. This signal is further enhanced by the off-\ndiagonal elements. The middle up-pointing peak is not\ncaused by the egpeaks of the tetragonal site, which are\ncompletely compensated by the t2gdown-pointing peak,\ninstead, it is caused by the off-diagonal elements. The\negandt2gof the two octahedral sites equally contribute\nto the third peak, which is slightly enhanced by the off-\ndiagonals. The small multiplet peaks between the L3and\nL2-edge are caused by a competition between the t2gof\nthe tetragonal and the octahedral sites. The signal at the\nat the L2-edge is mostly comprised of t2gsignal with the\negand off-diagonals reducing the signal causing the split\ninto two different peaks in the total signal.\nTheegandt2gprojected spectra of Co in CoFe 2O4can\nbe seen in Fig. 10. The first peak in the XAS comes from\nboth the egand the t2geven though the t2gorbital should\nbe at lower energies, however, they are nearly completely\ncompensated by the off-diagonal elements. After this the\nt2gdrops in value and has three more small peaks, which\nare overshadowed by the off-diagonals and the egsignal.\nThe double-peak structure is mainly formed by the off-\ndiagonals and the egsignals with similar intensities, just\nthat the off-diagonals have a higher intensity at the firstpeak, while the eghas the highest intensity at the second\npeak. The signal at the signal of the L2-edge is domi-\nnated by the t2g, which is lowered by the off-diagonals.\nIn the XMCD the first peak of the L2-edge is dominated\nby the eg-signal, while the main peak is mostly caused\nby the off-diagonal elements followed by the t2g. The\nsignal of the L2-edge is dominated by the t2gwhich is\nsuppressed by the off-diagonals.\nThe Fe-projected egandt2gspectra can be seen in\nFig. 11. The beginning of the L3-edge of the XAS is\ncharacterized by the competition of the off-diagonals and\nthet2gsignals, where the t2gsignal is slightly bigger than\nthe signal by the off-diagonal terms. The main peak is\nmainly caused by the egcontributions, but the t2galso\nshows a shoulder here, which has the same intensity as\nthe peak of the off-diagonals. At the L2-edge, most of the\nsignal is generated by the t2g. In the XMCD, we can see\nthat all the peaks of the L3-edge have different origins.\nThe first down peak is caused by the excitations into the\nt2g, the up-pointing peak by the off-diagonals and the\nlast peak by the eg.\nIn Fig. 12 we see the e g- and t2g-projected results of\nthe Ni site in NiFe 2O4. Here, nearly all the signal comes\nfrom the eg, because in Ni2+thet2gis fully occupied and\none can therefore only make electronic transitions to the\neg.\nFinally, the result for the Fe sites in NiFe 2O4are shown\nin Fig. 13. In the XAS, we see that the lowest energy\npeak is caused mostly by the t2g, however, the e gsig-\nnal also contributes and the off-diagonal elements reduce\nthe intensity. The main peak is caused by a combina-\ntion of all three signal sources with the biggest contri-\nbution coming from the t2g. At the L2-edge, one can\nsee that the two peak signals come mostly from the eg\nas the t2gis too broad the easily distinguish between\nthe peaks even though it has more intensity. While the\noff-diagonal reduces the intensity of the first peak and\nslightly increases the intensity of the second peak. The\nXMCD shows that the first down-pointing peak comes\nfrom excitations into the t2gorbitals which corresponds\nto the lowest lying states of the octahedral sites. At this\nposition, we also have the contributions from the egfrom\nthe tetrahedral site but the intensity is so small that is\nnot visible in the total spectrum. The middle peak that\nis caused by the tetrahedral site, is instead caused by\noff-diagonal elements and the t2gcontributions. The sec-\nond down-pointing peak is just as expected caused by\nthe higher lying egstates in the octahedral Fe site. At\ntheL2-edge, we can see that the initial dip in the total\nspectra is caused by the egand the off-diagonal signal\nand the peak after that is caused by the same.\n2. DMFT\nIn this section, we investigate the effects of using the\nlocal Hamiltonian and the hybridization function ob-\ntained from a DFT+DMFT calculation, instead of a14\n20\n 10\n 0 10 20\nEnergy (eV)5\n051015Intensity (arb. units)eg\nt2g\noffdiagonal\n20\n 10\n 0 10 20\nEnergy (eV)1.5\n1.0\n0.5\n0.00.51.0Intensity (arb. units)eg\nt2g\noffdiagonal\nFIG. 9: Calculated egandt2gL2,3XAS (top) and\nXMCD (bottom) of Fe 3O4(for details see text).\nDFT+ Ucalculation to construct the impurity Hamilto-\nnian of NiFe 2O4, while using the same Slater and SOC\nparameters as before. We used the method described in\nRef. [15], with one bath state per correlated orbital and\nthe ED impurity solver. Figs. 14 and 15 show the XAS\nand XMCD spectra of the Fe and Ni sites, respectively,\ncalculated using the same parameters as Figs. 5 and 6 ex-\ncept that the relative corelevel shift ∆ ϵpwas adjusted to\nthe values extracted from the DFT+DMFT calculation.\nHere, we can see that the spectra are very similar. In the\ncase of the Fe-projected XAS, we can see that the spec-\ntrum looks broader, which causes the L2-edge to be even\nmore overestimated. In the XMCD, we can see that the\nsecond peak and the oscillations after the third peak of\ntheL3-edge are better reproduced in the DFT+DMFT\ncalculation. Similarly, we can see improvements at the\nbeginning of the L2-edge, while the end is even more\noverestimated than in the DFT+ Ucalculation. In the\nNi-projected XAS, we can see that the L3-edge shoulder\nis more strongly pronounced in the DFT+DMFT calcu-\nlation. At the L2-edge however, we can see a double\npeak structure in the DFT+DMFT solution instead of\na pronounced first peak which we see in the experiment\n20\n 10\n 0 10 20\nEnergy (eV)3\n2\n1\n0123Intensity (arb. units)eg\nt2g\noffdiagonal\n20\n 10\n 0 10 20\nEnergy (eV)2.0\n1.5\n1.0\n0.5\n0.00.51.01.5Intensity (arb. units)eg\nt2g\noffdiagonalFIG. 10: Same for Co in CoFe 2O4.\nand DFT+ Ucalculation. In the XMCD, we can see that\nthe DFT+DMFT calculation reproduces the experiment\neven better than the DFT+ Ucalculation with a more\npronounced L3-edge shoulder and a better relative inten-\nsity of the peaks in the L2-edge. The similarities between\nthe results from DFT+ Uand DFT+DMFT are expected,\nbecause the sites mostly hybridize to the O sites, which\nare only treated using DFT. Another reason for the sim-\nilarities is that we included all exchange interaction in\nthe + U-term of the Hamiltonian (the so-called LDA+ U\napproximation, which is not identical to the LSDA+ U\napproximation, where exchange splitting is also included\nin the density functional), that guarantees that exchange\ninteractions are treated in the same way in the two ap-\nproaches.15\n20\n 10\n 0 10 20\nEnergy (eV)10\n5\n051015Intensity (arb. units)eg\nt2g\noffdiagonal\n20\n 10\n 0 10 20\nEnergy (eV)4\n2\n0246Intensity (arb. units)eg\nt2g\noffdiagonal\nFIG. 11: Calculated egandt2gprojection of the Fe L2,3\nXAS (top) and XMCD (bottom) of CoFe 2O4(for\ndetails see text).\n20\n 10\n 0 10 20\nEnergy (eV)1\n0123456Intensity (arb. units)eg\nt2g\noffdiagonal\n20\n 10\n 0 10 20\nEnergy (eV)1.5\n1.0\n0.5\n0.00.5Intensity (arb. units)eg\nt2g\noffdiagonalFIG. 12: Calculated egandt2gprojection of the Ni L2,3\nXAS (top) and XMCD (bottom) of NiFe 2O4(for details\nsee text).16\n20\n 10\n 0 10 20\nEnergy (eV)7.5\n5.0\n2.5\n0.02.55.07.5Intensity (arb. units)eg\nt2g\noffdiagonal\n20\n 10\n 0 10 20\nEnergy (eV)2.0\n1.5\n1.0\n0.5\n0.00.51.01.52.0Intensity (arb. units)eg\nt2g\noffdiagonal\nFIG. 13: Calculated egandt2gprojection of the Fe L2,3\nXAS (top) and XMCD (bottom) of NiFe 2O4(for details\nsee text).\n700 705 710 715 720 725 730\nEnergy (eV)0.02.55.07.510.012.515.017.5Intensity (arb. units)experiment\nLDA+DMFT\nLDA+U\n700 705 710 715 720 725 730\nEnergy (eV)2\n1\n01234Intensity (arb. units)experiment\nLDA+DMFT\nLDA+UFIG. 14: Calculated Fe L2,3XAS (top panel) and\nXMCD (bottom panel) edges in NiFe 2O4starting from\na converged DMFT calculation (red solid) and LDA+ U\n(blue dashed) compared to experiments (black solid).17\n840 845 850 855 860 865 870 875 880\nEnergy (eV)01234567Intensity (arb. units)experiment\nLDA+DMFT\nLDA+U\n840 845 850 855 860 865 870 875 880\nEnergy (eV)2.5\n2.0\n1.5\n1.0\n0.5\n0.00.5Intensity (arb. units)experiment\nLDA+DMFT\nLDA+U\nFIG. 15: Calculated Ni L2,3XAS (top panel) and\nXMCD (bottom panel) edges in NiFe 2O4starting from\na converged DMFT calculation (red solid) and LDA+ U\n(blue dashed) compared to experiments (black solid).18\n[1] M. 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Thun-\nstr¨ om, O. Eriksson, B. Sanyal, I. Di Marco, and Y. O.\nKvashnin, Phys. Rev. B 96, 245131 (2017).\n[34] S. J. Kim, W. C. Kim, and C. S. Kim, Journal of the\nKorean Physical Society 36, 430 (1999).\n[35] V. N. Antonov, B. N. Harmon, and A. N. Yaresko, Phys-\nical Review B - Condensed Matter and Materials Physics\n67, 244171 (2003).\n[36] H. L. Andersen, M. Saura-M´ uzquiz, C. Granados-\nMiralles, E. Can´ evet, N. Lock, and M. Christensen,\nNanoscale 10, 14902 (2018).\n[37] S. Klotz, G. Rousse, T. Str¨ assle, C. L. Bull, and\nM. Guthrie, Phys. Rev. B 74, 012410 (2006).\n[38] R. Knut, R. S. Malik, C. Kons, J. E. Shoup, F. Radu,\nC. Luo, Y. O. Kvashnin, A. Gupta, O. Karis, and D. A.\nArena, Journal of Physics: Condensed Matter 33, 225801\n(2021).\n[39] M. C. Richter, J.-M. Mariot, O. Heckmann,\nL. Kjeldgaard, B. S. Mun, C. S. Fadley, U. L¨ uders,\nJ.-F. Bobo, P. De Padova, A. Taleb-Ibrahimi, and\nK. Hricovini, Eur. Phys. J. Special Topics 169, 175\n(2009).\n[40] Y. K. Wakabayashi, Y. Nonaka, Y. Takeda, S. Sakamoto,\nK. Ikeda, Z. Chi, G. Shibata, A. Tanaka, Y. Saitoh,\nH. Yamagami, M. Tanaka, A. Fujimori, and R. Nakane,\nPhysical Review B 96, 104410 (2017).\n[41] Experimental data of nickel ferrite was kindly provided\nby Dario Arena of University South Florida, Ronny Knut\nand Susmita Saha of Uppsala University.\n[42] J. M. Wills, M. Alouani, P. Andersson, A. Delin,\nO. Eriksson, and O. Grechnyev, Full-Potential Elec-\ntronic Structure Method , edited by E. S. H. Dreysse and\nP. P. of Solids: Springer Series in Solid-State Sciences\n(Springer-Verlag, Berlin, 2010).\n[43] “RSPt - relativistic spin polarized toolkit,” http://\nfplmto-rspt.org/ (2017).\n[44] R. Eder, Multiplets in Transition Metal Ions (Verlag des\nForschungszentrum Julich, 2012)."    },    {        "title": "1511.01236v1.Numerical_Calculations_of_Wake_Fields_and_Impedances_of_LHC_Collimators__Real_Structures.pdf",        "content": "Numerical Calculations of Wake Fields and\nImpedances\nof LHC Collimators' Real Structures\u0003\nO. Frascielloy, M. Zobov, INFN-LNF, Frascati, Rome, Italy\nJuly 31, 2021\nAbstract\nThe LHC collimators have very complicated mechanical designs in-\ncluding movable jaws made of higly resistive materials, ferrite materials,\ntiny RF contacts. Since the jaws are moved very close to the circulat-\ning beams their contribution in the overall LHC coupling impedance is\ndominant, with respect to other machine components. For these reasons\naccurate simulation of collimators' impedance becomes very important\nand challenging. Besides, several dedicated tests have been performed\nto verify correct simulations of lossy dispersive material properties, such\nas resistive wall and ferrites, benchmarking code results with analytical,\nsemi-analytical and other numerical codes outcomes. Here we describe all\nthe performed numerical tests and discuss the results of LHC collimators'\nimpedances and wake \felds calculations.\n1 Introduction\nThe Large Hadron Collider (LHC) has a very sophisticated collimation system\nused to protect the accelerator and physics detectors against unavoidable regular\nand accident beam losses [1, 2]. The system has a complicated hierarchy com-\nposed of the primary (TCP), secondary (TCS) and tertiary (TCT) collimators\nand the injection protection collimators.\nSince the collimators are moved very close to the circulating beams they\ngive the dominant contribution in the collider beam coupling impedance, both\nbroad-band and narrow band. The electromagnetic broad-band impedance is\nresponsible of several single bunch instabilities and results in the betatron tunes\nshift with beam current, while the narrow band impedance gives rise to the\nmultibunch instabilities and leads to vacuum chamber elements heating.\nThe impedance related problem has been recognized already in the present\nLHC operating conditions [3] and is expected to be even more severe for the\nHigh Luminosity LHC upgrade [4], where one of the principal key ingredients\nfor the luminosity increase is the beam current increase. For this reason the\n\u0003Work supported by HiLumi LHC Design Study, which is included in the High Luminos-\nity LHC project and is partly funded by the European Commission within the Framework\nProgramme 7 Capacities Speci\fc Programme, Grant Agreement 284404.\nyoscar.frasciello@lnf.infn.it\n1arXiv:1511.01236v1  [physics.acc-ph]  4 Nov 2015correct simulation of the collimator impedance becomes very important and\nchallenging.\nIn order to simulate the collimators as close as possible to their real designs,\nwe used CAD drawings including all the mechanical details as inputs for the\nhigh performing, parallelizable, UNIX-platform FDTD Gd\fdL code [5]. A very\n\fne mesh, typically, of several billions mesh points, was required to reproduce\nthe long and complicated structures, described in huge .stl \fles, and to over-\ncome arising numerical problems. In order to be sure that the code reproduces\ncorrectly properties of lossy dispersive materials (resistive walls, ferrites) used\nin the collimators we have carried our several dedicated numerical tests compar-\ning the Gd\fdL simulations with available analytical formulae, other numerical\ncodes and semi-analytical mode matching techniques.\nThe only way to a\u000bord such a huge computational task was to use the Gd\fdL\ndedicated cluster at CERN, engpara, which has allowed us to study the wake\n\felds and impedances for several types of collimators without using any model\nsimpli\fcations: secondary collimators, new collimators with incorporated beam\nposition monitors and injection protection collimators. In such circumstances,\nGd\fdL wake \felds computation up to wake length of hundreds times the typical\ndevices lengths (\u00181m) took several days or two weeks at maximum.\nIn this paper we describe Gd\fdL tests of the resistive walls and ferrites sim-\nulations, discuss the calculated collimator impedances comparing the obtained\nresults with available experimental data.\n2 Resistive Wall Simulation Test\nOnly recently a possibility to carry out simulations with resistive walls (RW), im-\nplementing the impedance boundary conditions, was made available in Gd\fdL.\nSo it has been decided to perform a numerical test comparing the simulation\nresults with known analytical formulas. For this purpose we calculated both the\nlongitudinal and the transverse loss factors (the latter known also as kick factor)\nof a Gaussian bunch passing inside a round beam pipe having an azimuthally\nsymmetric thick resistive insert. The insert was enough long in order to be able\nto neglect the contribution of the insert ends, as shown in Fig. 1.\nFigure 1: Gd\fdL model for the azimuthally symmetric beam pipe with resistive\ninsert. The choosen length was L= 30 cm, the insert thickness a= 5 mm, the\npipe radius plus the insert thickness b= 10 mm, and the electrical conductivity\n\u001bc=7:69\u0001105S/m for Carbon Fiber Composite (CFC).\nIn this case the loss factors can be found analytically:\nkk=cL\n4\u0019b\u001b3=2\nzr\nZ0\u001a\n2\u0000\u00123\n4\u0013\n; (1)\n2for the longitudinal one and\nk?=cL\n\u00192b3r\n2Z0\u001a\n\u001bz\u0000\u00125\n4\u0013\n(2)\nfor the transverse one, where c= 2:997925\u0002108m/s is the speed of light, Lis\nthe length of the pipe, \u001a= 1=\u001bcis the electrical resistivity, \u001bzthe bunch length\nand \u0000 the Euler gamma function. Figure 2 shows a comparison between the\nanalytical formulas and the numerical data. As it is seen the agreement is quite\nsatisfactory.\nFigure 2: Loss and kick factors benchmark between Gd\fdL and analytical for-\nmulas Eq. (1) and Eq. (2).\nHowever, the loss factors are somewhat \\averaged\" values characterizing the\nbeam impedance. In order to check the impedance frequency behavior the RW\nimpedance of the insert has been calculated using the semi-analytical mode-\nmatching method (MMM) [6]. In turn, numerically the impedance till rather\nlow frequencies has been obtained by performing a Fourier transform of a long\n3wake behind a long bunch obtained by Gd\fdl, and also by CST for comparison.\nAs it is seen in Fig. 3 also the impedance frequency behavior is reproduced well\nby Gd\fdL.\nFigure 3: Dipolar transverse impedance benchmark between Gd\fdL, CST and\nMMM outcomes.\n3 Ferrite Material Simulation Test\nIn order to damp parasitic higher order modes (HOMs) in the new collimators\nwith embedded BPM pickup buttons, special blocks made of the TT2-111R lossy\nferrite material are used. For this reason we have carried out a comprehensive\nnumerical study to test the ability of Gd\fdL to reproduce frequency dependent\nproperties of the lossy ferrite in calculations of wake \felds, impedances and\nscattering matrix parameters [7].\nFor this purpose, we have a) simulated a typical coaxial-probe measurement\nof the ferrite scattering parameter S11; b) compared the computation results of\nCST, Gd\fdL and Mode Matching Techniques by calculating impedances of an\nazimuthally symmetric pill-box cavity \flled with the TT2-111R ferrite in the\ntoroidal region; c) benchmarked Gd\fdL simulations against analytical Tsutsui\nmodel for a rectangular kicker with ferrite insert [8, 9] and CST simulations for\nthe same device.\nAll the comparative studies have con\frmed a good agreement between the\nresults obtained by Gd\fdL and the results provided by other numerical codes,\nby available analytical formulas and by the mode matching semi-analytical ap-\nproach. As an example, Fig. 4 shows a simpli\fed sketch of a set-up for the\nferrite material properties measurements: just a coaxial line \flled with a ferrite\nmaterial under test. For such a simple structure the re\rection coe\u000ecient S11is\neasily measured and can be found analytically as in Eq. (3).\nS11=\u0001\u0001tanh(\rL)\u00001\n\u0001\u0001tanh(\rL) + 1; (3)\nwith\r=j!p\u000f\u0016and \u0001 =q\n\u0016r\n\u000fr. Figure 5 shows the S11coe\u000ecient calculated\n4for the TT2-111R material in a very wide frequency range, from 106to 1012Hz.\nAs it is seen, despite the complicated S11frequency dependence the agreement\nbetween Gd\fdL, HFSS and the analytical formula is remarkable.\nFigure 4: Coaxial probe measurement model for Gd\fdL S11simulations.\nFigure 5: Re\rection coe\u000ecient S11results for the arranged simulation setup.\nThe solid line is the analytical trend from Eq. (3).\n4 Impedance of LHC Run I TCS/TCT Collima-\ntors\nIn the 2012 LHC impedance model, collimators played the major role ( \u001890%)\nover a wide frequency range, both for real and imaginary parts, but the model\nwas essentially based on the resistive wall impedance of collimators, the resis-\ntive wall impedance of beam screens and warm vacuum pipe and a broad-band\nmodel including pumping holes, BPMs, bellows, vacuum valves and other beam\ninstruments. The geometric impedance of collimators was approximated only\nby that of a round circular taper [10].\n5However, several measurements were performed in 2012 of the total single\nbunch tune shifts vs. intensity, both at injection and at 4 TeV, the results\ncoming out to be higher than predicted ones with numerical simulations by a\nfactor of\u00182 at top energy and of \u00183 at injection [11]. This fact led to the\nneed for an LHC impedance model re\fning which, \frst of all, required a careful\ncollimator geometric impedance calculation. For this purpose, we carried out\nnumerical calculations of the geometric impedance of the LHC Run I TCS/TCT\ncollimator, whose design is shown in Fig. 6, and evaluated its contribution to\nthe overall LHC impedance budget.\nFigure 6: LHC Run I TCS/TCT collimator design.\nIn order to verify whether the geometric collimator impedance could give\na noticeable contribution to the betatron tune shifts, we suggested to compare\ntransverse kick factors due to the resistive wall impedance and the geometric\none, showing that the tune shifts are directly proportional to the kick factors\n[12].\nFigure 7: E\u000bective transverse impedances of theoretical Stupakov \rat taper\nmodel, Yokoya round taper model) and Gd\fdL simulations of TCS/TCT colli-\nmator, as a function of the jaws' half gap.\n6Figure 8: Comparison between geometrical kick factor and RW kick factors for\nCFC and W, as a function of the jaws' half gap.\nIn Fig. 7 and Fig. 8 the main results about the transverse broad-band\nimpedance and kick factors are reported, showing that the geometric impedance\nis better approximated by a \rat taper model than by a round taper one and\nthat the geometric contribution is not negligible with respect to the resistive\nwall one. In particular, for CFC made collimator, the geometrical kick starts\nto be comparable to resistive wall one at about 8 mm half gap. In turn, for W\nmade collimators, the geometrical kick dominates almost for all the collimator\ngaps.\nThe study contributed to the re\fnement of the LHC impedance model. It\nhas also been shown that the geometrical collimator impedance accounts for\napproximately 30% of the total LHC impedance budget, at frequencies close to\n1 GHz.\n5 Impedance of LHC Run II TCS/TCT Colli-\nmators\nDuring the last LHC Long Shutdown I (LSI), 2 TCS CFC and 16 TCT Tung-\nsten (W) collimators were replaced by new devices with embedded BPM pickup\nbuttons, whose design is shown in Fig. 9. RF \fngers were removed from the\nprevious LHC Run I TCS/TCT design and HOMs damping was entrusted to\nthe TT2-111R ferrite blocks. By means of Gd\fdL broad band impedance simu-\nlations of the new collimators' real structure, we gained the results for the kick\nfactors in Tab. 1, showing that an increase of about 20% is expected for the\ntransverse e\u000bective impedance, with respect to LHC RUN I type collimator's\ndesign.\nIn order to study the impedance behaviour of the new collimators and the\ne\u000bect of the ferrite blocks on HOMs, we performed detailed Gd\fdL wake \felds\nsimulations of the whole real structures. We set into Gd\fdL input \fle the \fnite\nconductivity of W and the frequency-dependent permeability of TT2-111R. As\n7Figure 9: LHC Run II TCS/TCT collimator with embedded BPM pickup but-\ntons.\nTable 1: Geometric Transverse Kick Factors Due to the Two TCS/TCT Ge-\nometries, Calculated at Di\u000berent Half Gap Values\nw/ BPM cavity w/o BPM cavity\nHalf gaps (mm) kT(V\nCm) kT(V\nCm)\n1 3 :921\u000110143:340\u00011014\n3 6 :271\u000110135:322\u00011013\n5 2 :457\u000110132:124\u00011013\na \frst result, an overall impedance damping feature was shown to be proper\nof the structure with resistive W jaws plus ferrite blocks at all frequencies [13],\nas clearly visible from the plot in Fig. 10. There, the red curve represents the\ncollimator simulated as a whole Perfect Electrical Conductor (PEC), without\nany resistive and dispersive material, while the black one represents the real\ncollimator with W jaws and ferrite blocks. The e\u000bect of ferrite results also\nin the shift of HOMs characteristic frequencies toward lower frequencies. As\nan example, the \frst HOM frequency shifts from \u001995 MHz to\u001984:5 MHz,\nat exactly the same frequency measured experimentally at CERN with loop\ntechnique [14]. It is clear that the computed impedance spectrum resolved very\nwell the low frequency HOMs, whose characteristic frequencies are in excellent\nagreement with those found experimentally. Moreover, under these simulation\ncircumstances, the computed shunt resistance of the \frst HOM at \u001984 MHz is in\nremarkably agreement, within a factor of 2, with that measured experimentally\nwith the wire technique at CERN [15], being Rsim\ns\u0019237 k\n=m andRmeas\ns\u0019\n152 k\n=m [16].\n6 Conclusions\nCalculations of wake \felds and beam coupling impedance have been performed\nfor the LHC TCS/TCT collimators, by means of Gd\fdL electromagnetic code.\nWe performed, for the \frst time in the \feld of impedance computations, a com-\nplete and detailed simulation campaign of collimators' real structures, including\nthe properties of real and lossy dissipative materials.\n8Figure 10: Real part of the impedance spectrum of LHC Run II TCS/TCT\ncollimators, the inset layer focusing on low frequency HOMs.\nFor LHC Run I collimators, the comparison of the transverse kick factors cal-\nculated for \fve di\u000berent jaws' half gaps, has shown that the geometric impedance\ncontribution is not negligible with respect to the resistive wall one. The study\nhas contributed to the re\fnement of the LHC impedance model, as a result of\nthe geometrical collimator impedance accounting for approximately 30% of the\ntotal LHC impedance budget, at frequencies close to 1 GHz.\nThe performed numerical tests have con\frmed that Gd\fdL reproduces very\nwell the properties of the lossy dispersive materials. The simulation test results\nfor the resistive walls and the lossy ferrites are in a good agreement with available\nanalytical formulae and the results of other numerical codes and semi-analytical\nmodels. The tests have made us con\fdent in the results of our impedance studies\ncarried out for the real structures of the new Run II TCS/TCT collimators with\nincorporated BPMs. Several important results have been obtained conducting\nthese studies. First, we found that there are no dangerous longitudinal higher\norder modes till about 1.2 GHz. This is important for the heating reduction\nof the collimators in the multibunch regime (for the nominal LHC bunches 7.5\nlong). Second, the TT2-111R ferrite resulted to be very e\u000bective in damping\nboth longitudinal and transverse parasitic modes for frequencies above 500 MHz.\nHowever, the modes at lower frequencies are less damped, residual transverse\nHOMs at frequencies around 100 MHz and 200 MHz with non-negligible shunt\nimpedances still existing. The calculated frequencies of the modes are in re-\nmarkable agreement with the loop measurements. The shunt impedances of the\nmodes obtained numerically agrees within a factor of 2 with the experimental\ndata of the wire measurements performed at CERN. Finally, the broad-band\ntransverse impedance of the new LHC Run II double taper collimators are eval-\nuated to be approximately by 20% higher with respect to that of the LHC Run\nI TCS/TCT collimators.\n7 Acknowledgments\nWe are grateful to W. Bruns for his invaluable support.\n9We would like to thank also the CERN EN-MME and BE-ABP departments,\nfor the providing of the collimators' CAD designs and S. Tomassini, of INFN-\nLNF, for his accurate handling and adjusting the CAD designs to serve as\ninputs for Gd\fdL simulations. In particular special thanks are addressed to the\nE. Metral, on behalf of the whole CERN LHC impedance group, for the support\nand pro\ftable discussions, to N. Biancacci for the MMM simulations' data and,\ntogether with F. Caspers, J. Kuczerowski, A. Mostacci and B.Salvant for the\ninformation on the collimators' impedance measurements side.\nReferences\n[1] R.W. Assman et al. Requirements for the LHC collimation system. Proceed-\nings of the 8th European Particle Accelerator Conference , pages 197{199,\n2002.\n[2] R.W. Assman et al. An improved collimation system for the LHC. Proceed-\nings of the 9th European Particle Accelerator Conference , pages 536{538,\n2004.\n[3] E. Metral et al. Transverse Impedance of LHC Collimators. Proceedings\nof Particle Accelerator Conference PAC07 , pages 2003{2005, 2007. CERN-\nLHC-PROJECT-REPORT-1015.\n[4] The HL-LHC collaboration. HL-LHC Preliminary Design Report Deliver-\nable: D1.5. Technical report, 2014. CERN-ACC-2014-0300.\n[5] W. Bruns. Gd\fdL web page. http://www.gdfidl.de .\n[6] N. Biancacci, V. G. Vaccaro, E. M\u0013 etral, B. Salvant, M. Migliorati, and\nL. Palumbo. Impedance studies of 2d azimuthally symmetric devices of\n\fnite length. Phys. Rev. ST Accel. Beams , 17:021001, Feb 2014.\n[7] O. Frasciello et al. Wake \felds and impedances calculations with Gd\fdL,\nMMM and CST for benchmarking purposes, 2014. Contributed talk at\nBE-ABP Impedance meeting, February 2nd, CERN, Geneva, Switzerland.\n[8] H. Tsutsui et al. Transverse Coupling Impedance of a Simpli\fed Ferrite\nLicker Magnet Model. Technical report, 2000. LHC-PROJECT-NOTE-234.\n[9] H. Tsutsui. Some Simpli\fed Models of Ferrite Kicker Magnet for Cal-\nculations of Longitudinal Coupling Impedance. Technical report, 2000.\nCERN-SL-2000-004 AP.\n[10] N. Mounet. The LHC Transverse Coupled-Bunch Instability. PhD thesis,\nEcole Polytechnique, Lausanne, Mar 2012.\n[11] N. Mounet et al. Beam stability with separated beams at 6.5 tev. In LHC\nBeam Operation Workshop Evian 17-20 December 2012 , 2012.\n[12] O. Frasciello et al. Geometric beam coupling impedance of LHC secondary\ncollimators. In Proceedings of IPAC 2014 .\n10[13] O. Frasciello et al. Present status and future plans of LHC collimators\nwake \felds and impedance simulations, 2015. Contributed talk at BE-ABP\nImpedance meeting, March 23rd, CERN, Geneva, Switzerland.\n[14] N. Biancacci et al. Impedance bench measurements on TCTP collima-\ntor with ferrite, 2014. Contributed talk at BE-ABP Impedance meeting,\nAugust 8th, CERN, Geneva, Switzerland.\n[15] N. Biancacci et al. Impedance bench measurements on TCTP and SLAC\ncollimators, 2014. Contributed talk at BE-ABP Impedance meeting, April\n14th, CERN, Geneva, Switzerland.\n[16] O. Frasciello et al. Beam coupling impedance of the new LHC collimators,\n2015. Contributed talk at 101st National Congress of the Italian Physical\nSociety (SIF), September 24th, Rome, Italy.\n11"    },    {        "title": "0807.4280v1.Do_the_Tellegen_particles_really_exist_in_electromagnetics_.pdf",        "content": "Do the Tellegen particles reall y exist in electromagnetics? \n \nE.O. Kamenetskii, M. Sigalov, and R. Shavit \n \nBen-Gurion University of the Negev, Beer Sheva 84105, Israel \n \nJuly 22, 2008 \n  \nAbstract \n \nIn 1948 Tellegen suggested that an assembly of  the lined up electric-m agnetic dipole twins can \nconstruct a new type of an electromagnetic materi al. Till now, however, th e problem of creation of \nthe Tellegen medium is a subject of strong disc ussions. An elementary symmetry analysis makes \nquestionable an idea of a simple combination of tw o (electric and magnetic) dipoles to realize local \nmaterials with the Tellegen particles as structural el ements. In this paper we show that in his search \nof sources with local j unctions of the electrical and magnetic  properties one cann ot rely on the \ninduced parameters of small electromagnetic scattere rs. No near-field electromagnetic structures and \nclassical motion equations for point charges give a physical basis to realize sources with the local \njunction of the electrical and magnetic prope rties. We advance a hypothesis that local \nmagnetoelectric (ME) particles should be the phys ical objects with eigenmode oscillation spectra \nand non-classical symmetry breaking effects.  Our studies convincingly prove this assumption. We \nshow that a quasi-2D ferrite disk with magnetic-di polar-mode oscillations is characterized by unique \nsymmetry features with topological phases resulting in appearance of the ME properties. The entire \nferrite disk can be characterized as a combined system with eigen electric and magnetic moments. \nThe fields near such a particle are dis tinguished by special sy mmetry properties. \n PACS numbers: 03.50.D e, 75.45.+j, 68.65.-k, 03.65.Vf \n  \nI. INTRODUCTION \n \nThe question on relations between magnetoelectricity and electro magnetism is a subject of a strong \ninterest and numerous discussions  in microwave and optical wave physics and material sciences. An \nidea about a local magnetoelectric (ME) effect in media goes back to Debye who sugge sted in 1925 \nthe possible existence of molecules which have a permanent electric dipole moment as well as a \npermanent magnetic dipole moment [1]. Tellegen considered an assembly of electric-magnetic dipole twins, all of them lined up in  the same fashion (either parallel  or anti-parallel) [2]. Since 1948, \nwhen Tellegen suggested such \"glued pairs\" as st ructural elements for composite materials, the \nelectrodynamics of these complex media was a subject  of serious theoretical studies (see, e.g. [3 – \n5]). Till now, however, the problem of creation of the Tellegen medium is a subject of strong \ndiscussions. The questions – How the \"glued pairs\"  of two small (electric and magnetic) dipoles can \nbe realized as local ME sources  for electromagnetic waves? and, more generally, Do the Tellegen \nparticles really exist in el ectromagnetics? – are still open. \n    The electric polarization is parity-odd and tim e-reversal-even. At the same time, the magnetization \nis parity-even and time-reversal-odd  [6]. These symmetry relationshi ps make questionable an idea of \na simple combination of two (electric and magnetic) small dipoles to realize local ME materials for \nelectromagnetics. If one supposes that he has crea ted a particle with the local cross-polarization  2effect one, certainly, should demonstrate a special ME  field near this particle . It means that using a \ngedankenexperiment with two quasistatic, electric and magnetic, point probes for the ME near-field \ncharacterization, one should observe  not only an electrostatic-poten tial distribution (because of the \nelectric polarization) and not only a magnetostatic -potential distribution (because of the magnetic \npolarization), one also should observe a special  cross-potential term (because of the cross-\npolarization effect). This fact c ontradicts to classical electrodynami cs. What will be an expression \nfor the Lorentz force acting between these particle s? This expression shoul d contain the \"electric \nterm\", the \"magnetic term\", and the \"ME term\". Such an expression is unknown in classical electrodynamics. One cannot consid er (classical electrodynamically ) a system of two coupled \nelectric and magnetic dipoles as local sources of the ME field a nd there are no two coupled Laplace \nequations (for magnetostatic and electrostatic pot entials) in the near-field region [6]. In a \npresupposition that a particle with th e near-field cross-polarization eff ect is really created, one has to \nshow that inside this particle there are internal  dynamical motion processes with special symmetry \nproperties.     Such well known \"electromagnetic ME  scatterers\" as sm all helices [4], Ω-paricles [7] and split-\nring-resonators (SRRs) [8] are, in fact, small de lay-line sections with distinctive inductive and \ncapacitive regions. In microwave experiments with these particles, no \"ME coupling\" was shown in \nthe standing-wave systems. For realization of th e effect of \"ME coupling\" in these special-form \nsmall scatterers one should have the propagating- wave behavior. Moreover, the \"ME response\" will \nbe dependable on the type of propagating electrom agnetic fields. For the plane, cylindrical, and \nspherical electromagnetic waves, there will be di fferent \"ME responses\". Th e question is still open: \nWhat kind of the near-field structure surrounding su ch an \"electromagnetic ME  scatterer\" should one \nexpect to see?  Any quasistatic theories (similar, fo r example, to the quasistatic Lorentz theory used \nfor artificial dielectrics [9]) are not applicable for such metallic-inclusion composites. In paper [10], authors declared about experimental  realization of the ferrite-based Tellegen particle in microwaves. \nThe particle, created as a small ferrite sphere comb ined with a small piece of a thin metal wire, was \nplaced in a rectangular waveguide cavity. The \"ME parameter\" of su ch a particle was estimated via \nan amplitude of a \"cross-polarized wave\". It is  well known, however, that for a microwave resonator \ncontaining enclosed gyrotropic-medium samples, th e electromagnetic-field eigenfunctions will be \ncomplex, even in the absent of dissipative losses. It means that one does not have the standing-wave \nfields in spite of the fact that the eigen freque ncies of a cavity with gyrotropic-medium samples are \nreal [11]. A microwave resonator with a ferrite in clusion acting in the proximity of the ferromagnetic \nresonance (FMR) is a nonintegrabl e system with the time-reversal  symmetry breaking effect. The \nwaves reflected from the ferrite boundary are ch aotic random waves in a cavity [12 – 14]. Since no \neigen oscillations inside a ferri te sphere are observed, the microw ave responses will be dependable \non a type of the exciting field. So for such a particle no definite and stable polarizability parameters characterizing the ME properties can be found. \n    The Tellegen proposition wa s about the particles wi th permanent coupled electric and magnetic \ndipole moments. In all the above \"ME scatterers\" we have the induced effects. It was stated in [15] \nthat so-called Janus particles can be considered as the Tellegen pa rticles with permanent (electric \nand magnetic) moments. Janus particles are bifaci al nanoparticles [16]. Th ey may be separately \nelectrically dipolar or ferromagne tic. The lack of centrosymmetry in synthetic Janus systems may \nlead to the discovery of novel material properties.  Probably, there can be the ME properties. These \nproperties, however, were observe d at DC fields and no wave dynamics effects are shown in the \nstudies [15]. On the other hand, the question on th e role of symmetry breaking in ME coupling is \nessential in ME crystals and piezo-composites.  There are we ll known ME materials which are \nsingle-phase non-centrosymmetric magnetic crystals or  composites that contai n a piezoelectric phase \n[17, 18]. One can change the material parameters  by a bias electric fi eld and observe the ME \nproperties for different wave dynami cs processes, even at microwav es [17 – 19]. At the same time, \nnatural magnetoelectric crystals and complex ferri te/piezoelectric structures are not particulate  3composites with ME (Tellegen-type) particles as st ructural elements. In a proposition of particulate \nME composites one may suppose that the unified ME fields originated from a point ME particle \n(when such a particle is created ) will not be the analytically-des cribed classical electromagnetic \nfields. It is known that in solids effective interactions of charges and spins are often quite different \nfrom the fundamental laws of electrodynamics, wh ich may give rise to unusual phenomena. One can \nexpect that the motion equations inside a conjectu ral local ME particle ma y lead to non-classical \nfields with special symmetry properties.      Recently it was found that magnetic-dipolar-mode  (MDM) oscillations in a quasi-2D ferrite disk \nare characterized by th e dynamical symmetry breaking effects [20] resulting in  the near-field \nstructures with unique topologi cal properties, which are reflect ed in microwave experiments by \nspecific responses [21 – 25].  Based on analytical and numerical studies,  in this paper we show that \nMDM ferrite-disk particles exhibit special topological effects and can  be observed (by the external \nnear fields) as local system s of two, electric and magnetic, eigen  moments. These ferrite ME \nparticles cannot be considered, however, as classical electromagnetic \"ME scatterers\".  \n    The paper will begin with Section 2 giving an analysis of different classical aspects in possible realization of local ME particles. Section 3 will be devoted to consideration of special mechanisms \nof generation of magnetoelectric ity by magnetic chirality. ME prope rties of quasi-2D ferrite disk \nparticles with magnetic-dipolar-mode oscillat ions will be shown in Section 4. From the \nelectromagnetic point of view, a ferrite ME particle behaves as a non-integrable object. Such an \nelectromagnetically chaotic system can be well  modeled, however, by numerical studies based on \nthe commercial numerical electromagnetic-simulati on programs. We present numerical verifications \nfor unique ME properties of thin-film ferrite disks. Section 6 is devoted to numerical studies of ME \nproperties of MDM ferrite disks with special geom etries. The paper will be concluded by a summary \nin Section 6 with an outlook to future developm ents of structures with  local ME properties. \n \nII. SEARCH OF CLASSICAL SOURCES WITH LOCAL JUNCTIONS OF THE \nELECTRICAL AND MAGNETIC PROPERTIES \n \nAs worthy argumentation, we should forestall our ma in analysis with nece ssary discussions on \npossible classical point sources wi th local junctions of the electrical and magnetic properties. \nPresently, there are numerous public ations regarding different classi cal ways of re alization of a \nTellegen particle. We will give now some basic as pects on local electromagnetic scatterers, near-\nfield structures and motion equations, which clearly prove our standpoints that  no particles with the \nlocal cross-polarization effect can be real ized from a classical point of view.  \n \nA. Local electromagnetic scatterers \n \nIn electromagnetics, local scatterers are system s whose individual dimensions are small compared \nwith a wavelength. It is evident that if a \"ME scatte rer\" exists as a classical object, the fields and \nradiation of such a conjectural s catterer should be the fields and ra diation of a localized oscillating \nsource in classical electrodynamics. In Maxwell equations, the potentials, fields, and radiation can be \nconsidered as being originated from a localized  system of charges and currents which vary \nsinusoidally in time:  \n                                                                   \n.)( ),(,)( ),(\ntiti\nerJtrJer tr\nωωρρ\nrrrr\n==\n                                                            \n(1) \n      4For a case when the wavelength is much bigger than sizes of a region occupied by charges and \ncurrents, the incident fields induc e electric and magnetic multipoles th at oscillate in definite phase \nrelationship with the incident wave. There are two lim it regions: (a) the near (static) zone and (b) the \nfar (radiation) zone. The near fiel ds are quasi-stationary, i.e. they are oscillating harmonically as \ntieω, but otherwise static in characte r. Since the fields are static in  character, no interactions between \nthe electric and magnetic multipoles are presumed. A general solution of Maxwell equations can be \nrepresented, for example, as a series for the He rtzian-vector solution [26]. A character of the \nexpansion depends on frequency and geometrical propert ies of the current distribution. The first term \nin the expansion describes the fi eld exciting by an oscillating elect ric dipole, while the second term \nin the expansion represents the field stipulated  by an oscillating magnetic dipole and electric \nquadrupole [26]. Only the lowest multipoles, usually  electric and magnetic dipoles, are important. \nThese induced dipoles can be calculated from sta tic or quasistatic boundary- value problems [6, 26]. \nThe electric dipole moment epr is defined by the electric polarizabil ity and an electric component of \nan incident field, while th e magnetic dipole moment mpr is defined by the magnetic polarizability \nand a magnetic component of an in cident field. For example, for a sm all dielectric (with a dielectric \nconstant ε) sphere of radius a one has [6] \n                  \n                                                             inceEar r3\n21p+−=εε.                                                                   (2) \n \nThere is no magnetic dipole moment.  For a sm all perfectly conducting sphere of radius a one has \n \n                                             inceEarr3p= ,                         incmBar r3\n21p−= .                                         (3) \n \nSo a small conducting obstacle exhibits an electric  dipole polarization as well as magnetic dipole \npolarization. One also has the induced electric and magnetic moments for a more complicated case \nof scattering of electromagnetic fields by a small gyr otropic sphere. In this case, as well, the electric \ndipole is induced by the electric component of the incident field and the magnetic dipole is induced \nby the magnetic component of the incident field [ 27]. From a classical point  of view, there is no \nphysical mechanism for interaction  between electric and magnetic dipol es in the near (static) zone. \nThe total field is a superposition of the partial fields originated from uncoupled  electric and magnetic \ndipoles and no cross polarization e ffects take place. Far away from the scatterer (in the radiation \nzone) the fields are found to be [28]  \n \n                                              ()[]\n002\n00 0\n  p p p ),(rreik k rErrik\nm e errr rrrr rrrr\n−×∇+∇⋅∇+=−\nω ,                                 (4) \n \n                                              ()[]\n002\n00 0\n  p p p ),(rreik k rHrrik\ne m mrrr rrrr rrrr\n−×∇−∇⋅∇+=−\nω ,                           (5) \n \nwhere c kω=0 . The role of electric dipole epr in the magnetic field structure as well the role of \nmagnetic dipole mpr in the electric field structure become  evident only in the far field zone.  \n    The fact that there is no physical mechanism for interaction between elec tric and magnetic dipoles \nin the near (static) zone and that only in the fa r (radiation) zone one can observe the effect of \"ME \ninteraction\" becomes evident not onl y for small scatterers with simple geometry but also for small  5scatterers with complicated geometry. This cl early follows from a classical multipole theory. \nMultipole expansions in electrodynamics pr ovide a powerful met hod of characterizing \nelectromagnetic fields [29]. A classical multipole th eory describes an effect of \"ME coupling\" when \nthere is time retardation between the points of the finite-region charge and current distributions and \nthis time retardation is comparable with time re tardation between the origin and observation points. \nIn such a case, an expression for the field cont ains combinations of both magnetic and electric \nmultipole moments [29]. One may obtain the EM-wave phase shift between the points of the finite-\nregion charge and cu rrent distributions, 1ϕ, comparable with the EM-w ave phase shift between the \norigin and observation points, 0ϕ, even for a very small scatterer. To obtain such an effect of \"ME \ncoupling\" one should make a scat terer in a form of a small LC delay-line section. In the far zone of \nthis scatterer we will observe \"ME coupling\". This can be explained with help of Fig. 1. Let a \ncharacteristic size of a scatterer be l and L l<<, where  L is a distance between the origin point and \nthe observation point P. Let 1k be the wavenumber of th e EM wave propagating in a  LC delay line \nand 0k be the wavenumber of the EM wave in vacuum. In a case when 0 1 k k>> , one may obtain \nLk kl0 0 1     =≈=ϕϕ . All the proposed \"electromagnetic ME scat terers\" [4, 7, 8] have a typical form \nof a delay-line section with dist inctive inductive and capacitive regions . In a series of experimental \npapers one can see that the \"M E coupling\" effect in these pa rticles was observed only in the \npropagation-wave behavior, without any near-field characterizations [30 – 34]. This fact has a clear \nexplanation. When a small special-form \"ME-coupli ng\" scatterer is located in a cavity, both phase \nshifts  1ϕ and 0ϕ become equal to zero. Thus, no \"ME coup ling\" takes place in the standing-wave \nsystems and the observed special properties of the fields scattered from a small \"ME particle\" are \ncaused, in any event, by the field retardation effects.  No such scatterers with local cross-polarization \neffects can be presumed from classical electr odynamics. This gives a clear evidence why the \nmultipole theory demonstrates the \"ME coupling\"  in a non-local medium [29]. It follows, for \nexample, that retrieval of the effective constituti ve parameters of bianisot ropic metamaterials from \nthe measurement of the S parameters [35] should be relied on ly on the far-field characterization. \n    A dense composite material means an artificia l structure with characteris tic sizes of structural \nelements and distances between them much less th an the electromagnetic-wa ve wavelength. One can \nrealize dense composite dielectric and magnetic materials. There is pu re static (quasi static) electric \ninteraction between neighboring metal lic rods in an artificial dielec tric [9] and there is pure static \n(quasistatic) magnetic interaction between neighbori ng dielectric resonators [36] and SRRs [37] in \nartificial magnetic materials. Physically, one can create a dense composite based on small delay-line-\nsection \"ME particles\" as structural elements, but th is will not be a composite with local (quasistatic) \ncross-polarization coupling and so no quasistatic theories (similar, for example, to the quasistatic \nLorentz theory used for artificial dielectrics) are a pplicable for such \"ME composites\". It is stated in \n[38] that the separation between the macroscopic and microscopic electromagnetic descriptions is \nnot quite as sharp in bianisotropic media as it is in pure dielectric s due to the fact  that the cross-\npolarization coupling vanishes in the long-wavelength limit. A suppos ition that one can realize dense \nparticulate composites based on \"e lectromagnetic ME scat terers\" raises also a question of boundary \nconditions. It is well known that for materials with  nonlocal properties there should be introduced so \ncalled additional boundary conditions (ABCs). The A BCs, being considered as supplementary to \nstandard electromagnetic boundary conditions, are derived from some motion equa tions in a medium \n[39, 40]. In a case of nonlocal structures composed by \"electro magnetic ME scatterers\" no motion \nequations can be presupposed and so no reliable soluti ons of the boundary-val ue problems can be \nobtained. The known \"electromagnetic ME composites\"  are, in fact, various diffracting structures, \nwhich do not have any inherent (dif ferent from pure electromagnetic) mechanism of ME interaction. \nIn a case of an \"electromagnetic ME particle\" one  has only imagination of the ME coupling in the \nfar-field region. All the proposed \"M E particles\" are, in fact, open electrical contours oscillating at a  6resonance frequency and interactin g with an external electromagne tic field as classical radiating \nsystems.      Let us suppose a priori  that (in spite of the above argumenta tions) a small electromagnetic particle \nwith the local (quasistatic) cross-polarization effect  has been created. When a small electric dipole \nlocalizes an electric field and a small magnetic dipole localizes a magnetic field, a small particle \nwith the quasistatic cross-polarization effect should have a special ME field in the near-field region. Our next question concerns the pr operties of the near-field elec tromagnetic structures: Can the \nknown near-field electromagnetic structures exhib it themselves as the fields with specific ME \nproperties? We will show that an an swer to this question is negative. \n \nB. Near-field electromagnetic structures \n  One of the main aspects attracte d the concept of metamaterials wa s a possibility for the near-field \nmanipulation [41]. In electrodynamics , the near-field EM fields are considered as the evanescent \n(exponentially decaying) modes. In such a sense, me tamaterials can be charact erized as structures \nwith tailored electromagnetic response. The importance of phenomena involving evanescent \nelectromagnetic waves has been r ecognized over the last years. The fact that evanescent waves are \nmore confined than the single tone sinusoid waves and hence contain wider range of spatial \nfrequencies indicates that it may be possible to have no theoretical  limit of resolution for the near-\nfield patterns. At present, the near-field ma nipulation becomes an important factor in new \napplications, such as near-field microscopy and ne w material structures. Physically, there can be \ndistinguished different cat egories of the near-fields [42]. For our purpose, we will consider three \ntypes of the near-field EM structures.      \n (1) Evanescent modes \n \n \nFrom a general point of view, th e near-field of evanes cent modes can be defined as the extension \noutside a given structure (sample)  of the field existing inside th is structure (sample). For the \nHelmhotz equation  \n                                                               \n2 2 2 2\n0 z y x k k k k ++=εµ ,                                                           (6)                    \n \nwhere  \n                                                                         22\n2\n0ckω= ,                                                                   (7)      \n \nz y x kkk ,,  are wavenumbers along x,y,z axes in a medium, two solutions are possible when y x k k  and  \nare real quantities. The first so lution corresponds to the case  \n     \n                                                                    2 2 2\n0 y xk k k+>εµ .                                                              (8)                    \n  \nIt shows that zk is a real quantity and one has, as a result, the 3D propagating EM process. The \nsecond solution takes place when  \n \n                                                                     2 2 2\n0 y xk k k+<εµ .                                                             (9)                    \n  7So zk is an imaginary quantity. One has the 2D propagating EM process along x and y axis and the \nevanescent-mode fields (the near fields) in z direction. The typical examples of such evanescent \nmodes can be demonstrated in the field structur es of closed below-cut-off microwave waveguides \nand open optical waveguides. The importance of evanescent electromagnetic waves was recently \nrealized in connection with emergence of near-field optics microscopy. Taking evanescent waves \ninto account prevents the use of any approximations and requires the detailed solution of the full set \nof Maxwell equations [43]. As an effective met hod to study evanescent electromagnetic waves, for \nsolving Maxwell's equations one can use expansion in multipoles where electromagnetic fields are \nconstructed from scalar-wave eigenfuncti ons of the Helmholtz equation [43, 44]. \n    It is evident that to get extremely big quantities of imaginary zk one should have extremely big \nquantities of real xk and yk. On the way to create a perfect lens  based on left-handed metamaterials \n[41], this fact may become the stronger limitation. Really, the concept of an  effective medium cannot \nbe used for a perfect-lens slab of a left-handed metamaterial illuminated by a point source, when \nevanescent waves have a transverse  wavelength of the order of or less than the dimensions of the \ninclusions or their spacings [45] . Misunderstanding of such a limitation can lead to serious flaws in \nphysics of new material propositions for perfect-lens slabs. As an example, we can refer to paper \n[46], where a material for a perfect-l ens slab is conceived as the dilu te mixture of he lical inclusions. \nThe fact that these nonlocal inclusi ons are not in each other near fiel d cast doubts on the vital effect \nof evanescent waves in such a slab lens.     The near fields of evanescent modes have a pure electromagnetic nature and, evidently, are not \nrelated to any specific ME fields.      (2) Quasistatic limit  \n \n   \nThe quasistatic limit means 00→kr\n and so 0 ,,→z y x kkk  as well. In this case, no-wave time-\ndependable quasistatic fields exist. Such quasistatic  electromagnetic fields can be realized only due \nto local sources: or local-capacitanc e alternative electric charges w ith surrounding pote ntial electric \nfields:                        \n                                                                         \nϕ−∇=Er\n,                                                                (10) \n  \nor local-loop conductive elec tric currents  with surroundi ng potential magnetic fields:  \n \n                                                                       ψ−∇=Hr\n.                                                                 (11)                   \n \nSpatial distributions of potential ϕ as well as potential ψ are described by the Laplace equation. \nExamples are the quasistatic fields surround ing tip-structure probes in modern microwave-\nmicroscopy devices [47]. Certainly, there is  no physical mechanism for possible ME coupling \nbetween such local electric  and magnetic sources.  \n (3) Quasistatic oscillations\n \n The symmetry between the electric and magnetic fields  is broken in finite temporally dispersive \nmedia. In this case, quasistatic oscillations may take place. For quasistatic oscillations  \n \n                                                                         \nl k 10<< ,                                                                 (12)                    \n  8where l is the characteristic size of a body. In such oscillati ons, there are no electromagnetic \nretardation effects since one negl ects or electric, or magnetic displacement currents. The following \nare some examples of quasistatic oscillations.     (a) Quasistationary EM fields in small metal samples. These fields are described by Maxwell equations with neglect of the el ectric displacement currents. Insi de a sample we have the \"heat-\nconductivity-like\" eq uation for the magnetic field: \n \n                                                                 \ntH\ncH∂∂=∇rr\n22 4πµσ.                                                           (13)                  \n \nOutside a sample there are the quasistatic-field equations:  \n                                                               \n0      ,0 =×∇=⋅∇ H Br r\n.                                                      (14)                    \n \nThe solutions correspond to imaginary numbers of  z y x k kk  and ,,  showing that there are non-\nstationary decaying fields [39].  \n    (b) Plasmon-oscillation fields are the fields due  to collective oscillations of electron density. When \none considers a metal or a semiconductor as a composite of positive ions forming a regular lattice and conduction electrons which move freely through th is ionic lattice, ther e can be longitudinal \noscillations of the electronic gas – the plasma oscillations. The inte rface between such a sample and \na dielectric may also support charge density osc illations – surface plasmons . In the case of surface \nplasmon modes, the surface plasmon field decays exponentially away from the interface. For the \nelectrostatic description (one neglects the magnetic displacement current), an electric field is the \nquasielectrostatic field (\nϕ−∇=Er\n). The plasmon oscillations may be characterized by electrostatic \nwave functions, which are eigenfunctions of th e Laplace-like equation. For negative frequency \ndependent permittivity, one can observe a discrete spectrum of propagating electrostatic modes [48] \nand electrostatic resonances [ 49]. Surface plasmons can interact  with photons (with the same \npolarization state) if th e momentum and energy condi tions are right. There is a link between near-\nfield focusing action and the existence of well-defined surface plasmons [41]. \n    (c) Magnetostatic (MS) oscillations are obs erved in small temporally dispersive ferromagnet \nsamples [11, 39]. For these quasistationary fields , a magnetostatic descri ption (one neglects the \nelectric displacement current) can be used. So a magnetic field is the quasimagnetostatic field: \nψ−∇=Hr\n. Inside a ferrite sample one has the Walk er equation for MS-potential wave function ψ:  \n \n                                                                 0] )( [ =∇⋅∇ ψωµt,                                                           (15)                  \n \nwhere µt is the permeability tensor. Outside a sample, there is the Laplace equation: \n \n                                                                        02=∇ψ .                                                                  (16)                     \n \nFor a negative diagonal component of  the permeability tens or, the solutions inside a sample may be \ncharacterized by real wave numbers for all three dimensions. In this case one can observe a discrete \nspectrum of propagating magnetostatic mode s and magnetostatic resonances [11]. \n      There are no classical physics mechanisms for internal coupling between the electrostatic and \nmagnetostatic oscillations. If one conceives realization of a sample with a simple combination of the \nplasmon and magnon sources, there will not be a lo cal ME particle surrounded by the unified ME \nfield.     9C. Classical motion equations for point charges  \n \nCould there be any kind of classical motion equati ons for point charges gi ving a physical basis to \nrealize sources with the local junction of  the electrical and magnetic properties? \n    As we all know (see e.g. [6, 39]), electromagne tic fields in a medium ar ise from the microscopic \nMaxwell equations written for the microscopic electric er and magnetic hr\n fields, microscopic \nelectric charge density ),(trrρ  and microscopic electric current density vr ρ: \n \n                                                        th\nce∂∂−=×∇r\nr 1,     πρ4=⋅∇er,                                                 (17)                    \n                                                         \n                                                        te\ncvch∂∂+=×∇rr r 1 4ρπ,     0=⋅∇hr\n.                                         (18)                     \n \nFor averaged fields one de fines electric polarization Pr\n and magnetization Mr\nas \n \n                                                         P EDrrr\nπ4+≡ ,        M B Hrrr\nπ4−≡ .                                           (19) \n \nFormally, it can be assumed that a medium is com posed with electric and magnetic dipoles. In this \ncase one can write the microscopic Maxwell equations with the micr oscopic electric charge density \n),(trerρ , electric current density e evr ρ , magnetic charge density ) ,(trmrρ  and electric current \ndensity m mvr ρ  as [50]: \n                                            th\ncvcemm\n∂∂− −=×∇r\nr r 1 4ρπ,      eeπρ4=⋅∇r,                                        (20) \n \n                                            te\ncvche e\n∂∂+ =×∇rr r 1 4ρπ,      mhπρ4=⋅∇r\n.                                          (21) \n \nAfter the averaging procedure, in this case one obtains the standard-for m macroscopic Maxwell's \nequations as well, but for averaged fields one has electric polarization ePr\n and magnetic polarization \nmPr\n: \n \n                                                    eP EDrrr\nπ4+≡ ,  mP HBrrr\nπ4+≡ .                                                    (22) \n \nAn analysis of both the above cases assumes th at the motion equations are local equations: the \naverage procedure for microscopic current densities takes place in scales much less than a scale of \nvariation of any macroscopic quant ity. At the same time, no ME couplings on the microscopic level \ncan be presupposed in these motion equations. In fram es of a classical descri ption, no helical loops \n(recursion motions) are possible for bound charges and no classical laws describe interaction \nbetween linear electric a nd linear magnetic currents. \n    Together with electric sources used in  the standard electromagnetism one can presume the \npresence of magnetic sources. An analysis made  based on the classical Hamilton principle shows \nthat there cannot be proper equations of motion in  which the fields origin ated from electrically \ncharged particles will exert forces on magnetically  charged particles, and vice versa [51]. From the \n\"electric\" Langragian density with  an electric current source, on e derives the standard set of \nMaxwell's equations with the fields defined as \n  10                                                    tA\ncE∂∂−−∇=rr 1ϕ ,       A Hrr\n×∇= ,                                             (23)     \n \nwhere ϕ and Ar\n are, respectively, the scalar and vector \"e lectric\" potentials. At the same time, from \nthe \"magnetic\" Langragian density with a magnetic current source, one deri ves Maxwell's equations \nwith the fields defined as  \n                                                      \ntC\ncH∂∂−−∇=′rr 1ψ ,    C Err\n×∇=′ ,                                          (24)     \n \nwhere ψ and Cr\n are, respectively, the scalar and vector  \"magnetic\" potentials. The Lorentz forces \nacting on an electric charge e and a magnetic charge g are defined, respectively, as \n \n                                                              ×+= HcvEe Ferrrr\n                                                            (25) \n \nand   \n                                                            \n\n\n′×−′= EcvHg Fmrrrr\n.                                                          (26) \n \nOne can derive the symmetrized set of Maxwell's equations from the summarized, \"electric\" plus \n\"magnetic\", Lagrangian density, but this will not result in the \"magnetoelectric\" Lorentz forces of the forms   \n                                                      \n()()\n\n′+×+′+= HHcvEEe Fe\nMErrrrrr\n                                               (27) \n \nand   \n                                                       \n()()\n\n′+×−′+= EEcvHHg Fm\nMErrrrr r\n                                              (28) \n \ngiving equations of motion in which the fields associated with elec trically charged particles will \nexert forces on magnetically charge d particles, and vice versa [ 51]. The symmetry properties of \nmagnetic charge and current densit ies under both spatial inversion and time reversal are opposite to \nthose of electric charge and current densities [6].  So coexistence of electric and magnetic charges \nmust involve some forms of pari ty violation which do not correspo nd to symmetries of classical \nlaws. One may expect realizing local ME couplin g only when dynamical symmetry breaking occurs.  \n    From the above analysis it follows that in his search of sources with  local junctions of the \nelectrical and magnetic properties – the ME particles – one cannot rely on the induced parameters of \nsmall electromagnetic scatterers (irr espective of material and geometry of these scatterers). No near-\nfield electromagnetic structures and no classical motion equations fo r point charges give a physical \nbasis to realize sources with the local junction of the electrical and magnetic properties. It becomes \nevident that the unified ME fields originated by local ME particles should appear with the near-field \nsymmetry properties distinguishing from th at of the electromagnetic fields.   11    While a local ME particle cannot be realized as a classical scatterer w ith the induced parameters, \nit can be created as a small magnetic sample with eigen magnetic oscillations having special symmetry breaking properties. In so me magnetic structures one can ob serve effective interaction of \nthe polarization and magnetization which is desc ribed by the laws quite different from the \nfundamental laws of electrodynamics. These pecu liar phenomena will constitute a basis of our \nsearch of local ME particles.   \nIII. MAGNETOELECTRICITY GENERA TED BY MAGNETIC CHIRALITY \n \nThe interplay between spin and char ge degree of freedom is one of the central issue in solid state \nphysics and the cross correlation between these two degr ees of freedom is of pa rticular interests. In \nsolid state structures, one can obser ve the ME effect from the viewpoi nt of the electric polarization \ninduced by the applied magnetic fi elds. This can show a proper wa y for realization of local ME \nparticles for electromagnetic composite materials.      In some magnetic structures, symmetry argumen ts are used to construct the coupling between the \nmagnetization vector and the electric polariza tion vector. For a magnetic crystal an electric \npolarization can arise in the vici nity of the magnetic  inhomogeneity [52, 53] . The polarization has \ndirectionality with inversion sy mmetry breaking. So the polariza tion can couple to magnetization if \nthe magnetization distribution shows the inversion sy mmetry breaking properties. This implies that \nthe chiral magnetic ordering can induce an electric  polarization. From general symmetry arguments, \none has the phenomenological coupling mechan isms between the electric polarization \nepr and \nmagnetization mr. The invariance upon the time reversal, t t−→ , which transformse ep prr→  and \nm mrr−→ , requires the ME coupling to be quadratic in mr. The symmetry with respect to the spatial \ninversion, r rrr−→ , upon which e ep prr−→  and m mrr→ , is respected when the ME coupling of a \nuniform polarization to an inhomogen eous magnetizati on is linear in epr and contains one gradient \nof mr.  \n    Within a continuum approximation for magnetic properties, the ME intera ctions responsible for a \nlong-range spatial modulations of magnetization contribute to the Land au-type free en ergies and are \nknown as Lifshitz invariants. In pa rticular, chiral struct ures in achiral magnetic systems can arise \nfrom the presence of the Lifshitz invariant in the free energy. Without  requirements of a special kind \nof a crystal lattice, the symmetry considerations  lead to a ME coupling term in the Landau free \nenergy of the form [53, 54]  \n                                                    \n()() [ ]m m m mp r Fe\nMEr rrrr∇⋅−⋅∇⋅∝)( .                                              (29) \n \nThe term on the right-hand side (RHS) of Eq. (29) is nonzero only if the magnetization mr breaks \nchiral symmetry, which is the canonical route towards a stro ng dependence between epr and mr. On \nphysical grounds, this term can readily be underst ood. The system sustaining a macroscopic electric \npolarization epr points out a particular direction in sp ace. Therefore, this polarization can only \ncouple to the magnetiza tion if and only if mr also has directionality a nd lacks a center of inversion \nsymmetry. One immediately understands  that this occurs when the magnetization is spiraling along \nsome axis. Based on standard vector-algebra transf ormations, it can be shown from Eq. (29) that the \nrelationship between the electric polarization and the chiral-order  magnetization is given by    \n \n                                                                   ()m m pe rrrr×∇×∝ .                                                           (30) \n  12It was shown, in particular [53], that magnetic  vortices in magnetically soft nanodisks are the \ninhomogeneous magnetization structur es with the induced electric- polarization properties.    \n    Let us consider a magnetically saturated ferrite disk with a normal bias magnetic field directed along z axis. For negligibly small magnetic losses, one has from the Landau-Lifshitz magnetization \nmotion equation the linear relation be tween the harmonically time-varied (~\n tωie) local precessing \nmagnetization mr and RF magnetic field Hr\n [11]:  \n \n                                                                      H mrtr⋅=χ ,                                                                   (31)                   \n \nwhere                 \n                                                           \n\n\n\n\n−=\n00 000\nχχχχ\nχaa\nii\nt                                                              (32) \n \nis the magnetic susceptibility tensor. Components of tensor χt are defined as                   \n2 2\n000 \nωωωγχ−=M and 2 2\n00 \nωωωγχ−=M\na , where 0H is a bias magnetic field, 0M is the saturation \nmagnetization, 0 0Hγω= , 0 4MMπγω= , and γis the gyromagnetic ratio. For such a structure, the \nvector relation (30) has the following components: \n \n                                            \n\n\n\n∂∂−∂∂=×∇×=ym\nxmKm m mK px y\ny x xe\n  \n  )] ( [ )(rrr r,                                 (33) \n \n                                            \n\n\n\n∂∂−∂∂−=×∇×=ym\nxmKm m mK px y\nx y ye\n  \n  )] ([ )(rrr r,                                (34) \n \n                                            \n\n\n\n∂∂+∂∂=×∇×=z  \nz  )] ([ )(y\nyx\nx z zemmmmK m mK prrr r,                             (35) \n \nwhere K is a coefficient of proportionality. \n    With representation of the in-plane components of a magnetization vector in a ferrite disk as  \n \n                                                             ti\nx ez yxA m  )( ),(ωξ≡                                                           (36) \n \nand  \n                                                             \nti\ny ez yxB m  )( ),(ωξ≡ ,                                                         (37) \n \none can rewrite Eqs. (33) – (35) for the real-time el ectric polarization components as                                      \n \n                                   \n\n\n\n\n\n\n\n\n∂∂−∂∂+\n\n\n\n∂∂−\n\n\n∂∂=ti\nxeeyABxBByABxBB K p 2* *\n     Re21)(ωξr,                   (38)  13 \n                                   \n\n\n\n\n\n\n\n\n∂∂−∂∂+\n\n\n\n∂∂−\n\n\n∂∂−=ti\nyeeyAAxBAyAAxBA K p 2* *\n     Re21)(ωξr,                 \n(39) \n \n                                   () [ ]ti\nzeeBB AA BB AAzK p 2 * * Re21)(ω ξξ +++∂∂=r.                                      (40) \n \n    Let us represent coefficients A and B as AieAAδ=  and BieBBδ= , respectively. It is evident that \nfor a case of a circularly polarized magnetization, when BA= and 2πδδ±=−A B , a component \nzep)(r will not be a time-varying quantity and one has precessing of vector epr around z axis with \nfrequency ω2. In a case of an elliptically polarized magnetization all three components of vector epr \nare time-varying.  \n    Let 0=z  corresponds to a middle plane of a ferrite disk and let media surrounding a ferrite disk \nhave identical parameters above and below di sk planes. If one assumes that function ) (zξ , giving a \ndistribution of magnetization along z axis, is a standing-wave functi on, even or odd with respect to \nthe disk thickness, one obtains from Eq. (40) that for 0>z  and 0<z  componentszep)(r have \ndifferent signs. So no total el ectric flux penetrates through the disk. When, however, media \nsurrounding a ferrite disk above and below the disk pl anes have different parameters, a total electric \nflux through the disk may occur.      The above consideration can be used for a case of  an external action of an electric field applied to \na sample with magnetic chirality. The ME interactions of ferrite samples with external fields can be observed if there are eigen chiral st ates in a particle. We will show  now that a quasi-2D ferrite disk \nwith magnetic-dipolar-mode oscillations is characterized by unique sy mmetry features with \ntopological phases resulting in appearance of eigen chiral st ates and ME properties.  \n  \nIV. ME PROPERTIES OF QUASI-2D FERRITE DISK PARTICLES WITH MDM \nOSCILLATIONS  \n \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 potential s are equivalent in all respects to  the Maxwell equations for fields \n[6]. The situation can become completely differe nt if one supposes to so lve the boundary problem \nfor electromagnetic wave processes in small samp les of a strongly temporally dispersive magnetic \nmedium [39]. For such magnetic samples with  the magnetostatic resonance behaviors in \nmicrowaves, the spectral problem for magnetic-dipol ar modes cannot be formally reduced to the \ncomplete-set Maxwell-equation representation and one  becomes faced with a sp ecial role of the MS-\npotential wave function ) ,(trrψ , which acquires a physical mean ing in the MDM spectral problem \n[20, 55]  and results in experi mental observation of energy shifts  of oscillating modes and eigen \nelectric moment properties [21 – 25, 56, 57].   14    In an assumption of separati on of variables for MS-pot ential wave functions in a quasi-2D ferrite \ndisk, a spectral problem in  cylindrical coordinates θ,,rz  is formulated with respect to membrane \nMS functions (described by coordinates θ,r) with amplitudes dependable on z coordinate. The main \nfeatures of the spectrum become evident from  boundary conditions on a lateral surface of a disk \n[20].  An orthogonal spectrum of os cillations is obtained when one solves a characteristic equation \nfor MS waves in an axially magnetized ferrite rod:   \n                                                                  \n0 )(21\n=′+′−\nνν\nννµKK\nJJ,                                                        (41) \n \nwhereν ννν K KJJ ′ ′  and , , ,  are the values of the Be ssel functions of an order ν and their derivatives \n(with respect to the argument) on a lateral cylindrical surface ( ℜ=r ),  together with a characteristic \nequation for MS waves in a nor mally magnetized ferrite slab: \n \n                                                                   ()µµβ+−−=12 tan h ,                                                      (42) \n \nwhere h is a disk thickness and β is a propagation constant along z axis. A quantity µ is a diagonal \ncomponent of the permeability tensor µt. A constant ν is an integer quantity. In these solutions one \nuses homogeneous boundary conditions for a dimens ionless membrane MS-pot ential wave function \n),(~θηr:  \n  \n                                                                0~ ~\n =\n\n\n∂∂−\n\n\n∂∂\n+ −ℜ= ℜ= r r r rη ηµ .                                           (43)                    \n \nA boundary condition (43) corresponds to a so-cal led essential boundary condition [58, 59]. It \nappears, however, that solutions for function η~, obtained based on Eqs. (41) and (43), do not satisfy \nconditions of continuity of the magnetic flux de nsity on a lateral surface of a disk. There exist \nanother type of a dimensionless me mbrane MS-potential wave function ϕ~ (different from the wave \nfunction η~), which satisfies the conditi ons of continuity of the ma gnetic flux density on a lateral \nsurface of a disk. For function ϕ~ one has the following boundary condition on a lateral surface of a \nferrite disk: \n \n                                                 \n− + −ℜ= ℜ= ℜ=\n\n\n∂∂\nℜ−=\n\n\n∂∂−\n\n\n∂∂\nra\nr rir r θϕµ ϕ ϕµ~ ~ ~\n,                                   (44) \n \nwhere aµ is an off-diagonal component of tensor µt. The boundary condition (44) is a so-called \nnatural boundary condition [58, 59 ]. From Eq. (44) it evidently fo llows that for a given sign of \nparameter aµ there are different functions, +ϕ~ and −ϕ~ corresponding to positive and negative \ndirections of an angle coordinate when πθ2 0≤≤ . So function ϕ~ is not a single-valued function. It \nchanges a sign when θ is rotated by π2. At the same time, function η~ is a single-valued function.  \n    For functions ϕ~ one has path dependent (that is non-inte grable) solutions. In such a case, one \ncannot use separation of variables in solving th e boundary problem for a MDM ferrite disk. This \ntakes place due to the time-reversal symmetry break ing effect (gyrotropy) on a surface of a ferrite \ndisk. It is well known that becau se of gyrotropy, electromagnetic waves incident on a ferrite- 15dielectric interface has reflection symmetry break ing [13, 60]. This effect takes place also for \nmagnetostatic waves propagating in a normally magneti zed ferrite film and incident on the film edge \n[61]. Fig. 2 illustrates this situation. As a result of reflection on the film edge at point A, \nmagnetostatic-wave rays 1 1 ′→→A  and 1 1→→′A  shown in Figs. 2 (a) and 2(b), respectively, \nacquire different phases. In a ferr ite disk, this results in appearance of azimuthally running waves. \nLet after a rotation around a disk, an azimuthal ly running magnetostatic wave acquires a phase 1Φ \nwhile for an opposite direction of a bias magnetic field such a phase is 2Φ. It is evident that \nΦ≡Φ=Φ2 1  and there should be p 22 1π=Φ+Φ  or  πp=Φ . Quantities p are odd integers. This \nfollows from the time-reversal symmetry breaking e ffect. A system comes back to its initial state \nafter a full π2 rotation. But this π2rotation can be reached if both pa rtial rotating processes, with \nphases 1Φ and 2Φ, are involved. So minimal 1 =p  and, generally, quantities p are odd integers. It \nmeans that for a given direction of  a bias magnetic field, function ϕ~ behaves as a double-valued \nfunction.  \n    Such a non-integrable problem has two diffe rent analytical solutio ns for the MDM spectral \nproblem. One of the analytical solutions is based on the concept of the path-dependent (or \ntopological) phase factor for orthog onal MS-potential wave functions. In this case, one uses Bessel \nfunctions of integer order ν as basis functions for the repr esentation of orthogonal functions η~ with \nan introduction of a special phase f actor on a lateral border of a ferri te disk. On a lateral border of a \nferrite disk one has the following corre spondence between double- valued functions ϕ~ and single-\nvalued functions η~[20]: \n \n                                                                   ()()− −ℜ=± ℜ=±=r rηδϕ~ ~,                                                     (45) \n \nwhere          \n                                                                      \nθδ±−\n±±≡iqef                                                                (46) \n \nis a double-valued function. The azimuth number ±q is equal to 21± and for amplitudes we have \n− +−=f f  and ±f = 1. Function ±δ changes its sign when θ is rotated by π2 so that 12−=±−π iqe . \nOne obtains the energy-eigenstate spectrum of MDM oscillations with topological phases \naccumulated by the boundary wave function δ. The topological effects become apparent through \nthe integral fluxes of the pseudo-electric fields [ 20]. There should be the positive and negative fluxes \ncorresponding to counterclockwise and clockwise e dge-function chiral rotati on. The different-sign \nfluxes are inequivalent to avoid can cellation. Every MDM in a thin fe rrite disk is characterized by a \ncertain energy eigenstate and two different-sign fluxes of the pseudo-electric fields which are \nenergetically degenerate. The spect ral theory developed based on or thogonal singlevalued membrane \nfunctions η~ and topological magnetic curre nts, shows the ME effect fr om a viewpoint of the Berry \nphase connection [20].  \n    Another type of an analyt ical solution for the MDM spectral pr oblem is to introduce an additional \nphase factor so that functions ϕ~ are represented as \"rotating Bessel functions\" with an integer \nazimuth number. This additional phase fact or appears when one considers function ϕ~ as an \"in-\nplane\" projection of a helical MS wave. Based on this  approach one obtains a pi cture of the fields of \noscillating MDMs derived fr om non-orthogonal functions ϕ~. An initial stage of studies of helical \nMS waves in a ferrite disk is given in Refs. [62] . In a case of rotating magnetic fields in a normally \nmagnetized ferrite disk the clockwise and countercl ockwise rotations are st rongly different. One has  16to distinguish the field rotating in  the same direction as the prece ssing spins rotate and the field \nrotating in an opposite directio n with respect to the spins. It is ev ident that energetica lly preferable is \nthe resonance interaction when the magnetic field ro tates in the same direction as the precessing \nspins.      The first-type MDM spectral problem so lutions (based on orthogona l MS-potential wave \nfunctions with topological phase factor s) we will conventionally call as the \nG-modes while the \nsecond-type MDM spectral problem solutions (b ased on \"rotating Bessel functions\") we will \nconventionally call as the L-modes. This arises from the fact that G-modes are described by the \nsecond-order differential operator Gˆ, while L-modes are described by th e first-order differential-\nmatrix operator Lˆ [20, 55]. In a case of G-modes, loop topological ma gnetic currents lead to \nappearance of eigen electric (anapole) moments. This clearly shows (bot h in theory [20] and \nexperiments [24, 25]) the ME proper ties of a MDM ferrite disk. The G-mode ferrite particle has \nspecial symmetry breaking properties and is not a classical scatterer with the i nduced parameters. In \nthis paper, we will mainly use rotating Bessel functions ( L-modes) for demonstration of the ME \neffect in a ferrite particle. In such a representa tion, the ME properties of  a ferrite disk can be \nanalyzed based on the Lifshitz i nvariant discussed in Section III of the pa per. Moreover, for L-modes \nthe structures of the fields inside  a ferrite disk and in the quasis tatic (near-field) region surrounding a \nferrite disk can be numerically modeled based on the HFSS-program studies [ 63] with a very good \ncorrespondence between the numeri cal and analytical results.  \n    Taking into account Eq. (42)  we write a solution for function ϕ~ in a form:      \n \n                                                                 θ\nµβϕl\nljerJ−\n\n\n\n\n−∝ ~,                                                      (47) \n \nwhere l is a positive integer quantity. For this type of a solution, a characteristic equation (44) is \nwritten as: \n \n                                                          0||)(21\n=ℜ−′+′−βµµl\nll\nll a\nKK\nJJ.                                                   (48) \n \nWhen a magnetic field is varied such that every point on a disk su rface describes a cl osed loop, then \nthe system should, at the end of this excursion, re turn to its original state. This will give a phase \nfactor θlie− for each eigenstate. For a very thin disk one can use separation of va riables. For a certain \nmagnetic-dipolar mode one has for MS- potential wave function [20, 55, 59]: \n \n                                                                  ) ,(~)(θϕξψ rzC= ,                                                          (49)  \n \nwhere ) (zξ  is an amplitude factor, C is a dimensional coefficient. Inside a ferrite disk \n( hz r ≤≤ℜ≤ 0  , ) one has: \n \n                                         ()θβ\nµβ\nµβθψl\nlljez zrJCzr−\n\n\n\n\n−+\n\n\n\n−=  sin1 cos ,, .                         (50)                    \n \nThis function satisfies characteristic equations ( 42) and (48). Based on such a MS-potential function \none defines the magnetic field ( ψ∇−=rr\nH ) inside a ferrite disk as the azimuth propagating wave: \n  17                            ()ti i\nr eez zrJ CtzrH  sin1 cos ,,,ωθβ\nµβ\nµβ\nµβθl\nl l−\n\n\n\n\n−+\n\n\n\n−′\n−=  ,               (51) \n   \n                             ()ti ieez zrJriCtzrH  sin1 cos ,,,ωθ\nθ β\nµβ\nµβθθl\nl l−\n\n\n\n\n−+\n\n\n\n−−= ,                   (52)    \n \n                              ()ti i\nz eez zrJ CzrH  cos1 sin  ,,ωθβ\nµβ\nµββθl\nll−\n\n\n\n\n−+−\n\n\n\n−= .                       (53)  \n \n    With use of Eqs. (51) – (53) we calcula ted the magnetic field distributions for a normally \nmagnetized ferrite disk. A ferrite disk has the following material parameters: saturation \nmagnetization is  1880  4=sMπ G and the linewidth is Oe 8.0=∆H . Disk diameter is 3=D mm and \nthickness 05.0=t mm. A disk is normally magnetized  by a bias magnetic field  49000=H Oe. We \nanalyzed the MDMs which are classified by numbers n – the numbers of zeros in the Bessel \nfunction, corresponding to different radial  variations – for the order parameter 1+=l . We \nconsidered the field distribu tions for first two modes ( n = 1, 2). The MDM resonance frequencies are \nobtained from solutions of Eqs. (42)  and (48). In Fig. 3 we show a ga llery of the analytically derived \nin-plane magnetic field distributions on th e upper plane of a ferrite disk for the 1st MDM ( f = 8.548 \nGHz) at different time phases. The ma gnetic field distributions for the 2nd MDM ( f = 8.667 GHz) at \nsome time phases are shown in Fig. 4. For the known mode magnetic fields we can calculate the \nvector ()m mrrr×∇× . Space orientations of vector ()m mrrr×∇×  on the upper and lower planes of a ferrite \ndisk at different time phases t ω are shown in Fig. 5 for the 1st MDM ( f = 8.548 GHz). It is very \ninteresting to find that elec tric polarization vectors [ ) ( m mprrrr×∇×∝ ] exhibit precessing behaviors \nThese precessing motions are in the same directi ons for the upper and lower halves of a disk. We \nalso have to note that whereas the magne tization precession occurs with frequency ω, the electric \npolarization precession is  with double frequency ω2. The z components of the electric polarization \nvectors are oppositely dir ected on the upper and lower planes of a disk.  \n    The MDMs in a thin ferrite disk can  be well studied based on the HFSS numerical \nelectromagnetic-simulation program [63]. Fig. 6 (a) shows the HFSS m odel of a short-wall \nrectangular waveguide section with a normally magne tized ferrite disk. The spectral characteristics \nfor a thin ferrite disk shown in Figs. 6 (b) a nd 6 (c) give a very good correspondence between the \nabsorption peak positions obtained from the numeri cal simulation and the calculated peak positions \nfor the G- and L-MDMs (for the order parameter 1+=l ). The HFSS-program magnetic field \ndistributions in a ferrite disk shown in  Figs. 7 and 8, respectively, for the 1st  (f = 8.52 GHz) and 2nd \n(f = 8.66 GHz) resonance states at different time phases are in a clear correspondence with the \nanalytically derived magnetic field distributions for the L-MDMs (see Figs. 3, 4). Based on the \nHFSS model we can study also  the electric fiel d distributions inside a ferri te disk. The electric field \ndistributions for the 1st (f = 8.52 GHz) and 2nd (f = 8.66 GHz) resonance st ates at different time \nphases are shown, respectively, in Figs. 9 and 10.       An analytical study of the electric field dist ributions for MDMs appears as a very complicated \nproblem. We can, however, use a simple analytical m odel. The assumptions used in this analytical \nmodel are not well defined. Nevertheless, the m odel shows a very good co rrespondence with the \nnumerical results of the electric field pictures. Formally, one can suppose that for the monochromatic \nMS-wave process there exis ts a curl electric field \nEr\ndefined by the Faraday-Maxwell law. One can \nrepresent the electric field as follows:   18                                                    mciHciBciEr r r rr\n 4   πωωω −−=−=×∇ ,                                          \n(54) \n \nwhere mr is RF magnetization. With the ×∇r\n differential operation for the left-hand and right-hand \nsides of Eq. (54) and taking into account that 0=×∇Hrr\n, one obtains: \n \n                                                                 mciEr r\n×∇=∇  42πω .                                                       (55) \n \nHere we used the relation 0=⋅∇Er\n which is relevant if one assumes that the permittivity of a disk is \ndescribed by a scalar. The electric  field can be formally represented as being originated from an \neffective electric current:   \n                                                                   \n)(\n22 4 ejciEr rπω=∇ ,                                                         (56) \n \nwhere   \n                                                                     m c j\nerrr\n×∇≡)(.                                                              (57) \n   \n    With use of Eqs. (31), (32) and (51) – (53) we can obt ain the components of magnetization mr and \nthen derive the components of vector mrr\n×∇ . The components of an effective electric current are the \nfollowing: \n \n              ()ti i a e\nr eez zrJrrJ icCtzrj 2\n)( cos1 sin    ,,,ωθβ\nµβ\nµβχβ\nµβ\nµβχθl\nl l ll−\n\n\n\n\n−−\n\n\n\n\n\n\n\n−+\n\n\n\n−′\n−= ,          (58) \n \n              ()ti i a eeez zrJrrJ cCtzrj 2\n)( cos1 sin    ,,,ωθ\nθ β\nµβ\nµβχβ\nµβ\nµχβθl\nl l ll−\n\n\n\n\n−−\n\n\n\n\n\n\n\n−+\n\n\n\n−′\n−= ,         (59) \n                 \n                  ()ti i a e\nz eez zrJ icC tzrj )( sin1 cos \n41,,,ωθβ\nµβ\nµβ\nµµ\nπθl\nl l−\n\n\n\n\n−+\n\n\n\n−−= .                                 (60) \n \n    The in-plane effective-elect ric-current distributions on the upper plane of a ferrite disk for the 1st \n(f = 8.548 GHz) and 2nd (f = 8.667 GHz) MDMs at different time phases are shown, respectively, in \nFigs. 11 and 12. As it follows from Eq . (56), the electric field should be o90  shifted with respect to \nthe effective electric current. So from the effec tive-electric-current dist ributions one can obtain \nqualitative pictures for the electric field distri butions. There is a very good correspondence of such \npictures of the electric fields w ith the electric field distributions obtained from numerical studies (see \nFigs. 9 and 10).      From the field structures of L-MDMs (as well as from the HFSS-program field structures) it \nbecomes evident that an entire ferrite disk does  not exhibit any electric -moment properties. It, \nhowever, shows the magnetic-m oment behavior. One can see that th e in-plane electric fields on the \nupper and lower planes of a ferrite disk are in opposite di rections at any time phase. Since the disk \nthickness is much, much less than the free-space electromagnetic wave length, the disk can be clearly  19replaced by a sheet with effective linear magne tic currents. A surface density of the effective \nmagnetic current is expressed as  \n                                                         \nm\nlower upper icE Enr rrr π4) ( −=−× ,                                                 (61) \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 Er r\n−= . Following the electric-field pi ctures, one can conclude that  there are rotating linear \nsurface magnetic currents. Non-zero current-line  divergence of such magnetic currents gives \nequivalent magnetic charges. As a result of this e quivalent representation, one has an evidence for an \nin-plane rotating magnetic dipoles for the entire ferri te disk with MDM oscillations. Due to such in-\nplane rotating magnetic dipoles  one has excitation of the L-MDMs in a quasi-2D ferrite disk shown \nin well-known experiments [56, 57].     Nevertheless, microwave experiments [24] gi ve also an evidence for the MDM excitation by RF \nelectric fields oriented normally to a ferrite disk . This excitation takes place because of the electric \n(anapole) moments of the G-MDMs. The mechanisms of an interaction of the MDM anapole \nmoments with the external RF electric fields are discussed in Ref. [20, 25]. Since the spectra of G- \nand L-MDMs are almost degenerate [see the peak positions of G- and L-MDMs in Fig. 6 (c)], the \nME properties of a ferrite disk ar e experimentally observed as the G-mode electric (anapole) \nmoment normal to a ferrite disk together with the L-mode magnetic moment rotating in the disk \nplane. Fig. 13 shows a MDM ferrite disk as a ME particle.        \nV. ME PROPERTIES OF MDM FERRITE DISKS WITH SPECIAL GEOMETRIES  \n \nWe consider now a MDM ferrite disk with special geometry properties. There will be a ferrite disk with a dielectric loading and a ferrite disk with  a wire surface metallization. Presently, for these \nstructures we do not have analytical models a nd so we will analyze the HFSS-program numerical \nsimulation results. It will be shown that in this case a MDM ferrite disk has ME properties which are \nwell verified by expe rimental results.  \n    As we showed above, when media surrounding a ferrite disk have identical parameters above and \nbelow disk planes, no elect ric-moment properties for L-modes for an entire ferrite disk take place. \nWhen, however, media surrounding a ferrite disk a bove and below th e disk planes have different \nparameters, distribution of the electric polarization vectors [ ) (m mprrrr×∇×∝ ] may lead to a total \nelectric flux through the sample . In Fig. 14 we show the HFSS-pr ogram electric field distributions \nfor the 1st resonance state for a sample composed as a th in ferrite disk (with the same parameters as \nin the above studies) loaded with a diel ectric disk with a dielectric constant 15=rε . One clearly sees \nthat in this case topological  electric charges appear and, as a result, one has an in-plane rotating \nelectric dipole. It is interesting to note that a rota ting electric dipole is in th e same orientation as an \nin-plane rotating magnetic dipole.       Fig. 15 shows the electric  field distributions for the 1\nst resonance state for a sample composed as a \nthin ferrite disk with a wire surface metallizati on. In this case one has an in-plane non-rotating \nelectric and in-plane rotating magne tic dipole. A concept of such a ME  particle was put forth in 1996 \n[64]. Then microwave experiments clearly verified the ME propertie s in such a structure [21 – 23]. \nOur present results give pictures for the fi eld distributions in these ME particles.  \n     \n \n  20VI. CONCLUSION \n \nUnderstanding a fundamental mechanism of a \"junc tion\" of electricity and magnetism in a point \nsource underlies comprehensive solution of a pr oblem of unification of  electromagnetism and \nmagnetoelectricity.  From our analysis it follows that in his search of sources with local cross-\npolarization properties one cannot rely on the induced  parameters of small electromagnetic scatterers \n(irrespective of material and geometry of these scat terers). No near-field electromagnetic structures \nand no classical motion equations for point charges give a physical ba sis to realize sources with the \nlocal magnetoelectricity. In the near-field region with local sources, an electromagnetic field falls \ninto an electric and a magnetic fields.  \n    It becomes evident that the unified ME fields originated by conjectural local ME particles should \nappear with the symmetry properties distinguishing from that of the electromagnetic fields. While a \nlocal ME particle cannot be realized as a classica l scatterer with the induced parameters, it can be \ncreated as a small magnetic sample with eigen magnetic oscillations having special symmetry \nbreaking properties. In some magn etic structures one can observe  effective interaction of the \npolarization and magnetization which is described by the laws quite different from the fundamental \nlaws of electrodynamics. These peculiar phenomena , being a consequence of broken time-reversal \nand space-inversion symmetries, may constitute a basis for search of local ME particles.      In thin-film ferrite disks with the MDM vortex structures one has the chiral states of magnetization which may provide us with a str ong dependence between el ectric polarization and \nmagnetization. Because of the spectral characteristic s with energy eigenstates, the MDM ferrite disks \nare the particles with strong dynamic localization of the ME fields. It  turns out that a MS-potential \nwave function stabilizes two-dimens ional states (the membrane states). At the same time, the quasi-\n2D MDM ferrite disk can be co nsidered as a \"chiral magnetic domain\". The chiral-magnetic-\nordering dynamics has the electr ic polarization properties. For given magnetization texture in a \nferrite disk, the in-plane distribution of the electri c polarization will have azimuth variations. It is \nevident that a disk will be characterized by such invariant as the electric flux only if the azimuth \nvariation of the electric pol arization is described by a non-singlevalued function.  \n    The MDM-ferrite ME fields are characterized by the topological-phase effects. There are not the \nTellegen-particle fields which are conceptualized as  the electric and magnetic fields of a system with \nstrong quasistatic (dipole-type) localizations. We  dispute the physical re alizability of Tellegen \nparticles as classical ME particles with cross-polari zation effects. The questions  raised in this paper \nare very important in a view of the present strong interest in electromagnetic artificial materials with ME properties. The properties of ME fields show n in this paper give very new insights into a \nproblem of local microwave ME composites.  \nReferences \n [1] P. Debye, Physica \n5, 377 (1925). \n[2] B. D. H. Tellegen, Philips. Res. Rep. 3, 81 (1948).  \n[3] E. J. 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Tretyakov, A. Sihvola, and L. Jylhä,  Photonics Nanostructures – Fundament. Appl. 3, 107  \n       (2005). [47] B.T. Rosner and D.W. van der Weide, Rev. Sci. Instr. \n73, 2505 (2002). \n[48] J.Q. Lu and A.A. Maradudin, Phys Rev. B 42, 11159 (1990).  \n[49] D.R. Fredkin and I.D. Mayergoyz, Phys. Rev. Lett. 91, 253902 (2003). \n[50] E.O. Kamenetskii, cond-mat/0601467 (2006).  22[51] D. Fryberger, Found. Phys. 19, 125 (1989). \n[52] V. G. Bar'yakhtar, V. A. L'vov, and D. A. Yablonskii, Pis'ma Zh. Eksp. Teor. Fiz. 37, 565  \n       (1983). [53] M. Mostovoy, Phys. Rev. Lett. \n96, 067601 (2006). \n[54] J. J. Betouras, G. Giovannetti, and J. van den Brink, Phys. Rev. Lett. 98, 257602 (2007). \n[55] E.O. Kamenetskii, Phys. Rev. E 63, 066612 (2001). \n[56] J.F. Dillon Jr., J. Appl. Phys. 31, 1605 (1960).  \n[57] T. Yukawa and K. Abe, J. Appl. Phys. 45, 3146 (1974). \n[58]  S. G. Mikhlin, Variational Methods in Mathematical Physics  (New York, McMillan, 1964). \n[59] E. O. Kamenetskii, M. Sigalov, and R. Shavit, J. Phys.: Condens. Matter 17, 2211 (2005).  \n[60] M. Sigalov, E.O. Kamenetskii, and R. Shavit, J. Appl. Phys. 103, 013904 (2008). \n[61] A. V. Vashkovsky and E. H. Lock, Phys. Usp. 49, 537 (2006). \n[62] E. O. Kamenetskii, J. Magn. Magn. Mater. 302, 137 (2006); arXiv:cond-mat/0505717. \n[63] The High Frequency Struct ure Simulator (HFSS) is the so ftware based on FEM method   \n        produced by ANSOFT Company  (www.hfss. com, Ansoft Corporation, Pittsburgh, PA 15219). \n[64] E. O. Kamenetskii, Microw. Opt. Technol. Lett. 11, 103 (1996). \n  \nFigure captions \n Fig. 1. Effect of \"ME coupling“: a small electromagnetic scattere r in a form of a LC delay-line \nsection has phase shifts Lk kl\n0 0 1     =≈=ϕϕ . \n \nFig. 2. Magnetostatic waves propagating in a norma lly magnetized ferrite film and incident on the \nfilm edge. As a result of reflect ion on the film edge at point A, magnetostatic-wave rays 1 1 ′→→A  \n(a) and 1 1→→′A  (b) acquire different phases. \n \nFig. 3. 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 = 8.548 GHz) at different time phases. (A \nqualitative picture). \n \nFig. 4. A gallery of the analytic ally derived in-plane magnetic fi eld distributions on the upper plane \nof a ferrite disk for the 2nd magnetic-dipolar mode ( f = 8.667 GHz) at some time phases. (A \nqualitative picture). \n \nFig. 5. Space orientations of vector ()m mrrr×∇×  on the upper (a) and lower (b) planes of a ferrite disk \nat some time phases for the 1st MDM ( f = 8.548 GHz). (A qualitative picture). \n Fig. 6. Spectral characteristics for a thin ferrite disk. (a) The HFSS model of a short-wall rectangular \nwaveguide section with a normally  magnetized ferrite disk; (b) ab sorption peak positions obtained \nfrom the numerical simulation; (c) the calculated peak positions for the G-MDM and L-MDM (for \nthe order parameter \n1+=l ).  \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 resonance state ( f = 8.52 GHz) at different time phases.  \n Fig. 8. A top view for the numerically modeled ma gnetic field distributions on the upper plane of a \nferrite disk for the for the 2\nnd resonance state ( f = 8.66 GHz) at different time phases.  \n  23Fig. 9. A top view for the numerically modeled electric field distributions  on the upper plane of a \nferrite disk for the for the 1st resonance state ( f = 8.52 GHz) at different time phases.  \n Fig. 10. A top view for the numeric ally modeled electric field dist ributions on the upper plane of a \nferrite disk for the for the 2\nnd resonance state ( f = 8.66 GHz) at different time phases.  \n Fig. 11. A gallery of the analytically derived in-p lane effective-electric-cu rrent distributions on the \nupper plane of a ferrite disk for the 1\nst magnetic-dipolar mode ( f = 8.548 GHz) at different time \nphases. (A qualitative picture).  Fig. 12. A gallery of the analytically derived in-p lane effective-electric-cu rrent distributions on the \nupper plane of a ferrite disk for the 2\nnd magnetic-dipolar mode ( f = 8.667 GHz) at some time phases. \n(A qualitative picture).  Fig. 13. The ME-particle model of a MD M ferrite disk. Since the spectra of G- and L-MDMs are \nalmost degenerate the ME properties of  a ferrite disk are observed as the G-mode electric (anapole) \nmoment \near normal to a ferrite disk together with the L-mode magnetic moment mpr rotating in the \ndisk plane.   \n Fig. 14. The electric fiel d distributions for the 1\nst resonance state for a sample composed as a thin \nferrite disk loaded with a dielect ric disk. One has an in-plane rota ting electric dipole with topological \nelectric charges. A rotating electric dipole is in th e same orientation as an in-plane rotating magnetic \ndipole.     Fig. 15. The electric fiel d distributions for the 1\nst resonance state for a sample composed as a thin \nferrite disk with a wire surface metallization. One has an in-plane  non-rotating electric dipole and an \nin-plane rotating magnetic dipole.   \n \n          \n  24 \n \nFig. 1. Effect of \"ME coupling“: a small electromagnetic scattere r in a form of a LC delay-line \nsection has phase shifts Lk kl0 0 1     =≈=ϕϕ . \n  \n   \n  \nFig. 2. Magnetostatic waves propagating in a norma lly magnetized ferrite film and incident on the \nfilm edge. As a result of reflect ion on the film edge at point A, magnetostatic-wave rays \n1 1 ′→→A  \n(a) and 1 1→→′A  (b) acquire different phases. \n \n          Small scatterer: the \nLC delay-line section\nPl\nL\nA A\nFerrite film Ferrite film\n(a) ( b)  25  \n                \n \n      \n  \n \n \n       \n  \n  \n     \nFig. 3. 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 = 8.548 GHz) at different time phases. (A \nqualitative picture).        °=0 tω  \n°=180 tω °=270 tω 26 \n        \n  \n  \n \n \nFig. 4. A gallery of the analytic ally derived in-plane magnetic fi eld distributions on the upper plane \nof a ferrite disk for the 2nd magnetic-dipolar mode ( f = 8.667 GHz) at some time phases. (A \nqualitative picture).               \n \n        \n  °=0tω °=90tω 27\n  \n  \n                          o0 2=tω                                                            o90 2=tω  \n                                                               ( a) \n  \n  \n  \n                          o0 2=tω                                                            o90 2=tω  \n  \n                                                               ( b) \n   \nFig. 5. Space orientations of vector \n()m mrrr×∇×  on the upper (a) and lower (b) planes of a ferrite disk \nat some time phases for the 1st MDM ( f = 8.548 GHz). (A qualitative picture). \n   \n                   28  \n  \n  \nFig. 6. Spectral characteristics for a thin ferrite disk. (a) The HFSS model of a short-wall rectangular \nwaveguide section with a normally  magnetized ferrite disk; (b) ab sorption peak positions obtained \nfrom the numerical simulation; (c) the calculated peak positions for the G-MDM and L-MDM (for \nthe order parameter 1+=l ).  \n  \n      \n                                                   \n(b) \n(a) \n(c)  29\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 resonance state ( f = 8.52 GHz) at different time phases.  \n          °=0 tω °=90 tω\n°=180 tω °=270 tω x y \nz 0Hr 30\n  \n  \n  \nFig. 8. A top view for the numerically modeled ma gnetic field distributions on the upper plane of a \nferrite disk for the for the 2nd resonance state ( f = 8.66 GHz) at different time phases.  \n    °=0t\nω °=90tω\n°=180tω °=270tωx y \nz 0Hr 31\n  \n  \n  \n Fig. 9. A top view for the numerically modeled electric field distributions  on the upper plane of a \nferrite disk for the for the 1\nst resonance state ( f = 8.52 GHz) at different time phases.  \n                          \n  \n °=0tω °=90tω\n°=180tω °=270tω x y \nz 0Hr 32\n  \n \n \nFig. 10. 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 resonance state ( f = 8.66 GHz) at different time phases.  \n \n  \n   \n \n \n              °=0tω °=90tω\n°=180tω °=270tωx y \nz 0Hr 33\n  \n  \n  \n  \n \nFig. 11. A gallery of the analytically derived in-p lane effective-electric-cu rrent distributions on the \nupper plane of a ferrite disk for the 1st magnetic-dipolar mode ( f = 8.548 GHz) at different time \nphases. (A qualitative picture).     \n \n         °=0tω °=90tω\n°=180tω °=270tω 34\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 ferrite disk for the 2\nnd magnetic-dipolar mode ( f = 8.667 GHz) at some time phases. \n(A qualitative picture).            Fig. 13. The ME-particle model of a MD M ferrite disk. Since the spectra of G- and L-MDMs are \nalmost degenerate the ME properties of  a ferrite disk are observed as the G-mode electric (anapole) \nmoment \near normal to a ferrite disk together with the L-mode magnetic moment mpr rotating in the \ndisk plane.   \n     \n °=0tω °=90tω\near\nmpr 35 \n \nFig. 14. The electric fiel d distributions for the 1st resonance state for a sample composed as a thin \nferrite disk loaded with a dielect ric disk. One has an in-plane rota ting electric dipole with topological \nelectric charges. A rotating electric dipole is in th e same orientation as an in-plane rotating magnetic \ndipole.     \n     \n0Hr\n0Hro90=tω\no270=tωFerrite disk \nFerrite disk \nDielectric Dielectric  36 \nFig. 15. The electric fiel d distributions for the 1st resonance state for a sample composed as a thin \nferrite disk with a wire surface metallization. One has an in-plane  non-rotating electric dipole and an \nin-plane rotating magnetic dipole.   \n   \n \n       \no0=tωo90=tω\no180=tωo270=tωWire surface \nmetallization \nFerrite disk "    },    {        "title": "1107.3623v1.Composition_Dependence_of_Structural_Parameters_and_Properties_of_Gallium_Ferrite.pdf",        "content": " 1Composition Dependence of Structur al Parameters and Properties of \nGallium Ferrite  \nSomdutta Mukherjee1, Vishal Ranjan2, Rajeev Gupta1,3 and Ashish Garg2* \n1Department of Physics \n2Department of Materials Science and Engineering \n3Materials Science Programme \nIndian Institute of Technology Kanpur, Kanpur 208016, India Abstract: \nWe show the effect of composition on structural  and magnetic characteristics of pure phase \npolycrystalline Ga\n2-xFexO3 (GFO) for compositions between 0.8 ≤ x ≤ 1.3. X-ray analysis \nreveals that lattice parameters of GFO exhibi t a linear dependence on Fe content in single \nphase region indicating manifestation of Vegard ’s law. Increasing Fe content of the samples \nalso leads to stretching of bonds as indicated by the Raman peak shifts. Further, low \ntemperature magnetic measurements show that the coercivity of the samples is maximum for \nGa:Fe ratio of 1:1 driven by a competition between decreasing crystallite size and increasing magnetic anisotropy.  Keywords : Gallium ferrite, Vegard’s Law, Ra man spectroscopy, Ferrimagnetism. \n \n                                                \n \n* Corresponding author; Email: ashishg@iitk.ac.in  2Recent years have witnessed a large number of investigations on magnetoelectric and \nmultiferroic materials due to their potential in novel device applications such as multi-state memory elements and spin field effect transistors.\n1-6  Gallium ferrite (GaFeO 3 or GFO) is a \nroom temperature piezoelectric and a ferrimagnet with transition temperature tunable to room \ntemperature or higher.7 The material also shows significant magnetoelectric coupling.8,9 \nInterestingly, site exchanges between Ga and Fe atoms which have size difference of ~4% \nlead to cationic site disorder resulting in ferrimagnetism in GFO.9,10 This characteristic also \nimplies that tuning of Ga:Fe ratio in the material could play an important role in tailoring its \nproperties.9 \nTypically, materials like GFO which demons trate large magnetoelectric coupling also \nshow substantial magnetostructural coupling making these attractive for a variety of applications.  In this context, it would be impor tant to revisit the effect of composition tuning \non the structure and phase evolution as well as magnetic behavior of GFO. GFO has a non-perovskite and a non-centrosymmetric orthorhombic structure with space group Pc2\n1n and its \nroom temperature lattice parameters are a ~ 8.75 Å, b ~ 9.40 Å, c ~ 5.08 Å.11 The unit-cell \nconsisting of eight formula units has two nonequivale nt Ga sites and Fe sites, marked as Ga1, \nGa2 and Fe1, Fe 2, respectively. Here, Ga1 is  tetrahedrally coordinated while Ga2, Fe1 and \nFe2 are octahedrally coordinated by oxygen. FeO 6 octahedra are significantly distorted and \nthe shift of Fe1 and Fe2 cations from the center of octahedra along b-axis renders GFO a \npolar material.9 As far as magnetism in GFO is concerned, Mossbauer   spectroscopy 10 and \nsubsequent neutron diffraction studies9 confirmed GFO as a ferrimagnet with cation site \ndisorder as source of magnetism. In this manuscr ipt, we show that structural parameters of \nGFO follow Vegard’s law very well. We also find that samples exhibit a rather broad range \nof composition where single phase GFO is st able. Further the low temperature magnetic  3measurements show that while remanent magne tization has linear com positional dependence, \ncoercive field attains a maximum for samples with x = 1.0.  \nPolycrystalline Ga 2-xFexO3 (0.7 ≤ x ≤ 1.4) ceramic samples were prepared using \nconventional solid-state-reaction method by mixing high purity β-Ga 2O3 and α-Fe 2O3 \npowders in relevant proportions and subse quently calcining the mixtures between 1300-\n1400oC in air. Powder X-ray diffraction (XRD) of the samples was conducted on Philips \nX’Pert Pro MRD X-Ray diffractometer using Cu Kα radiation. Raman spectra were acquired \nin backscattering geometry using a JY-Horiba micro Raman system equipped with a Ar+ \nLaser and Peltier cooled CCD detector. Temperature dependent magnetization measurements were carried out using a vibrating sample ma gnetometer in the temperature range 115-400 K \nwith a magnetic field of 500 Oe. Low temperat ure magnetic hysteresis measurements were \nperformed at 115 K. \n Room temperature XRD patterns for all the compositions of Ga\n2-xFexO3 (0.7 ≤ x ≤ \n1.4) are shown in Fig. 1(a). The compositions used were x = 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3 \nand 1.4 which correspond to 35, 40, 45, 50, 55, 60, 65 and 70 mol% Fe 2O3 in \n(Ga 2O3)(2-x)/2(Fe 2O3)x/2. First, the peaks in the XRD spectra were matched with GaFeO 3 \n(ICDD card no: 761005) and it was found that the samples with compositions from x = 0.8 to \n1.3 show single phase of GFO, while samples with two extreme compositions x = 0.7 and 1.4 \nshowed presence of unreacted β- Ga 2O3 (ICDD Card No. 871901) and α-Fe 2O3 (ICDD Card \nNo. 890599) respectively, in addition to GFO. This suggests that single phase GFO exists between 0.8 \n≤ x ≤ 1.3 and this range is also used for further analysis of the samples. Further \nwe carried out Rietveld analysis of th e XRD data using FullProf 2000 package12 and \nconsidering orthorhombic Pc2 1n symmetry for GaFeO 3 and monoclinic C2/m  symmetry for \nβ-Ga 2O3 and rhombohedral 3Rcsymmetry for α-Fe 2O3. The refinement results are plotted as \nsolid lines along with the raw data in Fig 1( a). The refinement yielded a goodness of fit  4parameter varying between 1 to 3 for the entire composition range. Further phase analysis of \nthe refined data suggests that the sample with x = 0.7 consists of ~80 mol% of the \northorhombic GFO and ~20 mol% as β- Ga 2O3. On the other hand, sample with x = 1.4 \nshows presence of 74 mol% GFO and ~26 mol% of α-Fe 2O3. These two compositions ( x ≥ \n0.8 and ≤ 1.3) mark the phase boundaries for the existence of single phase GFO. These \nresults are in excellent agreement with the previously reported data on the phase stability of \nGaFeO 3.13 The results also suggest that GFO is not a line compound as it can accommodate \ncation non-stoichiometry from x = 0.8 to 1.3 without formation of any secondary phases or \nany un-reacted ingredients. From the perspectiv e of Gibbs phase rule, it means that at room \ntemperature, the variation in composition does  always yield single phase GFO within these \ncompositional limits. \n Further, if we observe Fig. 1(b), we see that the peak position of the highest intensity \npeak (2 θ ~ 33.03 °) shifts towards lower 2 θ with increasing Fe content for single phase GFO \nindicating an increase in the lattice parameters with increasing Fe content. Similar trend was \nobserved for other peaks too. We then determin ed the lattice constants of GFO as a function \nof composition from the refined XRD data which are plotted in Fig 2. Here we see that the \nlattice constants ( a, b, and c) as well as the cell volume increase with increasing linearly Fe \ncontent. The data shown is only for the composition 0.8 ≤ x ≤ 1.3 corresponding to pure GFO \nphase. Such a linear variation of lattice constant s with composition in a binary system can be \nquantitatively expressed by Vegard’s law or a simple rule of mixtures. However, since any \nfurther change in composition beyond 0.8 ≤ x ≤ 1.3 leads to unreacted starting powders, the \nextrapolation of line does not fall an ywhere close to lattice constants of β-Ga 2O3 and α-\nFe2O3. Also, β-Ga 2O3 and α-Fe 2O3 have monoclinic and hexagonal structures respectively, \nsignificantly different than GFO’s structure. For a linear extrapolation until the end members, \none needs to use the lattice parameters  of isostructural forms of reactants i.e. κ-Ga 2O3 and ε- 5Fe2O3 as end members (for x = 0 and x = 2, respectively). The lattice parameters of these two \ncompounds were obtained from references 14-16. The lattice parameters of κ- Ga 2O3 were \ncorrected to room temperature values by using the thermal expansion coefficients of GFO.17 \nA comparison of these lattice constants with th ose obtained from the extrapolation of line in \nFig. 2 to x = 0.0 and 2.0 reveals an excellent correspondence and is a clear manifestation of \nVegard’s law. Hence, based on these observatio ns, a relation between the lattice parameters \nof κ- Ga 2O3, Ga 2-xFexO3 (0.8 ≤ x ≤ 1.3) and ε-Fe 2O3 can be expressed as  \n2- 3 2 3 2 3 (  = 1,2,3),Ga Fe O (  = 1,2,3), -Ga O (  = 1,2,3), -Fe O(2- ) = + 22xx ii ii iixxaa aκε  \nWith the exception of a few systems such as YMn 1-xGaxO318, well-studied \nmagnetoelectric materials have not been reported  to obey Vegard’s law. As far as perovskite \nstructured oxides are concerned, while solid solutions with structures close to cubic structure\n19 e.g. BaTiO 3-CaTiO 3, BaTiO 3-BaZrO 3, and SrTiO 3-BaZrO 3 follow Vegard’s law \nvery well, there have been many reports of deviations in some other perovskite oxides20,21. In \nthis context, it is indeed remarkable to observe Vegard’s law in the single phase region of \nGa2O3-Fe 2O3 system. Unlike simple perovskite oxides, GFO is a crystallographically \ncomplex oxide having orthorhombic structure wi th 40 atoms per unit-cell. With significantly \ndifferent lattice dimensions, the prediction of lattice constants as a function of composition \nallows one to tailor the material for epitaxial films for devices. The increase in the unit cell \nvolume is about ~2.5%, which is expected as the ionic radius of Fe3+ (~ 0.645 Å) is 4% larger \nthan that of Ga3+ (~0.62Å). This large change in the ionic radius also affects the polyhedral \ndistortion in the unit cell. This distortion can be quantified using Baur’s22 distortion index, \n.\n1 .() 1n\nia v g\ni avgll\nnl=−Δ=∑ where li is distance from the central atom to the ith coordinating atom, n \nis the number of bonds and lavg. is average bond length. The calculations suggest that while Δ \nis relatively large for Fe1 and Fe2 octahedra, change is maximum for Ga2 octahedra upon  6increasing the Fe content. In contrast, the distortion index of Ga1 tetrahedra is about an order \nof magnitude smaller and remains almost unchanged with varying Fe content. Latter also appears to substantiate the fact that Fe substitution predominantly occurs on octahedral Ga2 \nsites.\n9,23 \nTo further probe any subtle changes in the structure as a consequence of changes in \nthe composition, we obtained Raman spectra of our samples. As shown in figure 3, unpolarized Raman spectra of polycrystalline specimens yielded 24 Raman active modes in the spectral range of 80-870 cm\n-1 at room temperature. Although the observed number is \nmuch smaller than the number predicted by the group theory, it is consistent with earlier \nreports.17 From a cursory examination of these spectra, we observe that polycrystalline nature \nof our samples leads to suppression of some of the modes due to broadening. Further, a \ncomparison of these spectra for various compositions shows that number of observed Raman \nmodes remains same upon increasing the Fe content of the samples suggesting absence of any structural change upon changing the composition. However, a closer examination reveals that \nthe most of the modes soften with increasing Fe content, as clearly observed in the inset of Fig. 3. The inset show that the position of modes between 600-850 cm\n-1\n moves toward lower \nfrequency side as Fe content increases from x = 0.8 to 1.3.  \nChange in the mode frequency with increasing Fe content of the samples can be \ncorrelated to the changes in the bond strength, cation size and mass of cations. Ionic bond \nstrength of Ga-O and Fe-O bonds are expected to remain unchanged as electronegativities ( χ) \nof Ga and Fe are quite similar (Bond strength σ ∝ |χc-χa|, c: cation, a: anion).  However, the \nincrease in the cation size by ~4 % upon increasing Fe content (Fe3+ ~ 0.645 Å and Ga3+ ~ \n0.62Å) results in lattice expansion and hence, an increase in bond length as well as distortion \nindex as also shown by the XRD results. This increase in the bond length ( l) implies a \ndecrease in the phonon frequency ω 1\n2() l ω−∝  as observed in our experimental data. The  7contribution of the differences in the atomic wei ghts of Fe (56) and Ga (70) has a negligible \neffect on the phonon frequencies in the region 230-750 cm-1 as these involve motion of \noxygen polyhedra. On the other hand, the low frequency modes show no dispersion with \nincreasing Fe content. This is possibly due to two competing mechanisms: increase in lattice \nparameter leading to mode softening while a decrease in effective atomic mass (as Fe content is increased) would imply mode hardening. \nFollowing the structural characterization of GFO, we carried out magnetic \nmeasurements to investigate the magnetic parameters and transition temperature vis-à-vis \ncomposition. We carried out low temperature magnetization measurements and the results \nobtained at 115K are shown in Fig. 4. Further, to verify the quality of our samples and to compare the nature of composition dependence of the magnetic transition temperature with previous reports\n7,9, we also carried out temperature dependent magnetization measurements \nfor all compositions. Our results show that the transition temperature varies linearly with composition, and follow the trend reported earlier.\n7,9 We also observe that the remanent \nmagnetization of all the phase pure samples incr eases with increasing Fe content as shown in \nthe inset (a) of Fig 4. This is expected si nce only Fe and its occupancy on Ga2 site as a \nconsequence of site disordering is responsible for ferrimagnetism in GFO.9,23 Interestingly, \nthe coercive field (H c), plotted as a function of composition exhibits a maxima at x = 1.0 i.e. \nequal Ga and Fe content of the sample, as shown in  the inset (b) of Fig. 4. This result is in \ncontrast with that reported by Arima et al.9 on the single crystal samples of GFO who showed \na monotonic decrease in the coercivity upon in creasing the Fe content of the samples from x \n= 0.8 to x = 1.1. While Arima et al.9 attributed this decrease in the coercivity to occupation of \nGa sites by Fe ions leading to changes in the magnetic anisotropy, present trend appears to be \naffected by sample’s polycrystallinity. Microcrystallite sizes obtained from XRD data using the Debye Scherrer method (not shown here) suggest  a decrease in the crystallite size as the  8Fe content increases. While the increase in Fe content on one hand tends to increase the \nmagnetic anisotropy as also reported by Arima et al. ,9 the decreasing crystallite size and \nhence decreased size of magnetic domains would impede the magnetic moments and thus \nleading to the observed behavior. As a further study, it would be interesting to examine the \nmagnetic domains at a microscopic level using techniques such as magnetic force microscopy \nto further understand the contribution from these two factors.  \n In conclusion, XRD results clearly showed a phase pure region of GaFeO 3 in the \nrange 0.8 ≤ x ≤ 1.3 whilst unreacted Fe 2O3 and Ga 2O3 are retained outside this range, clearly \nmarking the phase boundary for pure GFO. Within the boundaries of the phase pure region, \nthe lattice constants vary linearly with the composition with isostructural end members falling \non this line suggesting that the system follows Vegard’s law excellently.  Raman spectroscopy measurements corroborated with the XRD data ruling out any structural change \nin the samples upon changing the composition. Moreover, shift in the mode frequencies to \nlow frequencies upon increasing Fe content are manifestation of bond stretching, ascribed to \na small but finite size difference between Ga an d Fe ions. Magnetic measurements revealed \nthat while material’s transition temperatures shows a predicted behavior as a function of \ncomposition, the coercivity of samples displays a rather different nature than that of single \ncrystals. Acknowledgements \nAuthors thank Council of Scientific and Indust rial Research (CSIR) and Department of \nScience and Technology (DST), Government  of India for the financial support. \n  9References: \n1 H. Schmid, Ferroelectrics 162, 317 - 338 (1994). \n2 J. F. Scott, Nat Mater 6, 256-257 (2007). \n3 G. A. Prinz, Science 282, 1660-1663 (1998). \n4 A. Zvezdin, A. Logginov, G. Meshkov, and A. Pyatakov, Bull. Russian Acad. Sci.: \nPhysics 71, 1561-1562 (2007). \n5 H. Schmid, Ferroelectrics 221, 9 - 17 (1999). \n6 W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature 442, 759-765 (2006). \n7 J. P. Remeika, J. App. Phys. 31, S263-S264 (1960). \n8 G. T. Rado, Phys. Rev. Lett. 13, 335 (1964). \n9 T. Arima, D. Higashiyama, Y. Kaneko, J. P.  He, T. Goto, S. Miyasaka, T. Kimura, K. \nOikawa, T. Kamiyama, R. Kumai, and Y. Tokura, Phys. Rev. B 70, 064426 (2004). \n10 R. B. Frankel, N. A. Blum, S. Foner, A. J. Freeman, and M. Schieber, Phys. Rev. Lett. \n15, 958 (1965). \n11 S. C. Abrahams and J. M. Reddy, Phys. Rev. Lett. 13, 688 (1964). \n12 J. Rodríguez-Carvajal, Physica B 192, 55-69 (1993). \n13 H. J. Van Hook, J. Am. Cer. Soc. 48, 470-472 (1965). \n14 R. Roy, V. G. Hill, and E. F. Osborn, J. Am. Chem. Soc. 74, 719-722 (1952). \n15 P. Kroll, R. Dronskowski, and M. Martin, J. Mater. Chem. 15, 3296-3302 (2005). \n16 E. Tronc, C. Chanéac, and J. P. Jolivet, J. Solid State Chem. 139, 93-104 (1998). \n17 S. Mukherjee, R. Gupta, and A. Garg, arXiv:1103.5541v1 [cond-mat.mtrl-sci] (2011). \n18 U. Adem, A. A. Nugroho, A. Meetsma, and T. T. M. Palstra, Phys. Rev. B 75, 014108 \n(2007). \n19 S. Lee, R. D. Levi, W. Qu, S. C. Lee, and C. A. Randall, J. Appl. Phys. 107, 023523-6 \n(2010).  1020 T. Mitsui and W. B. Westphal, Phys. Rev. 124, 1354 (1961). \n21 P. Ganguly, N. Shah, M. Phadke, V. Ramaswamy, and I. S. Mulla, Phys. Rev. B 47, \n991 (1993). \n22 W. H. Baur, Acta Crystallogr. : Sect. B 30, 1195-1215 (1974). \n23 M. J. Han, T. Ozaki, and J. Yu, Phys. Rev. B 75, 060404-4 (2007). \n  \n  11List of Figures: \nFig. 1  (a) Room temperature X-ray diffractio n data with Rietveld refined patterns of Ga 2-\nxFexO3. Pattern for x = 0.7 is refined with monoclinic C2/m  phase of β-Ga 2O3 and \northorhombic Pc2 1n phase of GaFeO 3 while spectra for x = 1.4 is refined using \nrhombohedral 3Rc−\n phase of α-Fe 2O3 and orthorhombic Pc2 1n phase of GaFeO 3; (b) \nspectra of all the compositions within 2 θ range 29.5 ° – 33.7 ° clearly showing the peak \nshift toward lower 2 θ with increasing Fe content (*: β-Ga 2O3 , $:α-Fe 2O3) \n \nFig. 2 Variation of lattice parameters of single phase GFO (0.8 ≤ x ≤ 1.3). Parameters of the \nend members ε-Fe 2O3 (Ref. 16) and κ- Ga 2O3 (Ref. 15) are represented by diamonds \nand circles.  \n \nFig. 3 Room temperature Raman spectra of single phase Ga 2-xFexO3. For clarity, only 18 \nmodes are marked. Inset shows clearly that the peaks between 600 to 850 cm-1 shift \ntowards lower frequency with increasing Fe concentration.  \n \nFig. 4 Magnetization (M) vs. applied field (H) pl ots for all the single phase compositions of \nGa2-xFexO3 measured at 115 K. Inset (a) and (b) show variation of remanent \nmagnetization (M R) and Coercive Field (H c), respectively with Fe content.  \n "    },    {        "title": "2008.00684v1.Solute_hydrogen_and_deuterium_observed_at_the_near_atomic_scale_in_high_strength_steel.pdf",        "content": "1 \n Solute hydrogen and deuterium observed at the near atomic \nscale  in high-strength steel \n \nAndrew J. Breen 1,2,3*, Leigh T. Stephenson 1, Binhan Sun 1, Yujiao Li 4, Olga Kasian 1,5, Dierk \nRaabe 1, Michael Herbig 1, Baptiste Gault 1,5* \n \n1 Max-Planck -Institut für Eisenforschung, Max -Planck -Straße 1, Düsseldorf, Germany  \n2 School of Aerospace, Mechanical & Mechatronic Engineering, The University of Sydney, \nSydney, NSW, Australia  \n3 Australian Centre for Microscopy & Microanalysis, The Universit y of Sydney, Sydney, NSW, \nAustralia  \n4 Zentrum für Grenzflächendominierte Höchstleistungswerkstoffe  (ZGH) , Ruhr-Universität \nBochum , Germany  \n5 Helmholtz -Zentrum Berlin, Helmholtz -Institute Erlangen -Nürnberg, 14109 Berlin, Germany  \n6 Department of Materials, Imperial College London, Royal School of Mine s, Exhibition Road,  \n* Corresponding author: andrew.breen@sydney.edu.au ,  b.gault@mpie.de   \nKeywords:  atom probe tomography;  hydrogen embrittlement ; pearlitic steel ; deuterium   \nAbstract  \nObserving solute hydrogen (H) in matter is a formidable challenge, yet , enabling quantitative \nimaging of H at the atomic -scale is critical to understand its deleterious influence on the \nmechanical strength of many metallic alloys that has resulted in many catastrophic fail ures of \nengineering parts and structures . Here , we report on the APT analysis of hydrogen (H) and \ndeuterium (D) within the nanostructure of an ultra-high strength steel with high resistance to \nhydrogen embrittlement . Cold drawn , severely deformed pearlitic steel wires ( Fe-0.98C -\n0.31Mn -0.20Si -0.20Cr -0.01Cu -0.006P -0.007S wt.%, 𝜀=3.1) contain s cementite \ndecomposed during the pre -deformation of the alloy and ferrite . We find H and D within the \ndecomposed cementite, and at some inte rfaces with the surrounding ferrite. To ascertain the \norigin of the H/D signal  obtained in APT , we explored a series of experimental workflows  \nincluding cryogenic specimen preparation and cryogenic -vacuum transfer from the \npreparation into a state -of-the-art atom probe. Our study points to the critical role of the \npreparation, i.e. the possible saturation of H -trapping sites during electrochemic al polishing, \nhow these can be alleviated by the use of an outgassing treatment, cryogenic preparation and \ntransfer  prior to charging. Accommodation of large amounts of H in the under -stoichiometric \ncarbide likely explains the resistance of pearlite agains t hydrogen embrittlement.  \n1 Introduction  \nH is the lightest and smallest of all the elements. It is also ubiquitous. Its presence within the \nstructure of metals cannot be avoided, partly due to its high mobility. The interactions between \nH and metals play an essential  role in the structure -property relationships of engineering alloys  2 \n [1]. Hydrogen’s most striking effect on materials is a sudden and often unpredictable decrease \nin ductility, toughness, and generally in the material’s resistance to fatigue -crack propagation \nknown as hydrogen embrittlement. This phenomenon was discovered as early as 1875 by \nJohnson but is still not fully understood to date [2]. A real -life example: the oil spillage in the \nGulf of Mexico in 2012 is attributed to the catastrophic failure of connector bolts due to \nhydrogen embrittlement [3]. Hydrogen embrittlement mostly affects structural, high -strength \nmaterials that find application in the e.g. automotive,  aeronautical, oil and gas and power \ngeneration industries.  \nHigh -strength steels are amongst those materials that are particularly affected by hydrogen \nembrittlement. As reviewed recently by Bhadeshia [4], limited  strategies have proven efficient \nto prevent such embrittlement. Among high -strength steels, pearlite exhibits both exceptional \nstrength  [5] and a high resistance to hydrogen embrittlement (HE) compared to other steels  [6]. \nPearlite cons ists of alternating layers of cementite and ferrite, formed through a diffusive \neutectoid decomposition reaction of the austenite phase. Cementite has the stoichiometry Fe 3C \nand its crystal structure is orthorhombic. Ferrite, has body -centered cubic struct ure and is Fe -\nrich. Upon severe wire drawing deformation, the cementite decomposes, driven by dislocations \npenetrating through it, leading to a reduction in the carbon content in the cementite and hence \nleaving carbon -vacancies within the structure  [7]. Trapping diffusive H at specific \nmicrostructural features (interfaces, precipitates) to prevent its accumulation and thereby \nmitigate its influence on the properties is a possible strategy .  \nDirect atomic -scale observation of H is limited , yet, of fundamental importance for \nunderstanding the mechanisms governing its detrimental effects in metals and the resultant \ninfluence on the macroscopic properties. Atom probe tomography (APT) is the on ly microscopy \nand microanalysis technique to allow for direct observation of H in three -dimensions at the near \natomic -scale, with high potential for possible quanti tative analysis of H . APT is poised to play \na key role in the investigation of H in materials. APT exploits an intense electric field to \nprogressively erode the specimen, almost atom -by-atom. Upon processing of the data, APT \nprovides 3D elemental mapping with near -atomic resolution in volumes routinely of 100 x 100 \nx 1000 nm 3 [8]. Steels were one of the prime targets for H -studies by APT, in particular seeking \nto reveal the trapping behaviour  of H/D at carbides [9–11]. \nAPT has the  intrinsic c apacity to detect H, however, although the analysis chamber is kept  at \nan ultra -high vacuum, residual H is detected during every experiment . This H likely originates \nfrom the chamber wall made of stainless steel in most instruments, as well as from adsorbed \nspecies on the surface of the specimen or the specimen holder. I t is however challenging to \ndetermine whether H ions originate from the specimen or from ionization of the residual gas \nfrom the analysis chamber . This severely hinders  the potential for deploying APT to investigate \nH in materials . One approach proposed to mitigate these issues is to use isotopic marking. The \nspecimen can be charged with deuterium (D or 2𝐻), which has a very low natural isotopic \nabundancy  and has been used in several studies [9,12 –14]. \nCharging can be performed through e ither electrochemical or gas -phase charging. For \ndeuteration to be successful, there must also be microstructural trapping sites available and the \nspecimen must not have been saturated with H. The high diffusivity of H/D in most materials \nmeans that the analysis must be performed quickly following charging  otherwise, the D desorbs \nfrom the specimen, as successfully demonstrated by Haley et al. [14]. If the specimen may be \ncryogenically cooled immediately  after charging, which slows down outward diffusion, the 3 \n likelihood of successful detection of D increases [9–11]. The distribution of D can then be \nstudied to reveal possible locations of segregation inside a material. With recent developments \nin cryogenic transfer dedicated to APT [15,16] , the potential for success h as increased \nsignificantly. In addition, forays into cryogenic specimen preparation by focused -ion beam \n(FIB) have also been reported [17], demonstrating how important this is to prevent the pick -up \nof H during FIB -preparation [18]. Yet much remains to be done to make the analysis of H in \nmetals routine, and quantitative.   \nResults on H/D analysis  using APT for solutes and stable hydrides remain limited but reports \ncan be found in the literature for multilayered materials [13,19] , Ti-based [20–23] or Zr-based \nalloys [24] for instance.  Deuteration is however not without ch allenges  since H can be detected \nboth as atomic ions, i.e. H +, as well as molecular ions, i.e. H 2+ and H 3+. The relative amount of \nthese different ions depends strongly on the strength of the electrostatic field during the APT \nanalysis  [25] making quantification  particularly complex [26]. \nHere, w e investigate the distribution of H in a pearlitic steel produced by severe cold drawing \nof a wire with hypereutectoid composition. We report on a uniquely thorough set of analys es \nof specimens prepared by electrochemical polishing  and focused -ion beam (FIB)  milling at \nroom and cryogenic temperature s, with and without cryogenic vacuum transfer, with and \nwithout D -charging.  Yet, this article is not about a mere methodology study since r eliably \ndetecting and discussing H/D and their traps in complex materials r equires to operate on the \nbasis of such a wide probing matrix , including many preparation and charging conditions , as \notherwise the H/D signal is too unreliable.  \nIndeed , we demonstrate that H in all APT datasets originates mostly from specimen preparation  \nwhen using electrochemical polishing  as well as the FIB. D-charging of specimens saturated \nwith H from the preparation is inefficient . Our study sheds light onto the origins of the \ncontamination by H, also that the H -distribution revealed by AP T can likely be used to highlight \nH-trapping sites within the bulk of a material, yet precise quantification still likely requires \nisotopic marking  and precise physics -informed statistical analyses . From a materials standpoint, \nboth H and  D are found segre gated to the partly -decomposed cementite , with a slight tendency \nto segregate at some interfaces . Our result rationalise the importance of the defects associated \nto the off -stoichiometric composition of the decomposed cementite as trapping sites for \nhydrog en, which help prevent hydrogen embrittlement.  \n2 Materials and methods  \n2.1 Materials  \nThe material used in this work was a severely deformed cold drawn pearlitic steel wire with \nhypereutectoid composition (Fe -0.98 C -0.31Mn -0.20Si -0.20Cr -0.01Cu0.006 P -0.007 S, wt. %). \nThe initial diameter of the wire was 0.54 mm with an interlamellar spacing of 67 nm. The wire \nwas then cold drawn to a true (logarithmic) strain of 3.10. A tensile test was then performed on \nthe wire at room temperature and at a constant strain rate of  2×10−3𝑠−1. The tensile strength \nof the wire was measured to be 3.85 ± 0.02 GPa. Further analyses were performed on the as -\ncold-drawn material. The performance of this wire, and a range of other wires with varying cold \ndrawn strain values, were previously reported in [7]. This wire was specifically chosen because \nit was likely t o contain numerous decomposed cementite and ferrite lamellae within one atom \nprobe dataset, enabling advantageous conditions to study the H trapping sites around these 4 \n interfaces and within the different phases. The wire also had a favourable geometry for \nspecimen preparation by electrochemical polishing.  \n2.2 Specimen  preparation  and imaging  \nNeedle -shaped specimens for APT were prepared, post -tensile test, by electrochemical \npolishing using a two -stage process: first rough polishing was performed in a solution of 25% \nperchloric acid (70%) in glacial acetic acid with 10 – 20 V until the specimen’s end was roughly \nbelow a micron in thickness; second, a solution of 5% perchloric in butoxyethanol with 10 – 20 \nV was used to finalize the shaping of the specimens. A du al beam scanning electron microscope \n/ Xe -plasma focused ion beam, FEI Helios PFIB, was used to sharpen electrochemically \npolished blanks with an end -size in the range of 20 µm. For transmission electron microscopy  \n(TEM ), specimens were prepared by in -situ lift-out from the cross -section of the wire, using a \ndual beam FEI Helios 600.  The inverse pole figure ( IPF) and image quality (IQ) maps were \ngenerated by n anobeam d iffraction  orientation mapping in a JEOL JEM -2200FS TEM operated \nat 200 kV using the commercial setup ASTAR by Nanomegas at 0.5 nm spot size, 1 nm step \nsize, 20 ms exposure times and without precession. An IPF map of only the ferritic phase was \ngenerated using the c ommercial indexing software by Nanomegas. All points above 150 \nindexing quality and 5 reliability were exported and further processed in the TSL 7.0.1. software \nby EDAX.  \n2.3 Specimen  charging  \nSpecimen charging and transfers were performed using the facilities from the Laplace project \nat the Max -Planck -Institut für Eisenforschung [16]. Specimens were charged with deuterium \nusing an electrochemical cell and 0.1 M NaOD in D 2O charging solution introduced in ref. [14]. \nFor simplicity, ease of handling an d more rapid transfer, some modifications to this previously \nreported setup were made. The Nafion ® membrane around the sample was removed to simplify \nhandling. A new sample holder was constructed at the working electrode to hold an APT \nmultiwire puck and 4  electropolished needles (see Figure 1a). Immediately after charging in \none of the explored workflow s (see section 3.2.) , the puck was removed from the charging cell \nand inserted into an Al puck holder block ( Figure 1b). Figure 1c shows the cell configuration \nand potentiostat attachment. For all charging experiments, a PalmSens ® EmStat3 ® potentiostat \nwas used. Figure 2c shows the charging cell and potentiostat connection assembly . \nComplementary testing of the charging setup was performed using a c ast ultra -fine-grained, \nfully pearlitic, high carbon steel of composition Fe -0.74C (w.%). The cold drawn wire was not \nused for these experiments primarily because the geometry was not suitable for thermal \ndesorption spectroscopy (TDS) . Unlike the pearlitic  wire, this material was not exposed to \nplastic deformation. However, the results still provide valuable information of how a fine \npearlitic microstructure behaves under the charging conditions used.  \nFigure 1d shows the resulting cyclic voltammetry curve for a test blank from the Fe -0.743C \nsample submerged approximately 5 mm into the charging solution with a cross section of 0.5 x \n0.5 mm.  The voltage was swept from -3 to +3 V. The hydrogen evolution reaction (HER) and \nthe oxygen evolution reaction (OER) regio ns can clearly be observed. A constant voltage of -\n1.2 V was found to generate the HER with measurable current response while suppressing \nbubble formation. The same charging solution and applied voltage had been previously reported \nby Haley et al. [14] in a similar  setup for charging high -Mn steel.  5 \n  \nFigure 1: Electrochemical H/D charging setup. (a) Schematic of the cell. 1. The Counter electrode \n(+). 2. The reference electrode (Ag/AgCl). 3. The working electrode ( -). 4. A Pt wire or foil ( which has \na surface area larger than the specimen/s being charged. 5. Glass Beaker which is filled with 0.1M \nNaOD in D 2O so that only the specimens and electrodes are in contact.  6. Special puck holder \nattachment (so that specimens can be charged directly  in puck). (b) Wire specimens in cryo multi -wire \npuck in Al block which could be precooled to LN 2 temperatures. (c) testing of electrochemical cell \nwith pearlite blank. (d) Cyclic voltammetry curve showing current response from pearlite test blank. -\n1.2 V  was found to be a good working voltage for a H/D charging regime.      \nThe possibility of electrochemical ly charging this pearlitic microstructure  with deuterium was \nconfirmed using  TDS. Two  14 x 14 x 0.5–1 mm plate s of the Fe -0.743C sample  were \nmechanica lly polished . A scanning electron microscope (SEM) image taken in a Zeiss Merlin \nat 3 kV acceleration voltage  using the inLens detector is shown in Figure 2a and confirm s the \nfine pearlitic microstructure similar to  that of the wires considered here .  \nOne plate  was then directly analysed for H/D content using TDS while the other was charged \nwith D for 90 mins , followed by TDS analysis.  Care was taken not to submerge the s upporting \nalligator clip into the charging solution so that the H/D evolution reaction would occur \nexclusively on the TDS sample not at the support. This meant that only approximately 50 % of \nthe TDS sample was submerged in the solution. The time interval between the end of D charging \nand the start of TDS experiments was less than 20 min. The TDS was performed on a home -\nbuilt ultra -high vacuum -based setup [27] at a constant heatin g rate of 26 C·min -1. The gases \nevolving during the continuous heating of a sample were detected by an MKS MicroVision 2 \nquadrupole mass spectrometer. The resulting spectra are displayed in Figure 2b, showing a clear \nindication of a peak associated to the charging of H inside the specimen at approx. 500 C. \nThe TDS analysis was performed only to indicate whether our electrochemical charging process \nwas effective. The posit ion of peaks in TDS  is highly dependent on the heating rates. Higher \nheating rates shift the TDS peaks to higher temperatures. Here, we use s a rather rapid heating \nrate (26 C/min) compared with more conventional, commercial TDS setups which are normall y \n6 \n operated at around 1~5 C/min . This TDS provides qualitative. Indeed, a  full Kissinger analysis \ncannot be performed here, since if would have required a thorough exploration of heating rates \nto determine the trapping energy associated to the peaks.  We still obtain a higher cumulative \namount of H/D upon charging, as readily visible on the plot. In addition,  it took approx. 20min  \nbetween the end of charging and the start of TDS, which might be already enough to los e some \nH from the most shallow traps . \n \nFigure 2: (a) A SEM image of the fine pearlite microstructure of the cast Fe -0.743C sample. (b) The \nTDS curve showing significantly higher D detection after charging , with the desorption rate on the left \naxis and the cumulative amount of H/D on the right axis .  \n2.4 Specimen  cryo -transfer  \nIn workflow W5, after charging, the block with puck and specimens was initially plunged into \nLN2 but the much higher temperature of the puck and samples, vigorous bubbling, along with \nthermal shock, was at risk of damaging the specimens. Instead, the puck and specimens were \n7 \n only passively cooled on the pre -LN2 cooled Al block. This was done in air and consequently \nice formed on the specimens. However, after loading the samples into the buffer chamber of \nthe PFIB docking station, this ice quickly sublimated under the high vacuum conditions. The \nsamples were then transferred directly into the Ferrovac ® suitcase. Within the suitcase, samples \ncould be kept at temperatures of approximately –180 C with a vacuum of 10 -8 - 10-7 mbar and \ntransferred directly to the buffer chamber of the LEAP 5000 XS/XR instruments.  \nWe did not succeed despite several attempts at charging and cryogenic cooling carried out \ninside a glovebox filled with N 2 and a highly controlled level of moisture, followed by direct \ntransfer into the at om probe via the cryo -vacuum transfer suitcase. Specimen yield was low, \ndespite starting to field  evaporate at low voltages, suggesting the specimens had successfully \nsurvived charging and transfer. Strong peaks in the mass spectrum pertaining to Cu and W \npeaks could be observed, which suggested contaminants from previous charging attempts \nfrozen onto the surface of the needles. Freshly prepared charging solution and more careful \ncleaning of the specimens with isopropanol before cryogenic cooling is highly necessary . \n2.5 Atom  probe experiments  \nMost of the APT data were acquired on a Cameca LEAP 5000 XR with a base temperature of \n60 K, in high -voltage pulsing mode with a pulse fraction of 15 -20% and at a repetition rate of \n200-250 kHz. Similar parameters were rep orted to yield accurate compositions in the analysis \nof these materials [5]. A detailed list of the experimental paramet ers for each representative \ndataset of the workflows is provided in the supplementary information. Voltage pulsing was \nchosen primarily to minimize the formation of H 2+ ions which complicate the quantification of \ndeuterium which was to be charged into the specimens. Voltage pulsing also had the added \nbenefit of improved spatial resolution which could be used to facilitate crystallographic \nindexing of the datasets  as shown in Figure 3. APT parameters were intentionally kept \napproximately constant so that changes to the H and D measurements could be more confidently \nattributed to the different experimental workflows  and not experimental running conditions . \nNevertheless,  some data was also collected on a Cameca LEAP 5000 XS and LEAP 3000 and \nusing laser pulsing so the influence of these changes to APT running conditions could be \nassessed. Data reconstruction and processing was performed in Cameca IVAS ® 3.8.2 and a \nrange o f specifically coded routines in Matlab.  \n3 Results  & Discussion  \n3.1 Material crystallographic characterization  \nNano -beam diffraction (NBD) and bright field scanning transmission electron microscopy (BF -\nSTEM) were performed on a  transverse  cross -section of a wire drawn to a true strain of   = 3.10  \nfor information on crystallographic texture . The same specimen was then analysed by APT.  \nFigure 3(a) is the inverse pole figure  (IPF)  map (Z) – where Z is the direction normal to the \nimaging surface and corresponds to the drawing direction of the wire  (only ferrite was indexed \nsince indexi ng cementite at these sizes and deformation states was not possible) .  Figure 3(b) is \nthe corresponding image quality (IQ) map on which the grain bounda ry misorientation angle  \nmap was superimposed .  Figure 3(c) is a n APT reconstruction of the wire (only Fe and C atoms \nare shown for clarity) . APT performed on this material in voltage pulsing mode yields high \nquality data in which partial crystallog raphic information  can also be  extracted . Although \ncrystallographic poles  were difficult to distinguish  from the  density variations , due to the 8 \n decomposed cementite lathes, regions containing clear lattice planes were observed ( Figure \n3(d)). Spatial distribution maps [28,29]  (SDMs) were calculated in the depth of a subset of the \nreconstructed data in the pole regions, confirming the presence of (110)  planes within the \nreconstruction, as shown in Figure 3 (e). This information was used for guiding reconstruction \ncalibration wherever possible by adjusting the reconstruction parameters to obtain appropriate \nplane spacings (measured from the peak -to-peak distances). These  snapshot s of the ultra fine \nand highly complex microstructure of the pearlitic wire highlight the need for high -resolution \ncorrelative structural and chemical characterization tools, as well as the inherent high quality \nof the coll ected APT data.  Only the use of such delicate probing protocols can provide clues as \nto the distribution of H and D and the associated trapping behaviour , and, potentially the \ninfluence of the crystallographic character of the interfacial regions.   \n \nFigure 3:microstructural analysis of the wire : (a) Inverse pole figure – Z from the  nanobeam \ndiffraction analysis of the  = 3.10 deformed wire showing microstructure and texture ; b \ncorresponding bright field scanning transmission electron micrograph, on which the g rain boundary \nmisorientation angle map  was superimposed . (c) representative APT reconstruction of the wire (only \nFe and C atoms are shown for clarity. (d) a zoomed in region of the reconstruction  showing  individual \nlattic e planes. (e) SDM of region in d showing clear lattice periodicity.   \n3.2 Experimental  workflows  for H and Deuterium charging and probing  \nIt is essential to develop and test protocols that can establish routine acquisition of high quality, \nreliable and reproduc ible data in order to inform atomic -scale H detection with well -\n9 \n determined, robust estimations of the measurement errors. Seven different workflows were \nfollowed between electropolished specimen preparation  to APT analysis, as laid out in  Figure \n4 and detailed below :  \n1. The specimen was directly inserted into the load -lock of the atom probe and ran in \nvoltage pulsing mode.  \n2. The specimen was directly inserted into the load-lock of the atom probe and ran in laser \npulsing mode.  \n3. The specimen was placed into the load -lock of the atom probe and heat treated under \nhigh vacuum conditions (below 10 -7mbar) for up to 4h at 150 C and ran in voltage \npulsing mode.  \n4. Following this heat treatment  for 4h at 150 C, the specimen was removed from the load -\nlock and electrochemically charged with D and transferred into the atom probe via the \nairlock, at room temperature, and ran in voltage pulsing mode.  \n5. Following heat treatment, the specimen was electrochemically charged with D, \npassively cooled on an LN 2 cooled block and transferred into the atom probe via a UHV \ncryogenically cooled suitcase and ran in voltage pulsing mode.  \n6. Following rough electropolishing  down to a size in the range of 2 –10 m, the specimen \nwas sharpened further at cryogenic temperatures (~ -130C) in the PFIB before being \ntransferred into the PFIB/UHV buffer chamber and then into the UHV cryogenically \ncooled suitcase. The details of the set -up and conditions are described in  [16]. \n7. Following rough electropolishing  down to a size in the range of 2 –10 m, the specimen \nwas sharpened further in the PFIB at room temperature before being transferred into the \natom probe in air. Samples were ran using voltage pulsing.  \nThe experimental workflows are non -exhaustive, other variations were attempted, but these 7 \nrepresent the workflows for which sufficient APT data was obtained and presented here. \nWorkflow 1 (W1) was designed as the basic ‘control’ experiment to wh ich the other workflows \nwere compared.  \n  \nFigure 4: Seven experimental  workflows (W1 – W7) from a n electropolished specimen to analysis in \nthe atom probe . D charged workflow numbers have been bolded. Cryogenic transfer experiments are \n10 \n coloured blue.  PFIB: Xe -plasma Focussed Ion Beam Instrument. LEAP: LEAP Atom Probe \nTomography Instruments.  \nA summary of the APT results obtained from  representative datasets  from each of the 7 \nexperimental workflo ws outlined previously are  shown  in Figure 5. Atom maps of the C and \nH/D are provided along with corresponding mass spectra (normalized by total atom counts) so \nthat the  change in H and D signal behaviour  between the different datasets can be assessed . The \nbulk C, H, D concentrations for each workflow are  also provided in Figure 6. In all datasets, H \nclearly partitions to the decomposed cementite lamellae, where C content is higher.  \n \n11 \n Figure 5: APT results obtained from the deformed pearlitic wi re after the different experimental \nworkflows . (a) – (g) correspond to the results from workflow s W1 – W7, respectively. (h) represents \nthe results from the same specimen after electropolishing ( workflow  W1) and then after heating in the \nload-lock for 1 hour ( workflow  W3). The APT reconstructions (H and C maps for clarity) and the \naccompanying H peaks in the mass spectrum are shown for each workflow . The results not only reveal \nthe strong link between specimen preparation , charging  and specimen transport but also the \nrobustness and quantification challenges  associated with H detection .  \n \nFigure 6: (a) bulk C, H and D  concentrations (at. %) for the respective workflows.  Bulk calculations \nwere background corrected on IVAS ® 3.8.2 . Composition profile through a decomposed cementite lath \nfor data obtained through W1 in (b) and W2 in (c).  \n3.3 High voltage  vs. laser pulsing  effects on H detection in atom probe \ntomography (W2)  \nIn Figure 5a, which represents the ‘control’ type experiment whereby the specimen was \nelectropolished and immediately analyzed using voltage pulsing, H exclusively evapora tes as \nthe atomic ion H+, detecte d in the peak at m/q = 1  Da. This behavior was  consistently  observed \nfor all datasets collected from workflow  W1. However, when laser pulsing was used in  \nworkflow  W2 (Figure 5b), a significant proportion of total H (32%) evaporated as a molecular \nion (H 2+). A much higher overall composition of H was also observed , i.e. 2.6% as compared \nto < 0.1% for all voltage puls ing measurements. This is evidenced by the difference in the base \nlevel of H + and H 2+ in the composition profiles calculated through a cementite lath in datasets \nobtained using workflows W1 and W2 shown respectively in Figure 6b and 4c.  In Figure 6c, \n12 \n the H -signal inside the lath even appears to drop, which can be explained by the local  increase \nin the electrostatic field, as explain below).  \nOur results from W2 suggest that laser pulsing promoted the detection of spurious H. This H \ncould originate from the field ionization of residual H from the vacuum chamber  in its gaseous \nform . It is also likely that the detection  of H comes , at least in part, from the field desorption of \nresidual H  that was before adsorbed on the specimen’s surface . This H can come from the \nchamber, the preparation routine , and potentially migrat ed along the specimen’s shank, driven \nby the gradient in electrostatic field . The higher electrostatic field conditions obtained in using \nvoltage pulsing (W1) lead to better screening of the specimen’s apex against contamination by \nH. These results  confirm once again that voltage pulsing is better for quantif ying H originating \nfrom within the specimen, in agreement with previous reports in the literature [11,14,23] . \nVoltage pulsing was therefore used in all remaining workflow s despite the higher chance of \nspecimen fracture  and lower yield . \n3.4 Effect of specimen pre-heating  (W3)  on H detection  \nIn experimental workflow  W3, the specimen was heated to ~150 ºC for approximately 4 hours \nin the load-lock of the atom probe . The heater used is the one typically used to clean the local \nelectrodes inside the vacuum chamber of the atom probe, and the temperature cannot be \nadjusted. Figure 4c indicates that the relative H content declined relative to w orkflow W1. This \nis suggest ive of  outgassing of H contained within the specimen.  \nHowever, a certain uncertainty still remained as to whether the reduction in H was due to a \ndifference  in the microstructure between specimens . Indeed, the relative volume fraction of \ndecomposed cementite and thickness of lamellae will influence bulk H detection [7]. We \nperformed another experiment, whereby the same specimen was an alysed  under workfl ow W1 \nand then workflow  W3 (Figure 5h). We confirmed that heating of the specimen, in this case for \nonly 1 hour, was sufficient to reduce the H detected by 83% without a ppreciable carbon \nredistribution or change in the underlying microstructure. The results further add proof to the \nargument that the majority of H being detected under these experimental conditions is in -fact \nfrom the sample and not from the chamber, otherw ise, such a drastic decline in H detection \nafter heating would not have been expected.  \n3.5 Effect  of D-charging  (W4) & cryogenic transport (W5)  \nCharging conditions are a critical consideration for successful D detection. Using our home -\nbuilt setup, we charged the specimens for 1 hour , which was sufficient to introduce D within \nthe sample, as confirmed by complementary Thermal Desorption Spectrosco py ( TDS ) \nexperiments. The conditions used , also did not appear to damage the specimens, although some \nsolution residue on the tip surface was suspected due to lack of cleaning after polishing. Several \nexperiments with charging times of only 5 minutes were conducted where no successful D was \nobserved.  \nA small peak at m/q = 2 Da was observed in some of the datasets obtained following workflow \nW4, as shown in Figure 5d. The ions corresponding to this peak were seen to segregate non -\nuniformly to the cementite/ferrite interfacial regions. Since H was expected to only evaporate \nas a single ion under these experimental conditions, this signal likely originated from D located \nat strong trapping sites at these positions. Further investigation into the local electric field, \nshown later in the manuscript, also provides evidence for this peak being D and not simply H 2 \nbrought about by a lower electric field. In workflow W5, after deu terium charging and 13 \n cryogenic transfer, an even stronger peak at m/q = 2 was observed (0.75 at. %), which has been \nattributed to D. The higher carbon content (2.17 at. %) suggests that a large decomposed \ncementite region was captured in this dataset , or, c onsidering the size, possibly a region of \nretained austenite, which is rather unexpected in these materials.  \n3.6 Cryogenic  vs. room temperature focussed ion beam preparation (W6 & W7)  \nFigure 5f and g show the results from PFIB specimen sharpening at cryogenic temperature in \nW6 and room temperature in W7. Bulk H composition in the specimen sharpened at cryogenic \ntemperature was only 0.07 at. % as opposed to 0.14 at. % when sharpened at room temperature, \neven though the cryogenic PFIB specimen captured a decomposed cementite region where H \nconcentration would usually be expected to be higher. The H composition was also compared \nto the decomposed cementite region in W1. An iso -composition surface of C = 3.05 at. % was \nused to approximately isolate the decomposed cementite regions and the H content was found \nto be 0.21 at. % - significantly higher than what was observed in the cryogenical ly PFIB \nprepared specimen. The results suggest that cryogenic PFIB suppresses H uptake into the \nspecimen. This supports the findings recently reported by Chang et al. [18], which focused on \nTi-based alloys and further suggests this phenomenon holds true for ferrous alloys also, albeit \nto a lesser  extent. Also interesting to note is  that Figure 5e and f both capture larger decomposed \ncementite regions under the same APT experimental parameters , yet the peak at m/q = 2 Da is \nonly seen in Figure 4e, further suggesting that this peak can be attributed to charged D and not \nH2. \n3.7 Influence  of local electric field and carbon concentration on H measurements  \nCare must be taken in the interpretation of APT data when comparing relative H/D \nmeasurements between the workflows because of numerous factors which influence these \nmeasurements. Reports exist in the literature on the relationship between the strength of the \nelectrostatic field and the detection of H either as atomic or molecular ions [25]. This also holds \ntrue for the carbon ions [30–32], in part because of th e complex nature of the field evaporation \nand dissociation processes for molecular ions [33]. The influence of the local electric field on \nthe carbon  and H detection must hence be considered. We track ed these metrics and compare \nbetween datasets by voxelis ing[34] each dataset and estimate  the local field, C and H contents \nin each voxel. Several different voxel sizes were tria lled and cuboidal voxel sizes from 2 3 nm 3 \n- 83 nm 3 were found to provide a good c ombination of resolution and signal.   \nTo measure the relative local field, the charge -state-ratio of the same atomic species of atom \ncan be compared. The ratio of the peaks will vary with changes to electric field with higher \ncharge states more likely to occur at higher fields, as predicted through post -ionization \ntheory [35]. The clearest occurrence of multiple ch arge states for the same ion species across \nthe datasets collected was for carbon. Peaks at m/q = 6 Da and m/q = 12 Da, containing 12C2+ \nand 12C1+ were observed. Some molecular C may have also been present in these peaks, but this \nwas expected to be low, p articularly in the voltage pulsed runs. Therefore, higher 12C2+/12C1+ \nratios can be attributed to relatively higher local electric fields.  \nThe distribution of C composition vs. relative electric field in the representative dataset \ncorresponding to workflow W1 is shown as an intensity histogram in Figure 7a. The 12C2+/12C1+ \nratio was calculated in each voxel and a 2D histogram of the results was plotted.  Most voxels \ncontain a very low carbon content of less than approximately 1 % , corresponding to the ferrite,  \nwhere the electric field is relative ly lower. T he 12C2+/12C1+ ratio increases sharply with only 14 \n small increases in C content up to approximately 2 %, corresponding to the decomposed \ncementite. Any further increase in C does not appear to influence the local electric field \nsignificantly. No region was f ound to contain C compositions close to the stoichiometric carbon \ncontent of cementite of 25%. Similar trends were observed for all representative datasets and \ncan be seen in the supplementary information.   \nThe relationship between the local C and H compo sitions is addressed in Figure 7b. The average \nH concentration for all voxels of a given C concentration in the range of 0 to 12 % in steps of \n2 % is plotted. The shaded  error bar represents the standard error with respect to the mean in \neach case. A clear linear upward trend is observed (X H(ave) = 0.017 X C + 0.12), confirming that \nthe H composition increases, on average, with increasing C composition. This agrees with th e \npartitioning behavio ur of H observed in Figure 5 and adds a level of quantification to the change \nof H solubility with C content. This would indicate that the decomposed cementite , which \ncontains a high density of crystalline defects, i.e. vacancies and dislocations , can indeed \naccommodate more H within its structure . In pristine cementite H solubility should be \nsignificantly lower.  \n \n \n \nFigure 7: (a) Relative  C conc. (at. %) vs. relative E -field intensity histogram (b) C conc. (at. %) vs. \naverage. H conc. (at. %)  line graph with shaded standard error of the mean (SEM) error zone.  \nCuboidal voxel size s of 83 nm 3 and 2 3 nm 3 were used for (a) and (b) respectively .  \n15 \n 3.8 Validating successful detection  of D \nIn order to  confirm the successful detection of deuterium in W4 and W5, the relative electric \nfield and concentration of ions originating from the m/q = 2 Da peak were  considered. \nMolecular ions are more likely to form at lower  electric  fields , so if the datasets corresponding  \nto W4 and W5 showed  regions of  significant ly lower field relative to the other datasets, this \ncould be an indication that molecular hydrogen  (H2) rather than deuterium was being detected.  \nThe grey -scale in the background of Figure 8a is a 2D intensity histogram of all non -charged \nvoltage -pulsed dat asets (W1, W3, W6, W7, W1 + 3 (1), W1 + 3 (3))  with, superimposed , scatter \nplots of the two deuterium charged datasets (W4 and W5)  in red and green . For all non -charged, \nvoltage pulsed runs, the 12C2+/12C1+ ratio is almost entirely between 0.4 – 2.3. The c alculated \n12C2+/12C1+ ratios in each voxel  for W4 and W5 also predominantly fall into this range . This is \nhighlighted in the corresponding histogram for W4 and W5 displayed to the side, which again \nshow no signs of 12C2+/12C1+ ratios outside the range observed in the non -charged datasets. The \naverage 12C2+/12C1+ ratios were 1.2 and 1.5 in W4 and W5 respectively. No clear clusters of \nvoxels with low field  are observed.  These observations point to similar field evaporation \ncondit ions and allows us to make direct comparisons between datasets. The scatter in W4 is \nsimilar to that observed in the non -charged voltage pulsed runs. The relative number of ions at \nm/q = 2 Da in workflow W5 is noticeably higher than that of the other volta ge pulsed datasets \nwith similar electric field , pointing to the importance of the cryogenic transfer to limit \noutgassing during specimen transfer.  \nOverall, the results suggest that a statistically significant number of ions originating from m/q \n= 2 Da wer e observed for the D-charged workflows at electric field levels very similar to that \nobserved in the non -charged datasets suggesting that the peaks were likely indicating successful \nD detection.  A region -of-interest  approximately 1800 nm 3 around the cluster of m/q = 2  Da ions \nin W4 (the ‘local’ region which was indicated in Figure 5d) returned 12C2+/12C1+ = 1.6 and a \nm/q = 2  Da concentration of 0.1 at. %.  Figure 8b shows the reconstruction obtained from the \nW4 dataset  as well as a cross sectional view of a thin slice through the tomogram showing the \ndistribution of C and D atoms. In the corresponding compositional map, the local higher \ncomposition of D appears higher at one of the interfaces between the ferritic matrix and the \ndecomposed cementite. Modelling work indicated a possible influence of the crystallographic \ncharacter of the interface on the H/D distributions [36].  16 \n  \nFigure 8: (a) ion concentration  at m/q = 2  vs. relative field ( 12C2+/ 12C1+). A 2D histogram of all non -\ncharged voltage pulsed runs (W1, W 3, W6, W7, W1 +3 (1), W1 +3(3)) was calculated from the 8 nm \nvoxelised reconstructions and the relative intensity plotted. Sca tter plots for each 8 3 nm 3 voxel in the D \ncharged voltage pulsed runs (W4 and W5) were then superimposed for comparison) . The \ncorresp onding histograms of the relative fields calculated for each of the 8 3 nm 3 voxels  in W4 and W5 . \n(b) C, H and D distribution in the tomographic reconstruction obtained from the data obtained \nthrough W4. (c) composition profile calculated along the pink arrow, and, as shown in the inset, H \nand D tend to be more highly concentrated in the region with sligh tly less C within the decomposed \ncementite.  \n3.9 Origins  of the H signal detected in APT  \nThe general perspective on studying H by APT was that it was impossible to distinguish H \noriginating either from the specimen or from the chamber. While heating experiment s suggested \nthat most of the H being detected in the initial experiment (W1) was, in -fact, from the sample \nitself, it is still difficult to confirm exactly where this H was introduced. If exposure to \natmosphere could lead to H ingress, likely to a small ex tent, H uptake could well occur during \nthe specimen preparation. However, recent evidence has pointed to the introduction of \nsignificant amounts of H in Ti and Ti -alloys during FIB -based specimen preparation [18,22] . \nThese reports agreed well with the literature that pointed to the formation of hydrides from \nmechanical and electrochemical polishing and ion beam -based pre paration of specimens for \nmicroscopic investigations [37–39]. In steel and ferrous -alloys, no such reports exist, but there \nis no reason that H ingress would not occur.  \n17 \n The bare metallic surface initially produced by the dissolution  of the metal or surface oxides  \noffers many new , potentially  very active , surface sites. Indeed, surface atoms are usually \nunsaturated and sterically more free to bond with other species. These atoms would  also be the \nfirst to leave the structure during dissolution. The dissolution of iron in an acidic solution should \nin principle follow these steps  \n(1) An anodic reaction such as 𝐹𝑒→𝐹𝑒2++2 𝑒−or  𝐹𝑒→𝐹𝑒3++3 𝑒−  leading to the \nformation of ferric or ferrous i ons in solution depending on e.g. the pH conditions;  \n(2) A cathodic reaction, likely involving the reduction of H and the release of gaseous H 2, \ni.e. 2𝐻++2𝑒−→𝐻2(𝑔) \nThe electrochemical polishing process is usually accompanied by the formation of gaseous \nspecies, often assumed to be originating from the oxygen evolution reaction, leading to \nbubbling of O 2 at the dissolving anode.  \nHowever, this dissolution reaction will be accompanied by a vast array of possible intermediate \nsteps reactions, includ ing the formation of oxides or hydroxides, for instance:  \n𝐹𝑒+𝐻2𝑂→𝐹𝑒𝑂 +2 𝐻++2𝑒−  \n2𝐹𝑒+3𝐻2𝑂→𝐹𝑒2𝑂3+6𝐻++6𝑒−    \n 𝐹𝑒+𝐻2𝑂→𝐹𝑒(𝑂𝐻)−+𝐻+  \nor  𝐹𝑒+2𝐻2𝑂→H𝐹𝑒O2−+3 𝐻++2𝑒− \nIn the case of the perchloric acid used here, the chlorine -containing ions can also react with the \nmetal : \n𝐹𝑒+2𝐻𝐶𝑙 𝑂4=𝐹𝑒(𝐶𝑙𝑂4)2+𝐻2 \n2𝐹𝑒+6𝐻𝐶𝑙 𝑂4→+3𝐹𝑒(𝐶𝑙𝑂4)3+𝐻2 \nUnder the conditions of very high potential used during the electrochemical polishing of the \nspecimen, all of these reactions, like ly only amongst others, can take place, albeit with different \nrates. No effort has ever been dedicated to controlling the conditions so as to select a specific \nreaction. In addition, literature points to the key role played by H in these reactions [40]. The \npresence of protons produced by these reactions at or near the surface can lead to the \nintroduction of H into the material.  \nImportantly, both the adsorption of water on the surface as well as the formation  of surface \noxide will lead to an over concentration of protons in the layer at the interface between the \nmetal surface and the solution. Some of these protons will also end up subject to chemisorption \nor physisorption at the metal surface.  This makes the ir absorption by the metal not only possible \nbut also likely. H penetration was shown to be energetically favourable by ab -initio simulations \nin the case of an iron surface in a sour gas environment [41], in conditions that would not be \ntoo dissimilar to those encountered here. H -ingress is also made more likely by the \noverconcentration of H + in the interfacial layer, i.e. the system will try to equilibrate the \nchemical potential of H in between the solut ion and the metal, leading to the introduction of H \nwithin the metal. This agrees with recent modelling effort by ab initio simulations of the \ncorrosion of Mg showed that it was possible that the hydrogen evolution reaction could take \nplace also at the ano de, but also that the splitting of water at the metal surface can lead to the \nintroduction of H within the metal [42].  18 \n There could be specimen preparation strategies optimised to limit or avoid  uncontrolled \nintroduction of H by using either highly controlled conditions of p otential  during \nelectrochemical polishing, i.e. maybe in the range of 1V . An alternative could be to find \nconditions for controlled polishing  in proton -free electrolytes or maybe in ionic liquids, \nincluding those containing H strongly bonded, i.e. part of C-H bonds for instance.  \n3.10 Characterising  and quantifying H by atom probe tomography   \nThe wide array of preparation and analysis protocols presented and discussed herein  enable us \nto identify adequate ways to probe H and D at the near-atomic scale and do so with clear \nmeasures of distinction between intruded, adsorbed and generically trapped H/D atoms. This \nleads us to define the pathway to finally overcoming the long -term pending challenge of how \nto measure single H and D atoms in complex microstructures and how to quantify the associated \nerrors.  \nTraditionally, the H peak observed at m/q = 1 Da in the mass spectrum of APT has been \nattributed to H present in the analysis c hamber which gets field evaporated in the presence of \nthe high electric field at the apex of the atom probe needle -shaped specimen.  However, on \ncloser inspection  of our experimental data from W3, it appears that at least 80% of the H \ndetected is  from the s pecimen itself , whether from within the core of the specimen or adsorbed \nat the surface . This observation agrees with recent studies on other alloy systems which have \nalso suggested that the H being detected at m/q = 1 Da most likely originat es from the \nspecimen [21,22] . It follows then that the H distribution in the datasets before heating is \nstatistically significant enough to highlight real microstructural segregation and partitioning \nwithin the alloy itself, albeit with no control over the conditions of H intake , i.e. the chemical \npotential tha t led to the ingress . The low H composition detected after heating (W3) can then \nbe attributed to a small amount of H from the chamber being field ionized or field desorbed  but \nalso to H in deep traps [43–45], unable to escape at the relatively low temperature (150 C) used \nfor the H -release treatment . \nThe c onditions during specimen preparation are highly out -of-equilibrium , which makes  the \ndetermination of the chemical potential of H difficult  if not impossible . Such conditions are not \nrepresentative of H-uptake during the service  life of a part . In addition, the baseline level of H \noriginating from residual gas from the chamber cannot be accurately determined or predicted, \nso D-charging is still particularly useful, not only because  it mitigates overlap , but also because \nit enables a more controlled source of D.  D-charging can likely only be performed reliably after \nelectrochemical polishing if a H -release heat treatment (W3)  is performed .  \nYet, under experimental conditions where th e electrostatic field is optimised to lead to \nsatisfactory analysis conditions, it is possible that the distribution of H, in particular correlated \nto regions of high electrostatic field, reveals the actual partitioning behaviour of H within the \nmaterial  [21]. Our experimental results obtained via W2 however indicate that results obtained \nfrom laser puls ed APT cannot be trusted. The heating of the specimen su bsequent to laser light \nabsorption enables field evaporation at electric fields that are too low to shield the apex from H \nmigrating along the specimen’s shank [8],  leading to excessive detection of H 2+, some of which \ncould dissociate into two H +, making the characterisation of H doubtful and its reliable \nquantification impossible . The additional heat can also assist the surface diffusion of adsorbed \natomic or molecular H at the surface towards the apex.  19 \n 3.11 Rationalising pearlite’s resistance against hydrogen  embrittlement  \nWith more stringent constraints on fuel efficiency for e.g. aut omotive vehicles, there has been \na surge of interest in developing high -strength steels. However, these materials are critically \nsubject to h ydrogen embrittlement  [46].  Most experimental studies  of hydrogen embrittlement \nhave focused on the mechanical degradation of the material in conjunction with indirect \nmeasurements of the effect of hydrogen on the mechanical behaviour of the all oy [6,47] . This \nwas often supported by estimations of the content of hydrogen within a material, via for instance \nTDS, and made assumptions on the nature of the energy traps associated  to specific \nmicrostructural features [4,47] . Amongst high -strength steels, a s-drawn pearlitic steel wire have \nbeen reported to exh ibit superior  resistance against hydrogen embrittlement  [6,48] . This was \nascribed to the high density of H trappi ng sites in pearlitic steel associated to the high density \nof defects in the off -stoichiometric cementite  [6], as well as possibly at interfaces between the \nferrite and the cementite [49].  \nThe results reported herein provide insights into the behaviour of H within the microstructure \nof these highly deformed pearlitic steel s and the influence that the experimental workflow has \non the final observed distribution.  Only a relatively low number of hydrogen ions were detected  \nin all datasets , even before any heating experiments, but the one obtained with laser pulsing \n(W2). This was expected due to the known  low solubility of H in cementite and ferrite in steel  \n[50]. The H and C atom maps , in e.g . Figure 5, indicate a clear partitioning behaviour of H to \nthe decomposed cementite regions, and we find a clear trend between H and C content ( Figure \n7). We thus find that the hydrogen (or deuterium) atoms are primarily trapped within the \ncementite. Among the different possible mechanisms for this preference the cementite’s \ndislocation content and C -vacancies are conceivable as trapping sites. D-charging after H -\nrelease support s the partitioning of D to the decomposed cementite, and a slight segregation is \nobserved at an interface between the cementite and the ferritic matrix.   \n4 Summary and conclusion  \nTo summarise, the distribution of  H/D distribution in heavily cold -drawn pearlitic steel wire of \nhypereutectoid composition (Fe -0.98 C -0.31Mn -0.20Si -0.20Cr -0.01Cu0.006 P -0.007 S, wt.%) \nwas studied using atom probe tomography (APT). Data from 7 different workflows were \ncompared to elucidat e hydrogen absorption and desorption behaviour  in these materials and \ngain an understanding how various experimental decisions can significantly influence measured \nresults.  \nThe main findings can be summarized as follows:  \n• > 80 % reduction in H concentratio n was observed after a low temperature heat \ntreatment in vacuum (150 °C for 1 hr) ( W3, W1+3) suggest ing that the majority of H \nbeing detected in the control  dataset ( W1) obtained via electropolishing and voltage \npulsing was  from within the sample itself an d not gaseous H in the chamber .  \n• Laser pulsing  (W2)  should, in principle, be avoided as it result s in enhanced ionization \nof residual gaseous H or desorption of residual H adsorbed on the cold surface of the \nspecimen , both leading to a higher overall H  level at 1  Da and 2  Da, inhibiting the \ndetection of H and/or D from the specimen ; \n• Electrochemical polishing , as well as FIB -based preparation , result in the introduction \nof H within the microstructure, to a much lesser extent using cryogenic spec imen \npreparation  (W5);  20 \n • Electrochemical charging successfully introduce d D into the microstructure , as \nconfirmed by compl ementary TDS  and observations of  a peak at 2 Da after W4 and W5. \nThe electric field conditions obtained for these workflows is not likel y to cause the \ndetection of molecular H 2 species ; more successful charging is achieved after a low \ntemperature annealing treatment to release H  and free up potential trapping sites . In W5, \nwhich captured a decomposed cementite region the highest bulk D concentration of 0.75 \nat. %  was observed.  \nTo conclude, APT can be used to analyse the distribution of H within steels but the influence \nof sample preparation and transport as well as  experimental running conditions must be taken \ninto account. We can conclude from our observations that W4 is the most appropriate workflow \nfor accurate analysis of H/D in pearlitic steel, and likely in other materials.  Only under such \noptimised experiment al conditions can the distribution of H from within the specimen be used \nto infer any microstructural information, in particular with regards to possible trapping sites, \nbearing in mind that the chemical potential of H during the electrochemical processing  is likely \nunique to these conditions and not representative of conditions faced by the material in service . \nFerrite and cementite were observed to both have a very low H solubility (bulk H concentrations \nin voltage pulsing runs, even after electrochemical  charging attempts, were typically less than \n0.1 at. %). H partitioned to the decomposed cementite regions , with some indication of a slight \ninterfacial segregation , and the local concentration of H increased linearly with C concentration.  \nOur observations  rationalise the good resistance of severely deformed pearlitic steels to \nhydrogen -embrittlement.  \n \n \n \n  21 \n Contributions  \nA.B., B.G., D.R., M.H.  designed the study . A.B. prepared specimens and performed APT . Y.L. \nand M.H. provided the pearlitic samples, helped with APT running conditions  and performed \nTEM. L.T.S. supported with charging, cryo experiments including  cryo-transfers and data \ninterpretation. B.S. performed TDS analyses. L.T.S. and O.K. contributed to discussions on \nelectrochemical pol ishing and charging. A.B. and B.G. processed the raw data, interpreted the \nAPT results, wrote routines for data extraction, prepared figures. A.B., B.G. and D.R. drafted \nthe manuscript. All authors discussed the results, had input and commented on the manu script.  \nAcknowledgements  \nAB and BS are  grateful for funding from the AvH Stiftung. MH acknowledges financial \nsupport from the German Federal Ministery for Research BMBF  through grant 03SF0535 . \nBG and LTS acknowledge financial support from the ERC -CoG-SHINE -771602.  The APT \ngroup at  MPIE is grateful to the Max-Planck Gesellschaft and the BMBF for the funding of \nthe Laplace Project. The authors thank Dr. H. Yarita, from Suzuki Metal Industry Co., Ltd., \nfor providing the cold drawn specimens.  Yanhong Chang, Uwe Tezins, Andreas Sturm  and \nChristian Bross are thanked for technical support with cryogenic transfer experiments. \nWaldemar Krieger, Daniel Haley,  Eason Chen  are thanked for help advising on  \nelectrochemical Deuterium charging experiments  and inter pretation of the electrochemical \nreaction occurring during charging. 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Lee, Hydrogen Delayed \nFracture Properties and Internal Hydrogen Behavior of a Fe –18Mn –1.5Al –0.6C TWIP \nSteel , ISIJ Int. 49 (2009) 1952 –1959. doi:10.2355/isijinternational.49.1952.  \n[48] D. Hirakami, T. Manabe, K. Ushioda, K. Noguchi, K. Takai, Y. Hata, et al., Effect of \naging treatment on hydrogen embrittlement of drawn pearlitic steel wire, ISIJ Int. 56 \n(2016) 8 93–898. doi:10.2355/isijinternational.ISIJINT -2015 -735. \n[49] S.H. Yu, S.M. Lee, S. Lee, J.H. Nam, J.S. Lee, C.M. Bae, et al., Effects of lamellar \nstructure on tensile properties and resistance to hydrogen embrittlement of pearlitic steel, \nActa Mater. 172 ( 2019) 92 –101. doi:10.1016/j.actamat.2019.04.040.  \n[50] A.H.M. Krom, A.D. Bakker, Hydrogen trapping models in steel, Metall. Mater. Trans. \nB Process Metall. Mater. Process. Sci. 31 (2000) 1475 –1482. doi:10.1007/s11663 -000-\n0032 -0. \n \n \n  25 \n  \n \n \n \n  26 \n Supplementary Information  \nExperiment inventory (note : experiments starting with R5096 were ran on a Cameca LEAP  \n5000 XR instrument  with reflectron  while experiments starting with R5076  were ran on a \nCameca LEAP 5000 XS with straight flight path , both at th e Max Planck Institute f ür \nEisenforschung ).  \n \nWorkflow  RunID  Ions \nreconstructed  Temp \n(K) Pulse \nfraction \n(%) Laser \npulse \nenergy \n(pJ) Pulse \nrate \n(kHz)  \n1 R5096_33494  1.48e8  60 20 N/A 200 \n2 R5076_33083  7.4e7  60 N/A 40 250 \n3 R5096_35040  9.5e7  60 15 N/A 200 \n4 R5096_33324  2.8e7  60 20 N/A 200 \n5 R5096_35122  5.9e6  60 15 N/A 200 \n6 R5096_37265  3.3e7  60 15 N/A 200 \n7 R5096_33582  6.9e6  60 20 N/A 200 \n1+3 (1)  R5096_37605  5.7e6  60 15 N/A 200 \n1+3 (2)  R5096_37611  8.8e6  60 15 N/A 200 \n \n \nFigure 9: C concentration vs. relative field ( 12C2+/ 12C1+) for each of the datasets from the different \nworkflows.  \n27 \n  \nMass spectrums  \n \n \n \n  \n28 \n  \n \n  \n29 \n  \n"    },    {        "title": "1309.2792v1.Fano_resonances_of_microwave_structures_with_embedded_magneto_dipolar_quantum_dots.pdf",        "content": "Fano resonances of microwave  structures with embedded \nmagneto-dipolar quantum dots \n \nE.O. Kamenetskii, G. Vaisman, and R. Shavit   \n \nMicrowave Magnetic Laboratory,  \nDepartment of Electrical and Computer Engineering, \nBen Gurion University of the Negev, Beer Sheva, Israel \n \nAugust 19, 2013 \n Abstract \n Long range dipole-dipole correlation in a ferroma gnetic sample can be treated in terms of \ncollective excitations of the system as a whole. Ferrite samples with linear dimensions smaller \nthan the dephasing length, but still much larger than the exchange-interaction scales are \nmesoscopic structures. Recently, it was shown that mesoscopic quasi-2D ferrite disks, distinguishing by multiresonance magneto-dipo lar-mode (MDM) spectra, demonstrate unique \nproperties of artificial atomic structures: ener gy eigenstates, eigen power-flow vortices and \neigen helicity parameters. Because of these properties, MDMs in a ferrite disk enable the \nconfinement of microwave radiation to subwavelength scales. In microwave structures with \nembedded MDM ferrite samples, one can obs erve quantized fields with topologically \ndistinctive characteristics. The use of a quasi- 2D ferrite-disk scatterer with internal MDM \nresonance spectra along the channel propagation direction could change the transmission \ndramatically. In this paper, we show that interaction of the MDM ferrite particle with its \nenvironment has a deep analogy with the Fano-resonance interference observed in natural and \nartificial atomic structures. We characterize the observed effect as Fano-resonance interference in MDM quantum dots. \n \nPACS numbers: 42.25.Fx, 76.50.+g, 78.70.Gq \n I. INTRODUCTION \n \nBeing originated in atomic physics [1], Fano resonances have become one of the most \nappealing phenomena in semiconductor quantum dots [2 – 4], different photonic devices [5 – \n9], and microwave structures [10 – 12]. An inte rest in observing and analyzing Fano profiles is \ndriven by their high sensitivity to the details of the scattering process. Since Fano parameters \nreveal the presence and the nature of different (resonant and nonresonant) pathways, they can \nbe used to determine the degree of coherence in  the scattering device.  Decoherence converts \nFano resonances into the limiting cases of the Breit-Wigner distribution (or Lorentz) \ndistribution. In these limiting cases, the Fano-resonance parameter q becomes equal to \nq \nor 0q. \n    The use of a finite-size scatterer with  internal resonance spectra along the channel \npropagation direction would change the transmissi on dramatically. In microwaves, the effects \nof short-range interactions between discrete eigenstates and the continuum have not been \nstudied sufficiently. In this connection, study of the effects observed in microwave structures \nwith embedded thin-film ferrite-disk particles ma y be of particular interest. Unique resonance \nproperties of these structures have long been kno wn, since experimental studies by Dillon [13] \nand Yukawa and Abe [14]. Recently, it was shown that such ferrite samples with a reduced \ndimensionality, distinguishing by multiresonance magneto-dipolar-mode (MDM) [or  2magnetostatic-mode (MS-mode)] spectra, bring in to play new effects which, being described \nbased on the quantized picture, demonstrate unique  properties of artificial atomic structures. \nThe MDM oscillations are characterized by ener gy eigenstates, eigen power-flow vortices and \neigen helicity parameters [15 – 21]. These os cillations can be observed as the frequency-\ndomain spectrum at a constant bias magnetic fields or as the magnetic-field-domain spectrum \nat a constant frequency. For electromagnetic waves irradiating a quasi-2D MDM disk, this small ferrite sample appears as a topological defect with time symmetry breaking.  \n    Long radiative lifetimes of MDMs combine strong subwavelength confinement of \nelectromagnetic energy with a narrow spectral line width and may carry the signature of Fano \nresonances. To the best of our knowledge, firs t experimental evidence of the Fano-resonance \nspectra in microwave structures with MDM ferrite-disk particles has been given recently, in Ref. [22]. In the present paper, we show that interaction of the MDM ferrite particle with its \nenvironment has a deep analogy with the Fano-resonance interference observed in natural [1] \nand artificial [2 – 4] atomic structures. In such a sense, we can characterize the observed effect \nas Fano-resonance interference in MDM quantum dots. Together with fundamental properties \nof this interaction, distinguishing by the time and space symmetry breakings, novel applications are very attractive. Perhaps th e most straightforward application of Fano \nresonances in MDM structures may concern the development of microwave sensors for \nchemical and biological objects with chiral properties [21, 23].  \n    The paper is organized as follows. In Sec. II, we analyze the main physical origin of energy \nquantization of MDMs in a ferrite disk. We show that such quantization may arise from \nsymmetry breaking for Maxwell electrodynamics. The quantized states of microwave fields in \na cavity with an enclosed MDM disk are st udied experimentally in Sec. III. The Fano-\nresonance phenomena in different microwave structure originated from MDM oscillations are shown in Sec. IV. In Sec. V, we discuss our findings and give conclusion on our experimental \nresults. \n \nII. ENEGRY QUANTIZATION OF MAGNETO-DIPOLAR MODES IN A FERRITE \nDISK  \nA. Quasistatic oscillations: Symmetry breaking for Maxwell electrodynamics \n \nSymmetry principles play an important role with respect to the laws of nature. Faraday’s law \ngives evidence for existence of a magnetic displacement current. To put into symmetrical shape the equations coupling together the electric and ma gnetic fields, Maxwell introduced an electric \ndisplacement current. Such an additive, introduced for reasons of symmetry, resulted in \nappearing a unified field: electromagnetic field. The dual symmetry between electric and \nmagnetic fields underlies the conservation of energy and momentum for electromagnetic fields \n[24]. What kinds of the time-varying fields can one expect to see when any (magnetic or electric) of these displacement currents is neglected and so the electric-magnetic field symmetry is broken \nin Maxwell equations? It is well known that in  a general case of small (compared to the free-\nspace electromagnetic-wave wavelength) samples made of media with strong temporal \ndispersion, the role of displacement currents in  Maxwell equations can be negligibly small and \nthe oscillating fields are quasist ationary fields [25]. For a case  of plasmonic (electrostatic) \nresonances in small metallic particles, one negl ects a magnetic displacement current and so has \nquasistationary electric fields. A dual situation is demonstrated for magneto-dipolar \n(magnetostatic) resonances in small ferrite particles, where one neglects an electric displacement \ncurrent and has quasistationary magnetic fields. As an appropriate approach for description of \nquasistatic oscillations in small particles, one can use a classical formalism where the material \nlinear response at frequency \n is described by a local bulk dielectric function – the permittivity  3tensor () – or by a local bulk magnetic function – the permeability tensor  (). With such \nan approach (and in neglect of a corresponding displacement current) one can introduce a notion \nof a scalar potential: an electrostatic potential  for electrostatic resonances (with the electric \nfield E\n) and a magnetostatic potential  for magnetostatic re sonances (with the \nmagnetic field H\n). It is evident that these potentials do not have the same physical \nmeaning as in the problems of \"pure\" (non-time-v arying) electrostatic and magnetostatic fields \n[24, 25]. Because of the resonant behaviors of small objects [confinement phenomena plus \ntemporal-dispersion conditions of tensors () or ()], one has scalar  wave functions: an \nelectrostatic-potential wave function (,)rt and a magnetostatic-potential wave function (,)rt, \nrespectively. The main note is that since we are on a level of the continuum description of media \n[based on tensors () or  ()], the boundary conditions for quasistatic oscillations should be \nimposed on scalar wave functions (,)rt or (,)rt and their derivatives, but not on the RF \nfunctions of polarization (plasmons) or magnetization (magnons). It means that in \nphenomenological models based on the effective-medium [the ()- or  ()- continuum] \ndescription, no electron-motion equations and boundary conditions corresponding to these \nequations are used.     In the spectral analysis of nanoparticle elec trostatic resonances, it was pointed out that for \nsuch oscillations one has a non-Hermitian eigenvalue problem with bi-orthogonal (instead of \nregular-orthogonal) eigenfunctions [26]. It mean s that electrostatic (plasmonic) resonance \nexcitations, existing for particle sizes significantly smaller than the free-space electromagnetic wavelength, can be described by the \nevanescent-wave  electrostatic-potential functions (,)rt. No \nretardation effects are presumed in such a description. The eigenvalue problem for magnetostatic \nresonances in subwavelength-size ferrite particle s looks quite different. A distinctive feature of \nMS resonances in small ferrite samples (in comp arison with electrostatic resonances in small \nmetal particles) is the fact that because of the bias-field induced anisotropy in a ferrite one may \nobtain the real-eigenvalue spectra for propagating-wave  scalar functions (,)rt. Such a regular \nmultiresonance spectrum in a quasi-2D ferrite disk , observed initially in experimental studies in \nRefs. [13, 14], was described later analytically as a quasi-Hermitian eigenvalue-problem solution \nfor oscillating  scalar functions (,)rt with energy eigenstates [15 –17]. This solution presumes \nexistence of non-electromagnetic  retardation effects in small ferrite samples.  \n \nB. Energy eigenstates of MDM oscillations  \n \nLong range dipole-dipole correlation in position of  electron spins in a ferromagnetic sample \ncan be treated in terms of collective excitations of the system as a whole. If the sample is \nsufficiently small so that the dephasing length phL of the magnetic dipole-dipole interaction \nexceeds the sample size, this interaction is non-local on the scale of phL. This is a feature of \nmesoscopic ferrite samples, i.e., samples with linear dimensions smaller than phL but still \nmuch larger than the exchange-interaction scales. \n    In a case of a quasi-2D ferrite disk, the quantized forms of these collective matter oscillations – magneto-dipolar ma gnons – were found to be quasiparticles with both wave-like \nand particle-like behavior, as expected for quantum excitations. The magnon motion in this \nsystem is quantized in the direction perpendicular to the plane. The oscillations are tailored by \na cylinder surface to form a sample, referred to as a magneto-dipolar quantum dot. The MDM \noscillations in a ferrite disk, analyzed as spectr al solutions for the MS-potential wave functions \n(,)rt, has evident quantum-like attributes [15 – 17]. For disk geometry, the energy-eigenstate  4oscillations are described by a two-dimensional (with respect to in-plane coordinates of a disk) \ndifferential G operator: \n \n                                                                    2 ˆ\n16qgG   ,                                                     (1)                          \n \nwhere 2\n is the two-dimensional Laplace operator,  is a diagonal component of the \npermeability tensor, and qg is a dimensional normalization coefficient for mode q. Operator \nˆG is positive definite for negative quantities . The normalized average (on the RF period) \ndensity of accumulated magnetic energy of mode q is determined as \n \n                                                                    2\n16  qq\nqzgE  ,                                                  (2) \n \nwhere \nqz is the propagation constant of mode q along the disk axis z. The energy eigenvalue \nproblem is defined by the differential equation: \n \n                                                                    ˆ\nqq qGE ,                                                          (3) \n \nwhere q is a dimensionless membrane (“in-plane”) MS-potential wave function. At a constant \nfrequency, the energy orthonormality for MDMs in a ferrite disk is written as: \n  \n                                                              ( ) 0qq q q\nSEE d S \n  ,                                               (4)                           \n \nwhere S is a cylindrical cross section of a ferrite disk. One has different mode energies at \ndifferent quantities of a bias magnetic field. From  the principle of superposition of states, it \nfollows that wave functions q ( 1,2,...q ), describing our quantum system, are vectors in an \nabstract space of an infinite number of dimensions – the Hilbert space. In quantum mechanics, \nthis is the case of so-called en ergetic representation, when the system energy runs through a \ndiscrete sequence of values. In the energetic re presentation, a square of a modulus of the wave \nfunction defines probability to find a system with a certain energy value. In our case, scalar-\nwave membrane function  can be represented as  \n \n                                                                     qq\nqa                                                           (5)                          \n \nand the probability to find a system in a certain state q is defined as \n \n                                                                 2\n2* qq\nSad S .                                                    (6)                           \n  \n    The statement that confinement phenomena for MS oscillations in a normally magnetized \nferrite disk demonstrate typical atomic-like prop erties of discrete energy levels can be well \nillustrated by an analysis of the experimental absorption spectra in Refs. [13, 14] obtained at a  5varying bias magnetic field and a constant operating frequency. The main feature of the multi-\nresonance line spectra in Refs. [13, 14] is the fact that high-order peaks correspond to lower \nquantities of the bias magnetic field. Phys ically, the situation looks as follows. Let \n() ()\n00 and  A BHH  be, respectively, the upper and lowe r values of a bias magnetic field \ncorresponding to the borders of a spectral region . We can estimate a total depth of a “potential \nwell” as:  () ()\n00 0 4 AB\nABUM H H d V   , where 0M is the saturation magnetization. Let \n(1)\n0H be a bias magnetic field, corresponding to the main absorption peak in the experimental \nspectrum (() ( 1 ) ()\n00 0B AHH H ). When we put a ferrite sample into this field, we supply it with \nthe energy: (1)\n00 4 MHd V. To some extent, this is a pumping-up energy. Starting from this \nlevel, we can excite the entire spectrum from the main mode to the high-order modes. As a \nvalue of a bias magnetic field decreases, the pa rticle obtains the higher levels of negative \nenergy. One can estimate the negative energies necessary for transitions from the main level to \nupper levels. For example, to have a transition from the first level (1)\n0H to the second level \n(2)\n0H (() ( 2 ) ( 1 ) ()\n00 0 0B AHHH H ) we need the energy surplus: \n (1) (2)\n12 0 0 0 4 UM H H d V   . The situation is very resembling the increasing a negative \nenergy of the hole in semiconductors when it “moves” from the top of a valence band. In a \nclassical theory, negative-energy solutions are rejected because they cannot be reached by a \ncontinuous loss of energy. But in quantum theory, a system can jump from one energy level to \na discretely lower one; so the negative-energy solutions cannot be rejected, out of hand. When \none continuously varies the quantity of the DC field 0H, for a given quantity of frequency , \none sees a discrete set of absorption peaks. It means that one has the discrete-set levels of \npotential energy. The line spectra appear due to  the quantum-like transitions between energy \nlevels of a ferrite disk-form particle. As a quantitative characteristic of permitted quantum \ntransitions, there is the probability, which define the intensities of spectral lines. The discrete nature of the MS-magnon states requires a mini mum of energy to excite a MS magnon, which \nis equivalent to having an energy gap. There are energy gap scales with the bias magnetic field \nat a given operating frequency. In paper [27], it was shown that because of the discrete energy \neigenstates of MDM oscillations resulting from st ructural confinement in a ferrite disk, one can \ndescribe the oscillating system as collective moti on of quasiparticles – the “light magnons”.  \n    From Eqs. (1) – (3) it follows that MDM re sonances in a ferrite disk correspond to discrete \nquantities of the permeability-tensor component \n. This component is defined as [28]  \n \n                                                           2\n00\n22 2\n01MH\nH,                                                         (7) \n \nwhere  is the gyromagnetic ratio. It is evident that discrete energy eigenstates of MDM \noscillations one can obtain also at a varying opera ting frequency and a constant bias magnetic \nfield. So, for given disk sizes and a give n quantity of saturation magnetization 0M, there are \ntwo different mechanisms of energy quanti zation: (i) quantization by a bias field 0H at a \nconstant signal frequency  and (ii) quantization by signal frequency  at a constant bias \nfield 0H. Let us consider a certain frequency f. For this frequency, there is a specific set of \nthe bias-field quantities for observation of the energy quantization levels: (1)\n0H, (2)\n0H,  6(3)\n0H, … On the other hand, for a given bias magnetic field, there is a specific set of the \nfrequency quantization levels. Fig. 1 illustrates correlation between the two mechanisms of \nenergy quantization. It becomes evident that there should exist a certain uncertainty limit \nstating that  \n \n                                                           0 fH uncertainty limit .                                            (8) \n \nThe uncertainty limit is a constant which depends on the disk size parameters and ferrite \nmaterial properties. It is evident that beyond the frames of the uncertainty limit (8), one has continuum of energy. The fact that there are different mechanisms of energy quantization gives \nus possibility to conclude that for MDM oscillations in a quasi-2D ferrite disk one can have \ndiscrete energy eigenstates as well as continuum of energy.  \n    It is worth noting that for different types of subwavelength particles, the uncertainty \nprinciple may acquire different forms. An inte resting variant of Heisenberg’s uncertainty \nprinciple was shown recently in subwavelength optics [29]. Applied to the optical field, this \nprinciple says that we can only measure the electr ic or the magnetic field with accuracy when \nthe volume in which they are contained is significantly smaller than the wavelength of light in \nall three spatial dimensions. As volumes smaller than the wavelength are probed, \nmeasurements of optical energy become uncertain, highlighting the difficulty with performing measurements in this regime.  \n     \nIII. QUANTIZED STATES OF THE CAVITY FIELDS ORIGINATED FROM MDM \nOSCILLATIONS \n  \nThe above analysis of energy eigenstates gives possibility for deeper understanding of the \nnature of the experimentally observed quantized fields in microwave structures with embedded \nferrite samples. As we will show, there are the fields with quantized to pological states. In our \nexperiments, we analyze the multiresonance spectrum of microwave oscillations in a \nmicrowave cavity originated from a MDM ferrite disk. The spectrum is obtained by varying a \nbias magnetic field at a constant operating fr equency, which is a resonant frequency of the \ncavity. For our studies, we use a disk sample of diameter mm 3 2 made of the yttrium iron \ngarnet (YIG) film on the gadolinium gallium ga rnet (GGG) substrate (the YIG film thickness \nmkm 6.49d , saturation magnetization G  1880  40M , linewidth Oe 8.0H ; the GGG \nsubstrate thickness is 0.5 mm). A normally magnetized ferrite-disk sample is placed in a \nrectangular waveguide cavity with the 102TE  resonant mode. The disk axis is oriented along the \nwaveguide E-field and the disk position is in a maximum of the RF magnetic field of the cavity \n[see Fig. 2 (a)]. Fig. 2 (b) shows an experimental multiresonance spectrum of modulus of the \nreflection coefficient obtained by varying a bias ma gnetic field and at a resonant frequency of   \n07.4731f GHz . The resonance modes are designated in succession by numbers n = 1, 2, 3, \n… The states beyond resonances we designate with small letters a, b, c, … \n    The shown multiresonance spectrum is, certainly, related to resonant variations of input \nimpedances of a cavity. In Fig. 3 (a) we show an equivalent electric circuit of our experimental \nsetup: A source with internal impedance 0Z  supplies a cavity with an embedded ferrite disk by \nmicrowave energy of frequency 0f. A load impedance , LZ, – an input impedance of a cavity – \nacquires discrete complex values  with variation of an external parameter – a bias magnetic field \n0H. Quantization of the cavity impedances due to MDM resonances of a ferrite particle can be \nwell illustrated by a Smith chart – a complex-plane nomogram designed for graphical display  7of impedance multiple parameters [30]. Based on our experimental studies we can obtain the      \nSmith-chart positions of complex impedances LZ corresponding to quantized states of the \ncavity fields originated from MDM oscillations. Fig. 3 (b) shows experimental results of the \nquantized-state impedances for modes 1 and 2 plotted on the complex -reflection-coefficient \nplane [31]. For an entire spectrum, impedances LZ are plotted schematically in Fig. 3 (c) as a \nset of circles on the complex-reflection-coefficient plane. The resonances are designated by \nnumbers while the states beyond resonances are designated by letters. At the peak-to-peak \nvariation, a reactive part of impedance LZ “oscillates” with changing a sign. Evidently, there \nexist quantized states with pure active quantities of the cavity impedance [see red dots in Fig. 3 \n(c)]. \n    Since we are working at a certain resonant frequency, the shown resonances are not the \nmodes due to quantization of the photon wave vect or in a cavity. So the question arises: What \nis the nature of the modes observed a cavity at a constant frequency? It is evident that the \ndiscrete variation of the cavity impedances and so the discrete states of the cavity fields are \ncaused by the discrete variation of energy of a ferrite disk, appearing due to an external source \nof energy – a bias magnetic field. Let us consider initially our microwave system at a quantity \nof a bias magnetic field above the 1st peak in the resonance spectrum. We designate this state \nby a capital letter A [see Fig. 2 (b)]. The corresponding bi as magnetic field, designated as \n()\n0AH , supplies a ferrite disk by energy: ()\n00 4 A\nAUM H d V. At this bias magnetic field, a \ncavity (with an embedded ferrite disk) ha s good impedance matching and can accumulate \ncertain RF energy. When we consider the state a (the state beyond resonances 1 and 2), a \ncavity has the same good impedance matching and the same level of accumulated RF energy. \nBut the energy supplied to a ferrite disk by a bias magnetic field is reduced by a quantity \n () ( )\n00 0 4Aa\nAa A aUU U M H H d V       . At a very narrow region of a bias magnetic field \ncorresponding to the 1st resonance-peak position, (1)\n0H, RF energy accumulated in the cavity is \nstrongly reduced because of increasing of the activ e-quantity cavity impedance [see Fig. 3 (c)]. \nThis reduction of the RF energy (designated as (1)\nRFu) must be equal in magnitude to quantity \nAaU . Such kind of relation between magnetic energy of a disk and RF energy of a cavity is \nexhibited also for other peaks in a spectrum. For the entire spectrum, in Fig. 4 we give \nqualitative pictures of potential energy of a ferrite disk and discrete states of the RF energy \naccumulated in the cavity with respect to a bias magnetic field. These states are shown in \ncorrelation with the spectral picture for the refl ection coefficient. From peak to peak one has \ndiscrete-portion reduction of the disk magnetic energy. Due to such a discrete-portion \nreduction of the disk magnetic energy we obser ve excitation of the RF resonance peaks.  \n    The quantized states of the cavity fields  are topological-state resonances. For understanding \nthe physical nature of these resonances, a more detailed study of analytical models for MDM \noscillations is necessary. For analytical soluti ons of the MDM spectral problem, two analytical \nmodels have been developed. There are the models based on so-called the G- and L-mode \nsolutions [15 – 21]. The G-modes are associated with a hermitian Hamiltonian for MS-\npotential wave functions ( , ) rt. These modes are related to the discrete energy states of \nMDMs, which we considered above in Sec. II of the paper. For the L-modes, one has a \ncomplex Hamiltonian for MS-potential wave functions ( , ) rt. For eigenfunctions associated \nwith a complex Hamiltonian, we have nonzero Berry potential (meaning the presence of \ngeometric phases). The main difference between the G- and L-mode solutions becomes clear \nwhen one considers the boundary conditions on a la teral surface of a ferrite  disk. In solving the \nenergy-eigenstate spectral problem for the G-mode states, the boundary condition on a lateral \nsurface of a ferrite disk, are expressed as   8 \n                                                            0\nrrrr\n      ,                                       (9) \n \nwhere ~ is the MS-potential membrane wave function (for the and G-mode solution)  is a \nradius of a ferrite disk. This boundary condition,  however, manifests itself in contradictions \nwith the electromagnetic boundary condition for the radial component of B\n on a lateral surface \nof a ferrite disk. Such a boundary condition, used in solving the spectral problem for the L-\nmode states, is written as  \n \n                                         ( ) ( ) ( )rr a rrrHH i H       ,                                            (10) \n \nwhere ()rH is the azimuth magnetic field on the border circle. In the magnetostatic \ndescription, this equation appears as  \n \n                                                  a r\nrrrr  \n\n       ,                               (11) \n \nwhere  is the MS-potential membrane wave function (for the and L-mode solution),  is the \nazimuth wave number, and a is a off-diagonal component of the permeability tensor. The \nspectral-problem solutions based on Eq. (9) are single-valued-function solutions. At the same \ntime, the spectral-problem solutions based on Eq . (11) are nonsingle-va lued-function solutions. \nBecause of dependence of the right-hand side of Eq. (11) on a sign of the azimuth wave \nnumber, the two (clock and counterclockwise) types of solutions exist. In the measurement, we \ndo not distinguish such clockwise or countercloc kwise types of oscillations. Microwave signals \nmeasured at the cavity ports are single-valued f unctions. It was shown that to get real-quantity \neigenstates of the L-mode solutions, a special differential operator acting on the boundary \nconditions should be introduced. As the eigenstate s of this operator, there are topological-phase \nmagnetic currents. It was also shown that due to these currents one can also satisfy the \nelectromagnetic boundary condition for the radial component of B\n in the G-mode solution. In \nfact, topological-phase magnetic currents may couple the G- and L-mode spectral solutions. \nThese magnetic currents result in appearance of fluxe s of gauge electric fields [17, 19 – 21].     \n        For L-mode solutions, there are no properties of parity and time-reversal invariance [19, \n20]. Because of these topologically distinguished properties, cavity-field oscillations are \nLorentz-violation solutions [32, 33] . Scalar wave functions of the L-mode solutions exhibit \nproperties of pseudo-scalar axi on-like fields [21, 34]. Axion electrodynamics is the standard \nelectrodynamics modified by an additional field – the axion field. This provides a theoretical \nframework for description of a possible violat ion of Lorentz invariance. Whenever a pseudo-\nscalar axion field  is introduced in the theory, the dual symmetry of electromagnetic fields is \nspontaneously and explicitly broken. An axion-electrodynamics term, added to the ordinary \nMaxwell Lagrangian [35]: \n \n                                                                EB\n ,                                                               (12)                    \n \nwhere  is a coupling constant, results in modified electrodynamics equations with the electric \ncharge and current densities replaced by [35, 36]   9                             () ()eeB    \n     and    () ()eej jB Et     \n.                (13) \n \nIntegrating Eq. (12) over a cl osed space-time with periodic boundary conditions, we obtain the \nquantization \n \n                                                                   4Sd x n  ,                                                  (14) \n \nwhere n is an integer. It is evident that S is a topological term. While S generically breaks the \nparity and time-reversal symmetry, both symmetries are intact at 0 and . The field  \nitself is gauge dependent. An axion-electrody namics term, added to the ordinary Maxwell \nLagrangian gives the Lorentz-violating extension of the Maxwell equations of the environmental \nRF electromagnetic fields.  \n    Together with the axion model, the interaction of the L-mode solutions with the cavity \nelectromagnetic fields can be analyzed as coupling between two types of vector fields:  the \nelectromagnetic (EM) and so-called magnetoelectric (ME) fields. The fields originated from the \nL-mode solutions are the states with specific spin and orbital rotational motion the field vectors \nand are characterized by eigen power-flow vortic es and helicity parameters [17 – 21, 34]. We \ncall these fields magnetoelectric (ME) fields. The ME fields (which are well described \nnumerically by the HFSS-program simulation) give evidence for spontaneous symmetry \nbreakings. Because of rotations of localized field configurations in a fixed observer inertial frame, coupling between EM and ME fields cause violation of the Lorentz symmetry of \nspacetime [32]. In such a sense, ME fields can be considered as Lorentz-violating extension of \nthe Maxwell equations. To characterize the ME-field singularities, the helicity parameter was \nintroduced.  In Refs. [21, 34], it was shown that in vacuum regions of a microwave structure with \nan embedded MDM ferrite disk, nonzero quant ities of a normalized helicity parameter \n \n                                                          *Im\ncosEE\nEE \n\n\n                                                (15) \n \ncan exist. When this parameter is not equal to zero, a space angle between the vectors E\n and \nE\n is not equal to 90. This is an evident violation of the Maxwell electrodynamics.  \n    One becomes evident now with the fact that due to properties of MDM oscillations, the \nfields of the cavity with an embedded ferrite disk  are characterized not only by discrete energy \nlevels, but also by specific topological egenstates. While the cavity fields related to the G-mode \nsolutions are the potential-energy eigenstates, the cavity fields related the L-mode solutions can \nbe conventionally characterized as kineti c-energy eigenstates. In Fig. 4, the a, b, c, … states are \npotential-energy states related to the G-modes. At these states, the disk virtually does not \ninteract with the cavity RF field, but accept energy from a bias-field source. The 1, 2, 3, … states can be characterized as kinetic-energy states, which are related to the L-modes. There are \ncoupled states of the ME (ferrite disk) and EM  (cavity) fields. Based on the HFSS-program \nnumerical solutions, we can observe the topological structures of the quantized states of the \ncavity fields at the 1, 2, 3, … resonances. In Figs. 5, 6 we show such topological structures for \nthe 1\nst and 2nd resonance modes. There are the pictures of the helicity-parameter distributions. \nThese pictures give evidence for an important fact that the quantized states of the fields are \nstrongly distinguished topologically. The regions where the helicity parameter, cos, is not  10exactly equal to zero are, in fact, the regions where the EM-ME field interaction takes place. \nMaximal quantities of cos corresponds to points or lines of phase singularities.  \n    The topological properties of the fields can be described by the current *Im( )j , \nwhich is a real vector field, analogous to local momentum, satisfying the continuity equation. It \nis the probability current density if ()r is a quantum mechanical wavefunction (the local \nexpectation value of momentum), and the Poynting vector in scalar theories of light [37]. In \nour case of MDM oscillations, the current j is the power-flow vector in vacuum for L-mode \ncomplex MS-potential scalar wave functions (,)rt[17, 19, 21]. In the HFSS solutions, the \ncurrent j is the power-flow of coupled  EM-ME fields [18, 20, 21]. In Fig. 7, we show such \ntopological structures of the power-flow  distributions in a cavity for the 1st and 2nd resonance \nmodes. The pictures were ob tained based on the HFSS nume rical solutions. The observed \ndiscrete topological states of the cavity fields are the closed-loop resonances. In the near-field \nregion, twisting excitations in vacuum form 2D closed orbits of the power-flow current \n* *\n16ij   . There are the power-flow vortices. A center of this closed orbit is \na point of a phase singularity of the L-mode complex function (,)rt. The orbit is threaded by \na flux of a gauge electric field. In points (lines) where cos is maximal, a modulus of a gauge \nelectric field is maximal as well. Phase singularities in complex scalar waves are lines in three \ndimensions on which the wave intensity vanishes and around which the phase changes by  \n(in our case, because of TRS breaking) times an integer (the strength of the singularity). For \nany wave in space, the set of its phase-singularity lines is a skeleton, supporting the phase \nstructure of the whole field. In general, the phase-singularity lines can be straight or curved, or \nform closed loops. In the region far from a MDM ferrite disk, one has 3D closed loops of \ncurrent j. It can be assumed that in the far-field region the knots of current j exist. It \nbecomes apparent that at the MDM resonances, the fields in a cavity are organized into \ntopologically independent loops. What are the forms of these helical-branch loops? Probably, \nthere are double-helical structures considered in Ref. [19]. What are topological invariants of \nthese double-helical configur ations? The answer on this question is unclear now. \n    It is worth noting that no topologically distinctive pictures of the cavity field structures are \nobserved at the a, b, c, … states. Since the G-mode and L-mode states are alternated, one can \nsay that topologically distinctive structures of th e 1, 2, 3, … resonances are also distinguished \nby the levels of magnetic potential energy.  \n \nIV. FANO RESONANCES IN A MICROWAVE CAVITY AND A MICROSTRIP \nSTRUCTURE WITH A MDM FERRITE DISK \n \nThe MDM excitations of a quasi-2D ferrite disk are strongly coupled to microwave fields. In a \nmicrowave cavity with an enclosed MDM ferrite disk, the disk acts as a key with discrete states which are switched by a bias magnetic field. Due to such a key, the supply of RF energy into a \ncavity by an external microwave source is quantized. A spectrum of MDM quantization in a \nsmall ferrite-disk particle as a function of a bias  magnetic field is very akin to a spectrum of \ncharge quantization in a small metallic or semiconductor particle (quantum dots) as a function \nof a bias voltage shown in Refs. [38, 39]. Fo llowing our previous discussions as well as \ndiscussions in Refs. [38, 39], we can say that both of them are artificial-atom spectra with \nenergy eigenstates. When a MDM disk is embe dded into a microwave structure, one has a \ndynamic system with mixed phase space. That is the phase space which comprises chaotic as  11well as regular regions. We have a stable-motion Hamiltonian system.  Extension of chaotic \ntrajectories is limited only by energy conservation. \n    In microwave scattering by a MDM ferrite particle one can observe the Fano-interference \neffects. There are the mesoscopic microwave phenomena. Let us consider the same structure as \nin Sec. III, that is, a rectangular  waveguide cavity operating at the 102TE  resonant mode with \nan embedded normally magnetized MDM ferrite disk. We will analyze the spectra obtained by \nvarying a bias magnetic field at constant operat ing frequencies, but contrary to experiment in \nSec. III, there will be experiments at frequencie s different from the cavi ty resonance frequency. \nOn the cavity resonant characteristics, thes e frequencies are shown in Fig. 8. A pair of \nfrequencies 1f and 2f as well as a pair of frequencies3f and 4f correspond to a certain level \nof the cavity reflection coefficient. At a non -resonant frequency, we have a wave-propagation \nbehavior in a cavity resulting in a certain phase difference between the forward and backward \nwaves. For frequencies 1f and 2f (as well as for frequencies 3f and 4f) these phase \ndifferences are of opposite signs.  \n    The Fano-type spectra of MDM quantization as  a function of a bias magnetic field shown in \nFigs. 9 and 10 correspond to frequencies 1f and 2f, respectively. In Figs. 11 and 12, one can \nsee such spectra at frequencies 3f and 4f, respectively. The Fano regime emerges when \ntunneling between a ferrite disk and a cavity ta kes place. An interaction is considered via \nevanescent exponential tails of eigen-wave-functi ons localized inside a ferrite disk and is \ndescribed by the overlap integral. The phase rela tion between forward and backward waves of \nthis radiation strongly influence on behavior of the Fano interference. In the Fano effect, two \npaths of the waves from the eigenstate of a system – one direct and one mediated by a \nresonance – to a state in an energy continuum interfere to produce an asymmetric absorption spectrum. Zero absorption occurs as the wavele ngth is scanned across the resonance, at a \nphoton energy corresponding to a 180 ° phase difference between the paths. The sign of the \ninterference (constructive or destructive) betw een wave paths depends on the phase difference \nbetween the paths. The observed Fano spectra of MDM quantization in a small ferrite-disk \nparticle as a function of a bias  magnetic field are very akin to the Fano spectra of charge \nquantization in a small metallic or semiconductor  particle (quantum dots) as a function of a \nbias voltage shown in Refs. [2 – 4]. We can sa y that both of them are artificial-atom Fano \nresonances with energy eigenstates. The statisti cs of transport through ferrite quantum dots can \nbe justified for non-interacting MDMs. However, real MDMs interact with each other. One of the mechanisms of this interaction is due to the fluxes of gauge electric fields [17, 19 – 21]. \n    In a cavity, we have discrete energy eigenstates as well as continuum of energy. The regular \nspectrum of the cavity quantized states is do minated by the closed loops of the power-flow-\nvector current in vacuum. In experiments, we see the eigenstates related to these loops. What \ncould be the nature of continuum chaotic trajectories in our structure? It is well known that in a \ncase of a lossless microwave resonator with non-gyrotropic media, the Maxwell-equation \nsolutions for the \nE\n and H\n fields have real amplitude coefficients. This corresponds to \nstanding-wave oscillations inside a cavity. Wh en, however, a microwave resonator contains an \nenclosed gyrotropic-medium sample, the electromagnetic-field eigenfunctions will be complex, \neven in the absent of dissipative losses. It means that electromagnetic fields of eigen \noscillations are not the fields of standing waves in  spite of the fact that the resonance frequency \nof a cavity with gyrotropic-medium samples is real [28]. In a case of such inclusions acting in \nthe proximity of the ferromagnetic resonance of  a ferrite material, the phase of the wave \nreflected from the ferrite boundary depends on the direction of the incident wave. This fact, \narising from special boundary conditions for the tangential components of the fields on the \ndielectric-ferrite interface, leads to the time-reversal symmetry breaking effect in microwave  12resonators with inserted  ferrite samples [40 – 42]. In genera l, microwave resonators with the \ntime-reversal symmetry breaking effects give an example of a nonintegrable system. The \nconcept of nonintegrable, i.e. path-dependent, ph ase factors is one of the fundamental concepts \nof electromagnetism. Presently, different noninte grable systems are the subject for intensive \nnumerical and experimental studies in microwave and optical resonator systems [40 – 45]. A \ndisk-shaped ferrite sample placed in a rectangular-waveguide cavity is a microwave billiard with the time-reversal-symmetry breaking. Due to the time-reversal symmetry breaking, in a \nmicrowave billiard with enclosed ferrite samples one may not observe single isolated \nresonances, but two coalescent resonances. There exists the T violating matrix element of the \neffective Hamiltonian which describes the coalescent resonances in a cavity with a ferrite \ninclusion [46].      The discussed above two different mechanis ms of energy quantizat ion of MDM oscillations: \n(i) quantization due to a bias field \n0H at a constant frequency  and (ii) quantization due to \nfrequency  at a constant bias field 0H, are well illustrated in observation of the Fano spectra. \nWhen a MDM ferrite disk is placed inside an endless microwave waveguide, the Fano \ninterference spectra is obtained by varying an ope rating frequencies at a constant bias magnetic \nfield. This effect of coupling between discrete states of MDM oscillations and waveguide EM fields is well observed numerically based on the HFSS-program sulutions [18, 20, 21, 34]. In \nRef. [20] it was shown analytically that the L-mode solutions for the MS-potential wave \nfunction \n give evidence for splitting resonance corresponding to the Fano-interference peaks \n[20]. There are coalescent resonances appearing due to the time-reversal symmetry breaking.  \n    Fano resonances originated from MDM quantum dots are exhibited in different microwave \nstructures, not only in a rectangular waveguide and in a rectangular-waveguide cavity. In a microstrip structure with an embedded MDM ferrite disk (see Fig. 13), we can observe \nexperimentally the Fano spectra both by varying a bias field \n0H at a constant frequency  and \nby varying frequency  at a constant bias field 0H. These spectra are shown in Figs. 14 and \n15. Contrary to the above studies, in a case of a microstrip structure we observe the Fano \ninterference for forward propaga ting waves. In the shown spectra we can see the peaks \noriginated both from the radial and azimuthal types of MDMs [16]. In the mode designation in \nFigs. 14 and 15, the first number characterizes a number of radial variations for the MDM \nspectral solution. The second number is a number of azimuthal variations for the MDM \nspectral solution [16]. A character of the Fano interference is different for azimuthal and radial \ntypes of MDMs. It is worth noting here that MDM resonances in a microwave cavity, shown in \nFig. 2 (b) and Figs. 9 – 12 are the radial-type resonances.  \nV. DISCUSSION AND CONCLUSION  \n \nA quasi-2D ferrite disk shows a confined structure which can conserve energy and angular \nmomentum of MDM oscillations. Because of these properties, MDMs in a ferrite disk enable the confinement of microwave radiation to subwavelength scales. In a vacuum subwavelength \nregion abutting to a ferrite disk one can obs erve quantized-state power-flow vortices and \nhelicity structures of the fields. We found that  the Fano interference in a microwave structure \nwith a MDM quantum dot appears when (a) the wave reflection/transmission is characterized \nby narrow resonances corresponding to localized states trapped on a vortex of stable power-flow motion with the field helicity property and (b) chaotic motion are typical for a microwave \nbilliard with the time-reversal symmetry breaking effects (arising from an inserted ferrite \nsample). The interference between these scattering processes gives rise to a variety of Fano-\nresonance shapes. In the present study, we did not make an analysis to show that the resonance  13line shapes can be cast by an expression similar to the standard Fano formula (but with a \ncomplex q parameter). This analysis is the purpose for our future publications.   \n    Tunneling of microwave radiation into a MDM ferrite disk is due to twisting excitations. In a \nway to understanding the properties of the obser ved spectral excitations, we have to refer to \nsome known physical notions. Momentum transfer between matter and electromagnetic field is \na subject of the longstanding Abraham-Minkowski controversy [24]. Could there be quantum vacuum contribution to the angular momentums of matter? As a certain answer to this \nquestion, we refer to a recent paper [47] wh ere it was shown that a nonzero matter angular \nmomentum can be induced by quantum electromagnetic fluctuations in homogeneous \nmagnetoelectric media. Photons, like other part icles, carry energy and angular momentum. A \ncircularly polarized photon carries a spin angular momentum [24]. Also, photons can carry additional angular momentum, called orbital angular momentum. Such photons, carrying both \nspin and orbital angular momentums are called twisting photons [48]. It is worth noting here \nthat from numerical study in Ref. [49], one ca n observe twisting of light around rotating black \nholes. Twisting photons are propagating-wave be haviors. These are \"real photons\". In near-\nfield phenomena, which have subwavelength-ra nge effects, and do not radiate through space \nwith the same range-properties as do electromagnetic wave photons, the energy is carried by \nvirtual photons, not actual photons. Virtual particles conserve energy and momentum. They are \nimportant in the physics of many processes, including Casimir forces. Can virtual photons \nbehave as twisting excitation s? Our studies give evidence  for such near-field twisting \nexcitations. There are subwavelength field stru ctures with quantized energy and angular \nmomentums.  \n    Microwave radiation can potentially couple to MDM oscillations if the ferrite sample in \nwhich the MS magnons reside shows a confined structure to satisfy conservation of energy and angular momentum. In this paper, we analyzed th e main physical origin of energy quantization \nof MDMs in a ferrite disk. We showed that  such quantization may arise from symmetry \nbreaking for Maxwell electrodynamics. The quantized states of microwave fields in a cavity \nwith an enclosed MDM disk have been studied experimentally. From peak-to-peak of the \nspectrum in a cavity, we have absorption of a single such near-field tw isting excitations. To a \ncertain extent, this is similar to the effect of  absorption of one electron by a semiconductor \nquantum dot [38, 39]. While a spectrum of MDM qua ntization in a small ferrite-disk particle is \na function of a bias magnetic field, a spectrum of charge quantization in a small metallic or \nsemiconductor particle (quantum dots) is a functi on of a bias voltage [38, 39]. It is worth note \nalso that the observed topologically distinctive energy eigenstates, appearing due to scattering of the cavity fields at the quasistatic-field MDM oscillations with the spin and orbital rotational \nmotions [21, 34], are the Lorentz-violation excita tions. The Lorentz violation is associated with \nrotations and boosts of localized field configurations in a fixed observer inertial frame [32]. In \nour sense, the ME-field excitations can be considered as Lorentz-violating extension of the \nMaxwell equations.     In this paper, we showed that interaction of the MDM ferrite particle with its environment \nhas a deep analogy with the Fano-resonance in terference observed in natural and artificial \natomic structures. We characterize the observed effect as Fano-resonance interference in MDM \nquantum dots. We showed the Fano-resonance phenomena in different microwave structure \noriginated from MDM oscillations. 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Anzolin, Nature Phys. 7, 195 (2011). \n \n \n \n \n  \nFigure captions \n Fig. 1. Correlation between two mechanisms of  energy quantization: quntization by signal \nfrequency and quantization by a bias magnetic field \n \nFig. 2. MDM resonances in a microwave cavity. (a) A \n102TE -mode rectangular waveguide \ncavity with a normally magnetized ferrite-disk sample; (b) An experimental multiresonance \nspectrum of modulus of the reflection coefficien t obtained by varying a bias magnetic field and \nat a resonant frequency of 07.4731f GHz . The resonance modes are designated in succession \nby numbers n = 1, 2, 3, … The states beyond resonances we designate with small letters a, b, c,  \n \nFig. 3. Quantized variations of input impedances of a cavity at MDM resonances. (a) An \nequivalent electric circuit of an experimental se tup. (b) Experimental results of the quantized-\nstate impedances for modes 1 and 2 plotted on the complex-reflection-coefficient plane; (c) \nThe entire-spectrum impedances shown schematically as set of circles on the complex-reflection-coefficient plane. Red dots show quantized states with pure active quantities of the \ncavity impedance. \n \nFig. 4. Quantized states of RF energy in a cavity and magnetic energy in a disk. (a) RF energy \naccumulated in a cavity; (b) Magnetic energy of a ferrite disk; (c) Multiresonance spectrum of modulus of the reflection coefficient.  \n \nFig. 5. Normalized helicity parameters of the mode fields near a ferrite disk shown in the xy \ncross section. The bounds of the green-color background correspond to the waveguide cross \nsection. (a) The helicity parameter for the 1\nst mode; (b) The helicity parameter for the 2nd \nmode.  \n  16Fig. 6. Normalized helicity parameters of the mode fields near a ferrite disk shown in the yz \ncross section. (a) The helicity parameter for the 1st mode; (b) The helicity parameter for the 2nd \nmode.  \n \nFig. 7. Power-flow distributions of the mode fields shown in the xz cross section situated at a \ndistance 0.3 mm above a ferrite disk. (a) The power-flow distribution for the 1st mode; (b) The \npower-flow distribution for the 2nd mode.  \n \nFig. 8. The cavity resonant characteristics. The quantities of the frequencies: 07.4731f GHz , \n17.4575f GHz , 27.49f GHz , 37.4081f GHz , 47.6188f GHz .  \n \nFig. 9. The Fano-interference effect in a 102TE - mode waveguide cavity with an embedded \nnormally magnetized MDM ferrite disk at frequency 17.4575f GHz . The MDM resonances \nare designated by numbers n = 1, 2, 3, … \n \nFig. 10. The Fano-interference effect in a 102TE - mode waveguide cavity with an embedded \nnormally magnetized MDM ferrite disk at frequency 27.49f GHz . \n \nFig. 11. The Fano-interference effect in a 102TE - mode waveguide cavity with an embedded \nnormally magnetized MDM ferrite disk at frequency 37.4081f GHz . \n \nFig. 12. The Fano-interference effect in a 102TE - mode waveguide cavity with an embedded \nnormally magnetized MDM ferrite disk at frequency 47.6188f GHz . \n \nFig. 13. A microstrip structure with an embedded MDM ferrite disk.  \nFig. 14. The Fano-interference effect in a microwave microstrip structure obtained by varying a \nbias field at a constant frequency 7.144\nf GHz . In the mode designation, the first number \ncharacterizes a number of radial variations fo r the MDM spectral solution. The second number \nis a number of azimuthal variations  for the MDM spectral solution [16]. \n Fig. 15. The Fano-interference effect in a microwave microstrip structure obtained by varying \nsignal frequency at a constant bias field \n04015 HO e . In the mode designation, the first \nnumber characterizes a number of radial variations for the MDM spectral solution. The second \nnumber is a number of azimuthal variations for the MDM spectral solution [16]. \n \n \n \n  \n \n \n \n  \n \n  17 \n                            \n  \n \nFig. 1. Correlation between two mechanisms of  energy quantization: quntization by signal \nfrequency and quantization by a bias magnetic field.  \n \n         \n  \n                                     (a)                                                                             (b) \n \nFig. 2. MDM resonances in a microwave cavity. (a) A 102TE -mode rectangular waveguide \ncavity with a normally magnetized ferrite-disk sample; (b) An experimental multiresonance \nspectrum of modulus of the reflection coefficien t obtained by varying a bias magnetic field and \nat a resonant frequency of 07.4731f GHz . The resonance modes are designated in succession \nby numbers n = 1, 2, 3, … The states beyond resonances we designate with small letters a, b, c, \n…                                 \n  18    \n    \n  \n                       (a)                                          (b)                                                (c) \n \nFig. 3. Quantized variations of input impedances of a cavity at MDM resonances. (a) An \nequivalent electric circuit of an experimental setup. (b) Experimental results of the quantized-state impedances for modes 1 and 2 plotted on the complex-reflection-coefficient plane; (c) \nThe entire-spectrum impedances shown schematically as set of circles on the complex-\nreflection-coefficient plane. Red dots show quantized states with pure active quantities of the \ncavity impedance. \n  \n \n \nFig. 4. Quantized states of RF energy in a cavity and magnetic energy in a disk. (a) RF energy \naccumulated in a cavity; (b) Magnetic energy of a ferrite disk; (c) Multiresonance spectrum of \nmodulus of the reflection coefficient.  \n  19\n \n \nFig. 5. Normalized helicity parameters of the mode fields near a ferrite disk shown in the xy \ncross section. The bounds of the green-color background correspond to the waveguide cross section. (a) The helicity parameter for the 1\nst mode; (b) The helicity parameter for the 2nd \nmode.  \n \n \n \nFig. 6. Normalized helicity parameters of the mode fields near a ferrite disk shown in the yz \ncross section. (a) The helicity parameter for the 1st mode; (b) The helicity parameter for the 2nd \nmode.  \n \n \n \nFig. 7. Power-flow distributions of  the mode fields shown in the xz cross section situated at a \ndistance 0.3 mm above a ferrite disk. (a) The power-flow distribution for the 1st mode; (b) The \npower-flow distribution for the 2nd mode.   20 \n \n \nFig. 8. The cavity resonant characteristics. The quantities of the frequencies: 07.4731f GHz , \n17.4575f GHz , 27.49f GHz , 37.4081f GHz , 47.6188f GHz .  \n \n \n \n \nFig. 9. The Fano-interference effect in a 102TE - mode waveguide cavity with an embedded \nnormally magnetized MDM ferrite disk at frequency 17.4575f GHz . The MDM resonances \nare designated by numbers n = 1, 2, 3, … \n  21\n \n \nFig. 10. The Fano-interference effect in a 102TE - mode waveguide cavity with an embedded \nnormally magnetized MDM ferrite disk at frequency 27.49f GHz . \n \n \n \nFig. 11. The Fano-interference effect in a 102TE - mode waveguide cavity with an embedded \nnormally magnetized MDM ferrite disk at frequency 37.4081f GHz . \n \n \n  22\n \n \nFig. 12. The Fano-interference effect in a 102TE - mode waveguide cavity with an embedded \nnormally magnetized MDM ferrite disk at frequency 47.6188f GHz . \n \n \n \n \nFig. 13. A microstrip structure with an embedded MDM ferrite disk. \n  23\n \n \nFig. 14. The Fano-interference effect in a microwave microstrip structure obtained by varying a \nbias field at a constant frequency 7.144f GHz . In the mode designation, the first number \ncharacterizes a number of radial variations fo r the MDM spectral solution. The second number \nis a number of azimuthal variations  for the MDM spectral solution [16]. \n \n \n \n \nFig. 15. The Fano-interference effect in a microwave microstrip structure obtained by varying \nsignal frequency at a constant bias field 04015 HO e . In the mode designation, the first \nnumber characterizes a number of radial variations for the MDM spectral solution. The second \nnumber is a number of azimuthal variations for the MDM spectral solution [16].  \n \n "    },    {        "title": "1309.7183v1.Constrained_non_collinear_magnetism_in_disordered_Fe_and_Fe_Cr_alloys.pdf",        "content": "Joint International Conference on Supercomputing in  Nuclear Applications and Monte Carlo 2013 (SNA + M C 2013) \nLa Cité des Sciences et de l’Industrie, Paris, Fran ce, October 27m31, 2013 \n  \n \n \nConstrained nonbcollinear magnetism in disordered Fe and F ebCr alloys \n \n \nD. NguyenmManh *, PuimWai Ma, M.Yu. Lavrentiev, S.L. Dudarev  \n \nEURATOM/CCFE Fusion Association, Culham Science Cen tre, Abingdon, Oxon, OX14 3BD, United Kingdom  \n* Corresponding Author, Emmail: duc.nguyen@ccfe.ac.uk  \n \nThe development of quantitative models for radiatio n damage effects in iron, iron alloys and steels, p articularly for \nthe high temperature properties of the alloys, requ ires understanding of magnetic interactions, which control the \nphase stability of ferriticmmartensitic, ferritic, and austenitic steels. In this work, disordered mag netic configurations \nof pure iron and FemCr alloys are investigated usin g Density Functional Theory (DFT) formalism, in the  form of \nconstrained nonmcollinear magnetic calculations, wi th the objective of creating a database of atomic m agnetic \nmoments and forces acting between the atoms. From a  given disordered atomic configuration of either pu re Fe or \nFemCr alloy, a penalty contribution to the usual sp inmpolarized DFT total energy has been calculated b y constraining \nthe magnitude and direction of magnetic moments. An  extensive database of nonmcollinear magnetic momen t and \nforce components for various atomic configurations has been generated and used for interpolating the \nspatiallymdependent magnetic interaction parameters , for applications in largemscale spinmlattice dyna mics and \nmagnetic MontemCarlo simulations. \n \n \nKEYWORDS:  firstmprinciples modeling, nonmcollinear magnetism,  disordered Fembased alloys, spinmlattice \ndynamics  \n \n \nI. Introduction  \nHighmtemperature properties of structural fusion ma terials, \nincluding those characterizing the response of mate rials to \nirradiation at high temperature, represent the key unknown \nentities critical to the development of viable fusi on reactor \ndesign, and are a source of major uncertainty in th e choice of \nreactor design strategy. Presently, significant exp erimental \nand modeling effort is devoted to understanding, te sting and \ninterpreting the available data on highmtemperature  \nproperties of fusion materials, and to interpolatin g the data to \nconditions being not accessible to experiments and tests.  \n  Ironmbased alloys and steels, including \nferriticmmartensitic, ferritic, and austenitic stee ls, owe their \nphase stability to magnetic interactions. This fact  is \nconfirmed by DFT calculations performed  in combination \nwith both Cluster Expansion (CE) and Magnetic Clust er \nExpansion (MCE) methods(1b6). Given the extensive use of \nsteels in ITER and DEMO, and the requirement that t hese \nmaterials are expected to operate at temperatures e xceeding \n350°C, and possibly approaching 750°C, there is a g enuine \nneed to develop accurate quantitative understanding  of \nradiation damage effects in iron, iron alloys and s teels \nspecifically in the high temperature limit. An addi tional \ncomplication associated with the treatment of magne tic \neffects in relation to radiation damage phenomena i s that the \ninitial development of radiation damage occurs thro ugh \nhighmenergy events, called collision cascades, wher e the \nenergy of a primary recoil atom is transferred, thr ough \ninteratomic interactions, to the local environment of the recoil atom and exciting electrons, resulting in th e local \nmelting of the lattice, shock wave events, and the formation \nof  vacancy and selfminterstitial atom defects, as well as \ndefect clusters. The challenge associated with mode ling a \ncascade event in a magnetic material stems from the  fact that \nnot only the temperature of the material in a casca de is high, \nbut also that the local atomic structure at any giv en moment \nof time no longer resembles that of a regular bodym centered \ncubic (or facemcentered cubic in the case of austen itic steels) \nlattice. To correctly understand the dynamics of ev olution of \na cascade, it is necessary to understand in detail the process \nof energy transfer between electronic, magnetic and  atomic \ndegrees of freedom in a quantitative form (7)  , including both \nthe conventional “scalar” interatomic interactions and \ndirectional intermmagnetic interactions (i.e. excha nge \ninteractions between magnetic moments) for an atomi c \nconfiguration disordered by a highmenergy collision  event. \nThe objective of this work is to develop an accurat e DFT \ndatabase for simulating highmenergy cascade events in \nFembased alloys by simultaneously taking into accou nt both \nmagnetic and atomic degrees of freedom. It also sho ws \nnonmcollinear magnetic excitations induces extra fo rces \ncomparing to collinear case. \n      \n \nII. Constrained SpinbVector DFT method  \n \nTo access nonmcollinear ground states properties of  complex \nmagnetic systems as well as for the prediction of \nfinitemtemperature properties from firstmprinciples ,  \n 2vectormspin density functional theory (DFT) has to be \napplied, which treats the magnetization density as a vector \nfield  (and not as a scalar field, as in collinear DFT \ncalculations). In 1972, von Barth and Hedin extende d this \nconcept to a spinmpolarized system (8) , replacing the scalar \ndensity by a Hermitian 2 x 2 matrix n(r).  From such \ncalculations it is possible to follow several direc tions. Like \nin molecularmdynamic simulations, spinmdynamics all ows to \nstudy the magnetic degrees of freedom either explor ing the \nground state or excited state properties. Or a mode l \nHamiltonian approach is used where magnetic interac tions \nare studied using parameters obtained from ab initio  \ncalculations. In both cases we introduce a discreti zation of \nthe vector magnetization density: in spinmdynamics,  the \nevolution of discrete spin vectors attached to cert ain atomic \npositions is monitored. Mapping the firstmprinciple s results \nonto a model Hamiltonian, which contains interactio ns \nbetween spins, also requires that it is possible to  assign a \ndefinite spin to an atom. In the vicinity of an ato m, e.g. \nwithin a sphere centered to the nuclei, it should b e therefore \npossible to define the magnetization density  \n \nM(r)=M iei                                     (1)  \n \nwhere  M i is the magnitude of magnetization and  ei is the \nmagnetization direction.  \n \nFor the manymbody Stoner model (3)   and the \nHeisenbergmLandau MCE Hamiltonian model (4) we \nformulated a theorem, closely related to the concep t of \ndynamic temperature of magnetic moments (9), stating that for \nan arbitrary nonmcollinear magnetic configurations with \narbitrary exchange fields, at energy extremum each magnetic \nmoment, Mi, associated with atom site i, is parallel to its \neffective exchange field, Hi, acting on it, i.e.,  \n \nMi×Hi=0                                       (2) \n \nFor a specific (constrained) direction of magnetic field, \n{Hi,c }, the nonmstationary solution of the total energy \nfunctional, E[n,{ Mi,c }] is given by,  \n \nE[n,{ Mi,c }]=E DFT + E p({ Mi},{ Mi,c })                   (3) \n \nHere, the first term is the usual DFT total energy and the \nsecond term represents energy penalty associated wi th the \nlocally constrained magnetic fields. { Mi,c } denotes the \nmagnitude and direction of the desired (constrained ) \nmagnetic moments whereas { Mi} is an ensemble of \nintegrated magnetic moments inside the sphere cente red at \nthe nucleus atom i. The constrained DFT developed by \nDederichs et al .(10 ) provides the necessary framework to deal \nwith arbitrary magnetic configurations, i.e. config urations \nwhere the orientations of the local moments are con strained \nto nonmequilibrium directions.  Nonmconstrained DFT  \ncalculations make it possible to find not only the spinmvector \nmagnetic moments { Mi} but also provide additional \ninformation about intermatomic forces  \n Fi=( δE[n,{ Mi,c }]/δ ri)                              (4) \n \nacting on atoms under constrained magnetic fields.        \n \nIII. Application  \n \nThe nonmconstrained DFT calculations were performed  using \nthe Vienna Abminitio Simulation Package (VASP ) (11,12) \nwithin the generalized gradient approximation (GGA)  with \nthe PerdewmBurkemEmzerhof exchange and correlation \nfunctional (13). For our calculations of nonmcollinear \nmagnetism, we used the spinminterpolation proposed by \nVosko, Wilk and Nusair (14). Solution of the KohnmShame \nequations have been carried out using a planemwave basis set \nwith an energy cutmoff of 400 eV and with the proje ctor \naugmented wave (PAW) pseudompotentials in which \nsemimcore electrons have been included. It is impor tant to \nstress that, like for all the other bcc transition metals, the \ninclusion of semimcore electrons through the use of  \npseudompotentials is important for predicting accur ately the \ndefect formation energies (15) in iron and ironmchromium \nbased alloys.  \n \nFrom Eq. (3), the penalty term in the total energy,   \nE[n,{ Mi,c }], is given by \n \nEp({ Mi},{ Mi,c })=∑ i λ[Mi B Mi,c (Mi,c .Mi)] 2              (5)  \n \nfor constraining the direction of magnetic moments.  The sum \nin Eq. (5) is taken over all atomic sites i,  Mi,c  is the desired \ndirection of the magnetic moment at site i, and Mi is the \nintegrated magnetic moment inside a sphere W i around the \nposition of atom i. The penalty term introduces an additional \npotential inside the sphere of each atomic site. Th is potential \nis determined by a function of Pauli spinmmatrices \nσ=(σ x,σ y,σ z), {Mi,c }, { Mi} and the weight λ of the penalty \nenergy. In the present study, a penalty functional is added to \nthe system which derives the integrated local momen ts from \nthe zmcomponent of magnetization (0 0 1) into the s pecific \ndirection (0 1 1).    \n \n1.  Nonbcollinear magnetic configurations of \ndisordered iron   \n \nThe initial disordered atomic samples used in the p resent \ninvestigation were generated by molecular dynamic \nsimulations for magnetic iron obtained from quenchi ng, \nusing conjugate gradient, of atomic structures at v ery high \ntemperature (~10000K) for three different atomic de nsities \ncorresponding to the bcc lattice constants of 2.75A°, 2.86A ° \nand 3.00A °, respectively. In these simulations, the atoms \ninteract via scalar manymbody forces calculated fro m a \nmagnetic interatomic potential (16)  as well as via spin \norientation dependent forces of the Heisenberg form . \nSpinmlattice dynamics simulations (9)  showing the complexity \nof finitemtemperature magnetic structure of a parti ally \ndisordered ensemble of iron atoms, have now made it  \npossible to explore the dynamics of magnetic moment s for \natomic configurations resembling those occurring in  a hot  \n 3liquid metal. However, the understanding of how mag netic \nmoments interact in an atomically disordered struct ure of \niron similar to that encountered in a collision cas cade, is still \nqualitative and rudimentary. Fig. 1 shows the nonmc ollinear \nmagnetic configurations obtained from the constrain ed \nspinmvector DFT calculations for disordered iron \nconfiguration with 250 atoms for the three consider ed atomic \ndensities. About 100 disordered configurations of p ure Fe \nhave been investigated and explored in the present work. \n \n \n  \na) d) \n \n  \nb) e) \n \n  \nc) f) \n \n  \nFigure 1 : Constrained nonbcollinear magnetic configurations i n \namorphous Fe obtained from DFT calculations in comb ination \nwith MD simulations for 250 atoms at three differen t atomic \ndensities with bcc lattice parameters: a) 2.75A °, b) 2.86A ° and c) \n3.00A °. The corresponding residual forces are shown in d) , e) \nand f), respectively. The magnitude of magnetic mom ents and \nresidual forces are represented by the respective c olours in \nlegends and by the lengths of arrows in figures.   \n \nThe constrained nonmcollinear magnetic configuratio ns \nshown in Fig. 1 were obtained from selfmconsistent \nelectronic structure calculations within the spinmv ector DFT \nformalism but at fixed atomic positions generated f rom \nmolecularmdynamic simulations. It is found that for  the high atomicmdensity configurations (a bcc =2.75A °), nonmcollinear \nmagnetic moments are strongly frustrated (Fig. 1a) with few \nof them flipping into antimferromagnetic configurat ions with \nnegative magnetization components. The correspondin g \nresidual forcemvectors (Fig. 1d) are unusually larg e at several \natomic sites reaching the maximum value of 1.6 eV/A °. The \nnonmcollinear magnetic configuration calculated at the \nequilibrium lattice constant for bcc iron (a bcc =2.86A °) shows \nthat all the magnetic moments are oriented along th e desired \npositive direction (0 1 1) (Fig. 1b) and the majori ty of \nresidual forcemvectors (Fig. 1e) are smaller in mag nitude in \ncomparison with the high atomic density case. Final ly, for \nthe case of low atomic density (a bcc =3.00A °), the constrained \nnonmcollinear magnetic moments are mainly positive with \nthe maximum magnitude of 3.0 µ B (Fig. 1c), whereas the \nresidual forcemvectors are mostly negligible except  for very \nfew atomic positions.   \n \nThe above nonmcollinear configurations are being us ed as a \nnew ab initio  database of magnetic and atomic structures to \ndevelop an understanding required for interpolating  the \nspatiallymdependent magnetic interaction parameters  between \ndissimilar structures, and applying them to largems cale \nspinmlattice dynamics simulations. In particular, t his database \nserves for constructing and deriving more reliable functional \nform of intermatomic potentials for iron with nonmc ollinear \nmagnetic moments based on the manymbody Stoner \ntightmbinding Hamiltonian (3,16,17 ).    \n \n2.  Nonbcollinear magnetic configurations for FebCr \nalloys \n \nThe magnetic origin of thermodynamic and kinetic ph ase \ntransformations, including nanomclustering as well as the \nbehaviour of point defects generated in irradiated FemCr \nalloys have been systematically investigated by usi ng a \ncombination of DFT calculations with statistical ap proaches \ninvolving cluster expansions and Monte Carlo \nsimulations (1,2,18) . Recently we have developed a new \napproach to the treatment of nonmcollinear magnetic  order in \nFembased alloys, the Magnetic Cluster Expansion (4.5,19 ). \nUsing this approach we were able to predict several  new \neffects, describing magnetic properties of nuclear steels, for \nexample the composition and microstructure dependen ce of \nthe Curie temperature, the nonmcollinearity of magn etic \nstructures found in bcc FemCr alloys, which has now  been \nconfirmed by experimental observations (20,21) . So far, the \ncalculations forming the DFT database used for cons tructing \nthe MCE model have been performed within the \nspinmpolarized collinear magnetic framework at T=0K .  \n \nIn this study, the constrained nonmcollinear magnet ic DFT \ndatabase is generated by using atomic configuration  samples \nof binary FemCr alloys obtained from exchange Monte mCarlo \nsimulations with decreasing temperature as describe d in \ndetails previously  and performed on small 4x4x4 bcc \nsupermcells(3,18 ).   These configurations were found by \ncooling down, from 2000 K to 0 K, an initially rand om binary \nalloy configuration. The lowest negative mixing ent halpy is  \n 4found at 6.25% Cr, which is consistent with our DFT  \nprediction of an ordered Fe15Cr structure (1 ). Importantly, \ndespite considering only small supercell, it is fou nd also that  \nthe mixing enthalpy changes sign as the Cr concentr ation \nincreases and is positive everywhere above 10–12% C r, \nwhere nanomclustering of the α’ phase (bccmCr) is o bserved, in \nexcellent agreement with both experimental and DFT data. \n \n \n  \na) c) \n \n  \nb) d) \n \n  \nFigure 2: Constrained nonbcollinear magnetic config urations in \nFebCr alloys obtained from DFT calculations in comb ination \nwith MC simulations for two different Cr compositio ns: a) \n6.25% and b) 37.5% and their corresponding residual  forces: c) \nand d), respectively . The orange and grey colours denote Fe \nand Cr atom, respectively. Magnitudes of magnetic m oments \nand residual forces are shown by colours indicated in the \nlegends.  \n \nHere we performed calculations for the constrained \nnonmcollinear magnetic configurations with two diff erent Cr \nconcentrations: 6.25% and 37.5% (Fig. 2). For the c ase with \nlow concentration, all the Cr atoms have their mome nts \naligned antimferromagnetically in comparison with t he \nmagnetic moments associated with all the Fe sites, although \nthe constraint of local magnetic field was imposed into (0 0 \n1) and ( 0 1 1) directions (Fig. 2a). The magnitude  of the \nconstrained nonmcollinear magnetic moments for Cr a toms \nvaries from 1.3m1.6 µ B in comparison with 1.7 µ B for \nselfmconsistent collinear magnetic calculations. Th e resulting \nresidual forcemvectors are relatively small (Fig. 2 c) with a \nmaximum value of 0.2 eV/A ° showing that the considered \nconfiguration of FemCr alloys is not far from equil ibrium \ncondition.   At high concentration (37.5%Cr), nanomclusters of C r atoms \nare formed due to the positive enthalpy of mixing r elated to \nthe segregation of α’ phase. It is shown that the C r cluster is \nbounded by the equivalent (110) planes interfacing with the \nbcc Fe matrice. The lowest enthalpy of mixing confi guration \nis thermodynamically consistent with the high densi ty of \n(110) Cr/Fe interfaces. More importantly, as it has  been \nanalyzed earlier, spinmpolarized DFT investigation \ndemonstrates that the clustering phenomena in FemCr  binary \nwith high Cr concentration have magnetic origin.  F ig. 2c \nshows that the constrained fields on each atom impl y the \norientation of magnetic moments of Cr atoms alignin g \nnonmcollinearly parallel to the (011) plane and \nantimferromagnetically orientated in comparison wit h those \nof Fe atoms. The magnitude of magnetic moments on C r \natoms is small (red colour) in comparison with high  \nmagnetic moments of 2.6 µB on Fe atoms (blue colour ). \nDespite complex and frustrated magnetic configurati ons at \nCr/Fe interfaces, the residual forces found for the  37.5%Cr \ncase (Fig. 2d) appear broadly similar to those eval uated for \nthe case of 6.25%Cr from our constrained nonmcollin ear DFT \ncalculations.     \n \nWe note that our present results of nonmcollinear m agnetisms \nat interfaces in ironmchromium alloys are alternati ves to \nthose investigated previously by DFT calculations f or the \ngroundmstate configurations (5,19) . For the (110) interfaces, the \nunconstrained DFT study showed that orientations of  \nmagnetic moments of Cr atoms  are also aligned almo st \nparallel to the interfaces but those of Fe atoms ar e oriented \nperpendicular to the interface. The constrained DFT  \nnonmcollinear magnetic calculations shown in Fig. 2 b and 2d \nprovided additional information about the magnitude  and \norientation of magnetic moments as well as about th e \ncorresponding components of force vectors acting on  each \natom from the equilibrium and groundmstate configur ations. \nThis information is important for the firstmprincip les \nunderstanding of displacement per atom (dpa) behavi our in \nironmbased alloys and steels under irradiation cond itions or \ncascade simulations, also offers an opportunity to extend the \nMCE formalism beyond the rigidmlattice approximatio n.  \n \n \n3.  Nonbcollinear magnetic configurations for \nselfbinterstitial atom (SIA) defects \n \nIt is now well established that magnetic effects ar e also \nresponsible for the fact that the atomic structure of radiation \ndefects in iron and steels is different from the st ructure of \ndefects formed under irradiation in nonmmagnetic \nbodymcentred cubic  metals, for example vanadium or \ntungsten (6,15,17,22b24).  Unlike vacancies, SIAs do not \nspontaneously form in materials at elevated tempera tures. \nSIAs generated in highmenergy collision cascades mi grate to \nsinks in the material, and this gives rise to swell ing, \nirradiation creep and radiation embrittlement. Our DFT \ncalculations of SIA defects in bcc transition metal s show that \nin all the nonmmagnetic bcc transition metals, incl uding \nbccmW, the most stable defect configuration has the  <111>  \n 5(crowdion) orientation. The pattern of ordering of SIA \nconfigurations is fundamentally different in ferrom agnetic \nbccmFe, where the <110> dumbbell configuration is f ound to \nhave the lowest formation energy. Spinmpolarized co llinear \nDFT calculations show that the two Fe atoms in the SIA \n<110> configuration have antimferromagnetically ali gned \nmagnetic moments of m0.25µ B in comparison with the positive \nferromagnetically ordered moments of 1.7 µB for the  four \nnearestmneighbour iron atoms. T hreemdimensional SIA \nclusters in iron formed directly in displacement ca scades can \nbe grown by capturing <110> dumbbells (25). \n \nFig.3a shows the result of DFT calculations within \nnonmcollinear magnetism, constraining the direction  of \nmagnetic moments for the <110> dumbbell defect in b ccmFe. \nIt is found that that the two iron atoms in a dumbb ell \nconfiguration maintain the opposite orientation (re d arrows) \nin comparison with the four nearest neighbour (grey  arrows) \nones although the direction of magnetic moments is different \ndue to the local constrained magnetic field.  The \ncorresponding residual forces (Fig.3c) are almost n egligible \nfor two dumbbell atoms as well as for the surroundi ng atoms. \nWe note that the initial configuration in this stud y was not the \nfully relaxed one obtained from collinear magnetic \ncalculations without ionic relaxation.    \n \nIf we have two Cr atoms in the <110> dumbbell confi guration \nin bccmFe, their interaction with iron atoms result s in a \nstabilized configuration in which the magnetic mome nts are \nlarger and negative in comparison with those of the  two iron \natoms in the dumbbell within a collinear magnetic s cheme (24).  \nFig.3c shows the corresponding result within the co nstrained \nnonmcollinear magnetic configuration, where Cr atom s are \nrepresented by grey colour in comparison with the o range one \nfor the Fe atoms. The orientation of magnetic momen ts on Cr \natoms is again opposite to those of Fe atoms and th e \nmagnitude of these moments is about 0.7 µ B compared with \nthe maximum value of 2.6 µ B for Fe atoms. Fig. 3d shows the \ncorresponding residual forcemvectors for CrmCr dumb bell \ndefect in bccmFe. It is interesting to note that wi thin the \nconstrained nonmcollinear magnetic calculations the  two Cr \natoms in a dumbbell attract, with forces acting bet ween the \natoms pointing towards each other, whereas the four  \nsurrounding Fe atoms have the force direction pulli ng them \naway from the SIA defect configuration. Such artificial \nmagnetic excitations induce extra forces comparing to \ncollinear case, which is an concrete evidence that show the \nimportant of considering thermal excitation in magn etic \nsubsystem, when we are calculating interatomic forc es in \nmagnetic metals.  \n \n \n \n \n \n \n \n \n \n \n  \na) c) \n \n  \nb) d) \n \n  \nFigure 3 : Constrained nonbcollinear magnetic configuration fo r \n<110> selfbinterstitial atom (SIA) defect configura tion obtained \nfrom DFT calculations in bccbFe: a) FebFe dumbbell,  b) CrbCr \ndumbbell and their corresponding residual forces: c ) and d), \nrespectively. The orange and grey colours denote Fe  and Cr \natom, respectively. The magnitude for the magnetic moments \nand residual forces is shown by colours in legends.     \n \nIV . Conclusion  \nWe have performed extensive constrained spinmvector  DFT \ncalculations for disordered Fe, FemCr alloy as well  as for \nirradiationminduced SIA configurations for point de fects with \nFemFe and CrmCr <110> dumbbells, for the purpose of  \nestablishing a database of magnetic and atomic prop erties \nrequired for the parameterization of spatiallymdepe ndent \nmagnetic interactions, with the subsequent applicat ion of the \nresult to largemscale spinmlattice dynamics simulat ions of \ncollision cascades. In addition to the DFT database  \ncontaining groundmstate configurations evaluated wi thin \ncollinear magnetic approximation, the constrained \nnonmcollinear calculations provide a much broader D FT data \nbase containing very rich information about magneti c \nmoments and forces (both treated as vectors with va riable \nmagnitude and direction in threemdimensional space)  for \ndisordered and defect configurations obtained under  various \nnonmequilibrium conditions. The present database is  being \nused to develop manymbody interatomic potentials fo r iron \nand ironmalloys with nonmcollinear treatment of mag netic \nmoments in these systems.  We also show forces can be \ninduced by magnetic excitations.  \n 6Acknowledgment \nThis work, partmfunded by the European Communities under \nthe contract of Association between EURATOM and CCF E, \nwas carried out within the framework of the Europea n \nFusion Development Agreement. To obtain further \ninformation on the data and models underlying this paper \nplease contact PublicationManager@ccfe.ac.uk . The views \nand opinions expressed herein do not necessarily re flect \nthose of the European Commission. This work was als o \npartmfunded by the RCUK Energy Programme under gran t \nEP/I501045. DNM would like to thank the Juelich \nsupercomputer centre for providing access to \nHighmPerformances Computer for Fusion (HPCmFF) faci lities \nas well as the International Fusion Energy Research  Centre \n(IFERC) for using the supercomputer (Helios) at \nComputational Simulation Centre (CSC) in Rokkasho \n(Japan). \n \nReferences  \n1)  D. NguyenmManh, M. Yu. Lavrentiev, S.L. Dudarev, “M agnetic \norigin of nanomclustering and point defect interact ion in FemCr \nalloys: An abminitio study”, J. ComputermAided Mate r. Des., 14, \n159m169 (2007).  \n2)  D. NguyenmManh, M. Yu. Lavrentiev, S.L. Dudarev, “T he \nFe–Cr system: atomistic modelling of  thermodynamics and \nkinetics of phase transformations”, C.R. Physique, 9, 379m388 \n(2008). \n3)  D. NguyenmManh, S.L. Dudarev, “Model manymbody Ston er \nHamiltonian for binary FeCr alloys”, Phys. Rev. B, 80, 104440 \n(1m11) (2009).  \n4)  M. Yu. Lavrentiev, D. NguyenmManh, S.L. 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NguyenmManh, A.P. Horsfield and S.L. Dudarev, \n”Selfminterstitial atom defects in bcc transition m etals: \nGroupmspecific trends”, Phys. Rev. B, 73, 020101R(1 m4) \n(2006). \n16)  S.L. Dudarev and P.M. Derlet “A 'magnetic' interato mic \npotential for molecular dynamics simulations”, J. P hys.: \nCondensed Matter, 17, 7097m7118 (2005) \n17)  M. Mrovec, D. NguyenmManh,  C. Elsasser and P. Gumbsch, \n“Magnetic BondmOrder Potential for Iron”, Phys. Rev . Lett., \n106, 246402 (1m4) (2011).   \n18)  M. Yu. Lavrentiev, R. Drautz, D. NguyenmManh, T. P.  C. \nKlaver, and S. L. Dudarev , “Monte Carlo study of \nthermodynamic properties and clustering in the bcc FemCr \nsystem”, Phys. Rev. B, 75, 014208(1m12) (2007). \n19)  M. Yu. Lavrentiev, D. NguyenmManh, J. Wrobel, S.L. 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Johnson  a,b \na Ames Laboratory, United States Department of Energy, Ames, Iowa 50011, USA  \nb Department of Materials Science & Engineering, Iowa State University, Ames, Iowa 50011, USA  \n \nAbstract  \n \nOrder -disorder transformation s hold an essential  place in chemically complex high-\nentropy ferritic -steels  (HEFSs)  due to their critical technological application. The \nchemical inhomogeneity arising from  mixing of multi -principal elements of varying \nchemistry can drive  property altering changes at the atomic scale , in particular short -\nrange order.  Using  density -functional theory  based  linear -response theory, we predict \nthe effect of compositional  tuning on the order -disorder transformation in ferritic steels  –\nfocusing on Cr-Ni-Al-Ti-Fe HEFSs . We show that Ti content in Cr-Ni-Al-Ti-Fe solid \nsolutions can b e tuned to modify short -range order  that changes  the order -disorder path \nfrom BCC -B2 (Ti atomic -fraction = 0) to BCC -B2-L21 (Ti atomic -faction >  0) consistent \nwith existing  experiments. Our study suggests that tuning degree of SRO through \ncompositional variation can be used as an  effective means to optimi ze phase selection  \nin technologically useful alloys.  \n \nKeywords:  High Entropy Alloys, Density -functional theory, Phase stability, Short -range order  \n \nIntroduction  \n High-entropy alloys, including metals and ceramics with near -equiatomic  \ncompositions with four and more elements [ 1-6], continue to  gain significant interest due \nto the unprecedented opportunity to explore large materials design space and  uncover \npotentially remarkable compositions with outstanding  structural and functional \nproperties [ 4-11]. The design strategy in high-entropy alloys has been to use the \nconcept of entropy to stabilize the single -phase solid -solution (e.g., face -centered cubic  2 \n (FCC phase)  or body -centered cubic  (BCC phase) ) [12] with  an attempt to find specific \nelectronic, thermodynamic, and microstructural properties [ 4,13-15]. While the progre ss \nover the last decade towards first -generation high-entropy alloys is remarkable, the \ncritical thermodynamic behavior  of these alloys indicates that only a little is known on \nthe effects of SRO  [16] and the associated lattice deformations [ 17] on electronic and/or  \nmechanical response  [18].  \n High-entropy f erritic steels (HEFSs) are one important class of multi -principal \nelement alloys  due to their  cost efficiency, low  thermal expansion, and good thermal \nconductivity compared to Ni -based superalloys and austenitic steels [19-21]. Similar to \nconventional allo ys, HEFSs  show chemistry and temperature dependent ordering  that \nmay undergo one or more phase transitions into less ordered phases. Precipitation \nhardening due to presence of ordered phases in HEFSs gives  excellent creep and \noxidation behavior  [22], which is analogous  to the  presence of 𝛾′ phase in Ni -based \nsuperalloys. Unlike L1 2 phase s in austenitic steels, presence of ordered B2 or B2/L2 1 \nphases in body -centered cubic matrix  [23,24 ] may provide similar mechanical effects \n[25]. Therefore, a detailed  understanding of order -disorder transformation s and \nprecipitat e formation  along with  compositional control in HEFSs  can be  of fundamental \nimportance.   \n Here  we present a systematic study on the effect of compositional tuning of Ti on \norder -disorder behavior in Cr-Ni-Al-Ti-Fe HEFSs using density -functional theory (DFT) \nmethod s in combination with configuration al averaging  [26,27 ]. The linear response \ntheory was used for calculating short -range order in the disordered Cr-Ni-Al-Ti-Fe \nHEFSs [25]. We show that degree of SRO can be controlled using Ti content , which 3 \n modifies order -disorder pathway from BCC -B2 to BCC -B2-L21 [28]. Our finding s are  in \ngood agreement with recent observations  of Wolf -Goodrich et al.  [29], who report \ncombination s of BCC /B2 and BCC /B2/L2 1 phases depending on Ti composition (at.% Ti) \nin Cr-Ni-Al-Ti-Fe HEFSs . The linear -response  theory for SRO analyzed by \nconcentration wave method was used to seed the fully self-consistent KKR-CPA \ncalculation in the broken symmetry case, which, unlike Monte -Carlo methods [ 30], does \nnot rely on fitted interactions.  We also discussed the phase stability (formation energy ) \nand electronic -structure origin of disord ered and (partial)  order ed phases  for selected \nHEFS compositions . We found that SRO can be a key structural feature for optimizing \nphase selection and mechanical response . \nComputational detail s: \nDensity -functional based linear -response theory:  Phase stability and electronic -\nstructure  were addressed using an all-electron, Green’s function based Korringa -Kohn -\nRostoker (KKR) electronic -structure metho d [26]. The configurational averaging to \ntackle chemical disorder is handled using the coherent -potential approximation (CPA) \n[27], and the screened -CPA was used to  address  Friedel -type charge screening [31]. \nValence electrons and shallow lying core electrons affected by alloying are addressed \nvia a s calar -relativistic approximation (where spin -orbit terms only are ignored) \n[26,27,31 ], whereas deep lying core are address using the full Dirac solutions.  \nElectronic density of states (DOS) and Bloch -spectral function (BSF) were calculated \nwithin the atomic sphere approximation (ASA) with periodic boundary conditions. The \ninterstitial electron contributions to Coulomb energy are incorporated using Voronoi \npolyhedra. The generalized gradient approximation to DFT exchange -correlation was 4 \n included us ing the  libXC  opensource code [32]. Brillouin -zone integrations for self -\nconsistent charge iterations were performed using a Monkhorst -Pack k-point mesh [33]. \nEach BSF was calculated for 300 k -points along high -symmetry lines in an irreducible \nBrillouin zone.  \nThermodynamic linear -response theory for short -range  order : Chemical short -\nrange order and associated instabilities were calculated u sing KKR-CPA-based \nthermodynamic linear -response  theory  [28,34 -37]. The Warren -Cowley SRO (pair-\ncorrelation) parameters 𝛼𝜇𝜈(k;T) for μ -ν elemental pairs are calculated directly in Laue \nunits [28]. The necessary energy integrals over the Green’s functions were performed at \nfinite temperature by summing over Matsubara frequencies [ n = k BT(2n+1)] [28]. \nDominant pairs driving SRO are identified from th e chemical pair -interchange energies \n𝑆𝜇𝜈(2)(k;T) (a thermodynamically averaged quantity – not a pair interaction) , determined \nfrom an analytic second -variation of the DFT-based  KKR-CPA grand potential with \nrespect to concentrations fluctuations of 𝑐𝜇𝑖  at atomic site i and 𝑐𝜈𝑗 at atomic site  j [28]. \nThe chemical stability matrix 𝑆𝜇𝜈(2)(k; T) reveals the unstable Fourier modes with ordering \nwavevector k o, or clustering if at k o=(000) at spinodal temperature (T sp) [28]. Here, T sp is \nthe temperature where SRO diverges, i.e., 𝛂𝜇𝜈−1(ko;Tsp)=0, which signifies absolute \ninstability in alloy and provides an estimate of order -disorder  (ordering systems)  or \nmiscibility gap  (in cluster ing systems) . \nFormation energy calculation : Formation energy (E form) of the Cr-Ni-Al-Ti-Fe HEFSs \nwas estimated using Eform=EtotalCr−Ni−Al−Ti−Fe(ci)− ∑ciEi i=1,N , where EtotalCr−Ni−Al−Ti−Fe is \nthe total energy, ci is elemental composition, Ei is the energy of alloying elements, and \n‘i’ labels elements BCC (Cr, Fe) FCC (Ni, Al), HCP Ti.  5 \n Temperature Estimate s: The Curie (or ferromagnetic ordering) temperature of Cr-Ni-\nAl-Ti-Fe HEFSs was assessed using  mean -field Heisenberg -like model [ 38]. The mean -\nfield relation for Tc [=2\n3[𝐸𝐷𝐿𝑀−𝐸𝐹𝑀]/𝑘𝐵] is proportional to the energy difference \nbetween paramagnetic  (PM)  and ferromagnetic  (FM) states , with the PM state \napproximated by the disordered local moment (DLM) state  – randomly oriented \n(uncorrelated) local moments easily represented with a separate CPA condition for \nmoment orientations [ 39]. However,  as discussed by  Sato et al.  [38] for dilute magnetic \nsemiconductor , it is appropriate to consider a slightly modified relation as an upper \nbound, i.e., 𝑇𝑐=2\n3∙[1\n1−𝑐]∙[𝐸𝐷𝐿𝑀−𝐸𝐹𝑀]/𝑘𝐵, given the Cr-Ni-Al-Ti-Fe HEFSs has a non -\nmagnetic element (Al) with concentration c. \nConcentration (Fourier) Wave analysis : Fourier analysis or concentration wave \napproach was used to interpret the (partial)long -range order observed in the SRO \ncalculations, where normal modes ( 𝑒𝑖𝜎) in Gibbs’ space were obtained from chemical \nstability matrix in linear -response theory  [28]. The occupation probabilities [ 𝑛𝑖(𝒓)] at site \n𝒓𝒊 are identical to element al compositions  [𝑐𝑖] in disorder phase of the alloy , which \ndepends on  type of order in long -range order ed phase. Here, ‘i’ is the index for the type \nof elements.  The occupation  probabilities  in N-component system, i.e., HEFSs, can be \nexpanded in a s Fourier series, i.e., concentration wave , which can be  written in terms of \nnormal modes as \n[    𝑛1(𝒓)\n𝑛2(𝒓)\n𝑛3(𝒓)\n⋯\n𝑛𝑁−1(𝒓)]    \n    =    \n[    𝑐1\n𝑐2𝑐3\n⋯\n𝑐𝑁−1]    \n    +  ∑𝜂𝜎𝑠\n𝑠,𝜎\n[    𝑒1𝜎(𝒌𝒔)\n𝑒2𝜎(𝒌𝒔)\n𝑒3𝜎(𝒌𝒔)\n⋯\n𝑒𝑁−1𝜎(𝒌𝒔)]    \n×∑𝛾𝜎(𝒌𝒋𝒔)𝑒𝑖𝒌𝒋𝒔 .𝒓\n𝑗𝑠  Eq. ( 1) 6 \n For a given atomic  position 𝒓𝒊, 𝑐𝑖 is the composition vector of  order (N–1) component, \nrelative to “host” element N. The sum in Eq. (1) runs over the star of inequivalent \nwavevectors ‘s’ that defin es the order, σ is eigenvector branch of the free -energy \nquadric, and js are equivalent wavevectors in star s. The 𝜂𝜎𝑠 (0≤𝜂(𝑇)≤1) is long -range \norder parameter of star `s’ and branch  `𝜎; where 𝑒𝑖𝜎(𝒌) is eigenvector of the normal \nconcentration mode for branch  𝜎, and symmetry coefficients 𝛾𝜎(𝐤𝒋𝒔) found by \nnormalization condition and lattice geometry. Note that the elements of vectors 𝑛𝑖(𝒓) \nand 𝑐𝑖 conserve probab ility and must add to 1 ( ∑𝑐𝑖=1𝑁\n𝑖=1 ), i.e., by the sum rule  first (N \n– 1) elements should add to final elements .  \nResults and Discussion  \nPhase stability , structural , and magnetic property  analysis for Cr-Ni-Al-Ti-Fe HEFSs , \nrelated to  recent experimental work of Wolf-Goodrich et al.  [29], are shown in Table 1 . \nWe calculated the formation energy (Eform) of each HEFSs in BCC , FCC and HCP \nphases . The calculated formation energy in HCP  phase  for each alloy was a large \npositive number compared to BCC and FCC phases, therefore, not discussed . Our \nphase stability  analysis Table. 1 indicates  that the BCC phase  is energetically more \nfavorable in Cr -Ni-Al-Ti-Fe HEFSs . This is the reason , we mainly  focused on  BCC  Cr-\nNi-Al-Ti-Fe HEFSs . The DFT calculated formation energies  in Table  1 show increase in \nEform with increase in  Ti+Al composition , where Cr0.05Ni0.15Al0.30Ti0.15Fe0.35 HEFS with \nTi+Al= 0.45 at.-frac. was found energetically more favorable compared to other alloys .  \n The trends in magnetization (cell moment) and Curie -temperature (estimated \nusing mean -field Heisenberg -like model [ 38]) show increase with decreasing Ti+Al +Cr 7 \n composition  in Table 1  (Cr composition was  included with Ti+Al in our magnetization \nanalysis as anti -ferromagnetic character of Cr is well known to impact the magnetic  \nbehavior of the alloy ). The total moment was found to decrease with increasing \nTi+Al+Cr composition from 0.56 𝜇𝐵 (Ti+Al+Cr= 0.50 at. -frac.) to 0.10𝜇𝐵 (Ti+Al+Cr= 0.60 \nat.-frac.). To understand this better, a detail ed local moment analysis was performed on \nCr0.20Ni0.10Al0.30Fe0.40  (no Ti) and Cr0.20Ni0.10Al0.25Ti0.10Fe0.35  (with Ti) HEFS s. The local \nmoment at Cr was found to increase from  -0.25 𝜇𝐵 (no Ti) to -0.41𝜇𝐵 (with Ti), \nrespectively . The sign of  local moment shows that Cr prefers  to align anti -\nferromagnetically (AFM) irrespective of the initial orientation  (FM or AFM) compared to \nother magnetic species  such as Ni and Fe.  The frustrated moment at Cr  (AFM \narrangement)  subsidizes the overall cell moment in HEFS s with increasing Cr \ncomposition. We found that  higher T i composition weakens the AFM (frustration) nature \nof Cr, which  reduces  the overall magnetic strength .  \n Regarding structural property, for example, bulk moduli (K), no  significant change \nwas observed with Ti+Al  composition variation.  We observed in our calculations that  \nAl+Ti  or Ti  stabilize s the BCC phase over FCC phase , which is obvious from the \ncalculated formation energ ies in Table. 1 . For example, equiatomic quaternary \nCr0.25Ni0.25Al0.25Fe0.25  HEFSs (with no Ti ) shows positive formation enthalpy , whereas \nalloys in presence of Ti show improve d BCC phase stability.  \n \n \n \n \n 8 \n Table 1. Lattice -constant  (Å), formation energy  (meV/atom) , bulk-moduli  (GPa) , magneti zation  \n(𝝁𝑩/cell) , and Curie temperature  (K) for various Cr-Ni-Al-Ti-Fe HE FSs.  \n \nHEFSs  \na [Å] Eform [meV/atom ] KBCC \n[GPa\n] Mag.  \n[𝝁𝑩/cell] Tc  \n[K] BCC  FCC  \nFM PM FM \nCrNiAlTiFe 2.92 -0.41 -0.27 80.01 162.5  0.10 1.8 \nCr0.20Ni0.10Al0.25Ti0.10Fe0.35 2.89 -12.27 8.78 83.21 173.8  0.38 250.5  \nCr0.05Ni0.15Al0.30Ti0.15Fe0.35 2.93 -82.79  -59.26  19.28 173.8  0.43 330.9 \nCr0.20Ni0.10Al0.30Fe0.40 2.87 -21.42  10.13  58.14  169.7  0.56 348.7  \nCrNiAlFe 2.86 7.82 19.92  58.75  174.8  0.33 124.9  \n \n The thermodynamic stability of multicomponent alloys is an important criterion  to \nunderstand relative phase stability  with respect to alloying element, which requires non -\ntrivial sampling over infinitely large  configurations  in disorder phase [ 26,27 ]. Recently, \nSingh et al [34,37] extended the Hume -Rothery criteria [ 36,40] by including DFT (KKR -\nCPA) calculated  formation energie s (includ ing proper configuration averaging  [26,27 ]) \nalong with  (i) size -effect, (ii) lattice structure, (iii) valence -electron composition (VEC), \nand (iv) electronegativity difference. Instead of empirically estimated  formation -energ ies, \nthe inclusion of 𝐸𝑓𝑜𝑟𝑚 (𝐷𝐹𝑇) has made the design criteria of predicting phase stability \nmore  robust.  To better understand the effect of alloying elements on thermodynamic \nstability and structural behavior of Cr -Ni-Al-Ti-Fe HEFSs, we calculated and present \nEform (in Fig. 1 ) and (bulk -moduli (K), volume (V), lattice constant (a) in Fig. 2a -c) with \nrespect to each alloying elements as a line plot, i.e ., Cr x(NiAlTiFe) 1-x, Ni x(CrAlTiFe) 1-x, \nAlx(NiCrTiFe) 1-x, Tix(NiAlCrFe) 1-x, and Fe x(NiAlTiCr) 1-x. 9 \n  \nFigure 1. The f ormation energ ies (Eform; in mRy/atom)  of Crx(NiAlTiFe) 1-x, Nix(CrAlTiFe) 1-x, \nAlx(NiCrTiFe) 1-x, Tix(NiAlCrFe) 1-x, and Fex(NiAlTiCr) 1-x HEFSs . The e quiatomic high -entropy \ncomposition is shown by vertical dashed line.   \n \n In Fig. 1 , we plot E form in Cr -Ni-Al-Ti-Fe HEFSs, where shaded zone below \nhorizonal line shows the energetically stable alloy compositions, while the vertical \ndashed line is the equiatomic high -entropy composition. The solid black line shows the \nvariation of E form with respect to Cr composition . The solubility limit of Cr in Ni -Al-Ti-Fe \nHEFSs is 0 -0.20 atomic -fraction (at.-frac.) beyond which the alloy becomes \nenergetically unstable. On the other hand, it was found that the solubility limit of Ni in \nCr-Ni-Al-Ti-Fe with two different zones, (i) 0 -0.15 at. -frac., and (ii) 0. 32-0.85 at. -fac., i.e., \nNi in composition range 0.15-0.32 at. -frac. remains w eakly stable or unstable for \nforming BCC HEFSs.  The solubility limit (energy stability) of Al was found  from 0.25-to-\n0.80 at. -frac., which shows that  stability of BCC   Alx(NiCrTiFe) 1-x increases with \nincreasing Al at. -frac. We also found that, although Ti solubility range is from 0 -0.65 at. -\n10 \n frac. has very weak effect on energy stability, i.e., no major benefits of adding excess Ti \nin alloy.  Interestingly, Fe is more peculiar here, because varying Fe at. -frac. in Cr -Ni-Al-\nTi-Fe HEFSs shows strong solubility of Fe from 0.60 -1 at.f -rac. Based on our stability \nanalysis, we found that optimal composition ranges for elements like Cr, Ni, Al , and Ti  \nare (0 -0.20), (0 -0.15), (0.25-0.80), and (0 -0.20) at.-frac., respectively.  The opti mal range \nfound in our phase stability analysis clearly matches with compositions experimentally \nsynthesized by Wolf-Goodrich et al.  [29], where author’s report the formation of different \ntype of precipitates in BCC alloys such as B2/L2 1 depending on specific elemental \ncompositions.   \n \nFigure 2.  (a) Lattice constant (Å), (c) volume (V, in Å3-atom-1), and (c) bulk -moduli (K, in GPa) of \nCrx(NiAlTiFe) 1-x, Nix(CrAlTiFe) 1-x, Alx(NiCrTiFe) 1-x, Tix(NiAlCrFe) 1-x, and Fex(NiAlTiCr) 1-x HEFSs. \nThe K in (c) was fitted with volume and total -energy using Birch -Murnaghan equation of state.  \n \nFor small strains, the relation between lattice constant (a) and volume (V) with \nbulk moduli (K) can be defined in terms of change in lattice constant with respect to a \n(a/a), change in pressure ( P) required for volume change per unit volume, i.e.,   Δ𝑎\n𝑎=\nΔ𝑃\n3𝐾. The expression suggests that change in lattice constant inversely related to change \nin volume or lattice constant as shown in Fig. 2a-c. We can see in Fig. 2a-b that `a’ and \n`V’ increases with increase in Al/Ti compositions whereas decreases for Cr/Ni/Fe. \n11 \n Clearly , bulk moduli in Fig. 2c clearly decreases for Al/Ti  whereas increa ses for \nCr/Ni/Fe cases that agrees well with relationship  between a/V with K in   Δ𝑎\n𝑎=Δ𝑃\n3𝐾. \nTo elucidate chemical ordering in Cr-Ni-Al-Ti-Fe HEFSs, we calculated  SRO on \nTi-rich and Ti -poor compositions, i.e., Cr 0.20Ni0.10Al0.25Ti0.10Fe0.35 and \nCr0.20Ni0.10Al0.30Fe0.40 (no Ti).  While KKR -CPA Eform determines ground -state stability of \nBCC versus FCC, our linear -response SRO calculations indicate [28] directly the \nchemical instabilities, i.e., clustering or ordering  modes , inherent in a given high-entropy \nalloys [28,34,37 ], and the likely low-temperature long -range order [ 28], including its \nelectronic origin.  \nIn Fig. 3a -d, we show SRO and 𝑆𝛼𝛽(2)(k, T) for Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 and \nCr0.05Ni0.15Al0.30Ti0.15Fe0.35 HEFSs, respectively. Figure 3a,c reveal s a decrease in local \nchemical order to from 6 Laue for Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 (Fig. 3a) to 3 Laue for \nCr5Ni15Al30Ti15Fe35 (Fig. 3 c) with decrease in  Ti (0.15 to 0.10 at. -frac.) composition. The \nchemical stability  matrix  [𝑆𝛼𝛽(2)(k, T=1.15T sp)] plot in Fig. 3 b,d reveals the atomic pairs  \nand modes  driving  SRO that are manifest ed in SRO pairs in Fig. 3a,b . The SRO at \nwavevector k=k o reveals maximal  diffuse intensities above the spinodal temperature  \nT>T sp, i.e., 895 K for 0.10 Ti at.-frac. in Cr 0.20Ni0.10Al0.25Ti0.10Fe0.35 and 815 K  for 0.15 Ti \nat.-frac. in Cr 0.05Ni0.15Al0.30Ti0.15Fe0.35. The SRO  in BCC phase  for both the alloys in Fig. \n3a,b shows maximal peak at H=(111) point  that indicat es B2 type ordering  with a  \npotentially  secondary ordering mode at P=(½ ½ ½) . The p resence of  combined H+P \ntype ordering peaks indicat e L21-type ordering  [37]. Below Tsp, the SRO predicts \npossible phase decomposition into B2 of disordered Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 and \nCr0.05Ni0.15Al0.30Ti0.15Fe0.35 HEFSs, and the B2 phase may undergo secondary phase 12 \n transformation into L2 1 on further lowering the temperature.  The atomi c pairs in 𝑆(2)(k, \n1.15T sp) in Fig. 3 b,d shows that the  Al-Fe and Al -Ti pairs are the strongest pairs  driving  \nphase decomposition  below spinodal temperature, i.e., Tsp of 895 K for \nCr0.20Ni0.10Al0.25Ti0.10Fe0.35 and 810 K for Cr 0.05Ni0.15Al0.30Ti0.15Fe0.35, respectively . We \nnote that driving mode s in 𝑆(2) (k, 1.15T sp) at H- and P-point s are different,  however, the \ndominant SRO modes at (H+P) -point in both Fig. 3a and Fig. 3c are the same, i.e., Ni -\nAl and Al-Ti pairs . This asymmetry atomic pairs in SRO and 𝑆(2) pairs  occur  due to the \nconservation of the sum rule  [28].  \n \nFigure 3. (a,c) SRO (in Laue ), and (b,d) chemical stability matrix 𝑆𝛼𝛽(2)(k, T=1.15T sp) [in Rydberg ] \nfor Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 and Cr0.05Ni0.15Al0.30Ti0.15Fe0.35 HEFSs , respectively,  along \nhigh-symmetry directions  (-P-N--H) of BCC Brillouin zone .  Peak s at H=(100)  indicate B2 -\ntype SRO  (dominated by Al-Ni pairs ). Secondary peaks at Γ and P suggest possible \nsegregation (dominated by Cr-Ni pair ) and weaker L2 1-type SRO (dominated by Al-Ti pair ). \n \nThe Block -spectral function  (BSF ) and partial density of states  (PDOS)  for BCC \nCr0.20Ni0.10Al0.25Ti0.10Fe0.35 are shown in Fig. 4a,b. The BSF in Fig. 4a shows large \ndisorder broadening (at)near Fermi energy (E Fermi), where scale on right shows weak \n(black) to strong (red) disorder effect arising from mixing of different alloying species \n13 \n [33]. For in -depth understand ing of the alloying  effect , we plot the PDOS of \nCr0.20Ni0.10Al0.25Ti0.10Fe0.35 in Fig. 4b. The four distinct energy regions are shaded  at (i) -\n0.052 mRy, (ii) -0.094 mRy, (iv) -0.20 mRy , and (i v) -0.275 mRy below E Fermi in PDOS. \n(i) The energy region shade d in pink  show s strong hybridization among overlapping \npeaks of Cr -3d, Fe-3d, and Ti -3d states, which also coincides with strongly diffused BSF \nalong -H in Fig. 4a. Similarly, regions (ii -iv) in Fig. 4b show strong hybridization among \n(ii) Cr-3d, Fe-3d, Al-2p in blue region , (iii), Ni-3d, Al-2p in orange region , and ( iv) Cr-3d, \nFe-3d, Ni-3d, Al-2p in green region ), respectively. We some obvious  instances  in \nregion s ii-iv, where the transition -metal d-states and the Al -p states were found to show \nstrong hybridization . The  strongly diffuse d bands (in red) in BSF  in the energy range (ii -\niv) also agrees well with PDOS analysis . The strong hybridization found between Ni -3d \nand Al -2p bands at -0.20 mRy in Fig. 4b directly connects with dominant SRO pair with \nH-point ordering in Fig. 3. \n \n14 \n Figure 4. (a) Bloch -spectral function s (i.e., electronic dispersion , with broadening du e to \nchemical disorder ), (b) partial density of states, and (c) short -range order (in Laue) along high -\nsymmetry directions  (-P-N--H) of a BCC Brillouin zone. Peak s at H=(111)  indicate B2 -type \nSRO  dominated by Al -Ni pair. The possib ility of  secondary ordering  phase arises due to \npresence of stronger ordering peak at H  in Ni -Al pair  and weaker ordering peak at P  in Al-Ti \npair. The SRO peaks at H+P are suggestive of L2 1 phase [ 25].  \n \nIndeed, the expectation of low-temperature order ing due to increased \nhybridizat ion among alloying elements was also confirmed by the presence of strong \nSRO peaks at H -point  (indicating B2 -type mode) and H+P -point  (indicating L2 1-type \nmode)  for Cr 0.20Ni0.10Al0.25Ti0.10Fe0.35  HEFS in Fig. 4c. The maximal SRO peak at H -\npoint in Fig. 4c shows Ni -Al dominated B2 -type ordering. The strong Ni -Al peak at H -\npoint is followed by Fe -Al and Ni -Ti SRO pairs . A fairly strong Fe-Al SRO can be \nattributed to the larger solubility of Fe than Cr at low-temperature in ordering phases. \nAlso, a weaker Cr-Ni peak  in SRO at -point  in Fig. 4c  indicat es the tendency of \nsegregation, i.e., energetically Cr and Ni do not prefer same neighborin g environment \n[28]. Note  that a weak secondary Al -Ti peak at P -point as shown in inset of Fig. 4c is \nindicative of  Ti enriched B2 phase. The presence of a weak ordering  peak  at P along \nwith strong ordering peak  at P is consistent with coexist ant B2 and L2 1 phases as \nreported by Wolf -Goodrich et al.  [29] in nearly same composition as \nCr0.20Ni0.10Al0.25Ti0.10Fe0.35  HEFS .  \nTo explore the possibility of coexistent phases, we extracted H- and P -point \neigenvectors from the  SRO analysis above phase decomposition temperature for \nanalytically solving CW  Eq. (1) [37]. The CW analysis shows that partially  ordered B2 \nand L2 1 phases of Cr0.20Ni0.10Al0.25Ti0.10Fe0.35  HEFS exhibit lower energies than BCC by \n–35.22  meV-atom-1 and –53.72  meV-atom-1, respectively. Notably, Amalraj et al.  [41] \nalso observed L2 1 peaks in Cr 0.20Ni0.10Al0.25Ti0.10Fe0.35. Hence, the phases (and their 15 \n estimated energy gains) initially indicated by the SRO in Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 in \nFig. 3a  (same in Fig. 4c ) shows a good agreement with recent experiments [ 29,41 ]. \n The SRO in Ti-rich ( Cr0.20Ni0.10Al0.25Ti0.10Fe0.35) and no -Ti (Cr 0.20Ni0.10Al0.30Fe0.40) \ncases in Fig. 5a -b were compared to better understand the effect of Ti. We found that \npronounced secondary -ordering peak at P -point (Al -Ti pair; see inset) in Fig. 5a  \ndisappears when Ti is reduced to 0 at. -frac. in Cr0.20Ni0.10Al0.30Fe0.40 HEFS in Fig. 5b . \nUnlike Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 (with strong B2 (H -point)  ordering  along with possible \nL21 ordering  in Fig. 5a ), the Cr 0.20Ni0.10Al0.30Fe0.40 HEFS only shows possible B2 \nordering with no -sign of L21 (in strong agreement with observations of Wolf-Goodrich et \nal. [29]). \n \nFigure 5. The short -range order ing pairs  for (a) Cr0.20Ni0.10Al0.25Ti0.10Fe0.35  and (b) \nCr0.20Ni0.10Al0.30Fe0.40 HEFSs are shown a long high -symmetry directions ( -P-N--H) of BCC \nBrillouin zone. The p otential secondary  ordering peak at P (L2 1) in (b) disappears at Ti=0 \natomic -fraction .  \n16 \n  The direct energy calculation of low temperature ordering phases (B2 and L2 1) \nand comparing them with disorder phase (BCC) will allow us to establish the fact that \npredicted incipient long -range order phases may exist [ 28]. However, the determination \nsublattice occupation (on phase decomposition of disorder phase) is needed for energy \ncalculation of ordering phases  using DFT, which remains unk nown . The concentration \nwave in Eq. (1) [ 42,44 ] was used to estimate the occupation probabilities of each \nalloying element in possible ordering phases (B2 and L2 1 are two such possibilities in \ndisorder BCC phase) of Cr-Ni-Al-Ti-Fe HEFSs, which is required by DFT for direct \nenergy  calculation.   \n The CW Eq. (1) was rewritten using the  eigen -vector information extracted from \nSRO analysis of Cr0.20Ni0.10Al0.25Ti0.10Fe0.35  HEFS below the phase decomposition \ntemperature as  \n[𝑛(𝐶𝑟)\n𝑛(𝑁𝑖)\n𝑛(𝐴𝑙)\n𝑛(𝑇𝑖)]=[0.20\n0.10\n0.25\n0.10]+𝜂\n2[ 0.3550\n    0.3955   \n 0.7603\n−0.3736]exp[2×𝜋×𝑖×𝒓×(111) ]      Eq. (2)  \nhere, the CW equation is solved for unknown order parameter ` 𝜂’. The factor 𝛾=1\n2 \ncomes from the symmetr y of the BCC cell with two lattice cites, i.e., r= (000) and (½ ½ \n½). The eigen -vector, 𝑒(H)=(0.3550,  0.3955,  0.7603 ,−0.3736), required to solve Eq. \n(1) related to H -point ordering were extracted from SRO calculation at 1.15T sp. The \neigen -vectors were taken at 1.15 times above spinodal temperature (T sp), where SRO of \none of the dominant pairs diverges or `inverse -SRO’ goes to zero, i.e., 𝛂𝜇𝜈−1(ko;Tsp)=0.  \n The estimated  occupation at r1=(000) and r2= (½ ½ ½) lattice positions  in B2 \nphase using Eq. (2) are (Cr= 0.30, Ni=0.20, Al=0.45, Fe=0 .05) and (Cr= 0.10, Al=0.05, 17 \n Ti=0.20, Fe= 0.65) at. -frac., respectively. Similarly, the occupation probability in  L21 \nphase can be estimated at three symmetry inequivalent sites, namely,  r1=(000), r2=(½ ½ \n½), and r3=r4=(¼ ¼ ¼) & (¾ ¾ ¾)   as (Cr=0.25254, Ni= 0.17051, Al= 0.31826, Ti= 0.20, \nFe=0.05869), (Cr=0.14746, Ni= 0.02949, Al= 0.18174, Fe= 0.64131), and (Cr= 0.200, \nAl=0.10.0, Ni=0.250, Ti= 0.100, Fe= 0.350) at. -frac., respectively .  \n The concentration -wave analysis combined with direct DFT calculations reflects \nthe energy stability of B2 and L2 1 phases in BCC Cr0.20Ni0.10Al0.25Ti0.10Fe0.35. The \nformation energy difference of B2 and L21 phases with respect to BCC phase is  \n∆Eform(B2−BCC)= –35.22 meV-atom-1 and ∆Eform(L21−BCC)=–53.72 meV-atom-1, \nrespectively , which  shows  that B2 and L2 1 phases are energetically more stable than \nBCC.  Notably, L2 1 phase is the energetically most stable phase of all three . The SRO \ncalculations in Fig. 3a  predicted the possibility of L21 phase,  which was recently \nobserved by Wolf -Goodrich et al.  [29]. The H= [111] point instability in \nCr0.20Ni0.10Al0.25Ti0.10Fe0.35  shows B2 -type ordering in Fig. 5a , which is dominated by Ni -\nAl SRO pair. The ordering behavior arises from the filling of bonding states that results \ninto strong hybridization as shown in Fig. 4a  through diffused BSF near Fermi -level. \nThe stronger hybridization in BSF indicates increased charge -fluctuations among \nvarious alloying elements in Cr0.20Ni0.10Al0.25Ti0.10Fe0.35  HEFS. The increased charge -\nfluctuations at lower temperatures can lead to the development of short -range order, \ne.g., B2 and L2 1 in BCC alloys, which enhances the hybridization among various \nalloying elements of complex alloy systems.  \n On the other hand, the formation energy  difference of  B2 and L2 1 ordering \nphases  with respect to disorder (BCC) phase  of Cr0.20Ni0.10Al0.30Fe0.40 HEFS , i.e., 18 \n ∆Eform(B2−BCC)= –23.53 meV-atom-1 and ∆Eform(L21−BCC)=–23.55 meV-atom-1, \nshows that ordering phases are energetically degenerate.  This further establishes that \nadding Ti to Ni -Cr-Al-Fe HEFSs plays a crucial role in stabilizing L2 1 phase.  \n The total density of states (TDOS)  is also a  good indicator of alloy stability, for \nexample, peak or high density of electronic -states at Fermi -level leads to instability in \nalloy whereas valley (pseudo -gap) or very -low densities suggest stability [ 36]. We \nperformed electronic -structure calculations of Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 and \nCr0.20Ni0.10Al0.30Fe0.40 HEFSs and  show the TDOS of disorder ( BCC ) and partially ordered \n(B2/L2 1) phases in Fig. 6a -b. The TDOS of  Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 in Fig. 6a shows \nthat both majority -spin and minority -spin channel  have a pseudo -gap at the Fermi \nenergy, indicative of increased energy stability [ 36]. The predicted Eform of -12.27 meV-\natom-1 (BCC), -47.49 meV-atom-1 (B2), and -65.99 meV-atom-1 (L2 1) in disorder and \npartially ordered phases also confirms our analysis. In Fig. 6b,  the BCC phase shows \npseudo -gap region both in up -spin and down -spin channel, which suggests strong \nstability . The low E form of -21.42  meV-atom-1 also confirms our hypothesis . The TDOS of \nB2 and L2 1 phases are identical and show strong pseudo -gap region in up -spin \nchannel, however, down -spin channel shows a peak structure just below the Fermi -\nlevel . Unlike TDOS of Cr0.20Ni0.10Al0.25Ti0.10Fe0.35  HEFS in Fig. 6a, the presence of large \nelectronic density of states in Cr0.20Ni0.10Al0.30Fe0.40 HEFS in Fig. 6b  at Fermi -level leads to \nweaker change in energy stability in B2 (-23.53 meV -atom-1) and L2 1 (-23.55 meV -atom-\n1) phase s compared to disorder phase . 19 \n  \nFigure 6. Total density of states  of (a) Cr0.20Ni0.10Al0.25Ti0.10Fe0.35, and (b) Cr0.20Ni0.10Al0.30Fe0.40 \nHEFSs in BCC (grey region) , B2 (blue lines) , and L2 1 (red lines) phases.  \n \nConclusion  \nIn summary, the density -functional theory based linear -response theory was \nused to directly calculate the short -range -order for all atomic pairs simultaneously \nrelative  to the homogeneously disordered BCC phase . We show ed that the order -\ndisorder transformation,  i.e., BCC -to-B2 and BCC -B2-L21, can be controlled by \ncompositional tuning. The proposed hypothesis of SRO -controlled ordering \ntransformation  was exemplified  in Cr-Ni-Al-Ti-Fe based ferritic -steels , and we show that \nthe predicted ordering pathways are in good agreement with existing experiments . Our \ncalculation s also indicate the possibility of coexistence of  order ing phases such as  B2 \nand L21 below phase decomposition temperature . This study further emphasizes that  \n20 \n SRO is important both from fundamental and application point of view  as it is known to  \naffect  phase selection [ 16] and mechanical response [ 18]. Therefore, the tunability of  \nSRO in multi -principal element alloys using purely chemistry  provides unique insights \nfor controlling phase transformation, which shows the usefulness of our theory guided \ndesign of next generation high -entropy ferritic steels.  \nAcknowledgements  \nPS would like to thank Dr. Michael Gao (NETL) and JMR for the invitation to  contribute \nto the  JMR Early Career Scholars  in Materials Science 2022 . We thank Dr. Marshal \nAmalraj at Aachen  University for fruitful  discussion s. Work at Ames Laboratory was \nsupported by the U.S. Department of Energy ( DOE ) Office of Science,  Basic Energy \nSciences, Materials Science & Engineering Division. Research was performed at Iowa \nState University and Ames Laboratory, which is operated by ISU for the U.S. DOE \nunder contract DE -AC02 -07CH11358.  \n \nReferences  \n \n1. J.-W. Yeh, S. -K. Chen, S. -J. Lin, J. -Y. Gan, T. -S. Chin, T. -T. Shun, C. -H. Tsau and \nS.-Y. Chang. 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Staunton, and H. Winter . A first -principles \ntheory of ferromagnetic phase transitions in metals . J. Phys. F. Met. Phys. 15, 1337 -\n1386 (1985).  \n40. W. Hume -Rothery, R.W. Smallman, and C.W. Haworth, The Structure of Metals and \nAlloys. The Institute of Metals, London, 1969 . \n41. M. Amalraj et al. Effect of synthesis temperature on the phase formation of \nNiTiAlFeCr compositionally complex alloy thin films . J. Alloy Compd. 854, 155178 \n(2021).  \n42. A. Khatcharuryan, Theory of Structural Transformation in Solids  (Wiley, New York, \n1983).  \n43. A. Khachaturyan, Soviet Phys. JETP 36, 753 (1972).  \n44. A.K.D. Badalyan, A.G. Khachaturyan, Theory of order -disorder phase transformation \nin molecular crysta ls. 2, Soy. Phys. Crystallogr. 14, 333 (1969).  "    },    {        "title": "1010.1881v1.The_High_Field_Magnetization_in_the_RF_Sputter_Deposited_Copper_Ferrite_Thin_Films.pdf",        "content": "THE HIGH FIELD MAGNETIZATION IN THE RF SPUTTER \nDEPOSITED COPPER FERRITE THIN FILMS \n \n         Prasanna D. Kulkarni and Shiva Prasad               \n         Department of Physics                                        IIT Bombay, Mumbai 400076                             N. Venkataramani                                       \n         ACRE IIT Bombay                                                    \n         Powai, Mumbai 400076                                       R. Krishnan           Laboratoire de Magnetisme et d’op tique de Versailles,CNRS, 78935 Versailles, France  \n          Wenjie Pang, Ayon Guha, R.C. Woodward, R.L. Stamps \nSchool of Physics, M013, The University of Western Australia, 35 Stirling Hwy, Crawley WA \n6009 (Australia) \n  \n        ABSTRACT \nCopper ferrite thin films were deposited on amor phous quartz substrates. The as deposited films \nwere annealed in air and either quenched or sl ow cooled. Magnetization st udies were carried out \non the as deposited as well as annealed films using a SQUID magnetometer. The M-H curves were measured up to a field of 7T, at temperat ures varying from 5K to 300K. The magnetization \nin the films did not saturate , even at the highest fiel d. The expression, M(H)= Q(1- a /H\nn) fitted \nthe approach to saturation best with n=1/2, for a ll films and at all temperatures. The coefficient  a \nwas the highest for the as deposited film and was the smallest for the quenched film. In the case of as deposited film, the value of coefficient a increased with increasing temperature, while for \nthe annealed films, the value of a showed a decrease as temperature increases.  \n \nINTRODUCTION \nThe magnetization of nanocrystalline ferrite films does not saturate even at very high fields. This \nphenomenon is called the High Field Susceptibili ty (HFS). Margulies et al. have reported non-\nsaturation in magnetic fields upto 8T in Fe\n3O4 samples1, at room temperature. A detailed study of \nHFS in LiZn ferrite films at room temper ature is recently published by Dash et al.2. They attribute \nthis phenomenon to the def ects in the thin films, even though their exact or igin and nature are not \ndiscussed. In order to understand the reason for th is phenomenon, it is necessary to get more data \non this behavior.   In the present paper, a study of  the temperature dependence of the HFS on copper ferrite thin \nfilms is reported. Copper ferrite can  be stabilized in two different phases in thin film form, viz., a \ncubic and a tetragonal phase\n3. Quenching the copper ferrite films after carrying out a post \ndeposition annealing stabilizes the cubic phase. The slow cooling of the films after the annealing, \non the other hand, results in the tetragonal phase. High field magnetization studies are carried out \non both cubic and tetragonal phases of  the copper ferrite films.    \nEXPERIMENTAL \nCopper ferrite films were deposited using a Leybold Z400 rf sputtering system on amorphous \nquartz substrates. No heating or cooling was carried out dur ing sputtering. The rf power \nemployed during the deposition was 50W. The thic kness of the films was ~2400 Å. The films \nwere also annealed at 800 °C for 2 hours, followed by either quenching or slow cooling. The \nmagnetization was measured using a SQUID magneto meter for as deposited, quenched and slow \ncooled films in a field up to 7T and at various temperatures between 5K and 300K.  \n \nRESULTS  \nFig.1(a), shows the magnetization M( H) as a function of field H for the as deposited (Asd), slow \ncooled (SC) and quenched (Que) film at 300K . One can notice from the figure that the \nmagnetization of the Que film is the highest. This is because of  the cubic nature of the film3. One \nalso notices clearly that the magne tization of the films does not satu rate even at the highest field \nfor all the three films.                    Fig. 1. (a) M Vs H, (b) M(H)/M(7T) Vs H, for 50W copper ferrite thin films at 300K.  \n To assist the comparison of HFS,  the data of Fig.1(a) is replot ted in Fig.1(b), with M(H)/M(7T) \non the y-axis. One can very clearly see from these figures that the Asd film has the least tendency to saturate, while the Que film seems to get saturated comparatively easily.  \n In order to see the effect of temperature, M(H)/M(7 T) has been plotted in Fig. 2 (a) (b) and (c), as \na function of field at 5K and 300K for the Asd, SC and Que films. From Fig. 2, one observes an \ninteresting feature related to the temperature de pendence of high field su sceptibility. While the \nHFS increases upon lowering the temp erature in SC and Que films, it decreases for the Asd film. \n    \n0 20000 40000 60000050010001500200025003000\n Quenched\n Slow Cooled\n As Deposited\n  M (gauss)\nH (Oe)(a)\n0 15000 30000 45000 60000 750000.40.60.81.0 Quenched\n Slow cooled\n As deposited\n  M(H)/M(7T)\nH (Oe)(b) \n        \n \n                    Fig. 2. M(H)/M(7T) as a function of magnetic fi eld at 5K and 300K for (a) As deposited, (b) \nSlow cooled, (c) Quenched, 50W copper ferrite thin films.  \n DISCUSSION The approch to saturation is discussed by Chikazumi\n4. In the Chikazumi expression, the \nmagnetization at a given applied fi eld is written in the following way,               \n \n                     4 πM = Q (1 – a/H1/2 – b/H – c/H2 - ….) + eH                                                           (1) \n \nHere 4πM is the actual value of magnetization that is observed at a field H, and Q, a, b, c, and e \nare constants. The value of Q s hould correspond to a value of 4 πM in the infinite field. The last \nterm, eH, is caused by an increas e in spontaneous magnetization by the external field and is \ngenerally assumed to be  negligible. The c/H2 term is related to the presence of restoring torque \ndue to the magnetic anisotropy. Chikazumi has derived the expression for the coefficient c for the \ncase when the torque due to external field coun terbalances the restoring torque due to magnetic \nanisotropy. This coefficient c is related to the anisotropy c onstant and magnetization of the 0 15000 30000 45000 60000 750000.40.50.60.70.80.91.0\n 5K\n 300K\n  M(H)/M(7T)\nH (Oe)(a)\n0 15000 30000 45000 60000 750000.40.50.60.70.80.91.0\n 5K\n 300K\n  M(H)/M(7T)\nH (Oe)(b)\n0 15000 30000 45000 60000 750000.40.50.60.70.80.91.0\n 5K\n 300K\n  M(H)/M(7T)\nH (Oe)(c)0 50 100 150 200 250 30012162024283236404448 As deposited\n Slow cooled\n Quenched\n  a (Oe1/2)\nT (K)material. The observed b/H term can be explaine d if the restoring torque increases with the \napproach of magnetization to satu ration. If the magnetization is fixe d firmly due to point defects, \nthe magnetization surrounding these defects will form  transition layers that are similar to the \nordinary domain wall. The thickness of these transi tion layers will decrease with the increase in \nfield strength. In this case, the change of magnetization will be first proportional to the 1/H1/2 \nterm and finally to the 1/H2 term. Dash et al2 used an expression of the type a/H1/2 to fit their \nmagnetization data at high field and found that it fits thei r data best up to th e highest field. Based \non that they concluded that hi gh field magnetization in their f ilms is caused by point defects. \nIn order to quantify the high field susceptibility we  have fitted the magnetization data from  0.8T \nto 7T field to an expression involving just the a/H1/2 term in eq. (1). The value of a thus found is \nplotted in Fig. 3 as a function of temperature fo r Asd, SC and Que films. One can see that the \nvalue of a is the smallest for the Que films, which are cubic. \n                 Fig. 3. The variation of the coefficient a of H\n1/2 term with temperature for 50W films.  \n If the HFS phenomenon is due to the presence of defects as envisaged by Chikazumi, one indeed \nexpects that the slope a would be largest in Asd film, as these films are likely to be most \ndefective. Annealing is expected to cause reduc tion in the defects, which can explain why the \nslope decreases in the case of  Que and SC film. Even in a nnealed films, the value of a is smaller \nfor Que film than SC film. The Qu e film being cubic has lesser anis otropic field than the SC film, \nwhich is in tetragonal phase. This indicates a could be related to magneto crystalline an isotropy. If \nit is so a is also expected to increase with d ecreasing temperature which is observed.  \n It is known that the sputter deposited  ferrite films are nano-crystalline\n5. The grain sizes in the \nfilms are so small that an isolated grain of th at size would be superparamagnetic. However a M-H \nloop is observed in all these materials indicating the presence of magnetic order. This is most \nlikely because of the inter-granular interac tions, which may substantially modify the \nsuperparamagnetic behavior. Nevertheless the film may contain grains, which are \nsuperparamagnetic in addition to the one that are magnetically ordered. These superparamagnetic \ngrains may also contribute to the HFS.   If HFS is due to superparamagnetism, then  also one can understand the lower value of  a in \nannealed films. The fraction of th e superparamagnetic particles is likely to be less in the SC and \nQue film than in the Asd film. This is due to th e increase in the grain sizes  of the material with \nannealing. The annealed films still remain nano-crystalline5. However, in the case of \nsuperparamagnetic particles, the susceptibility  increases with the d ecrease of temperature \nfollowing Curie’s law6.  \nIf superparamagnetism is dominant in Asd films, a is expected to increase with lowering \ntemperature. The fact that this is not observed coul d be either because some grains get frozen into \nmagnetic order upon lowering the temperature or b ecause of some effect involving intergranular \ninteraction. The value of the coefficient a for the Que and the SC f ilms, on the other hand, shows \na behavior, which is somewhat similar to the sus ceptibility of superparam agnetic particles. But \nthese are the films in which the grain sizes are expected to be bigger, and they are likely to contain lesser number of superparamagnetic gr ains. The present results, thus, can not be \nunderstood purely in terms of superparamagneti c grains. A combination of superparamagnetic \nand anisotropic effects may be requi red to explain the HFS phenomenon. \n \nCONCLUSIONS \nThe coefficient of a/H\n1/2 term has the highest value for as deposited film and is lowest for \nquenched film. The temperature dependence of a shows that the value of a decreases with \ndecreasing temperature for as deposited film. For annealed films, however, a increases with \ndecreasing temperature. This can not be unde rstood purely on the basis of presence of \nsuperparamagnetic grains. A combination of  defects and superparamagnetism could be \nresponsible.  \nACKNOWLEDGEMENT \nThe author Prasanna D. Kulkarni thanks  the CSIR, India for financial support.  \n \nREFERENCES \n       1D.T. Margulies, F.T. Parker, F.E. Spada, R.S.  Goldman, J. Li, R. Sinclair, A.E. Borkowitz, \n“Anomalous moment and an isotropy behavior in Fe 3O4 flims,” Physical Review B,  53[14] 9175-\n9187 (1996-II). \n       2J. Dash, S. Prasad, N. Venkatara mani, R. Krishnan, P. Kishan, N.  Kumar, S.D. Kulkarni, S.K. \nDate, “Study of magnetization and crystallization in sputter deposited LiZn ferrite thin films,” \nJournal of Applied Physics , 86[6] 3303-3311 (1999). \n       3M.Desai, S. Prasad, N. Venkataramani, I. Sa majdar, A.K. Nigam, R. Krishnan, “Annealing \ninduced structural change in s putter deposited copper ferrite thin  films and its impact on magnetic \nproperties,” Journal of Applied Physics , 91[4] 2220-27 (2002). \n      4S. Chikazumi, S.Charap, “Law of approach to saturation,”; pp. 274-279 in Physics of \nMagnetism ,  John Wiley & Sons, New York (1964). \n       5M. Desai, J. Dash, I. Samajdar, N. Venkatarama ni, S. Prasad, P. Kishan, N. Kumar, “A TEM \nstudy on lithium zinc ferrite thin  films and the microstructure correlation with the magnetic \nproperties,” Journal of Magnetism and Magnetic Materials,  231 108-112 (2001).  \n       6C. P. Bean, J.D. Livingston, “ Superparamagnetism,” Journal of Applied Physics,  30[4] 120S-\n129 (1959). "    },    {        "title": "1603.05153v1.Molecular_dynamics_simulation_of_nanoindentation_on_nanocomposite_pearlite.pdf",        "content": "1 \n \n Molecular dynamics simulation of nanoindentation on nanocomposite pearlite  \nHadi Ghaffarian1,2, Ali Karimi Taheri1, Seunghwa Ryu*2 and Keonwook Kang*3 \n1 Departme nt of Materials Science and Engineering, Sharif Un iversity of Technology, Tehran, Iran \n2 Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology \n(KAIST), Daejeon, Korea 34141  \n3 Department of Mechanical Engineering, Yonsei University, Seoul, Korea  03722  \n*Corresponding Authors:  ryush@kaist.ac.kr , kwkang75@yonsei.ac.kr  \n \nPACS 87.10.Tf – Molecular dynamics simulation  \nPACS 61.72.Lk - Linear defects:  dislocations  \n \nABSTRACT -We carry out molecular dynamics simulations of nanoindentation to investigate the \neffect of cementite size and temperature on the deformation behavior of nanocomposite pearlite  \ncomposed of alternating ferrite and cementite layers . We find that, i nstead of the coherent transmission, \ndislocation propagates by forming a widespread plastic deformation in cementite layer. We also show \nthat increasing temperature enhances the distribution of plastic strain in the ferrite layer, which redu ces \nthe stress acting  on the cementite layer . Hence, thickening cementite layer or increasing temperature \nreduces the likelihood of dislocation propagation through the cementite layer. Our finding shed s a light \non the mechanism of dislocation blocking by c ementite layer in the pearlite . \n \nIntroduction - Pearlitic  phase, a lamellar structure composed of alternating layers of ferrite and \ncementite, plays an important role in determining the mechanical properties of steels,  such as toughness, \nstrength and formability  [1]. Compared to ferrite, pearlitic structure is k nown as a harder phase with \nhigher strength due to the presence of cementite lamellae, and its mechanical properties are \nsignificantly affected by the cementite microstructure  [2-3]. It has been found that the fine pearlite with \nsmall interlamellar spacing  (~100 nm) and narrow cementite lamellae (~10 nm) shows higher ductility \nthan coarse pearlite during plastic deformation [4 -8]. Cementite layer act as a  hard obstacle in front of \nferrite dislocations and causes to dislocation pile up at the ferrite/cementi te interface  [9]. However,  to \nthe best of our knowledge,  there has been no direct experimental or simulation study  on the mechanism \nof the dislocation blockage by the cementite layer.  \nNanoindentation is a mechanical test  which uses an indenter with a known  geometry to plunge into a \nspecific site of the specimen by applying an increasing load [ 10]. It is widely used to determine the \nmechanical properties of thin films to clarify  the effect of geometric confinement on mechanical 2 \n \n properties [1 1-14]. It can also be used to elucidate the cementite size effect on the deformation \nbehavior of pearlite phase at nano scales, while complex equipment  setup as well as the difficulty in \npreparing a sample with a desired cementite thickness  make s the test costly and time -consuming. These \ndifficulties can be circumvented by employing  atomistic simulations.  \nMD simulation has been widely used to virtually perform nanoindentation in order to investigate the \nelastic and plastic deformation mechanisms [1 5-20]. In this study, we per form a series of \nnanoindentation tests of ferrite -cementite nanocomposites using MD simulations to investigate  the role \nof cementite in blocking dislocation propagatio n in pearlite structure at various temperatures.  \n \nFig. 1. Illustration of  pearlite nanocomposite samples; (a) pure Fe, (b) P1, (c) P3 and (d) P5 samples. \nThe position of cementite layers have been pointed by the back arrows.  \nSimulation methods - Four ferrite -cementite nanocomposite samples with different thickness of \ncementite ( P=0, 1, 3,  and 5 layers of cementite  unit cells in the x-z plane) were prepared as depicted in \nfig. 1 to investigate the cementite size effect and the temperature effect on  the deformation behavior  of \npearlit e under indentation .  \nWe chose                ,   ̅              and     ̅            for the interface orientation \nrelationship between ferrite and cementite according to Bagaryatsky [2 1] (with regard to cementite \n3 \n \n lattice constants a= 5.05 Å, b=6.69 Å and c=4.49 Å).  Sample dimensions of pure Fe are 15 nm by  15 \nnm in the x-z plane and 8 nm in the y axis. The thickness  of Fe layers  is kept constant as 4 nm in \npearlite samples, while the thickness of cementite layer is varied as 0.67, 2.0 and 3.35 nm in P1, P3 and \nP5 samples, respectively. The interatomic poten tial of Fe -C developed by Liyanage et al. [22], based \non a modified embedded atom method (MEAM), was used to describe  interatomic forces. We found \nthat g eneralized stacking fault energy curves obtained from the MEAM potential are consistent with \nthose obta ined from ab initio calculations, as presented in Supplementary Materials . We also have used \nthe potential to investigate the origin of brittle -to-ductile transition of nano -crystalline cementite [2 3].  \nThe MD simulations were carried out using a parallel MD code, LAMMPS [2 4] with periodic \nboundary conditions in x and z dimensions at 100, 300 and 700 K. For each temperature, samples were \nrelaxed for 100 ps under zero pressure with Nose -Hoover isobaric -isothermal (NPT) ensemble. All \nsamples were indented wit h a constant velocity of 10 m/s along y direction by frictionless spherical \nindenter . We used three different indenter radii ( R=3 nm, 5 nm and 7 nm)  in order to investigated the \neffect of indenter radius on the dislocation propagation through the cementite  layer. The force on each \natom was calculated by:  \n  {             \n      (1) \nwhere K=10 (eV/Å3) is the specified force constant and r is the distance from the atom to the center of \nthe indenter. To prevent translation motion during indentation, we fixed the atoms located in the region \nwithin 7 Å from  the bottom of sample. The atomic arrangements of the samples, including dislocations, \nwere then visualized using centrosymmetry parameter  [25]. The atomic shear strain was also calculated \nusing the method by Shimizu et al. [26]. \nResult and Discussion - Figure 2(a) shows the load -displacement curves of nanoindentation for pure Fe  \nsample indent ed by 3nm radius indenter  at different temperature s. We find that the indentation load \ngradually increases with  the indenter movement up to a certain local maximum and suddenly drops due \nto dislocation emission from the ferrite region below the indenter tip. In fig. 2(a), the elastic responses \nare almost identical at different temperatures of 100 to 700 K. Our calculations show that t he Young’s \nmodul us of ferrite along the loading direction does not change significantly, when tem perature rises \nfrom 100 K to 700 K (see Supplementary Materials  for details).  This thermal insensitiveness is \nconsistent  with t he experimental data that the elastic constants of single crystal Fe change only by ~10 % \nfrom 100 to 700  K [27-28]. On the other  hand, the load required for the 1st dislocation emission 4 \n \n decreases with the increase in temperature . Also, t he increase in temperature leads to the reduction of \nthe indentation load in plastic flow region after  the first dislocation emission, and advances  the \nemission of subsequent dislocations due to the thermal softening.  \n \nFig. 2. Load -displacement curves from  nanoindentation test s by indenter with 3 nm radius; (a) P0 \nsample  at different temperatures , (b) pearlite structure with different cementite thickness at 100 K  and \n(c) P3 sample indented by different indenter radius at 300 K . The atomistic prediction of load -\ndisplacement and Hertzian contact theory are in  good agreement  until the 1st dislocation nucleation . \nArrows indicate first dislocation emi ssion s. \n5 \n \n The presence of cementite does not significantly affect the load -displacement curve s in Fig. 2(b), \nbecause the elastic moduli of ferrite and cementite are not appreciably different [ 6]. We calculate the \nindentation load for all samples by the Hertzian theory for semi -infinite thin films,  \n  ( \n )    ⁄   ⁄ (2) \nwhere E is the Young’s modulus, R is indenter radius and h is indentation  depth . The calculated \nYoung’s modul i of ferrite and cementite along the loading direction are 203 and 159 GPa at 100 K , \nrespectively (see Supplementary Materials  for details).  We use the Reuss model to estimate an effective \nmodulus of pearlite elasticity:  \n   \n  \n     \n   (3) \nwhere ρf and ρc are the volume fraction s of ferrite and cementite layer s in pearlite. From eq. (3), the \nvalue of effective modulus changes from 198 for P1 to 188 GPa for P5, respectively.  As depicted in fig. \n2(b), our MD simulation results are in  good agreement with prediction from the Hertzian  theory before \nthe first dislocation emission.  Similar load -displacement behaviors for P1, P3 and P5 samples are \nobserved at 300 K and 700 K.  \nFigure 2(c) illustrates the load -displacement curves of  the P3 sample indented b y different indenter \nradii at 300 K. The larger indenter develops higher indentation load, due to much larger contact area. \nOn the other hand, the increasing indenter radius causes to postpone dislocation emission due to the \nreduction of stress concentration below the indenter. Similar res ult has been reported by Ruestes et al. \nfor tantalum indentation [29].  \n \n6 \n \n Fig. 3. Dislocations evolution after 30 Å indentation  of indenter with 3 nm radius  in pure Fe,  P1, P3 \nand P5 samples  at (a) 100 K, (b) 300 K and (c) 700 K . Out of three pearlite samples, only in P1 , \ndislocations are formed in the bottom ferrite layer . Atoms were colored according to the \ncentrosymmetry parameter values between 1 (blue) and 25 (red) with number of neighbors of  14. Thus, \natoms on a perfect BCC  lattice were not visualized.  Cementite layers are illustrated by black arrows . \nWe present  atomic configurations  showing dislocation evolution  after 30 Å indentation  by indenter \nwith 3 nm radius  in pure Fe, P1, P3 and P5 samples at different temperatures  in fig. 3, where the \ncentrosymmetry parameter [2 5] was applied to screen out atoms on a perfect BCC lattice  site. We \nobserve  dislocation  formation  in the bottom ferrite layer only in P1 sample  at 100 K , while in P3 and \nP5 samples dislocations  are complete ly trapped in the top ferrite layer regardless of imposed \ntemperature.  \nDepend ing on the structural details dislocations can either transmit across the interface or be blocked  at \nthe interface. In the case of ferrite -cementite composite, dislocations could rarely pass through the \nincoherent bcc/orthorhombic interface of Bagaryatski ’s orientation relationship.  The likelihood of \ndislocation transmission is carefully studied by investigating transmission pathway index   [30]: \n  {   ( \n  \n  )   ( \n  \n  )                   \n              (4) \nwhere θ is the minimum angle between the intersection lines that each slip plane in ferrite and \ncementite makes with the interface, κ is the minimum angle between Burgers vector s of slip systems in \nferrit e and cementite , and θc and κc are limiting  angles of θ and κ for transmission . If the χ of a certain \npair of slip systems is equal to or close to unity, the dislocation transmission is favorable at the \ninterface  between the two slip systems  [31]. According to eq. (4), when either θ or κ exceeds their \ncorresponding thresholds ( θc=45° and κc=15° [ 30]), transmission is impossible. Considering  main slip \nsystems in ferrite ( {   }〈   ̅〉 and {   }〈   ̅〉) and cementite ( {   }〈   〉), there are only 5 distinct \ngeometrically efficient pathways with  > 0.9  among 24×6=144 possible combinations of slip system  \npairs (see Table 1). We find that the (110) plane is the slip plane of newly formed  dislocation loops in \nthe bottom ferrite layer of P1 sample . Acco rding to Table 1, the (110) slip plane  does not belong to any \npossible transmission pathway  on ferrite/cementite interface , and the scenario of dislocation \ntransmission can be excluded out.  Besides, the higher GSF energy in cementite [3 2-33] makes the \ndislocation nucleation  in cementite  harder than that in ferrite.  \nTable 1.  Efficient t ransmission pathways for the ferrite/cementite interface  of Bagaryatsky orientation \nrelationship.  7 \n \n No. Slip system in \nferrite  Slip system in \ncementite  χ \nvalue  \n1         ̅   \n[100](001)  1 2        ̅    \n3         ̅   \n4         ̅   \n5        ̅    \nThe incoherent  crystal structure of cementite layer acts as a barrier to dislocation transmission , which \nin turn develops stress concentration in cementite layer.  Thus, in P1, the developed stress is high \nenough to deform the thin cementite layer and nucleate s new dislocation loops in the bottom layer of \nferrite . In contrast, in P3 and P5 samples, dislocations were not formed  below the cementite layer due \nto smalle r compliance of thicker cementite layers to transmit stress to bottom ferrite layer .  \n \nFig. 4. Dislocations evolution after 30 Å indentation  of indenter with 5 nm radius  in pure Fe,  P1, P3 \nand P5 samples  at (a) 100 K, (b) 300 K and (c) 700 K . All conditio ns are same as Fig. 3.  \nIn order to investigate the effect of indenter radius on dislocation evolution during nanoindentation, \nadditional simulation tests were performed with 5 and 7 nm indenter radius. Figure s 4 and 5  illustrate \nthe dislocation structure  in nanocomposite  sample s by indenters with 5 and 7 nm radius, respectively . \nWe found higher dislocation activity in pure Fe sample when the indenter radius increases. Basically, \n8 \n \n the volume occupied by the indenter creates geometrically necessary dislocatio ns (GNDs). The volume \noccupied by the indenter with 3, 5 and 7 nm radius at 30 Å indentation depth is 18 π, 36π and 54 π nm3, \nrespectively.  Hence , the larger indenter generates higher GND density at the same indentation depth. In \nother words, larger indenter  radius induces higher stress on the cementite layer, which leads to \ndislocation nucleation below the thicker cementite layer. However, we find that no dislocation loops \nare formed  in the bottom ferrite layer  of P5 sample  regardless of  imposed temperatures  during \nindentation by indenter with 5 nm radius, while the 7 nm radius indenter generates new dislocations \nbelow the  cementite  layer  in all nanocomposite samples and at all temperatures. Thus, w e can conclude \nthat the dislocation blocking capability increases with the thickness of the cementite.  \n \nFig. 5. Dislocations evolution after 30 Å indentation  of indenter with 7 nm radius  in pure Fe,  P1, P3 \nand P5 samples  at (a) 100 K, (b) 300 K and (c) 700 K . Dislocations are formed in the bottom ferrite \nlayer  in all deformed samples. All conditions are same as Fig. 3.  \nComparison of figs. 3, 4 and 5 reveal the effect of temperature on the dislocation activity in the ferrite \npart. It  is interesting to note that the dislocation nucleation in bottom ferrite layer decreases as \ntemperature rises, while the activation of new slip systems are more facile in the ferrite at elevate \ntemperature [34]. As temperature  rises, the dislocations in  top ferrite part (or  pure Fe sample) become \nmore distributed,  and more frequently  annihilate  with each other.  This implies that the cementite layer \nexperiences  less stress concentration, implying smaller chance for dislocation nucleation in the bottom \nferrite layer.  \n9 \n \n  \nFig. 6. Percent of atoms with shear strain more than 0.1 in deformed samples; (a) temperature effect, (b) \nindenter radius effect and (c) cementite thickness effect. The indenter radius in (a) is 5 nm and the \ntemperature in (b) and (c) is 100 and 700 K, respectively.  \nIn order to clarify the deformation mechanism in nanocomposite pearlite structure, the fraction of \nhighly strained atoms in deformed samples  was count following the approach by Shimizu et al. [26]. \nFigure 6(a) shows the effect of temperature on the percent of atoms with shear strain more than 0.1 in \ndeformed samples after indentation of 30 Å by indenter with 5 nm radius  (see Supplementary Materials  \nfor details  of shear strain distribution during nanoindentation) . As temperature rises, the percent of high \nstrained atoms increases gradually for all deformed samples, implying more distributed deformation at \nelevated temperature. Same behavior was observed for other indenter radiuses.  \nOn the other hand, fig. 6(b) reveals the effect of indenter radius on the perce nt of high strained atoms. \nAn increase in indenter radius in fig. 6(b) increases the percent of high strained atoms dramatically, due \nto higher density of GNDs. However, we find in fig. 6(c) that the increasing cementite thickness \n10 \n \n significantly decreases t he percent of high strained atoms for all indenter radii at a constant temperature. \nIn other words, thicker cementite layer is more effective in trapping dislocation propagation and act as \nan obstacle for preventing plastic deformation of the bottom ferrit e layer. In contrast, at low er \ntemperature , thinner cementite layer is less likely  to sustain the localized stress  by the dislocation pile -\nup at the interface during the indentation.  \n \nConclusions -In this work, we performed molecular dynamics simulations  of the nanoindentation test to \ninvestigate the  role of cementite in blocking dislocation propagatio n in pearlite structure.  The load -\ndisplacement curves were found to change with temperature mainly in plastic region  and the presence \nof cementite layer do es not affect the elastic response  significantly.  However, the indentation load rises \nsignificantly with larger indenter radius size.  We find that the cementite layer acts as a hard obstacle to \ndislocation propagation in pearlite. We also show that increasin g temperature enhances the distribution \nof plastic strain in the ferrite layer, which reduces the stress acting  on the cementite layer. Thus, we \nfind that the dislocation propagation  is more likely to form  for a thin cementite layer at low temperature  \nwith large indenter radius.  \n \nAcknowledgements  H.G. and A.K.T. acknowledge the Research Board of Sharif University of \nTechnology, Tehran, Iran.  K. K.  and S. R. acknowledge  the financial support from Basic Science \nResearch Program through the National Research Foundation of Korea  (NRF) funded by the Ministry \nof Education (2013R1A1A2063917)  and (2013R1A1A 1010091) , respectively . \n \nReferences:  \n[1] Dollar  M. et al., Acta Metall.,  36 (1988 ) 311. \n[2] Umemoto M. et al., Mater. Sci. Forum,  426-432  (2003 ) 859. \n[3] Wang L. et al., Mater. Metall.,  4 (2005 ) 155. \n[4] Embury  J. D. et al, Acta Metall. , 14 (1966 ) 147. \n[5] Hong  M. H. et al., Metall. Mater. Trans . A, 30 (1999 ) 717. \n[6] Langford  G., Metall. Trans. A,  8 (1977 ) 861. \n[7] Pepe  J.J., Metall. 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A,  8 (1977 ) 1689.  \n Supplementary materials for  \n“Molecular dynamics simulation of nanoindentation on nanocomposite \npearlite” \nHadi Ghaffarian1,2, Ali Karimi Taheri1, Seunghwa Ryu*2 and Keonwook Kang*3 \n1 Department of Materials Science and Engineering, Sharif University of Technology, Tehran, \nIran \n2 Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology \n(KAIST), Daejeon, Korea 305-701 \n3 Department of Mechanical Engineering, Yonsei University, Seoul, Korea 120-749 \n*Corresponding Authors: ryush@kaist.ac.kr , kwkang75@yonsei.ac.kr \n \n1. Generalized stacking fault energy of (010) plane in cementite:  \nIn order to test the ability of MEAM empirical potential to reproduce  plasticity, \ndislocations, and the slip system of the cementite, we calculated generalized stacking fault (GSF) \nenergy of the (010) plane in cementite and compared with the ab initio  results. The (010) planes \nare grouped into two types; one type is a plane between two neighboring layers of only Fe atoms (labeled as type I hereinafter) while the other is the one between Fe and Fe/C layers (labeled as \ntype II). The stacking fault (SF) energy along th e [100] and [001] directions on both types of \n(010) planes was calculated by introducing relative slip across the slip plane in the crystal and accomplishing relaxation at T=0 K as introduced in [ 1]. For ab initio  calculations, we used the \nVienna ab initio simulation package (VASP) [ 2] with the local density approximation (LDA). A \nsimulation cell of [100] ⅹ3[010]ⅹ[001] containing 48 atoms (36 Fe atoms and 12 C atoms) was employed under periodic boundary conditions imposed along [100] and [001].  The k-points are \nsampled by a 4 ⅹ1ⅹ4 Monkhorst-Pack method and the cut-off energy for the plane wave is \n38.22 Ry. The results of molecular statics (MS) and ab initio  simulation are shown in fig. S1. \nEvery local minimum in fig. S1(a) indicates formation of a partial dislocation along the [100] in \n(010) slip planes in cementite. Both MS and ab initio  agree that partial formation is more \npreferable in type I plane or a plane between only Fe-atom layers. On the other hand, in fig. \nS1(b), no local minima is observed along the [001 ] direction on type II cross section in both MS \nand ab initio .  A plateau region is predicted on type I cross section by both MS and ab initio , and \nthe ab initio  curve has wider plateau with small local minima. However, at finite temperature, the \neffect of such a small minima would be negligible. Overall, the MEAM potential captures the characteristic features in GSF curves qualitatively well, and is considered as a reliable model to \nsimulate plastic event in cementite. Stable stacking fault energy and the corresponding \ndisplacement are summarized in Table S1.  \nBesides, our results are consistent with experimental observation, where the embedded \ncementite particles in ferrite matrix are deformed through the movement of stacking faults, \nwhich characterized by an α[100](010) partial dislocation [3-4]. The GSF energy of (010) \ncementite slip plane was also calculated using ab initio method. Our results are also consistent \nwith ab initio results, where the local minima on ly observed along [100] direction on both cross \nsection of (010) cementite plane. \n \nTable S1.  Stable stacking fault energy \n and corresponding displacement x in the \n(010)[100] slip system. In ME AM MS, the lattice constants are a = 5.05, b=6.69, and c=4.49 (Å) \nand those in ab initio are a = 4.926, b = 6.623, c =4.377 (Å).  MEAM MS Ab initio \n (eV/Å2) x  (eV/Å2) x \nSection I 0.079 0.4[100] 0.059 0.3[100] \nSection II 0.096 0.49[100] 0.082 0.3[100] \n \n \nFig. S1. Stacking fault energy of (010) plane along (a) [100] and (b) [001] directions with \ndifferent types of cross section. Circles and triangles indicate MS results and ab initio results, \nrespectively.  The local minima indicate favorable displacement of atoms. \n2. Calculation of elastic mo duli of ferrite and cementite \nIn order to obtain the Young’s moduli of ferr ite and cementite, we calculated the elastic \nconstants of ferrite and cementite at 100 K, 300 K and 700 K using Liyanage’s MEAM potential \n[5]. The each component of elastic modulus tensor was computed from the stress-strain curves obtained by MD simulations at a given temperature in the NVT ensemble. Due to the complex \nscreening function of the MEAM potential used in this study, elastic softening was observed \nonly at the temperatures higher than 300 K. Such anomaly in elastic response would not have a considerable effect on the plastic response observed in this study. \nܥ\nିଵ ൌ\nۏێێێێێێێۍܿଵଵ ܿଵଶ ܿଵଶ 000\nܿଵଶ ܿଵଵ ܿଵଶ 000\nܿଵଶ ܿଵଶ ܿଵଵ 000\n000 ܿସସ 00\n0000 ܿସସ 0\n0 0000 ܿସସےۑۑۑۑۑۑۑې\nൌ\nۏێێێێێێۍ220 147 147 000\n147 220 147 000\n147 147 220 000\n000 126 00\n0000 126 0\n0 0000 126 ےۑۑۑۑۑۑې\n \n(1) \nܥିଵ ൌ\nۏێێێێێێێۍܿଵଵ ܿଵଶ ܿଵଷ 000\nܿଵଶ ܿଶଶ ܿଶଷ 000\nܿଵଷ ܿଶଷ ܿଷଷ 000\n000 ܿସସ 00\n0000 ܿହହ 0\n0 0000 ܿےۑۑۑۑۑۑۑې\nൌ\nۏێێێێێێۍ308 142 164 000\n142 233 119 000\n164 119 325 000\n000 24 00\n000 0 99 0\n0 00 0 0 57ےۑۑۑۑۑۑې\n \n(2) \nܥିଷ ൌ\nۏێێێێێێێۍܿଵଵ ܿଵଶ ܿଵଶ 000\nܿଵଶ ܿଵଵ ܿଵଶ 000\nܿଵଶ ܿଵଶ ܿଵଵ 000\n000 ܿସସ 00\n0000 ܿସସ 0\n0 0000 ܿସସےۑۑۑۑۑۑۑې\nൌ\nۏێێێێێێێۍ222 145 145 000\n145 222 145 000\n145 145 222 000\n000 127 00\n0000 127 0\n0 0000 127 ےۑۑۑۑۑۑۑې\n \n(3) ܥିଷ ൌ\nۏێێێێێێێۍܿଵଵ ܿଵଶ ܿଵଷ 000\nܿଵଶ ܿଶଶ ܿଶଷ 000\nܿଵଷ ܿଶଷ ܿଷଷ 000\n000 ܿସସ 00\n0000 ܿହହ 0\n0 0000 ܿےۑۑۑۑۑۑۑې\nൌ\nۏێێێێێێێۍ280 145 158 000\n145 240 121 000\n158 121 308 000\n000 28 00\n000 0 96 0\n0 00 0 0 61ےۑۑۑۑۑۑۑې\n \n(4) \nܥି ൌ\nۏێێێێێێێۍܿଵଵ ܿଵଶ ܿଵଶ 000\nܿଵଶ ܿଵଵ ܿଵଶ 000\nܿଵଶ ܿଵଶ ܿଵଵ 000\n000 ܿସସ 00\n0000 ܿସସ 0\n0 0000 ܿସସےۑۑۑۑۑۑۑې\nൌ\nۏێێێێێێۍ214 136 136 000\n136 214 136 000\n136 136 214 000\n000 126 00\n0000 126 0\n0 0000 126 ےۑۑۑۑۑۑې\n \n(5) \nܥି ൌ\nۏێێێێێێێۍܿଵଵ ܿଵଶ ܿଵଷ 000\nܿଵଶ ܿଶଶ ܿଶଷ 000\nܿଵଷ ܿଶଷ ܿଷଷ 000\n000 ܿସସ 00\n0000 ܿହହ 0\n0 0000 ܿےۑۑۑۑۑۑۑې\nൌ\nۏێێێێێێێۍ260 150 143 000\n150 244 112 000\n143 112 287 000\n000 29 00\n000 0 93 0\n0 00 0 0 59ےۑۑۑۑۑۑۑې\n \n(6) \n \nFerrite and cementite have body centered cubic (BCC) and orthorhombic structures, \nrespectively. For arbitrary loading direction, the Young’s moduli in the cubic and orthorhombic symmetry crystals are given by the following equations, respectively [6]: \n1ܧ\n௧ ൌܵ ଵଵሾܵସସെ2ሺܵଵଵെܵ ଵଶሻሿሺܽଵଵଶܽଵଶଶܽଵଶଶߙଵଷଶܽଵଷଶܽଵଵଶሻ ⁄  \n(7) \n1ܧ ௧௧⁄ ൌܵ ଵଵܽଵଵସܵ ଶଶܽଵଶସܵ ଷଷܽଵଷସܽଵଵଶܽଵଶଶሺ2ܵଵଶܵ ሻ\n                                  ܽଵଵଶܽଵଷଶሺ2ܵଵଷܵ ହହሻܽଵଶଶܽଵଷଶሺ2ܵଶଷܵ ସସሻ (8) \nwhere, E is Young’s modulus, Sij are compliance elastic constants ( S=C-1), and  a1i are direction cosines of the arbitrary tensile direction represented in the symmetry axes. Table S2 shows the \ncalculated Young’s moduli of ferrite and cementite along  ሾ112തሿf and [010] c at 100, 300 and 700 \nK: \nTable S2.  Calculated Young’s moduli of ferrite and cementite along ሾ112തሿf and [010] c at 100, \n300 and 700 K . \n \nTemperature (K) Eferrite along  ሾ112തሿf (GPa) Ecementite  along [010] (GPa) \n100 203 159 \n300 209 158 \n700 208 154 \n \nThe effective elastic moduli can also be obtained from Ruess model for P0, P1, P3 and P5 \nsamples: Table S3.  The effective Young’s moduli of P0, P1, P3 and P5 samples along y direction at 100, \n300 and 700 K . \nSample Name Elastic modulus \nat 100 K (GPa) Elastic modulus \nat 300 K (GPa)Elastic modulus \nat 700 K (GPa) \nP0 203 209 208 \nP1 198 203 202 \nP3 192 196 194 \nP5 188 191 188 \n \n 3. Atomic Shear Strain Analysis \nTo further investigate the deformation mechanisms in detail, an atomic sh ear strain analysis was \nperformed following the approach by Shimizu et al. [7]. Figure S2 presents the atoms with shear \nstrain more than 0.1 in deformed samples at 30 Å indentation depth by 3 nm radius indenter at different temperatures. In pure Fe sample, shear strain spread laterally at 100 K, and increasing \ntemperature causes more depth-wise spreading of the strained region. However, due to the \nexistence of cementite layer, different deformation behavior is observed in P1, P3 and P5 samples, where lateral spreading of the strained region is more pronounced at elevated \ntemperature as shown in fig. s2(b)-(c). \nFigures S3 and S4 also represent the shear strain distribution in deformed sample by indenter with 5 and 7 nm radius, respectively. Comparison of shear strain distribution in figs. S2, S3 and \nS4 reveals the effect of indenter radius on the strain distribution in cementite layer. While the \nshear strain is almost trapped by the cementite layer in fig. S2 (3 nm indenter radius), the strained region spreads below the thicker cementit e layer in more samples when indenter radius \nincreases in figs. S3 and S4 due to higher dislocation activities.  \nFigure S2 . Atomic shear strain distribution of Fe and C atoms after 30 Å indentation of 3 nm \nindenter radius at (a) 100 K, (b) 300 K and (c) 700 K. Only atoms with shear strain more than 0.1 \nhave been showed. Dashed lines illustrate cemen tite layers and red color points indicate the \natoms with a shear strain equal or more than 1. \n \nFig. S3 . Atomic shear strain distribution of Fe and C atoms after 30 Å indentation of 5 nm \nindenter radius at (a) 100 K, (b) 300 K and (c) 700 K. Only atoms with shear strain more than 0.1 \nhave been showed.  \n \nFig. S4 . Atomic shear strain distribution of Fe and C atoms after 30 Å indentation of 7 nm \nindenter radius at (a) 100 K, (b) 300 K and (c) 700 K. Only atoms with shear strain more than 0.1 \nhave been showed.  \n \nReferences: \n[1] Kang K. et al., Philos. Mag.,  87 (2007) 2169. \n[2] Kresse G. et al., Phys. Rev. B,  54 (1996) 11169. \n[3] Inoue A. et al., Metall. Trans. A,  8 (1977) 1689. \n[4] Kar’kina L.E. et al., phys. Met. Metallogr.,  114 (2013) 234. \n[5] Liyanage L.S.I. et al., Phys. Rev. B,  89 (2014) 094102. \n[6] Lee D.N. et al., Solid Mech. Mater.  Eng.,  6 (2012) 323. \n[7] Shimizu F. et al., Mater. Trans., 48 (2007) 2923.  \n"    },    {        "title": "2303.17070v1.Towards_Quantitative_Analysis_of_Deuterium_Absorption_in_Ferrite_and_Austenite_during_Electrochemical_Charging_by_Comparing_Cyclic_Voltammetry_and_Cryogenic_Transfer_Atom_Probe_Tomography.pdf",        "content": " 1 Towards Quantitative Analysis of Deuterium absorption in Ferrite and Austenite during Electrochemical \nCharging by Comparing Cyclic Voltammetry and Cryogenic Transfer Atom Probe Tomography.  \nDallin J. Barton1, Dan-Thien Nguyen1, Daniel E. Perea2, Kelsey A. Stoerzinger3, Reyna Morales Lumagui1, \nSten  V. Lambeets1, Mark G. Wirth2, Arun Devaraj1* \n1Physical and Computati onal Sciences Directorate, Pacific Northwest National Laboratory, Richland, USA  \n2Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, USA  \n3School of Chemical, Biological and Environmental Engineering, Oregon State University, Corvallis, OR \n97331, United States of America  \nCorresponding author: arun.devaraj@pnnl.gov  \nAbstract  \nHydrogen embrittlement mechanisms of steels have  been studied for several decades. Understanding \nhydr ogen diffusion behavior  in steels  is crucial towards both developing predictive models for hydrogen \nembrittlement and identifying mitigation strategies. However, because  hydrogen has a  low atomic mass , \nit is extremely challenging to detect by most analytical methods. In recent years, cryogenic -transfer atom \nprobe tomography  (APT)  of electro chemically -deuterium -charged steels has provided i nvaluable \nqualitative analysis of nanoscale deuterium traps such as  carbides, dislocations, grain boundaries and \ninterfaces between ferrite and cementite.  Independently, cyclic voltammetry (CV) has provided valuable \nanalysis of bulk hydrogen diffusion in steels.  In this work , we use a combination of CV  and cryogenic -\ntrans fer APT for quantitative analysis of deuterium pickup in electrolytically charged  pure Fe  (ferrite)  and \na model austenit ic Fe18Cr14Ni alloy  without any second phase or defect trap sites . The high solubility and \nlow diffusivity of hydrogen in austenite versus ferrite are  highlighted to result in clear observable \nsignatures  in CV and cryogenic -transfer APT results . The remaining challenges and pathway for enabling \nquantitative analysis of h ydrogen pick up in steels is also discussed.  \n \n   2 1. Introduction  \nHydrogen embrittlement (HE) is the reduction in  the ductility  of an alloy due to elevated  internal H \nconcentration . HE occurs  in high strength ferriti c/martensitic steels and  face -centered cubic  (FCC)  \naustenitic stainless steels (γ -SS). Fractography with visible -light microscopy shows that γ -SS experiences a \nductile -to-brittle transition with increasing H concentration [1, 2] . American Iron and Steel Institute (AISI) \nstandardized  304 stainless steels  can achieve over 80 % strain before failure, but cannot achieve a 60  % \nstrain after being charged with H [3 -7]. Other  stainless steels such  as AISI 316, 321, and 347 all suffer a \nsimilar degradation in ductility after hydrogen charging [8 -11]. This reduction of ductility in γ -SS leads to \npremature fracture in environment s and processing conditions that encourage hydrogen absorption  and \nlimits engineering applications.  Therefore, u nderstanding the hydrogen distribution in steels is important \nto develop strategies to mitigate the deleterious effects of hydrogen on the mechanical properties.  \nTwo commonly attributed mechanisms of HE is  hydrogen enhanced decohesion (HEDE) and hydrogen \nenhanced local plasticity (HELP). HEDE suggests that elevated H concentration in materials lowers the \ncohesive strength causing brittle failure [12, 13] . HELP on the other hand suggests that hydrogen \ninterstitials enhance  dislocation mobility, and hence, plasticity [14-16]. In the case of stainless steel, both \nHEDE and  HELP have been identified as the main contributors to HE [17 -24]. A predictive model \nattempting to combine the mechanisms of HEDE and HELP has suggested a strong dependence on both \nhydrogen conc entration and strain [25] . Additional mechanisms such as adsorption -induced dislocation - \nemission  (AIDE) and hydrogen enhancement of the strain -induced generation of vacancies  (HESIV ) are \nalso postulated to be alternate mechanisms [26, 27] . Any unified theory of HE will require experimental \nconfirmation including the spatial measurement of H distribution in materials.  \nTo verify which hydrogen embrittlement mechanism  is responsible for HE of specific steels, hy drogen \nconcentration must be measured with high spatial resolution and compositional sensitivity. Methods such \nas desor ption analysis  (thermal [28-34] and electron -induced [35]), and e lectrochemical methods \nincluding cyclic voltammetry  (CV)  and chronoamperometry  [36-39] have been used to measure bulk \ndissolved hydrogen concentration in alloys . Among these , CV has  recently emer ged as a valuable  \nensemble average technique  for analyzing hydrogen diffusivity in steels [36-39]. CV involves applying a \nvariable potential difference between the working electrode and reference electrode  and measures the \nresulting current as the potential is varied to study the reduction and oxidat ion reactions of materials , \nproducing a voltammogram.  Features  in the voltammogram before and after electrochemical hydrogen \ncharging can be used to analyze hydrogen incorporation in  steels  [36-39].  3 Mass spectrometry techniques such as  secondary ion mass spectrometry  (SIMS ) and atom probe \ntomography  (APT) can provide  spatially resolved analysis of hydrogen concentration  in materials.  SIMS \nhas been used to confirm elevated deuterium  (2H) concentrations at crack tips of 2H-charged γ -SS [40], as \nwell as higher  2H in FCC phases of γ- SS than body -centered cubic  (BCC)  phases [41, 42] . APT can provide \nchemical and isotopic  information  in three -dimensional space  at resolutions smaller than 1 nm.  Isotope \nseparation allows APT to separate a deliberate  charg ing of  2H from artificial 1H introduced through sample \npreparation or analysis [43]. \nIn materials with  deeper  trap sites  for hydrogen,  observable deuterium can remain  even  after room -\ntemperature transfer . For example,   deuterium  in poly -crystalline Si has been unambiguously measured \nto remain trapped  at dislocations and high -angle grain boundaries  even after room -temperature vacuum \ntransfer [44]. In the case of Fe, the type of defect can influence the binding energy creating a range of \npotential traps. Bulk -measured and first -principles calculated hydrogen binding energies of s olid solution \nmetal additions usually have a  hydrogen binding energy  of -Eb < 30 kJ/mol [45, 46] . Carbide/Fe  [47, 48] , \noxide/Fe  [49], and sulfide/Fe  [50] interfaces have  stronger  hydrogen  binding energies , -Eb > 70 kJ/mol.  \nHydrogen binding ener gies in grain boundarie s, voids,  and dislocations  are roughly found in between \nthese two ranges  (18 < -Eb (kJ/mol) < 78) [51-53].  \nConsidering the broad range of H binding energies  in a material, hydroge n would be expected to persist \nat high binding energy  defects even after hydrogen in lower binding energy sites has diffused away (as \noccurs during e.g. vacuum transfer followed by APT ). Cooling down the sample immediately after \nhydrogen charging and keepi ng the sample at cryogenic temperatures during transfer can reduce \nhydrogen diffusivity, enabling its observation even and lower binding energy sites.  \nUsing the cryogenic plunge freezing process, deuterium has been qualitatively observed in medium and \nhigh binding energy traps in various steels. In the last decade, 2H charging -APT studies have been \nperformed with ferritic -martensitic dual- phase steels, precipitation strengthened steels, bainite- austenite \nand ferrit e-austenit e dual phase steels [5 4-66]. These studies, however, ca nnot yet deliver quantitative \ninformation relating specific processing conditions to thermodynamic or kinetic behavior  because \ndeuterium is rarely seen at the no or low- energy binding sites . In almost all reports , there is no deuterium \nfound in the matrix of the material after charging,  only in higher binding energy locations. To  \nquantitatively characterize specific spatial behavior of traps, a standardized method must be created to  \ncompare the relationship between low- energy and high -energy binding sites .   4 This work  combin es CV and cryogenic -transfer APT to assess both total and spatially resolved hydrogen \nuptake . Electropolished needle  samples of body centered cubic ( BCC) pure Fe and  FCC model Fe -18Cr -\n14Ni (wt. %) alloy s were electrochemically  charged with  deuterium and  brought  to saturation . Following \nan accelerated room -temperature vacuum transfer method and a cryogenic freezing, frost removal, and \ncryogenic transfer method , they were then quantitatively analyzed via APT . This hydrogen quantification \nfrom APT was then compared with CV to quantitatively analyze  the H -pick up in  ferrite and austenite .  \nExperimental methods  \nFe-18.0 Cr-14.0Ni wt.  % (Fe-19.2Cr -13.2Ni at. % , shortened to Fe18Cr14Ni ) alloys were induction melted \nfrom high -purity elements, then cast and homogenized by remelting five times. The alloys were then  cold \nrolled  to a 50% reduction in area and  recrystallized to 3  mm thick sheets via annealing at 900  °C for 4  h. \nPrevious crystal structure characterization s hows that the austenitized Fe18Cr14Ni was FCC  stabilized  \n[67]. Pure Fe  rods  (99.99 %) were purchased from Goodfellow  inc. Bars with a square cross section with \nan area of 1 mm2 were sectioned from the fabricated pure Fe and Fe18Cr14Ni alloy via electric discharge \nmachining (EDM). The bars were then ground and polished on all sides at decreasing grit size down to 1 \nµm diamond suspension.  \nElectrochemical characterizations  were conducted  using Gamry Potentiostat (Interface 1000). The \nelectrolyte was 0.1 M NaOH (98%, Alfa Aesar) in ultrapure deionized  water (Sigma Aldrich). The \nelectrochemical cell consisted of 1 mm2 square cross section bar s of Fe-18Cr -14Ni  or Fe as working \nelectrode, A g/AgCl reference electrode ( 0.281 V vs.  Standard Hydrogen Electrode (SHE) ), and Pt wire as \ncounter electrode. 1 cm length of metal bar  was dipped into electrolyte solution. The experimental \nprocedure consisted of 3 stages as follows . First, the electrode was submitted to ten consecutive \npretreatment CV cycles between - 1.25 V and +0.75 V vs Ag/AgCl at a scan rate of 100 mV  s-1 (final scan  to \n-1.25 V) to obtain a reproducible sample surface. Next, one cycle of CV between - 1.25 V and 0.15 V  was \nrecorded before hydrogen charging. This is followed by hydrogen charging by polarizing the sample at a \nconstant voltage of −1.25 V ( vs. Ag/AgCl) , whi ch co -occurs with  water reduction on the working electrode. \nAfter hydrogen charging, the working electrode is instantly submitted to 5 consecutive CV cycles starting \nfrom -1.25 V to 0.15 V ( referred to as “after hydrogen charging ”). Electrochemical data wa s processed \nusing Gamry Echem Analyst.  \nFabricated bars  from the same parent material as  measured with CV  were then sharpened into needles \nusing a two-step electropolishing technique. The first coarse polishing solution was 25 % perchloric acid  5 in glacial a cetic acid. The second fine polishing solution was 2  % perchloric acid in 2 -butoxyethanol. A \nconcluding polish was conducted in a Thermo Fisher Hydra plasma FIB with Xe plasma. The voltage for \nmilling was 30 kV at decreasing currents with final milling at 2 kV. APT was performed using a local \nelectrode atom probe (LEAP) 4000XHR (Cameca). All runs were conducted in voltage pulsing mode, with \nvoltage pulse fraction at 20  %. The temperature of every sample during data collection was 40 K. Data \nwere  reconstruct ed using Interactive Visualization and Analysis Software (IVAS 3.8, Cameca).  \n2. Results and Discussion  \n3.1 Cyclic voltammetry of Hydrogen Charged  Fe and Fe18Cr14Ni   \nPure Fe and Fe18Cr14Ni (1 mm2 square cross section rods) were electrochemically charged at -1.25 V \nversus Ag/AgCl for 5 minutes  in 0.1  M NaOH. C yclic voltammograms  of Fe and Fe18Cr14Ni electrodes \nbefore and after the charging steps are shown in Figure 1 (a-b). For Pure Fe, the CV before hydrogen \ncharging is shown  in black in Figure 1(a) along with the CV after 5 min hydrogen charging  (1st and 5th cycles) \nshown in different shades of red. For Fe18Cr14Ni, the CV before charging and after 5 min hydrogen \ncharging are shown in black and green  (1st and 5th cycles  after charging shown different shades of green ) \nrespectively in Figur e 1(b). The current density vs . time during the 5 min charging of pure Fe and \nFe18Cr14Ni are shown in Figure 1(c). The cyclic voltammograms of Fe and Fe18Cr14Ni before hydrogen \ncharging shows a small shoulder at -0.8 V in cathodic scan , which  is probably due to the reduction of  γ-\nFe2O3 and/or α-FeOOH  to Fe3O4 (Equation 1-2). A broad cathodic peak (c1 , c1’) at -1 V to -1.2 V is attributed \nto the reduction of Fe 3O4 to Fe(OH) 2 and/or FeO  (Equation 3-4) and possibly to Fe0 (Equation 5-6). The \nanodic peak (a1 , a1’ ) is attributed to the oxidation of Fe(OH) 2 /FeO to Fe 3O4 (Equation 3-4) [68]. After 5 \nmin hydrogen charging of pure Fe , a new anodic  peak  appears at - 0.85 V  (marked as a2 in Figure 1 (a)), \nwhich is assigned to H -oxidation  [36, 37] . It should be noted that the signal at -0.85 V also appears in the \nCV before hydrogen charging bu t only at much lower intensity. In addition, a new anodic peak (a3) and a \nhigh anodic current above -0.4 V is also noticeable  in the CV curve from hydrogen charged pure Fe given \nin Figure 1(a) . Both a3  (and a3’)  and anodic current above - 0.6 V declined wit h repeated CVs scans.  \nTherefore, we hypothesized that this observation is also related to the oxidation of hydrogen.  In contrast, \nin the hydrogen charged Fe18Cr14Ni, no distinct peak emerged near - 0.85 V  but overall background \ncurrent density increased nea r the hydrogen oxidation  peak location as evident in Figure 1(b).  \nFurthermore,  the anodic peak a3’  current density increased to a higher value in comparison to  a1’ and \nshifted to higher potential. The higher anodic current is also observed above - 0.6 V in anodic scan, like  \nthat observed in Fe electrode. With repeated CV scans,  the anodic current between - 0.8 V to - 0.9 V,  6 corresponding to hydrogen oxidation, decreased.  Both a3’ and anodic current above - 0.6 V  also declined \nand is similar to  the CV curve before  hydrogen charging. This trend is consistent with behaviors observed \nin the Fe electrode.  \n2Fe 3O4 + 2OH-  3 γ-Fe2O3 + H 2O + 2e- (1) \nFe3O4 + OH- + H 2O 3α-FeOOH  + e- (2) \n3Fe(OH) 2 + 2OH-  Fe3O4 + 4H 2O + 2e- (3) \nFeO + 2OH-  Fe3O4 + H 2O + 2e- (4) \nFe + 3H 2O  FeO +2H 3O+ + 2e- (5) \nFe + 4H 2O  Fe(OH) 2 + 2H 3O+ + 2e-  (6) \n \n \nc d 7 Figure 1: Cyclic voltammetry of (a) pure Fe and (b)  Fe18Cr14Ni before and after 5 minutes hydrogen \ncharging at - 1.25 V vs.  Ag/AgCl (1st and 5th cycle). (c) Chronoamperometric curves for 5 minutes  charging \nand (d) charge calculated from curves in (c). \n \nThe charge  calculated from chronoamperometric curves for 5 minute s hydrogen charging (Figure 1 (d)) is \nrecorded to evaluate current response and to determine the amount of charge as a function of time. The \nFe electrode shows  higher current in the first few cycles of  the charging process, corresponding to higher \namount of charge transfer during hydrogen charging.  The difference s in the cyclic voltammogram of Fe \nand Fe18Cr14Ni electrode s are related to the hydrogen  traps present and  diffusion from the bulk to the \nsurface , and hydrogen oxidation kinetics (related to the surface composition) . We hypothesi ze that the \nabsence of an overall elevation of current density wit h no apparent hydrogen oxidation peak ( a2) in \nFe18Cr14Ni is due to  hydrogen diffusivity being lower in austenite in comparison to much faster diffusion \nout of pure Fe . This might lead to the current from  hydrogen  oxidation occur over a broader voltage range \nand the present fast sweep rate . We also do not rule out the role of native Cr 2O3 on Fe18Cr14Ni as a \nhydrogen permeation barrie r as reported in literature  [69-72]. First principles calculation s determined the \ndiffusion coefficient of hydrogen in Cr 2O3 is 5.03 x 10-10 cm2/s at 500 ° C, which is threefold slower than \nthat in 316L stainless steel [73]. A more detailed understanding of the hydrogen oxidation  mechanism in \npure Fe and  Fe18Cr14 Ni requires  intensive experiment investigation in interfacial reactions and will be \nreported in a future work.   \nIn this stud y, we focused on the  difference in hydrogen diffusion kinetics and storage properties between \nBCC pure Fe and FCC Fe18Cr14Ni, which can lead to more hydrogen being retained within the FCC lattice \nof Fe18Cr14Ni . To further verify this  hypothesis, atom probe analysis  was performed to quantitatively \nanalyze the  hydrogen pick up  in the  electrochemically charged  Fe18Cr14Ni  and pure Fe . \n3.2 Establishing a baseline  for Hydrogen  detected  in Fe and Fe18Cr14Ni  in Atom Probe \nTomography before hydrogen charging  \nIn APT mass -to-charge spectra, hydrogen  background  peak s may be observed  either from  hydrogen that \nmay be introduced into materials during APT sample preparation, or desorption of adsorbed hydrogen on \nthe walls and other components of the  atom probe  analys is chamber made out of stainless steel [74]. To \nminimize  the detected background  hydrogen signal, we perform ed a parametric study of different \nvariables known to influence hydrogen detection including voltage pulse frequency (kHz) and detection \nrate (event/pulse). Pulse frequency  and detection rates were chosen between 50 -200 kHz and 0.5 -3.0 %  8 respectively. Both  a higher detection rate and higher voltage pulse frequency  results in lower background \nhydrogen detection in both pure Fe (Figure 2a) and Fe18Cr14Ni (Figure 2(b)) . The lowest recorded \nconcentration of hydrogen was present at a pulse frequency  of 200 kHz and a detection rate of 3.0%, \nwhich were the highest detection parameters used. No molecular peaks of hydrogen, at 2 Da, 3 Da, or 4 \nDa, was observed in either material at all combinations of detection rate and v oltage pulse frequencies  \n(mass -to-charge spectra shown in Figure 2 (c) for pure Fe and Figure 2 (d) for Fe18Cr14Ni) . \n \nFigure 2: APT analysis parameter dependence on hydrogen  quantification.  Concentration of hydrogen \nmeasured by atom probe tomography as a function of voltage pulse frequency  (PF) and detection rate \n(DR) at 20% pulse fraction for (a) pure Fe and (b) Fe18Cr14Ni. All the data for both materials  are collected \nwhile analyzing the same needle specimen  and parameters were changed during the same APT \nexperiment. The mass -to-charge ratio spectra (m/ q) from 0 -5 Da given in (c)  Pure Fe at the parameters of \n(ci) DR 0.5% PF 50 kHz (cii) DR 0.5% PF 125 kHz, (ciii)  DR 0.5% PF 200 kHz, (civ)  DR 1.0% PF 50 kHz, (cv) DR \n1.0% PF 125 kHz, (cvi)  1.0% PF 200 kHz, (cvii)  DR 3.0% PF 50 kHz, (cviii)  DR 3.0% PF 125 kHz, (cix)  DR 3.0% \nPF 200 kHz, and (d) Fe18Cr14Ni at the parameters of (di) 0.5% PF 50 kHz, (dii)  DR 0.5% PF 200 kHz, (diii)  \nDR 3.0% PF 50 kHz, (div)  DR 3.0% PF 200 kHz. Counts for these mass -to-charge spectra are normalized to \nthe h ighest count recorded, which is 56Fe2+. \n \nIf the lowest concentration of hydrogen is desired, pulse frequency  and detection rate should be \nmaximized; however, this can result in premature fracture of APT needle samples . All Fe18Cr14Ni needles \nfracture d during field evaporation at 3.0 % detection rate and 200 kHz pulse rate before 2 million ions \ncould be collected. While this setting produced the lowest H signal, it was impractical to run at these \nconditions. A closer inspection of the mass spectra shows that  a lower pulse frequency  (50 kHz)  at a high \ndetection rate (3.0 %)  also produce d a low hydrogen signal  with low background  while still resulting in a \nhigher sample yield . Therefore, all subsequent atom probe samples were analyzed at a detection rate and \n 9 pulse frequency  of 3.0 % and 50 kHz to maximize the combination of decreased artificial hydrogen count, \nlower background, and  maintain  high sample yield .  \n3.3 Atom Probe Tomography Analysis of Deuterium charged Fe and Fe18Cr14Ni  \n \nPure Fe and Fe18Cr14Ni , needle  samples  were analyzed in APT to get the baseline APT result s before \ncharging. Then the APT experiments were stopped, and samples were removed to charge 2H \nelectrochemically. The samples fixed in a Cu holder were  placed in 23 mL of a 0.1 M solution of sodium \ndeuteroxide (NaOD) in deuterium oxide  (D2O, 99.9 % pure) . The counter electrode wire was Pt. The \nschematic of the deuterium charging setup is shown in F igure 3(a). A constant direct current bias of - 2.2 \nV was generated by a power supply onto the system  for 2 minutes . The steady state (achieved in 10 \nseconds) reference electrode measurement was -1.6 V vs AgCl for the pure Fe needle and - 2.0 V for the \nFe18Cr14Ni needle.  Here, a -2.2 V is not measured against a working or reference electrode but  was the \nvoltage from the power supply on the system. Similar APT works published so far indicate that at this \nsystem voltage  and time , we can ensure H saturation in the material without producing too many bubbles \nthat could prevent H absorption [54, 65] . In all experiments, <1 bubble per 5 seconds were observed using \na visible light stero microscope.   10  \nFigure 3: Demonstration  of 2H charging and cryogenic transfer process. (a) Charging setup  diagram  with \nthe sample anode in a Cu holder, Pt wire cathode, and solution. (b)  Wire attached to the Cu puck modified \nfor thermal isolation against the transfer rod and manipulation by the quorum stage. (c)  Puck and needle \nplunging into liquid nitrogen . (d) Photograph of c ryogenic  stage in a FIB/SEM modified to accept APT \npucks.  (e) SEM image of deuterium charged needle sample  with frozen condensed water  after quenching. \n(f) needle sample after final sharpen ing. (g) Blue arrows indicate path of the sample  from the transfer \nshuttle into the environmental transfer hub ( ETH), buffer cham ber, and finally, the APT analysis chamber.     \nAfter  2H charging, the samples were immediately removed from the charging apparatus, dipped three \ntimes in a container of pure D 2O to remove any NaOD , blow dried with N 2 gas, and placed in an atom \nprobe puck , Figure 3(b). The puck with the deuterium -charged wire sample was then either transferred  \nusing a vacuum shuttle  at room temperature or at cryogenic temperatures  into the atom probe analysis \nchamber . The time from removing the needle from electrochemical charging solution to plunge freezing \nwas less than 180 seconds.  \nTo achieve cryogenic  transfer, the deuterium -charged sample  was first plunge d into liquid nitrogen  (Figure \n3(c)). The plunge frozen sample was then transferred into the cryogen ic stage of a Thermo Fisher Scientific \nQuanta FIB/ SEM ( Figure 3d) using a Quorum shuttle . A minor amount of frost form ed during the sample \ntransfer process (Figure 3(e)) which was removed by using 2kV Ga ion beam milling to get a frost -free \n 11 needle sample ( Figure 3(f)). The final needle sample was  then transferred into the APT using the Quorum \nshuttle device, through the cryo genic  stage on the environmental transfer hub  (ETH), which was \nmaint ained at high vacuum and temper ature below  -150 ˚C. The sample is t hen transferred from the ETH \nonto a thermal insulative puck holder on the buffer chamber caro usel and then transferred into the APT \nanalysis chamber. This transfer workflow allows the sample to remain at cryogenic temperatures while \nalso maintaining  high  vacuum, preventing any additional frost formation  during transfer process .  \nA comparison of the mass -to-charge spectra peaks at 0 -5 Da from pure Fe before charging (Figure 4(a)) \nand after 2 -minute  2H charging and room temperature transfer (indicated by the sun  schematic)  (Figure \n4(b)) show a small but distinct new peak emerging at 2 Da. The composition profile plotted along the z-\naxis of the APT reconstruction of the 2  min charged -room temperature transferred pure Fe sample is given \nin Figure 4(c)  where 1H is <0.1 at. % and 2H concentration is < 0.05 at. % .  On the other hand, t he \ncomparison of APT mass -to-charge spectra peaks of pure Fe before (Figure 4( d)) and after charging \nfollowed by cryogenic transfer (highlighted by a blue snowflake schematic ) (Figure 4( e)), show a significant \nincrease in the intensity of the 2  Da peak.  The corresponding composition profile from the d euterium \ncharged and cryogenic transferred sample given in Figure 4(f) highlights the increased concentration of 1H \nand 2H to 0. 2 at. %. The distribution of Fe, 1H and 2H within the cryogenically transferred d euterium \ncharged pure Fe is shown in Figure 4(g) where a minor tendency for 1H and 2H to segregate to the poles \nin the reconstruction is evident. These results highlight the v alue of cryogenic transfer in retaining more \n2H within the BCC structured pure Fe APT sample.    12  \n \n 13 Figure 4: Atom probe tomography  results  of pure Fe and Fe18Cr14Ni charged in 0.1M NaOD in D 2O at -\n2.2 V for 2 min followed by either room temperature transfer  or cryogenic transfer into the APT. APT \nmass -to-charge ratio spectrum between 0 -5 Da for pure Fe (a) before charging and ( b) after 2 min 2H \ncharging and room temperature transfer ( c) composition profile along the z axis of the reconstr uction for \ndeuterium charged and room temperature transferred pure Fe. APT mass -to-charge ratio spectrum \nbetween 0 -5 Da for pure Fe ( d) before charging and ( e) after 2 min 2H charging and cryogenic transfer (f) \ncomposition profile along the z axis of the reconstruction for deuterium charged and cryogenic transferred \npure Fe. (g)  distribution of Fe, 1H and 2H within the APT reconstruction in the d euterium  charged and \ncryogenic transferred pure Fe. APT mass -to-charge ratio spectrum between 0 -5 Da for Fe18Cr14Ni (h) \nbefore charging and ( i) after 2 min 2H charging and room temperature transfer ( j) composition profile \nalong the z axis of the reconstruction for deuterium charged and room temperature transferred \nFe18Cr14Ni. APT mass -to-charge ratio spectrum between 0 -5 Da for Fe18Cr14Ni (k) before charging and \n(l) after 2 min 2H charging and cryogenic transfer (m) composition profile along the z axis of the \nreconstruction for deuterium charged and cryogenic transferred pure Fe. (n) distribution of Fe, Cr, Ni, 1H \nand 2H within the APT reconstruction in the d euterium charged and cryogenic transferred Fe18 Cr14Ni.   \n \nThereafter, we conducted a similar comparison of the effectiveness of room temperature transfer versus \ncryogenic transfer for Fe18Cr14Ni samples. The comparison of the mass -to-charge spectra peaks at 0 -5 \nDa from Fe18Cr14Ni before charging (Figure  4(h)) and after 2 min deuterium charging and room \ntemperature transfer (Figure 4(i) ) show a distinct new peak emerging at 2 Da. The composition profile \nplotted along the z -axis of the APT reconstruction of the 2 min charged -room temperature transferred \nFe18Cr14Ni sample is given in Figure 4(j) where 1H concentration is <0.25 at. % and 2H concentration is \n<0.05 at. %. The comparison of 0 -5 Da APT mass -to-charge spectra peaks of Fe18Cr14Ni before (Figure \n4(k)) and after charging followed by cryogenic transfe r (Figure 4(l)), show an even higher intensity for the \n2 Da peak. The corresponding composition profile from the deuterium charged and cryogenic transferred sample given in Figure 4(m) show the \n1H concentration remained close to 0.25 at. % but the 2H \nconce ntration increased to approximately 1.5 at. % highlighting a significantly high 2H pickup and \nretention within the cryogenically transferred sample. The distribution of Fe, Cr, Ni, 1H, and 2H within the \ncryogenically transferred deuterium charged Fe18Cr14N i is shown in Figure 4(n) where a rather uniform \ndistribution of 2H in the matrix was observed. Here again in Fe18Cr14Ni cryogenic transfer process after \n2H charging led  to retention of more 2H within the Fe18Cr14Ni versus room temperature transfer. One \nadditional observation is much higher amount of 2H concentration within austentic Fe18Cr14Ni versus \npure Fe charged for the same time. This difference in observed 2H concentration is presumed to  be arising \nfrom the differences in the solubility and diffusivity of hydrogen within the BCC Fe versus austentic \nFe18Cr14Ni.   14 Gas permeation  studies , galvanostatic measurement, thermal desorption spectroscopy, and first \nprinciples calculation s estimate that the hydrogen diffusion  coefficient  in ferrite  is generally  higher than  \naustenite  [75-86]. Additionally, the hydrogen solubility in ferrite is  lower than  in austenite  [87]. Therefore, \nmore hydrogen can be diffused into γ -SS than pure Fe  during electrochemical charging . Once hydrogen \nenters the lattice because of  electrochemical charging, the subsequent outward  diffusion of hydrogen will \nbe much faster from pure Fe than γ -SS. The differences in  H solubility and diffusion rate  are demonstr ated \nin the  CV results of γ-SS compared to pure Fe. The higher diffusion of hydrogen out of the pure Fe leads \nto a CV peak  for stripping hydrogen absorbed onto the steel (a2) whereas  no distinct hydrogen stripping  \nfeature  is present in the austenitic  Fe18Cr 14Ni . The lower diffusion and higher solubility of hydrogen in \naustenitic  Fe18Cr14Ni  allows for the increased current density without a distinct peak at a2. Additionally , \nthis also explained the detection of  high er concentration of 2H using APT in Fe18Cr14 Ni versus pure Fe , \ncharged for the same time and transferred cryogenically into the atom probe analysis chamber .    \nAs a result of this advancement in cryotransfer- APT for quantitative detection of  deuterium in the matrix \nof the alloys, we can develop  future systematic studies to quantitatively understand deuterium  behavior  \nin materials . Electrochemistry, including solution chemistry, potential, time , and temperature may be \nvaried. Differences between trap  sites  may be measured with greater rigor. D iffusion  time constants  may \nbe assessed by varying times to quench after charging. Spatial diffusion may be measured as a function of \ndistance enabled by mi lling the tip via cryo -FIB.   \n3. Conclusions:  \nIn summary, using CV and cryogenic -transfer APT, we approach  quantitative analy sis of  the hydrogen \npickup in body -centered cubic structured pure Fe and face centered cubic austen itic Fe18Cr14Ni model \nalloy. CV results showed clear differences in hydrogen oxidation from hydrogen charged pure Fe being \nfaster than from Fe18Cr14Ni alloy. Using d euterium as a tracer, cryogenic transfer versus room \ntemperature transfer to APT after  deute rium charging, always resulted in detecting a higher concentration \nof 2H in both pure Fe and Fe18Cr14Ni alloy. The overall higher concentration of 2H detected after char ging \nand cryogenic transfer to APT in Fe18Cr14Ni was attributed to the lower hydrogen d iffusion coefficient \nand higher solubility of hydrogen in austenite than ferrite  and potentially the native chromium oxide on \nsurface acting as a diffusion barrier  for hydrogen. We believe correlating  CV and cryogenic transfer atom \nprobe tomography can be highly informative in understanding the  diffusion and trapping of hydrogen in \nFe-based alloys with and without defects and trap ping sites .  \n4. Acknowledgements   15 This research was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, \nMaterials Sciences and Engineering Division  as a part of the Early Career Research program  (FWP # 76052) . \nThis work was also supported in part by the U.S. Department of Energy, Office of Science, Office of \nWorkforce Development for Teachers and S cientists (WDTS) under the Science Undergraduate Laboratory \nInternships Program (SULI) . KAS was supported by the U.S. Department of Energy, Office of Science, Basic \nEnergy Sciences, Chemical Sciences, Geosciences, & Biosciences Division as a part of the Early Career \nResearch program ( DE-SC0022970 ). The APT was conducted using facilities at Environmental Molecular \nSciences Laboratory (EMSL), which is a DOE national user facility funded by Biological and Environmental \nResearch Program located at Pacific North west National Laboratory. The authors would like to \nacknowledge Jack Grimm for his assistance in charging and cryo genic  transfer and the EMSL machine shop \nfor sample fabrication.   \n \nReferences  \n[1] C.L. Briant, Hydrogen Assisted Cracking of  Type 304 Stainless Steel, Metallurgical Transactions A 10A \n(1979) 181 -189.  \n[2] E. Herms, J.M. Olive, M. Puiggali, Hydrogen Embrittlement of 316L type Stainless Steel, Materials \nScience and Engineering A  (1999) 279 -283.  \n[3] Y.H. Fan, B. Zhang, J.Q. Wang, E.H. Han, W. Ke, Effect of grain refinement on the hydrogen embrittlement of 304 austenitic stainless steel, Journal of Materials Science & Technology 35(10) (2019) 2213- 2219.  \n[4] M. Martin, S. Weber, W. Theisen, T. Michler, J. 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Yu, Diffusion coefficient of hydrogen interstitial atom in α -Fe, γ-Fe and ε -Fe \ncrystals by first -principle calculations, International Journal of Hydrogen Energy 42(44) (2017) 27438 -\n27445.  \n[87] S. Kou, Welding Metallurgy, John Wiley & Sons, Hoboken, NJ, 2003.  \n "    },    {        "title": "1904.01271v1.Magnetoelectrically_driven_catalytic_degradation_of_organics.pdf",        "content": "     \n1 \n Magnetoelectrically driven catalytic degradation of organics  \n \nFajer Mushtaq*, Xiang -Zhong Chen*, Harun Torlakcik, Christian Steuer, Marcus Hoop, \nErdem Can Siringil, Xavi Marti, Gregory Limburg, Patrick Stipp, Bradley J. Nelson and \nSalvador Pané*  \n \nF. Mushta q, Dr. X. -Z. Chen, H. Torlakcik, Dr. M. Hoop, E. C. Siringil, G. Limburg, P. Stipp, \nProf. Dr. B. J. Nelson, Dr. S. Pané   \n \nMulti -Scale Robotics Lab (MSRL), Institute of Robotics and Intelligent Systems (IRIS), ETH \nZurich, CH -8092 Zurich, Switzerland.  \n \nE-mail: fmushtaq@ethz.ch, chenxian@ethz.ch, vidalp@ethz.ch  \n \nDr. C. Steuer  \nInstitute of Pharmaceutical Sciences, ETH Zurich, CH -8093 Zurich, Switzerland.  \n \nDr. Xavi Marti  \nInstitute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnick´a 10, 162 00 \nPraha 6, Czech Republic  \nIGS Research Ltd., Calle La Coma, Nave 8, 43140 La Pobla de Mafumet, Tarragona, Spai n \n \n \nKeywords : magnetoelectric, multiferroic , bismuth ferrite, catalysis, organics degradation  \n      \n2 \n Abstract  \n \nHere, we report the catalytic degradation  of organic compounds by exploiting the \nmagnetoelectric (ME) nature of cobalt ferrite -bismuth ferrite (CFO -BFO) core -shell \nnanoparticles. The combination of magnetostrictive CFO with the multiferroic BFO gives rise \nto a magnetoelectric engine that purifies  water under wireless magnetic fields via advanced \noxidation processes, without involvement of any sacrificial molecules or co -catalysts. \nMagnetostrictive CoFe 2O4 nanoparticles are fabricated using hydrothermal synthesis, \nfollowed by sol -gel synthesis to c reate the multiferroic BiFeO 3 shell. We perform theoretical \nmodeling to study the magnetic field induced polarization on the surface of magnetoelectric \nnanoparticles. The results obtained from these simulations are consistent with the \nexperimental findings  of the piezo -force microscopy analysis, where we observe changes in \nthe piezoresponse of the nanoparticles under magnetic fields. Next, we investigate the \nmagnetoelectric effect induced catalytic degradation of organic pollutants under AC magnetic \nfields and obtained 97% removal efficiency for synthetic dyes and over 85% removal \nefficiency for routinely used pharmaceuticals. Additionally, we perform trapping experiments \nto elucidate the mechanism behind the magnetic field induced catalytic degradation of o rganic \npollutants by using scavengers for each of the reactive species. Our results indicate that \nhydroxyl and superoxide radicals are the main reactive species in the magnetoelectrically \ninduced catalytic degradation of organic compounds.  \n \n      \n3 \n Magnetic nano structures have been widely used as magnetically recoverable catalysts \nor as carriers for catalytic materials .[1] While magnetic nanomaterials have found widespread \napplications in tuning catalytic processes, a majority of the employed strategies focus on their \nmotion to enhance the reagents’ mass transport.[2] A next lev el of control can be achieved by \nforcing such magnetic nanoparticles to interact[3] or, alternatively, by coupling them to \ncatalysts to enhance the  reaction performance.[4] In previous investigations, magnetic fields \nonly support the c atalytic chemical conversion but never act as the ultimate trigger. Direct \ncontrol of causality is fundamental in realistic scenarios, where the choice of the precise \nmoment of actuation is critical. In this direction, magnetically induced heating has been  \ndemonstrated to initiate chemical catalysis on demand.[5] Here, we demonstrate a localized \ntrigger for catalytic reactions via the direct magnetoelectric (ME) effect on the surface of \nmultiferroic nanoparticles. Ou r ME nanocatalysis is able to decompose organic contaminants, \nsuch as dyes and various pharmaceuticals, without the involvement of any sacrificial \nmolecules or co -catalysts. Our experiments revealed a rich interdependence between the \napplied magnetic field  parameters and the reaction speed, which significantly improves the \nperformance of catalytic reactions for environmental remediation.  \nOrganic pollutants such as pharmaceuticals, pesticides and industrial chemicals are \npersistent compounds that are resista nt to degradation through conventional processes and \nbio-accumulate in ecosystems, causing severe impacts on human health and the \nenvironment.[6] Towards this effect, billions of dollars have been invested annually to study \nnew approaches capable of efficiently combating this global crisis.[7] Surface charges that can \ninitiate redox reactions an d form hydroxyl and superoxide radicals, can serve as an attractive \napproach for non -selective degradation of such organic pollutants. The occurrence of surface \ncharges, thus, becomes the trigger to ignite such reactions and shifts the spotlight to any \nmechanism that can generate such surface charges on demand. The magnetoelectric (ME) \neffect not only fulfills this requirement, but, also allows for wireless operation while      \n4 \n restricting the targeted regions to areas where the ME materials are deployed. While \ntheoretically possible, experimentally realizing this has been hindered by the dearth of ME \nmaterials and nanostructures that actually can be fabricated. Many research efforts have \nresponded to this call by investigating the growth, characterization and op eration of ME \nnanoparticles (NPs) in -depth.  \nOur main results are summarized in Figure 1 . We have investigated the capability of \nour core -shell ME NPs to initiate electrochemical processes under the application of \nalternating magnetic fields (Figure 1a) by  studying the degradation of a model organic \npollutant, rhodamine B (RhB). Degradation curves obtained for RhB under alternating \nmagnetic fields for CFO -BFO and controls are presented in Figure 1b. From this figure we \ncan observe that the control sample (w ithout any NPs) and the CFO NPs sample displayed a \nnegligible response under alternating magnetic fields. BFO NPs, showed a slight decrease in \nRhB concentration (22%), which can be attributed to the weak coupling between the \nferromagnetic and ferroelectric ity of BFO at room temperature.[8] In contrast, core -shell CFO -\nBFO NPs demonstrated an elevated RhB degradation efficiency of 97% within 50 min (Figure \n1b). We were also successful in extending our novel approach fo r degradation of a cocktail of \nfive commonly used pharmaceuticals, Carbamazepine, Diclofenac, Gabapentin, Oxazepame \nand Fluconazole[6b, 9]. Figures 1c and S1 show that all five pharmaceuticals were degraded \nwith ove r 80% efficiency. These results confirm our hypothesis that our ME NPs can be used \nto simultaneously target a wide variety of organic compounds in a non -selective approach. \nThis further highlights the advantage of using our approach over conventional ozone  \ntreatments for pharmaceutical removal in wastewater treatment plants, which is known to \ndisplay a negligible reactivity towards a vast variety of ozone -resistant such as Gabapentin, \nOxazepame and Fluconazole.[10] \n \n      \n5 \n The magnetoelectric nanocatalysts used in this work consist of magnetostrictive cobalt \n(II) ferrite (CoFe 2O4, CFO) cores coated with multiferroic bismuth (III) ferrite (BiFeO 3, BFO) \nshells.  CFO NPs were f abricated by a hydrothermal synthesis approach by carefully tuning \nthe growth conditions to obtain single -crystalline and phase -pure NPs. A co -precipitation \nmethod was employed to form CFO NPs using NaOH solution to form precipitates of iron \nand cobalt hyd roxides.[11] CTAB was used as the surfactant to control the nucleation and \nshape of the CFO NPs.[12] This mixture was then sealed in an autoclave and placed in an oven \nat ele vated temperature for hydrothermal treatment. Core -shell magnetoelectric (ME) CFO -\nBFO NPs were fabricated by coating the CFO NPs with a BFO precursor via a sol -gel \napproach, followed by annealing the NPs to crystallize the BFO shell.[13] (Figure S2). CFO \nNPs fabricated in this study are octahedrons and have an average size of 30 nm. (Figure S3 \nand S4a). The core -shell CFO -BFO NPs have an average size of 42 nm ( Figure 2 a, S4b and \nS5). Energy -dispersive X -ray (EDX) map pings confirm the presence of a shell composed of \nbismuth, iron and oxygen around the cobalt ferrite core (Figure 2b, S6 and S7).  \nThe crystalline structure of the CFO and CFO -BFO NPs was analyzed using \ntransmission electron microscopy (TEM) and X -ray diff raction (XRD).  XRD patterns (Figure \n2c) showed that for CFO NPs all peaks can be assigned to the pure Fd3m structure of \nCoFe 2O4 (JCPDS No. 01 -1121), indicating a cubic spinel structure . Similar analysis \nperformed on CFO -BFO NP sample shows that, in additi on to the CoFe 2O4 peaks, they \npossess new peaks that can be assigned to the pure phase of BiFeO 3 (JCPDS No. 71 -2494), \nindicating a rhombohedral perovskite structure with the space group R3c. A high resolution \nTEM (HRTEM) analysis performed on a single CFO NP sample is presented in Figure 2d, \nfeaturing an intact and orderly structure. The planes with interplanar d -spacing of 0.295 nm \nmatches the (220) crystal face of CFO. Its corresponding selected area electron diffraction \n(SAED) pattern is presented in Fig ure 2e, which indicates the occurrence of a single -\ncrystalline CFO structure. The spots in the SAED pattern have been indexed according to the      \n6 \n pure cubic spinel (Fd3m) structure of CoFe 2O4. The HRTEM image obtained for the BFO \nshell shows the presence of a n intact, orderly, single -crystalline structure (Figure 2f). The \nplanes with interplanar d -spacing of 0.198 nm match the (024) crystal face. Its corresponding \nSAED pattern is presented in Figure 2g and shows that the BFO shell is also single crystalline. \nThe spots in the SAED pattern have been indexed according to the R3c structure of BiFeO 3.  \nThe ferroelectricity and magnetoelectricity of a single core -shell NP was directly \nprobed using piezoresponse force microscopy (PFM) under an external magnetic field.  A \nconductive cantilever tip was used in contact mode to apply an alternating voltage to the \nCFO -BFO NP and induce piezoelectric surface oscillations, which were sensed through the \ncantilever deflection. To investigate the ferroelectric and magnetoelectric  coupling effect in \nour CFO -BFO NPs, local piezoresponse hysteresis loops were obtained at random locations of \nan NP ( Figure 3 a, b) by sweeping the applied DC bias, while simultaneously measuring the \nphase and amplitude response. The excitation voltage wav eform was programmed to be a \nstepwise increasing pulsed DC voltage that was superimposed on a small AC voltage. In order \nto minimize the possible interference caused by electrostatic forces, the AC response signal \nwas acquired only during the off -phase of the voltage pulse sequence.[14] From the phase loop \npresented in Figure 3a it can be clearly observed that the BFO shell exhibits polarization \nreversibility both with and without the application of the external magn etic field. From this \nimage it is clear that BFO’s polarization directions can be switched at both polarities of the tip \nDC-bias voltage. Both piezoresponse phase loops are horizontally shifted, a trend that can \nalso be observed from the amplitude curves w ith asymmetric butterfly shape (Figure 3b)[15] \nThis asymmetry in the loops can be attributed to many factors, such as the imprint effect, \ninternal bias fields inside the materials, and/or due to a work function diff erence between the \ntop, Pt -coated Si probe and the bottom gold electrode.[16] The coercive voltages for the BFO \nshell measured without magnetic field are -4.69 V and 2.65 V, respectively. When the \nmagnetic field was  applied, the coercive voltages changed to -3.47 V and 3.06 V, respectively.      \n7 \n This smaller coercive voltage obtained under a magnetic field indicates that the strain \ngenerated in the magnetostrictive CFO core was effectively transferred to the shell, \nfacili tating the polarization reversal process in BFO. This is indirect evidence of strain \nmediated magnetoelectric effect in the core -shell CFO -BFO NPs. The positive coercive \nvoltage change (0.41 V) is smaller than the negative coercive voltage change (1.22 V).  The \nasymmetric change indicates that there is an offset of the center of the piezoresponse loop \nunder the magnetic field, which is caused by an electric field generated by the magnetoelectric \neffect.[14b, 17] The m agnetoelectric coupling coefficient is defined as,  \n \nwhere ∆H is the change in magnetic field and ∆E is the change in the electric field caused by \nthe external magnetic field. For our CFO -BFO NPs, the ∆E i.e. the offset of center of the loop \nupon application of magnetic field can be estimated to be (1.22 – 0.41 V)/2/10 nm = 40.5 MV \nm-1. Hence, the local ME coefficient can be estimated as 40.5 × 104 mV cm-1 Oe-1. This value \nis in the same order of magnitude as those reported for some core -shell magnetoelectric \nnanostructures such as FeGa@P(VDF -TrFE), CoFe 2O4-PbZr0.52Ti0.48O3, CoFe 2O4@BaTiO 3 \nand CoFe 2O4@BiFeO 3, which were evaluated by similar methods . [18] In order to further \ninvestigate this magnetoelectric coupling observed in our CFO -BFO NPs, we performed finite \nelement  simulations on a single core -shell NP under static magnetic fields. Figures 3c and S7 \npresents the strain distribution generated on the BFO shell when the core -shell NP was placed \nin an external magnetic field of 15 mT. From Figures 3d and S8, we can obse rve the \ncorresponding electric potential gradient induced on the surface of the BFO shell, which is \ndetermined by the magnetoelectric coupling and the compliance matrix of the BFO shell. \nFrom these simulations we can observe that, when subjected to externa l magnetic fields, a \nCFO -BFO NP can generate a local surface potential in the µV range, which can be exploited \nto initiate certain chemical reactions.       \n8 \n  \nTowards this effect, we used our core -shell NPs to study the degradation trend of the \norganic pollutant RhB dye under magnetic fields. These degradation curves were already \npresented in Figure 1b and proved that only the core -shell CFO -BFO NPs were capable of \nremoving 97% of the organic pollutant. This enhanced organic pollutant removal performance \nof CFO -BFO NPs can be attributed to the ME effect induced redox reactions that are \nresponsible for the catalytic degradation of RhB. Quantitative analysis on ME effect -induced \nRhB degradation was performed by comparing the reaction rate constants k, which can be \ndefined by,  \n \nwhere Co is the initial RhB concentration and C is the RhB concentration at time t. This \ncalculation is based on the assumption that the kinetics of RhB degradation reaction catalyzed \nby the CFO -BFO nanostructures are (pseudo) -first-order reac tions.[19]  \nThe effect of varying magnetic field strengths and frequencies on RhB degradation \nrate is shown in Figure 4 a and b. We observed that increasing the magnetic field strength and \nfrequency have a positive e ffect on RhB degradation rate. These results are also supported by \nthe simulations performed on the core -shell NPs (Figure S10). Based on these results, a \nmagnetic field strength of 15 mT and a frequency of 1.1 kHz were chosen as our preferred \nmagnetic fie ld parameters for further degradation experiments ( k-value of 0.0725 min-1). At \nthese chosen parameters we also investigated the effect of concentration of BFO sol -gel \nprecursor solution on the degradation efficiency of RhB to find the optimal CFO -BFO \nmorp hology for further experiments (Figures S11 -12). \nIn order to elucidate the magnetoelectric effect -induced RhB degradation mechanism, \nwe performed trapping experiments of the prominent reactive species that are responsible for \ndecomposition of organic pollu tants.  For this, degradation of RhB dye was carried out under      \n9 \n the optimized magnetic field parameters in the presence of CFO -BFO NPs and different \nreactive species scavengers (Table 1). It can be seen from Figure 5 a that the catalytic \ndegradation efficien cy decreases with the addition of scavengers, proving that they all \nparticipate in the degradation of RhB. Addition of the electron scavenger, AgNO 3 and the \nhole scavenger, ethylene diamine tetraacetic acid (EDTA)[20] lowers the reaction -rate \nconstants. Trapping superoxide radical O 2•- with benzoquinone (BQ) or the hydroxyl radical \nOH• radical with tert -butyl alcohol (TBA) suppressed the degradation of RhB greatly. These \nresults reveal that the predominant reactive s pecies for magnetoelectrically -induced RhB \ndegradation were the radicals. In addition to the trapping experiments, we confirmed the \nformation of hydroxyl radicals in our degradation experiments by using terephthalic acid as a \nphotoluminescent OH• trapping agent. Terephthalic acid readily reacts with OH• radicals to \nproduce a fluorescent product, 2 -hydroxyterephthalic acid, which emits a fluorescent signal at \n425 nm.[20b] From the results of this experiment (Figure 5b) we can observe an increase in \nfluorescence intensity at 425 nm with increasing piezo -photoc atalytic reaction time, which \noffers further proof of OH• formation during the catalytic reaction.  \nIn this work, we have successfully fabricated core -shell magnetoelectric CoFe 2O4-\nBiFeO 3 NPs and demonstrated their use for catalytic degradation of organic compounds. In a \ntwo-step process, magnetostrictive CFO NPs were fabricated using a hydrothermal approach, \nfollowed by a sol -gel method to form a BFO shell. The magnetoelectric nature of the NPs has \nbeen evidenced by the observed modulation of their piezoel ectric response upon the \napplication of magnetic fields. Such findings are consistent with our theoretical modeling of \nstrain mediated magnetoelectric effect in core -shell NPs. Assisted by magnetic fields, our NPs \nhave been able to degrade, first, RhB poll utant with an efficiency of 97% within 50 min and, \nlater, a cocktail comprising routinely employed micro -pollutants in the pharmaceutical \nindustry. To understand the mechanism behind this magnetoelectric effect -induced \ndegradation, trapping experiments hav e been performed using scavengers for each of the      \n10 \n reactive species. Our results indicate that OH• and O2•- radicals are the main reactive species in \nthe magnetoelectrically induced catalytic degradation of organic compounds.  \n \n \n \nAcknowledgements  \nAuthor Con tributions ： Conceptualization: F.M., X.C. and S.P.; methodology: F.M., X.C., \nM.H.; investigation: F.M., X.C., H.T., C.S., E.C.S, G.L., P.S.; writing, original draft: F.M and \nS.P.; writing, review and editing: F.M., S.P., X.C., X.M., B.N., M.H., H.T., C.S., E.C.S., G .L., \nand P.S.; supervision: X.C., S.P. and B.N.; funding acquisition: S.P.  \nThis work has been financed by the European Research Council Starting Grant \n“Magnetoelectric Chemonanorobotics for Chemical and Biomedical Applications \n(ELECTROCHEMBOTS)”, by the ER C grant agreement no. 336456. 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Niu, S. Wang, L. Qin, Y. Huang, Scientific Reports  2016 , 6, \n26467; b) J. Wu, W. Mao, Z. Wu, X. Xu, H. You, A. X. Xue, Y. Jia, Nanoscale  2016 , 8, \n7343.  \n      \n13 \n \n \n \nFigure 1. (a) Scheme showing ma gnetoelectric (ME) effect induced catalytic degradation of \norganic pollutants using core -shell CFO -BFO NPs under magnetic fields. (b) Catalytic \ndegradation curves obtained for model organic dye, RhB, under 15 mT and 1 kHz magnetic \nfields (n=5).  (c) Removal  efficiency of a cocktail of five common pharmaceuticals using the \ncore-shell NPs (n=4).       \n14 \n \n \n \nFigure 2. Structural characterisation of core -shell CFO -BFO NPs. (a) TEM image showing \nmany overlapped CFO -BFO NPs. (b) HAADF STEM image obtained for some overlapp ed \nCFO -BFO NPs and its corresponding EDX maps obtained for Co, Fe, O and Bi, with the \nsuperimposed images clearly showing core -shell CFO -BFO NPs. (c) XRD patterns obtained \nfor core -shell NPs. (b) HRTEM image of a single CFO NP (c) and its corresponding SAE D \npattern. (d) HRTEM image of a core -shell NP showing the BFO shell region and (e) its \ncorresponding SAED pattern. (scale bars: (a) 100 nm, (b) 30 nm, (d) 2 nm, (e) 4 nm-1, (f) 2.5 \nnm and (g) 3 nm-1).      \n15 \n  \n \n \n \nFigure 3.  Ferroelectric and magnetoelectric char acterisation of core -shell CFO -BFO \nnanostructures. (a) Amplitude response of a single core -shell NP obtained with and without \nmagnetic field and (b) the corresponding phase response. (c) COMSOL simulations \nperformed on a CFO -BFO nanopartcile under a magnet ic field of 15 mT showing the (c) \nstrain generated on the BFO shell due to the magnetostrictive CFO core and (d) the \ncorresponding electric potential induced on the surface of the BFO shell.  \n      \n16 \n \n \n \nFigure 4.  Magnetoelectric effect induced catalytic degradati on of organic pollutants using \ncore-shell CFO -BFO NPs under alternating magnetic fields. (a) Comparison of RhB \ndegradation rate contants obtained by using CFO -BFO NPs under different magnetic field \nstrengths at 1.1 kHz field frequency (n=5)  and (b) under d ifferent field frequencies and at a \nmagnetic field strength of 15 mT (n=5) .       \n17 \n \n \n \nFigure 5 . Magnetoelectric effect induced catalytic degradation mechanism of RhB using \ncore-shell CFO -BFO NPs under 15 mT and 1.1 kHz magnetic fields. (a) Trapping \nexperiments demonstrating the effect of the four reactive species on degradation efficiency of \nRhB, where, -, +, O2•- and OH• refer to negative charge carriers, positive charge carriers, \nsuperoxide and hydroxyl radicals, respectively.  (b) Reaction of terephthalic acid  with OH• \nradicals to produce increasing amounts of fluorescent 2 -hydroxyterephthalic acid with peak \nintensity at 425 nm.  \n      \n18 \n  \nTable 1.  List of all the scavengers used in the trapping experiments and the reactive species \nthey quench.   \n \nScavenger  Reactive spe cies quenched  \nAgNO 3 e- \nEthylene diamine tetraacetic acid (EDTA)  h+ \nTert-butyl alcohol (TBA)  OH• \nBenzoquinone  (BQ)  O2•- \n \n \n      \n19 \n  \nOrganic compounds are degrada ded by the magnetoelectric (ME) nature of cobalt \nferrite -bismuth ferrite (CFO -BFO) core -shell nan oparticles. The combination of \nmagnetostrictive CFO with the multiferroic BFO gives rise to a magnetoelectric engine that \npurifies water under wireless magnetic fields via advanced oxidation processes, without \ninvolvement of any sacrificial molecules or co -catalysts.  \n \nmagnetoelectric, multiferroic , bismuth ferrite, catalysis, organics degradation  \n \nF. Mushtaq*, X. -Z. Chen*, H. Torlakcik, C. Steuer, M. Hoop, E. C. Siringil, Xavi Marti, G. \nLimburg, P. Stipp, B. J. Nelson, S. Pané*  \n \nMagnetoelectrically driven catalytic degradation of organics  \n \n \n      \n20 \n  \n \nSupporting Information   \n \nMagnetoelectrically driven catalytic degradation of organics  \n \nFajer Mushtaq*, Xiangzhong Chen*, Harun Torlakcik, Christian Steuer, Marcus Hoop, \nErdem Can Siringil, Xavi Marti, Gregory Limbur g, Patrick Stipp, Bradley J. Nelson and \nSalvador Pané*  \n \nExperimental  Section  \n \nFabrication of core -shell CoFe 2O4-BiFeO 3 nanostructures  \nCoFe 2O4 (CFO) nanoparticles (NPs) were fabricated by a hydrothermal synthesis approach. \nFor the fabrication of CFO NPs, 0 .14 M hexadecyltrimethylammonium bromide (CTAB), \n0.092 M FeCl 3·6H 2O and 0.046 M CoCl 2 were dissolved in DI water under continuous \nmechanical stirring. Next, a 6 M NaOH solution was added to the above solution under \nvigorous mechanical stirring followed by ultrasound. Finally, the above solution was \ntransferred to a sealed, Teflon -lined steel autoclave and heated at 180 ˚C for 24 h. The \nobtained black powder was washed with DI water and ethanol and dried overnight at 80 ˚C. \nNext, a precursor of BiFeO 3 (BFO) was prepared by dissolving 0.011 M Bi(NO 3)·5H 2O and \n0.01 M Fe(NO 3)·9H 2O in ethylene glycol. CoFe 2O4-BiFeO 3 core-shell nanostructures were \nprepared by dispersing 0.1 g of dried CFO nanoparticles into 60 mL of the BFO precursor \nsolution and sonicated for 2 h. This solution was then dried at 80 ˚C overnight , followed by \nannealing the dried powder at 600 ˚C for 2 h at a hea ting ramp rate of 10 ˚C min-1.  \nMaterial characterization  \nMorphology of the resulting CFO NPs was studied by transmission electron microscopy \n(TEM, FEI F30), and scanning transmission electron microscopy (STEM, FEI F30). \nDistribution of elements along the nanoparticles were studied by energy -dispersive X -ray \n(EDX) mapping using HAADF STEM (FEI Talos F200X). The crystallographic structure of \nthe nanostructures was analyzed by X -ray diffraction (XRD) on a Bruker AXS D8 Advance      \n21 \n X-ray diffractometer, equipped w ith a Cu target with a wavelength of 1.542 Å. Local \ncrystallographic structure was studied by selected area electron diffraction (SAED). \nPiezoresponse force microscopy (PFM) investigations were performed on a commercial \natomic force microscope (NT -MDT Nteg ra Prima). PtIr -coated Si probes, i.e. FMG01/Pt \n(spring constant k ~ 3 N m-1), were used, and the imaging contact force set -points were \ncarefully controlled. To acquire local piezoresponse loops, ac signals (V AC = 0.5 V) were \nsuperimposed on triangular sta ircase wave with DC switching from -10 V to 10 V. To study \nchange in piezoelectric response of the sample under magnetic field, an in -plane magnetic \nfield of 1000 Oe was applied to the sample.  \n \nMultiphysics simulation of ME CFO -BFO NPs  \nThe simulations were  implemented in the commercially available software COMSOL \nMultiphysics based on similar examples from literature. 1,2 The physics of our COMSOL \nmodel used to describe the ME effect was divided into magnetic fields , solid mechanics and \nelectrostatics. Influences from the surrounding medium on the induced electrical surface \npotential  are neglected and the relative permittivity and permeability is assumed to be 1. For \nthese simulations presented in Figure 3 c -d, octah edral CFO NP with a diameter of 30 nm and \na BFO shell with a thickness of 5 nm were selected as input parameters. An epitaxially grown \nBFO shell on the CFO core´s [111] plane was considered and implemented accordingly in the \nmodel.3,4 The magnetic field strength was fi xed at 15 mT and applied on the boundaries of the \nmedium along the global z -axis. Since the NPs are free to move in the surrounding medium, it \nwas assumed that under magnetic field they align with the excitation field and hence, \nmagnetostriction along the easy axis was considered. COMSOL was used to compute the \nmagnetization gradients within the material by using the applied magnetic field and its \ncorresponding magnetization values from the measured VSM hysteresis curve for CFO NPs      \n22 \n (Figure S6). The internal  strain generated in a CFO NP under magnetic fields was governed \nby the following equation,  \n \nwhere,  \n  is the strain along the z -axis, \n  the magnetostriction parameter ( λs,CFO= -273 ppm, \nλs,BFO=-0.002 ppm) , \n  the magnetization along the z -axis and \n  the saturation \nmagnetization of the CFO core.5-7 The strain transfer from CFO core to the BFO shell was \nassum ed to be ideal.1 This strain in the BFO shell is converted into electric polarization on the \nsurface of B FO. BFOs’ piezo -electric coupling and compliance matrix  with R3c symmetry are \ngiven by  \n \n \n \n \nand were derived from literature.2,8-10 \nThe remaining elastic and electric properties of the CFO core and BFO shell were  also \ndetermined from literature.2,11-15  The mechanical boundary condition was set in the middle \nplane of the CFO core through fixing the vertices. The electrical ground was applied on the \nboundaries of the medium . For the study of the induced electrical surface potential as a \nfunction of BFO shell thickness, shell thickness values were selected from 2.5 nm to 20 nm.  \n \nMagnetoelectric effect induced RhB degradation measurement       \n23 \n Experiments were performed to study th e degradation of RhB dye in the presence of our \nCFO -BFO NPs using AC magnetic fields. An RhB concentration of 2 mg L-1 was chosen to \nperform degradation experiments. 20 mg of CFO -BFO NPs were dispersed in 20 mL of RhB \nsolution and placed inside the custom -built magnetic set -up and subjected to various magnetic \nfields and frequencies under constant agitation, after the adsorption -desorption equilibrium \nwas reached. A UV -Vis  spectrophotometer (Tecan Infinite 200 Pro) was used to obtain the \nfluorescent spectr a of RhB over time by taking aliquots of irradiated RhB solution every 10 \nminutes for 50 minutes.  \n \nTrapping experiments  \nTo investigate the degradation pathway behind magnetoelectric effect -induced  catalysis, we \nperformed trapping experiments by using diff erent scavengers. AgNO 3 (2 mM), ethylene \ndiamine tetraacetic acid (EDTA, 2 mM), tert -butyl alcohol (TBA, 2 mM) and benzoquinone \n(BQ, 0.5 mM) solutions were prepared in a 2 mg L-1 RhB solution. For the catalysis \nexperiments, 2 mg of CFO -BFO NPs were dispers ed in 2 mL of RhB solution and placed \ninside the custom -built magnetic set -up. To probe the formation of OH• radicals, 0.5 mM \nterephthalic acid solution was prepared and subjected to AC magnetic fields with CFO -BFO \nNPs, after which the solution’s intensity  was monitored at 425 nm every 30 mins.  \n \nMagnetoelectric effect induced micro -pollutant degradation measurement  \nExperiments were performed to study the degradation of five common pharmaceuticals in the \npresence of our CFO -BFO NPs using AC magnetic fields.  The pharmaceuticals chosen for \nthis study were Carbamazepine, Diclofenac, Gabapentin, Oxazepam and Fluconazole, each \nhaving a concentration of 50 µg L-1. 50 mg of CFO -BFO NPs were dispersed in 55 mL of \nabove solution and placed inside the custom -built mag netoelectric set -up and subjected to the \noptimised magnetic field conditions under constant agitation, once the adsorption -desorption      \n24 \n equilibrium was reached. After magnetic treatment, the NPs were removed from the solution \nusing centrifugation and magneti c separation. To compare the efficiency of the magnetic \ntreatment, a second sample set was prepared without being subjected to any magnetic \ntreatment. Sample preparation for evaluating concentration of pharmaceuticals was done as \ndescribed by Dasenaki et a l. .16 Briefly, 50 mL of samples were used. Samples were stored at \n4 °C and were analyzed within 5  days. All samples showed a pH value between 1.9 and 2.1 \nand hence,  samples were not acidified as reported before.16 Before loading, Strata -XL \ncartridge were first conditioned with 6 mL of methanol and afterwards with 6 mL of ultrapure \nwater. Conditioning process was done under gravity. Samples were loaded on the cartridges, \nalso under gravity. After loading, cartridges were washed with 6 mL of pure water and \nsubsequently dried under reduced pressure for 30 min. Target analytes were eluted with 6 mL \nof methanol. Solvent was evaporated under a gentle steam of nitrogen at 40 °C. Dried \nresidues were disso lved in 100 µL of 0.1% formic acid in water. Five replicates of each \nsample were analyzed. Analysis was performed using a LTQ -XL linear ion trap (Thermo \nScientific, San Jose (CA), United States ) mass spectrometer coupled to a Waters Acquity™ \nUPLC system ( Milford (MA), United States). Gradient elution was done on a Waters BEH \nC18 column (2.1 x 50 mm, 1.7 µm). The mobile phase consisted of 0.1% formic acid in water \n(eluent A) and formic acid (0.1%) in acetonitrile (eluent B). The flow rate was set to 0.5 \nmL/min and the injection volume was 10 µL. Dwell volume of the UPLC system was 0.7 mL. \nThe final LC gradient was as follows: 0 -2 min 2% B, 2 -12 min to 90% B, 12 -15 90% B, 15 -\n15.5 min to 2% B,15.5 -20 min 2% B. The column oven and the autosampler was set to 30  °C \nand 10 °C, respectively. MS settings were used as reported by Wissenbach et al.17 and are as \nfollows. Analysis of carbamazepine, fluconazole, gabapentine and oxazepam was performed \nin the positive ionization mo de. For diclofenac, negative ionization mode under same solvent \nconditions was used. The LTQ -XL was equipped with a heated ESI II source set to 150 °C. \nSheath gas 40 arbitrary units (AU), auxiliary gas 20 AU: source voltage 3.00 kV; ion transfer      \n25 \n capillary 300 °C. capillary voltage 31 V; tube lens voltage 80 V. Automatic gain control was \nset to 15000 ions for full scan and 5000 for MSn. Collision induced dissociation (CID) - MSn \nexperiments were performed on precursor ions selected for MS1. Using information \ndependent acquisition, MS1 was performed was performed in full scan mode ( m/z 100-1500). \nMS2 and MS3 were performed in the IDA mode: four IDA MS2 experiments were performed \non the four most intensive signal from MS1 and additionally eight MS3 scan filters were \nchosen to record the most and second most ions from MS2. Removal efficiency was evaluated \nby comparing peak areas of the respective drugs before and after treatment.  \n \n \nFigure S1.  MS spectra obtained for the five pharmaceuticals (a) before and (b) af ter treatment \nwith ME CFO -BFO NPs under magnetic fields, showing drastically reduced concentrations \nafter treatment.  \n \n      \n26 \n \n \nFigure S2. Fabrication scheme of CFO -BFO nanoparticles using hydrothermal and sol -gel \nsynthesis.  \n \n \n \n   \n      \n27 \n Figure S3. Characterisation o f CFO NPs uisng SEM and TEM. (a) SEM image showing many \noctahedron CFO NPs. (b) STEM and (c) TEM image of octahedral CFO NPs. (d) HRTEM \nimage showing a is single crystalline CFO NP.  \n \n \n \n \n \nFigure S4. Size distribution of (a) CFO NPs with an average size of 30 ± 6 nm and (b) CFO -\nBFO NPs with an average size of 42 ± 6 nm.  \n      \n28 \n \n \nFigure S5. TEM images showing core -shell CFO -BFO NPs after sol -gel coating and \nannelaing.  \n \n \n \nFigure S6. EDX maps obtained for HRTEM image showing CFO -BFO core -shell NPs, where \nthe distrib ution of elements clearly highlights the presence of a BFO shell formed around a \nCFO core.  \n      \n29 \n \n \nFigure S7. EDX spectra obtained during TEM analysis performed on core -shell CFO -BFO \nNPs showing the presence of O, Fe, Co and Bi elements. (peaks from Cu and Si a re \norigination form the TEM sample holder).  \n \n \n \n \n \n \n \n \n \n \n      \n30 \n \n \n \n \nFigure S6. Magnetic characterisation of as -fabricated CFO NPs (without using magnetic \nseparation) to investigate their initial magnetisation and magnetic saturation M s values.  \n \n      \n31 \n \n \n \nFigure S7. COMS OL multiphysics simulations on CFO -BFO NPs for a BFO shell thickness \nof 5 nm and an applied magnetic field of 15 mT. (a,c) volumetric strain distribution on the \nsurface of CFO -BFO NP and (b.d) the corresponding strain distribution in cross -section.  \n \n \n      \n32 \n \n \nFigure S8. COMSOL multiphysics simulations on CFO -BFO NPs for a BFO shell thickness \nof 5 nm and an applied magnetic field of 15 mT. Potential generated (a) in the cross -section of \nCFO -BFO NP, (b -d) on the surface of BFO shell viewed from different angles sh owing a \ngradient potential distribution with opposite polarities on the extreme sides.  \n \n \n \n      \n33 \n \n \nFigure S9. Fluorescent spectra of RhB collected every 10 minutes and measured at a peak of \n554 nm during magnetoelectrically induced catalytic degradation of RhB.  From this plot it \ncan be seen that with increasing reaction time, RhB’s concentration reduces dramatically.  \n \n      \n34 \n Figure S10. COMSOL simulations performed on CFO -BFO NPs to study the potential \ngenerated on BFO shell under increasing magnetic field strengths . This plot clearly shows that \nincreasing the magnetic field leads to an increase in the induced surface potential on BFO \nshell, a trend that supports the RhB degradation rates presented in Figure 4b.  \n \nFigure S11. COMSOL simulations performed on CFO -BFO N Ps with varying shell thickness \nshow that increasing shell thickness of BFO led to a lower absolute strain generation in the \nCFO core, presumably due to the clamping effects.  \n      \n35 \n \n \nFigure S12. COMSOL simulations performed on CFO -BFO NPs with varying shell th ickness \nand its influence on the potential generated on the BFO shell. An optimal BFO shell thickness \nwas identified at 7.5 nm, which is close to the shell thickness of our fabricated core -shell \nstructures, where the maximum electrical potential of 5.24 μV could be generated.  \n \n \n \n \n \n \n \n \n      \n36 \n References  \n (1) Betal, S.; Shrestha, B.; Dutta, M.; Cotica, L. F.; Khachatryan, E.; Nash, K.; \nTang, L.; Bhalla, A. S.; Guo, R. Scientific Reports  2016 , 6, 32019.  \n (2) Aimon, N. M.; Liao, J.; Ross, C. A.  Applied Physics Letters  2012 , 101. \n (3) Madigou, V.; Souza, C. P. D.; Madigou, V.; Souza, C. P. D.; Con, C. L. 2014 . \n (4) Apc International, L. Piezoelectric ceramics: principles and applications ; \nAPC International, 2002.  \n (5) Chikazumi, S. Physics of fer romagnetism ; 2nd ed., r ed.; Oxford : Oxford \nUniversity Press, 1999; Vol. 94.  \n (6) Khaja Mohaideen, K.; Joy, P. A. Journal of Magnetism and Magnetic \nMaterials  2014 , 371, 121.  \n (7) Kumar, A.; Scott, J. F.; Martínez, R.; Srinivasan, G.; Katiyar, R. S. Physic a \nStatus Solidi (A) Applications and Materials Science  2012 , 209, 1207.  \n (8) Graf, M.; Sepliarsky, M.; Machado, R.; Stachiotti, M. G. Solid State \nCommunications  2015 , 218, 10. \n (9) Wang, Y. L.; Wu, Z. H.; Deng, Z. C.; Chu, L. Z.; Liu, B. T.; Liang, W. H.; Fu, \nG. S. Ferroelectrics  2009 , 386, 133.  \n (10) Nye, J. F. Physical properties of crystals : their representation by tensors and \nmatrices ; [Reprinted ed.; Oxford : Clarendon Press, 2009.  \n (11) Catalan, G.; Scott, J. F. Advanced Materials  2009 , 21, 2463.  \n (12) Bueno -Baques, D.; Corral -Flores, V.; Morales -Carrillo, N. A.; Torres, A.; \nCamacho -Montes, H.; Ziolo, R. F. Mater. Res. Soc. Symp. Proc  2017 . \n (13) Gujar, T. P.; Shinde, V. R.; Kulkarni, S. S.; Pathan, H. M.; Lokhande, C. D. \nApplied Surface Science  2006 , 252, 3585.  \n (14) Vasundhara, K.; Achary, S. N.; Deshpande, S. K.; Babu, P. D.; Meena, S. S.; \nTyagi, A. K. Journal of Applied Physics  2013 , 113. \n (15) Gujar, T. P.; Shinde, V. R.; Lokhande, C. D. Materials Chemistry and Physics  \n2007 , 103, 142.  \n (16) Dasenaki, M. E.; Thomaidis, N. S. Anal. Bioanal. Chem.  2015 , 407, 4229.  \n (17) Wissenbach, D. K.; Meyer, M. R.; Weber, A. A.; Remane, D.; Ewald, A. H.; \nPeters, F. T.; Maurer, H. H. J. Mass Spectrom.  2012 , 47, 66. \n "    },    {        "title": "1902.07188v1.Magnetic_interaction_and_anisotropy_axes_arrangement_in_nanoparticle_aggregates_can_enhance_or_reduce_the_effective_magnetic_anisotropy.pdf",        "content": "arXiv:1902.07188v1  [physics.app-ph]  19 Feb 2019Magnetic interaction and anisotropy axes arrangement in na noparticle\naggregates can enhance or reduce the effective magnetic anis otropy\nV. R. R. Aquino1, L. C. Figueiredo2, J. A. H. Coaquira2, M. H. Sousa3, A. F. Bakuzis1,*\n1Instituto de F´ ısica, Universidade Federal de Gois, 74690- 900, Goinia-GO, Brazil\n2Instituto de F´ ısica, Nucleo de F´ ısica Aplicada, Universi dade de Bras´ ılia, 70910-900, Bras´ ılia-DF, Brazil\n3Faculdade de Ceilndia, Universidade de Bras´ ılia, 72220-1 40,Bras´ ılia-DF, Brazil\n*Corresponding author: bakuzis@ufg.br\nAbstract\nThe magnetic response of nanostructures plays an important role on biomedical applications being strongly\ninfluenced by the magnetic anisotropy. In this work we invest igate the role of temperature, particle concentra-\ntion and nanoparticle arrangement forming aggregates in th e effective magnetic anisotropy of Mn-Zn ferrite-based\nnanoparticles. Electron magnetic resonance and coercivit y temperature dependence analyses, were critically com-\npared for the estimation of the anisotropy. We found that the temperature dependence of the anisotropy follows\nthe Callen-Callen model, while the symmetry depends on the p article concentration. At low concentration one\nobserves only an uniaxial term, while increasing a cubic con tribution has to be added. The effective anisotropy was\nfound to increase the higher the particle concentration on m agnetic colloids, as long as the easy axis was at the\nsame direction of the nanoparticle chain. Increasing even f urther the concentration up to a highly packed condition\n(powder sample) one observes a decrease of the anisotropy, t hat was attributed to the random anisotropy axes\nconfiguration.\nKeywords: magnetic anisotropy, electron magnetic resonance, dipola r interaction, random anisotropy, magnetic\nhyperthermia.\nI INTRODUCTION\nThe magnetic anisotropy constant have a great impact in the magne tic response of nanoparticles. In non-interacting\nsystems it defines (together with the particle size) if the nanopart icle is at the blocked or superparamagnetic (SP)\nregime. SP particles are believed to play an important role in biomedical applications, spanning from contrast agents\nforMRI, heat generatorsinmagnetichyperthermia, cellsortinga pplicationsdue tomagnetophoresisproperties, among\nothers. [1–4] Moreovernanoparticles can form aggregates, for example linv ear chains or spherical nanostructures (even\nat the SP state due to van der Waals interactions), [5] where in this c ase the intraparticle interactions (mainly due to\nmagnetic dipolar interactions) can modify the effective magnetic anis otropy. The knowledge of this effect is of great\nfundamental and technological importance due to the several ap plications of magnetic nanoparticles (even beyond the\nbiomedical field), since it directly correlates with the relaxation of th e magnetization.\nIn the literature there is still a strong debate about the effect of d ipolar interaction, where some authors claim it\ndecreases the effective anisotropy, while others point to the oppo site effect. [5–10] It is also curious to notice that few\narticles discuss about the task of anisotropy axes arrangement, the anisotropy temperature dependence or the particle\ninteraction role on the anisotropy energy symmetry, which again ca n impact the magnetic relaxation. [11]\nNowadays several techniques are applied to determine the magnet ic anisotropy, as for instance ZFC/FC mag-\nnetization curves, [12,13] coercivity temperature dependence a nalysis, [14,15] electron magnetic resonance, [16–18]\nMossbauer spectroscopy, [6] among others. In some cases one fi rst determine the blocking temperature and then\ncalculate the magnetic anisotropy, where some methods assume th e anisotropy to be temperature independent. This\nis obvious incorrect since several anisotropy contributions are te mperature dependent, such as the magnetostatic or\nmagnetoelastic terms. In particular, for the ZFC/FC curves, [12] generally, it is not clear that the determination of\nanisotropy via blocking temperature is an estimated anisotropy ref ering to the value at this temperature. For several\nstudies this might not be that important, however there is a great in terest on using magnetic nanoparticles for cancer\ntherapy through magnetic hyperthermia. [3,4,19,20] Here temp erature effects can have a great impact on the clinical\noutcome. Furthermore this biomedical application showed a strong dependence upon particle arrangement, [10,21–26]\nand therefore understanding the role of particle interactions and nanoparticle arrangements is also crucial. It might\nalso be relevant to notice the experimental condition that the effec tive anisotropy is determined, for example in some\nworks the magnetic anisotropy of the nanoparticles is estimated us ing a powder sample, while the relevant application\nproperty is analysed for the colloidal suspension. In the powder sa mple the nanoparticles are highly packed and one\nmight assume that the anisotropy axes are randomly arranged, wh ile in the colloidal suspension distinct aggregate\n1formations can arise, as for instance linear chains that are expect ed to have their anisotropy axes arranged along the\nchain (longitudinal configuration). [27]\nIn this work, we investigate the role of particle arrangement on the effective magnetic anisotropy by critically\ncomparing data using the coercivity temperature dependence ana lysis and electron magnetic resonance (EMR). In the\nHcvsTmethod powder samples were analysed, while in the former magnetic c olloids at distinct particle concentra-\ntions were investigated. Mn-Zn ferrite nanoparticles surface-co ated with citric acid of distinct sizes were compared,\nallowing us to determine the value of the effective magnetic anisotrop y from the non-interacting condition (highly\ndiluted magnetic colloid) up to a particle volume fraction of around 0.64 , that corresponds to the packing fraction of\nmonodisperse spherical particles. [28] Here, we demonstrate tha t magnetic anisotropy is strongly temperature depen-\ndent, and that its behavior is well represented by the Callen-Callen m odel. [29] The room temperature anisotropy is\nfound to increase the higher the particle concentration in the colloid , and above a critical concentration it shows a\ncubic anisotropy symmetry contribution that was not reported be fore in other works. We show that the experimental\nresult is in accordance with the theoretical prediction that the exis tence of linear chains will influence the anisotropy\nby means of an additional uniaxial contribution. [5] On the other han d, the existence of the cubic anisotropy term\nsuggests a multipolar contribution. Increasing even further the c oncentration up to a highly packed condition (powder\nsample), we observed a decrease of the anisotropy and relate it to the random anisotropy axes configuration for this\nparticle arrangement situation. The result might be useful on unde rstanding some contradictory reports in the litera-\nture regarding the effect of particle interactions, and might impact not only the magnetic hyperthermia field, but also\nmagnetic particle imaging [30,31] and magnetic nanothermometry, [3 2,33] among others.\nThe article is organized as follows: In section II, we present the the oretical background for both methods, coer-\ncivity temperature dependence analysis and electron magnetic res onance, used to determine the magnetic anisotropy.\nDifferent from most works of the literature the effect of the tempe rature dependence of the anisotropy is explicitly\ntaken into account in the HcvsTmethod, that also determines the sample blocking temperature dist ribution. On the\nother hand, EMR extracts the anisotropy field. Where from the te mperature dependence magnetization studies of the\nsamples one can determine the effective magnetic anisotropy as fun ction of temperature and particle concentration.\nSection III discuss the experimental procedures, i.e. synthesis a nd characterization techniques. Finally, section IV\npresents the results and discussions, while the conclusions are sho wn in section V.\nII THEORETICAL BACKGROUND\nA Coercivity temperature dependence method\nAccording to this method, the coercivity temperature dependenc e/angbracketleftHC/angbracketrightTof the sample can be modeled using, [34,35]\n/angbracketleftHC/angbracketrightT=MR(T)\nχsp(T)+MR(T)\nHCB(T), (1)\nwhereMR=αMs/integraltext∞\nTP(TBl)dTBlis the remanent magnetization, χsp=25M2\nS\n3Kef/integraltextT\n0TBlP(TBl)dTBlthe superparamag-\nnetic susceptibility, P(TBl) is the lognormal distribution function of the blocking temperature s (TBl). For randomly\noriented particle systems α= 0.48, whileHCBis described by the following equation\nHCB(T) =α2Kef(T)\nMs/parenleftBigg\n1−/parenleftbiggT\n/angbracketleftTBl/angbracketrightT/parenrightbigg3/4/parenrightBigg\n, (2)\nwith/angbracketleftTBl/angbracketrightT=/integraltext∞\nTTBlP(TBl)dTBl/integraltext∞\nTP(TBl)dTBla temperature dependent parameter related to the distribution o f the blocking tem-\nperatures at a given temperature T, while the exponent 3 /4 corresponds to the random anisotropy case. [35] More\nimportant, different from previous works in the literature, the effe ctive anisotropy constant is assumed to be temper-\nature dependent. This dependence is shown to experimentally follow the Callen-Callen model, [29] which establishes\na relationship between magnetization and anisotropy via the equatio n:\nK(T)\nK(0)=/bracketleftbiggMs(T)\nMs(0)/bracketrightbiggl(l+2)/2\n, (3)\nFor the uniaxial case l= 2 andK(0) is the anisotropy at T= 0K. Thus, this equation can be rewritten using\nthe Bloch model, Ms(T) =Ms(0)/parenleftbig\n1−bT3/2/parenrightbig\n, whereMs(0) is the saturation magnetization at 0 Kandbis the\nBloch constant, that are determined from magnetization measure ments. Therefore, the temperature dependence of\nthe effective uniaxial magnetic anisotropy constant can be written as\nKef(T) =K(0)/parenleftBig\n1−bT3/2/parenrightBig3\n. (4)\n2B EMR method\nThe other technique that will be used to estimate the magnetic aniso tropyis the Electron Magnetic Resonance (EMR).\nFor spherical particles the resonance field condition is given by [16,1 7]\nHR=ω\nγ√\n1−α2−2Kef\nMs, (5)\nwhereωis the angular frequency, γis the gyromagnetic ratio and αis the damping term (usually much lower than\n1). Note that the effective anisotropy constant can be expanded in terms of spherical harmonics such as Kef=/summationtext\nl/summationtext\nmKlPl(cosθ)eimϕ, wherePlis the Legendre polynomial and θis the angle between the applied magnetic field\nand the nanoparticle anisotropy axis. [16] For spherical particles, m= 0, while considering only the uniaxial case,\nl= 2 term, reveals an uniaxial anisotropy field 2 Ks\n2/MSP2(cosθ). For the longitudinal case ( θ= 0) this expression is\nthe well known uniaxial anisotropy field 2 Ks\n2/MS, as expected. On the other hand, more complex cases can appear\nby including other values of l, that reflect the symmetry of the anisotropy. For example, the a dditional existence of a\ncubic contribution, l= 4, results in the condition:\nHR=ω\nγ√\n1−α2−2K2\nMsP2(cosθ)−2K4\nMsP4(cosθ). (6)\nSo, the behavior of Hras a function of the angle can indicate the anisotropy symmetry, wh ere including only the\nl= 2 term, reflects an uniaxial anisotropy symmetry, while the neces sity of other contributions reveals a more complex\nsituation.\nC Dipolar interaction contribution to the anisotropy\nIn 2013, Bakuzis et al. [5] mathematically showed that the dipolar inte raction between nanoparticles, forming small\nagglomerates in a linear chain, has a uniaxial contribution due to the p article-particle interaction term. The linear\nchain model was applied to two situations, namely fanning and cohere nt. In the coherent case, it is assumed that the\nmagnetic moments of the particles are all in the same direction, rota ting coherently in the direction of the magnetic\nfield. In the fanning structure, the magnetic moments of the adja cent particles rotate in opposite directions. In both\ncases, fanning and coherent, a uniaxial contribution to the energ y density of the particle ( l= 2 term) is noted. [5]\nThese results can be generalized for the case of a chain containing Qparticles. Below, we present the contribution to\neffective anisotropy constant in the fanning case, [5,10]\nKfanning\ndip=µ0\n4πM2\nSVp/parenleftbig¯D+dss/parenrightbig3×\nQ/summationdisplay\ni=odd(Q−i)\nQ(i)3+3Q/summationdisplay\ni=pair(Q−i)\nQ(i)3\n,\nwhereµ0is a magnetic permeability of the vacuum, Dis the nanoparticles mean diameter and VPthe particle volume.\nThe nanoparticles centers are distant from a value r=¯D+dss, wheredssis the distance between the surfaces of the\nnanoparticles. The coherent calculation can be found in. [5]\nIII EXPERIMENTAL PROCEDURE\nAmong the various methods for synthesizing magnetic nanoparticle s, with new morphologies and dimensions, the\nhydrothermal technique has been extensively explored. This synt hetic route allows working with temperatures above\nthe boiling point of the chosen solvent, changing the crystallization/ recrystallization conditions in the synthetic envi-\nronment. [36] For the preparation of magnetic nanoparticles base d on manganese-zinc ferrite with the stoichiometry\nMn0.75Zn0.25Fe2O4, we mixed 3 .75mmolof manganese chloride tetrahydrate (MnCl 24H2O), 1.25mmolof ions zinc\nchloride (ZnCl 2) and 10mmolof ferric chloride hexahydrate (FeCl 36H2O) from 1mol/lstock solutions. Thus, 50 ml\nof 8.0wt% aqueous methylamine (CH 3NH2) was poured into the metal’s solution under magnetic strring for abo ut\n5minat room temperature. The mixture was sealed in a teflon-lined autoc lave and maintained at 160◦C for 5hinside\nan oven.\nAfter this time, the formed magnetic material was washed three tim es with distilled water. To the resulting\nmagnetic material was added 300 mlTo prepare the citrate-capped nanoparticles, 2 gof trisodium citrate was added to\nthis dispersion under magnetic stirring at 80◦Cfor 30minutes. The precipitate was magnetically separated, washed\nwith acetone and redispersed in water. The pH of dispersion was car efully adjusted to form a stable magnetic fluid\nat pH∼7. To obtain samples with different mean diameters, we used a size-so rting method based on the increase\nthe ionic strength of the sol – through the addition of NaCl – which in duces a phase transition in the colloid. [37,38]\nTypically, NaCl is added to the sol with a magnet placed at the bottom of the flask until visual colloidal separation.\nAfter magnetic separation, the supernatant (with smallest nanop articles) and precipitate (with larger nanoparticles)\nare washed with acetone in order to resuspend nanoparticles in aqu eous solution at pH ∼7.\n3Once the magnetic fluid was achieved, they were diluted in different vo lumes fractions, which were checked by\nexpression: φ≈Mfluid/Ms, whereMfluidandMsare the saturation magnetization of the fluid and the powder,\nusing a VSM (vibrating sample magnetometer, 2 Tesla). After obtaining the magnetic fluid, the nanoparticles were\ncharacterized by several techniques, such as energy dispersive spectroscopy (EDS), obtaining images for compositional\nanalysis; X-ray diffraction (XRD), to obtain the crystalline phase an d the average size of the crystallite; transmission\nelectron microscopy (TEM), for the calculation of the distribution o f diameters and shape of the particles; Supercon-\nducting Quantum Interference Device (SQUID) magnetometer, n ecessary to achieve saturation magnetization and the\ncoercive field at low temperatures and electron magnetic resonanc e (EMR), for anisotropy field determination.\nMeasurements of the chemical composition of the samples were car ried out using energy dispersive spectroscopy\n(EDS) using the transmission electron microscope (JEOL model JEM -2100), operating in EDS mode at 15 kV. For\nthe determination of particle sizes, we dry a part of the colloid to obt ain the powder and perform the analysis by\nXRD (Shimadzu 6000). Images of the nanoparticles were obtained u sing the transmission electron microscope (Jeol\nmodel JEM-2100) operated at 200 kV, with resolution of 25 ˚A. For the characterization of magnetic properties at low\ntemperatures, we used a VSM-SQUID (Quantum Design PPMS3) with a DC field ranging from -70 to 70 kOe, and\ntemperatures ranging from 5 to 300 K. On the other hand the calcu lation of the particle volume fraction at room\ntemperature was obtained using a VSM (ADE Magnetic, EV-9 model) w ith a DC field ranging from -20 to 20 kOe.\nFinally, EMR measurements were perfomed with a spectrometer EMX -Plus model Bruker, where the magnet had a\nmagnetic field amplitude up to 14 kGand X-band microwave bridge tuned around 9.5GHz. The EMR proced ure to\nextract the anisotropy field was the same of Refs. [16,17,39]. Bas ically, at room temperature, one apply the highest\nexternal magnetic field to the magnetic fluid sample with the objectiv e to orient the nanoparticle magnetic anisotropy\naxis along the field direction. With the field on, the sample is frozen to 1 00K. The procedure blocks the nanoparticle’s\nanisotropy in a specific direction. A goniometer device allow rotation o f the sample with respect to the applied field\nduring the EMR experiment. So, EMR spectra at distinct angles, for a given temperature and particle concentration,\ncan be obtained to determine the value and symmetry of the effectiv e anisotropy constant.\nIV RESULTS AND DISCUSSION\nFigure 1(a) shows the XRD data for both nanoparticles confirming t he cubic spinel structure. Crystallite sizes of 10.3\nand 11.4nmwere obtained using the Scherrer equation, i.e. DXRD=κλ/βcosψ , whereκ= 0.89 is the Scherrer\nconstant,λ= 0.154nmis the X-ray wavelength, βis the line broadening in radians obtained from the square root of\nthe difference between the square of the experimental width of th e most intense peak to the square of silicon width\n(calibration material), and ψis the Bragg angle of the most intense peak. The inset of Fig. 1(b) sh ows an image of a\nfilm made of 10nm nanoparticles, while somespots showplaces where w ereperformed the EDS-TEM analysis. Fig. 1b\nshows the EDS analysis of this sample, where one can clearly observe the existence of Mn, Zn and Fe, as expected (Cu\nsignal is due to the TEM microgrids). Fig 1(c) shows the size distribut ion for both samples obtained from the analysis\nof TEM pictures, while the inset shows TEM images of the 10nm size nan oparticles revealing spherical-like particles.\nFrom the fit of the histogram using the lognormal distribution size (m edian and size dispersion parameters) one can\ncalculate the mean diameters and the standard deviation, 10 ±2 and 11 ±2nm. Magnetic characterization is shown\nin Figure 1(d) for the 10nm particle size, while the inset shows hyster esis curves at low field range. Measurements\nwere performed at a wide temperature range 5 to 300 K for a powde r sample. The saturation magnetization value is\nobtained from the analysis at the high field limit, i.e. from extrapolation of data ofM×1/Hwhen 1/Htends to\nzero. Fig. 1(e) shows the temperature dependence of the satur ation magnetization for both samples. The symbols\nrepresent the experimental data, while line is the best fit using the B loch’s law, that revealed the Bloch constants,\n7.5×10−5and 6.6×10−5K−3/2for 10 and 11 nmdiameters and the saturation magnetization value at 0 K, 585 and\n565emu/cm3, respectively. On the other hand, the particle concentration of t he magnetic fluids was obtained by the\nanalysis of the magnetization at room temperature using a 2T VSM. F igure 1(f) shows the magnetization curves of\nthe same sample but now at different particle volume fractions ( φ). The estimation of φarises from the ratio of the\nsaturation magnetization of the sample to the saturation measure d for the nanoparticle (powder sample) at the same\nexperimental condition, that for our samples were found to be 293 emu/cm3and 303emu/cm3for particles of 10 and\n11nm.\nA Blocking temperature distribution\nFigure 2(a) shows the temperature dependence of the coercivity for both samples, while in the inset is shown the low\nfield hysteresis curves for the 11nm sample. Symbols correspond t o experimental data, while the lines correspond to\nthe best fit using the theoretical model discussed in section IIA, t hat included the effective anisotropy temperature\ndependence by using Kef(T) =K(0)/parenleftbig\n1−bT3/2/parenrightbig3. It is clear in this model that only the value of the anisotropy\nat 0Kbecomes one of the parameters of adjustment of the coercive fie ld data, where the other fitting parameter is\nrelated to the blocking temperature distribution, Tm\nBl. Note that because of lognormal distribution properties, one\ncan assume that the dispersion of blocking temperatures (that is p roportional to the particle volume) is related to\n42E(degree)30 40 50 60\n  10.3 nm\n  11.4 nm\n-20 -10 0 10 20-8-4048\nfluid volume fraction\n  I\u0003 \u0003\u0016\u0003\b\n  I\u0003 \u0003\u0014\u0003\b\n  I\u0003 \u0003\u0013\u0011\u0014\u0003\b\n  I\u0003 \u0003\u0013\u0011\u0013\u0015\u0017\u0003\b\n  I\u0003 \u0003\u0013\u0011\u0013\u0014\u0015\u0003\b10 nm\n0 50 100 150 200 250 300350400450500550600 10 nm\n 11 nm\n model\n Diameter (nm)\nH (kOe) H (kOe) T (K)EDS - TEMIntensity (a.u)\nIntensity (a.u)\nn° of particles (a.u)M emu/cm&\nM emu/cm&Ms (emu/cm&a) b) c)\nd) e) f)\n-60 -30 0 30 60-600-400-2000200400600\n-200 -100 0 100 200-8-4048M (emu/cm3)\nH (Oe)10 nm 5 K\n 50 K\n 100 K\n 150K\n 200 K\n 250 K\n 300 K\n  \n6 9 12 15 18  10 ± 2 nm\n           G\u0003 \u0003\u0013\u0011\u0015\u0014\n  11 ± 2 nm\n           G\u0003 \u0003\u0013\u0011\u0014\u001c  \n Lognormal\n \n4 6 8\nkeV10 12 14\nFigure 1: (a) XRD data of the MnZn samples. Crystaline sizes are det ermined using Scherrer equation. (b) EDS\nanalysis of the 10nm sample showing the existence of Mn, Zn and Fe in t he nanoparticle composition. The inset\nshows a TEM picture and the position of the EDS analysis. (c) Size dist ribution obtained by TEM. (d) Hysteresis\ncurves at distinct temperatures for the 10nm sample. The inset sh ows the magnetization curves at low field range.\n(e) Saturation magnetization temperature dependence study. S ymbols represent data, while lines are the Bloch law\nmodel. (f) Magnetization data of the 10nm-based colloid for particle volume fraction determination.\nTable 1: Effective anisotropyconstantsand averageblocking temp eratures of the Mn 0.75Zn0.25Fe2O4samples according\nto the coercivity temperature dependence analysis.\nD\n(nm)Kn\nef\n(erg/cm3)Tn\nBl\n(K)Kef(0K)\n(erg/cm3)Kef(300K)\n(erg/cm3)TBl\n(K)\n10 1 .5×105110±76 1 .5×1053.4×105122±85\n11 1 .4×105130±80 1 .4×1054.1×104141±87\nthe size dispersity (obtained from TEM analysis) through σB= 3σTEM. Therefore, the mean blocking temperature\nof the sample can be calculated using the equation TBl=Tm\nBlexp[(σB)2/2]. Fig 2(b) shows the blocking temperature\ndistribution obtained from the analysis of /angbracketleftHc/angbracketrightTvs T. The dashed lines indicate the position of the mean blocking\ntemperature for both samples. As expected, higher value was fou nd for the larger particle size.\nTable 1 summarizes the parameters obtained from the coercive field analysis, /angbracketleftHc/angbracketrightTvs Tmethod, namely TBl\nandKef(0K). For comparison we also included the values estimated using the non -temperature dependent model,\nKn\nefandTn\nBl. Curiously the value obtained is very close to Kef(0K), whileTn\nBlis slightly lower than TBl. As\nfor instance, the 10nm sample showed TBl= 122K, while for the non-temperature dependent model Tn\nBl= 110K.\nThe difference with other models from the literature [34] are that: ( i) firstly, in our case, we can easily identify the\ntemperature correspondence of this anisotropy; (ii) Second, he re there is no necessity to fit ZFC/FC curves to extract\nthe blocking temperature distribution, a procedure that dependin g on the nanoparticle is not easily performed; (iii)\nIn the present model, one can estimate the anisotropy constant a t room temperature, where it is found that both\nsamples shows room temperature anisotropy values on the order o f∼104erg/cm3. However, it is very important\nto notice that so far we made the asumption that the anisotropy te mperature dependence follows the Callen-Callen\nmodel. In the next section we demonstrate that this is indeed a very good approximation.\nB Anisotropy temperature dependence and the Callen-Callen model\nThe inset of Fig. 2(c) shows EMR spectrum at different angle position s for the 10nm sample with a particle volume\nfraction of 3%. Similar experiments were performed at distinct part icle concentrations for both samples. Fig. 2(c)\nshows the resonance field position as function of the angle between the applied field and the anisotropy axis for the\n50 50 100 150 200 250 30036912151821\n volume fraction\n I\u0003\u0003 \u00033 %\n I\u0003\u0003 \u0003\u0013\u0011\u0013\u001c3 %\n I\u0003\u0003 \u00030.018 %\n I\u0003\u0003 \u00030.005 %\n model  11 nm\n \n0 50 100 150 200 250 30036912151821\n10 nmvolume fraction\n I\u0003 \u00033 %\n I\u0003 \u00031 %\n I\u0003 \u00030.1 %\n I\u0003 \u00030.024 %\n I\u0003 \u00030.012 %\n   model\n0 20 40 60 80 100 120 140 160 1802.02.22.42.62.8volume fraction\n I\u0003\u0003 \u00033 %\n I\u0003\u0003 \u00030.093 %\n I\u0003\u0003 \u00030.018 %\n I\u0003\u0003 \u00030.005 %\n  uniaxial model\n  uniaxial + cubic model   11 nm\n100 K0 50 100 150 200 250 300 350 400\n 10 nm\n11 nm-6( x sr8\u0000?I7;\n-6( x sr8\u0000?I7;Hr (kOe)\nHr (kOe)P  (\u0017Fj) < Hc >T  Oe\n\u0017Fj(K) T(K)\nE(graus)E(graus)\nT(K) T(K)a) b) c)\nd) e) f)0 100 200 300050100150200\n-40 -20 0 20 40-8-4048\n11 nm 300 K\n 250 K\n 200 K\n 150 K\n 100 KM (emu/cm3)\nH (Oe)\n  10 nm\n  11 nm\n  model\n0 20 40 60 80 100 120 140 160 1802.02.22.42.62.8\n  100 K\n10 nm\nI\u0003 \u00033%\n ANGULAR POSITION\n                      0° \n                      20°\n                      40°\n                      60°\n                      90°10 nm\n100 K uniaxial model\n uniaxial \n        + cubic modelvolume fraction\n I\u0003 \u0003\u0016\u0003\b\n I\u0003 \u0003\u0013\u0011\u0014\u0003\b\n I\u0003 \u0003\u0013\u0011\u0013\u0015\u0017\u0003\b\n I\u0003 \u0003\u0013\u0011\u0013\u0014\u0015\u0003\b\nFigure 2: (a) Coercive field as function of temperature. Symbols ar e data, while lines correspond to the model.\nThe inset shows the the magnetization curves at low field range for t he 11nm sample. (b) Blocking temperature\ndistributions obtained from the model for both samples. In (c) and (d) EMR field as function of the angle between\nthe applied field and the anisotropy axis at 100K and distinct particle c oncentrations, respectively for the 10nm and\n11nm samples. Solid and dash lines are the adjustments of the exper imental data according to the models discussed in\nthe text. The inset in (c) shows typical EMR spectra at distinct ang les. In (e) and (f) symbols represent the uniaxial\nanisotropyterm K2as function of temperature and particle concentration, respect ively for 10 and 11nm samples, while\nlines are the best fit using the Callen-Callen model.\n10nm sample at different particle concentrations, while Fig. 2(d) sho ws similar data but for the 11nm sample. It is\nclear that there is a shift in the resonance field position increasing th e angle up to 90 degrees. Above this value the\nresonance field position decreases returning to its initial position fo r an angle of 180 degrees. The EMR measurements\nshown here were performed at 100 K, but other experiments at te mperatures 150, 200 and 250K were also obtained.\nHigher temperatures were not analysed because the nanoparticle s were dispersed in water, and we wanted to mantain\nthe nanoparticles in the frozen matrix and avoid the solid-liquid trans ition. Symbols represent EMR data, while solid\nlines corresponds to the uniaxial case (only l= 2 term - Ks\n2) and dashed lines are the multiaxial case (Eq.(6)), where\none can obtain the values of K2andK4. Note that, for low concentrations, the uniaxial contribution alon e (Ks\n2) is able\nto explain the behavior of the resonance field. In particular, it is fur ther noted that the increase in the difference of the\nHr variation is a result of a higher anisotropy when the concentratio n increases. However, in the higher concentrations\nthe adjustment of the experimental data considering only the l= 2 term was not so good. Therefore the analysis\npresented here corresponds to the one using Eq. (6), i.e. one obt ain bothK2andK4. According to the theoretical\nmodels the difference between the resonance field at 0 and 90 degre es is related to the anisotropy field. Since the\nsaturation magnetization temperature dependence was determin ed before (see Fig. 1(e)), using the anisotropy field\nwe extract the temperature dependence of the anisotropy cons tants. Figs. 2(e) and 2(f) shows the effective anisotropy\ntemperature dependence for the 10 and 11 nm samples, respectiv ely. Here we are showing only the K2value, but\nTable 2 summarizes all the other parameters analysed in this study ( including the Ks\n2for the uniaxial case). Symbols\nrepresent experimental data, while solid lines correspond to the be st fit using the Callen-Callen model. It is obvious\nfrom this analysis the excellent aggreement with the data, justifyin g the assumption on the later section. On the\nother hand, K4is only relevant at high particle concentrations, although its value is a round one order of magnitude\nlower than K2(see Table 2). Room temperature anisotropy values can be found u sing the Callen-Callen model, and\nrevealed an increase the higher the particle concentration. Above a critical concentration a cubic anisotropy symmetry\ncontribution has to be added in the analysis, suggesting a possible mu ltipolar contribution to the anisotropy. Possibly,\nthis is the first experimental evidence of the existence of a multipola r contribution at high concentrations. It is possible\nthat such term is the result of a more complex organization of nanop articles than only isolated linear chains. In favor\nof this argument is the fact, well known in the literature, that the m agnetic fluid has a liquid-solid transition increasing\nthe concentration of particles. [40,41] This phenomenon results in the formation of complex self-organized structures,\nfor example in the formation of hexagonal columnar structures. [4 1] This might be different from the case of isolated\nlinear chains, that due to the dipolar interaction between nanopart icles showed only a uniaxial contribution term. [5]\n6Table 2: Anisotropy constants of Mn 0.75Zn0.25Fe2O4of 10 and 11 nmfor each concentration and at different temperatures. Thus, fr om 100 to 250 KEMR data, while\n0 and 300Kcorresponds to extrapolation values using Callen-Callen model.\nφ\n(%)D\n(nm)0K\nK2\n(×104erg/cm3)100K\nK2−K4Ks\n2\n(×104erg/cm3)150K\nK2−K4Ks\n2\n(×104erg/cm3)200K\nK2−K4Ks\n2\n(×104erg/cm3)250K\nK2−K4Ks\n2\n(×104erg/cm3)300K\nK2-K4\n(×104erg/cm3)\n3 10 16 .5 14 .7 3.3 13.8 13.6 3.0 12.8 12.0 2.6 11.2 10.3 2.3 9.6 8 .8 1.70\n1 10 14 .2 12 .5 2.2 11.3 11.4 2.3 10.5 10.0 1.8 9.1 8.4 2.3 7.9 7 .1 1.19\n0.1 10 13 .8 12 .1 2.1 10.6 10.8 1.8 9.8 9.4 1.3 8.7 7.9 1.1 7.6 6 .5 0.98\n0.024 10 9 .9 8 .7 1.0 8.4 7.6 0.8 7.1 6.6 0.8 6.4 5.4 0.6 5.4 4 .4 0.46\n0.012 10 7 .8 6 .6 0.3 6.5 5.7 0.2 5.6 4.7 0.3 4.8 4.0 0.3 3.9 3 .2 0.22\n3 11 17 .3 15 .2 3.6 14.6 14.3 3.3 13.3 12.6 2.7 12.0 10.6 2.2 10.1 9 .5 1.77\n0.093 11 13 .9 12 .2 1.9 14.6 11.2 1.7 10.6 9.9 1.4 9.5 8.3 1.0 8.2 7 .2 0.76\n0.018 11 10 .9 9 .4 0.8 9.1 8.4 0.6 7.9 7.3 0.4 7.1 5.9 0.2 5.8 4 .9 0.13\n0.005 11 7 .5 6 .4 0.3 6.5 5.9 0.2 5.8 5.3 0.1 5.1 4.2 0.1 4.1 3 .6 0.04\n7C The role of the magnetic interaction and axes arrangement o n the anisotropy\nThe excellent agreement between the experimental data and the C allen-Callen model allowed us to calculate the\nanisotropy constants by estrapolation at 0 Kand 300Kfor both samples. Table 2 summarizes all the results obtained\nas function of temperature and concentration. Also, since the ma in anisotropy contribution arises from the uniaxial\nterm for most samples, from now on we will focus on the room temper ature concentration dependence of K2. Figure\n3(a), presents the data at the ambient temperature of the effec tive anisotropy constant (considering only the uniaxial\nterm -K2), for different samples, as a function of the particle volume fractio n. Open symbols corresponds to the\nanalysis obtained from the EMR data of the colloids, while solid symbols c orrespond to the coercivity temperature\ndependence analysis (of the powder). According to the literature , [28] for monodisperse spherical nanoparticles one\ncan assume that the packing of the nanoparticles in the powder con figuration correspondsto a particle volume fraction\nclose toφ= 0.64 (although the nanoparticles are not perfectly spherical or mon odisperse this is considered to be a\ngood approximation). Recall that unlike the case of the magnetic flu id in which the anisotropic axes are oriented\ntowards the freezing field (longitudinal case [27]), in the case of the powdered sample the axes are estimated to be in\nthe random configuration. For the colloid one can observe an increa se of the effective magnetic anisotropy the higher\nthe particle concentration. This result is in agreement with the theo retical prediction for the case of nanoparticles\nforming a linear chain. [5]\nOn the other hand, for the higher concentration, a lower value is ob served for the anisotropy, which at first can\nsuggest the existence of a maximum as a function of the concentra tion. Indeed, the random anisotropy model, already\napplied even for magnetic dipolar fluids, [42] suggests a decrease of the effective anisotropy, such that for the case of\nNparticles interacting collectively, the anisotropy can be represent ed asKef=Kint/√\nN. HereKintrefers to the\nanisotropy of an individual particle (interacting or non interacting) . Therefore, the transition from a longitudinal to\nrandom condition may explain such a decrease, at least qualitatively. As discussed previously, to explain the existence\nof the cubic term in the anisotropy, it is known in magnetic fluids that in creasing the concentration of particles occurs\na liquid-solid transition. [40,41] In this case, formation of complex se lf-organized structures can break the longitudinal\ncondition, favoring a situation with randomly organized anisotropy a xes. The consequence of such an effect may be\nthe decrease in anisotropy. In the case of samples of magnetic fluid s such an effect was not observed, but may appear\nonly in the case of higher concentrations, which would allow the existe nce of a maximum. As this did not happen for\nthe fluids in our experimental condition, we decided to theoretically d etermine, via the linear chain model, [5] if the\nincrease in the size of the chain is able to explain the value of the anisot ropy observed experimentally.\nFigure 3(b) presents the calculations of the increase of the effect ive magnetic anisotropy for a 11nm nanoparticle of\nMn0.75Zn0.25Fe2O4, for fanning (squares) and coherent (circles) cases, assuming a surface-to-surface distance of 1.1nm\n(estimated for citric acid molecules). An increase in the effective anis otropy of the nanoparticle is clearly evident with\nthe increasein the number ofparticles in the chain. It is alsoobserve dthe effective anisotropyvalue tends to saturation\nfor a large number of particles in the chain, being higher for the cohe rent case. The role of particle size is shown in\nthe inset of Fig. 3(c) considering only the fanning case. From the th eoretical estimations of Fig. 2(b), by comparison\nwith the values extracted from the EMR analysis, it is possible to estim ate the chain size for any magnetic particle\nconcentration of the colloids. For example, for φ= 0.03 we found an anisotropy value of Kef= 9.5x104erg/cm3that\nfor the fanning case reveals a mean chain size of 5.3. The arrows indic ate the average chain size (x-axis) that has\nthe experimental value of the anisotropy (y-axis). Thus, for the volumes fractions of 0 .005, 0.018, 0.093 and 3 %, we\nobtain as average: 1 .1, 1.5, 2.8 and 5.3 particles in a chain for the case fanning and 1 .0, 1.1, 1.3 and 1.6 particles for\nthe coherent case, in which it is clear the increase of the size of the a ggregate as a function of the concentration of\nparticles. Further smaller chains are observed for the coherent c ase when compared to the fanning system. It is known,\nhowever, that in the situation of lower energy the fanning case is mo re favorable. Figure 3(c) shows the increase in\nthe number of aggregates with the increase of the particle volume f raction for both samples, considering the fanning\ncase. Similar trend is found for both particle sizes and suggest that the increase in the effective magnetic anisotropy\nis related to the formation of larger linear chains the higher the part icle concentration. This behavior is in accordance\nwith the theoretical model of Refs. [5,10] that states an increas e of the effective anisotropy due to the dipole-dipole\nparticle interaction.\nFinally, it is tempted to discuss what are the implications of for cancer hyperthermia. Firstly, if the heat gener-\nation is governed by the Nel relaxation mechanism, then is obvious th at the results presented here have important\nconsequences, since the relaxation magnetization depends expon entially on the effective anisotropy. So, one could in\nprinciple tune the effective anisotropyin orderto maximize the heat g eneration. If one needs to increasethe anisotropy\nthen linear chains are interesting options. Indeed, magneto-bact eria chains have been shown to heat efficiently at high\nfield conditions. On the other hand, if one needs to decrease the eff ective anisotropy, then arranging the nanoparticles\nas spherical aggregates seems an interesting approach because the random anisotropy axes configuration could lead\nto this goal. Behind this effect one can find the explanation for sever al apparently contradictory results regarding the\nrole of the magnetic particle interaction on the heat efficiency.\n81E-4 1E-3 0.01 0.1123456\n \n 11 nm\n 10 nm\nö(volume fraction)1E-4 1E-3 0.01 0.1 12468101214\n300 K\n  Kef= KEMR\n2  - 10 nm\n Kef= KEMR\n2  - 11 nm\n Kef= KHc x T - 10 nm\n Kef= KHc x T - 11 nm\nö(volume fraction)\nQ(mean chain size)\n2 4 6 8 10 12468theoretical\nParticles diameter\n      10 nm\n      15 nm\n      20 nm\n      30 nm\nQ (mean chain size)\u000ecd:Hsr8\u00007; \u000e6:Hsr8\u00007; Q(mean chain size)\n\u000ecd:Hsr8\u00007;a)\nb)\nc)0 1 2 3 4 5 6 70369121518\n11 nm Fanning\n Coherent\nMn0.75Zn0.25Fe2O4\n9.5 x104\n7.2 x104\n4.9 x104\n3.6 x104\n  \nFigure 3: (a) Room temperature uniaxial anisotropy constant as f unction of particle volume fraction for both samples.\nOpen symbols corresponds to dat obtained by EMRmethod, while solid symbols are obtained from the HcvsT\nmethod. (b) Theoretical anisotropy constant calculation as func tion of chain size. Circles corresponds to coherent,\nwhile squares are the fanning case. Here we used dss= 1.1nmandMs= 366emu/cm3. Arrowsindicate the chain size\ncorresponding to the experimental effective anisotropy value (da sh lines). (c) Mean chain size as function of particle\nvolume fraction for both samples assuming the fanning case. The ins et shows the same as (b) for the fanning case,\nbut including distinct particle sizes.\n9V CONCLUSION\nElectron magnetic resonance analysis showed that the anisotropy field of these nanoparticles at low particle concen-\ntration show uniaxial symmetry that grows with increasing particle v olume fraction, thus confirming the influence of\nthe dipole-dipole interaction. At very high particle concentrations, a cubic symmetry term for the anisotropy has to\nbe added to the model suggesting the possibility of multipolar contrib utions. It was also possible to prove that the\nanisotropyofthesenanoparticlesisstronglytemperaturedepen dent, andcanbeadjustedwiththeCallen-Callenmodel.\nThe room-temeprature magnetic anisotropy values obtained from EMR data analysis for the Mn 0.75Zn0.25Fe2O4based\ncolloids as a function of the concentration were the linear chain mode l with anisotropy axes aligned in the chain di-\nrection, longitudinal configuration. This analysis indicates that the increase of the anisotropy with the concentration\nis related to the existence (and formation) of chains in the magnetic colloid, and that the size of them increases with\nthe concentration. In particular, for the 11 nmsample the anisotropy changed from 3 .6×104erg/cm3, corresponding\nto an average chain size of only 1 .1 for the most dilute case, to an anisotropy value of 9 .5×104erg/cm3, which in\nthe model corresponds to a chain containing on average 5 .3 for the most concentrated sample (3% volume fraction).\nOn the other hand, the anisotropy of the powder sample (estimate d to have a particle volume fraction around 64%),\nthat was evaluated from the coercive field temperature dependen ce analysis, showed a reduction of the anisotropy\nto 4.1x104erg/cm3, probably due to the random distribution of the anisotropy axes in t his experimental condition.\nThe results indicate that the effective magnetic anisotropy is stron gly dependent on the magnetic interaction between\nthe particles and the arrangement of the anisotropy axes, which m ight explain some contradictory discussions in the\nliterature since one might enhance or decrease the effective anisot ropy depending on the specific situation.\nACKNOWLEDGMENTS\nThe authors would like to thank financial support from the Brazilian a gencies CNPq, CAPES, FAPEG, FAPDF and\nFUNAPE.\nReferences\n[1] Y. 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E , 91:042317,\n2015.\n12"    },    {        "title": "1310.6667v1.Electronic_structure_and_optical_band_gap_determination_of_NiFe2O4.pdf",        "content": "arXiv:1310.6667v1  [cond-mat.str-el]  24 Oct 2013Electronic structure and optical band gap determination of NiFe2O4\nMarkus Meinert1,∗and G¨ unter Reiss1\n1Center for Spinelectronic Materials and Devices,\nDepartment of Physics, Bielefeld University, D-33501 Biel efeld, Germany\n(Dated: October 25, 2013)\nIn a theoretical study we investigate the electronic struct ure and the band gap of the inverse spinel\nferrite NiFe 2O4. The experimental optical absorption spectrum is accurate ly reproduced by fitting\nthe Tran-Blaha parameter in the modified Becke-Johnson pote ntial. The accuracy of the commonly\napplied Tauc plot to find the optical gap is assessed based on t he computed spectra and we find\nthat this approach can lead to a misinterpretation of the exp erimental data. The minimum gap of\nNiFe2O4is found to be a 1.53eV wide indirect gap, which is located in t he minority spin channel.\nToday, DFT is the main tool to obtain the electronic\nstructure of solids.1,2A long-standing problem of elec-\ntronic structure theory is the description of transition\nmetal oxides. These exhibit strong electron-electron cor-\nrelation, which is not properly accounted for by the\ndensity functional theory (DFT) with approximate local\nfunctionals. Here we focus on NiFe 2O4, a ferrimagnetic\ninverse spinel ferrite,3,4which poses an example of such\ndifficult todescribesystems. Experimentalinvestigations\nmostlybasedonopticalabsorptiononthismaterialfound\nband gaps between 1.5eV and 5eV.5–13Theoretical in-\nvestigationson the electronic characteristicsofbulk NFO\nusing a self-interaction-corrected local spin-density ap-\nproximation (SIC-LSDA) approach,14or by including a\nHubbard correction in terms of the DFT+ Umethod15,16\nhave predicted a bandgap of around 1eV. Sun et al.per-\nformed band structure calculations using DFT+ Uand a\nhybrid functional (HSE06).13They obtained a bandgap\nof 2.7eV with HSE06 and 1.6eV for the DFT+ Ucom-\nputations. Thus, there is still a lot of controversy on the\nband structure and the gap of NiFe 2O4.\nThe appropriateframeworkto discuss electron correla-\ntions and band structures is the many-body perturbation\ntheory, e.g., within the GWapproximation.17,18Unfortu-\nnately, this approach is computationally very expensive.\nTran and Blaha recently proposed an alternative, sim-\nilarly accurate and computationally cheaper method to\nobtainthe band gapdirectlyasdifferences ofKohn-Sham\neigenvalues: they modified the Becke-Johnson exchange\npotential19with a parameter c, so that it reads20\nvmBJ\nx,σ(r) =cvBR\nx,σ(r)+(3c−2)1\nπ/radicalbigg\n5\n12/radicalBigg\n2tσ(r)\nnσ(r),(1)\nwherenσ(r) is the spin-dependent electron density\nandtσ(r) is the spin-dependent kinetic-energy density.\nvBR\nx,σ(r) is the Becke-Roussel potential, which models the\nCoulomb potential created by the exchange hole.21Due\nto the kinetic-energy dependent term in the mBJ po-\ntential, it reproduces the step-structure and derivative\ndiscontinuity of the effective exact exchange potential of\nfree atoms.22The parameter cwas proposed to be de-\ntermined self-consistently from the density and is related\nto the dielectric response of the system.23,24cincreaseswith the gap size and has a typical range of 1.1–1.7.20\nThe mBJ potential has been proposed to be combined\nwith LDA correlation (mBJLDA). Its particular merits\nand limits have been reviewed by Koller et al.25\nIn recent publications, the performance of mBJLDA\nfor complete band structure calculations rather than just\nband gap predictions has been discussed. For simple\nsemiconductorsit wasfoundthat thebandwidthsaretoo\nsmall ifcis adjusted to get the correct band gap.26It is\nalso unsuitable to describe half-metals.27However, mBJ-\nLDA predicts the unoccupied band structure of NiO and\noptical spectra of TiO 2with good accuracy.28,29In this\ncommunication, we compare optical absorption spectra\nof NiFe 2O4thin films with computational results using\nthe mBJLDA potential.\nThe calculations in this work are based on the full-\npotential linearized augmented-plane-wave (FLAPW)\nmethod and were done with the elkcode.30The mBJ\nexchange potential is available through an interface to\ntheLibxclibrary.31A 10×10×10k-point mesh with\n171 inequivalent points was used for the Brillouin zone\nintegration. The muffin-tin radii were set to 1.8bohr\nfor the transition metals and 1.7bohr for O. The mBJ\nexchange potential was coupled with the Perdew-Wang\nLDA correlation.32We have used the experimental lat-\ntice constant of a= 8.33˚A,4and relaxed the internal\natomic coordinates using the PBE functional.33The di-\nelectric function was computed in the independent par-\nticle approximation. In the inverse spinel structure, the\ntransition metals sites surrounded by O tetrahedra are\noccupiedwithFe, whiletheoctahedralsitesarerandomly\noccupied with Fe and Ni. We have to use an an ordered\ncellinsteadofthe properdisorderedunit cell inthe calcu-\nlation, sothe symmetryis artificiallyreduced from Fd¯3m\ntoImma. Thus, we take the observable macroscopic di-\nelectric function to be εM(ω) = 1/3Trεij(ω) to restore\nthe full symmetry. The spectra were broadened with an\n80meVwideLorentzian. Theeffectofexcitonsontheab-\nsorption spectrum was investigated with time-dependent\nDFT (TDDFT) using the bootstrap kernel.34\nIn Fig. 1 a) we compare the experimental op-\ntical absorption spectra of high-quality pulsed laser\ndeposited thin films of NiFe 2O4with our mBJLDA\ncalculation.13,35,36The Tran-Blaha parameter c= 1.442\n/s55\n/s54\n/s53\n/s52\n/s51\n/s50\n/s49\n/s48/s97/s32/s40/s49/s48/s55/s32/s109/s45/s49/s32/s41\n/s54 /s53 /s52 /s51 /s50 /s49 /s48\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s109/s66/s74/s76/s68/s65\n/s56\n/s54\n/s52\n/s50\n/s48/s97/s32/s40/s49/s48/s55/s32/s109/s45/s49/s32/s41\n/s54 /s53 /s52 /s51 /s50 /s49\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s80/s66/s69/s43/s85\n/s32/s109/s66/s74/s76/s68/s65\n/s54 /s53 /s52 /s51 /s50 /s49\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s109/s66/s74/s76/s68/s65/s32/s43/s32/s84/s68/s68/s70/s84/s97/s41\n/s98/s41 /s99/s41\nFIG. 1. a): Optical absorption spectrum of a NiFe 2O4thin\nfilm from Ref. 36 and mBJLDA calculated absorption spec-\ntrum. b): Comparison of mBJLDA and PBE+ Ucalculated\nabsorption spectra. c): Comparison of noninteracting and\nbootstrap TDDFT mBJLDA absorption spectra. The arrow\nmarks the mBJLDA calculated fundamental gap.\nhasbeen chosensuch that the computed absorptionspec-\ntrum matches the experimental spectrum between 2 and\n2.5eV. We note that the density-based cfrom the origi-\nnal Tran-Blahapaper is just slightly smaller, cTB= 1.42.\nThe overall agreement between experiments and mBJ-\nLDA calculation is remarkably good. The agreement is\nremarkable in view of the fact that an accurate absorp-\ntion spectrum requiresa good descriptionofboth valence\nand conduction states. However, some spectral weight\naround 5eV is missing in the calculation. The agree-\nment up to 4.5eV is somewhat better than for a PBE+ U\ncalculation, which we show in Fig. 1 b). Here, the Hub-\nbard parameters37have been chosen as UFe,Ni= 4.5eV\nandJFe,Ni= 0.9eV.UFegoverns the size of the gap;\nchanging it leads to a rigid shift of the absorption spec-\ntrum up to 5eV. It was chosen to match the mBJLDA\ngap. The choice of UNiis not critical and only leads to\nmodifications of the absorption spectrum above 5eV. We\nattribute the good reproduction of the absorption spec-\ntrum by mBJLDA to a more accurate description of the\nOpstates, which are more localized in the mBJLDA cal-\nculation. WhilethePBE+ Ucalculationonlycorrectsthe\ntransitionmetal dstates(withrespecttoaplainPBEcal-\nculation), mBJLDAallowsforanimproveddescriptionof\nall electrons. This will be discussed in more detail later.\nBound excitons play no significant role for the optical\nproperties of NiFe 2O4as is shown in Fig. 1 c). In the/s49/s46/s50\n/s48/s46/s56\n/s48/s46/s52\n/s48/s46/s48/s40/s97/s69/s41/s48/s46/s53/s32/s40/s49/s48/s52/s32/s101/s86/s48/s46/s53/s109/s45/s48/s46/s53/s32/s41\n/s52/s46/s48 /s51/s46/s48 /s50/s46/s48 /s49/s46/s48\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s51/s46/s48\n/s50/s46/s53\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48/s40/s97/s69/s41/s50/s32/s40/s49/s48/s49/s54/s32/s101/s86/s50/s109/s45/s50/s32/s41\n/s52/s46/s48 /s51/s46/s48 /s50/s46/s48 /s49/s46/s48\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s109/s66/s74/s76/s68/s65\n/s97/s41 /s98/s41\nFIG. 2. a): Tauc plot of ( αE)0.5for indirect gaps. b): Tauc\nplot of (αE)2for direct gaps.\nTDDFT calculation, the absorption is enhanced by 20 to\n40% (at odds with experiment), but the spectral features\ndonot shift to lowerenergies. Due to the small band gap,\nthe screening is strong: the ion-clamped static dielectric\nconstant is εmBJLDA\n∞= 5.4. Thus, no localized Frenkel\nexcitons are expected to show up, as is confirmed numer-\nically by the TDDFT calculation. While the bootstrap\nkernel does well in describing Frenkel excitons, it fails\nto describe the delocalized Wannier excitons.38These,\nhowever, are typically rather weak with binding ener-\ngies of less than 0.1eV for materials with similar gaps\nand dielectric constants.39As we will show later, due to\nthe particular localization of conduction band minimum\n(CBM) and valence band maximum (VBM) states, also\nthe binding energies for Wannier excitons are expected\nto be small.39\nA common way to extract the indirect and direct gaps\nfrom optical absorption spectra is the Tauc plot, which\nis based on the assumption that the energy-dependent\nabsorption coefficient α(E) can be expressed as40,41\nα(E)E=A/parenleftbig\nE−Edirect\ng/parenrightbig0.5\n+B/parenleftbig\nE−Eindir\ng±Ephon/parenrightbig2(2)\nwith two parameters AandB, the indirect and direct\ngapsEg, and the phonon energy Ephon. Thus, straight\nline segments in ( αE)2indicate direct gaps and straight\nline segments in ( αE)0.5indicate indirect gaps. In Fig. 2\nwe show that particularly the ( αE)0.5-plot does not indi-\ncate an indirect gap in NiFe 2O4: both the experimental\naswellasthetheoreticalTaucplotshowidenticalstraight\nline segments. However, the theoretical Tauc plot can by\nno means indicate indirect transitions, because these are\nnot included in the calculation. Furthermore, the com-\nputed fundamental gap on which the theoretical spec-\ntrum is based is 1.53eV, while the Tauc plots indicate an\nindirect gap of about 1.65eV.\nHaving established that mBJLDA provides a good de-\nscription of the electronic structure of NiFe 2O4, we go\ninto moredetail. Table I summarizesthe calculated mag-\nneticspinmomentsandvalencechargesinsidethemuffin-\ntin spheres. The data for Ni are in good agreement with3\n/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s32/s47/s32/s101/s86/s32/s47/s32/s102/s46/s117/s46/s41\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s78/s105/s32/s79/s104\n/s32/s70/s101/s32/s84/s100\n/s32/s70/s101/s32/s79/s104\n/s32/s79\nFIG. 3. Site-projected density of states of NiFe 2O4obtained\nfrom an optimized mBJLDA calculation. Majority states are\nshown on positive, minority states are shown negative scale .\nThe energy is set to zero at the valence band maximum.\nan ionic Ni2+configuration. For Fe, the valence charges\nare actually too large and the magnetic moments are too\nsmallfortheanticipatedFe3+configuration.4However,it\nhas been shown for Fe 3O4that the nominal Fe3+species\nhave a somewhat larger charge (lower oxidation state),\nwhich agrees with our calculation for NiFe 2O4.42A sub-\nstantial amount of charge (8 .84e−/f.u.) is in the intersti-\ntial region between the muffin-tin spheres and accounts\nfor the missing charge of the O2−ions. This number\nis larger in the PBE+ Ucalculation (9 .63e−/f.u.), indi-\ncating the weaker localization of the O states discussed\nearlier. In Fig. 3 we show the site-projected density of\nstates (DOS), which reveals that the fundamental gap is\nin the minority states. The VBM is mostly composed\nof Ni and O states and has a small exchange splitting\nof 0.08eV. This is in contrast to DFT+ Uand HSE06\ncalculations, which predict a significant exchange split-\nting of the VBM.13Still, the overall shape of the DOS is\nvery similar to the HSE06 calculation. The conduction\nstates below 6eV are composed of the transition metal d\nstates, which hybridize weakly with the O atoms. While\nthe states of the two Fe species have about the same en-\nergy,theNistatesareclearlysetoff. Thisleadstothedip\naround 5eV in the computed absorption spectra which\nis much less pronounced in the experiment. Thus, the\nunoccupied Ni dstates are actually about 0.5eV lower in\nenergy. Due to the different localization of VBM (mostly\non Ni and O) and CBM (mostly on Fe and O), electron-\nhole pairs generated in photoabsorption are well sepa-\nrated, which leads to a vanishing binding energy of Wan-\nnier excitons.39This spatial separation and correspond-\ninglysmallwavefunctionoverlapofthestatesdefiningthe\nband gap is also responsible for the tiny optical absorp-\ntion below 2eV, which makes the optical determination\nof the fundamental gap difficult.\nFig. 4 displays the band structure plots calculated\nwith mBJLDA. The high-symmetry points RandTcor-/s51\n/s50\n/s49\n/s48\n/s45/s49\n/s45/s50/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s84 /s71 /s82/s109/s105/s110/s111/s114/s105/s116/s121/s49/s46/s53/s54/s32/s101/s86/s51\n/s50\n/s49\n/s48\n/s45/s49\n/s45/s50/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s84 /s71 /s82/s109/s97/s106/s111/s114/s105/s116/s121/s50/s46/s50/s54/s32/s101/s86\nFIG. 4. Band structure plots of NiFe 2O4calculated with\nmBJLDA. The high-symmetry points RandTcorrespond\nto theXpoint of the disordered unit cell of the inverse spinel\nstructure with full cubic symmetry.\nrespond to the Xpoint of the cubic cell with full symme-\ntry. In the real, disordered case, the dispersion along the\nΓ−Xpath will smear out and form intermediate states\ndefined by the Γ −Rand Γ−Tdispersions. The minor-\nity gap of NiFe 2O4is found to be a 1.53eV wide indirect\ngap between Tand Γ. However, it is only 0.03eV smaller\nthan the minimum direct gap in the minority states at\ntheTpoint and is thus expected to play no significant\nrole, particularly at room temperature. The minimum\ngap of the majority states is a 2.26eV wide direct gap at\nΓ. The Tauc plot in Fig 2 b) indicates two direct gaps\nat 2.35eV and 2.8eV. The first one could correspond to\nthe onset of majority absorption, but could equally well\nbe due to the onset of absorption into the second unoc-\ncupied minority band. Moreover, there is no gap in the\nband structure that could correspond to the 2.8eV Tauc\ngap. Thus, the ( αE)2Tauc plot erroneously assigns a\nstructure in the absorption spectrum to a gap, which in\nfact has its origin in the particular features of the band\nstructure. Consequently, this type of plots is unsuitable\nto determine the band gap of NiFe 2O4and its use may\nhavecontributedtothebroadrangeofexperimentalband\ngaps found in the literature.\nIn conclusion we have shown that the mBJLDA po-\ntential is well suited to describe the electronic struc-\nture of NiFe 2O4. Based on the computed optical ab-\nNi Fe(T d) Fe(O h) O(1) O(2)\nm 1.75 -3.87 4.08 0.09 -0.01\nnV7.77 5.25 5.42 5.68 5.68\nTABLE I. 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B 74, 094407 (2006)."    },    {        "title": "2010.14895v1.On_the_happiness_of_ferroelectric_surfaces_and_its_role_in_water_dissociation__the_example_of_bismuth_ferrite.pdf",        "content": "On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite AIP/123-QED\nOn the happiness of ferroelectric surfaces and its role in water dissociation: the\nexample of bismuth ferrite\nIpek Efe,1Nicola A. Spaldin,1and Chiara Gattinoni1,a)\nMaterials Theory, Department of Materials, ETH Zürich, Wolfgang-Pauli-Strasse 27,\n8093, Zürich, Switzerland\n(Dated: 29 October 2020)\nWe investigate, using density functional theory, how the interaction between the ferro-\nelectric polarization and the chemical structure of the (001) surfaces of bismuth ferrite\ninfluences the surface properties and reactivity of this material. A precise understanding of\nthe surface behavior of ferroelectrics is necessary for their use in surface science applica-\ntions such as catalysis as well as for their incorporation in microelectronic devices. Using\nthe (001) surface of bismuth ferrite as a model system we show that the most energetically\nfavoured surface geometries are combinations of surface termination and polarization di-\nrection that lead to uncharged, stable surfaces. On the unfavorable charged surfaces, we\nexplore the compensation mechanisms of surface charges provided by the introduction of\npoint defects and adsorbates, such as water. Finally, we propose that the special surface\nproperties of bismuth ferrite (001) could be used to produce an effective water splitting\ncycle through cyclic polarization switching.\na)Electronic mail: chiara.gattinoni@mat.ethz.ch\n1arXiv:2010.14895v1  [cond-mat.mtrl-sci]  28 Oct 2020On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\nThe following article has been submitted to the Journal of Chemical Physics.\nI. INTRODUCTION\nTransition metal oxides occupy a prominent place in heterogeneous catalysis, and are nowadays\nthe most used industrial catalyst type1. A variety of industrially relevant processes, for example\nwater splitting or the degradation of pollutant molecules, however, still lack an efficient catalyst. In\nthe search of novel catalytic materials, the Sabatier principle, which states that effective catalysis\noccurs when the adsorption between a molecule and a surface is of intermediate strength, is a\nlimiting factor2. However, the adsorption strength between a molecule and the surface can be\ncontrolled by utilizing oxides with tunable functionalities, such as piezo- and ferroelectricity3,\nand research in this field is flourishing4–7. In particular, there is great potential for the use of\nferroelectric thin films8or nanoparticles5,6,9in electricity generation, water remediation or drug\ndelivery4,7.\nFerroelectric materials present a spontaneous switchable bulk polarization, and their surfaces,\nwhere reactions occur, are complex. In particular, the ferroelectric polarization results in surface\nbound charges which need to be compensated in order to avoid a polar discontinuity10. Thus, the\nsurface structure of a ferroelectric and, as a consequence, its reactivity are largely determined by\nthe interplay between bound charges and compensation mechanisms11,12.\nMuch progress has been made in our understanding of ferroelectricity at a material’s surface.\nIndeed, it is now well understood that compensation of the ferroelectric bound charges at a sur-\nface occurs preferentially through adsorbates and defect formation rather than by electronic re-\nconstructions12–15. It has also been shown that switching of the surface polarity can be used to\npromote catalysis for molecular dissociation3. The precise structure of the surface has also been\nshown to influence the strength and direction of the ferroelectric polarization in thin films, and\nengineering of surface stoichiometry has been used to manipulate the polarization on ferroelectric\nsurfaces11,16–20.\nThere are still, however, many open questions regarding the surface science of ferroelectrics8.\nIn particular, how the ionic charge in the layers of ferroelectric perovskites interact with the ferro-\nelectric polarization, and the effect of this interplay on the surface structure, is still poorly under-\nstood. Here, we investigate this question in bismuth ferrite (BFO), a material which has a robust\n2On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\nferroelectric polarization at room temperature and, in the (001) direction, neighboring positively-\ncharged Bi3+O2−and negatively-charged Fe3+O2−\n2layers (see Fig. 1a). It is also an especially\npromising catalyst for applications in water remediation9, water splitting6,21and nanoscale drug\ndelivery5. In the following, we investigate the stability of the (001) surface of BFO, including the\ninteraction of the polarization with defects and water molecule adsorbates. Our findings allow us\nto propose a catalytic cycle for efficient water splitting taking advantage of the special properties\nof BFO (001) surfaces.\nII. METHODS\nDensity functional theory calculations were performed within the periodic supercell approach\nusing the V ASP code22–25. The optB86b-vdW functional26, a revised version of the van der Waals\n(vdW) density functional of Dion et al.27, was used throughout, as it has been shown to describe\nwell molecular adsorption on transition metal oxides28–30. Core electrons were replaced by pro-\njector augmented wave (PAW) potentials31, while the valence states (5e−for Bi, 8e−for Fe and\n6e−for O) were expanded in plane waves with a cut-off energy of 500 eV . In all calculations we\nused slabs with a√\n2a×√\n2asurface area and 4 aheight (shown in Fig. 1a), where ais the lattice\nparameter of the pseudocubic unit cell. Using the optB86b-vdW functional the pseudocubic lattice\nparameter was calculated to be a=3.95 Å, with the γangle in the rhombohedral structure being\nγ= 90.23◦. The difference of the calculated lattice parameters with respect to the experimental\nstructure is below 0.5%32. A Monkhorst-Pack k-point grid of (5×5×1) was used for all calcula-\ntions. An antiferromagnetic G-type ordering was imposed, which gave a magnetic moment of 4.2\nµBper Fe ion in the bulk. The BFO (001) slabs had a thickness of four cubic unit cells and were\nseparated from their periodic repetitions in the direction perpendicular to the surface by ∼20 Å\nof vacuum. Upon testing we found that this thickness was sufficient to converge the adsorption\nenergies of the water molecules (see Table S1). A dipole correction along the direction perpendic-\nular to the surface was applied, and geometry optimizations were performed with a residual force\nthreshold of 0.01 eV/Å. BFO has a large intrinsic polarization, P, whose experimental value is\n∼0.9 C/m2along the (111) direction33; we calculated P with the formula:\nP=e\nVN\n∑\nm=1Qmum, (1)\n3On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\nwhere eis the charge of the electron, Vthe unit cell volume, Nthe number of atoms in the unit\ncell and uthe atomic displacements from the high symmetry positions. We obtained a value of\nP=0.86 C/m2when using the formal charges for Qand of P=1.19 C/m2when using the Born\neffective charges.\nAdsorption energies for the water molecules, E ads, were calculated as:\nEads=/parenleftbig\nEwater /BFO−EBFO−n×Ewater/parenrightbig\n/n, (2)\nwhere E BFO, Ewater and E water /BFOare the total energies of the relaxed bare slab, an isolated gas\nphase water molecule and a system containing nwater molecules adsorbed on the slab, respec-\ntively. Negative values of the adsorption energy indicate favorable (exothermic) adsorption. Wa-\nter coverages varying between 1/2 and 1 monolayer (ML) — where 1 monolayer is one water\nmolecule per surface metal atom — were considered.\nTo calculate the charge density differences of Fig. 4 we first obtained the real-space charge\nfor the slab/water system ( ρall) and for the isolated slab ( ρslab) and water molecules ( ρwater). The\ndifference was then obtained as:\nρdiff=ρall−ρslab−ρwater. (3)\nIII. RESULTS AND DISCUSSION\nBFO (001) has interesting surface properties when we consider the interplay between layer\ncharge and ferroelectric polarization, and they are schematically shown in Fig. 1. The formal\ncharges of Bi3+, Fe3+and O2−result, in the (001) direction, in alternating positively charged BiO\n(+1 C/m2) and negatively charged FeO 2(−1 C/m2) layers, see Fig. 1a. This surface charge re-\nquires a compensating charge of opposite sign and half the magnitude10,34(∼±0.5 C/m2, negative\nfor BiO and positive for FeO 2) to obtain surface stability. Remarkably (and coincidentally), the\n(001) component of the ferroelectric polarization in BiFeO 3has the value P∼±0.5 C/m2(re-\nsulting in the surface charge density of ∼±0.5 C/m2), positive when the polarization is directed\ntowards the surface and negative when away from the surface. Thus, the interplay of these two\ncontributions of equal magnitude can result either in fully self-compensating surfaces with a total\nsurface charge density σ=0 C/m2in which the surface polarization compensates the layer charge,\nor highly uncompensated surfaces in which both the layer charge and the surface polarization are\ncontribute to a non-zero surface charge.35The self-compensating case, shown in Fig. 1b, occurs\n4On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\nFeO2-BiO+-+PBiO+FeO2--+PFeO2neutral\nb)a)\nBiFeO[001]BiOBiOFeO2FeO2√2aBiOpos\nBiOneutralFeO2neg!= 0!= 0!=1!=-1c)\nFIG. 1. (a) Unit cell of bismuth ferrite used in this work, with the axes oriented along the (001)and, in\nplane, the (110)and (110)crystallographic directions. The formal charges of the atoms are Bi+3, Fe+3,\nO−2. Purple represents Bi, gold Fe and red O atoms. (b) The favorable polarization direction which creates\ncharge-compensated surfaces points from the BiO to the FeO 2termination. (c) The unfavorable polarization\ndirection which creates polar surfaces points from the FeO 2towards the BiO surface termination. σis the\nsurface charge density in C/m2units.\nin BiO surfaces with the polarization pointing away from them (we will refer to these surfaces\nas BiOneutral) and FeO 2surfaces with the polarization pointing towards them (FeOneutral\n2). The\nhighly uncompensated surfaces, shown in Fig. 1c, are, instead, the BiO (FeO 2) surfaces with the\npolarization pointing towards (away from) them and we will refer to these surfaces as BiOposand\nFeOneg\n2.\nIn this work we study the two stoichiometric (001) systems shown in Fig. 1b and c. In panel b\nthere is a fully compensated BFO (001) slab which we we will refer to as the “happy” system, since\nthe full surface charge compensation means that there is no polar discontinuity at the surface and\nthe polarization is stable. The uncompensated slab of panel c will be referred to as the “unhappy”\nsystem, because the non-zero surface charge density results in an unphysical polar discontinuity,\nand the surface charge needs to be compensated to render the surface stable10.\n5On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\nIn the following we explore ways to stabilize the polarization in the unhappy system both\nwith defect engineering and molecular adsorption. In particular, we investigate how the different\nsurface electronic properties of the two slabs—their “happiness”, if you will—affect the geometry\nand adsorption strength of water. We show that the resulting polarization-dependent dissociation\nbehavior has great potential for catalytic applications.\nA. Achieving surface stability through point defect engineering\nIt is known that point defects and adsorbates can provide charge compensation to ferroelectric\nsurfaces3,11,12. As already remarked, the self-compensating surfaces of the happy system have\nno polar discontinuity and do not require any further compensation. Indeed, our calculated unit\ncell by unit cell polarization plotted in Fig. 2a shows that the ferroelectric polarization is stable\nthroughout the slab thickness. Upon geometry relaxation, the unhappy slab also relaxes into the\nstructure of Fig. 2a, meaning that in order to avoid the polar discontinuity at the surface, the\npolarization direction reverses, resulting in a happy system. This indicates that the polarization\ndirection in the unhappy system cannot exist without a means to compensate the surface charges.\nTo stabilize the unhappy system we consider Bi and O adatoms and vacancies: the positively\ncharged Bi adatom and O vacancy compensate the negatively charged FeOneg\n2surface, the nega-\ntively charged Bi vacancy and O adatom compensate the positive BiOpossurface. Bi defects, rather\nthan Fe ones, are considered here as they are seen to occur more often in experiments36.\nFig. 2b shows the geometry-optimized structure and the unit cell by unit cell polarization in the\nunhappy slabs compensated with point defects at both surfaces. We note that, indeed, compen-\nsation of the surface charges by vacancies and adatoms is effective in stabilizing the downward-\npointing polarization direction in the unhappy slab. We also observe surface enhancements of\nthe polarization above the bulk value of 90 µC/cm2, especially at the surfaces where O and Bi\nadatoms are present, which are driven by the surface chemistry. In particular the bonding between\na surface Bi and the O adatom (top of Fig. 2b) pulls the Bi atom away from the surface, enhancing\nthe polarization at the BiO surface of the slab.\nWe also investigated partial compensation of the slab, by including point defects on only one\nsurface, rather than both. This allows us to understand whether compensation from one surface\nonly is sufficient to ensure stable polarization throughout the slab thickness, and also to separately\ninvestigate the BiOposand FeOneg\n2surfaces. The results are shown in Fig. 2c.\n6On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\n-190-100-1080-190-100-1080\n-190-100-1080P [µC/cm2]a) Happy b) Compensatedc) Half-compensatedO vacancy\nBi vacancy-190-100-1080P [µC/cm2]-190-100-1080P [µC/cm2]Bi adatom\nBi vacancy\nO vacancy\nbO adatomP [µC/cm2]P [µC/cm2]\nFIG. 2. Calculated structures and corresponding layer-by-layer polarization for a range of a four unit cell-\nthick BFO (001) slabs. a) Happy system. P is constant throughout the slab thickness without further\ncompensation. b) Unhappy slabs with surface charges compensated by O (top) and Bi (bottom) defects. c)\nunhappy slab with partial compensation of the surface charges. Compensation of the FeO 2surface with an\noxygen vacancy (top) and of the BiO surface with a Bi vacancy (bottom). The bond between surface Ti and\nsubsurface O atoms is shown as b. The purple shading indicates areas of polarization reversal. The black\narrows indicate the polarization direction.\nOn the non-compensated side of the slab polarization reversal occurs, confirming that compen-\nsation on both surfaces is needed to obtain a robust polarization throughout the thickness. In the\nuncompensated BiOpostermination (top of Fig. 2c), polarization reversal occurs only in the out-\nermost BiO surface layer (shown in purple shading), the polarization pointing away from the BiO\nsurface. For an uncompensated FeOneg\n2surface (bottom of Fig. 2c) the polarization reversal (purple\nshading) involves the topmost two unit cells and the polarization points towards the surface. Thus,\nboth uncompensated surfaces become happy by this reversal of the polarization. The problem of\ncharge compensation now occurs within the slab, where the positive (in the top of Fig. 2c) and\nthe negative (in the bottom of Fig. 2c) ends of the polarization meet creating a polar discontinuity.\n7On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\nCharge compensation in the bulk forces a metallic layer at the site of the polar discontinuity, which\nrequires band bending. The energy cost of the band bending is however offset by the favorable\n— happy — surface configuration. However, also the local surface chemistry drives this surface\nstructure. On BiOposthe cation has a lone pair of electrons which orients towards the vacuum,\npushing the ion towards the subsurface (here, FeO 2) layer and creating a ferroelectric polarization\npointing away from the surface, and, as a consequence a BiOneutralsurface. A similar behavior is\nobserved for the PbO surface of lead titanate11which also has a lone pair of electrons. On FeOneg\n2\nthe bond labelled bin Fig. 2c is shorter than in the bulk, as it is generally the case for atomic bonds\nbetween the two topmost layers of a slab37. This shorter bond bforces the Bi lone pair downwards\nand the ion upwards, thus imposing a polarization which points towards the surface, which per-\nsists, to a lesser degree, in the unit cell below. Note that in the previous example of lead titanate,\nno polarization inversion is observed for the TiO 2termination with the polarization pointing away\nfrom it11. The difference in behavior between these two ferroelectric perovskites is probably due\nto the higher relative polarizability of Ti4+compared to Fe3+.\nHaving shown how intrinsic point defects can stabilize the ferroelectric polarization in the un-\nhappy systems, we now investigate how stability can be obtained through adsorbates, by examining\nthe behavior of water on BFO (001).\nB. Achieving stability through adsorbates: the example of water\nAs well as intrinsic surface defects, adsorbates can play an important role in shaping the sur-\nface structure of a ferroelectric12. The interaction of a surface with water is especially important\nbecause of water’s ubiquity in air and in solutions, and also because of the potential for applica-\ntions which arise from the interaction between water and functional materials. In the following,\nwe analyse the behavior of water adsorbed on the surfaces of the systems in Fig. 1b and c, and\nreveal how water can stabilize the unhappy system and, in turn, how surface charges affects the\nwater adsorption energy and propensity for dissociation.\n1. Water adsorption on a happy surface\nWe identified the most stable sites for water adsorption on the surfaces of the happy system,\nand they are shown in Fig. 3. On FeOneutral\n2, the most favorable configuration for molecular H 2O\n8On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\n(b)       Eads=-0.67 eV(a)Eads=-0.80 eV\n(d)      Eads=-0.64 eV(c)Eads=-0.45 eVFeO2neutralBiOneutralMolecular adsorptionDissociative adsorption\nFIG. 3. The most favorable adsorption sites for the adsorption of a water molecule, their adsorption energies\nand the bond distances between the water molecule and the surface ions for the two happy terminations.\nMolecular adsorption of a water molecule on the (a) FeOneutral\n2 and (b) BiOneutraltermination. Dissociative\nadsorption of a water molecule on (c) FeOneutral\n2 and (d) BiOneutraltermination. Purple indicates Bi, gold Fe,\nred O and white H atoms.\nadsorption is parallel to the surface with the formation of a 2.17 Å Fe-O bond and of a 1.93 Å\nH-O surf(Osurfis a surface oxygen) bond (see Fig. 3a). For the BiOneutraltermination, the water O\natom sits at the bridging site between two Bi atoms, aligned perpendicularly to the surface. This\nconfiguration permits only one hydrogen bond of length 1.54 Å (Fig. 3b). Indeed, charge density\ndifference calculations, presented in Fig. 4b, show that minimal charge transfer between the water\nO and the Bi surface atom occurs. The water-surface binding is stronger on the FeOneutral\n2termi-\nnation than on the BiOneutralby∼0.13 eV , since in the former molecular adsorption is established\nby a strong ionic bond and a hydrogen bond (Fig. 4a).\nFor a dissociated water molecule, the favored binding sites for the hydroxyl groups are a sur-\nface Fe for the FeOneutral\n2termination (see Fig. 3c) and the Bi-Bi bridging site for the BiOneutral\ntermination (see Fig. 3d), similar configurations to the molecularly adsorbed water. Also, the ro-\ntation of the hydroxyl with respect to the surface is similar to that of the intact water molecule:\n9On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\n  \n(a) (b)\nFIG. 4. Charge density differences ρdiff, calculated using Eq. 3, for molecular H 2O adsorption on the (a)\nFeO 2and (b) BiO termination. Light blue represents electron density reduction while yellow represents the\nelectron density increase. The isosurface level is 0.01 e/volume.\nparallel to the surface on FeOneutral\n2and perpendicular on BiOneutral. In both cases the H ion binds\nto an O surf(Fig. 3c-d).\nThe adsorption energies in Fig. 3 and Table I show that dissociation of the water molecule\nis disfavoured on both compensated terminations of a happy BFO (001) slab, by ∼30 meV for\nBiOneutraland∼350 meV for FeOneutral\n2.\nIt is worth noting that in all systems the polarization throughout the film is bulk-like and mini-\nmally affected by the adsorption of either molecular or dissociated H 2O.\n2. Water adsorption on an unhappy surface\nWe next turn our attention to the adsorption of water on the unhappy slab with FeOneg\n2and\nBiOpossurfaces, and we find that on this system dissociative water adsorption is favored.\nSince the unhappy slab is unstable, calculations of the BFO/water system in this section are\nperformed with a “frozen” BFO slab: we kept the ionic positions of the inner layers of the slab\nfixed at the bulk values and allowed only the adsorbed molecules and topmost surface layer, where\nadsorption occurs, to relax. The “frozen” layers are shown in blue shading in Fig. 5a. We refer to\nthe adsorption energies with respect to this “frozen” substrate as Efrozen\nads.\nWe simulated molecularly and dissociatively adsorbed water on the FeOneg\n2and BiOpossur-\nfaces, and we observed similar adsorption geometries as on the happy slab, both in the preferred\n10On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\nadsorption sites and bond lengths (the structures are shown in Fig. S1). Indeed, on FeOneg\n2the\nwater molecule and the hydroxyl adsorb parallel to the surface; on BiOposthey adsorb perpendic-\nularly to the surface in the Bi-Bi bridge site. However, the energy trends are significantly different\nfrom the happy case, and dissociative adsorption is favorable on these uncompensated surface ter-\nminations. Indeed, the values in Table I show that dissociative adosrption is favoured by 290 meV\non the FeOneg\n2surface and by 160 meV on BiOpos.\nIt is worth noting that, despite the similarities in the adsorption geometries, there are significant\ndifferences between the electronic structures of the happy and unhappy systems, as highlighted in\nthe local density of state ( lDOS) of the surface layers and adsorbates in Fig. 5a. In the happy sys-\ntem (left), the lDOS of both FeOneutral\n2and BiOneutralsurface layers presents an insulating behavior\nwith a∼2 eV-wide band gap around the Fermi level, as in bulk BFO. The conduction and valence\nband edges on both FeOneutral\n2and BiOneutralare at the same energy, showing that the ferroelectric\npolarization is well screened and no band bending occurs through the slab. Conversely, the FeOneg\n2\nand BiOpossurfaces in the unhappy system (right of Fig. 5a) are metallic, since the large surface\ncharge of±1 C/m2is compensated by electrons (BiOpos) and holes (FeOneg\n2)10.\nWhen water is molecularly adsorbed on happy and unhappy surfaces alike (dashed lines in\nthe graphs in Fig. 5a), the lDOS of the water molecules and of the surface overlap away from\nthe Fermi level and the position of the band edges have only a small effect on the water/surface\ninteraction. This can explain the similar adsorption energies for intact water on the happy and\nunhappy system, see Table I. The increased stability in the adsorption energy of dissociated water\non the unhappy system can instead be related to a partial compensation of the surface charge by\nthe OH−(on BiOpos) and H+(on FeOneg\n2), which results in change in the lDOS at the Fermi level\nfor the unhappy surfaces (especially on FeOneg\n2).\nSystemFeO 2 BiO Adsorption on both sides\nIntact Dissociated Intact Dissociated Intact Dissociated\nHappy -0.80 -0.45 -0.67 -0.64 -0.75 -0.60\nUnhappy (frozen slab) -0.90 -1.19 -0.57 -0.73 -0.60 -0.92\nStabilized (frozen slab) — — — — — -3.16\nStabilized (relaxed slab) — — — — — -0.47\nTABLE I. Adsorption energies for 0.5 ML of water adsorbed on the BiO and FeO 2surfaces of BFO (001)\n11On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\nThe molecular adsorption of water on an unhappy slab does not provide adequate charge trans-\nfer to stabilize the unfavorable polarization direction, and neither does the co-adsorption of hy-\ndroxyl and H on the same surface. Indeed, for these structures, when we allow the ions in the\n“frozen” slab to relax into their energetically favorable position, we obtain a happy system. How-\never, stabilization of the polarization in the unhappy system can be achieved when 1 ML of OH−is\nadsorbed on the positively charged BiOpostermination and 1 ML of H+on the negatively charged\nFeOneg\n2termination, thus fully compensating the surface charges of ±1 C/m2. The adsorption\nstructure is, as expected, at a Bi-Bi bridge site for the hydroxyl groups and atop an O surfatom\nfor the H atoms. We will refer to this configuration as the “stabilized system” and it is shown in\nFig. 5b.\nThe stabilized system is the most stable among the computed water structures on unhappy\nBFO. Indeed, the adsorption energy of dissociated water in the stabilized system, with respect to\nthe frozen substrate is Efrozen\nads=−3.16 eV/mol, much larger than the values (reported in Table I)\nfor the structures examined in Fig. 5a.\nSince the unhappy ferroelectric polarization (from FeO 2to BiO) in the stabilized system in\nFig. 5b is now fully compensated, we can relax the ionic positions of the whole slab. We obtain\nan adsorption energy for the fully relaxed stabilized system (with respect to relaxed happy slab)\nof E ads=−0.47 eV/mol. In comparison, water adsorbed on the happy system leads to a more\nnegative (by 0 .2 eV/mol) adsorption energy, see Table I, and thus to a more energetically favorable\nstructure. This comparison tells us that the system in Fig. 5b, despite being stable, will not occur\nspontaneously, but will be reached by switching the polarization with an external electric field.\nC. Discussion\nThe results presented in this work show the complex coupling between the surface chemistry\nand the ferroelectric polarization at the (001) surfaces of BFO. The BiOneutraland FeOneutral\n2termi-\nnations of BFO (001) surfaces are charge neutral, thanks to the interaction between layer charge\nand ferroelectric bound charges. Upon reversal of the polarization, both surface terminations\nBiOposand FeOneg\n2present a large surface charge that can be effectively compensated by point\ndefects or by dissociated water molecules. Upon growth of ferroelectric films and nanocrystals, it\nis very difficult to obtain defect-free surfaces, and these results point to which defects are likely\nto occur. Moreover, the surface defect engineering could be important during thin-film growth of\n12On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\n-5-4-3-2-101234-5-4-3-2-101234\nP\n1.922.252.192.51\n1.72(OH)-\n(H)+b)a)BiOFeO2H2O\nH2OE [eV]Unhappy\nH2OBiOFeO2H2OHappy\nBiOposFeO2negStabilized systemdissociativemolecularE [eV]\nFIG. 5. (a) Local density of state ( lDOS) for the surface layers and water molecules for the \"happy\" (top)\nand \"unhappy\" (bottom) systems. The lDOS for water is in black, for the BiO layer in purple and for the\nFeO 2layer in orange. The black vertical line at E=0 eV is the Fermi level. The dashed (solid) lines are\nthelDOS for molecularly (dissociatively) adsorbed water. A representative system is shown in the middle.\nThe light blue shading indicates the BFO layers which are kept frozen in the calculations of the unhappy\nsysttem. (b) Stabilized system with two water molecules adsorbed dissociatively with OH groups on the\nBiO termination and H atoms on the FeO 2termination.\nunhappy ferroelectric surfaces, where defect formation could be engineered to stabilize or even\nenhance the surface polarization11,38.\nNow we focus on water adsorption and dissociation. Our calculations reveal that the adsorption\nmode of water on stoichiometric BFO (001) is highly dependent on the combination of polarization\ndirection and surface termination. Indeed, we find that compensated surfaces favor the molecu-\nlar adsorption of water, uncompensated ones dissociative adsorption. The obtained adsorption\nstructures are in good agreement with previous theoretical work on (001) perovskites, such as\nstrontium ruthenate39, barium hafnate40, barium zirconate40, strontium titanate41and strontium\nzirconate42. All these materials are non ferroelectric and have charge-neutral surfaces, therefore,\ndespite similar adsorption structure, we do not necessarily expect the same behavior and the same\nenergy ordering between the intact and dissociated structures. Indeed, the literature (summarized\nin Table II in the SI) shows that water can show a range of different adsorption behavior on the sur-\n13On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\nfaces of these complex materials. Together with the surface structure, the interplay of many other\nfactors including neutrality of the crystal ionic layers, lattice parameters43and dielectric/metallic\ncharacteristics of the materials should be considered to understand the interface chemistry and\nstructure stability.\nWe believe that this polarization dependence of water dissociation in bismuth ferrite could have\ninteresting ramifications for catalysis. We propose that the opposite affinity towards water dissoci-\nation of the happy and unhappy systems could also be utilized for the creation of a water splitting\ncatalytic cycle, by exploiting the ferro- and piezoelectric properties of BFO as illustrated in Fig-\nure 6. The cycle starts with a happy BFO (001) slab which favours molecularly adsorbed water on\nboth BiOneutraland FeOneutral\n2terminations (Figure 6, panel 1). Upon switching of the polarization,\nwe obtain an unhappy slab with charged surfaces. The resulting system favours dissociation of\nthe adsorbed water molecules on BiOposand FeOneg\n2surfaces (Figure 6, panel 2). Our calculations\nthen indicate that selective desorption of the H+ion from the BiOpostermination and OH−group\nfrom the FeOneg\n2termination is favorable as it stabilizes the polarization by compensating the sur-\nface charges (Figure 6, panel 3). By further switching of the polarization back into its initial happy\ndirection, competitive adsorption would favor the further removal of dissociation products and the\nadsorption of molecular H 2O (Figure 6, panel 4). Thus, in principle, cyclical switching of the\npolarization in a BFO (001) slab immersed in water could efficiently produce H and OH species,\nwhich can then by used directly in, e.g. the degradation of pollutants9, or for H 2production to-\ngether with a metal cathode. Polarization switching in nanoscale BFO can be obtained not only\nwith an electric field, but also through mechanical strain44. It could thus be economically achieved\nwith, for example, sound waves9. We hope that this thought experiment can pave the way for the\ncreation of an effective BFO-based water splitting device.\nACKNOWLEDGMENTS\nC. G. is supported by the European Union’s Horizon 2020 research and innovation programme\nunder the Marie Skłodowska-Curie grant agreement No. 744027. N.A.S. acknowledges funding\nfrom the European Research Council (ERC) under the European Union’s Horizon 2020 research\nand innovation programme grant agreement No 810451. I. E. acknowledges the use of the Euler\ncluster managed by the HPC team at ETH Zurich. C. G.’s work was supported by a grant from the\nSwiss National Supercomputing Centre (CSCS) under project ID s889.\n14On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\nFIG. 6. Demonstration of the proposed cyclic process for a water splitting device. (1) Water is adsorbed\nmolecularly on both terminations when the slab is spontaneously polarized in the favorable polarization\ndirection. (2) Switching of the polarization via an external electric field forces the molecularly adsorbed\nmolecules to dissociate on the surface. (3) Selective desorption of the functional groups carrying the same-\nsign charge with the surfaces. 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Schmid, “Structure of a ferroelectric and ferroelastic monodomain crystal of the\nperovskite BiFeO 3,” Acta Crys. B 46, 698–702 (1990).\n33T. Rojac, E. Khomyakova, J. Walker, H. Ursic, and A. Bencan, “BiFeO 3ceramics and thick\nfilms: Processing issues and electromechanical properties,” in Magnetic, Ferroelectric, and Mul-\ntiferroic Metal Oxides , Metal Oxides, edited by B. D. Stojanovic (Elsevier, 2018) pp. 515 – 525.\n34A. Ohtomo and H. Y . Hwang, “A high-mobility electron gas at the LaAlO 3/SrTiO 3heterointer-\nface,” Nature 427, 423–426 (2004).\n35For a discussion in terms of the Modern Theory of Polarization see Ref. 45.\n36L. Jin, P. X. Xu, Y . Zeng, L. Lu, J. Barthel, T. Schulthess, R. E. Dunin-Borkowski, H. Wang, and\nC. L. Jia, “Surface reconstructions and related local properties of a BiFeO 3thin film,” Scientific\nreports 7, 39698 (2017).\n37K. Kern, “Restructuring at surfaces,” in Surface Science: Principles and Applications , edited\nby R. F. Howe, R. N. Lamb, and K. Wandelt (Springer Berlin Heidelberg, Berlin, Heidelberg,\n1993) pp. 81–94.\n38N. Strkalj, C. Gattinoni, A. V ogel, M. Campanini, R. Haerdi, A. Rossi, M. D. Rossell, N. A.\nSpaldin, M. Fiebig, and M. Trassin, “Bilateral interface control of nanoscale ferroelectricity,” in\npreparation (2020).\n39D. Halwidl, B. Stöger, W. Mayr-Schmölzer, J. Pavelec, D. Fobes, J. Peng, Z. Mao, G. S. Parkin-\nson, M. Schmid, F. Mittendorfer, J. Redinger, and U. Diebold, “Adsorption of water at the SrO\nsurface of ruthenates,” Nature Mat. 15, 450–455 (2016).\n40A. V . Bandura, R. A. Evarestov, and D. D. Kuruch, “Hybrid HF-DFT modeling of monolayer\nwater adsorption on (001) surface of cubic BaHfO 3and BaZrO 3crystals,” Surf. Sci. 604, 1591–\n1597 (2010).\n18On the happiness of ferroelectric surfaces and its role in water dissociation: the example of bismuth ferrite\n41H. Guhl, W. Miller, and K. Reuter, “Water adsorption and dissociation on SrTiO 3(001)revisited:\nA density functional theory study,” Phys. Rev. B 81, 155455 (2010).\n42R. A. Evarestov, A. V . Bandura, and V . E. Alexandrov, “Adsorption of water on (001) surface\nof SrTiO 3and SrZrO 3cubic perovskites: Hybrid HF-DFT LCAO calculations,” Surf. Sci. 601,\n1844–1856 (2007).\n43X. L. Hu, J. Carrasco, J. Klimeš, and A. Michaelides, “Trends in water monomer adsorption and\ndissociation on flat insulating surfaces,” Phys. Chem. Chem. Phys. 13, 12447–12453 (2011).\n44L. Chen, Z. Cheng, W. Xu, X. Meng, G. Yuan, J. Liu, and Z. Liu, “Electrical and mechanical\nswitching of ferroelectric polarization in the 70 nm BiFeO 3film,” Sci. Rep. 6, 19092 (2016).\n45C. Gattinoni, I. Efe, and N. Spaldin, “The half polarization quantum in BiFeO 3and its conse-\nquences for thin films and heterostructures,” in preparation (2020).\n19Supplementary information AIP/123-QED\nSupplementary information\nIpek Efe,1Nicola A. Spaldin,1and Chiara Gattinoni1,a)\nMaterials Theory, Department of Materials, ETH Zürich, Wolfgang-Pauli-Strasse 27,\n8093, Zürich, Switzerland\n(Dated: 29 October 2020)\na)Electronic mail: chiara.gattinoni@mat.ethz.ch\n1arXiv:2010.14895v1  [cond-mat.mtrl-sci]  28 Oct 2020Supplementary information\nI. CONVERGENCE TEST FOR SLAB THICKNESS\nCalculations with the most favorable adsorption sites are carried out for the slab thicknesses of\n2–6 unit cells (u.c.) for a water molecule on the charge-compensated slab of Fig. 1b. All atoms\nwere allowed to relax. On the BiO surface the adsorption energy is converged for 4 u.c. On the\nFeO 2surface, the absolute values of the adsorption energies are still varying for a slab thickness of\n6 u.c., however relative energies between the intact and dissociated structure are already converged\nat 4 u.c.\nAdsorption Energies (eV)\nFeO 2Surface BiO Surface\nThickness Intact Dissociated Intact Dissociated\n2 u.c. -0.788 -0.424 -0.664 -0.617\n3 u.c. -0.593 -0.560 -0.456 -0.425\n4 u.c. -0.804 -0.451 -0.672 -0.642\n5 u.c. -0.787 -0.407 -0.654 -0.624\n6 u.c. -0.784 -0.428 -0.607\nTABLE I. Adsorption energy of a water molecule on the compensated surfaces of Fig. 1b with slabs of\nvarying thickness.\n2Supplementary information\nII. WATER ON “UNHAPPY” SLAB\nStructure of a single water molecule adsorbed on an “unhappy” frozen substrate, Fig. 1.\nFIG. 1. Most favorable adsorption geometries and the corresponding adsorption energies for a water\nmolecule on the surfaces of frozen unhappy slab.\n3Supplementary information\nIII. COMPARISON OF WATER ADSORPTION BEHA VIOR ON PEROVSKITES\nReference MaterialLattice\nconstant (Å)H2O on AO H2O on BO 2Charged layers DFT Functional\nhappy\nsystemBiFeO 3 3.95 Molecular Molecular Yes optB86b-vdW\nunhappy\nsystemBiFeO 3 3.95 Dissociated Dissociated Yes optB86b-vdW\n1SrTiO 3 3.94 DissociatedDissociated/\nMolecularNo GGA-PBE\n2Sr2RuO 4 3.9 Dissociated N/A No opt86-vdW\n3SrTiO 3 3.94Equal\npreferenceMolecular No B3LYP XC\n3SrZrO 3 4.19 Dissociated Molecular No B3LYP XC\n4BaHfO 3\nBaZrO 34.19 Dissociated Dissociated No PBE0-XC\nTABLE II. Comparison of the adsorption modes on different systems and their properties in the literature\nand this work\nREFERENCES\n1H. Guhl, W. Miller, and K. Reuter, “Water adsorption and dissociation on SrTiO 3(001 )revisited:\nA density functional theory study,” Phys. Rev. B 81, 155455 (2010).\n2D. Halwidl, B. Stöger, W. Mayr-Schmölzer, J. Pavelec, D. Fobes, J. Peng, Z. Mao, G. S. Parkin-\nson, M. Schmid, F. Mittendorfer, J. Redinger, and U. Diebold, “Adsorption of water at the SrO\nsurface of ruthenates,” Nature Mat. 15, 450–455 (2016).\n3R. A. Evarestov, A. V. Bandura, and V. E. Alexandrov, “Adsorption of water on (001) surface\nof SrTiO 3and SrZrO 3cubic perovskites: Hybrid HF-DFT LCAO calculations,” Surf. Sci. 601,\n1844–1856 (2007).\n4Supplementary information\n4A. V. Bandura, R. A. Evarestov, and D. D. Kuruch, “Hybrid HF-DFT modeling of monolayer\nwater adsorption on (001) surface of cubic BaHfO 3and BaZrO 3crystals,” Surf. Sci. 604, 1591–\n1597 (2010).\n5"    },    {        "title": "2112.00916v1.Strain_Engineering_of_Magnetic_Anisotropy_in_Epitaxial_Films_of_Cobalt_Ferrite.pdf",        "content": "\"This is the pre-peer reviewed version of the following article: Onoda, H., Sukegawa,\nH., Inoue, J.-I., Yanagihara, H., Strain Engineering of Magnetic Anisotropy in Epitaxial\nFilms of Cobalt Ferrite. Adv. Mater. Interfaces 2021, 2101034., which has been published\nin \fnal form at https://doi.org/10.1002/admi.202101034 . This article may be used\nfor non-commercial purposes in accordance with Wiley Terms and Conditions for Use of\nSelf-Archived Versions.\"\nStrain engineering of magnetic anisotropy in epitaxial \flms of\ncobalt ferrite\nHiroshige Onoda,1Hiroaki Sukegawa,2Jun-ichiro Inoue,1and Hideto Yanagihara*1, 3\n1Department of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan\n2National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0047, Japan\n3Tsukuba Research Center for Energy Materials Science (TREMS),\nUniversity of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan\u0003\nAbstract\nPerpendicular magnetic anisotropy (PMA) energy up to Ku= 6:1\u00060:8 MJ m\u00003is demonstrated\nin this study by inducing large lattice-distortion exceeding 3% at room temperature in epitaxially\ndistorted cobalt ferrite Co xFe3\u0000xO4(x = 0.72) (001) thin \flms. Although the thin \flm mate-\nrials include no rare-earth elements or noble metals, the observed Kuis larger than that of the\nneodymium-iron-boron compounds for high-performance permanent magnets. The large PMA is\nattributed to the signi\fcantly enhanced magneto-elastic e\u000bects, which are pronounced in distorted\n\flms with epitaxial lattice structures upon introducing a distortion control layer of composition\nMg2\u0000xSn1+xO4. Surprisingly, the induced Kucan be quantitatively explained in terms of the\nagreement between the local crystal \feld of Co2+and the phenomenological magneto-elastic model,\nindicating that the linear response of induced Kuis su\u000eciently valid even under lattice distortions\nas large as 3.2%. Controlling tetragonal lattice deformation using a non-magnetic spinel layer for\nferrites could be a promising protocol for developing materials with large magnetic anisotropies.\nKeywords: Magnetic anisotropy, Thin \flms, Magneto-elastic e\u000bect, Spinel ferrite, Strain\n\u0003yanagihara.hideto.fm@u.tsukuba.ac.jp\n1arXiv:2112.00916v1  [cond-mat.mtrl-sci]  2 Dec 2021I. INTRODUCTION\nSaturation magnetization, Curie temperature, and magnetic anisotropy (MA) are the\nfundamental properties of ferromagnets. Materials used for fabricating modern permanent\nmagnets require large coercivity and high saturation magnetization to generate su\u000ecient\nmagnetic \rux. The mechanism of coercivity is very complicated; unfortunately, no universal\nmodel for the coercivity of ferromagnets has yet been reported. However, previous studies\nhave revealed that the maximum value of coercivity is limited by MA.[1, 2]In addition, mag-\nnetic thin \flms with large perpendicular magnetic anisotropies (PMAs) are crucial for appli-\ncation in functional spintronic devices and high-density magnetic recording technology.[3{7]\nTo explore or synthesize magnetic materials with high coercivity, materials with large MAs\nare required for obtaining high-performance permanent magnets. Because MA is indirectly\nproportional to the symmetry of the crystal, materials with lower crystal symmetry are\npromising for application as high-MA compounds. Generally, the physical origin of large\nMAs is spin-orbit interaction (SOI), calculated as the inner product of orbital ( L) and spin\nangular momenta ( S) with an interaction constant \u0015. For a given spin moment, the mag-\nnitude of SOI is determined by \u0015and the expected value of the orbital angular momentum.\nMagnetic materials composed of noble metals often exhibit large MAs, attributed to the\nlarge\u0015values of noble metals. In contrast, the large MAs of permanent magnets com-\nposed of rare-earth elements are attributed to their large values of \u0015and orbital angular\nmomenta.[3, 8{13]\nIn transition metal alloys or compounds that do not contain heavy or rare-earth elements,\nLis often quenched in a crystal \feld. Consequently, magnetic materials with large MAs\nare rare. In contrast, a certain amount of orbital angular momentum is sometimes retained\nin oxides, owing to the localized character of the wave functions of transition metal ions\nin the crystal \feld. The magnitude of the orbital angular momentum is in\ruenced by the\nelectronic con\fgurations of the magnetic ions; therefore, the MAs of magnetic oxides can\nbe induced/enhanced by introducing asymmetry, such as lattice deformations. This phe-\nnomenon can be considered to be magneto-elastic in nature because the change in magnetic\nstate is induced by lattice deformation. Furthermore, a large uniaxial MA can be realized\nby uniaxial lattice deformation. The epitaxial distortion arising from the lattice mismatch\nbetween oxide thin \flms and their substrates can be employed to e\u000bectively induce lattice\n2distortion.\nCoxFe3\u0000xO4(CFO) has a cubic lattice, as shown in Figure 1(a) , and exhibits a large\ncubic MA with a N\u0013 eel temperature of 769 K for x= 2.0 and has been reportedly used as\npermanent magnets.[14]Extensive magneto-elastic e\u000bects have been reported, along with\nthe existence of a large orbital moment in Co2+.[15{26]The large cubic MA of bulk CFO\nhas been elucidated theoretically using a single-ion model; the cubic and local trigonal\nlattice symmetries split the down-spin t2gstate into a singly occupied and doubly degenerate\nstate with a magnetic quantum number `\u0018 \u0006 1; moreover, the degeneracy is lifted by\nSOI.[27, 28]Once uniaxial lattice distortion is introduced as shown in Figure 1(b) , a large\nPMA should arise because of the magneto-elastic e\u000bects. This can be inferred from a\nphenomenological magneto-elastic model that predicts that induced MA is proportional\nto the lattice distortion.\nRecently, Tainosho et al. observed that CFO \flms grown on an MgAl 2O4(001) substrate\nsu\u000bering 3.6% in-plane compressive stress exhibit a large negative Kuof\u00005 MJ m\u00003, which\nwas con\frmed by magneto-torque measurements. The induced Kucould be quantitatively\nexplained by the phenomenological magneto-elastic e\u000bects, despite the relatively large dis-\ntortion of 3.6%.[24]They also reported the occurrence of lattice relaxation caused by mis\ft\ndislocations in the early stages of growth because of the large lattice mismatch between the\n\flms and substrates. Based on these results, further enhancement of Kucan be expected by\ninducing a 3{4% tensile distortion. Moreover, the introduction of uniformity is desirable in\ncases where the lattice mismatch with the substrate is large. We have recently found that\nMg2SnO 4(001) (MSO), an oxide with a spinel structure, is suitable for the application of\nlarge tensile stresses in CFO thin \flms.[25, 26]In this study, we attempted to quantitatively\ncontrol the lattice distortion of CFO thin \flms to induce large positive Kuvalues in the\nabsence of platinum group or rare-earth elements. Two methods were adopted to introduce\nthe distortion into CFO thin \flms on MSO bu\u000ber layers: control of the CFO thickness and\ncontrol of the MSO lattice constants. Notably, both the methods produce signi\fcantly large\npositiveKuvalues (perpendicular MA). The latter method is preferable over the former one\nfor arbitrary control of the uniform distortion of the thin \flms. We con\frmed the largest\nvalue of the Kuto be 6:1 MJ m\u00003, consistent with that expected from the phenomenological\nmagneto-elastic e\u000bects.\n3II. RESULTS AND DISCUSSION\nFigure 1 (c-e) show the representative re\rection high-energy electron di\u000braction (RHEED)\npatterns of MgO substrates, post-annealed MSO thin \flms, and CFO thin \flms in [100]\nazimuth. The RHEED patterns of MSO and CFO thin \flms exhibit typical di\u000braction\npatterns from the (001) plane of the spinel structure. The RHEED patterns of the post-\nannealed MSO thin \flms displayed a spot-like pattern, whereas that of the CFO displayed\na streak-like appearance. Figure 1 (f) shows the cross-sectional transmission electron mi-\ncroscopy (TEM) images. The interface between the MSO thin \flms and MgO substrate is\nrough; this is because post-annealing was performed after the MSO thin \flm deposition. In\ncontrast, the interface between the CFO and MSO thin \flms appears \ratter compared to\nthe interface of the MSO thin \flms and MgO substrate. Figure 1 (g) shows a fast Fourier\ntransform (FFT) image corresponding to the red region in Figure 1 (f);x\u0000andz\u0000denote\nthe in-plane and the out-of-plane directions, respectively. An FFT pattern constitutes fun-\ndamental and superlattice spots (red circles in Figure 1 (f)) owing to the diamond glide of\nthe spinel space group ( Fd3m).[29]This result is consistent with the RHEED results. As\nshown in Figure 1 (g), the spots are split into two along the z\u0000direction, implying that\nthe CFO and MSO thin \flms have identicalin-plane lattices and mismatched out-of-plane\nlattices. This indicates that epitaxial distortion is introduced in the CFO thin \flm.\nTo determine the lattice constants of the distorted thin \flms, reciprocal space map (RSM)\nmeasurements were performed. Figure 2 and3(a-d) display the RSM around the CFO and\nMSO (115) prepared through Methods A (CFO thickness control)and B (MSO composition\ncontrol), respectively. The vertical and horizontal axes represent the reciprocal lattice vec-\ntors parallel to the out-of-plane and in-plane directions, respectively. The di\u000braction peaks\nof the thin \flms can be obtained by \ftting with a 2D Gaussian function. In the samples\nprepared through Method A, when the CFO thin \flms are thinner than 10 nm, the peak\npositions of the CFO and MSO along the in-plane direction are equal within the experimen-\ntal error. This indicates that the CFO thin \flms are epitaxially locked to the MSO lattice.\nWhen the thickness of the thin \flms exceeds 10 nm, lattice relaxation occurs, and the lat-\ntice constants advance to the bulk value. Contrastingly, in the samples prepared through\nMethod B, because the thickness of the CFO thin \flms is 5 nm throughout the entire PSn,\nthe peak positions of the CFO and MSO along the in-plane direction are equal within the\n4experimental error. This implies that the CFO thin \flms are epitaxially locked to the MSO\nlattice. The PSn-dependence of the lattice constants of the thin \flms are shown in Figure\n3(e). The in-plane lattice constants of the CFO and MSO thin \flms are identical for each\nPSn. The lattice constant of the MSO thin \flms advance to the bulk value (8 :64/RingA) as the\nPSnincreases. This implies that the distortion introduced into the CFO thin \flms increases\nas thePSnincreases. The in-plane, out-of-plane, and the total distortion of the CFO thin\n\flms are de\fned as \"ip= (aip\u0000a0)=a0,\"oop= (aoop\u0000a0)=a0, and\u001f=\"ip\u0000\"oop, respectively.\nHere,aip,aoop, anda0are the lattice constants of the in-plane, out-of-plane, and bulk CFO\n(a0= 8:38/RingA), respectively. The total distortion \u001fdecreases as CFO thickness increases, as\nis evident from Figure 2 (f). Similarly, \u001falso increases as PSnincreases, as seen in Figure\n3(f). The maximum \u001fis 0.032 in both the methods and is twice as large compared to\nthe value when the deposition occurs on the MgO substrates.[23]According to the magneto-\nelastic e\u000bect, the induced uniaxial MA Kucan be expressed as Ku=B1\u001f. Here,B1is the\nmagneto-elastic constant, which is 0 :14 GJ m\u00003for bulk CFO ( B1= 3=2\u0015100(C12\u0000C11),\nwhere\u0015100=\u0000590\u000210\u00006,C11= 273 GPa, C12= 106 GPa).[27, 30, 31]Therefore, we can\nestimate that the induced Kuis approximately 4 :5 MJ m\u00003for 3.2% distorted CFO thin\n\flms.\nThe valence state of the cations in the CFO thin \flms were investigated using X-ray\nabsorption near edge structure (XANES) spectra for Fe and Co K-edges, as demonstrated\ninFigure 4 (a) and (b), respectively. For comparison, the spectra of FeO[32]in the presence\nof Fe2+, Fe 3O4in the presence of Fe2+and Fe3+, and\r-Fe2O3in the presence of Fe3+, are\nincluded in Figure 4 (a). Additionally, the spectra of CoO in the presence of Co2+, Co 3O4\nin the presence of Co2+and Co3+, and Co 2O3[33]in the presence of Co3+, are shown in\nFigure 4 (b). Furthermore, except for FeO and Co 2O3, all samples were prepared in our\nlaboratory. The spectrum of Fe in the CFO \flm is close to that of \r-Fe2O3, indicating\nthat Fe3+is dominant, and the vacancies at the octahedral sites exist as in \r-Fe2O3[34]\nowing to the Fe-rich composition of the CFO \flm ( x= 0:72). In the XANES spectrum of\nCo in the CFO \flms, the strongest peak of the CFO thin \flms appeared at approximately\n7728 eV, corresponding to Co2+; however, the peak position was slightly shifted to the\nhigh-energy region due to the di\u000berence in structure. The structure of the CoO is rock\nsalt, while the CFO exhibits the spinel structure. Therefore, we can conclude that the\nvalence of Co in the CFO thin \flms was Co2+. Consequently, the cations are distributed\n5as [Fe3+]A(Co2+\n8=11Fe3+\n13=1121=11)BO4in the CFO thin \flms. Here, [ ] Aand ( ) Brepresent the\nA-sites and B-sites of the spinel structure, respectively, and 2represents the vacancies.\nFigure 5 displays the out-of-plane MH curves of the CFO thin \flms at room tempera-\nture. The maximum applied \feld was \u00067 T. In both the methods, the coercivity tends to\nincrease as the introduced distortion increases. All the values of the saturation magnetization\nMSof the CFO thin \flms were comparable to or larger than the bulk value (425 kA m\u00001).[35]\nThe values of the MAs were determined by magnetic torque measurements. Figure\n6(a) shows the torque curves for the CFO/MSO thin \flms prepared through Method B\nunder\u00160H= 9 T. The magnetic torque T(\u0012) around the [001]-direction is represented by\nT(\u0012) =\u0000Ke\u000b\nusin 2\u0012. Here,\u0012is the angle between the magnetization and the [001]-direction.\nAs evident from Figure 6 (a), the magnitude of the torque curve increased as PSnincreased,\nindicating that the MA increases with an increase in the lattice distortion. Even under\nan applied \feld of 9 T, the shapes of all the torque curves deviated considerably from the\nsinusoidal curve, and rotational hysteresis was observed at the crossing of the magnetic hard\naxis. This suggests the presence of an exceedingly large uniaxial MA with a preferential axis\nnormal to the \flm plane. Because the anisotropic magnetic \feld is larger than the applied\nmagnetic \feld, the intrinsic Ke\u000b\nuwas derived using the extrapolation method. When the\nmagnetic torque Lmeasured under the applied \feld His plotted in the direction of 45 °to the\n\flm surface as a function of ( L=H )2, it becomes linear.[36]From the extrapolation, the Ke\u000b\nu\nwas estimated to be 5 :9\u00060:8 MJ m\u00003at its maximum. Considering the shape anisotropy\nof the \flm form, the MA induced by the epitaxial distortion was 6 :1\u00060:8 MJ m\u00003. This\nvalue was considerably large for a ferromagnetic oxide and larger than the value of rare-earth\ncompounds, such as Nd 2Fe14B (\u00184:9 MJ m\u00003).[13]Figure 6 (b) shows the total distortion\n\u001f-dependence of the induced Kuof all the samples. Although there are variations in Kufor\n\u001f\u00180:03, theKuvalues of the samples prepared through either method is proportional to\nthe distortion \u001f. The slope corresponding to the magneto-elastic constant B1was 0:148\u0006\n0:006 GJ m\u00003. This value is approximately equal tobulk value of 0 :14 GJ m\u00003, indicating that\nthe phenomenological theory of the magneto-elastic e\u000bect is still valid even under the large\n\u001fvalue of 0.032. Moreover, it is evident from the results that larger Kucan be expected by\nintroducing further distortion.\n6III. CONCLUSION\nCFO epitaxial thin \flms were grown by RF magnetron sputtering on MSO thin \flms\nand the MA induced by lattice distortion was investigated. Two methods were adopted to\ngrow CFO thin \flms on MSO bu\u000ber layers: control of the CFO thickness and control of\nthe MSO lattice constants. We found both the methods to be extremely productive with\nlarge positive Kuvalues. The induced Kuwas approximately 6 MJ m\u00003larger than the\nMA of Nd 2Fe14B. Furthermore, the induced Kucould be explained quantitatively based on\nthe phenomenological magneto-elastic theory, indicating that larger Kucould be realized\nby introducing further distortion. Since a large MA is essential for improving spintronic\ndevices, the strain engineering technique demonstrated in this study could prove promising\nfor the induction of large Kuin the absence of heavy metals or rare-earth elements.\nIV. EXPERIMENTAL SECTION\nThe thin \flms were deposited by reactive RF magnetron sputtering by a multi-cathode\nsystem. MSO thin \flms (thickness 10 nm) were grown at 600 °C, employing Mg and Sn metal\ntargets. The thus-grown MSO thin \flms were post-annealed at 1000 °C for 10 min in air.\nCFO thin \flms were grown on the post-annealed MSO thin \flms using a CoFe (1:3 atomic\nratio) metal alloy. The introduction of distortion in the CFO thin \flms was attempted\nby two methods: either by altering the thicknesses of the CFO thin \flms (Method A), or\nby changing the lattice constants of the MSO thin \flms (Method B). In Method A, the\nCFO thicknesses were varied in the range from 4.9 to 22 :6 nm. In Method B, the CFO\nthickness was set to 5 nm to prevent lattice relaxation; instead, the lattice constants of MSO\nwere controlled by changing the power input to a Sn target (the power input to the Mg\ntarget was constant at 100 W, while the power input to the Sn target PSnwas varied as PSn\n= 16, 18, 20 and 22 W). The surface structure of the thin \flms was observed by RHEED\nimmediately after growth. The \flm thickness, determined by X-ray re\rectivity (XRR) using\na Rigaku Mini\rex two-axis X-ray di\u000bractometer, was (Cu K \u000b;\u0015=1:5418/RingA). In-plane and\nout-of-plane lattice constants of the CFO thin \flms, determined using a Rigaku SmartLab\nfour-axis X-ray di\u000bractometer, were (Cu K \u000b1;\u0015=1:5405/RingA). All the magnetic measurements\nwere conducted at room temperature. Magnetization was measured using a superconducting\n7quantum interference device, SQUID-VSM (Quantum Design, MPMS) in \felds up to \u00067 T.\nThe MAs were evaluated from the magnetic torque curves recorded over the range from zero\nto 9 T using a torque magnetometer (Quantum Design, PPMS Tq-Mag).\nSupporting Information\nSupporting Information is available from the Wiley Online Library or from the author.\nAcknowledgments\nThis research was supported by the Japan Science and technology Agency (JST) under\nCollaborative Research Based on Industrial Demand High Performance Magnets: Towards\nInnovative Development of Next Generation Magnets (JPMJSK1415). This study was per-\nformed with the approval of the Photon Factory Program Advisory Committee (Proposal\nNo. 2015G655, 2017G602, 2016S2-005 and 2019S2-003). H. O. was supported by Grant-in-\nAid for JSPS Fellows (19J12384).\n[1] E. C. Stoner, E. P. 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Sawa, K. Siratori, Journal ofthe\nPhysical Society ofJapan, 2006 , 75(5):054708.\n[35] B. D. Cullity, C. D. Graham, Introduction tomagnetic materials. John Wiley & Sons, 2011 .\n[36] H. Miyajima, K. Sato, T. Mizoguchi, Journal ofApplied Physics, 1976 , 47(10):4669{4671.\n10FIG. 1. RHEED images and cross-sectional TEM images of CFO/MSO. (a)Spinel structure. (b)\nEpitaxial distortion and the determination of total distortion \u001f. RHEED images in [100] azimuth\nof the (c)MgO substrate, (d)Mg2SnO 4(MSO), and (e)Co-ferrite thin \flms (CFO). (f)Cross-\nsectional TEM image. (g)FFT image corresponding to the red square region in (d). The FFT\npattern shows fundamental spots and superlattice spots (red circles) due to the diamond glide of\nthe spinel space group ( Fd3m).\n11FIG. 2. Structural properties of CFO/MSO prepared through Method A. Reciprocal space map of\nCo-ferrite (CFO) and (Mg, Sn) 3O4(MSO) (115). The thicknesses of the CFO thin \flms are (a)\n4:9 nm, (b)6:8 nm, (d)14:9 nm, and (d)22:6 nm. The black dashed and chained lines represent\nQip, which are the same for CFO and MSO thin \flms. This implies that the CFO thin \flms are\nepitaxially locked on to the MSO thin \flms. (e, f) CFO-thickness-dependence of in-plane and\nout-of-plane lattice constants and the distortion of the CFO thin \flms.\n12FIG. 3. Structural properties of CFO/MSO. (a-d) Reciprocal space map of Co-ferrite (CFO) and\n(Mg, Sn) 3O4(MSO) (115) prepared through Method B. The thicknesses of the CFO and MSO\nthin \flms are 5 nm and 10 nm, respectively. Q ipis the same for CFO and MSO thin \flms, implying\nthat the CFO thin \flms are epitaxially locked on to the MSO thin \flms. (e)Input power into\nthe Sn target ( PSn)-dependence of the in-plane and out-of-plane lattice constants of the CFO and\nMSO thin \flms. (f)PSn-dependence of the CFO distortion.\n13FIG. 4. K-edge XANES spectra of CFO/MSO. (a)Fe K-edge XANES of \r-Fe2O3, Fe 3O4, and\nMSO/CFO thin \flms. The CFO on the MSO \flm is similar to that of \r-Fe2O3.(b)Co K-edge\nXANES of Co 3O4, CoO and MSO/CFO thin \flms. The spectrum of CFO on MSO is similar to\nthat of CoO. ( \r-Fe2O3, Fe 3O4, Co 3O4, and CoO were prepared in our laboratory.)\nFIG. 5. Magnetization processes of CFO/MSO. The out-of-plane magnetization curves of the CFO\nthin \flms prepared in (a)Method A and (b)Method B at room temperature. The maximum\napplied \feld was \u00067 T.\n14FIG. 6. Magnetic anisotropy of CFO/MSO. (a)Torque measurement curves applied \u00160H= 9 T\nfor CFO/MSO thin \flms at room temperature. (b)Total distortion \u001f-dependence of the induced\nmagnetic anisotropy Ku. The straight line \fts the result in the form of a linear function. The\nmagneto-elastic constant can be obtained from the slope as B1= 0:148\u00060:006 GJ m\u00003.\n15Supplementary Material\nSAMPLES PREPARED IN METHOD A\nMethod A was employed to grown samples twice at di\u000berent times (Sample A-1 and\nSample A-2). The structural parameters and magnetic properties of the fabricated samples\nare summarized in Table I.\nTABLE I. Structural parameters and magnetic properties of CFO samples prepared via Method A\nThickness a c \"ip\"perp Total distortion \u001f M SKu\n(nm) ( \u0017A) ( \u0017A) (10\u00002) (10\u00002) ((\"ip\u0000\"oop)10\u00002) (kA m\u00001) (MJ m\u00003)\nSample A-1 4.9 8.521 8.253 1.68 -1.51 3.19 580 6.06\n6.8 8.518 8.248 1.65 -1.58 3.23 540 5.16\n14.9 8.470 8.280 1.07 -1.19 2.26 580 3.50\n22.6 8.442 8.299 0.740 -0.968 1.71 555 2.43\nSample A-2 5.39 8.492 8.301 1.30 -0.96 2.27 394 2.40\n10.7 8.460 8.278 0.978 -1.21 2.18 441 2.45\n15.8 8.445 8.293 0.8 -1.04 -1.12 497 1.68\n21.0 8.397 8.314 0.23 -0.79 1.02 415 1.44\n40.9 8.366 8.352 -0.18 -0.16 -0.34 439 0.788\n51.2 8.354 8.365 -0.32 -0.19 -0.13 415 0.577\n16The reciprocal space maps for CFO and MSO(115) are displayed in Figure 7 and Figure 8,\nrespectively. The vertical and horizontal axes represent the reciprocal lattice vector parallel\nto the out-of-plane and in-plane directions, respectively. The di\u000braction peak positions for\neach thin \flm were obtained by \ftting with a 2-dimensional Gaussian function. The same\nin-plane indices of CFO and MSO were observed for samples less than 10 nm in thickness.\nFurthermore, lattice relaxation was observed in samples with thicknesses exceeding 10 nm.\nFIG. 7. Reciprocal space map of CFO and MSO(115) for CFO samples with thicknesses (a) 5 nm,\n(b) 7 nm, (c) 15 nm, and (e) 23 nm in Sample A-1.\n17FIG. 8. Reciprocal space map of CFO and MSO(115) for CFO samples with thicknesses (a) 5.39\nnm, (b) 10.66 nm, (c) 15.76 nm, (d) 40.9 nm, and (e) 51.2 nm in Sample A-2.\n18Figure 9 displays the CFO thickness t-dependence of the areal saturation magnetization\nMS\u0001tfor Samples A-1, A-2, and CFO/MgO. Additionally, linear \ftting lines were shown in\nthe Figure 9. In case of CFO/MgO, the intercept of the \ftting line is in the positive region\nof the horizontal axis, indicating the presence of a magnetic dead layer with a thickness of\n3\u00060.5 nm. The dead layer can be attributed to the introduction of antiphase boundaries\ncaused by the di\u000berence in the crystal structures between the CFO and MgO structures[1{\n6]. In CFO/MSO, the intercept of the horizontal axis is -0.1 \u00060.4 nm and -0.8 \u00061.1 nm\nfor Sample A-1 and A-2, respectively, indicating that the \ftting line almost passes through\nthe origin. Therefore, the dead layer thickness is negligible for CFO/MSO, implying that\nthe formation of antiphase boundaries near the CFO interface is e\u000bectively suppressed by\nthe use of a spinel MSO bu\u000ber. Figure 10 shows the total distortion \u001f-dependence of the\ninducedKufor Sample A-1 and Sample A-2. The max value of Kuwas 6.1 MJ m\u00003.\nFIG. 9. Thickness-dependences of MS\u0001t.\n19FIG. 10. Total distortion \u001f-dependence of the induced Ku.\n20TORQUE ANALYSIS (45\u000eMETHOD)\nWhen the anisotropy \feld of the sample is large, the 45\u000emethod proves e\u000bective, wherein\nthe torque Lis plotted as a function of ( L=(\u00160HV))2at a \fxed angle. When plotting the\nmagnetic torque Lthat is measured under the applied \feld H in the direction of 45\u000eto the\n\flm surface, the plot becomes linear. Several examples of 45\u000emethod are given in Figure\n11.\nFIG. 11. 45\u000eplot for (a) Sample A-1, (b) Sample A-2, and (c) the samples prepared in Method B.\n21[1] P.A.A. van der Heijden, P.J.H. Bloemen, J.M. Gaines, J.T.W.M. van Eemeren, R.M. Wolf, P.J.\nvan der Zaag, and W.J.M. de Jonge. Magnetic interface anisotropy of mbe-grown ultra-thin\n(001) fe3o4 layers. Journal ofMagnetism andMagnetic Materials, 159(3):L293 { L298, 1996.\n[2] A.R. Ball, H. Fredrikze, D.M. Lind, R.M. Wolf, P.J.H. Bloemen, M.Th. Rekveldt, and P.J. van\nder Zaag. Polarized neutron re\rectometry studies of magnetic oxidic fe3o4/nio and fe3o4/coo\nmultilayers. Physica B:Condensed Matter, 221(1):388 { 392, 1996. Proceedings of the Fourth\nInternational Conference on Surface X-ray and Neutron Scatterin.\n[3] A. R. Ball, A. J. G. Leenaers, P. J. van der Zaag, K. A. Shaw, B. Singer, D. M. Lind,\nH. Fredrikze, and M.Th. Rekveldt. Polarized neutron re\rectometry study of an exchange\nbiased fe3o4/nio multilayer. Applied Physics Letters, 69(10):1489{1491, 1996.\n[4] W. Eerenstein, T. T. M. Palstra, T. Hibma, and S. Celotto. Origin of the increased resistivity\nin epitaxial fe 3o4\flms. Phys. Rev. B, 66:201101, Nov 2002.\n[5] J.-B. Moussy, S. Gota, A. Bataille, M.-J. Guittet, M. Gautier-Soyer, F. Delille, B. Dieny, F. Ott,\nT. D. Doan, P. Warin, P. Bayle-Guillemaud, C. Gatel, and E. Snoeck. Thickness dependence of\nanomalous magnetic behavior in epitaxial fe 3o4(111) thin \flms: E\u000bect of density of antiphase\nboundaries. Phys. Rev. B, 70:174448, Nov 2004.\n[6] A. V. Ramos, J.-B. Moussy, M.-J. Guittet, A. M. Bataille, M. Gautier-Soyer, M. Viret, C. Gatel,\nP. Bayle-Guillemaud, and E. Snoeck. Magnetotransport properties of fe3o4 epitaxial thin \flms:\nThickness e\u000bects driven by antiphase boundaries. Journal ofApplied Physics, 100(10):103902,\n2006.\n22"    },    {        "title": "1704.03314v1.Magnetic_Dipole_Interaction_and_Total_Magnetic_Energy_of_Lithium_Ferrire_Thin_Films.pdf",        "content": " \n1 \n Magnetic Dipole Interaction and Total Magnetic Ener gy of Lithium Ferrire Thin Films \n \nP. Samarasekara and C. K.D. Sirimanna \n Department of Physics, University of Peradeniya, Pe radeniya, Sri Lanka \n \nAbstract \nThe total magnetic energy of Lithium ferrite thin f ilms was determined using the classical \nHeisenberg Hamiltonian. The short range magnetic di pole interactions between spins within one \nunit cell and the interactions between spins in two  adjacent unit cells have been determined in order \nto find the total magnetic energy of lithium ferrit e films. Only the spin pairs with separation less \nthan cell constant have been taken into account to calculate dipole interaction and spin exchange \ninteraction. Theoretically several easy and hard di rections of lithium ferrite film were found for one  \nset of energy parameters included in our modified H eisenberg Hamiltonian. The dependence of total \nmagnetic energy of a lithium ferrite film on number  of unit cells, spin exchange interaction, dipole \ninteraction, second order magnetic anisotropy, four th order magnetic anisotropy, internal applied \nmagnetic field and stress induced magnetic anisotro py has been explained in this manuscript. \n \n1. Introduction: \n              Lithium Ferrite (Li 0.5 Fe 2.5 O 4) is a potential candidate in applications of magne tic memory \ndevices, monolithic microwave integrated circuits a nd microwave devices. The magnetic properties \nof films depend on the stress of the films induced within cooling and heating process. The crystal \nstructure of lithium ferrite has been explained pre viously 1. Magnetic properties of lithium ferrite \nnanoparticles with a core/shell structure have been  investigated 2. Infrared spectral properties of \nMagnesium and Aluminum co-substituted Lithium ferri tes have been studied 3. Influence of \nsubstrate on the octahedral site order of Structura l and magnetic properties of Lithium ferrite thin \nfilms have been investigated 4. The effect of Dipole and Anisotropic Hyperfine Fi elds on NMR of \nFe in Lithium Ferrite has been studied 5. \n              The short range magnetic dipole inter action of Lithium ferrite thin films have been \nexplained in this manuscript. Previously the magnet ic dipole interactions of barium ferrite 6 films \nhave been described by us. However, cobalt belongs to the ferromagnetic category. In all above \ncases, the total magnetic energy was also determine d using the classical Heisenberg Hamiltonian \nmodified by introducing fourth order magnetic aniso tropy, stress induced anisotropy and \ndemagnetization factor. In addition, spin exchange interaction, magnetic dipole interaction, second \norder magnetic anisotropy constant and applied magn etic field terms can be found in this classical \nHeisenberg Hamiltonian. The investigations of bariu m ferrite films were restricted to unperturbed \nHeisenberg Hamiltonian. However, the studies of fer rite and ferromagnetic films were extended to \nthird order perturbed Heisenberg Hamiltonian 7-15 . In addition, the total magnetic energy of cobalt \nfilm was determined using second order perturbed He isenberg Hamiltonian 15 . The technique used \nto calculate magnetic dipole interaction in our pre vious reports was employed to find the magnetic \ndipole interaction of Lithium ferrite films explain ed in this manuscript. Only the interactions \nbetween iron ions were taken into account, since th e net magnetic moment of each lithium and \noxygen ions is zero. \n \n  \n2 \n 2. Model:  \nThe total energy of a magnetic thin film is given b y following modified Heisenberg Hamiltonian 6.  \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, strength of \nlong range dipole interaction, azimuthal angle of s pin, second and fourth order anisotropy constants, \nin plane and out of plane internal magnetic fields,  stress induced anisotropy constant and spin plane \nindices, respectively. When the stress applies norm al to the film plane, the angle between m th  spin \nand the stress is θm. Therefore the ground state energy will be calcula ted per spin with Z-axis \nnormal to film plane. The total magnetic energy per  unit spin of Fe 3+  will be determined. The \ndemagnetization factor was not considered in this s tudy. In Lithium ferrite structure, only the Fe 3+  \nions contribute to magnetic moment. Due to the unav ailability of unpaired electrons, Li + ions don’t \nhave any net magnetic moment. Fe 3+  ions occupy octahedral and tetrahedral sites of th e lattice as \ngiven in table 1. \n Layer Atomic number Coordinates \nx y z \nA1 A1 1 0 0 0 \nA1 2 1 0 0 \nA1 3 0 1 0 \nA1 4 1 1 0 \nA1 5 0.5 0.5 0 \nB1 B1 1 0.625 0.125 0.125 \nB1 2 0.375 0.875 0.125 \nB1 3 0.125 0.625 0.125  \n3 \n A2 A2 1 0.25 0.25 0.25 \nA2 2 0.75 0.75 0.25 \nB2 B2 1 0.625 0.375 0.375 \nB2 2 0.375 0.625 0.375 \nB2 3 0.875 0.125 0.375 \nA3 A3 1 0.5 0 0.5 \nA3 2 0 0.5 0.5 \nA3 3 1 0.5 0.5 \nA3 4 0.5 1 0.5 \nB3 B3 1 0.125 0.125 0.625 \nB3 2 0.875 0.875 0.625  \nB3 3 0.375 0.375 0.625  \nA4 A4 1 0.75 0.25 0.75 \nA4 2 0.25 0.75 0.75 \nB4 B4 1 0.625 0.875 0.875 \nB4 2 0.875 0.625 0.875  \nB4 3 0.125 0.375 0.875  \nA5 A5 1 0 0 1 \nA5 2 1 0 1 \nA5 3 0 1 1 \nA5 4 1 1 1 \nA5 5 0.5 0.5 1 \nTable 1: Coordinates of Fe 3+ ions in a unit cell corresponding to A (tetrahedral ) and B \n(octahedral) sites. \n           First the spin exchange interactions bet ween spins were determined as following. Only the \ninteractions between spins separated by a distance less than the cell constant were considered. \nSpin exchange interactions were calculated for near est spin neighbors within one cell and two \nadjacent cells as given in table 2.  \n4 \n  \n Interaction Number of nearest neighbors \nWithin unit cell Between adjacent cells \nNA A-A 70 12 \nNB B-B 63 47 \nNAB  A-B 230 63 \n \nTable 2: Number of nearest neighbors for each type of interaction. \n \n       By using the values given in table 2 and the  first term in equation 1, the exchange interaction  \nenergy within a unit cell was found as 87J. Similar ly the exchange interaction energy between \ntwo adjacent unit cells can be expressed as 4J. If there are N number of unit cells in the Lithium \nferrite film in z direction, there are (N-1) adjace nt cell combinations within the film. Therefore, \nthe total exchange interaction energy can be writte n as 87NJ+4(N-1)J. \nThe magnetic dipole interaction energy between near est spins (S i and S j) was determined using \nfollowing equation and matrix. \nj ij i SrWS Err\n). (.ω=                     (2)  \nHere \n\n\n\n\n− − −− − −− − −\n=\n222\n3\nˆ31ˆ ˆ3 ˆ ˆ3ˆ ˆ3 ˆ31ˆ ˆ3ˆ ˆ3 ˆ ˆ3 ˆ31\n1) (\nz z y z xy z y y xx z x y x\nr r r r rr r r r rr r r r r\nrrW                         (3)  \nand 32\n0\n4aπµµω=  \n         Only the interactions between spins separa ted by a distance less than the cell constant were \nconsidered. A film with (001) orientation of Lithiu m ferrite cell has been considered. As an \nexample, the calculations of some elements of matri x appeared in equation 3 are given below. In \nlayer A1 (bottom layer of the unit cell), five Ferr ic ions occupy A1 1(0, 0, 0), A1 2(1, 0, 0), A1 3(0, \n1,0), A1 4(1, 1, 0) and A1 5(0.5, 0.5, 0) sites 3. Because the interactions bet ween spins with \nseparation less than the cell constant have been co nsidered, only the A1 1A1 5, A1 2A1 5, A1 3A1 5 \nand A1 4A1 5 spin interactions have been taken into account.  \n             The spins of Fe ions in the second spi n layer located at a distance “a/8” above the botto m \nlayer of Lithium ferrite unit cell occupy B1 1(0.125, 0.625, 0.125), B1 2(0.375, 0.875, 0.125) and \nB1 3(0.625, 0.125, 0.125) sites 3. Because of the separations between following inte ractions are \nless than the cell-constant, the spin interactions between A1 1B1 1, A1 1B1 2, A1 1B1 3, A1 2B1 1, \nA1 3B1 2, A1 3B1 3, A1 4B1 1, A1 4B1 2, A1 4B1 3, A1 5B1 1, A1 5B1 2 and A1 5B1 3 were considered for \nthese calculations. \n         In a similar method, all the matrix elemen ts for adjacent spin layers were calculated, and  \nthe dipole interactions for next nearest layers in the cell were calculated. All W 11 , W 12 , W 13 , W 21 , \nW22 , W 23 , W 31 , W 32  and W 33  matrix elements were calculated from equation 3 us ing the same \nmethod described in one of our previous publication s 8. This matrix was found to be symmetric.  \n5 \n After substituting these matrix elements in the mat rix given in equation (3), the dipole matrix can \nbe obtained. By substituting that dipole matrix in equation (2), the dipole interaction energy in \neach case can be found as following. \nThe total dipole moment for the spins within A laye r can be given as \n \nEdipoleA = -11.31371 ω(0.25+0.75cos2 θ) \n \nSimilarly for all B type and A-B nearest neighbor i nteractions within the unit cell, the total \ndipole interaction energy can be given as \n \nEdipoleB = 20.174631 ωsin2 θ \nEdipoleAB = 26.44766 ωsin2 θ \n \nFinally the total dipole interaction energy within the unit cell can be given as \n \nEdipole-unit cell = ω(46.52229sin2 θ−2.82843−8.48528 cos2 θ)                             (4) \n \nSimilarly for all A type, B type and A-B nearest ne ighbor interactions between two adjacent unit \ncells in z direction, the total dipole interaction energy can be given as \n \nEdipoleA-adj = -26.06992 ω(0.25+0.75cos2 θ) \nEdipoleB-adj = -ω(20.60867+ 22.42482sin2 θ+66.26353 cos2 θ) \nEdipoleAB-adj = ω(60.1526-25.85484sin2 θ+180.45779 cos2 θ) \n \nTherefore the total dipole interaction energy betwe en two adjacent unit cells in z direction can be \ngiven as  \n \nEdipole-adj-cells = ω(33.02645-48.28966sin2 θ+94.64182 cos2 θ)                            (5) \n \nIf there is N number of unit cells in the Lithium f errite film in z direction, then the total dipole \ninteraction energy can be given as \n \nEdipole-total = NE dipoleunit-cell + (N-1)E dipole-adj-cells                                                                                      (6) \n \nBy substituting equations (4) and (5) in equation ( 6), the total dipole interaction energy of a \nLithium ferrite film with thickness Na (height of N  cubic unit cells) in z direction can be given \nas, \nEdipole-total = N ω(46.62229sin2 θ −2.82843 −8.48528 cos2 θ) \n                       +(N-1) ω(33.02645-48.28966sin2 θ+94.64182 cos2 θ)               (7) \n \nThen by substituting equation (7) in equation (1), the total energy per unit spin can be given as, \n \n \n  \n6 \n E( θ)=(91N-4)J+ ω{N(46.62229sin2 θ-2.82843-8.48528cos2 θ) \n                          +(N-1)(33.02645-48.28996s in2 θ+94.64182cos2 θ)} \n         )2sin cos sin (4 cos cos ) 4 (\n1 14 ) 2 ( 2θ θ θ θ θs out in mN\nmN\nmm K H HN D D + + + − −∑ ∑\n= = \nHence \n+ − =ω ωθ JNE) 4 91 () ( N(46.62229sin2 θ-2.82843-8.48528cos2 θ) \n                          +(N-1)(33.02645-48.28996s in2 θ+94.64182cos2 θ) \n               )2sin cos sin (4 cos cos ) 4 (\n1 14 ) 2 ( 2θ θ θ θ θs out in mN\nmN\nmm K H HN D D + + + − −∑ ∑\n= = \nHence Following 2-D and 3-D graphs have been plotte d using above final energy equation. \n \n3. Results and discussion: \nThe 3-D graph of ωθ) (E versus ω) 2 (\n1mN\nmD∑\n= and θ is given in figure 1. The variation of energy is \nmost likely periodic. Values of other parameters we re kept at N=1000, \n5 10 ) 4 (\n1= = = = =∑\n=\nω ω ω ω ωmN\nm s out in D\nand K H HJ. \n \nFigure 1: 3-D graph of ωθ) (E versus ω) 2 (\n1mN\nmD∑\n= and θ   \n7 \n By using the ω) 2 (\n1mN\nmD∑\n= values for each of the maxima and minima of the 3D  graph, the 2-D \ngraphs of ωθ) (E versus θ were plotted in order to find easy and hard direct ions of magnetization. \nOne of those graphs is shown in figure 2 for ω) 2 (\n1mN\nmD∑\n== 44 (one of the maximum energy points) \nand another graph is shown in figure 3 for ω) 2 (\n1mN\nmD∑\n== 55 (one of the minimum energy points). \nAccording to figure 2, the hard directions are 0.2,  3, 6.4 radians ----etc for given values of energy \nparameters. The easy directions are 1.6, 4.5, 7.9 r adians ------etc from figure 3.  \n \n \nFigure 2: The graph of ωθ) (E versus θ at ω) 2 (\n1mN\nmD∑\n== 44 (one of the maximum energy points in \n3-D plot). \n  \n8 \n \n \nFigure 3: The graph of ωθ) (E versus θ at ω) 2 (\n1mN\nmD∑\n== 55 (one of the minimum energy points in 3-\nD plot). \n \n4. Conclusion: \n   The total magnetic energy obtained using modifie d Heisenberg Hamiltonian varies periodically \nwith energy parameters. The magnetic easy and hard directions were determined using 2-D and \n3-D plots. The variation of energy is somewhat simi lar for ω) 2 (\n1mN\nmD∑\n=and ω) 4 (\n1mN\nmD∑\n=. Higher \nvariations could be observed for larger values of ω ω ωs out in Kand H H, . When the number of layers \nor ωJ increases, the peak value of energy also increases . Changing ω) 2 (\n1mN\nmD∑\n=, ω) 4 (\n1mN\nmD∑\n=, \nωJ,ωsK or N does not change the easy axis or hard axis ve ry much. But the angle of the hard axis \nincreases with ωin H, and the angle of the easy axis decreases with ωout H .  \n9 \n References: \n1. A. I. Smolentsev, A.B. Meshalkin, N.V. Podberezs kaya and A.B. Kaplun, 2008. Refinement   \n     of LiFe 5O8 crystal structure. Journal of Structural Chemistry  49 (5), 953. \n2. Natasa Jovic, Bratislav Antic, Gerardo F. Goya a nd Vojislav Spasojevic, 2012. Magnetic   \n    properties of lithium ferrite nanoparticles wit h a core/shell structure. Current Nanoscience   \n    8(5), 651-658. \n3. K.B. Modi, J.D. Gajera, M.P. Pandya, H.G. Vora a nd H.H. Joshi, 2004. Far-infrared spectral   \n    studies of magnesium and aluminum co-substitute d lithium ferrites. Pramana Journal of \n    Physics 62 (5), 1173-1180. \n4. Cihat Boyraz, Dipanjan Mazumdar, Milko Iliev, Ve ra Marinova, Jianxing Ma, Gopalan \n     Srinivasan and Arunava Gupta, 2011. Structural  and magnetic properties of lithium ferrite  \n    (LiFe 5O8) thin films: Influence of substrate on the octahed ral site order. Applied Physics    \n    Letters 98, 012507. \n5. V.D. Doroshev, V.A. Klochan, N.M. Kovtun and V.N . Seleznev, 1972. The effect of dipole  \n     and anisotropic hyperfine fields on NMR of Fe 57  in lithium ferrite Li 0.5 Fe 2.5 O4. Physica Status   \n     solidi (a) 9(2), 679. \n6. P. Samarasekara and Udara Saparamadu, 2013. Easy  axis orientation of Barium hexa-  \n      ferrite films as explained by spin reorientat ion. Georgian Electronic Scientific Journals:   \n      Physics 1(9), 10-15. \n7. P. Samarasekara and William A. Mendoza, 2010. Ef fect of third order perturbation on   \n      Heisenberg Hamiltonian for non-oriented ultra -thin ferromagnetic films. Electronic Journal   \n      of Theoretical Physics 7(24), 197-210.  \n8. P. Samarasekara, M.K. Abeyratne and S. Dehipawal age, 2009. Heisenberg Hamiltonian with \nSecond Order Perturbation for Spinel Ferrite Thin F ilms. Electronic Journal of Theoretical \nPhysics 6(20), 345-356. \n9. 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.  \n10. P. Samarasekara and N.H.P.M. Gunawardhane, 2011 . Explanation of easy axis orientation    \n     of ferromagnetic films using Heisenberg Hamilt onian. Georgian electronic scientific  \n     journals: Physics 2(6): 62-69.   \n10 \n 11. P.  Samarasekara. 2008. Influence of third orde r perturbation on Heisenberg Hamiltonian of  \n      thick ferromagnetic films. Electronic Journal  of Theoretical Physics 5(17): 227-236. \n12. P. Samarasekara and Udara Saparamadu, 2013. In plane oriented Strontium ferrite thin  \n      films described by spin reorientation.  Research & Reviews: Journal of Physics-STM  \n       journals 2(2), 12-16. \n13. P. Samarasekara, 2008. Four layered ferromagnet ic ultra-thin films explained by second  \n     order perturbed Heisenberg Hamiltonian. Ceylon  Journal of Science 14, 11-19  \n14. P. Samarasekara and B. I. Warnakulasooriya, 201 6. Five layered fcc ferromagnetic films  \n      as described by modified second order perturb ed Heisenberg Hamiltonian. Journal of   \n      science of the University of Kelaniya Sri Lan ka 11, 11-21.   \n15. P. Samarasekara and Amila D. Ariyaratne, 2012. Determination of magnetic properties of  \n     Cobalt films using second order perturbed Heis enberg Hamiltonian. Research &     \n     Reviews: Journal of Physics-STM journals 1(1),  16-23. \n \n \n \n \n \n \n \n \n \n \n "    },    {        "title": "1111.5453v1.Magnetic_and_structural_characterization_of_nanosized_BaCo_xZn__2_x_Fe__16_O__27__hexaferrite_in_the_vicinity_of_spin_reorientation_transition.pdf",        "content": " \n \n \n  \n \nMagnetic and structural characterization of nanosized \nBaCo xZn 2-xFe16O27 hexaferrite in the vicinity of spin \nreorientation transition \nA Pasko1, F Mazaleyrat1, M LoBue1, V Loyau1, V Basso2, M Küpferling2, C P \nSasso2 and L Bessais3 \n1 SATIE, ENS Cachan, CNRS, UniverSud, 61  av President Wilson, F-94235 Cachan, \nFrance \n2 INRIM, Strada delle Cacce 91, I-10135 Torino, Italy \n3 ICMPE, CNRS, Université Paris 12, 2-8 rue Henri Dunant, F-94320 Thiais, France \nE-mail: pasko@satie.ens-cachan.fr \nAbstract . Numerous applications of hexagonal ferrites are related to their easy axis or easy \nplane magnetocrystalline anisotropy configurations. Certain W-type ferrites undergo spin \nreorientation transitions (SRT) between differe nt anisotropy states on magnetic field or \ntemperature variation. The transition point can be tuned by modifying the chemical composition, which suggests a potential application of hexaferrites in room temperature \nmagnetic refrigeration. Here we present the resu lts of structural and magnetic characterization \nof BaCo\nxZn2–xFe16O27 (0.7 ≤ x ≤ 2) doped barium ferrites. Fine powders were prepared using a \nsol-gel citrate precursor method. Crystal stru ctures and particle size distributions were \nexamined by X-ray diffracti on and transmission electron microscopy. The optimal synthesis \ntemperature ensuring complete formation of single W-phase with limited grain growth has \nbeen determined. Spin reorientation transitions were revealed by thermomagnetic analysis and \nAC susceptibility measurements. \n1.  Introduction \nHexaferrites have been the subject of intensive studies due to an appealing combination of good \nmagnetic properties and low cost. This large family of oxides with hexagonal crystal structure contains \nferrimagnetic compounds with easy axis of magnetization ( e.g. M-type ferrites) and easy plane of \nmagnetization ( e.g. Y-type ferrites). Hence, hexaferrites have  been widely adopted in two distinct \nfields: permanent magnets and microwave technolog y components [1]. On the other hand, W-type \nferrites Ba M2Fe16O27 (M = Mg, Mn, Fe, Co, Ni, Cu, Zn) can unde rgo spin reorientation transitions \n(SRT) between different anisotropy configurations (easy plane ↔  easy cone ↔ easy axis) induced by \nchange of temperature or applied magnetic field [2–9]. The transition temperatures can be tuned by \nmodifying the chemical compositi on (substitution of bivalent metal M ). Moreover, some SRT are \nexpected to be of the first order, which suggests a potential application of W-type ferrites in room \ntemperature magnetic refrigeration [10]. \nConventional ceramic method of hexaferrite synthesis is efficient, but requires elevated \ntemperatures for solid state reaction to occur between premixed  powders. Alternative production \nroutes (aerosol pyrolysis, chemic al co-precipitation, glass crystallization, hydrothermal synthesis, etc.)  \n \n \n  \n \nare intended to improve mixing of initial component s down to atomic level and thereby to facilitate \ndiffusion. In particular, a sol-gel technique enabl es to obtain sufficiently homogeneous precursors for \nlow-temperature synthesis of nanosized simple (s pinel) ferrites. However, hexagonal ferrites with \ncomplex layered crystal structures still require re latively high temperatures to form because of \nthermodynamic stability conditions. Trying to avoid rapi d growth of grains, we have used this soft-\nchemistry approach for production of W-type ferrit es. An important goal was to determine the heat \ntreatment regimes ensuring complete transformation of a precursor into the smallest particles of single \nW-phase. In this paper the results of structural and magnetic characterization of BaCo xZn2-xFe16O27 \npowders are presented. The chemical compositions were chosen so that SRT occur near room \ntemperature ( x = 0.7, 0.75, 0.8) or much higher (x  = 2). \n2.  Experimental \nHexaferrite powders have been prepared by a so l-gel citrate precursor method [11–13]. High purity \niron(III) nitrate, barium hydroxide, cobalt(II) oxide, zi nc oxide and citric acid were used as starting \nmaterials with the molar ratio of citrate to metal io ns 2:1. Iron(III) nitrate was dissolved in deionized \nwater and quantitatively precipitated with excess of ammonia solution as iron(III) hydroxide. The \nprecipitate was filtered and washed with water until neutrality. Then the obtained iron(III) hydroxide \nwas dissolved in a vigorously stirred citric acid solution at 60–70 °C. Barium hydroxide and other \nmetal sources were added according to stoichiometr y. At this stage 3 samples of each composition \nwere separated: (A) pH value of the solution was adjusted to 6 for better chelation of cations [13]; (B) \nno modification was done [12]; (C) ethylene glycol was added to increase viscosity by \npolycondensation reaction [11]. Water was slowly ev aporated at 80–90 °C with continuing stirring \nuntil a highly viscous residue is formed. The gel was dried at 150–170 °C and heat treated for 2 h at \n450 °C for total elimination of organic matter. Fi nally, the inorganic precursor with homogeneous \ncationic distribution was calcined at temperatures up to 1300 °C for 2 h with heating/cooling rate \n200 K/h to synthesize a hexaferrite phase. Details of the procedure will be discussed elsewhere. \nCrystal structures were examined by PANalytical X’Pert Pro X-ray diffractometer (XRD) in Co-K α \nradiation with X’Celerator detector for rapid data  acquisition. Magnetization curves were recorded on \nLake Shore vibrating sample magnetometer (VSM). Direct observations of powder particles were \ncarried out by FEI Tecnai G2 F20 transmission electron microscope (TEM) operating at 200 keV. \nThermogravimetric analyzer (TGA) PerkinElmer Pyris 6 equipped with a permanent magnet was used \nfor thermomagnetic measurements above room temperature. AC magnetic susceptibility at low \ntemperatures was studied in Quantum Design PPMS. Ri etveld analysis of XRD spectra was performed \nusing MAUD software [14]. \n3.  Results and discussion \nCharacterization had a twofold purpose: to find a correlation between the calcination temperature and \nthe final product properties (phase composition, mean grain size, coercivity); to confirm the existence \nof SRT at expected temperatures in fine hexaferrite powders. \n3.1.  Structural characterization and phase analysis \nFigure 1 represents XRD patterns from BaCo 2Fe16O27 powder synthesized at different temperatures. \nW-ferrite becomes the major phase at 1300 °C, while lower calcination temperatures lead to a mixture \nof W-ferrite, M-ferrite (both are hexagonal) and S-ferr ite (with spinel structure) in the final product. \nTraces of α-Fe 2O3 are sensitive to the method of preparation and dissolve completely with increase of \ncalcination temperature; however, other secondary phases may appear together with W-ferrite. The \nresults of quantitative phase analysis based on Rietveld method of XRD full profile fitting are shown \nin figure 2. At 900 °C only M-ferrite can be fo rmed, with excess of cobalt giving also S-ferrite. \nIncrease of temperature creates favorable conditions for W-ferrite synthesis, directly from the precursor components (nanocrystalline or amorphous  oxides) or through the intermediate reaction \nBaFe\n12O19 + 2 CoFe 2O4 → BaCo 2Fe16O27  \n \n \n  \n \nThe proportion between 3 ferrite phases depends on calcination temperature and production route \n(as the precursor composition is affected, for inst ance, by pH value of the solution). However, at \n1300 °C M-ferrite and S-ferrite almost completely disappear. \n \nFe2O3 (hematite)CoFe2O4 (spinel)M-type Ba ferriteW-type Ba ferrite\nHT 1300 °C\nHT 1250 °C\nHT 1200 °C\nHT 900 °C\n20 25 30 35 40 45 50 55\nDiffraction angle 2 θ [°]Intensity [arb. unit]\n \nFigure 1. XRD patterns of BaCo 2Fe16O27 powder A taken after different heat treatments. \n \nCBA\n0102030405060708090100\n850 950 1050 1150 1250 1350\nTemperature [°C]Mass Fraction [%]\n CBA\n0102030405060708090100\n850 950 1050 1150 1250 1350\nTemperature [°C]Mass Fraction [%]\n M-phase W-phase \n(a) (b) \nFigure 2. Phase composition of BaCo 2Fe16O27 powders A, B, C (from Rietveld fitting of XRD spectra) \nvs. calcination temperature: (a) M-ferrite fraction; (b) W-ferrite fraction.  \n \n \n  \n \n3.2.  Morphology and average size of particles \nThe size of diffracting crystallites in W-ferrite powde rs is derived from XRD data by analysis of line \nbroadening using Rietveld refinement as shown in  figure 3(a). It gives a minimum estimate for \naverage particle size (a particle can contain several crystallites). The mixture of M-ferrite and S-ferrite \nformed at 900 °C have a mean size of crystallites ~ 50 nm. With increase of calcination temperature \nthe calculated values slightly decrease, this can reflect contribution to line broadening caused by \ngeneration of defects in the transforming mixture of phases. At 1300 °C, when W-ferrite phase rapidly \ngrows and becomes dominant, the apparent size of  crystallites significantly increases. In addition, \nXRD spectra of the powders synthesi zed at this temperature exhibit a specific texture characteristic for \nplate-like particles pressed in a holder. \nMagnetic measurements give an illustration to th ese phase transformations. Figure 3(b) shows that \nafter calcination at 900 °C coercivity is high due to the presence of M-ferrite. When W-ferrite forms, \nthe coercivity decreases and reaches ~ 15 mT at 1300 °C, a typical for this phase value. \n \nCBA\n20406080100120140160180200\n850 950 1050 1150 1250 1350\nTemperature [°C]Crystallite Size [nm]\n A\nBC\n0306090120150180210240270300\n850 950 1050 1150 1250 1350\nTemperature [°C]Coercivity [mT]\n W-phase M-phase \nM-phase \nW-phase \n(a) (b) \nFigure 3. BaCo 2Fe16O27 powders A, B, C calcined at different temperatures: (a) mean size of \ncrystallites (from XRD line broadening); (b) coercivity (from VSM measurements). \n \nDirect observations of W-ferrite particles were  performed by TEM. Powders calcined at 900 °C \nconsist of aggregates of clean and almost round particles not exceeding ~ 100 nm, an example is shown in figure 4(a). The shape of crystallites re flects their hexagonal crystal structure. Powders \ncalcined at temperatures high enough to form W-fe rrite exhibit absolutely different morphology as \nshown in figure 4(b). The particles have a wide  size distribution from 50 nm to 2000 nm and are \nmostly plate-like. This morphology is observed not only for 1300 °C when W-ferrite is almost single \nphase, but also for lower temperatures when W-ferrite  appears in the precursor. The average particle \nsize of BaCo\nxZn2-xFe16O27 (x = 0.75) powder estimated by laser light scattering analysis is ~ 700 nm, \nwhich agrees with TEM observations.  \n \n \n  \n \n \n  (a) (b) \nFigure 4. Typical TEM images of powder particles: (a) BaCo 2Fe16O27 precursor after calcination at \n900 °C (mixture of ferrites); (b) W-type BaCo xZn2–xFe16O27 (x = 0.75) ferrite synthesized at 1275 °C. \n \n3.3.  Spin reorientation transitions \nThermomagnetic measurements we re performed on loose powders in a weak (1–10 mT) magnetic \nfield with cycling from room temperature up to 600 °C. An example of magnetization as a function of \ntemperature for W-type BaCo xZn2-xFe16O27 ferrite with x  = 0.8 is shown in figure 5(a). Heating and \ncooling segments do not match (typical hysteretic be haviour), and the first heating differs from others \nbecause of particular magnetization history. Magne tic susceptibility is higher in the easy plane state \nthan in the easy axis one, therefore both spin reorientation (at ~ 55 °C) and ferrimagnetic-\nparamagnetic (at ~ 430 °C) transitions are well reso lved. Small amplitude of the anomalies observed \nabove W-ferrite Curie point confirms that frac tions of magnetic secondary phases are negligible. \n \n2nd cycle1st cycle← Curie pointSRT →\nheating →← cooling\n0 100 200 300 400 500\nTemperature [°C]Magnetization [arb. unit]\n × 10–4\nχ′\nχ″\n← SRT →× 10–6\n2.533.544.555.566.5\n-250 -200 -150 -100 -50 0 50\nTemperature [°C]Susceptibility (Re) [m3·kg–1]\n01234567\nSusceptibility (Im) [m3·kg–1]\n (a) (b) \nFigure 5. Magnetic transitions in W-type BaCo xZn2–xFe16O27 ferrites: (a) magnetization measurements \nfor x = 0.8 (higher temperatures); (b) AC susceptibility measurements for x  = 0.7 (lower temperatures). \n  \n \n \n  \n \nOn the first heating, a sharp drop of magnetization related to SRT is preceded by a distinct peak. \nThe spin reorientation temperature of polycrystallin e powder is also characterized by inflection point \nof magnetization dependence [15]. Another reference can be the point where subsequent cooling and \nheating curves converge (clearly seen in figure 5(a) on the left of magnetization hump). Unlike [6], the \nlatent heat of SRT has not been reliably detected  by differential scanning calorimetry (DSC), while \nCurie point is well visible. \nAC magnetic susceptibility as a function of temperature for W-type BaCo\nxZn2-xFe16O27 ferrite with \nx = 0.7 is shown in Fig. 5(b). The real part of susceptibility has a maximum at ~ 6 °C followed by a \nsharp drop attributed to SRT. Its imaginary part has a minimum at ~ 27 °C which can also be taken as \nthe spin reorientation temperature. Moreover, the mi nimum of imaginary part turns out to approach the \ninflection point of the real part. \n4.  Conclusions \nSingle-phase W-type ferrite submicron powders with different magnetocrystalline anisotropy \nconfigurations are synthesized through a sol-gel ci trate precursor route. The effect of calcination \ntemperature and other production parameters on th e phase composition, average particle size and \nmagnetic properties of the powders is established. Chemical compositions of  substituted hexaferrites \nundergoing spin reorientation transitions near room temperature are determined. \nAcknowledgements \nThis work is supported by EC Seventh Framework Programme fundi ng (project SSEEC, contract FP7-\nNMP-214864). The authors are grateful to P. Au debert (ENS Cachan) for assistance in sol-gel \ntechnique and G. Wang (ICMPE) for TEM observations. \nReferences \n[1] Smit J and Wijn H P J 1959 Ferrites  (Eindhoven: Philips Technical Library) \n[2] Asti G, Bolzoni F, Licci F and Canali M 1978 IEEE Trans. Magn.  14 883–5 \n[3] Albanese G, Calabrese E, Deriu A and Licci F 1986 Hyperfine Interact.  28 487–9 \n[4] Paoluzi A, Licci F, Moze O, Turilli G, Deriu A, Albanese G and Calabrese E 1988 J. Appl. \nPhys.  63 5074–80 \n[5] Samaras D, Collomb A, Hadjivasiliou S, Achille os C, Tsoukalas J, Pannetier J and Rodriguez J \n1989 J. Magn. Magn. Mater. 79 193–201 \n[6] Naiden E P, Maltsev V I and Ryabtsev G I 1990 Phys. Status Solidi A 120 209–20 \n[7] Naiden E P and Ryabtsev G I 1990 Russ. Phys. J.  33 318–21 \n[8] Sürig C, Hempel K A, Müller R and Görnert P 1995 J. Magn. Magn. Mater.  150 270–6 \n[9] Turilli G and Asti G 1996 J. Magn. Magn. Mater.  157–158  371–2 \n[10] Naiden E P and Zhilyakov S M 1997 Russ. Phys. J.  40 869–74 \n[11] Licci F and Besagni T 1984 IEEE Trans. Magn.  20 1639–41 \n[12] Srivastava A, Singh P and Gupta M P 1987 J. Mater. Sci.  22 1489–94 \n[13] Zhang H, Liu Z, Yao X, Zhang L and Wu M 2003 J. Sol-Gel Sci. Technol.  27 277–85 \n[14] Lutterotti L 2000 Acta Crystallogr.  A 56 s54 \n[15] Boltich E B, Pedziwiatr A T and Wallace W E 1987 J. Magn. Magn. Mater.  66 317–22 "    },    {        "title": "0811.3677v1.Flexomagnetoelectric_effect_in_bismuth_ferrite.pdf",        "content": "Flexomagnetoelectric effect in bismuth ferrite \n \nA.K. Zvezdin1, A.P. Pyatakov1,2 \n \n1) A.M. Prokhorov General Physics Ins titute, 38, Vavilova st., Russia, 119991 \n2) M.V. Lomonosov Moscow State University, Leninskie gory, MSU, Russia, 119992 \n \nThere is a profound analogy between inhomogeneou s magnetoelectric effect in multiferroics and \nflexoelectric effect in liquid crystals . This similarity gives rise to the flexomagnetoelectric  polarization induced by \nspin modulation. The theoretical estimations of flexomagnetoelectric polarization agree with the value of jumps of \nmagnetoelectric dependences (~20 μC/m2) observed at spin cycloid suppression  at critical magnetic field 200kOe.  \n \nIntroduction \nThe last few years marked the great progre ss in the field of magnetoelectric and \nmultiferroics materials [1-3]. The interest to th em was triggered by the discovery of so-called \nspiral multiferroics in which polarization was in duced by spin modulation [4-8] and the reports \non inverse effects of electrically induced spin modulation [9-12].  \nThese magnetoelectric phenomena can be  simply explained by inhomogeneous \nmagnetoelectric interaction proposed in 1980-ies [13] that stems fr om relativistic exchange of \nDzyaloshinskii-Moriya type. However the microsc opic mechanism of this coupling is still not \nclear, and the models proposed recently [14, 15] are being questioned. The supporters of \nnonrelativistic scenario based on Heisenberg ex change interaction [16] doubt whether weak \nrelativistic coupling is relevant to the electric polarization observed in multiferroics. \nIn this context the fact of existence of spatially modulated  spin structure in bismuth \nferrite BiFeO 3 gains particular importance. The long range  spin cycloid with the period 62nm is \nknown to be induced by spontaneous  electric polarization due to the relativistic inhomogeneous \nmagnetoelectric interaction [17] . On the other hand spin modul ation may induce an additional \nelectric polarization ΔP due to the same relativistic mechanism [13].  \n The comprehensive view of magnetoelectric in teraction in bismuth ferrite is not only of \nfundamental but also of pr actical importance as BiFeO 3 is the most promising material for \npractical application (see reviews [3] and [18] and reference therei n). It has record high electric \npolarization [19,20], and room temperature multifer roic properties. The electric field induced \nmagnetization switching was implemented in BiFeO 3/CoFe exchanged coupled structure [21]. \nThere also has been some pr ogress in integrating BiFeO 3 with silicon, desirable for Si-CMOS \n(complementary metal oxide semiconductor) room temperature electronics applications [3]. \nIn this paper the relation be tween electric polarization and spin modulation in bismuth \nferrite is analyzed: to what extent the electric pol arization is intrinsic and to what extent it is \ninduced by spin modulation. The anomalies of ferroel ectric properties observ ed in [17] near the \nmagnetic field induced phase transition are reexamin ed. It is shown that the jump of polarization \nat phase transition from spin modulated to homoge neous state is the mani festation of additional \nelectric polarization ΔP due to the relativistic mechanism. The profound analogy between \ninhomogeneous magnetoelectric effect in multiferroic s and flexoelectric effect in liquid crystals \nis emphasized.   \n \nSpatially modulated spin structure  \nTo find out the role of magneto electric interaction in the orig ins of spin cycloid structure \nlet us examine the contributions to the ther modynamic potential relevant to the magnetic \nstructure. The total expression for th e free-energy density has the form  \nan L exch F F FF + + = ,      ( 1 )  \nwhere     () () () ( ) ∑\n=∇ + ∇ = ∇ =\nzyxii exch A l A F\n,,2 2 2 2sin ϕθ θ    (2) \nis the exchange energy, l is the unit antiferromagnetic (AFM) vector, A the constant of \ninhomogeneous exchange (or exchange stiffness), and  θ and φ are the polar and azimuthal angles \nof the unit antiferromagnetic vector ( )θ ϕ θ ϕ θ cos, sin sin, cos sin=l  in the spherical coordinate \nsystem with the polar axis aligned with the principal axis c, where the second term corresponds \nto the inhomogeneous magnetoelectric effect in the form of Lifs hitz-like invariant: \n( ) L s x xz y yz z xx z yy F Pl l l l l l l lγ=⋅ ∇ +∇ − ∇ − ∇ .   (3) \nwhere γ is a constant of inhomogeneous magnetoelectric effect, P s is spontaneous polarization, \n∇ is the differential operator.   \n and, finally, the third term:  \nθ2cosu an K F −=       ( 4 )  \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 anisotropy gives for the functions (, ,)xyz θ  and (, ,)xyz ϕ   \n0 ()y\nxqconst arctgqϕ== ; yqxqy x+ =0θ ;   (5) \nwhere q is the wave vector of the cycloid. Equation (5) 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 [22,23]:  \nθθ 2cos 1mmAK\ndxdu−⋅=       (6a) \n()\nuKmAmK⋅=14λ ;       ( 6 b )  \nwhere ()∫−=2\n021cos 1π\nθθ\nmdmK  is an elliptical integr al of the first kind, and m the modulus \nparameter of the elliptical integral that is f ound by minimization procedur e of the free-energy (1) \n[23]. For an anisotropy constant mu ch smaller than the exchange energy 2\nuKA q<< , the \nmodulus parameter m tends to zero, and solution (6a) becomes harmonic with a linear \ndependence of θ on coordinates (5). By substituting (5) into (1), one can obtain the volume-\naveraged free-energy density for a harmonic cycloid approximation, as \n()22 u\nsKqP Aq F − − = γ .       ( 7 )  \nThe wave-vector corresponding to the energy minimum is then \nAPqs\n22\n0⋅= =γ\nλπ.        ( 8 )  \nThus inhomogeneous magnetoelectric term (3) gives rise to the spin modulation. The period (6 \nb) in unperturbed state of cycloi d (e.g. in zero external field) can be estimated by formula (8). Magnetic field-induced cycloid transformation \n Application of high external magnetic fields will result in changes of the effective \nanisotropy )(H Keff  and may disturb or even  suppress spin cycloid. \nH||c axis geometry  \nConsider the case of  H||c -axis. It is convenient to use dimensi onless units of magnetic field:  \n2\n02AqHh⊥=χ;       ( 9 )  \nwhere ⊥χthe magnetic susceptibility  in the direc tion perpendicular to the antiferromagnetic \nvector  l, q0 (8) the value of the wavev ector of the cycloid correspond ing to the minimum of the \nfree energy (1) in the absence of extern al fields and negl ecting anisotropy.  \nThe free energy density can be conv eniently written as the sum  \nan L exch f f ff + + = ,      ( 1 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     ( 1 1 )  \ndxd\nq AqFfL\nLθ⋅ −= =\n02\n02      ( 1 2 )  \n() θ2coshk fan−= ;        ( 1 3 )  \nwhere () () β− − = =2\n2\n0h k\nAqK\nhkueff is the dimensionless effective anisotropy constant, which takes \ninto account the effect of magnetic field  h, and weak ferromagnetism induced by spontaneous  \nelectric polarization: [] l P M × =sα0   that is expressed by the parameter2\n02\n0\n2AqM\n⊥=χβ . For the \nhomogeneous state with an  antiferromagnetic vector cl⊥ (2πθ=) we have   \n0=⊥f .        ( 1 4 )  \nIn the spin cycloid state, we can obtain the expression for the total free energy (10) \naveraged over the period ()∫=2\n04π\nλθθθλdddxf f  by taking into account (6a):  \n() ()\n()()\nmhk\nK mKmK\nmhkf\n1 1221π\nλ−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− −= ;   (15) \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.  \n  \nFig 1. The transformation of the cycloid profile Lx(x) in the high magnetic field: a) h||c b) h ⊥c. \nTo each field strength h there corresponds a modulus m of the elliptic integral for which \nthe energy is minimum. Physically, this means that the cycloid’s profile changes under applied \nfield (fig 1 a). Under st rong field, its shape diffe rs significantly from that  of a harmonic profile, \nbecoming similar to a function describing a peri odic structure of doma ins separated by walls \n(solitons) whose widths are considerably smaller than the domain width. It follows from (14) \nthat, upon the transformation to the phase with the antiferromagnetic vector l ⊥c, the energy of \nthe domain walls λfchanges sign and the spat ially modulated spin stat e becomes energetically \nunfavorable. \nH⊥ c axis geometry  \nIn magnetic field H⊥ c in free energy (10) the additional term appears that corresponds to \nZeeman energy in magnetic field:  \nθ βθsin 2sin\n2\n00hAqHMfZeeman −= −=        ( 1 6 )  \n \nThe expressions (6 a) for the spin dist ribution in cycloid is also modified:  \n θ θθsin~cos 12m mmAK\ndxdu− −⋅= ,        ( 1 7 )  \nwhere )(~\nhkhmm⋅=β is the parameter that characterizes asymmetry of the spin cycloid: the \nin-plane directions of antiferromagnetic vector are not equivalent any more. The direction of l \nthat corresponds magnetoelectr ically induced magnetization [] l P M × =sα0  oriented parallel to \nthe external magnetic field is energetically mo re preferable than th e one with magnetization \noriented antiparallel to the field.  \nFormulas for the period (6b), and for the ex change (11), inhomogeneous magnetoelectric \ninteraction (12) and anisotropy (1 3) contributions as well as aver aged total energy remain valid \nprovided that the elliptic  integrals are replaced with the following ones:  \n2\n1\n2 01\n41c o s 2 s i n()xdK\nmmhkhπθ\nθ βθ=\n−−��      ( 1 8 a )  \n2\n2\n2\n011c o s 2 s i n4( )xmKm h dkhπ\nθ βθ θ =− − ⋅∫,     (18b) \nand the energy of homogeneous state (14) :  \nh f β2−=⊥        ( 1 9 )  \nIn high field h ⊥c the shape of the cycloid differs si gnificantly from that of a harmonic \nprofile. Unlike the case of h||c (fig 1a) the domains are not equal: the ones parallel to the external \nmagnetic field shrink while those ones that have the magnetization antiparallel to the external \nfield inflate (fig 1 b).  \na) b)Flexomagnetoelectric effect \nThere is a profound analogy between spa tially modulated structures in a \nferroelectromagnet and waves of the director vector a nematic liquid crystal [13,24,25]. This \ncorrelation formally manifests itself in the sim ilarity of the expressi on for the inhomogeneous \nmagnetoelectric interaction (3) in a multiferroic a nd the one for the flexoelectric effect in liqiud \ncrystals:  \n()( )Flexo electricF γ− =⋅ ∇ − ⋅ ∇ ⎡ ⎤⎣ ⎦En n n n     (20) \nIt can be easily shown that ( 20) is isotropic form of (3) pr ovided that the director vector n \nstands for antiferromagnetic one  l and external electric field E stands for spontaneous electric \npolarization P s. This is gives us the grounds to name the energy term (3) as flexomagnetoelectric \ninteraction and additional polarization ∆P induced by spin modulation as flexomagnetoelectric \none: \ndxd\nEFPL θγκ=∂∂−=Δ        (21) \nwhere κis electric susceptibility of the material: E Pκ= , E is electric field.  \nFor the averaged over the period flexomagne toelectric polarization we obtain simple \nexpression: \nγκλπθθ λπ2)(12\n0= Δ =Δ ∫dddxxP P      ( 2 2 )  \nwhere λ is a period of the cycloid determined by (6b).  \n In figure 2 the numerically calculated wave length (6b) and electric polarization (22) in \ndimensionless units are given.     \n0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,00,91,01,11,21,31,41,5λ\nh\n   0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,00,00,20,40,60,81,0Pfme\nh\n \na )       b )  \nFig 2 a) the magnetic field dependence of the cycloid period normalized on its zero field value λ0=62nm b) the \ndependence of normalized electric polarization near pha se transition to homogeneous state calculated from the \nmagnetic field dependence of cycloid period.  Black line is for h||c axis, red is for h ⊥c axis \n \nFrom (8) and (22) we can estimate the fl exomagnetoelectric polarization that should \nresult in anomaly in magnetic field dependence of electric polarization: \nsPAqP22κ=Δ ;       ( 2 2 )  \nTaking into account P s~1C/m2 (3 105 CGS) [19,20], 7103−⋅=A erg/cm, 6\n010=q cm-1, \n314≈− =πεκ  we obtain for ∆P~2 10-5 C/m2 (6 CGS) that is very close to the value of \npolarization jump observed in magnetoelectric dependence ∆Pc(Hc) near the field h=1.9 \n(H~200kOe) of phase transition from spin  modulated to homogeneous state [17].  \n References \n \n1. Manfred Fiebig, “Revival of the magneto electric effect”, J. Phys. D: Appl. Phys. 38, R123–R152 (2005).  \n2 W. Eerenstein, N. D. Mathur & J. F. Scott, Multiferroic and magnetoelectric materials, Nature 442, 759 (2006).  \n3 R. Ramesh, N. A. Spaldin, Multiferroics: progress  and prospects in thin films, Nature Materials 6, 21 (2007).  \n4. Kimura T., T. Goto, H. Shintani, K. Ishizaka, T. Arima, Y. Tokura, Nature 426, 55 (2003). \n5. A. M. Kadomtseva, Yu.F. Popov, G.P. Vorob’ev, K. I. Kamilov, A. P. Pyatakov, V. Yu. Ivanov, A. A. Mukhin, \nA. M. Balbashov, Specifity of ma gnetoelectric effects in new GdMnO\n3, magnetic ferroelectric, JETP Letters 81, 19 \n(2005).  \n6. Yoshinori Tokura, Multiferroics as Quantum Electromagnets, Science 312 , 1481 (2006).  \n7. S.-W. Cheong, M. Mostovoy, Mu ltiferroics: a magnetic twist for Ferroelectricity, Nature Materials 6, 13 (2007).  \n8. Shintaro Ishiwata, Yasujiro Taguchi, Hiroshi Mura kawa, Yoshinori Onose, Yoshinori Tokura, Low-Magnetic-\nField Control of Electric Polarizati on Vector in a Helimagnet, Science 319, 1643 (2008). \n9. E.V. Milov, A.M. Kadomtseva, G.P. Vorob’ev, Yu.F. Popov, V.Yu. Ivanov, A.A. Mukhin and A. M. Balbashov, \nSwitching of spontaneous electric polarization in the DyMnO\n3 multiferroic, JETP Letters, 85, 503 (2007).  \n10. Y. Yamasaki, H. Sagayama, T. Goto, M. Matsuura, K. Hi rota, T. Arima, and Y. Tokur a, Electric Control of Spin \nHelicity in a Magnetic Ferroelectric, PRL, 98, 147204 (2007). \n11. A.S. Logginov, G.A. Meshkov, A.V. Nikolaev, A.P. Pyatakov, Magnetoelectric Control of Domain Walls in a \nFerrite Garnet Film, JETP Letters, 86, 115 (2007). \n12. A.S. Logginov, G.A. Meshkov, A.V. Nikolaev, E.P. Nikolaeva, A.P. Pyatakov, A.K. Zvezdin Room temperature \nmagnetoelectric control of micromagnetic struct ure in iron garnet films, Appl. Phys. Lett.   93, 182510 (2008) \n13. V.G. Bar'yakhtar, V.A. L'vov, and Yablonskiy, D.A. Inhomogeneous magneto-electric effect. JETP Lett. 37, 673 \n(1983). \n14. Hosho Katsura, Naoto Nagaosa, and Alexander V. Balatsky,  Spin Current and Magnetoelectric Effect in \nNoncollinear Magnets, PRL 95, 057205 (2005). \n15. I. A. Sergienko and E. Dagotto, Role of the Dzyaloshin skii-Moriya interaction in mu ltiferroic perovskites, Phys. \nRev. B 73, 094434 (2006). \n16. A.S. Moskvin, S.-L. Drechsler, Microscopic mechanisms of spin-dependent electric polarization in 3d oxides, Phys. Rev. B 78, 024102 (2008). \n17. Yu. F. Popov, A.M. Kadomtseva, G.P. Vorob’ev, A.K.  Zvezdin, Discovery of the li near magnetoelectric effect \nin magnetic ferroelectric BiFeO\n3 in a strong magnetic field, Ferroelectrics  162,  p.135 (1994). \n18. A. M. Kadomtseva, Yu.F. Popov , A.P. Pyatakov, G.P. Vorob’ev, А.К. Zvezdin, and D. Viehland, Phase \ntransitions in multiferroic BiFeO3 crys tals, thin-layers, and ceramics: Enduring potential for a single phase, room-\ntemperature magnetoelectric ‘holy grail’, Phase Transitions   79,1019 (2006). \n19. Kwi Young Yun, Dan Ricinschi, Takeshi Kanashima, Minoru Noda and Masanori Okuyama, Giant \nFerroelectric Polarization Beyond 150 μC/cm2 in BiFeO 3 Thin Film, The Japan Society of Applied Physics 43, No. \n5A, L 647–L 648 (2004). \n20. D. Lebeugle, D. Colson, A. Forget, M. Viret, P. Bo nville, J. F. Marucco, and S. Fusil, Room-temperature \ncoexistence of large electric polari zation and magnetic order in BiFeO 3 single crystals, Phys. Rev. B 76, 024116 \n(2007). \n21. Ying-Hao Chu, L. W. Martin, M. B. Holcomb, M. Gajek, Shu-Jen Han, Qing He, N. Balke, Chan-Ho Yang, D. \nLee, Wei Hu, Qian Zhan, Pei-Ling Yang, A. Fraile-Rodrígu ez, A. Scholl, Sh. X. Wang , R. Ramesh, Electric-field \ncontrol of local ferromagnetism using a magnetoelectric multiferroic, Nature Materials 7, 478 (2008). \n22. M.M. Tehranchi, N.F. Kubrakov, A.K. Zvezdin, Spin-flop and incommensurate structures in magnetic \nferroelectrics, Ferroelectrics 204, 181 (1997). \n23. A. G. Zhdanov, A. K. Zvezdin, A. P. Pyatakov, T. B.  Kosykh, and D. Viehland, Th e influence of electric field \non magnetic incommensurate-commensurate phase transitions in multiferroic like BiFeO 3, Phys.Solid State 48, 88 \n(2006).  24. A. Sparavigna, A. Stri gazzi, A.K. Zvezdin, Electric -field effects on the spin-density wave in magnetic \nferroelectrics, Phys. Rev. B 50, 2953 (1994).  \n25. I. Dzyaloshinskii, Magnetoelectricity in ferromagnets EPL, 83, 67001 (2008). "    },    {        "title": "1204.5844v1.Peculiarities_of_timing_and_spectral_diagrams_of_magnetic_video_pulse_excitation_influence_on_NMR_spin_echo_in_magnets.pdf",        "content": "Peculiarities of t iming and spectral diagrams of magnetic video-pulse excitation  \ninfluence on NMR spin-echo in magnets\nG.I. Mamniashvilia, T.O. Gegechkoria, A.M. Akhalkatsib, T.A. Gavashelib,\nE.R. Kuteliac, L.G. Rukhadzec, D.I. Gventsadzec\naAndronikashvili Institute of Physics of I.Javakhishvili Tbilisi State University,\n  6 Tamarashvili str., 0177. Tbilisi, Georgia.\nbI.Javakhishvili Tbilisi State University, 3 Chavchavadze av. 0128. Tbilisi, Georgia.\ncGeorgian Technical University, 69, Kostava str. Tbilisi, 0175, Georgia\nzviadadzemichael@yahoo.com\nAbstract\nWe present the first systematic study  of timing  and spectral diagrams  of magnetic  \nvideo-pulse influence on the NMR two-pulse echo in a number of magnets (ferromagnets,  \nferrites, half metals, intermetals). It is shown that the timing diagrams showing the dependence  \nof two-pulse echo intensity on the temporal location of a magnetic video-pulse in respect to  \nradio-frequency pulses and the spectral diagrams of this influence are defined mainly by the  \nlocal hyperfine field anisotropy and domain walls mobility. These diagrams could be used for  \nthe identification of the nature of NMR spectra in multidomain magnetic materials and to  \nimprove the resolution capacity of the NMR method in magnets.\n1. Introduction\nThe possibilities  of  using  different  methods  of  nuclear  spin-echo  spectrometry  for  \nstudying properties of magnetically ordered substances were analyzed in a large number of  \nworks [1]. One of such widely employed methods is based on the introduction of additional  \npulses of a dc magnetic field into the system of exciting radio-frequency (RF) pulses; these  \npulses were called magnetic video-pulses (MVPs), since they lack the filling frequency. Thus,  \nin [2-4] the MVPs were used to  investigate the properties  of domain  walls (DW) in the  \neuropium iron garnet Eu 3Fe5O12, in ferrites with a spinel structure, thin magnetic films  and  Y 2-\nxGdxCo17 compounds with the substitution of Gd for Y ions. In these works, the different role  \nof MVPs influence during their symmetrical and asymmetrical position with respect to the  \nsecond RF pulse in the two-pulse echo (TPE) procedure was revealed. These differences make  \nit  possible  to  find  the  coercive-force-related  distribution  of  DW  upon  the  symmetrical  arrangement of MVPs, and the anisotropy of the hyperfine field (HF) at the nuclei in the case  \nof  its  asymmetrical  arrangement.  In  particular,  the  detection  [3]  of  the  inhomogeneous  \ninfluence  of  MVPs  on  different  sections  of  the  frequency  spectrum  of  59Co  NMR  in \n(Y0.9Gd0.1)2Co17 made it possible to reveal those crystallographic positions that prove to be  \npreferable upon the substitution of Gd for Y. \nPreviously, such a procedure was used [2] for determining the magnetic-field strength  \nwhich shifts a DW by a distance equal to its thickness. The scheme of the experiment is shown  \nin Fig. 1, borrowed from [4].\nThe authors of [4] investigated the influence of MVPs on the intensity of TPE. They  \nshowed that due to comparatively small hyperfine field anisotropy on 57Fe in lithium ferrite the  \nmaximum suppression effect is achieved when the MVP coincides with one of the RF pulses.  \nThis is caused by the fact that the action of a MVP on the multidomain ferromagnetic material  \nis in essence reduced to a reversible (in weak magnetic fields) displacement of DW. A MVP  \nsymmetric in respect to the second RF pulse has the effect that both the first and second RF  \npulses  excite  nuclei  whose  positions  within  the  DW  change.  These  positions  define  the  \nresonance frequency as well as the factor of enhancement of the RF field η inside DW [1]. If \nthe HF field inhomogeneity is small and the excitation pulses are sufficiently short then the  \nchange  of  resonance  frequency  can  be  neglected.  In  this  case  the  change  in  the  RF  \nenhancement factor η reduces the echo intensity [2], as it is seen for lithium ferrite, Fig.  1. The \napplication of a MVP asymmetrically in the interval between the RF pulses can significantly  \ninfluence the intensity of TPE only in the case of anisotropic HF interactions [2].\nFig. 1. Timing diagrams of the relative intensity I/Imax dependence of the two-pulse echo ( •) on \nthe temporal location of a MVP of H d=5 Oe, for 57Fe NMR in lithium ferrite at: τ1 = τ2 = 0.8 \nµs,  Δτ = 21  µs,  τd  = 3  µs, fNMR=74.0 MHz, Hd=0.  τ1, τ2, Δτ,  τd are rf pulse durations, time  \ninterval between them and magnetic pulse duration, correspondingly.2So, the great importance is such characteristics of magnetic materials as the mobility of  \nDW and HF based anisotropy, which are different in magnetically soft cubic lithium ferrite and  \nmagnetically harder material as uniaxial cobalt.\n2. Experimental results and their discussion\nTo study the above noted properties of the magnetic materials in more detail, we carried  \nout experimental concerning the effect of a MVP with an amplitude of the magnetic field up to  \nHd=500 Oe and durations equal to several microseconds on the signals of TPE in several  \nmagnetic substances, namely,  polycrystalline cobalt-thin magnetic films (TMF), half metal  \nCo2MnSi, Co-Cu and MnSb ferromagnetic alloys. The half metals are regarded as promising  \nmaterials for spintronics [5]. The NMR spectrometer and MVP excitation technique, as well as  \nthe procedures of sample preparations are described in [4, 6-9].\nAs it follows from Figs. 1- 7, the essential difference is seen for MVP action on the TPE  \nin different magnets. The dependences of TPE intensities at application of a MVP for  59Co \necho signal in Co TMF (Fig. 2),  59Co echo in Co-Cu (Fig.  3) and Co2MnSi (Fig. 4,5a, 6a) \ndiffers considerably from closer-to-each-other dependences for  57Fe echo in lithium ferrite  \n(Fig. 1) and 55Mn echo in Co2MnSi (Fig.5b, 5c, 6b) and MnSb (Fig.7).\nFig. 2. Timing diagrams of the intensity dependence of the two-pulse echo on the temporal  \nlocation of a MVP with duration τd and amplitude Hd in\n(a) cobalt thin magnetic films at: τ1 = τ2 = 1.5 µs, Δτ = 9 µs, τd = 3 µs, Hd=10 Oe, fNMR=216.5 \nMHz, Io – echo amplitude at H d=0.\n(b) NMR spectrum of Co film ( •) and frequency spectra diagrams for MVP influence for  \nsymmetric (\n) and asymmetric (\n) application at: τ1 = 1.3 µs, τ2 = 1.5 µs, Δτ = 9 µs, τd = 3 µs, \nHd=10 Oe.3Fig. 3. NMR spectra (•) for Co (a) and Co-Cu alloys (b,c) for comparisons (b – 5 % Cu, c –  \n10 %) and spectral diagrams for MVP influence in case of symmetric  (\n ) and asymmetric (\n) \nMVP application with amplitude H d=350 Oe.\nFig. 4 Two-pulse echo intensity amplitude dependence on magnetic video-pulse amplitude (\n) \nand asymmetric (\n) influence in polycrystalline cobalt at: τ1 = 1.3 µs, τ2 = 1.5 µs, Δτ = 9 µs, τd \n= 3 µs, fNMR=218 MHz (1) and cobalt thin magnetic films at: τ1 = 1.3 µs, τ2 = 1.5 µs, Δτ = 9 µs, \nτd= 3 µs, fNMR=218 MHz (2).\nFig. 5. Timing diagrams of the intensity dependence of the two-pulse echo on the temporal  \nlocation of a MVP with duration τd and amplitude Hd in Co2MnSi\n(a) for 59Co NMR at: τ1 =1.1 µs, τ2 =1.4 µs, Δτ=10 µs, τd=2 µs, fNMR=145.5 MHz, Hd=550 Oe;\n(b) for 55Mn NMR at: τ1=τ2=3 µs, Δτ = 7 µs, τd = 2 µs, Hd=300 Oe, fNMR=354 MHz, Io – echo \namplitude at Hd=0.\n(c) amplitude diagrams of the TPE intensity dependence on the MVP value with duration τd \nand amplitude Hd in Co2MnSi for symmetric (\n ) and asymmetric (\n) influence for:\n1) 59Со NMR spin echo at: τ1 = τ2 = 2 µs, Δτ = 10 µs, τd = 3 µs, fNMR=145 MHz. \n2) 55Mn NMR spin echo at: τ1 = 0.8 µs, τ2 = 0.9 ��s, Δτ = 8 µs, τd = 1.6 µs, fNMR=353 MHz, Io – \necho amplitude at Hd=0.4Fig. 6. Frequency dependence of the effect of the symmetric (\n) and asymmetric (\n) MVP on \nthe two-pulse echoes intensities in Co 2MnSi for 59Co NMR (a) and 55Mn NMR (b) at:\n(a) τ1 = 1.1 µs, τ2 = 1.4 µs, Δτ = 10 µs, τd = 2 µs, Hd=550 Oe;\n(b) τ1 = 0.8 µs, τ2 = 1 µs, Δτ = 13 µs, τd = 4 µs, Hd=190 Oe;  Io – echo amplitude at H d=0.\nFig. 7 (a) Timing diagrams of the intensity dependence of the two-pulse echo on the temporal  \nlocation of a MVP of H d, lasting for a time interval τd in MnSb at: τ1 = 1.2 µs, τ2 = 1.6 µs, Δτ = \n17 µs, τd = 2 µs, Hd=280 Oe, fNMR=257 MHz. Io – echo amplitude at H d=0.\nNMR spectrum of MnSb ( •) and spectral diagrams of MVP influence for symmetric (\n ) and \nasymmetric (\n)) application for (b) Mn0.5Sb0.5 at: τ1 = 1.2 µs, τ2 = 1.5 µs, Δτ = 10 µs, τd= 1.6 \nµs, Hd=280 Oe and (c) for Mn0.52Sb0.48 at:  τ1  = 0.4 µs, τ2  = 0.5 µs, Δτ = 10 µs, τd  = 1.6 µs, \nHd=90 Oe.\nThe influence of MVPs on echo intensity along the NMR frequency spectra was firstly  \nstudied in [2,3]. \nIn Fig. 2 we present the timing and spectral diagrams of MVP influence on the  59Co \nTPE in TMF cobalt samples. The MVP spectral influence on 59Co in Co1-xCux system (x=0.5 \n%, 10 %) is given in Fig.  3. Corresponding MVP influence dependences for TPE in Co 2MnSi \nand MnSb samples are presented in Figs 5-7. Besides it, in Fig.  4 and Fig. 5c it is presented the  \namplitude dependences of TPE intensities for 59Co and 55Mn spin echoes on the amplitude of  \nMVP Hd in Co and TMF, and Co 2MnSi, correspondingly.5Note the difference in shape of the timing  and order of spectral diagrams of MVP  \ninfluence for the 59Co and 55Mn nuclei in Co2MnSi (Fig. 5,6), correspondingly. This apparently  \nreflects  difference  in  the  anisotropies  of  HF  fields  for  these  nuclei  what  affects  the  \ndependences of the intensity of the TPE on the time of application of MVPs: the symmetric  \nMVPs which coincide in time with the RF pulses comparatively less suppress the signals of  \necho by  59Co nuclei and comparatively more reduce TPE of  55Mn nuclei as compared with  \nasymmetric MVPs. So, the type of timing or spectral diagrams is defined mainly by the HF  \nfield anisotropy of corresponding nuclei which is small for 57Fe and 55Mn as compared for that  \nof 59Co positions. This rule holds for both magnetically soft and hard samples, Fig.  4. This also \nis reflected in the reversed order of spectral diagrams of MVP action for two type positions.\nThe reason for this it could be understood also from Fig.  5c, where the reduced echo  \nintensity dependences on the MVP amplitude are shown for  55Mn and  59Co positions in the  \nsame Co2MnSi sample. It is seen that for these positions amplitude dependences of MVP  \ninfluence for asymmetric action of MVP differ much stronger for both nuclear types than ones  \nfor symmetric MVP influences.\nSo, the timing diagrams type is defined mainly by the anisotropic part of HF interaction  \nwhile the degree of suppression of echo signals by MVP strongly depends on the DW mobility.\nLet us note also that timing and spectral diagrams in case of half metallic Co 2MnSi give \na visual picture describing different HF field anisotropies at the 59Co and 55Mn sites. Note also  \nthat the large difference for DW mobility does not change the type of timing and frequency  \ndiagrams.\nThe shape of frequency diagrams of MVP influence shows the inhomogeneous degree  \nof  MVP  influence  (both  symmetric  and  asymmetric)  through  a  spectrum.  The  frequency  \ndiagram  of  MVP  influence  for  asymmetric  action  is  arranged  below  the  one  for  the  \ncorresponding symmetric MVP action diagram for anisotropic sites and vice versa for isotropic  \nones.\nThe analysis of these diagrams shows that in correspondence with [2] they could play  \nrole of additional characteristics of the magnetic materials, as example, for the characterization  \nof nature of the NMR spectra in magnets.\nActually, the appearance of the frequency diagram of MVP action in cobalt is close to  \nthe shape of the NMR spectrum. As it is known [10], the NMR spectrum peak at 217 MHz in  \ncobalt corresponds to the nuclei arranged in center of DWs of face-centered cubic (fcc) phase,  \nbut at 220 MHz – to nuclei arranged in stacking faults of crystal lattice.\nFrequency  measurements  of  MVP  influence  in  cobalt-copper  alloy  system  (Fig.  3) \nshowed a ununiform degree of influence of a MVP on the echo signal in different parts of the  659Co NMR spectra in this system. The influence was weakest near 217 MHz corresponding to  \nnuclei located in the DWs centers of fcc phase.\nSo,  the  MVP  influence  technique  allows  one  to  make  a  direct  experimental  \ndetermination of the domain wall center resonances similar to that demonstrated by the NMR  \nspin-echo decay envelope enhanced modulation effect resulting from the application of a small  \nlow-frequency alternating (ac) magnetic field [11]. Accordingly model presented in [11], the  \ndepth of modulation should be minimal for nuclei located at the DW centers and their edges.\nThe two techniques are related because the ac magnetic field influence on the echo  \nsignal  intensity  could  be  mainly  approximated  by  two  MVPs  because  to  the  first  \napproximation, the amplitude of additional ac magnetic field is important in the process of  \necho formation only in those instants when RF pulses also affect the sample [1 2]. But to our \nopinion the MVPs influence technique is more convenient and direct and offers additional  \nopportunities to characterize magnetic materials.\nThese considerations holds for NMR peaks arising from a single site in magnetically  \nordered material containing DWs like Co and its alloys with transition metals.\nThe appearance of frequency diagrams of MVP influence in Co corresponds to this  \nsupposition as the minimal influence is observed for nuclei arranged DWs centers frequencies  \nand maximal – in the range of stacking fault frequencies (~ 220 Hz).\nThe obtained data for MnSb system suggest that it could be also right for this material  \nand observed NMR spectrum and MVP influence frequency diagram s point to NMR spectrum  \nfrom nuclei arranged in DW centers and split by the quadrupolar interaction. The characteristic  \npeculiarity of the nuclear spin echo spectrum in the stoichiometric MnSb composition is the  \npresence of well resolved quadrupolar structure in the 55Mn NMR spectra which is caused by  \nthe specific properties of DW structure in these magnets [9]. In particular, into the all range of  \nDW the magnetization vector is perpendicular to the electric field gradient (EFG). As result,  \nthe  spectral  transitions  frequencies  do  not  depend  on  the  mutual  orientation  of  the  \nmagnetization vector and EFG what stipulates comparatively rare possibility of the observation  \nof the resolved quadrupolar structure in the NMR spectra of polycrystalline magnets. The other  \nspecific peculiarity of the investigated system is the relatively small value of HF anisotropy  \nfacilitating the interpretation of NMR spectra.\nThe  55Mn  NMR  spectrum  of  stoichiometrically  pure  multidomain  polycrystalline  \nferromagnet MnSb is presented by the resonance line in the frequency range of 250-260 MHz  \nsplitted by the quadrupolar interaction on five spectral component with widths ~ 1.6 MHz and  \ndistances between their centers of the order of 2.0 MHz. At observation of the resonance in the  \nDW of ferromagnet due to the rotation of local magnetization the spread of dipolar fields  7results in the broadening and splitting of the inhomogeneously broadened NMR lines. The  \ninfluence of dipolar fields on the resonance spectrum is determined  both by the value of  \ndipolar  shifts  and  rotation  angles  under  the  action  of  exciting  pulses.  Accordingly  the  \nassessment [13] the dipole interaction influence in the investigated system is not so large as to  \ndisturb significantly the quadrupolar structure of the NMR spectra, but it contributes into the  \nwidth  and  shape  of  the  separate  components  of  spectrum  splitted  by  the  quadrupolar  \ninteraction. At the violated stoichiometry of MnSb it appears the HF shifts near the excess ions  \n[14] resulting in the gradual disappearance of the quadrupolar splitting.\nAs the consequence of two these factors is apparently the fact that the MVP action is  \nmost effective in the intervals between quadrupolar maximums of NMR spectra what could  \nstipulate of the observed shape of the spectral diagram of MVP influence.\n3. Conclusion\nIn this work it is carried out the first systematic study of timing and spectral diagrams \nof MVP influence on TPE in a number of magnets with different anisotropy of HF fields and  \nDW mobility. It is shown that these timing and spectral diagrams are defined by local HF field  \nanisotropy and DW mobility and could be used for additional identification of the nature of  \nNMR lines in multidomain magnetic materials and thereby to improve the resolution capacity  \nof the NMR method for magnets.\nAcknowledgements\nThe  authors  thank  Professor  T.N.  Khoperia  for  kindly  providing  us  with  Co  TMF  \nsamples. The work was supported by the Shota Rustaveli National Science Foundation  Short-\nterm Individual Travel Grant  2012_tr_237.\nReferences\n1. M.I. Kurkin and E.A. Turov. NMR in magnetically ordered substances and its application.  \nMoscow.: Nauka, 1990 (244 p).\n2. L.A. Rassvetalov, A.B. Levitski. Influence of a pulsed magnetic field on the nuclear spin  \necho in some ferromagnets and ferrimagnets. Sov. Phys. Solid State, 23, 3354- 3359 (1981).\n3. E. Machowska, S. Nadolski. Echo defocusing in Y 2Co17. Solid State Commun. 68, 215-217  \n(1988).\n4. I.G. Kiliptari, V.I. Tsifrinovich. Single-pulse nuclear spin echo in magnets.  Phys. Rev. B: \nCondens. Matter. 57, 11554-11564 (1998).\n5. J.S. Moodera. Half metallic materials. Phys. Today 54, 39-44 (2001).86.  A.M. Akhalkatsi,  T.O. Gegechkori,  G.I. Mamniashvili,  Z.G.  Shermadini,  A.N. Pogorely, \nO.M. Kuz'mak. Magnetic video-pulse action on the nuclear spin echo in multidomain magnetic  \nmaterials. Phys. Met. Metallogr . 105, 351-355 (2008).\n7.  T.N. Khoperia. Investigation of the substrate activation mechanism and electroless Ni-P  \ncoating ductility and adhesion. Microelectron. Eng.  69, 391-398 (2003).\n8. T.N. Khoperia. Electroless deposition in nanotechnology and ULSI. Microelectron. Eng.  69, \n384-390 (2003). \n9. T.Sh. Abesadze, A.M. Akhalkatsi, I.G. Kiliptari, M.G. Melikiya, T.M. Shavishvili. \nMultiquantum effects in NMR of 55Mn in an Mn1+δSb system. Sov. Phys. JETP 69, 103-106 \n(1989).\n10. H.P. Kunkel, C.W. Searle. Experimental identification of domain-wall-center and domain-\nwall-edge NMR resonances in magnetically ordered materials. Phys. Rev. B: Condens. Matter.  \n23, 65-68 (1981).\n11. C.W. Searle, H.P. Kunkel, S. Kupca, I. Maartense. NMR enhancement of a modulating field  \ndue to the anisotropic component of the hyperfine field in HCP Co and YCo 5. Phys. Rev. B: \nCondens. Matter. 15 , 3305-3308 (1977).\n12. A.M. Аkhalkatsi, G.I. Mamniashvili. On the role of pulse edges in the single-pulse spin  \necho technique. Phys. Met. Metallogr. 81, 632-635 (1996).\n13. I.G. Kiliptari, A.M. Akhalkatsi. Calculation of the dipole field at 55Mn nuclei in MnSb.  \nSov. Phys. Solid State 32, 744-745 (1990).\n14. T.M. Shavishvili, A.M. Akhalkatsi, I.G. Kiliptari. Calculation of hyperfine fields and 55Mn \nNMR spectra in non-stoichiometric Mn 1+δSb. Sov. Phys. Solid State 31, 1382-1387 (1989).9"    },    {        "title": "1606.05782v1.Magnetization_reversal_in_mixed_ferrite_chromite_perovskites_with_non_magnetic_cation_on_the_A_site.pdf",        "content": "arXiv:1606.05782v1  [cond-mat.mtrl-sci]  18 Jun 2016Magnetization reversal in mixed ferrite-chromite perovsk ites with non magnetic\ncation on the A-site\nOrlando V. Billoni,1,∗Fernando Pomiro,2,†Sergio A. Cannas,1,‡\nChristine Martin,3,§Antoine Maignan,3,¶and Raul E. Carbonio2,∗∗\n1Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba and\nInstituto de Física Enrique Gaviola (IFEG-CONICET), Ciuda d Universitaria, 5000 Córdoba, Argentina.\n2INFIQC (CONICET – Universidad Nacional de Córdoba),\nDepartamento de Fisicoquímica, Facultad de Ciencias Quími cas,\nUniversidad Nacional de Córdoba, Haya de la Torre esq. Medin a Allende,\nCiudad Universitaria, X5000HUA Córdoba, Argentina.\n3Laboratoire CRISMAT, UMR 6508 CNRS/ENSICAEN/UCBN,\n6 Boulevard Marechal Juin, 14050 Caen cedex, France.\n(Dated: October 2, 2018)\nIn this work, we have performed Monte Carlo simulations in a c lassical model for RFe 1−xCrxO3\nwith R=Y and Lu, comparing the numerical simulations with ex periments and mean field calcu-\nlations. In the analyzed compounds, the antisymmetric exch ange or Dzyaloshinskii-Moriya (DM)\ninteraction induced a weak ferromagnetism due to a canting o f the antiferromagnetically ordered\nspins. This model is able to reproduce the magnetization rev ersal (MR) observed experimentally\nin a field cooling process for intermediate xvalues and the dependence with xof the critical tem-\nperatures. We also analyzed the conditions for the existenc e of MR in terms of the strength of DM\ninteractions between Fe3+and Cr3+ions with the x values variations.\nPACS numbers: 75.10.Hk,75.40.Mg,75.50.Ee,75.60.Jk\nKeywords: Classical spin models, Perovskites, Magnetizat ion reversal, DM interactions\nINTRODUCTION\nSome magnetic systems when cooled in the presence of\nlow magnetic fields show magnetization reversal (MR).\nAt high temperatures the magnetization points in the\ndirection of the applied field while at a certain tem-\nperature the magnetization reverses, becoming opposite\nto the magnetic field in a low temperature range. In\nparticular, this phenomenon has been observed in or-\nthorhombic (space group: Pbnm ) perosvkites like RMO 3\nwith R=rare earth or yttrium and M=iron, chromium or\nvanadium1–8. These materials exhibit a weak ferromag-\nnetic behavior below the Néel temperature (T N), arising\nfrom a slight canting of the antiferromagnetic backbone.\nThe weak ferromagnetism (WFM) observed in these com-\npounds can be due to two mechanisms related with two\ndifferent magnetic interactions: antisymmetric exchange\nor Dzyaloshinskii-Moriya interaction (DM) and single-ion\nmagnetocrystalline anisotropy9,10. In particular, in or-\nthochromites RCrO 3and orthoferrites RFeO 3the WFM\nis due mainly to DM interactions9.\nMR was also observed in several ferrimagnetic systems\nsuch as spinels11,12, garnets13, among others. In these\nmaterials, MR has been explained by a different temper-\nature dependence of the sublattice magnetization arising\nfrom different crystallographic sites, as predicted by Néel\nfor spinel systems. However, this explanation cannot\nbe applied to the orthorhombic perovskites with formula\nRM1−xM′\nxO3where R is a nonmagnetic ion (for example\nY3+or Lu3+), because the magnetic ions occupy a single\ncrystallographic site. In the case of YVO 3, the origin of\nMR has been explained based on a competition betweenDM interaction and single-ion magnetic anisotropy14.\nSome years ago, the presence of MR was also re-\nported in polycrystalline perovskites with two magnetic\ntransition ions randomly positioned at the B-site and\nnon magnetic R cation at the A site. Some examples\nare BiFe 0.5Mn0.5O3, LaFe 0.5Cr0.5O3, YFe0.5Cr0.5O3and\nLuFe0.5Cr0.5O315–18. In a work by Kadomtseva et al.\n[1] the DM interactions were successfully used to ex-\nplain the anomalous magnetic properties of single-crystal\nYFe1−xCrxO3with different Cr contents. They showed\nthat these compounds are weak ferrimagnets with a\nmixed character of the DM interaction. Moreover, the\ncompeting character of DM interactions is used in a mean\nfield (MF) approximation by Dasari et al. [6] to explain\nthe field cooling curves of polycrystalline YFe 1−xCrxO3\nfor0≤x≤1. In their work the dependence of mag-\nnetization as a function of temperature, for the entire\nrange of composition, is explained from the interplay of\nDM interactions of the Fe–O–Fe, Cr–O–Cr and Cr–O–Fe\nbounds. At intermediate compositions (x=0.4 and 0.5)\nMR is also reported in this work.\nNumerical simulations have been proved to be useful to\nmodel magnetic properties of perovskites. Several stud-\nies of magnetic perovskites have been performed using\nMonte Carlo simulations (MC)19–21, for instance, to char-\nacterize the critical behavior in yttrium orthoferrites19\nand in La 2/3Ca1/3MnO320,21. However, to the best or\nour knowledge, MR has not been studied using MC sim-\nulations. In the case of solid solutions, MC simulations\ncan take into account fluctuations in the distribution of\natomic species and thermal fluctuation that cannot be\nconsidered in mean field models.2\nIn this work we have performed MC simulations using\na classical model for RFe 1−xCrxO3with R = Y or Lu,\ncomparing the numerical simulations with experiments\nand mean field calculations6,22. We also adapted MF ap-\nproximations to test our MC simulations. This model is\nable to reproduce the magnetization reversal (MR) ob-\nserved in a field cooling process for intermediate xvalues\nand the dependence on xof the critical temperature.\nWe also analyzed the conditions for the existence of\nMR in terms of the strength of DM interactions between\nFe3+and Cr3+and the chromium content.\nI. METHODS\nNeutron diffraction studies have shown that the mag-\nnetic structure of RFe 1−xCrx03compounds with a non-\nmagnetic R ion (space group : Pbnm ) isΓ4(Gx,Ay,Fz)\nin the Bertaut notation23. In this structure the mo-\nments are oriented mainly in an AFM type-G arrange-\nment along the x-direction. A nonzero ferromagnetic\ncomponent along the z-axis (canted configuration) and\nan AFM type-A arrangement along the y-axis are allowed\nby symmetry9,24.\nWe model the RFe 1−xCrxO3perosvkites, with R =Lu\nor Y using the following Hamiltonian of classical Heisen-\nberg spins lying in the nodes of a cubic lattice with\nN= (L×L×L)sites,\nH=−1\n2/summationdisplay\n/an}bracketle{ti,j/an}bracketri}ht[Jij/vectorSi·/vectorSj+/vectorDij·(/vectorSi×/vectorSj)] (1)\n−/summationdisplay\niKi(Sx\ni)2−H/summationdisplay\nimiSz\ni,\nwhere/an}bracketle{t.../an}bracketri}htmeans a sum over the nearest neighbor sites\nand/vectorSiare unitary vectors. Jij<0takes into ac-\ncount the superexchange interaction and /vectorDijthe anti-\nsymmetric Dzyalshinskii-Moriya interactions, where this\nvector points in the ˆjdirection. Due to the collective\ntilting of the (Fe,Cr)O 6octahedra the DM interaction\nis staggered. Hcorrespond to the external applied field\nand is expressed as, H=BµFe/kB, whereBis the ex-\nternal field and µFe=gµBSFewithg= 2the gyro-\nmagnetic factor constant, µBthe Bohr magneton and\nSFe= 5/2is the total spin of Fe 3+ion –equivalently\nSCr= 3/2for Cr3+ion. Then, mi= 1 for then\nFe3+ions and mi=SCr/SFe= 0.6for the Cr3+ions.\nBoth interactions, JijandDij, depend on the type of\nions (Fe3+or Cr3+) that occupy sites iandj, so each\npair interaction can take three different values. Suppose\nthat site 1is occupied by Cr3+and site 2by Fe3+ion,\nthen the super-exchange interaction couplings are J22=\n2S2\nFeJFeFe/kB,J12=J21= 2SFeSCrJFeCr/kBand\nJ11= 2S2\nCrJCrCr/kB, whereJαβ– withα,β=Cr or Fe –\nare the exchange integrals. In the case of DM interactions\nD22=S2\nFeDFeFe/kB;−D12=D21=SFeSCrDFeCr/kBandD11=S2\nCrDCrCr/kB. Finally, the single site in-\nteractions corresponding to the uniaxial anisotropy are\nK1=SCrKCr/kB=K2=SFeKFe/kB>0which point\nin thexdirection. kBis the Boltzmann constant. For\nsimplicity we consider the same anisotropy for Cr3+and\nFe3+ions.\nA. Monte Carlo methods\nWe performed Monte Carlo simulations using a\nMetropolis algorithm. Along this work we considered\na cubic lattice with N= 40×40×40sites and open\nboundary conditions. In order to simulate RFe 1−xCrxO3\ncompounds the sites of the cubic lattice are occupied by\nCr3+ions with probability x, and with probability (1−x)\nby the Fe+3ions. Since all the super-exchange interac-\ntions are antiferromagnetic the system can be divided\ninto two sublattices AandB, each one ferromagnetically\nordered in the ˆidirection and opposite to the other. We\ncomputed the sublattice magnetization,\n/vector mα=1\nN/summationdisplay\n/vectorSi∈{/vectorSα}/vectorSi, (2)\nwhere{/vectorSα}withα=A,Bis the set of spins belonging\nto sublattice Aor sublattice B, and the susceptibility,\nχα= (N\nkBT)(/an}bracketle{tm2\nα/an}bracketri}ht−/an}bracketle{tmα/an}bracketri}ht2), (3)\nwhere/an}bracketle{t.../an}bracketri}htmeans a thermal average. At each tempera-\nture we equilibrated the system using 105Monte Carlo\nsteps (MCS). After that we get the thermal averages us-\ning another 105MCS, measuring the quantities (e.g. the\nmagnetization) every 102MCS. From the peak of the sus-\nceptibility we obtained the critical temperature as func-\ntion of the Chromium content for x= 0,0.1,0.2,...,1. As\nwe will explain later, the values of the J11andJ22inter-\nactions were chosen in order to reproduce the Néel tem-\nperature TNof the pure compounds, RCrO 3and RFeO 3,\nrespectively. J12were considered as a free parameter,\nto be fitted from the experiments. The value used for\nK1=K2in all the simulations was K1= 7×10−3J2219.\nIn the case of LuFe 1−xCrxO3we used the following val-\nues for the DM interactions D11= 0.74×10−2J22and\nD22= 2.14×10−2J22, taken from Refs.[25] and [26],\nrespectively. There is no estimation of D12in the lit-\nerature, so we assume as a reference the value D12=\n−1.7×10−2J22to obtain the critical temperatures, con-\nsidering that a similar value was obtained by Dasari et\nal6fitting YFe 1−xCrxO3data. Since DM interactions\nare considerably lower than super-exchange interactions,\nsmall variations of this interactions does not substantial ly\naffect the antiferromagnetic ordering temperatures.3\nB. Effective model\nIn order to get a deeper physical insight about the low\ntemperature behavior of these systems, we compared the\nMC results against an effective model that generalizes\nsome ideas introduced by Dasari et al6. In this model a\nsiteiis occupied with probability P1[x] =PCr[x] =x\nby a Cr3+ion and with probability P2[x] =PFe[x] =\n(1−x)by a Fe3+ion. The energy, in a two-sublattice\napproximation, is then given by\nE=z[J11/vectorMA\n1·/vectorMB\n1+J22/vectorMA\n2·/vectorMB\n2+\n+J12(/vectorMA\n1·/vectorMB\n2+/vectorMA\n2·/vectorMB\n1)+\n+D11·(/vectorMA\n1×/vectorMB\n1)+D22·(/vectorMA\n2×/vectorMB\n2)+\n+D12·(/vectorMA\n1×/vectorMB\n2+/vectorMA\n2×/vectorMB\n1)]+\n− K1[(MA\n1x)2+(MB\n1x)2]−K2[(MA\n2x)2+(MB\n2x)2)]+\n− H1[MA\n1z+MB\n1z]−H2[MA\n2z+MB\n2z], (4)\nhere/vectorMα\n1, withα=AorBis the total magnetization\nof the chromium ions belonging to the sublattice, Aor\nB, respectively, and /vectorMα\n2is the total magnetization of the\niron ions belonging to the sublattice, AorB, respectively.\nJij=Pi[x]Pj[x]Jij,Dij=Pi[x]Pj[x]Dij,Ki=Pi[x]Ki,\nandHi=miPi[x]Hwithm1= 0.6andm2= 1. Letφ\nandθthe canting angles of M1andM2respectively (see\nfigure 1). zis the number of nearest neighbors.\nFor small canting angles, disregarding constant terms,\nthe energy is\nE= 2J11φ2+2J22θ2+J12(θ+φ)2−2D11φ−2D22θ+\n+ 2D12(θ+φ)+2K1φ2+2K2θ2−2H1φ−2H2θ.(5)\nThe minimum energy configuration is obtained from\n∂E\n∂θ= 4J22θ+2J12(θ+φ)−2D22+D12+4K2θ+\n−2H2= 0\n∂E\n∂φ= 4J11φ+2J12(θ+φ)−2D11+D12+4K1φ+\n−2H1= 0.\nSolving these two equations for θandφwe can obtain\nthe magnetization per site as function of xas:\nMz(x) =µcrP1[x]φ(x)+µFeP2[x]θ(x). (6)\nIn order to compare with MC simulations we define the\nreduced magnetization mz=Mz/µFe. Forx= 0,P2= 1\nandP1= 0we have\nmz=D22+m2H\n2(J22+K2)m2, (7)\nand forx= 1,P2= 0andP1= 1, soAM1\nBM1\nBM2AM2θ\nθφφ\nzx\nD\nFIG. 1. Sketch of the configuration of the two sublattice mag-\nnetization.\nmz=D11+m1H\n2(J11+K1)m1. (8)\nThese are the zero temperature –weak– magnetization for\nthe pure compounds, LuFeO 3and LuCrO 3, respectively.\nC. Experiments\nLuFe1−xCrxO3(x= 0.15,0.5and0.85) samples were\nprepared in polycrystalline form by a wet chemical\nmethod. A very reactive precursor was prepared start-\ning from an aqueous solution of the metal ions and citric\nacid. Stoichiometric amounts of analytical grade Lu 2O3,\nFe(NO 3)3·9H2O and Cr(NO 3)3·9H2O were dissolved\nin citric acid and some drops of concentrated HNO 3,\nto facilitate the dissolution of Lu 2O3. The citrate so-\nlution was slowly evaporated, leading to an organic resin\nthat contained a homogeneous distribution of the in-\nvolved cations. This resin was dried at 120oC and then\ndecomposed at 600oC for 12 h in air, with the aim\nof eliminate the organic matter. This treatment pro-\nduced homogeneous and very reactive precursor mate-\nrials that were finally treated at 1050oC in air for 12\nh. LuFe 1−xCrxO3compounds were obtained as orange,\nwell-crystallized powders as shown in Ref.[18]. The mag-\nnetic measurements were performed using a commercial\nMPMS-5S superconducting quantum interference device\nmagnetometer, on powdered samples, from 5 to 400K,\nand for the 300 to 800K measurements the VSM option\nwas used in the same MPMS.4\nII. RESULTS\nA. Antiferromagnetic ordering temperature\nThe analysis of the solid solution Néel temperature\nTN(x)of LuFe 1−xCrxO3as a function of the Cr con-\ncentration allowed us to estimate the coupling constants\nof the model as follows.\nThe critical temperature obtained from MC in our\nmodel for x= 0(LuFeO 3) isTN\nJ22= 1.44. Consider-\ning that the measured Néel temperature27isTN= 628\nK, then the value for the superexchange interaction be-\ntween the Fe+3ions that reproduces the experimental\nresult in our model is: J22= 436 K. Similarly, for x= 1\n(LuCrO 3),TN= 115 K28and then J11= 79.8K. In our\nanalysis the value of J12is a fitting parameter and it will\nbe extracted from the approach of MC simulations and\nthe experimental results. Namely, we choose the value of\nJ12which provides an MC curve TN(x)that minimized\nthe sum of the mean square deviations respect to theavailable experimental results.\nPrevious estimations of the solid solution Néel tem-\nperature TN(x)in this kind of compounds were based\non mean field approximations, in which DM interactions\nwere neglected6,22. For instance, Dasari et al.6obtained\nTN=z\n3\n2/summationdisplay\ni=1,j=1J2\nijPi[x]2Pj[x]2\n1\n2\n, (9)\nwherezis the number of nearest neighbors, JijandPi[x],\nwithi,j= 1,2where already defined in Sections I A and\nI B. In a cubic lattice z= 6, so for the pure compounds\n(x= 0orx= 1)TNi=z\n3Jiiand therefore\nTN=/radicalBig\nT2\nN1P1[x]4+8J2\n12P1[x]2P2[x]2+T2\nN2P2[x]4.\n(10)\nIn a different mean field approximation Hashimoto22ob-\ntained the expression\nTN=1\n2/bracketleftBigg\nP1[x]TN1−P2[x]TN2+/radicalbigg\n(P1[x]TN1+P2[x]TN2)2+4P1[x]P2[x](4z2\n9J2\n12−TN1TN2)/bracketrightBigg\n. (11)\nThe dependence of the Néel temperature on the Cr\ncontent obtained from experiments in polycrystals is\nshown in Fig. 2. The values x= 027,x= 128were\ntaken from the literature, and x= 0.15,0.5and0.85\nwere sinthetized in our experiments. In this figure we\nalso compare the best fittings of the experimental results\nobtained from the MC simulations and using Eqs. (10)\nand (11).\nFrom Hashimoto and Dasari expressions very different\nvalues of J12are obtained, 11K and162K respectively.\nThe value derived from the MC simulations is J12= 106\nK, which is in between the values obtained from Eqs.(11)\nand(10). Considering that the value of the exchange inte-\ngralJFeCr=J12/(2SCrSFe)whereS2\nCr=SCr(SCr+1)\nandS2\nFe=SFe(SFe+ 1)we obtain JFeCr= 28,3K,\nJFeCr= 1.9K andJFeCr= 9.25K from Eq. (11),\nEq. (10) and MC simulations, respectively. The value\nofJFeCrreported by Dasari et al [6] for YFe 1−xCrxO3\n(JFeCr= 24.3K) is comparably to the value we have\nfound for LuFe 1−xCrxO3using the same equation. The\nvalue reported by Kadomtseva et al. for YFe 1−xCrxO3\nmonocrystals using Eq. (11) ( JFeCr= 6,64K) is consid-\nerably lower than the value reported by Dasari et al. in\nthe same compound.\nIn order to test the mean field approximations, we fit-\nted the MC results with the corresponding expressions\n(10) and (11). In figure 3 we show a fit of the Néel tem-\nperatures obtained from MC simulation using Eq.(10).\nFrom this fit we obtained J12= 306 K which is con-siderably greater than the value used in MC simulations\nJ12= 106 K. For comparison we included in Fig. 3 a\nplot of the mean field expression using the MC simula-\ntion value ( J12= 106 K). One can observe that with\nthis value Eq.(10) clearly departs from the results of MC\nsimulations. We concluded that a fit with expression (10)\nalways overestimates the value of the exchange interac-\ntionJ12. Similarly, fitting the MC results using Eq.(11)\nsystematically underestimates J12. The accuracy of MF\nmodel is expected to be good in low and high Cr content;\nat intermediate concentrations the effect of the distribu-\ntion of the interactions is important. Then, a fit which\ntakes into account all the concentration range somehow\nbiases the value of the J12interaction.\nIn figure 4 we show experimental data for the criti-\ncal temperature of YFe 1−xCrxO3as a function of the\nchromium content reported by Dasari et al.6. We\nalso show the data obtained from MC simulations with\nJFeCr= 9.25K tuned to get the best fit with the ex-\nperimental points. We also include a plot of the mean\nfield expression derived by Dasari et al.6using the value\nJFeCr= 24.0K which is the value reported by these au-\nthors. Finally, a plot of Hashimoto’s expression22Eq.(11)\nusing the value of JFeCr= 6.64K reported by Kadomt-\nseva et al1is also included. In this last work the samples\nstudied were single monocrystals of the YFe 1−xCrxO3\ncompound. MC approach gives a very good agreement\nwith the experiments in all the range of concentrations\nand the value obtained for JFeCr= 9.25K is higher5\n 100 200 300 400 500 600 700\n 0  0.2  0.4  0.6  0.8  1TN (K)\nCr substitution (x)\nFIG. 2. (Color online) Néel temperature as function of the\nfraction of Cr. Full red circles correspond to the experimen ts\nwith LuFe 1−xCrxO3, and open triangles to MC simulations.\nThe lines correspond to fits of the experiments using Eq.(10)\n(blue dashed lines) and Eq.(11) (green dot dashed lines), gi v-\ningJ12= 162 K andJ12= 11K, respectively. The parameter\nvalues of the MC simulation were J22= 436 K,J11= 79.8K,\nandJ12= 106 K.\n 0 100 200 300 400 500 600 700\n 0  0.2  0.4  0.6  0.8  1TN (K)\nCr substitution (x)\nFIG. 3. (Color online) Néel temperature as function of the\nfraction of Cr for LuFe 1−xCrxO3. Open triangles correspond\nto MC simulations and the blue dotted line corresponds to a\nfit of the MF expression Eq. (10). The parameter that results\nfrom the fit is J12= 306 K. The full line corresponds to a plot\nof Eq. (10) using the value of MC simulations ( J12= 106) .\nthan the reported by Kadomtseva et al. and lower than\nthe reported by Dasari et al. Finally, the values JFeCr\nobtained from MC for both compounds YFe 1−xCrxO3\nand LuFe 1−xCrxO3are the same, indicating that the ex-\nchange integral is not substantially affected by the sub-\nstitution of yttrium by lutetium. 100 300 500 700\n 0  0.2  0.4  0.6  0.8  1TN (K)\nCr substitution (x)\nFIG. 4. (Color online) Néel temperatures as function of the\nfraction of Cr for6YFe1−xCrx03(red circles) and MC simu-\nlations (open triangles). The blue dashed line corresponds to\nEq.(10) using the value of J12reported by Dasari et al [6] and\nthe green dot dashed line corresponds to Eq. (11) using the\nvalue of J12reported by Kadomtseva et al [1]\nB.T= 0magnetization\nIn Fig.5 we show the modulus of the canted magneti-\nzation in the z direction ( mz) atT= 0as function of\nthe Cr content which is obtained from MC simulations\nin a ZFC process. We also plot the modulus of mzob-\ntained using Eq. (6) with the physical constants used\nin MC simulations. We see a good agreement between\nMC and the effective model at low and high chromium\ncontents where the model is expected to work better.\nThe local maximum at intermediate concentrations ob-\nserved in MC simulations is related to a change in the\nsign of the magnetization. Like in the mean field model\nat intermediate concentrations the effect of the distribu-\ntions of the DM bonds is important, and for this reason\nin this concentration range the effective model departs\nfrom MC simulations, in fact the effective model takes\ninto account only averaged values in the distribution of\nthe DM interactions. In addition, the effective canting\ndue to the DM interactions can be approached using the\nfollowing expression for the magnetization as function of\nthe chromium content\nmz=1\n2[m22P1[x]2+2m12J2\n12P1[x]P2[x]+m22P2[x]2]\n(12)\nwheremijis the averaged canted magnetization contri-\nbution of a pair of spins interacting through the DM in-\nteraction. Then, m11= 2α11m1,m12=α12(m1+m2),\nandm22= 2α22m2. Hereαijare the average canting an-\ngles between ions of type iandj(assuming low angles).\nAccording to Eq. (6), αij≃Dij\n2Jijfori=j, andα12is\nan effective parameter to be fitted. One can estimate as6\n 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014\n 0  0.2  0.4  0.6  0.8  1|m|\nCr substitution (x)MC\nMF\nfit\nFIG. 5. (Color online) Module of the canted magnetization as\nfunction of the Cr content at T= 0and zero applied magnetic\nfield. Monte Carlo simulations (MC): red circles. Full lines\ncorrespond to the effective model Eq. (6) using the same\nparameters as in the simulation. The dashed lines correspon d\nto a fit using Eq. (12).\nα12=D12\n2J21which turn in α12=−0.0345. The fixed pa-\nrameters are α11= 0.0203andα22= 0.0107. Fitting Eq.\n(12) to the MC data we obtain α12=−0.0258, showing\nthe consistency of Eq.(12). Moreover, the negative mag-\nnetization can be understood from Eq.(12) as an effect of\nthe negative sign of the DM interactions between Fe and\nCr ions, which favors the canting of both ions in the neg-\native z direction. Since Fe-Cr pairs are the majority at\nintermediate values of the Cr concentration x, the term\nin Eq.(12) associated with m12<0is dominant and mz\nbecomes negative.\nThis local maximum has been reported in experiments\ncarried out in single crystals of the YFe 1−xCrxO3com-\npounds1where the low temperature magnetization is\nmeasured as function of the chromium content.\nC. Magnetization reversal\nIn Fig. 6 we show the mzcomponent of the to-\ntal magnetization as function of the temperature for\nx= 0.4chromium content obtained by MC simulations\nusing the parameters of the LuFe 1−xCrxO3compound al-\nready obtained in section II A. The cooling is performed\nunder three different applied fields in the zdirection;\nh=H/J22= 5×10−4,1×10−3, and2×10−3. The\narrow indicates the Néel temperature for this composi-\ntion. We can observe reversal in the magnetization for\nthis composition for the three applied fields. The mag-\nnetization increases with the applied field at low tem-\nperatures, although the compensation temperature ap-\npears to be almost independent of h, at least in the small\nrange of values. In this case, x= 0.4, the compensa--0.0025-0.0005\n 0  0.2  0.4  0.6  0.8  1mz\nT/J22 h = 5x10-4  \n h = 1x10-3  \n h = 2x10-3  \nFIG. 6. Field cooling curves under different applied magneti c\nfields in the zdirection for x= 0.4. The arrow indicates the\nordering temperature obtained from the peak of the suscep-\ntibility. The applied field is in units of J22i.e.h=H/J22.\ntion temperature is clearly smaller than the Néel tem-\nperature. We do not observe magnetization reversal for\nx= 0.5and in fact in the composition range x >0.4the\nmagnetization reversal is unstable. However, the shape\nof the curve obtained for x= 0.4qualitatively repro-\nduces the curves reported in the experiments for yttrium5\n(YFe0.5Cr0.5O3) and lutetium18(LuFe0.5Cr0.5O3) com-\npounds. Moreover, the ratio between the compensa-\ntion and Néel temperatures obtained in our simulation\nTcomp/TN= 0.76compares well with the experimental\nvalue18Tcomp/TN= 0.83.\nIn Fig. 7 we show FC magnetization curves with\nh= 2×10−3forx= 0.4, and curves obtained through a\nmean field approach. Here we have measured in MC sim-\nulations separately the temperature dependence of the\ntotal magnetization of the Fe3+ions and that of the Cr3+\nions. We can see that below the Néel temperature the\nmagnetization due to the Fe3+ions aligns in the direc-\ntion of the magnetic field, while the magnetization due\nto the Cr3+ions is opposite to the field. In this way,\nwhen the field breaks the inversion symmetry along the\nzaxis the Zeeman energy is reduced due to the coupling\nof the larger magnetic moments of the Fe3+ions. The dif-\nferent temperatures dependencies in the magnetization of\nFe3+and Cr3+ions turns into the magnetization reversal.\nFor lower compositions ( x≤0.3) the Fe3+ions are also\naligned in the direction of the applied field but magneti-\nzation reversal is not observed because the contribution\nto the magnetization of Fe3+is dominant.\nIn this figure we also show mean field curves which are\nobtained through Eq. (4) using a molecular field approx-\nimation for the dependence of the sublattice magnetiza-\ntion on the temperature. This approximation agrees very\nwell with MC results for the sublattice magnetization. In\nthe calculation of the mean field curves showed in Fig.7\n-0.008-0.006-0.004-0.002 0 0.002 0.004 0.006\n 0  0.2  0.4  0.6  0.8  1mz\nT/TNmzmz:Cr\nmz:Fe\nFIG. 7. (Color online) Field cooling curves with an ap-\nplied fields in the zdirection for x= 0.4. The applied field\n(h= 0.002) is the same for all the curves. The symbols with\nlines correspond to MC simulations and lines to MF results.\nMC simulations: chromium magnetization (black triangles) ,\niron magnetization (red circles) and total magnetization ( blue\nsquares). MF calculations: chromium magnetization (black\ndotted line), iron magnetization (red dashed line) and tota l\nmagnetization (blue continuous line). The arrows indicate the\ncompensation temperatures in both cases, and the inset is a\nzoom of MF curves close to the ordering temperature.\nFig. 7 we used the same parameters than in MC results.\nThese curves reproduce the features observed in MC re-\nsults. However, in this case the compensation tempera-\nture (see inset) is much closer to the Néel temperature.\nFrom an analysis of the different energy term contribu-\ntions, Eq. (4), we observed that close to the Néel tem-\nperature the Zeeman term is the most important hence\nthe coupling with the field at high temperatures rules the\nmagnetization process and induces the symmetry break-\ning. In the lower temperature range DM interactions\nprevail and convey the reversal of the magnetization.\nIII. DISCUSSION\nMonte Carlo simulations using the proposed micro-\nscopic classical model reproduce the whole phenomenol-\nogy of both LuFe 1−xCrxO3and YFe 1−xCrxO3com-\npounds as the chromium content is varied. From these\nsimulations it turns out that the superexchange interac-\ntion between Cr3+and Fe3+ions is lower than the super-\nexchange interaction between Fe3+and Fe3+, and greater\nthan the super-exchange between Cr3+and Fe3+ions i.e.\nJ22> J12> J11. From the fit of our experimental results\nwith different mean field expressions, Eqs. (10) and (11)\nwe obtained J12= 162 K andJ12= 11K, respectively.\nThese results show a big dispersion depending of the\nexpression used to fit the experiments. In particular, thevalue obtained from MC simulations, J12= 106 K, is\nin between this two values. The values of J12available\nin the literature for Y perovskites YFe 1−xCrxO3, also\nshow an important dispersion. For instance, J12= 139\nK, when Eq. (10) is used in polycrystals6andJ12= 38\nK has been reported in single crystals1using Eq. (11).\nSuch large sensitivity to the details of the particular mean\nfield approximation is not surprising in a solid solution,\nwhere the interplay between thermal fluctuations and the\ninherent disorder of the solution is expected to be very\nrelevant to determine thermal properties. Consistently,\nthe experimental results are better described by the MC\nsimulations than by the MF expressions. Hence, we ex-\npect our estimation of J12to be more reliable than the\nprevious ones. In addition, our results suggest that the\nexchange constant (and therefore the general behavior) is\nnot substantially affected by the substitution of yttrium\nby lutecium.\nThe zero temperature magnetization obtained from\nMC simulations in a ZFC process, which is due to the\ncanting of the AFM spins in the zdirections, is well ap-\nproached by an effective coarse grain model in the range\nof low and high chromium contents as expected. A bump\nin the magnetization is observed in MC simulations at\nintermediate concentrations which is a signature of the\nmagnetization reversal. This bump is also observed in the\ncoarse grain approach but is less pronounced. The differ-\nence between MC simulations and the effective model at\nintermediate concentrations is expected since the coarse\ngrain approach does not take into account information\non the distribution of the ions in the lattice which is im-\nportant at intermediate Cr concentrations.\nMagnetization reversal is observed at intermediate\nchromium contents x= 0.4in a ZFC process depend-\ning on the value of D12(≃1.7×10−2J22). When the field\nis increased above a certain threshold MR disappear, and\nthe magnetization points in the direction of the applied\nfield in whole temperature range. The presence of mag-\nnetization reversal is very sensitive to the value of the\nDM interaction between Cr3+and Fe3+ions and also\nto the value of superexchange J12interaction. We do\nnot observe magnetization reversal in MC simulations at\nx= 0.5such as is observed in experiments. This could\nbe due to size effects which are particularly important in\nsystems that includes disorder. In fact, the fields we used\nto obtain the ZFC curves (e.g. h= 0.001correspond to\nB≃0.13T) are much greater that the ones used in ex-\nperiments (e.g. B∼0.01T). A reduction of the fields in\nMC is only possible in larger systems.\nSummarizing, Monte Carlo simulations based in a\nHeisenberg microscopic classical model reproduce the\ncritical temperatures observed in experiments. Besides\nthis is a classical model, MC fit can provide a better\nestimation of J12since in this model the random occu-\npation of the Cr3+and Fe3+ions is taken into account.\nRegarding the phenomena of magnetization reversal, we\nfound it for appropriated values of the superexchange\nand the Dzyaloshinskii-Moriya interactions at interme-8\ndiate Cr concentrations. However, the mechanism for\nthe appearance is subtle and further investigations are\nneeded to shed light on this point.\nACKNOWLEDGMENTS\nThis work was partially supported by CONICET\nthrough grant PIP 2012-11220110100213 and PIP 2013-11220120100360, SeCyT–Universidad Nacional de Cór-\ndoba (Argentina), FONCyT and a CONICET-CNRS co-\noperation program. F. P. thanks CONICET for a fel-\nlowship. A. M. gratefully acknowledges a collaboration\nproject between CNRS and CONICET (PCB I-2014).\nThis work used Mendieta Cluster from CCAD-UNC,\nwhich is part of SNCAD-MinCyT, Argentina.\n∗billoni@famaf.unc.edu.ar\n†fernandopomiro@gmail.com\n‡cannas@famaf.unc.edu.ar\n§christine.martin@ensicaen.fr\n¶antoine.maignan@ensicaen.fr\n∗∗rcarbonio@gmail.com\n1A. M. Kadomtseva, A. S. Moskvin, I. G. Bostrem, B. M.\nWanklyn, and N. A. Khafizova, Sov. Phys. JETP 45, 1202\n(1977).\n2K. Yoshii and A. Nakamura, Journal of Solid State Chem-\nistry155, 447 (2000).\n3K. 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Khomskii, A. A. Nugroho,\nA. A. Menovsky, and G. A. Sawatzky, Physical Review B62, 6577 (2000).\n15J. Mao, Y. Sui, X. Zhang, Y. Su, X. Wang, Z. Liu,\nY. Wang, R. Zhu, Y. Wang, W. Liu, and J. Tang, Ap-\nplied Physics Letters 98, 192510 (2011).\n16P. Mandal, A. Sundaresan, C. N. R. Rao, A. Iyo, P. M.\nShirage, Y. Tanaka, C. Simon, V. Pralong, O. I. Lebe-\ndev, V. Caignaert, and B. Raveau, Physical Review B 82,\n100416 (2010).\n17a.K. Azad, a. Mellergård, S.-G. Eriksson, S. Ivanov,\nS. Yunus, F. Lindberg, G. Svensson, and R. Mathieu,\nMaterials Research Bulletin 40, 1633 (2005).\n18F. Pomiro, R. D. Sánchez, G. Cuello, A. Maignan, C. Mar-\ntin, and R. E. Carbonio, submitted (2016).\n19A. K. Murtazaev, I. K. Kamilov, and Z. G. Ibaev, Low\nTemperature Physics 31, 139 (2005).\n20E. Restrepo-Parra, C. Bedoya-HincapiÃľ, F. Jurado,\nJ. Riano-Rojas, and J. Restrepo, Journal of Magnetism\nand Magnetic Materials 322, 3514 (2010).\n21E. Restrepo-Parra, C. Salazar-EnrÃŋquez, J. 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Fennie1,\u0003\n1School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA\n2Department of Physics and Astronomy, Rutgers University, Piscataway, NJ, USA\n(Dated: October 30, 2018)\nImproper ferroelectricity (trimerization) in the hexagonal manganites RMnO 3leads to a network\nof coupled structural and magnetic vortices that induce domain wall magnetoelectricity and mag-\nnetization ( M) neither of which, however, occurs in the bulk. Here we combined \frst-principles\ncalculations, group-theoretic techniques, and microscopic spin models to show how the trimeriza-\ntion not only induces a polarization ( P) but also a bulk Mand bulk magnetoelectric (ME) e\u000bect.\nThis results in the existence of a bulk linear ME vortex structure or a bulk ME coupling such that\nifPreverses so does M. To measure the predicted ME vortex, we suggest RMnO 3under large\nmagnetic \feld. We suggest a family of materials, the hexagonal RFeO 3ferrites, also display the\npredicted phenomena in their ground state.\nTwo themes at the forefront of materials physics are\nthe cross-coupling of distinct types of ferroic order1{4\nand topological defects in systems with spontaneous bro-\nken symmetry5{8. Common to both are a plethora of\nnovel phenomenon to understand, and new properties\nand functionalities to exploit for novel applications. Mul-\ntiferroics9,10are an ideal platform to realize both themes\nin a single material. In this regard, an exciting develop-\nment is the discovery of a topologically protected vortex-\ndomain structure in one of the most extensively studied\nclass of multiferroics, the hexagonal (hexa) rare-earth\nmanganites. Here, antiphase structural (`trimer') do-\nmains are clamped to ferroelectric domain walls (and vice\nversa)11{14forming a `clover-leaf' pattern, Fig. 1a. These\ntrimer domains have a particular phase relationship that\nresult in the appearance of structural vortices, which in\nturn induce magnetic vortices15,16, strongly coupled anti-\nferromagnetism and the polarization at the domain wall.\nThis domain wall magnetoelectric phenomenon produces\na magnetization localized at the wall15,16.\nThe key to realizing these unusual e\u000bects is the im-\nproper nature of ferroelectricity. Here the polarization\n(P) which is stable in the paraelectric (PE) P6 3/mmc\nstructure, is induced by a zone-tripling structural distor-\ntion,Q\b\nK316{18. The latter, referred to as the trimer dis-\ntortion, is associated with a 2-up/1-down buckling of the\nR-planes and tilting of the MnO 5bipyramids, Fig. 1b. It\nis nonlinearly coupled to the polarization,\nFtrimer\u0018PzQ3\nK3cos(3\b) (1)\nthe form of which implies that a nonzero trimer distortion\ninduces a nonzero P. There are three distinct \b domains\n(\u000b,\f, and\r) corresponding to one of the 3 permutations\nof 2-up/1-down. Also there are two tilting directions,\neither towards (+) or away from ( \u0000) the ~2caxis, i.e.,\n1-up/2-down or 2-up/1-down, respectively. This results\nin six P6 3cm structural domains. A consequence of the\nimproper origin of ferroelectricity is that the sign of Pde-\npends on the direction of Q\b\nK3. This simple fact leads to\nthe nontrivial domain structure of the hexa manganites,\nFig 1a11{13,16.Our focus here is on elucidating a remarkable inter-\nplay of this trimerization, magnetism, and polarization\nin the hexa manganite structure. We show from \frst\nprinciples that the trimer structural distortion not only\ninduces a P, but can also induce both a bulkmagnetiza-\ntion,M, and a bulklinear ME e\u000bect. We make this clear\nby connecting an exact microscopic theory of spin-lattice\ncoupling to a simple phenomenological theory. This not\nonly brings additional insight to known experiments, but\nleads us to discover entirely new bulk phenomena, not\npreviously seen in a multiferroic such as 1) the existence\nof a linear magnetoelectric (ME) vortex structure and 2)\na bulk coupling of ferroelectric (FE) and ferromagnetic\ndomains such that if Preverses 180\u000eso does M.\nWe show that the former is widely accessible in most\nhexa RMnO 3manganites under large magnetic \felds,\nwhile the latter is realizable in the ground state of a new\nfamily of materials, the hexa RFeO 3ferrites. Recently\nthin \flms of RFeO 3have been epitaxially stabilized in\nthe hexa P6 3cm structure19{21. By performing a detail\ncomparison of the ferrites to the manganites we explain\nwhy the ground state of all ferrites display an intrinsic,\nbulkMandbulklinear ME e\u000bect.\nI. FIRST-PRINCIPLES CALCULATIONS ON\nSWITCHING \bBY 180\u000e\nGeometric frustration of the strongly antiferromag-\nnetic (AFM) nearest neighbor Mn (or Fe) spins leads to\na planar 120\u000enon-collinear order which can be described\nby two free parameters, \t 1and \t 2, as shown in Fig. 1c.\nFour principle con\fgurations, denoted A 1(\u0000\u0019=2;\u0019=2),\nA2(\u0019;0), B 1(0;0) and B 2(\u0019=2;\u0019=2), have been pre-\nviously de\fned, among which it is well-known that only\nthe A 2(magnetic space group P6 3c0m0) and intermediate\nspin con\fgurations that contain a component of A 2allow\na net Malong thezaxis. What hasn't been appreciated\nin the past is that this known symmetry-allowed Mis in\nfact induced by the trimer distortion, the consequences\nof which are quite profound.\nWe now begin to make this clear by performing \frst-arXiv:1302.1099v2  [cond-mat.mtrl-sci]  1 Aug 20132\na b \nc e f g h \nQK (Angstrom) 3 Mz (µB/Mn) Mz (µB/Mn) P (µC/cm2) Energy (eV) \n!1Q1 Q6 Q3 Q2 Q5 Q4 d \n!1!2L1 L6 L3 L2 L5 L4 \n!2\nFIG. 1: First Principles: Improper structural ferroelectricity inducing weak-ferromagnetism, M z, in hexagonal\nstructure from \frst principles. Results are shown for ErMnO 3, however, they are true for all hexa manganites and RFeO 3\nferrites in the A 2phase that we have considered a, Six structural domains of the primary \\trimer\" Q\b\nK3distortion and the\nsecondary polarization P. Here\u000b+() \b = 0 and a counterclockwise rotation corresponds to domains di\u000bering by \b = + \u0019=3.\nArrows indicate the direction of the trimer distortions, only distortions around the ~2caxis are shown for clarity. b, crystal\nstructure of the ferroelectric phase ( Q0\nK3(\u000b+) domain). c, The spin angles \t 1and \t 2describing all possible 120\u000enon-collinear\nantiferromagnetic (AFM) spin con\fgurations (see Supplemental). Li(i21 to 6) denotes transverse component of the ithspin.\nWe de\fne L1= cos \t 1^x+ sin \t 1^y,L2= cos \t 2^x+ sin \t 2^yand remaining four vectors can be created by the crystal symmetry\noperations. d, Represents directions of local trimer distortions. The local trimer distortion at site 1 ( Q1) and 2 ( Q2) are de\fned\nas,Q1= cos \b 1^x+ sin \b 1^yandQ2= cos \b 2^x+ sin \b 2^y, respectively. Crystal symmetry implies \b 1\u0011\b and \b 2=\u0019\u0000\b1.\ne, Energy, f,P,gandh,Mzas a function of QK3, allowing only \b = 0 or \u0019trimer domains. Notice that P=0 when Q\b\nK3= 0, suggesting that a proper ferroelectric mechanism in not likely. Note that for higher magnitude of QK3,P/QK3and\nMz/QK3, thus provide linear ME coupling.\nprinciples calculations for a speci\fc example, ErMnO 3in\nthe A 2phase (which is the spin con\fguration realized ex-\nperimentally under an applied magnetic \feld22). In \frst-\nprinciples calculations of the hexagonal manganite struc-\nture it is easy to reverse the trimer distortion, and hence\nP, via a structural change from a 1-up/2-down buckling\nand tilting `in' of the R-planes and MnO 5bipyramids,\nrespectively, to a 2-up/1-down and tilting `out', thus re-\nmaining in the same distinct domain, e.g., \u000b+!\u000b\u0000.\nHaving performed these calculations we \fnd that the\ntrimer distortion not only induces ferroelectricity, but\nalso weak-ferromagetism23,24and the linear ME e\u000bect,\n\u000bzz=@Mz=@Ez\u0019@QMz\u0003@QPz(see Supplemental),\nas shown in Figures 1e through 1h. Furthermore, no-\ntice that reversal of the trimer distortion reverses either\nthe direction of the M, Fig. 1g, or the sign of the linearME tensor, Fig. 1h, where both situations are symmetry\nequivalent and correspond to whether or not the AFM\nspin con\fguration remains \fxed, respectively. This re-\nsult is true for all hexa manganites and RFeO 3ferrites\nin the A 2phase that we have considered and as we prove\nbelow is a general property of the A 2-phase.\nWe pause now to stress the point that real switch-\ning will occur via a rotation to a neighboring trimer do-\nmain16, e.g.,\u000b+!\f\u0000or\r\u0000. These \frst-principles re-\nsults, however, contain all of the unique ME physics, that\nis, if the polarization is reversed either the bulk magne-\ntization will reverse or the antiferromagnetic order will\nchange in such a way that the sign of the bulk magne-\ntoelectric tensor changes sign. To understand the con-\nsequences of these two choices we next derive a simple\nphenomenological theory { valid for any trimer domain,3\n\b, and any spin con\fguration { starting from a micro-\nscopic model.\nII. PHENOMENOLOGY THEORY FROM\nMICROSCOPIC MODEL: GENERALIZING THE\nFIRST-PRINCIPLES RESULTS TO SWITCHING\n\bBYn\u0019\n3\nWe start by deriving a spin-lattice model from an ef-\nfective spin Hamiltonian25\nH=X\nijJijSi\u0001Sj+X\nijDij\u0001Sj\u0002Sj+X\niSi\u0001^\u001ci\u0001Si(2)\nwhere the Jij's are the symmetric exchange interac-\ntions and Dij's are the Dzyaloshinskii-Moriya (DM) an-\ntisymmetric exchange vectors, and ^\u001ciis the single-ion\nanisotropy (SIA) tensor. (Note that our calculations re-\nveal that a dominant DM interaction, Dij, exists only\nbetween nearest neighbor spins within the triangular\nplanes.)\nIn the PE structure the DM vector has only a zcom-\nponent, which further acts to con\fne the spins within\nthexyplane, while the SIA tensor is diagonal. In the FE\nstructure, however, the trimer distortion induces a trans-\nverse component of the DM vector, Dxy\nij, parallel to the\nxyplane and o\u000b-diagonal components of the SIA tensor.\nThese induced interactions are key and therefore are the\nfocus in the remaining discussion (all other interactions\ncan be safely ignored).\nThe e\u000bective DM and SIA interactions for a\nsingle layer of spins. Let us \frst consider a single layer\nof spins, denoted as layer I. We derive the relationship\nbetween the local structural distortions and the induced\nDM and SIA interactions by considering a single triangle\nof spins ( S1,S3andS5), in the\u000b+and\u000b\u0000domains,\nFig. S3c (the exact mapping from a single layer of spins\nto a single triangle is proved in the Supplemental).\nNote that the induced DM and SIA interactions cant\nthe spins out of the plane, but all spins in a single plane\nhave to cant in the same direction, we can therefore write\nSi=Li+MI, where Li\u0011Sxy\ni, i.e., the component of\nthe spin lying in the xyplane (Fig. 1c), while MI\u0011Sz\ni\nis the net magnetic moment per spin of layer I. The total\ncanting energy per spin can be written as\nEcanting\nI =EDM\nI+ESIA\nI=1\n3X\ni=1;3;5de\u000b\ni\u0001[Li\u0002MI]:(3)\nwhere de\u000b\ni=dDM\ni+dSIA\niis the e\u000bective DM-like vector\ninduced by the tilting of the bipyrimid. This e\u000bective in-\nteraction includes contributions from both the transverse\nDM interactions and the o\u000b-diagonal elements of SIA ten-\nsor (see Supplemental for derivation). Considering that\nMI=Mz\nI^zandMz\nIde\u000b\ni\u0001[Li\u0002^z] =Mz\nILi\u0001\u0002\n^z\u0002de\u000b\ni\u0003\n, the\ncanting energy per spin can be compactly rewritten as\nEcanting\nI =1\n3jdjMz\nIX\ni=1;3;5Li\u0001^Qi=jdjMz\nI(LI\u0001^QI) (4)where we have de\fned ^Qisuch thatjde\u000b\nij^Qi\u0011^z\u0002de\u000b\ni,\nand where by symmetry jde\u000b\nij=jde\u000bj\u0011jdjandL1\u0001^Q1=\nL2\u0001^Q2=L3\u0001^Q3\u0011LI\u0001^QI.\nIt is interesting that ^Qiis the direction of the in-plane\ndisplacement of the apical oxygen that lies directly above\nspinSi. It is zero in the PE phase and is in opposite di-\nrections in the \u000b\u0006domains. It behaves in every aspect\nas an order parameter that de\fnes the local trimer dis-\ntortion. In fact, one of the ^Qi's is the atomic distortion\nthat Mostovoy has used to de\fne a particular trimer do-\nmain16. If we had considered a di\u000berent domain, e.g.,\n\f+, the ^Qi's rotate appropriately and in fact have sim-\nilar transformational properties as the trimer structural\ndistortions, Q\b\nK3. This suggests (and we prove in the\nSupplemental) that ^Qiis a local trimer distortion, which\ninduces the local DM-like interaction, de\u000b\ni, and subse-\nquently cants the spins.\nIt is now clear that 1) if the relative phase between\nthe local AFM spin, LI, and the local trimer distortion,\n^QI, changes sign, the net magnetic moment per spin of a\nlayer, MI, reverses, i.e,MI/(LI\u0001^QI)^z, and 2) canting\noccurs only in a FE phase (where ^QI6= 0) and only when\nthere is a nonzero projection of a spin along the direction\nof the local trimer distortion. This is why there is no\ncanting for the A 1and B 2spin states (where LI\u0001QI= 0).\nThe real structure { the stacking of two layers.\nThe real hexa unit cell has two triangular planes, layer I\n(which includes sites 1, 3, 5) and layer II(which includes\nsites 2, 4, 6), stacked along the zaxis, as shown in Fig. 1c\nand d. The canting energy per spin is\nEcanting =jdjh\nMz\nI(LI\u0001^QI) +Mz\nII(LII\u0001^QII)i\n(5)\nThis can be alternatively written as\nEcanting =jdjMz[LI\u0001^QI+LII\u0001^QII]\n+jdjLz[LI\u0001^QI\u0000LII\u0001^QII] (6)\nwhereMz\u0011(Mz\nI+Mz\nII)=2 andLz\u0011(Mz\nI\u0000Mz\nII)=2 are\nthe total and stagger magnetic moment per spin respec-\ntively. Although the speci\fc sites we choose to de\fne I\nandIIare arbitrary, it is convenient to associate Iwith\nsite1andIIwith site 2. Because symmetry implies\n\b1=\u0019\u0000\b2, where \b 1(\b2) is the local trimer angle at\nsite1(2), a single trimer angle, \b \u0011\b1, can be de\fned.\nThis single angle was used to de\fne the trimer domains\nin Fig. 1a.\nNote that Eq. 6 is the exact result we derived from\nLandau theory (see Supplemental) and explains our \frst-\nprinciples calculations displayed in Fig. 1; in the A 2state\nLI\u0001^QI= +LII\u0001^QII, and therefore Mz\nI=Mz\nIIleading to\na net magnetization as we previously showed from \frst\nprinciples. For completeness note that in the B 1state,\nhowever, the projection has the opposite sign in adjacent\nlayers, LI\u0001^QI= -LII\u0001^QII. The spins in each plane still\ncant but since the projection changes sign in adjacent\nlayers no net magnetization exists, Mz\nI=\u0000Mz\nII. We call\nthis weak-antiferromagentism, wAFM.4\na b c +αME\t\r -‐αME\t\r +αME\t\r -‐αME\t\r +αME\t\r -‐αME\t\r +αME\t\r +αME\t\r +αME\t\r +αME\t\r +αME\t\r +αME\t\r \nFIG. 2: Predicted clamping of trimer and magnetic\ndomains and electric \feld response. At the lowest en-\nergy trimer domain wall \b changes by \u0019=3, which will rotate\nthe AFM spin vectors, L1andL2, by either a,\u0019=3 and\u0000\u0019=3\norb,\u00002\u0019=3 and 2\u0019=3. In the former case Mzremains con-\nstant while in the latter Mzreverses at each trimer domain\nwall, i.e., ferroelectric, ferromagnetic, and trimer domains are\nmutually clamped. c, The schematic diagram displaying the\nelectric-\feld switching of ferromagnetic domains (note, \fgure\ninspired by Ref.11).\nIII. IMPLICATIONS: TESTABLE\nPREDICTIONS\nNote that if Pswitched via rotating \b by \u0019=3, e.g.,\n\u000b+!\f\u0000,Lmust rotate by either j\u0019=3j(L\u0001^Q\u000b+=\u00001!\nL\u0001^Q\f\u0000=\u00001) orj2\u0019=3j(L\u0001^Q\u000b+=\u00001!L\u0001^Q\f\u0000= +1).\ns\nPrediction 1{ In the former case the expected domain\ncon\fgurations, Fig. 2a, are such that at trimer domain\nwalls di\u000bering by \u0001\b = \u0019=3 the AFM spins rotate\n\u0001\t1;2=j\u0019=3j. In this case the direction of the mag-\nnetization remains the same across the domain wall, i.e.,\nalthough Pswitches, Mis not reversed similar to that\nshown in Fig 1h. Still, there exist a bulk linear ME e\u000bect\n\u000bzz/cos(3\b) ( LI\u0001^QI+LII\u0001^QII) (7)\n(see Supplemental), which in this case leads to a presence\nof a remarkable linear ME vortex structure as the pro-\njections, Li\u0001^Qi, are equal in all domains, and therefore\n\u000bzzis of opposite sign in neighboring trimer domains.\nPrediction 2{ Note that this seemingly lower energy\nAFM switching pathway results in a homogeneous mag-\nnetization across the entire material. In zero \feld, this\nhas to be unstable towards the formation of ferromag-\nnetic domains. But what kind of domains? They can\noccur within the bulk of a trimer domain, i.e., a free\nAFM domain. There is, however, an energy cost to form\nthis domain wall. An alternative path to minimize thetotal energy of the system is considered in Fig. 2b, where\nat trimer domain walls that di\u000ber by \u0001\b = \u0019=3 the AFM\nspins now rotate \u0001\t 1;2=j2\u0019=3j. In this case, the mag-\nnetization direction reverses with the polarization sim-\nilar to that shown in Fig 1g. Therefore, even though\nthis domain con\fguration at \frst appears less likely than\nFig 2a, it provides an avenue for the system to minimize\nthe magnetostatic energy without having to introduce\nfree domains. Additionally, in Fig. 2c we sketch the ex-\npected response of the domains to electric-\feld poling.\nIn this process the positive electric \feld, E, e.g., chooses\nthe (+ P,+M) state and therefore reversing of the direc-\ntion of electric \feld will not only switches the direction\nof polarization, but also reverses the direction of magne-\ntization.\nIV. DISCUSSION: POSSIBLE REALIZATIONS\nOF PREDICTIONS\nRealization 1: The hexagonal Manganites .\nIn the hexagonal manganites, e.g., ErMnO 3, an A 2\nphase appears under the application of external magnetic\n\feld22. Here, as the magnetic \feld is swept from zero to\na large (for example) positive value, a ME vortex struc-\nture is expected to appear. Additionally, as the magnetic\n\feld is scanned to negative values the sign of the ME vor-\ntex structure should switch. This is precisely what our\npreliminary imaging of the ME vortex structure shows\nusing a new technique called Magnetoelectric Force Mi-\ncroscopy, as shown in the supplemental.\nRealization 2: The A 2ground state in hexa-\nRFeO 3. Are there materials in which this physics is\nrealized in the ground state? Recently thin \flms of\nRFeO 3have been epitaxially stabilized in the hexa P6 3cm\nstructure19,20. These hexa ferrites exhibit ferroelectric-\nity above room temperature, but with con\ricting results\nas to its origin26,27. Additionally there is evidence of a\nmagnetic transition around \u0018100K, at which Mbecomes\nnonzero21,26{28, however, the signi\fcance of this or even\nif it is an intrinsic or bulk e\u000bect is not previously known.\nOur calculations suggest strongly that ferroelectricity\nin the hexa ferrites is of the improper structural type\nwhere the trimer distortion induces P(see Supplemen-\ntal for full discussion), and therefore a similar topologi-\ncal domain structure should exist as in the manganites.\nThe di\u000berence in electronic structure between mangan-\nites and ferrites, however, requires Fe spins of any hexa\nferrite to order in the A 2spin con\fguration in ground\nstate. Additionally, the much stronger exchange interac-\ntions leads to the possibility of spin ordering above room\ntemperature, as recently suggested by the experiments\nof Ref. 21.) Therefore, both scenarios displayed in Fig. 2\nare possible. We now discuss our \frst-principles calcula-\ntions indicating that the ground state of any hexagonal\nferrite will indeed have the A 2magnetic con\fguration.\nMagnetic structure. In addition to the principle mag-\nnetic con\fgurations we also considered the four known5\na\nb\nFIG. 3: The noncollinear magnetic and electronic\nstructure from \frst principles. First-principles calcula-\ntions of a, Noncollinear magnetic energies and b, density of\nstates (collinear). Note that the di\u000berence in orbital occu-\npancy is the fundamental reason why ferrites, top, and man-\nganites, bottom, have di\u000berent magnetic ground states and\nsubsequently the key to the predicted bulk coupling of polar-\nization and magnetism in ferrites. Insets: crystal \feld split-\nting and occupancy of the TM majority 3 dchannel.\nintermediate magnetic structures (see Fig.S7). The re-\nsults of our total energy calculations for LuFeO 3, LFO,\nand LuMnO 3, LMO, are presented in Fig. 3a (for clar-\nity we limit our discussion to these two compounds). In\nagreement with non-linear optical measurements29, we\n\fnd that LMO stabilizes in the wAFM B 1state. In con-\ntrast, LFO stabilizes in the wFM A 2state giving rise to\na net canted spin moment Mz= 0.02\u0016B/Fe along the z\naxis. Note, however, that the A 1state, where the net\nmagnetic moment is equal to zero in each layer by sym-\nmetry, is also close in energy.\nElectronic structure. In the PE phase the crystal \feld\nat the TM site has a D 3htrigonal point symmetry, which\nsplits atomic 3 dlevels into three sets of states as shown\nin the insets of Fig.3b. The density of states (DOS) plots\ncalculated for LFO and LMO in the FE phase are shown\nin Fig. 3b. LFO is a charge-transfer insulator with the\nconduction band formed by minority Fe 3 dstates and\nthe valence band composed of O 2 pstates, below which\nare the \flled majority Fe 3 dbands. In the case of LMO\nmajority 3dbands are partially \flled with d4electronic\ncon\fgure, while minority 3 dlevels are completely empty.\nThe importance of these di\u000berences will be made clear\nwhen discussing the magnetic interactions.\nSymmetric exchange. Although it is the DM interac-\ntions and SIA that drives spin canting, it is the symmet-\nric exchange that determines the magnetic con\fguration\ntype. There are two important symmetric exchange in-\nteractions. The \frst is a strong, AFM superexchangeinteraction between in-plane nn spins, Jnn. Its magni-\ntude is much larger for LFO compared with LMO, sug-\ngesting a substantially larger magnetic ordering tempera-\nture for ferrites (within mean \feld theory the calculated\nCurrie-Weiss temperature for LFO in the FE phase is\n\u0002CW= 1525 K, while in LMO \u0002 CW= 274 K30).\nThe second is a weak, super-superexchange interaction,\nJc, which couples consecutive spin planes via a TM-O-\nLu-O-TM exchange pathway (see Fig.S8a). In the PE\nstructure, a spin in one layer is connected to three spins\nin a consecutive layer. Each of these degenerate spin-spin\ninteractions has two equivalent exchange pathways. We\n\fnd that the interlayer exchange is AFM for LFO but\nFM for LMO. Although the strength of this interaction\nis relatively weak, this sign di\u000berence turns out to be key.\nIn the PE structure symmetry implies that the relative\norientation of the spins in consecutive spin planes is arbi-\ntrary. The trimer distortion, however, splits the three de-\ngenerate interactions into: a single J11\ncinteraction, medi-\nated by two equivalent TM-O-Lu1-O-TM exchange path-\nways, and two J12\ncinteractions, where each interaction is\nmediated by a TM-O-Lu1-O-TM and a TM-O-Lu2-O-\nTM exchange pathway (see Fig.S8b). This remarkably\nintroduces an extra contribution to the energy\nESE\ninter= 2\u0001Jccos(\t 1\u0000\t2) (8)\nwhere \u0001Jc=J11\nc\u0000J12\nc, the sign of which is key in deter-\nmining the spin con\fguration type: A-type (\t 2=\t1+\u0019)\nfor \u0001Jc>0 or B-type (\t 2=\t1) for \u0001Jc<0.\nA simple structural analysis shows that the super-\nsuper exchange mediated through the Lu 2ion is always\nweaker than that mediated through the Lu 1ion, and in-\ndeed our calculations show that the magnitude of J11\ncis\nalways larger than J12\nc(see Supplemental). We therefore\nsee that the choice between A-type and B-type in the FE\nstructure is in fact determined by the sign of Jcin the\nPE structure. This is important. In ferrites, the AFM\nnature of the interlayer exchange is uniquely determined\nby the orbital occupancy, it is always AFM and therefore\nferrites will always prefer A-type magnetic con\fgurations\nand the wFM ground state. (Although the interlayer ex-\nchange in LMO is FM, which explains why it prefers B-\ntype magnetic con\fgurations, it is not universally so; a\ndiscussion is given in the Supplement).\nV. SUMMARY\nIn this paper we have discussed an intriguing con-\nsequence of improper ferroelectricity in the hexagonal\nmanganite-like systems. We have shown for the \frst time\nthat a non-polar trimer structural distortion not only\ninduces an electrical polarization but also induces bulk\nweak-ferromagnetism and a bulk linear magnetoelectric\nvortex structure. It is a universal feature of A 2-type hexa\nsystems in which the trimer distortion mediates an intrin-\nsic bulk trilinear-coupling of the polarization, magnetiza-\ntion, and antiferromagnetic order.6\nNote that it was recently inferred from neutron di\u000brac-\ntion21that LFO orders above room temperature (high for\na frustrated magnet) in an AFM state with M= 0, and\nat a lower temperature undergoes a reorientation tran-\nsition to the A0phase inducing a M6= 0. As shown in\nFig. 3a, the A 1(M= 0 by symmetry) and A0(\fnite\nMallowed) states, lie energetically very close to ground\nstate in LFO, which support such a picture.\nThere is, however, an intriguing alternative scenario\ninvolving a crossover from a state in which several mag-\nnetic order types are degenerate to the A 2spin con\fgura-\ntion ground state, driven by the trimer distortion. Note\nthat the symmetry of the PE structure not only implies\nthat the A and B spin con\fgurations are degenerate, but\nin fact that all of the principle spin con\fgurations are\ndegenerate as there can be no in plane anisotropy. The\ntrimer distortion lifts the degeneracy between the A and\nB spin con\fgurations as\n\u0001Jc/Q2\nK3(9)\nwhile the in plane anisotropy due to the e\u000bective DM-\nlike interaction (the trimer distortion induced in plane\nanisotropy of the SIA tensor is negligible) always favors\nphases with canted spins and is lifted as\nde\u000b/QK3 (10)\nas we show in the Supplemental. In this picture, as tem-\nperature is lowered and the trimer distortion increases in\nmagnitude, there is a smooth crossover to the A 2state.\nVI. METHOD\nThe \frst principles calculations were performed\nusing the DFT+U method31with the PBE form of\nexchange correlation functional32. We considered Lu\n4fstates in the core and for TM 3 dstates we chose\nU= 4:5 eV andJH= 0:95 eV. Structural relaxations,\nfrozen phonon and electric polarization calculations\nwere performed without the spin-orbit coupling (SOC)\nusing the projected augmented plane-wave basis based\nmethod as implemented in the VASP33,34. We used\na 4\u00024\u00022 k-point mesh and a kinetic energy cut-o\u000b of\n500 eV. The Hellman-Feynman forces were converged\nto 0.001 eV/ \u0017A. The electronic and magnetic properties\nwere studied in the presence of SOC. We additionally\ncross-validated the electronic and magnetic properties\nusing the Full-potential Linear Augmented Plane Wave\n(FLAPW) method as implemented in WIEN2K code35.\nVII. SUPPLEMENTARY MATERIALS\nA. Landau Theory\nIn this section we use a phenomenological Landau the-\nory to show the existence of a trilinear coupling betweenthe antiferromagnetic order, the trimer distortion and the\nmagnetization.\nThe unit cell contains two triangular layers of transi-\ntion metal (TM) ions. Within the layer \u000b(\u000b=I, II) a\nlocal magnetic structure is a combination of 120\u000eanti-\nferromagnetic order in the xyplane and a magnetization\nalong thezaxis,Mz\n\u000b. The former can be represented by\na complex order parameter\nL\u000b=Lxyei\t\u000b(11)\nwhere the angles \t Iand \t IIdescribe in-plane spin di-\nrections of two reference TM ions, as we consider S1and\nS2respectively, from adjacent layers connected by the ~2c\naxis that remains in the FE phase, see Fig. 4b. Therefore\n\tI\u0011\t1and \t II\u0011\t2. Note that because of symmetry\nwe only need to consider one of the three spins in each\nlayer.\nThe layer magnetizations can be alternatively repre-\nsented by the zcomponents of the net magnetization\n(Mz\u0011(Mz\nI+Mz\nII)=2) and staggered magnetization\n(Lz\u0011(Mz\nI\u0000Mz\nII)=2).\nThe trimer distortion corresponds to the condensation\nof the zone-boundary K3mode. While the small repre-\nsentation of K3is one-dimensional, the star contains two\nwavevectors ( kand\u0000k) and therefore the trimer distor-\ntion can be described by a complex order parameter\nQI=QK3ei\b1(12)\nIt turns out that \b 1\u0011\b transforms as the phase angle\nthat describes the in-plane displacement of some refer-\nence apical oxygen in the FE phase which we choose to\nlie directly above the reference TM ion used for de\fning\n\t1, , see Fig. 4b and c. While the order parameter QI\nfully describes the trimer distortion, it is convenient to\nintroduce additional trimer order parameter\nQII=QK3ei\b2(13)\nwith \b 2=\u0019\u0000\b1. As shown in Fig. 4c, \b 2describes the\nin-plane displacement of the apical oxygen lying directly\nabove the reference TM ion used for de\fning \t 2. The\nintroduction of the second trimer order parameter allows\nus to represent the structural distortion in an analogous\nway as the magnetic ordering which will lead to a par-\nticularly transparent form of coupling between structure\nand magnetism.\nThe character table below shows the transformation\nproperties of the L\u000bandQ\u000border parameters as well as\ntheir complex conjugates with respect to symmetry op-\nerations of the P6 3/mmc1' reference structure (included\nare only the symmetry elements that are broken by the\nmagnetic and/or the trimer orderings). In addition, the\ntransformation properties of the following combinations\nof these order parameters are shown\nX\u0011=QIL\u0003\nI+\u0011QIIL\u0003\nII (14)\nwhere\u0011=\u0006. Note that X\u0011are the only bilinear combi-\nnations of antiferromagnetic and trimer order parameters7\nL1 L2 !1!2!1!2a b c \nFIG. 4: Magnetic and trimer order parameters. a \"Clover-leaf\" structural domain pattern, where antiphase structural\ndomains are clamped to ferroelectric domain walls. bThe spin angles \t 1(\u0011\tI) and \t 2(\u0011\tII), describing the in-plane spin\ndirections for two reference TM spins S 1and S 2respectively. cPhase angles \b 1(\u0011\bI) and \b 2(\u0011\bII) that describe the local\ntrimer distortion for layer I and II respectively, where \b 1= \b and \b 2=\u0019\u0000\b1.\nthat are invariant under translation. The transformation\nproperties of Mz,Lz, and thezcomponents of the elec-\ntric polarization ( Pz) are also shown.\nTABLE I: The character table showing the transformation\nproperties of L\u000bandQ\u000border parameters as well as their\ncomplex conjugates with respect to symmetry operations of\nthe P6 3/mmc1' reference structure. In addition the trans-\nformation properties of X\u0011,Pz,Mz, andLzare shown. We\nde\fned\u0011=\u0006and\u001e=e\u0000i2\u0019=3\n.\nSx~2cmxyIR\nPz+Pz+Pz+Pz\u0000Pz+Pz\nQI\u001eQI\u0000QII+Q\u0003\nI+QII+QI\nQ\u0003\n1\u001e\u0003Q\u0003\nI\u0000Q\u0003\nII+QI+Q\u0003\nII+Q\u0003\nI\nQII\u001e\u0003QII\u0000QI+Q\u0003\nII+QI+QII\nQ\u0003\nII\u001eQ\u0003\nII\u0000Q\u0003\nI+QII+Q\u0003\nI+Q\u0003\nII\nLI\u001eLI\u0000LII\u0000L\u0003\nI+LII\u0000LI\nL\u0003\nI\u001e\u0003L\u0003\nI\u0000L\u0003\nII\u0000LI+L\u0003\nII\u0000L\u0003\nI\nLII\u001e\u0003LII\u0000LI\u0000L\u0003\nII+LI\u0000LII\nL\u0003\nII\u001eL\u0003\nII\u0000L\u0003\nI\u0000LII+L\u0003\nI\u0000L\u0003\nII\nX\u0011+X\u0011\u0011X\u0011\u0000X\u0003\n\u0011\u0011X\u0011\u0000X\u0011\nMz+Mz+Mz\u0000Mz+Mz\u0000Mz\nLz+Lz\u0000Lz\u0000Lz\u0000Lz\u0000Lz\nFrom the above table it is clear that the following two\nfree energy invariants are allowed:\nFMz\ntri\u0018<[X+]Mz (15)FLz\ntri\u0018<[X\u0000]Lz (16)\nwhere<denotes a real part. These invariants can be\nalternatively written as\nFMz\ntri/(QI\u0001LI+QII\u0001LII)Mz (17)\nFLz\ntri/(QI\u0001LI\u0000QII\u0001LII)Lz (18)\nwhere we de\fned two-dimensional real vectors L\u000b=\nLxy(cos \t\u000b;sin \t\u000b) and Q\u000b=QK3(cos \b\u000b;sin \b\u000b). As\na consequence of this trilinear coupling, presence of\nthe trimer distortion and the 120\u000eantiferromagnetic or-\nder leads to weak (anti)ferromagnetism with (staggered)\nmagnetization given by\nMz/(QI\u0001LI+QII\u0001LII) (19)\nLz/(QI\u0001LI\u0000QII\u0001LII) (20)\nNote that due to equivalence of TM ion layers we have\njMz\nIj=jMz\nIIjwhich in turn implies that the proportional-\nity coe\u000ecients in Eqs. (19) and (20) are equal. Therefore,\nby adding and subtracting Eqs. (19) and (20) we obtain\nMz\n\u000b/Q\u000b\u0001L\u000b (21)\nwhere the proportionality coe\u000ecient doesn't depend on\n\u000b. The above equation clearly shows that for a given TM\nion layer canting appears when a projection of an xy-\nspin along the direction of the local trimer distortion is\nnonzero which is for any spin con\fguration except A 1and\nB2structures. Further, if the projection have the same\nsign for adjacent layers (A spin con\fgurations) there is\na net magnetization while if for neighboring layers the8\nprojections have opposite signs (B spin con\fgurations)\nwe have weak anti ferromagnetism.\nIn order to make a connection of the phenomenological\ntheory with our microscopic model let's de\fne the vector\nde\u000b\n\u000bbyQ\u000b=^z\u0002de\u000b\n\u000b. Then invariants (17) and (18) can\nbe written as\nFMz\ntri/(de\u000b\nI\u0001(LI\u0002^z) +de\u000b\nII\u0001(LII\u0002^z))Mz(22)\nFLz\ntri/(de\u000b\nI\u0001(LI\u0002^z)\u0000de\u000b\nII\u0001(LII\u0002^z))Lz(23)\nwhile the layer magnetization is given by\nMz\n\u000b/de\u000b\n\u000b\u0001(L\u000b\u0002^z) (24)\nWhere de\u000b\nI\u0011de\u000b\n1andde\u000b\nII\u0011de\u000b\n2. We can thus in-\nterpret de\u000b\n\u000bas an e\u000bective Dzyaloshinskii-Moriya (DM)\nvector for layer \u000b. We will show in the next section\nthatde\u000b\n\u000boriginates from the transverse component of the\nDM interaction and o\u000b-diagonal elements of the single-\nion anisotropy (SIA) tensor which are both induced by\nthe trimerization distortion.\nB. Spin-Lattice coupling from the\nDzyaloshinskii-Moriya interaction and Single-Ion\nAnisotropy\nThe Dzyaloshinskii-Moriya interaction of a sin-\ngle layer of bipyramids: We considered only in-plane\nnearest-neighbor (nn) DM interactions which are medi-\nated by TM-O eq-TM paths (here O eqdenotes an equato-\nrial oxygen atom). In the paraelectric phase all nn DM\nvectors are equivalent and only their zcomponents are\nnonzero. Note that the \u001bzmirror plane requires DM vec-\ntors for adjacent TM-TM bonds to be opposite, see Fig.\n5a. Physically this is a result of di\u000berent chiralities of\nTM-O eq-TM hopping paths for these DM vectors. The z\ncomponents of the DM vectors con\fne the spins within\nthexyplane and don't contribute to canting. They are\nthus ignored in the following discussion.\nIn the ferroelectric phase we have two nonequivalent\nequatorial oxygens: O1\neqand O2\neq. Consequently, the nn\nDM vectors split into two nonequivalent types: one medi-\nated by a TM-O1\neq-TM path and the other mediated by a\nTM-O2\neq-TM path. In addition, both types of DM vectors\nacquire a nonzero transverse ( xy) component which is\nperpendicular to the corresponding TM-TM bond. Note\nthat for DM vectors mediated by TM-O2\neq-TM path the\ncomponent parallel to the TM-TM bond is, in general,\nallowed by symmetry due to di\u000berent orientations of api-\ncal oxygen displacements for the two TM ions. How-\never, since apical oxygens have minor e\u000bect on TM-TM\nhopping, the parallel component is small which was con-\n\frmed by our \frst principles calculations. In the follow-\ning we thus neglected the parallel component to obtainclearer picture of DM interactions. We point out, how-\never, that the inclusion of the parallel component doesn't\na\u000bect our main conclusions. Transverse components of\nDM vectors between TM site 1 (see Fig. 5a) and its\n3\n5\n3 5QK3= 0\nQK3> 0\nQK3< 0x^y^τxz\nτxz\n−τ xz\n−τ xzD13D15 D13\nD15\nD13D15d1σz\n35a\n1b\nτxx\nxx\nzzτ 0\n0 00 0\n0τ\nτ\nτ\nτxx\nyy\nzz0\n00\n0\nτ\nτ\nτxx\nyy\nzz0\n00\n0\nFIG. 5: Dzyaloshinskii-Moriya interactions and Single-\nIon Anisotropy. a The in-plane nn DM vectors for a single\ntriangular layer of TM ions, bSIA tensors for the paraelectric\nphase and for two opposite directions of the trimer distortion\nin the ferroelectric phase. All six DM vectors acting on TM\nsite 1 and SIA tensor for this site are shown. DM vectors\nand SIA tensors for other bonds and sites can be generated\nby applying the appropriate symmetry operations of crystal\nspace group. In the paraelectric phase only the zcomponent\nof the nn DM vectors are non-zero since triangular layers are\nmirror planes. The cross (dot) mark represents direction of a\nDM vector along positive (negative) ^ zaxis. In the ferroelectric\nstructure, the trimer distortion lowers the symmetry, leading\nto two nonequivalent types of in-plane nn DM vectors: one\nmediated by TM-O1\np-TM path and the other mediated by TM-\nO2\np-TM path.\nnearest neighbors for di\u000berent trimer domains are shown\nin Table VII B.\nHere we derive the relationship between the local struc-\ntural distortions and the induced Dxy\nij. Let's consider the\nlayer\u000b= I. The DM interaction energy (per spin) is given\nby (see notation in Fig. 5a)9\nTABLE II: Transverse components of DM vectors between TM site 1 (see Fig. 5a) and its nearest neighbors for di\u000berent trimer\ndomains.\n\b Dxy\n13 Dxy\n130 Dxy\n1300\n0Dxy(\u00001\n2;\u0000p\n3\n2)D0\nxy(\u00001;0)D0\nxy(1\n2;\u0000p\n3\n2)\n\u0019=3D0\nxy(\u00001\n2;\u0000p\n3\n2)D0\nxy(1;0)Dxy(1\n2;\u0000p\n3\n2)\n2\u0019=3D0\nxy(1\n2;p\n3\n2)Dxy(1;0)D0\nxy(1\n2;\u0000p\n3\n2)\n\u0019Dxy(1\n2;p\n3\n2)D0\nxy(1;0)D0\nxy(\u00001\n2;p\n3\n2)\n4\u0019=3D0\nxy(1\n2;p\n3\n2)D0\nxy(\u00001;0)Dxy(\u00001\n2;p\n3\n2)\n5\u0019=3D0\nxy(\u00001\n2;\u0000p\n3\n2)Dxy(\u00001;0)D0\nxy(\u00001\n2;p\n3\n2)\n\b Dxy\n15 Dxy\n150 Dxy\n1500\n0Dxy(1\n2;\u0000p\n3\n2)D0\nxy(\u00001\n2;\u0000p\n3\n2)D0\nxy(1;0)\n\u0019=3D0\nxy(1\n2;\u0000p\n3\n2)D0\nxy(1\n2;p\n3\n2)Dxy(1;0)\n2\u0019=3D0\nxy(\u00001\n2;p\n3\n2)Dxy(1\n2;p\n3\n2)D0\nxy(1;0)\n\u0019Dxy(\u00001\n2;p\n3\n2)D0\nxy(1\n2;p\n3\n2)D0\nxy(\u00001;0)\n4\u0019=3D0\nxy(\u00001\n2;p\n3\n2)D0\nxy(\u00001\n2;\u0000p\n3\n2)Dxy(\u00001;0)\n5\u0019=3D0\nxy(1\n2;\u0000p\n3\n2)Dxy(\u00001\n2;\u0000p\n3\n2)D0\nxy(\u00001;0)\nEDM\nI=2\n3(Dxy\n13+Dxy\n130+Dxy\n1300)\u0001(S1\u0002S3) +2\n3(Dxy\n35+Dxy\n3500+Dxy\n305)\u0001(S3\u0002S5) +2\n3(Dxy\n51+Dxy\n501+Dxy\n5001)\u0001(S5\u0002S1)\n=2\n3\u0016D13\u0001(S1\u0002S3) +2\n3\u0016D35\u0001(S3\u0002S5) +2\n3\u0016D51\u0001(S5\u0002S1) (25)\nwhere we de\fned the bar DM vectors\n\u0016D13\u0011Dxy\n13+Dxy\n130+Dxy\n1300=\u0016Dxy(cos(\b\u00002\u0019=3);sin(\b\u00002\u0019=3)) (26)\n\u0016D35\u0011Dxy\n35+Dxy\n3500+Dxy\n305=\u0016Dxy(cos(\b);sin(\b)) (27)\n\u0016D51\u0011Dxy\n51+Dxy\n501+Dxy\n5001) =\u0016Dxy(cos(\b + 2\u0019=3);sin(\b + 2\u0019=3)) (28)\nwith \u0016Dxy=Dxy+D0\nxy. The above expressions can be\neasily obtained from Table VII B or Fig. 5a. Note that\nthe bar DM vectors have magnitude \u0016Dxyand form a 120\u000e\nangle with each other.\nTherefore, a much simpler picture emerges; the rela-\ntionship between the local structural distortions and the\ninduced Dxy\nijcan be derived by considering a single tri-angle of spins ( S1,S3andS5) interacting by the bar DM\nvectors, see Fig. 6 for the case of the \u000b+and\u000b\u0000domains.\nSince all spins cant in the same direction we can write\nSi=Li+MIwhere Liare de\fned in Fig. 1d of the\nmain manuscript and MIis the parallel to the zaxis\nlayer magnetization. We then obtain\nEDM\nI=2\n3(\u0016D13+\u0016D15)\u0001(L1\u0002MI)+2\n3(\u0016D35+\u0016D31)\u0001(L3\u0002MI)+2\n3(\u0016D51+\u0016D53)\u0001(L5\u0002MI) =1\n3X\ni=1;3;5di\u0001[Li\u0002MI] (29)\nwhere the di's, e.g., d1\u00112(\u0016D13+\u0016D15), are the e\u000bective, transverse DM-interactions. Using A\u0001(B\u0002C) =B\u0001(C\u0002A)\nandMI=Mz\nI^ zthe DM energy can be rewritten as\nEDM\nI=1\n3Mz\nIX\ni=1;3;5Li\u0001[^ z\u0002di] =2p\n3\n3\u0016DxyMz\nIX\ni=1;3;5Li\u0001^Qi (30)\nwhere we used 2p\n3\u0016Dxy^Qi=^ z\u0002diwith ^Qi's being unit vectors de\fned in Fig. 1d of the main manuscript. The10\nrelation between diand^Qican be straightforwardly ob-\ntained from Eqs. 30 but more physical insight into this\nrelation can be gained by noting that Dxy\nij/^ rij\u0002uOeq\nwhere ^ rijis the unit vector pointing from site itowards\njanduOeqis the displacement of the equatorial oxygen\naway from the plane (e.g., due to the tilting of the bipyra-\nmid) which is zero in the PE phase and parallel to the z\naxis in the FE phase. For the \u000b\u0006trimer domains we \fnd\nd1/\u0006^ z\u0002(^ r13+^ r15) (31)\nAs seen from Fig. 6b, ^ r13+^ r15=^Q1giving\n^Q1/\u0006^ z\u0002d1 (32)\nOther dican be found by cyclic permutations: 1 !3,\n3!5, and 5!1. The same results can be obtained for\nother trimer domains.\nNote that\nL1\u0001^Q1=L3\u0001^Q3=L5\u0001^Q5\u0011LI\u0001^QI=Lxycos(\b\u0000\t1)\n(33)\nWe thus obtain\nEDM\nI= 2p\n3\u0016DxyMz\nILI\u0001^QI=Mz\nIL1\u0001^ z\u0002d1 (34)\nwhich leads to\nMz\nI/L1\u0001^ z\u0002d1= 2p\n3\u0016DxyLI\u0001^QI (35)\nin agreement with Eqs.21 obtained from the Landau\ntheory. It is now clear that the microscopic origin of the\nlayer magnetization is the the trimer induced transverse\ncomponents of the DM interactions which cant spins\naway from the xyplane. As we will see in the next\nsection, however, there is also another contribution to\nthe canting that originates from the single-ion anisotropy\n(SIA).\nThe Single-ion anisotropy of a single layer of\nbipyramids: In the paraelectric phase the crystal \feld\nhas the same orientation for all TM ions so the single-ion\nanisotropy (SIA) tensor, ^\u001ci, does not depend on magnetic\nsite indexi. A global coordinate system (see Fig. 5) can\nbe thus chosen in which ^\u001cis diagonal with elements \u001cxx,\n\u001cyy,\u001czz. A uniaxial site symmetry and the zero-trace\ncondition lead to \u001cxx=\u001cyy=\u0000\u001czz=2.\nOn the other hand, in the ferroelectric phase the crys-\ntal \feld may have di\u000berent orientations for di\u000berent TM\nions and therefore ^\u001cidoes depend on i. Even though\nall TM ions remain equivalent and thus SIA tensors for\ndi\u000berent magnetic sites are related by symmetry, in any\nglobal coordinate system SIA tensor for some magnetic\nions have o\u000b diagonal components. In addition, the uni-\naxial site symmetry is lost in the ferroelectric phase lead-\ning to the in-plane anisotropy ( \u001cxx6=\u001cyy). In the coor-\ndinate system as in Fig. 5a the SIA tensor for site 1 in\nthe \b = 0 trimer domain is given by\n✕\nS1 S3 S5 \n¤\t\r S1 S3 S5 \nS5 S3 S1 ˆr13ˆr15ˆr13+ˆr15\nˆDij!(\"ˆz#ˆrij)ˆd1![(\"ˆz)#(ˆr13+ˆr15)]ˆDij!(+ˆz\"ˆrij)ˆd1![(+ˆz)\"(ˆr13+ˆr15)]=![(!ˆz)\"(ˆr13+ˆr15)]\n¤\t\r \n!Oep displacing || -z ¤\t\r !\nOep displacing || +z \nM \nM \nL1 L3 L5 \nQ1 Q3 Q5 L1 Q1 L3 Q3 Q5 L5 !+(!=0)a !!(\"=\")b c \nd !D15!D53!D31!D15!D53!D31FIG. 6: Microscopic: Basic ingredients to describe\nconnection between microscopic spin-lattice model\nand a simple phenomenological model. a andb, local\ntrimer distortion around S1from two di\u000berent projections.\nc, the transverse components of DM vectors are shown for\na single triangle in a single layer. Cross and dot signs de-\nnote displacement of the equatorial oxygen along -^ zand +^z\nrespectively. d, demonstrates the relation Mz/(L:Q)^z.\n^\u001c=2\n64\u001cxx0\u001cxz\n0\u001cyy0\n\u001cxz0\u001czz3\n75 (36)\nFor a general trimer domain the SIA tensor for site 1\nis given by R\b^\u001cR\u00001\n\bwhereR\bis a rotation matrix\nR\b=2\n64cos \b\u0000sin \b 0\nsin \b cos \b 0\n0 0 13\n75 (37)11\nThe e\u000bect of trimer distortion on the components of\nthe SIA tensor can be understood if we assume that the\ncrystal \feld for a given TM ion is determined solely by\nits oxygen bypyramid. In this case the components of ^\u001c\nin Eq. (36) can be expressed in terms of the tilting angle\n\u0012and the value of \u001czzin the paraelectric phase (hereafter\ndenoted by \u001c0\nzz). For site 1 we have:\n\u001cxx=\u0000\u001c0\nzz=2 (38)\n\u001cyy=\u0000cos2(\u0012)\u001c0\nzz=2 + sin2(\u0012)\u001c0\nzz (39)\n\u001czz= cos2(\u0012)\u001c0\nzz\u0000sin2(\u0012)\u001c0\nzz=2 (40)\n\u001cxz=\u00003 sin(2\u0012)\u001c0\nzz=4 (41)First principles calculations show that the in-plane\nanisotropy is very small (see Table. V). Indeed, as seen\nfrom Eqs. (41) this di\u000berence is proportional to sin2\u0012\nwhich is a very small quantity. On the other hand, the\no\u000b-diagonal component, \u001cxzis proportional to sin 2 \u0012and\nis correspondingly substantially larger and plays impor-\ntant role in the canting.\nLet's consider the SIA contribution to the canting en-\nergy (per spin) for the layer \u000b= I\nESIA\nI=1\n3X\ni=1;3;5Si\u0001^\u001c1\u0001Si=2\n3\u001cxzMz\nIX\ni=1;3;5Li\u0001^Qi= 2\u001cxzMz\nILI\u0001^QI (42)\nwhere we kept only the terms proportional to Mz\nI. The\nabove equation has a similar form as Eq. 30. Indeed, we\ncan de\fne a DM-like vector dSIA\nias 2\u001cxz^Qi=^ z\u0002dSIA\ni.\nWe then get\nESIA\nI=1\n3X\ni=1;3;5dSIA\ni\u0001[Li\u0002Mz\nI] (43)\nThe total energy and magnetization of a single\nlayer of bipyramids: Combining Eqs. (30) and (43)\nwe obtain\nEcanting\nI =EDM\nI+ESIA\nI=1\n3X\ni=1;3;5de\u000b\ni\u0001[Li\u0002MI] (44)\nwhere de\u000b\ni=di+dSIA\niis the e\u000bective transverse DM\nvector with the magnitude jde\u000bj= 2jp\n3\u0016Dxy+\u001cxyj.\nAgain using A\u0001(B\u0002C) =B\u0001(C\u0002A) and MI=Mz\nI^ z,\nEcanting can be rewritten as\nEcanting\nI =Mz\nIX\ni=1;3;5Li\u0001[^ z\u0002de\u000b\ni]\n=jde\u000bjMz\nILI\u0001^QI (45)so that the layer magnetization due to canting is given\nby\nMz\nI\u0019(p\n3\u0016Dxy+\u001cxz)\n6JnnLI\u0001QI (46)\nwhereJnnis the nn exchange interaction.\nWe thus recovered the result from Landau Theory.\nThe real structure: the stacking of two layers:\nLet us consider now a real hexa structure which is com-\nposed of two layers \u000b= I and\u000b= II, each with a, in\nprinciple di\u000berent, layer magnetization, Mz\nIandMz\nIIre-\nspectively. The canting energy is\nEcanting = (Ecanting\nI +Ecanting\nII )=2 =jde\u000bj[Mz\nI(LI\u0001QI) +Mz\nII(LII\u0001QII)]=2 (47)\nThis result shows clearly that the B 1state displays weak-\nantiferromagentism, wAFM, i.e., there is a canting of the\nspins out of each spin plane, but since the projectionchanges sign in adjacent layers, i.e., LI\u0001QI= -LII\u0001QII,\nno net magnetization exists, Mz=Mz\nI+Mz\nII= 0. In\nthe A 2phase, however, the projection has the same sign12\nin adjacent layers, LI\u0001QI= +LII\u0001QII, and therefore\nMz\nI=Mz\nIIleading to a net magnetization along the z\naxis. This result explains our \frst-principles calculations\ndisplayed in Fig.1 of the main manuscript and provides\na microscopic justi\fcation for the results of our simple\nLandau theory.\nThe above results can be rewritten in terms of the\ntrimer phase and the spin angles\nEA2\ncanting/MzQK3Lxy[cos(\t 1\u0000\b)\u0000cos(\t 2+ \b)] (48)\nEB1\ncanting/LzQK3Lxy[cos(\t 1\u0000\b) + cos(\t 2+ \b)] (49)\ndescribing weak-ferromagnetism for the A2phase\nand weak-antiferromagnetism for the B1phase re-\nspectively. Notice that if Pswitched via rotating \b\nbyj\u0019=3j, e.g.,\u000b+!\f\u0000,Lmust rotate by either\nj\u0019=3j(L\u0001Q\u000b+=\u00001!L\u0001Q\f\u0000=\u00001) orj2\u0019=3j,\n(L\u0001Q\u000b+=\u00001!L\u0001Q\f\u0000= +1).\nC. Magnetoelectric e\u000bect\nThe ferroelectric phase in the A 2magnetic structure\nhas P6 3c'm' space group. The corresponding point groupis 6m'm' which allows for magnetoelectric (ME) e\u000bect\nwith magnetoelectric susceptibility tensor,\n^\u000b=2\n64\u000b?0 0\n0\u000b?0\n0 0\u000bk3\n75 (50)\nIn order to understand the origin of this ME cou-\npling we consider Landau expansion with respect to the\nP63/mmc10reference structure. The part of the free en-\nergy that depends on Mzcan be written as\nF(Mz) =1\n2aMM2\nz\u00001\n2ctrQK3Mz(^QI\u0001LI+^QII\u0001LII) (51)\nwhere we de\fned ^Q\u000b=Q\u000b=QK3. Minimizing with re-\nspect toMzwe \fnd an equilibrium magnetization\nMz=1\n2ctr\naMQK3(^QI\u0001LI+^QII\u0001LII) (52)\nAssuming the in-plane spin components are rigid (this as-\nsumptions is rigorous in the A 2phase) thezzcomponent\nof the ME susceptibility is\n\u000bk=@Mz\n@Ez\f\f\f\f\nEz=0=1\n2ctr\naM(^QI\u0001LI+^QII\u0001LII)@QK3\n@Ez\f\f\f\f\nEz=0(53)\nIn order to \fnd@QK3\n@Ez\f\f\f\nEz=0we consider the free energy as a function of PzandQK3\nF(Pz;QK3) =1\n2aPP2\nz+1\n2aQQ2\nK3+1\n4bQQ4\nK3\u0000dPzQ3\nK3cos 3\b +1\n2d0P2\nzQ2\nK3\u0000EzPz (54)\nIn aboveMzwas integrated out resulting in renormalization of the aQcoe\u000ecient. Minimizing with respect to Pzwe\nobtain\nPz=dQ3\nK3cos 3\b +Ez\naP+d0Q2\nK3(55)\nWe assume that we are well below the trimerization transition and QK3is large and satis\fesd0\naPQ2\nK3>>1. Then the\nabove equation simpli\fes to\nPz\u0019d\nd0QK3cos 3\b +1\nd0Q2\nK3Ez (56)\nMinimization of Eq. (54) with respect to QK3leads to\naQQK3+bQQ3\nK3\u00003dPzQ2\nK3cos 3\b +d0P2\nzQK3= 0 (57)\nSubstituting (56) into (57) we obtain\naQQK3+~bQQ3\nK3\u0000d\nd0Ezcos 3\b +O(E2\nz) = 0 (58)\nwhere ~bQ=bQ\u00002d2=d0and we took into account that within any trimer domain cos 3\b = \u00061. Taking derivative\nwith respect to EzatEz= 0 we obtain\naQ@QK3\n@Ez\f\f\f\f\nEz=0+ 3~bQQ2\nK3(Ez= 0)@QK3\n@Ez\f\f\f\f\nEz=0\u0000d\nd0cos 3\b = 0 (59)13\nFrom (58) we obtain Q2\nK3(Ez= 0) =\u0000aQ=~bQleading to\n@QK3\n@Ez\f\f\f\f\nEz=0=\u00001\n2aQd\nd0cos 3\b (60)\nTherefore, the magnetoelectric susceptibility becomes\n\u000bk=\u00001\n4ctrd\naMaQd0(^QI\u0001LI+^QII\u0001LII) cos 3\b (61)\nNote that from ( 52) and ( 56) it follows that,\n\u000b/@Mz\n@QK3\u0003@Pz\n@QK3\u0011@QMz\u0003@QPz (62)\nFew comments are in order. First, \u000bkis nonzero only\nwhen ( ^QI\u0001LI+^QII\u0001LII) is nonzero which is exactly\nthe condition for existence of weak ferromagnetism that\nrequires that the magnetic con\fguration has a nonzero\nA2component. Second, if ( ^QI\u0001LI+^QII\u0001LII) is \fxed\n(i.e., the projections ^Q\u000b\u0001L\u000b) are equal in all domains),\nthen the sign of \u000bkswitches as we go from prime (\b =\n\u0019;\u0006\u0019=3) to nonprime (\b = 0 ;\u00062\u0019=3;) trimer domains.\nIn other words, the domains with parallel MzandPz\nhave an opposite sign of \u000bkthan domains with Mzand\nPzantiparallel.\nD. Implications of the trilinear coupling\nThe trilinear coupling of Eq. 47 is quite remarkable.\nIt implies that in the A 2phase the trimer distortion\nnot only induces a polarization but also mediates a non-\ntrivial bulk P-Mcoupling. To make this clear, we con-\nsider a thought experiment in which an electric \feld, E\napplied along the zaxis can switch Pto any one of the\nthree trimer domains with \u0000P(more in the Discussion).\nLet the system be initially in the \u000b+domain with polar-\nization + PandQ\u000b\u0001L\u000b=\u00001, Fig. 7b. Then there are\ntwo possible scenarios:\nf\u000b+!\u000b\u0000g. In a proper FE like PbTiO 3the struc-\nture of the + Pdomain is related to the \u0000Pdomain by\na reversal in the direction of the polar distortions w.r.t\nthe PE structure, e.g, the Ti4+ion moving from up to\ndown. The analogous situation in the hexa systems corre-\nsponds to a structural change from a 2-up/1-down buck-\nling and tilting `out' of the R-planes and bypyramids,\nrespectively, to a 1-up/2-down and tilting `in', while re-\nmaining in the same distinct domain, e.g., \u000b+. This cor-\nresponds to switching Pvia rotating \b by \u0019(Fig. 7c).\nIn this\u000b\u0000domain, because of Eq. 47, either L\u000bhas to\nrotate 180\u000e(Q\u000b\u0001L\u000b=\u00001) or the small canting angle has\nto change sign ( Q\u000b\u0001L\u000b= +1). It is not unreasonable\nto expect the latter to be more favorable, leading to a\nreversal of M.\nf\u000b+!\f\u0000g. The improper nature of ferroelectricity,\nhowever, o\u000bers an even more interesting possibility is that\nthere exists three distinct and accessible domains ( \u000b,\f,\nand\r). As an example let Pswitch via rotating \b by \u0019=3\nβ−domain Φ=π/3 α−domain Φ=π\nβ−domain Φ=π/3Q.L ~ −1α+domain Φ=0\nx^y^x^y^\nΦ2\n. ..β−domain Φ=π/3\nΦ1\nL switches by 60OP = +P \nQ= −Q  Initial configuration\nImmediately after switching Q to\nA long time after switching to \nTwo possible equilibrium magnetic domains\nL switches by −120\n= −Q \nQ.L= +L \n= −M (M ~        )= +P Q.L ~ −1\nP \nQ\nL\nM= −Q \nQ.L = +M (M ~        )= +P Q.L ~ +1\nP \nQ\nL\nM= −Q \nQ.L = +M (M ~        )= +P~ +1\nP \nQ\nL\nM= −L \n= −L QP \nL\nM= −M (M ~        ) Q.La b\nc d\ne fout of equilibriumBy symmetry\nΦ1 = π−Φ2   Φ\nO= −P \n= +Q \n= −L\nQ.L −1 < Q.L <0 FIG. 7: Bulk cross-coupling of polarization and mag-\nnetization in A 2phase. Thought experiment to elucidate\nthe cross-coupling of ferroelectricity and magnetism mediated\nby the trimer distortion for the predicted A 2magnetic ground\nstate. a,De\fnition of local trimer angles \b 1and \b 2,b,Initial\nequilibrium \u000b+domain\u0011f-P,+Q, -L xy, -Mzg. Immediately\nafter switching c,to the\u000b\u0000domain,Lxystill in ground state,\ntherefore -M z!+Mz;\u000b\u0000domain\u0011f+P,-Q, -L xy, +Mzg,\nd,\f\u0000domain,Lxyis not in a ground state and must rotate\neither, e,60\u000e, therefore -M z!-Mz;\f\u0000domain\u0011f+P,-Q,\n+Lxy, -Mzg, orf,120\u000e, therefore -M z!+Mz;\f\u0000domain\u0011\nf+P,-Q, -L xy, +Mzg.\nand consider the con\fguration immediately after, Fig. 7d.\nIn this\f\u0000domain\u00001<Q\u000b\u0001L\u000b<0, implying the system\nis not in equilibrium, and therefore L\u000bmust rotate by\neitherj\u0019=3j, Fig. 7e, orj2\u0019=3j, Fig. 7f. In the former case\nQ\u000b\u0001L\u000b=\u00001 as in the initial \u000b+con\fguration, therefore\nMis not reversed, while in the latter Q\u000b\u0001L\u000b= +1 and14\nTABLE III: Calculated frequencies of relevant phonon modes\nfor the high-temperature phase of RTMO 3(TM=Fe, Mn). In\naddition, electric polarization ( P) for the fully relaxed low-\ntemperature structures are listed.\nSystem !(cm\u00001)Ps\n\u0000\u00001\n2K1K3(\u0016C/cm2)\nHoFeO 373 276 114i 9.0\nErFeO 375 271 113i 9.2\nTmFeO 378 268 112i 9.5\nYbFeO 3102 298 97i 9.6\nLuFeO 384 263 108i 9.8\nHoMnO 365 249 125i 6.9\nErMnO 361 245 128i 7.0\nTmMnO 356 240 130i 7.2\nYbFeO 392 266 127i 7.4\nLuFeO 354 232 132i 7.4\nMswitches 180\u000e.\nE. Discussion: Possible realizations of predictions\nRealization 1: The hexagonal Manganites:\nFig. 8 shows Piezoelectric force microscopy (PFM) and\nMagnetoelectric force microscopy (MeFM) images. The\nMeFM images, were aligned to PFM image by the topo-\ngraphic landmarks and are in the same color scale, 9.18\nmHz.\nRealization 2: The A 2ground state in hexa\nRFeO 3:\n1. Origin of ferroelectricity in hexaferrites\nExperimental studies of hexagonal ferrites RFeO 3\nindicates that at room temperature these materials\nhave a polar P6 3cm structure19,20while at high tem-\nperature they crystallize in the paraelectric P6 3/mmc\nphase26.According to the group theory analysis there are\nthree possible phase transition sequences connecting the\nhigh-temperature phase with the low-temperature P6 3cm\nphase18: (1) a direct transition between the two phases\nby freezing-in a K 3zone boundary phonon mode, (2) a\ntransition to the intermediate P6 3mc ferroelectric phase\nby softening of the \u0000\u0000\n2zone-center polar mode followed\nby the transition to the low-temperature phase, and (3) a\ntransition to the intermediate non-polar P6 3/mcm phase\nby freezing-in a zone boundary K 1mode followed by tran-\nsition to the low temperature phase.\nCalculated phonon frequencies in the high-temperature\nphase for the above mentioned phonon modes are pre-\nsented in Table. III both for manganites and ferrites\nwith di\u000berent R ions. As seen, for all compounds theK3phonon mode is unstable while both \u0000\u0000\n2and K 1\nmodes are stable. These results indicate that for fer-\nrites systems, similarly as for manganites, the high-\ntemperature P6 3/mmc phase transforms directly into the\nlow-temperature P6 3cm structure by softening of the K 3\nzone boundary mode. The non-linear coupling between\nK3and the polar \u0000\u0000\n2mode induces polarization into the\nsystem. The polarization is thus a secondary order pa-\nrameter and the ferroelectricity is improper. Indeed, for\na high-temperature structure with frozen-in K 3mode\nthe calculated polarization is very small (less than 0.1\n\u0016C/cm2) and originates only from an electronic contri-\nbution. Only when full ionic relaxations are done the\npolarization of the order of 10 \u0016C/cm2develops (see Ta-\nble. III).\nThe above analysis indicates that the origin of ferro-\nelectricity in RFeO 3is the same as in manganites. There-\nfore, an intermediate P6 3mc phase which was proposed\nto explain the two-step polarization decay observed for\nYbFeO 327is rather unlikely. Indeed, the presence of such\nintermediate phase would typically lead to soft \u0000\u0000\n2mode\nwhich is not the case according to our calculations. In\nfact, we found that if we freeze-in the \u0000\u0000\n2mode and\nperform full structural relaxations, the amplitude of the\nmode goes to zero. We also found that the \u0000\u0000\n2mode re-\nmains stable for a wide range of epitaxial strain, see Fig.\n9.\n2. Non-collinear magnetic spin con\fgurations\nWe have considered 8 non-collinear magnetic con\fgu-\nrations, as depicted in Fig. 10, to determine the magnetic\nground state of the hexagonal manganite and ferrite sys-\ntems.\n3. Symmetric exchange interaction Determine the magnetic\ncon\fguration\nTABLE IV: Exchange parameters and the corresponding\nCurie-Weiss temperatures (\u0002 CW) for LFO and LMO. In the\nlast row (LMO-II) the exchange parameters obtained for LMO\nusing the lattice parameters of LFO are given.\nSystemJnnJnnnJ11\ncJ12\ncJ21\ncJ22\nc\u0002CW\nmeV meV meV meV meV meV K\nLFO 6.307 0.457 0.942 0.557 0.088 0.044 1525\nLMO 2.573 -0.254 -0.369 -0.322 -0.152 0.050 274\nLMO-II 3.152 -0.452 -0.730 -0.687 -0.417 0.113\nWe considered in-plane nearest neighbor (nn), Jnn,\nand next-nearest neighbor (nnn), Jnnn, exchange inter-\nactions as well as two di\u000berent inter-layer interactions\n(J1\ncandJ2\nc), see Fig. 11. While in the ferroelectric15\na b c \nFIG. 8: Magnetoelectric force microscopy (MeFM) images .aPiezoelectric force microscopy (PFM) image taken under\nambient condition. bandc, show MeFM images taken under the application of 6 and 0.2 T external magnetic \feld, respectively.\n-3 -2 -1 0 1 2 3\nMisfit strain (%)708090100110120130140ω (cm-1)\nFIG. 9: E\u000bect of epitaxial strain. Calculated phonon fre-\nquency of \u0000\u0000\n2mode as a function of epitaxial strain.\nphaseJnnsplits into two nonequivalent exchange inter-\nactions, the di\u000berence between them is very small and\ntherefore this e\u000bect was neglected. The same applies to\nJnnn. Similarly, J1\nc(J2\nc) splits into J11\ncandJ12\nc(J21\ncand\nJ22\nc) interactions. In this case, however, this splitting is\nessential for determination of the magnetic ground state\nand therefore it was included. The exchange interactions\nwere calculated by \ftting ab initio energies of several\ncollinear magnetic con\fgurations. The results are pre-\nsented in Table IV.\nAs seen,Jnnn is order of magnitude smaller than\nJnnand has a negligible in\ruence on magnetic ordering.\nTherefore, it was neglected in the main manuscript. On\nthe other hand, J2\ncis not always signi\fcantly smaller than\nJ1\nc. Nevertheless, its qualitative e\u000bect is quite analogous\ntoJ1\ncand therefore for clarity it was not included in the\nmain text. Here, however, this interaction is included on\nequal footing with J1\nc.\nThe dominant exchange parameter is Jnn. It is the\nS1 S2 S4 S3 S6 S5 \n!A(P63)\n!B(P63)\nI1(P3c)\nI2(P3c)P P \nP P I I \nI \nI \n!2!1\nFIG. 10: Non-collinear magnetic structures. 120\u000espin\nordering with spin directions directed by blue arrows. Light\nand dark gray balls represent magnetic TM ions situated at\nlayer I and II, respectively. The angles \t 1and \t 2de\fne non-\ncollinear spin con\fgurations. The values of angles [\t 1;\t2]\nin \b = 0 structural domain are given. Principle magnetic\nstructures, those are compatible with the crystal symmetry,\nare denoted as A 1, A2, B1and B 2. Among four principle\nmagnetic structures A 2allows weak ferromagnetism (wFM).\nThe intermediate states that connect A2 with A1 and B2,\ndenoted as A0and I 2, respectively, also allow wFM.16\nLu1Lu2Oap1Oap2Op1Op2Lu X O XJnn\nJc1Jc11\nJc12\nJc21\nJc22\nJc2Ferroelectric phasebParaelectric phasea\nFIG. 11: Symmetric exchange paths a Symmetric exchange paths in paraelectric phase of Jnn,J1\ncandJ2\ncare denoted by\nsolid black, cyan and pink lines, respectively. bThe trimer distortion splits six fold degenerate nn exchange interaction Jnninto\ntwo fold degenerate J1\nnn(mediated via TM-O1\np-TM exchange path) and four fold degenerate J2\nnn(mediated via TM-O2\np-TM\nexchange path). As the nn exchange interactions are of superexchange in nature and the in-plane distortions in terms of bond\nlengths and bond angles are not signi\fcant, we considered J1\nnn=J2\nnn=Jnn. Similarly trimer distortion splits six fold degenerate\ninter-layer exchange interactions J1\nc(J2\nc) into two fold degenerate J11\nc(J21\nc) and four fold degenerate J12\nc(J22\nc), respectively.\nWhileJ11\ncis mediated by two equivalent X-O-Lu1-O-X exchange pathways, J12\ncinteraction is mediated by one X-O-Lu1-O-X\nand one X-O-Lu2-O-X exchange pathways. J21\ncandJ22\ncare mediated via X-O-Lu1-O-X and X-O-Lu2-O-X exchange pathways,\nrespectively.\nAFM superexchange interaction mediated by equatorial\noxygens, O eq. For LFO this exchange originates primar-\nily from the hopping from O 2 pstates into minority Fe\n3dstates. Since the TM-O eq-TM angle is 120\u000ethe FM\nsuperexchange involving two perpendicular porbitals is\nexpected to be much smaller than the AFM superex-\nchange involving a single orbital. This mechanism is also\npresent for LMO but here the hopping from occupied\ninto empty 3 dorbitals is equally important. In the lat-\nter case the dominant in-plane hoppings are between e0\nande00orbitals which favor antiparallel spins. As a re-\nsultJnnfor LMO is also AFM. The magnitude of Jnn\nis smaller for LMO. This is partially caused by a larger\nband gap for this compound as compared to LFO. How-\never, probably more important contribution comes from\nthe hopping from occupied 3 dorbitals into empty major-\nityd3z2states that mediates a FM coupling and partially\ncompensates the AFM contribution. Therefore, ferrites\nare expected to have in general larger Jnnthan mangan-\nites which indicates that magnetic ordering temperature\nin ferrites should be much higher; possibly close to the\nroom temperature.\nAs seen from Table. IV, for LFO the inter-layer ex-\nchange parameters are AFM. This can be understood by\nnoting that both exchange parameters originates primar-ily from virtual hopping of two pzelectrons of O apion\ninto 3dminority states of Fe and 5 dorbitals of R. These\nhoppings are the most e\u000bective for both apical oxygens\nin the exchange path only when spins on Fe ions are an-\ntiparallel. Otherwise, pzelectrons from di\u000berent O apions\ncannot hop on the same orbital of the R ion. The above\nmechanism of AFM interlayer coupling is also present for\nLMO. In this case however, similarly as for in-plane ex-\nchange interactions, there is another type of low-energy\nexcitations that can contribute to the interlayer exchange\ncoupling. They involve hopping from occupied 3 dMn\nstates into empty 3 dstates of Mn ion in the adjacent\nplane (note that this hopping is still mediated by O ap\nand R ions on the exchange path). Clearly, the most im-\nportant hopping in this case is between \flled majority\ne0and empty majority d3z2which prefers parallel spins.\nThis leads to FM coupling which competes with AFM\nexchange interaction from the above paragraph. As a\nresult either sign of interlayer exchange couplings is ex-\npected for LMO.\nImportantly, we \fnd that jJk1\ncj>jJk2\ncj(k= 1;2) both\nfor LMO and LFO. In fact this is a generic feature for\nthese systems. Indeed, consider three di\u000berent nn inter-\nlayer interactions: J12,J14, andJ16(see Fig. 11 for\nnotation). In the paraelectric phase they are all equal to17\nJ1\ncand their change under the trimer distortion can be written in the following way (up to second order in QK3):\nJ1j\u0019J1\nc+\u0012@J1j\n@QI\u0013\n0QI+\u0012@J1j\n@Q\u0003\nI\u0013\n0Q\u0003\n1+\u0012@2J1j\n@Q2\nI\u0013\n0Q2\nI+\u0012@2J1j\n@(Q\u0003\nI)2\u0013\n0(Q\u0003\nI)2+\u0012@2J1j\n@QI@Q\u0003\nI\u0013\n0QIQ\u0003\nI (63)\nwherej= 2;4;6, subscript 0 indicates that the deriva-\ntives must be evaluated in the paraelectric phase, and\norder parameters QI,Q\u0003\nIwere introduced in the previous\nsection. Note that the derivatives must be invariant un-\nder the symmetry operations of the paraelectric phase. In\nparticular, this condition requires \frst order derivatives\nto be zero and the mixed second order derivatives to be\nindependent on index j. Further it imposes relations be-\ntween remaining second order derivatives such that we\ncan write\nJ12\u0019J1\nc+cQ2\nK3+ 2Xcos(2\b)Q2\nK3(64)\nJ14\u0019J1\nc+cQ2\nK3+ 2Xcos(2\b + 2\u0019=3)Q2\nK3(65)\nJ16\u0019J1\nc+cQ2\nK3+ 2Xcos(2\b\u00002\u0019=3)Q2\nK3(66)\nwherecis a constant and X=\u0010\n@2J12\n@Q2\nI\u0011\n0. In the\u000b(\b = 0)\nphase we have\nJ12=J11\nc\u0019J1\nc+cQ2\nK3+ 2XQ2\nK3(67)\nJ14=J16=J12\nc\u0019J1\nc+cQ2\nK3\u0000XQ2\nK3(68)\nIt is reasonable to assume that Xhas the same sign as\nJ1\ncwhich leads tojJ11\ncj>jJ12\ncj. Using similar considera-\ntion we can also prove that jJ21\ncj>jJ22\ncj.\n4. DM\nWe calculated all six DM interactions acting on TM 1\nion (see Fig. 5 for notation). We considered ap3\u0002p3\u00021\nsupercell with 18 RXO 3(X=Mn,Fe) formula units and\nfollowed the procedure described in Ref.36. This proce-\ndure is described as: (1) select one particular interaction\nby replacing selected X+3ions with Al+3 ions, (2) do to-\ntal energy calculation by constraining the direction of the\nspin moments. The zcomponent of the DM interaction\nbetween ithand jthTM ions is then given by,\nDz\nij=1\nS2(E[^Si\nx;^Sj\ny]\u0000E[^Si\n\u0000x;^Sj\ny]) (69)\nSimilarly one can calculate the other components.\nIn Table V we show the calculated magnitudes of zand\nxycomponents of both types of DM vectors for LFO and\nLMO compounds. The directions of the zcomponents\nin the paraelectric phase are shown in Fig. 5. They are\nthe same for both compounds and they are not altered inthe ferroelectric phase. The directions of the transverse\ncomponents for LFO are shown in Fig. 5. For LMO the\ntransverse components are opposite.\n5. SIA\nWe evaluated the components of SIA tensor by replac-\ning all TM ions by Al+3ions except TM 1 and calculating\ntotal energy by constraining spin directions along [100],\n[010], [001] and [101]. The o\u000b-diagonal component \u001cxzis\ngiven by,\n\u001cxz=1\n2S2(2E[^Sx;0;^Sz]\u0000E[^Sx;0;0]\u0000E[0;0;^Sz]) (70)\nThe diagonal components were determined by solving\nthe following equations,\n\u001cxx+E0=1\nS2E[^Sx;0;0] (71)\n\u001cyy+E0=1\nS2E[0;^Sy;0] (72)\n\u001czz+E0=1\nS2E[0;0;^Sz] (73)\n\u001cxx+\u001cyy+\u001czz= 0 (74)\nwhereE0is the total energy without any spin-spin in-\nteractions.\nCalculated components of SIA tensor in Eq. (36) are\nshown in Table V. The most important component is \u001czz\nwhich is positive for LMO and negative for LFO. Di\u000ber-\nent signs of \u001czzfor these materials stems from di\u000berent\nnumber of 3 delectrons for these materials. Indeed, in the\nsecond order perturbation theory the energy decrease due\nto SOC can be written as \u0001 E=\u0000P\nn6=m\u00152jhmj^ s\u0001^ljni\n\u000fm\u0000\u000fn.\nHerejniandjmiare, respectively, occupied and empty\nsingle-electron eigenstates of 3 dcharacter with \u000fnand\n\u000fmbeing the corresponding eigenenergies, \u0015is the SOC\nconstant, and ^ s(^l) is single-electron spin (orbital) an-\ngular momentum operator. We choose ^lquantization\naxis alongz. For LFO with half-\flled 3 dshell we have\nonly spin-\rip excitations which are induced by a trans-\nverse component of ^ s. In addition, the most impor-\ntant excitations (corresponding to the smallest energy\ndenominators) involve also the change of the magnetic\norbital quantum number, \u0001 ml=\u00061 which requires a18\nTABLE V: Magnitudes of zandxycomponents of nn DM interactions as well as components of SIA tensor for LFO and LMO\ncompounds. D0\nzis the magnitude of the zcomponent of DM vector in the paraelectric phase while Dz(Dxy) andD0\nz(D0\nxy)\nare magnitudes of the z(xy) components of DM vectors in the ferroelectric phase mediated by TM-O1\np-TM and TM-O2\np-TM\npaths, respectively. Notation for components of the SIA tensor is given in the text. Unite are meV.\nSystem DxyD0\nxyDzD0\nzD0\nz\u001cxx\u001cyy\u001czz\u001cxz\u001c0\nzz\nLFO 0.095 0.097 0.061 0.047 0.072 0.081 0.083 -0.164 0.018 0.181\nLMO -0.047 -0.050 0.027 0.033 0.042 -0.077 -0.079 0.156 -0.013 -0.160\ntransverse component of ^l. Therefore, the largest energy\ngain is achieved when ^ sand^lare parallel (so that their\ntransverse components couple with each other) prefer-\nring spins along zaxis and leading to \u001czz<0. On the\nother hand, for LMO the most important excitation is thetransition from majority e0to majority d3z2. Here spin\nis conserved but the magnetic orbital quantum number\nchanges which is possible only when ^ sand^lare perpen-\ndicular to each other, i.e., when spins lie in the xyplane\nleading to\u001czz>0.\n\u0003Electronic address: fennie@cornell.edu\n1E. Bousquet, M. Dawber, N. Stucki, C. Lichtensteiger,\nP. Hermet, S. Gariglio, J.-M. Triscone, and Ph. Ghosez,\nNature 452, 732 (2008).\n2Y. Tokunaga, N. Furukawa, H. Sakai, Y. Taguchi, T.-h.\nArima, and Y. Tokura, Nat Mater 8, 558 (2009).\n3J. H. Lee, L. Fang, E. Vlahos, X. Ke, Y. W. Jung, L. F.\nKourkoutis, J.-W. Kim, P. J. Ryan, T. Heeg, M. Roeck-\nerath, et al., Nature 466, 954 (2010).\n4Y. Tokunaga, Y. Taguchi, T.-h. Arima, and Y. Tokura,\nNat Phys 8, 838 (2012).\n5N. D. Mermin, Rev. Mod. Phys. 51, 591 (1979).\n6N. Balke, B. Winchester, W. Ren, Y. H. Chu, A. N. Mo-\nrozovska, E. A. Eliseev, M. Huijben, R. K. Vasudevan,\nP. Maksymovych, J. Britson, et al., Nat Phys 8, 81 (2012).\n7D. Meier, J. Seidel, A. Cano, K. Delaney, Y. 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Schwarz (Technische Universitat^ a Wien, Vi-\nenna, 2001).\n36C. Weingart, N. Spaldin, E. Bousquet, Phys. Rev. B 86,\n094413 (2012).\nAcknowledgments. We acknowledge discussions with\nD.G:Schlom, Julia Mundy, Charles Brooks, P. Schi\u000ber,\nMaxim Mostovoy, and J.Moyer. This work started at19\nthe suggestion of D.G :S. and J.M. H.D. and C.J.F.\nwere supported by the DOE-BES under Award Number\nDE-SCOO02334. A.L.W. was supported by the Cornell\nCenter for Materials Research with funding from NSFMRSEC program, cooperative agreement DMR-1120296.\nY.G. and W.W. were supported by the U.S. DOE-BES\nunder Award Number de-sc0008147."    },    {        "title": "1202.3387v1.Surface_phase_transitions_in_BiFeO3_below_room_temperature.pdf",        "content": "Surface phase transitions in BiFeO 3 below room temperature \nR. Jarrier1, 2, X. Marti3, J. Herrero-Albillos4, 5, P. Ferrer6,7, R. Haumont1,2, P. Gemeiner2, G. \nGeneste2, P. Berthet1, T. Schülli8, P. Cvec9 , R. Blinc9, Stanislaus S. Wong10,11, Tae-Jin \nPark10,12, M. Alexe13, M. A. Carpenter14, J.F. Scott15, G. Catalan16, B. Dkhil2,* \n \n1 Laboratoire de Physico-Chimie de l'Etat Solide, ICMMO, CNRS-UMR 8182, Bâtiment 410 - \nUniversité Paris-Sud XI, 15 rue George s Clémenceau 91405 Orsay Cedex, France \n2 Laboratoire Structures, Propriétés et Modélisa tion des Solides, CNRS-UMR8580, Ecole Centrale \nParis, Grande Voie des Vignes, 92295 Chatenay-Malabry Cedex, France \n3 Department of Physics, Charles University, Prague \n4 Helmholtz-Zentrum Berlin für Materialien und En ergie GMBH, Albert-Einstein-Straße 15, 12489 \nBerlin, Germany. \n5 Centro Universitario de la Defensa, Academi a General Militar-Universidad de Zaragoza, 50090 \nZaragoza, Spain \n6 SpLine (BM25), ESRF, Grenoble, France \n7 Instituto de Ciencia de Materiales de Madrid ICMM-CSIC, Madrid, Spain \n8 ESRF Beamline ID01, Grenoble, France. \n9 Jozef Stefan Institute, Jamova 39, Ljubljana 1000, Slovenia \n10 Department of Chemistry, State University of New York at Stony Brook, Stony Brook, NY 11794-\n3400, USA \n11 Also at Condensed Matter Physics and Materials Sc iences Department, Brookhaven National \nLaboratory, Building 480, Upton, NY 11973, USA \n12 Also at Korea Atomic Energy Research Ins titute (KAERI), 989-111 Daedoek-daero, Yuseong, \nDaejeon, Korea 305-353 \n13 Max Planck Institute for Microstructural Physics, Halle, Saale, Germany \n14 Dept. of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, U. K. \n15 Dept. Physics, Cavendish Lab., Cambridge Univ., Cambridge CB3 0HE, U. K.  \n16 ICREA and CIN2 (CSIC-ICN), Universitat Au tonoma de Barcelona, Bellaterra 08193, Spain \n \n \nKeywords: multiferroics, BiFeO 3, surface phase transition, low temperature anomalies \n *brahim.dkhil@ecp.fr\n \n PACS : 76.50.+g 77.80.Bh 81.30.Dz \nWe combine a wide variety of experiment al techniques to analyze two heretofore \nmysterious phase transitions in multiferroic bismuth ferrite at low temperature. Raman \nspectroscopy, resonant ultrasound spectroscopy, EPR, X-ray latt ice constant measurements, \nconductivity and dielectric response,  specific heat and pyroelectric  data have been collected  \nfor two different types of samp les: single crystals and, in or der to maximize surface/volume \nratio to enhance surface phase transition effects, BiFeO\n3 nanotubes were also studied. The transition at T=140.3K is shown to be a surface phase transition, with an associated sharp \nchange in lattice parameter and charge density  at the surface. Meanwhile, the 201K anomaly \nappears to signal the onse t of glassy behaviour.  \n \n \n1. Introduction \nBismuth ferrite BiFeO 3 (BFO) is one of the most popular research materials in \ncondensed matter physics at present.[1, 2]. Despite the intense activity, however, there remain a \nnumber of unanswered questions  concerning its structure a nd phase diagrams. From the \nbeginning[3] a large number of phases were reported as  a function of temp erature, and more \nrecently[4,5,6,7,8,9] more as a function of hydrostatic pre ssure. The high temper ature end of the \nphase diagram is now resolved, based upon the neutron studies of Arnold et al.[10, 11] and \ninvolves an ambient rhombohedral  R3c phase, a first-order tran sition to orthorhombic Pbnm \nnear 1103K,[10] and an iso-symmetric Pbnm-Pbnm Mott- like metal-insulator transition near \n1220K.[11] The powder neutron results show that th e latter structure cannot be resolved by X-\nray studies, because only the oxygen ions are signi ficantly displaced; and it further shows that \nthere are no high-T monoclinic or tetragona l phases, contrary to claims elsewhere.[12, 13, 14] In \nsome non-powder thin-film specimens, a cubic Pm -3m phase is inferred a few degrees below \nthe melting temperature of ca. 1225K[13] but it still remains to be confirmed.[16]  \nAlthough the high-temperature phases have been identifie d, questions remain about \nlower-temperature anomalies. The true nature and stability of its l ong-period (ca. 63 nm), \nincommensurate cycloidic spin st ructure has been controversial[17-25] and there are a number \nof cryogenic phase transitions whose origin has not been clarified. For example, anomalies \nnear 140K and 201K[26, 27] have been interpreted as sp in reorientation transitions,[27] analogous \nto those in orthoferrites such as ErFeO 3, and evidence has also been reported for spin-glass \nbehaviour,[28, 29] with an Almeida-Thouless line (AT-line) terminating at 140K,[30] a clear separation of field-cooled and zero-field-cooled  susceptibilities beginning at 230K, additional \nmagnon light-scattering cross-section divergen ces near 90K and 230K, and a bump in the \ndielectric constant near 50K.[31-32] At ca. 30K, there are two anomalies: an extrapolated \nfreezing temperature from a Vogel-Fulcher analys is of data (29.4K). These studies therefore \nindicate up to six cryogenic anomalies at te mperatures at 30, 50, 90, 140, 201, and 230K. On \nthe other hand, neutron diffraction experiments a nd other bulk-sensitive probes, such as single \ncrystal magnetometry, show no indication of an y magnetic transition, with the spin cycloid \nseemingly unaltered all the way down to 4K.[18, 25] There is therefore a clear contradiction at \nthe core of all these results that needs to be resolved. \nProbably the most thoroughly studied transitions are th ose at 140.3K and 201 K, \nreported independently by Cazayous et al.[26] and Singh et al.[27] These transitions are manifest \nin magnon Raman scattering as divergences in  cross-section, but they have remained \ncontroversial because they do not appear in car eful measurements of bulk magnetometry or \nspecific heat, such as those in Figure 1. This  has led to speculation that these may be \nanomalies of extrinsic origin (e.g. second phase s, magnetic impurities, or simple artefacts). \nHowever, the measurement of Raman magnon linewidth narrowing[26, 27] rules out magnetic \nimpurities, as does the observation of cr itical exponents for Raman cross-sections[32, 33] and \nAlmedia-Thouless dependence for field-cooled and zero-field-cooled magnetization in thin \nfilm samples.[30] \nVery recently (2011), two papers have shed  additional light on this aspect. Marti et \nal.[34] have shown, using impedance analysis and grazing incidence x-ray diffraction, that the \nsurface-layer (“skin”) of BiFeO 3 has a surface-confined phase transition[34], and suggested \nthat some of the cryogenic anomalies of BiFeO 3 may also be confined  to its skin layer. \nMeanwhile, Ramazanoglu et al.[35] have shown that extremely small uniaxial pressures change \nthe magnetic domain structure strongly, and inferred from that that the low-T transitions (at 140K and 201K) may be linked to such pheno mena, which mimic magnetic reorientation transitions like those in orthofer rites. Certainly the fact that  the low temperature anomalies \ntend to be clearer in surface- sensitive probes such as back-scattering Raman experiments \nwould support the idea that these transitions are c onfined in the surface. At the same time, the \nstrong effect of stress on the magnetic configurati on suggests that if the surface is structurally \ndifferent from the interior, so will its magnetic behaviour. In the present paper, we show that \nthe 140K transition in BiFeO 3 is indeed that of a surface phase, and we characterize its \nstructural and electronic properties. \n2. Raman spectroscopy \nAs already mentioned, the low-T phase transitions were evidenced using Raman \nspectroscopy techniques, especially in the low- wavelength region through the analysis of the \nMagnetic Field Cooling (MFC) versus Zero Fi eld Cooling (MZFC) regime and the study of \nthe electromagnons.[22, 23] The Raman spectrum measured at  80K on a single crystal shows no \nsignificant change compared to th at at room temperature. Howeve r, it is known that any static \nand/or dynamic changes in the structure should, in  principle, lead to a variation in the phonon \nbehaviour, and the analysis of the wave-number,  intensity and/or linew idth evolution of the \nwhole spectra as a function of te mperature is expected to give insight into those changes. \nFigure 2 shows typical temperat ure dependences of the wavele ngth position for two different \nRaman bands. Several features are noteworthy.  First, all the Raman phonon modes, and not \nonly those related to the electromagnons,[22, 23] show changes in the low temperature regime: \nwhatever the mode, a change of slope occurs at  140K. However, the sign of the slope change \n(softening vs hardening) is different for differ ent modes. At higher temp erature, a new change \nof slope appears either at 200K or higher or une xpectedly at 180K. As an example, the E-type \nphonon mode position at around 80cm-1 (Figure 2a) is nearly constant at ~83cm-1 from 80K to \n140K and then continuously increases until 180K, reaching a value of ~97cm-1 that remains \nconstant until room temperature. Note that the same behaviour was found in several different \nsamples including single crystals and powders. 3. Elasticity \nPhonon frequencies are directly linked to inter- atomic forces, so the fact that all the \nRaman lines shift in the 140-200K range signals that changes in the elastic constants are \ntaking place. In order to test the extent to which these are related to changes in elastic \nproperties, single crystal and ce ramic samples have been inves tigated by Resona nt Ultrasound \nSpectroscopy (RUS).[36] Elastic resonances are domina ted by shearing motions and the \nmeasured elastic constants scale with f2 (where f are the frequencies of  resonance peaks). The \ninverse mechanical quality factor, usually given as Q-1 = Δf/f, is a measure of anelastic losses \nassociated with the application of a dynamic sh ear stress. For the present study, RUS spectra \nin a frequency range 0.1-2MHz were collected  in the temperature range ~10-295 K in a \nhelium flow cryostat descri bed by McKnight et al.[37,38] Results for f and Q-1 obtained from \ndifferent resonance peaks are given in Figure 3.  \nResonance frequencies, normalized to their value at 300K, all decr ease with increasing \ntemperature, consistent with thermal softeni ng of the lattice. A deviation from the linear \nthermal softening starts to appear around 150 K, with a steep increase (elastic stiffening) \nbetween ~175 and ~200 K. An equivalently st eep drop back to the ba seline occurs between \n~225 and ~250 K. The breaks in slope of resona nce frequencies of the single crystal sample \nnear 150 and 200 K coincide with breaks in  slope of the Raman data (Figure 2). \nFrequency data for resonances of the ceramic sample do not show these sharp features, \nbut data for Q-1 (elastic losses) from both the ceramic and single crystal samples show similar \nanomalies in the temperature range of interest: (i) there is a slight break in the slope of the baseline variation in the vicinity of 150 K, fr om relatively low and fairly constant losses at \nlow temperatures to a trend of increasing lo ss with increasing temperature; (ii) all the \nresonances show a peak in Q\n-1 centred on ~180 K, and (iii) there is a further peak or break in \nslope at ~240 K for the single crystal data, a nd less well resolved anomalies above ~225 K for \nthe ceramic sample. The break in the slope of Q-1 near 150 K is reminiscent of increasing dissipation due to disordering of protons during heating of the mineral lawsonite,[38] though \nthe magnitude of the effect is much smaller. If the analogy is correct, some element of \nstructural or magnetic disordering occurs within the samples above ~150 K. Increasing \ndissipation implied by the Q-1 data could be understood as impl ying that the structure stable \nabove ~150 K has more disorder (s tatic or dynamic) in comparison with the structure stable at \nlower temperatures.  \nAll in all, the RUS measurem ents indicate significant coupling of strain with the \nchanges in structural or magnetic properties id entified in other measurements, and suggest the \npresence of a dissipative –perhaps disordered - state in the temperat ure range 150K-250K. The \nmeasurements, however, do not allow discrimination between phase transitions which occur \nwithin the bulk of the sample from one whic h only occurs within the skin. We nevertheless \nnote that elastic properties probed by sound-pr opagation measurements (which are only \nsensitive to bulk as the sound wavelength is of  the order of hundreds of microns) show no \nanomalies at all in this temperature range.[39] \n4. X-ray diffraction \nIn order to gain furt her insight, x-ray diffraction (XRD) wa s also used. In particular, to \ndiscriminate between surface and bulk contributions, we will compare data collected in grazing incidence diffraction (G ID) and standard co-planar ge ometry. The grazing incidence \nmeasurements were performed in the ID01 line at the ESRF synchrotron in Grenoble. In \ncontrast to the bulk-sensitive coplanar diffractio n, GID allows tuning of the information depth \nby tuning the incidence angle and/or the pho ton energy. Following the approach of the \npreceding high-temperature study,\n[34] we monitored only the cha nges in the length of the \nreciprocal space vectors (modulus of q) rather than both their le ngth and direction (vector q). \nThis allows for evidence of structural changes confined in skin layers  while circumventing the \nalignment difficulties inherent to single cr ystals with strong twinning and mosaicity.  Figure 4 (open symbols) shows the relative change of | q| for the (202) reflection \nmeasured on a single crystal as a function of te mperature in the heating regime from 100K to \n300K. The bulk temperature dependence displays no hint of structural  change inside the \ncrystal in this temperature range, and only a subtle change in thermal dilatation coefficient from 6.4.10\n-6 K-1 to 9.4.10-6 K-1 at 180K –which is also the temperature of the first peak in \nanelastic loss. In contrast, the surface-sensitive data for the (101) peaks reveal an abrupt \nexpansion up to ~1 % between T = 140 K and T  180 K (Figure. 4, solid symbols). This \nanomaly was not detected in coplanar diffrac tion where the information depth surpasses few \nmicrons. This indicates that the structural chan ge is confined in a surface layer. The surface \nlayer thickness cannot be stated beyond the upper bound placed by the penetration depth of \nthe coplanar geometry (microns), but the sa mple is nevertheless the same for which a \ntransition at T*=550K was estimated to be with in the topmost 10nm. As  a matter of fact, it \nappears that in addition to the phase transi tion occurring at 550K, th e nearby-surface layer \novercomes at least another phase transition at  ~140K and thus has its own phase diagram. \n \n5. Impedance analysis and pyroelectric-like currents \nImpedance analysis is an effective tool to  probe the electronic properties of surface \nlayers. In particular, it ha s been noted that Maxwell-Wa gner behaviour usually arises \nwhenever there is a substantial difference in  conductivity between th e bulk and interfacial \nregions: at low enough frequencie s, the contact impedance dominates and the interfacial \nproperties are evidenced[40, 41, 42]. Indeed, this appears to be  the case also for our single \ncrystals (Figure 5). \nThe impedance shows a strong frequency depe ndence typical of tw o lossy dielectric \ncomponents in series.[40,41,42] However, a sudden drop of the impedance (which is equivalent \nto a sudden increase of the capacitance) is also observed at 140K, which is frequency-\nindependent and thus corresponds to a true phas e transition. The fact that the jump in the impedance is bigger for lower frequencies is c onsistent with the phase transition occurring at \nthe interface of the crystal, in a behaviour an alogous to that observed in the interfacial T* \ntransition at 550K.[34] The surface transition appears to be  first order, as attested by the \nsharpness of the jump and by the difference in the critical temperatures on cooling and heating \nregime (Fig 5 right). This is also consistent  with the change in the unit cell volume observed \nin the grazing incidence diffraction results  shown in Figure 4. Recently, Ashok Kumar et al. \n[43] have discovered abrupt onsets of in-pla ne dielectric loss at 550K and 201K by using \ninterdigital electrodes, which are more sens itive to in-plane surface impedance. This \ncomplements our data and supports our interpretation as surface transitions. \nIn order to test the electroni c properties of the surface, we  also performed pyroelectric \nmeasurements. These are shown in Figure 6. The results show a very sharp and sudden peak \nin pyroelectric current near 140K. We measured  the current discharge both for zero-field-\ncooling-and-heating and for zero-field-heating-af ter-field-cooling regimes; in addition to the \n140K anomaly, the field-cooled sample show s a further broad anomaly around 200K, plus a \nsharp jump at 280K. The last anomaly is ascribed  to artefact during the measurement. It is \nworth mentioning that the anomalies at ~ 140K and ~200K are observed in very many \ndifferent samples included unprocessed surfaces indicating that those an omalies are intrinsic \nand not related to chemical etchi ng or mechanical polishing effects. \nUnlike in a classical ferroelectric phase tr ansition, where the pyroelectric peak position \nin temperature does not depend on the poling history of the sa mple, here the field-cooled \nsample has the peak at a significant lower te mperature (5K less) than the zero-field-cooled \none. This shift of peak position toward lower temperatures is a fingerprint of a thermally stimulated current: a current that is generated by emission of trapped charge from a trap level \nin the forbidden gap of BFO.\n[44] So, while pyroelectric currents are often related to changes in \npolarization, here we believe that the current we measured is not due to ferroelectricity but to charge injection and thermally stimulated emi ssion from trapping centres. When the skin layer \nundergoes the phase transition, th e Fermi level is likely to experience an abrupt re-\narrangement. As a result, interfacial defect states below the Fermi level might cross over \nabove it and release their charge , causing the abrupt thermally-sti mulated peak in current. The \nelectronic mechanism for the 140K pyrolectric-like anomaly is also  consistent with electron \nparamagnetic resonance results in la rge surface-to-volume samples (BiFeO 3 nanotubes), \ndiscussed in the next section. \n \n6. Electron Paraelectric Resonance (EPR) and magnetism in BiFeO 3 nanotubes \nBecause the analyses above emphasize surface phase transitions, it seems useful to \nprepare samples which maximize th e surface to volume ratio. To this aim, we prepared BFO \nnanotubes and characterized them via EPR, whic h is sensitive to relatively small volumes. \nDetails of the fabrication process are given in  the appendix. The EPR curves are fitted with a \nLorentzian line shape when the sample is pur ely insulator and then the line is perfectly \nsymmetric, or with a Dysonian type function when the line is an asymmetric reflecting \nconduction component: EPR Dysonian  = Absorption cos() + Dispersion sin(). The \nasymmetry is described by the parameter  and its value for insulator is zero and 1 for a full \nconductor.[Erreur ! Source du renvoi introuvable. ]  \nIt is clear from the EPR data (Figure 7) that the sample’s conductivity is maximum at \n~140K and that the conductivity behaviour changes again, less ab ruptly, at ~200K . Therefore, \nthe EPR data for the nanotubes also indicate an  increase in surface charge density at 140K, \nconsistent with detrapping trap levels; the charge released at 140K causes the large \npyroelectric-like current observed in Figure 6.  \nAlso relevant to the results here, we note that BiFeO 3 samples with large surface to \nvolume ratios (e.g., nanocrystals) have been shown to display spin-glass behaviour.[46] Glassy effects are well known for small particles of antif erromagnets in general,  where the lack of \nspin compensation at the surface is thought to frustrate the long range magnetic order (e.g. NiO\n[47,48]). Three new EPR observations (Figure 8) are similar to those known in other spin \nglasses, especially that in Cd 1-xMn xTe:[49] (1) The gyro-magnetic ratio is g > 2 above the \napparent spin-glass transition at T sg = 140K and g = approximately 2.0 below; (2) the decrease \nin g-value at 140 K is rather abrupt with temper ature and nearly 1% in magnitude; (3) there is \na divergence in EPR linewidth that satisfies a dependence  = 0 + 1 exp [-T/T f], as shown \nin Figure 8, with an extrapolated freezing temperature T f = 33±3 K that is in good agreement \nwith that measured independently as 29.4±0.2 K.[28]  \nWe note also that the EPR susceptibility of the nanotubes is increased between 125 K \nand 200 K, which is essentially the same temp erature range (bearing in mind the sample \ndifference between the nanotubes and the single cr ystals) where structural disorder has been \ninferred from elastic spectroscopy. This suggests that  the structural disorder has its replica in \nthe magnetic behaviour. We note also that the gl assy fitting to the EPR linewidth (Figure 8-\nmiddle) departs from the actual data below th e skin transition temperature, suggesting a \ntransition from a glassy or magnetically soft st ate to a more rigidly or dered configuration, the \ndetails of which are at this point unknown. We nevertheless emphasize that it is not easy to \ndistinguish magnetoelectric spin-glasses from crystals  with domain-wall pinning,[50, 51] both of \nwhich would be consistent with the magnetic and elastic results. An extremely fine pattern of \ndomains has in fact been observed in the near-surface region of BiFeO 3[52] so this is not out of \nthe question. \n \n7. Electronic structure \nTo gain further insight on the origin of th e surface properties, we also performed some \nfirst-principle density-functional calculations by introducing some  defects which can exist at \nthe surface of BFO samples. Due to bismuth volatility, the most likely defects are Bi vacancies. The calculations  have been performed with the SIESTA code.[53,54] Two \napproximations for the exchange-correlation en ergy have been tested: the Local Density \nApproximation (LDA) and the Generalized Gradie nt Approximation in the form of Perdew, \nBurke and Ernzerhof (GGA-PBE).[55] Troullier-Martins pseudopoten tials have been used. \nSemicore electrons (3p for Fe, 5d for Bi) are explicitly treated as valence electrons. The \nequivalent plane wave cutoff for the grid is 200 Ry in the LDA case and 400 Ry in the GGA \ncase. The excitation energy defining the range of the atomic orbitals is  0.01 Ry. The periodic \nparts of the Kohn-Sham wave functions are expa nded on a basis of numerical atomic orbitals \nof double zeta type (plus polarizat ion orbitals). A single-zeta 7s type orbital is added in the \nbasis set of Bi.  \nWe first optimized bulk BFO in its R3c phase (both in LDA and GGA) and used the \nlattice constant obtained to build a 2x2x2 supe rcell (thus containing 80 atoms), whose shape \nand volume was kept fixed. Then Bi  vacancies were introduced in  the supercell, in different \ncharge states (0, -1, -2, -3). In the case of charged defects, neutrality is insured by adding a \nbackground compensating jellium. Finally the at omic geometries were optimized by using a \nconjugate-gradient scheme, so as to obtain Cart esian components of atomic forces below 0.04 \neV/Angstrom. Figure 9 shows the electronic Density of Stat es (DOS) versus energy for \nstoichiometric (black curve) a nd non-stoichiometric BFO supercel ls containing Bi vacancies.  \nThe calculated LDA energy band-gap fo r stoichiometric BFO is 0.8 eV, an \nunderestimate compared with th e experimental value 2.74 eV;[13] this underestimate is typical \nof LDA calculations. The main effect of Bi vacanc ies, whatever the charge  of the defect, is to \nintroduce energy levels within the band-gap. These levels can explain the trapping-detrapping \nprocess suggested by our electrical measurements.  \nIt is also interesting to remark that Bi vacancies modify the magnetism of the system. \nElectrons are rearranged giving rise in some  cases to a net magnetic moment probably associated with a hole polaron. This rearrangement which can occur within the surface may \nthen explain the concomitant magnetic anoma lies observed in the magnon spectra and in the \nEPR susceptibility. The possibility of the exis tence of polaron within the close surface is \nreinforced because this charge-phonon coupling w ould also explain the anomalies observed in \nXRD, Raman and RUS data. \n8. Discussion \nThe grazing incidence XRD re sults show unambiguously th at the anomaly at 140K \ncorresponds to a surface phase transition. Its ke y features are an abrupt change in unit cell \nvolume (which expands by 1% on heating, Figure 4), and a concomitant change in electronic \nstructure, with an impurity level crossing the Fe rmi level and releasing charge, as signalled by \nthe field-dependent pyroelectric- like discharge (Figure 6) a nd increased conductivity inferred \nfrom AC impedance (Figure 5) and EPR analysis (Figure 7). Ab initio calculations show that \nBi vacancies might be at the origin of the im purity levels (Figure 9). The Raman spectra also \nshow that the 140K anomaly is strongest in the magnon peaks, and EPR confirms that this \ntransition affects the magnetic stru cture, as also suggested by th e first-principl e calculations.  \nNow the question is what could be the origin of these anomalies. In the perovskite \nstructure, two structural degr ees of freedom can be consid ered; either atomic (polar) \ndisplacements or oxygen octahedral tilts. In th e case of magnetic materials, as is BiFeO 3, a \nthird degree of freedom is the spin. Rama n spectroscopy is very sensitive to oxygen \noctahedral rotations,[56] and yet the number of peaks in th e Raman spectra was not observed to \nchange as a function of temperature --though surface-sensitive UV Raman would be desirable \nto confirm this. The abrupt change in the latt ice volume, as indicated by the grazing incidence \nXRD, points out instead to a change in atomic distances without a change in symmetry. Given \nthat the in-plane lattice parameters of the surf ace must be coherent with those of the skin, \nmeans that a strong uniaxial strain is develope d at 140K, which is relevant because uniaxial \nstrain has recently been reporte d to have a very strong effect on the magnetism of BiFeO 3.[35] The change in unit cell volume is the likely cu lprit of the crossover of a shallow impurity \nlevel across the Fermi line, re sulting in charge release. \n9. Conclusions \nThe data presented here confirm the interpretation of the 140K anomaly in BiFeO 3 as a \nsurface phase transition, with surface effects detected in very many different bulk single \ncrystals and exacerbated in na notubes owing to their very hi gh surface to volume ratio. The \nmain features of this phase transition are a sharp volume cha nge without actual change of \nsymmetry; sharp emission of charge at 140K ( pyroelectric-like curr ent) and maximum in \nconductivity (peak in the  parameter of the EPR dysonian lineshape), consistent with a \ncrossover between an im purity level and the Fermi level, a nd structural and magnetic disorder \nbetween 140K and c.a. 200K. \nAs was argued for its high temperature T* counterpart, the surface phase transition at \n140K is likely to be aided by the inherent complexity of the pha se diagram of BiFeO 3, which \nis very sensitive to even small perturbations[57] such as surface tension or local strain fields \naround vacancies and defects. A melting of th e Bi sublattice has been reported for BiFeO 3 \npowders with a radius smaller than 9nm:[58] if there were Bi vacanci es at the surface layer, \nthese would be able to provide both electronic im purity levels and local strain fields capable \nof explaining the electronic, ma gnetic and structural changes. \nMore generally, these results indicate that it is not appropriate in general to treat BiFeO 3 \nas a homogeneous material. Its skin layer is ra ther different from the bulk, having its own \nstructural, electronic and magnetic  properties, and its own phase diagram that already includes \nat least two confirmed phase tr ansitions at 140K and 550K, as we ll as a probable glassy state \nbetween 140K and 230K. The surf ace is at least as importan t as the bulk for functional \ndevices, as it determines key properties such as magnetic exchange bias and conductive barrier height. It is th erefore of utmost importance that its nature and properties be fully understood.  \n10. Appendix \nBiFeO 3 (BFO) single crystal growth \nBFO single crystals were grown using a met hod similar to the original method proposed \nby Kubel and Schmid.[59] Adjusting the cooling rate allows growing of millimeter diameter \nof rosette-like pyramidal crystals, as described by Burnett et al.[60] All crystals were polished \nparallel to the surface, which in rosette crysta ls is the (100) crystallographic plane. Samples \ntypically larger than 1x1 mm2 area and 300 µm thick were obtai ned. Optical quality crystal \nsurfaces were obtained by polishing using 0.25 µm diamond paste. The remaining damaged \nsurface layer and polishing scratches were removed by chemical mechanical polishing (CMP). \nCMP was performed usually for 30 min using SiO 2 colloidal solution (Syton) diluted with \nwater in a 1:1 ratio. \nBFO nanotube growth \nThe nanotubes were prepared via wet-chemis try synthesis: In a typical synthesis, \nBi(NO 3)3•5H 2O was initially added to ethylene glycol to ensure complete dissolution \nfollowed by Fe(NO 3)3•9H 2O to yield a molar ratio in so lution of Bi:Fe as 1:1.[39,61] The \nresulting mixture was stirred at 80 ºC for 1 h, after which a transparent sol was recovered \nupon evaporation of the excess et hylene glycol. Droplets of th e sol were deposited using a \nsyringe onto a porous anodic al umina (AAO) template (Whatm an Anodisc®) surface with \napplication of pressure.[40,41] AAO membranes with different pore sizes, such as 200 nm \nand 100 nm, have been used. The resultant sa mples of AAO templates containing the BiFeO 3 \nprecursors were subsequently oven-dried at 100°C for an hour and then preheated to 400°C \nfor three separate runs at a ramp-rate of 5° C/min in order to get rid of excess hydrocarbons \nand NO x impurities. The sample was further annealed at 600°C for 30 min. BiFeO 3 nanotubes were isolated after removal of the AAO temp late, following its imme rsion in 6M NaOH \nsolution at room temperature for 24 h. Thereaf ter, to retrieve reasonable quantities of \nnanotubes, the base solution was di luted in several steps with dist illed water and lastly ethanol. \nTubes were collected by centrifugation. The t ubes were shown to be ferroelectric, with \nswitching hysteresis. The nanotubes were subj ected to electrical switching by applying a \nvoltage across a single tube, with the Ir/Pt tip of an atomic force microscope serving as the top \nelectrode. The measured pi ezoelectric constant hysteresis is quite large (about 2/3 the value of \nPZT). \nGrazing incident diffraction (GID) \nFor the GID experiments (performed at ID 01 beamline at ESRF), we chose 7 keV and \n0.2 degrees as incidence angles, thereby limitin g our information depth to few nanometers. \nThe BiFeO 3 crystal was cooled using the Oxford Cr yojet blowing cold nitrogen gas on the \nsample, while temperature was measured in the gas stream and by a thermocouple attached at \none side of the crystal (1 mm thick). The crys tal employed was the same one as in the GID \nhigh temperature study [34]. \nAcknowledgements \nResearch at Stony Brook and Brookhaven National Laboratory (including support for \nTJP and SSW as well as for synthesis experiment s) was supported by the U.S. Department of \nEnergy, Basic Energy Sciences, Materials Scien ces and Engineering Division under contract \nnumber DE-AC02-98CH10886. 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Specific heat at 0 and 1 T and ac magnetic susceptibility at 0 T (inset) as a function of \ntemperature for BFO single crystal. \n   \n  \nFigure 2.  : Positions of two Raman peaks as a function of temperature, showing shifts beginning at 140K, \nwith sharp anomalies at 180K, and further changes of slope at 200K. \n \n0 50 100 150 200 250 300051015 \n 10-3 Q-1\nT (K)0.00.20.40.60.81.0\n (f(T)-f(300K))/f(0K)\n \n norm frec (439kHz)\n fo = 560 kHz (SC) (558kHz)\n norm frec (659kHz)\n fo = 390 kHz (Ceram.)\n fo = 930 kHz (Ceram.)\n \nFigure 3.  (top)RUS results for a BFO single crystal (full symbols) and a ceramic sample (open symbols): \nthe resonant frequencies increase between ~140 and ~240 K, indicating a hardening of the lattice between these \ntwo temperatures. (below)The elastic loss (inverse of th e quality factor) shows gradual increase above ~150 K, \nwith peaks at ~180 K and at 220-240 K depending on the sample. \n  \nFigure 4.  Relative change of the reciprocal lattice vector |q| as a function of temperature probing the bulk \nof the crystal (blue open symbols) and the top-most su rface (red solid symbols). The surface data show a rapid \nexpansion of the lattice parameter upon heating above 140K, and this feature is absent from the bulk. \n \nFigure 5.  (Top) Z modulus vs temperature and frequency; (bottom): Z modulus and impedance phase \nangle as a function of temperature on heating and cooling showing a first-order phase transition at ~140K.  \nFigure 6.  Discharge current anomalies in BiFeO 3 single crystals. (left) pr istine samples show two clear \nanomalies at 140K and 200K, though in subsequent runs (right) only the 140K anomaly is clear, although the \n200K anomaly is still visible for field-cooled samples. Th e field-cooling dependence of the peak temperature for \nthe 140K anomaly indicates that this py roelectric-like current is due the sudden  carrier emission from trap levels \ntriggered by the surface phase transition. \n \n \nFigure 7.  Alpha parameter reflecting the asymmetry of the EPR curves. 1601802002202406080100\n50 100 150 200 250 3001.9901.9952.0002.0052.010(G) EPR(arb. units) gyro-magnetic ratio\nT (K)\n \nFigure 8.  (top panel) Electron paramagnetic susceptibility, showing an increase between ∼125K and \n200K, which is close to the temperat ure range where elastic softening and anomalous skin expansion has been \ndetected; (middle panel) an exponential fitting to the EP R linewidth yields an extra polated freezing temperature \nof c.a. 33K; the experimental data departs from the gla ssy fit for temperatures below 140K; (lower panel) the \ngyromagnetic factor also shows a rather abrupt drop below the skin transition temperature.  The EPR \nsusceptibility shows a sharp drop at 201K, whereas the g- value (bottom panel) shows a broad maximum near this \ntemperature. \n  \n \nFigure 9.  Density Of States versus energy from ab initio calculations for pure BFO (black) and BFO with \nVBi vacancies in different charge states (V Bix, V Bi’, V Bi’’, V Bi’’’).  \n EF "    },    {        "title": "1804.02458v2.Effect_of_Substrate_Temperature_on_Structural_and_Magnetic_Properties_of_c_axis_Ori_ented_Spinel_Ferrite_Ni0_65Zn0_35Fe2O4__NZFO__Thin_Films.pdf",        "content": "1 \n \nEffect of substrate temperature on s tructural and magnetic p roperties of c -axis oriented \nspinel ferrite Ni0.65Zn0.35Fe2O4 (NZFO) thin f ilms \nDhiren K. Pradhan1,a), Shalini Kumari2,5 , Dillip K. Pradhan3, Ashok Kumar 4, Ram S. Katiyar \n5, and R. E. Cohen1,6,a) \n1Extreme Materials Initiative, Geophysical Laboratory, Carnegie Institution for Science , \nWashington, DC 20015,  USA \n2Department of Physics and Astronomy, West Virginia University, Morgantown, WV 26506, \nUSA \n3Department of Physics and Astronomy, National Institute of Technology, Rourkela -769008, \nIndia  \n4National Physical Laboratory  (CSIR), Delhi -110012 , India.  \n5Department of Physics and Institute of Functional Nanomaterials, University of Puerto Rico, \nSan Juan, PR 00936,  USA  \n6 Department  of Earth - and Environmental Sciences, Ludwig Maximilians Universit y, Munich \n80333, Germany  \nABSTRACT  \nVarying the substrate temperature changes structural and magnetic properties of spinel ferrite \nNi0.65Zn0.35Fe2O4 (NZFO) thin films. X-ray diffraction of films grown at different temperature \ndisplay only (004) reflection s, without any secondary peaks , showing growth orientation \nalong the c-axis. We find  an increase in crystalline quality of these thin films  with the rise of \nsubstrate temperature. The surface topography of the thin films grown on various growth \ntemperature s conditions reveal that these films are smooth with low roughness; however,  the \nthin films grown at 800 oC exhibit lowest average and root mean  square  (rms) roughness \namong all thin films . We find iron and nickel to be more oxidized (greater Fe3+ and Ni3+ con-\ntent) in films grown and annealed at 700  oC and 800  oC, compared to those  films  grown at \nlower temperature s. The magnetic moment is observed to increase with an increase of su b-\nstrate temperature  and all thin films possess high saturation magnetization and low coercive \nfield at room temperature. Films grown at 800 oC exhibit a ferrimagnetic –paramagnetic phase \ntransition well above room temperature. The observed large magnetizations with soft magne t-\nic behavior in NZFO thin films above room temperature suggest potential application s in \nmemory, spintronics, and multifunctional devices.  \n \nKeyword s: Ferrites, Thin films, Magnetic p roperties . \n \n Author to whom correspondence to be addressed. Electronic mail:  dhire n-\nkumarp@gmail.com  (Dhiren K. Pradhan), rcohen@carnegiescience.edu  (R. E. Cohen).  \n \n1. Introduction  \n     With the rapid development of new devices miniaturizing technology, magnetic structures \nare being pushed to nanoscale dimensions for potential applications in memory,  spintronics, \nand other multifunctional devices  [1-7]. Among the various magnetic materials, the most \nnoteworthy and important materials are strongly correlated spinel oxides of AB 2O4 structure, \nparticularly the spinel ferrites having general formula AFe 2O4 [1-2],[7-10]. Many spinel  fer-\nrites exhibit magnetic transition temperature  well above room temperature and strong ma g-\nnetic properties along with high magnetostriction, high resistivity,  high dielectric permittiv i-2 \n \nty, good thermal, chemical, and structural stabilities  [7-8]. These ferrites also provide a non -\nzero magnetic moment along with spin -dependent band gaps  [7-10].  Ferrimagnetic materials \nare also widely used as the magnetic candidate in multiferroic composite structures to achieve \nhigh magnetization, magnetic phase tr ansition temperature s and strong magnetoelectric (ME) \ncoupling above room temperature  [11-15]. Most of the single -phase materials exhibit mu l-\ntiferroicity at cryogenic temperatures and weak magnetoelectric cou pling  [14-15]. Comp o-\nsites of ferrimagnetic  materials with ferroelectric perovskites provide strong ME coupling with \nboth ferroelectric and magnetic transition above room temperature with low leak age current and \ndielectric loss  [14-16]. Nickel -zinc ferrites are the most promising ferrimagnetic mate rials, \npossess ing high saturation magnetization, low coercive field, high resistivity, high magnet o-\nstriction, high dielectric constant, and low dielectric losses with magnetic transition temper a-\nture well above room temperature[ 7-8],[17-19]. Here we study Ni0.65Zn0.35Fe2O4  (NZFO) , \nsince  this optimized composition of Ni and Zn shows highest saturation magnetization  in the \nentire Ni -Zn series , considerable magnetostriction ( λ100 = - 24 ×10-6 , λ100 = -13.2 ×10-6  ) and \nmagnetic transition temperature s above 600 K  [17-20]. \n        Growing high-quality , single crystalline thin films with atomically smooth surface are  a \nchallenge. Many attempts have been made to grow high -quality spinel ferrite thin films by \ndifferent techniques such as molecular beam epitaxy  (MBE) , rf sputtering, pulse d laser dep o-\nsition  (PLD) , spin-spray, sol -gel, electrochemical deposition, direct liquid phase precipitation \nand hydrothermal growth  [13], [21-26]. The structural and magnetic properties of NZFO are \nstrongly dependent on growth parameters, such as substrate temperature, oxygen partial pre s-\nsure, substrate to tar get distance, laser fluence etc  [13], [27]. Among all the growth param e-\nters, substrate temperature plays a vital role during growth , but studies on effects of growth \ntemperature on structural and magnetic properties of c -axis oriented NZFO thin films grown \nby PLD are limited . Here, we have systematically studied the evolution of structural, topo g-\nraphy and magnetic properties of NZFO thin fi lms as a function of growth temperature.  \n \n2. Experimental details  \n     A high purity single phase ceramic target of Ni0.65Zn0.35Fe2O4 (NZFO) was prepared in air \natmosphere by standard solid -state reaction method s; detail s are reported elsewhere.12 We \nfabricat ed highly oriented NZFO thin films on LaNiO 3 (LNO) buffered (LaA-\nlO3)0.3 (Sr2AlTaO 6)0.7 (LSAT)  (100) substrates u sing optimized pulsed laser deposition (PLD) \nwith an excimer laser (KrF, 248 nm). The PLD chamber was initially evacuated to a base \npressure of 1.0×10-6 Torr. Then 60 nm thin LNO buffer layer s were  grown on LSAT su b-\nstrate s at 700 °C under an oxygen pressure of 200 mTorr . The film s were then anneal ed in a \noxygen pressure of 300 Torr for 30 minutes at  the same temperature and then slowly cooled \nto room temperature. The LNO buffer layer used as bottom electrode for electrical measur e-\nments. After the deposition of buffer layer, NZFO  thin films were deposited at temperature s \nof 500, 600, 650, 700, 750 and 800  °C under an oxygen pr essure of 150 mTorr using a laser \ndensity of ~ 1.5 J/cm2 at a deposition  frequency of 5 Hz. All the NZFO thin films were then \nannealed at their respective growth temperature in an oxygen atmosphere of 300 Torr for 30 \nminutes and later cooled slowly to room  temperature . The phase purity and orientation of the \nNZFO thin films annealed at different temperatures were evaluated by X -ray diffraction \n(XRD) using a Rigaku Ultima III with Cu  Kα1 radiation (λ=1.5405Å) operated at a scan rate \nof 1o/min at room temperature. The film thicknesses were checked using XP -200 profilom e-\nter and filmetrics. Room -temperature topography images of NZFO thin films annealed at \ndifferent temperatures  were collected by Atomic force microscopy (AFM) (Veeco) operated \nin contact mode. High -resolution X -ray photoemission spectroscopy (XPS) was utilized for \nthe elemental analysis of NZFO thin films annealed at different temperatures . Room temper a-3 \n \nture magnetic hysteresis (M vs H) of all the samples was obtained using a vibrati ng sample \nmagnetometer (VSM Lakeshore Model 142A).  Temperature dependence of magnetic prope r-\nties of thin films grown and annealed at 800 oC was measured utilizing a PPMS DynaCool \n(Quantum design) in a wide temperature range of 300 -800 K.  \n \n3. Results  \n3.1 Structural characterization  \n     The phase purity and crystalline quality of all our NZFO thin films grown at different \ntemperature were investigated by high -resolution x -ray diffraction (HRXRD) measurements. \nFig. 1(a) shows the XRD patterns of NZFO th in films grown at a different temperature on \nconducting LaNiO 3 (LNO)  / (LaAlO 3)0.3 (Sr2AlTaO 6)0.7 (LSAT) (100) substrates.  These \nfilms showed only the ( 00c) (c = 4 for NZFO and 1, 2 and 3 for LSAT/LNO) diffraction \npeaks over a large angle x -ray scans (20°  to 80°) , confirm ing that these films are highly c - \naxis oriented. We did not observe any diffraction peaks from secondary phases in the spectra \nindicating high purity and high -quality growth of the films. The sharpness of diffraction \npeaks of NZFO is foun d to increase with increasing substrate temperature , indicating an i n-\ncrease of crystalline quality of these films [ 21-23].            \n      \n3.2 Morphological characterization  \n     To investigate the quality of growth, distribution of grains and roughness of all NZFO thin \nfilms grown at a different temperature s, surface topography images were captured in contact \nmode on a scan size of 3 × 3 μm2 (Fig. 2). The surface of all the thin films at different growth \ntemperature s is smooth and homogeneous, free of microcracks, pores or holes. Smooth su r-\nfaces with low roughness are necessary for the required physical properties for device appl i-\ncations. We computed t he average  roughness  (Ra) and root mean square ( rms) roughness  (Rq) \nof each thin film from the atomic force micrographs  from  the AFM line profiles (Fig. 3).  The \nthin films grown at 500  and 600  oC contain  grains of different shape , so the variations in the \nsurface height  is more  (2.54 to 8.47  nm for film grown at 600 oC), whereas the thin films \ngrown at higher temperatures have densely packed grain s of approximately the same  shape,  \nhence exhibit ing low roughness  (Ra = 0.23 nm for film grown at 800 oC). We did not observe \nany systematic variation of the roughness with growth temperature . We find that the thin \nfilms grown at 800 oC exhibit lowest average and rms roughness among all thin films.  \n \n3.3 Elemental analysis  \n      We performed high-resolution X -ray photoelectron spectroscopy (XPS) measurements of \nfilms grown and annea led at 500, 600, 700, and 800  oC to confirm the presence of all indivi d-\nual elements and measure their oxidation states . Fig. 4 shows the XPS spectra of the film \ngrown at 600  oC. The core -level spectra of Ni  2p, Zn 2p, Fe 2p and O  1s confirm  the presence \nof all individual elements . The electron excitation energies of Ni, Zn, Fe and O observed in \nthese thin films are in good  agreement with existing literature, and are characteristics of \nNZFO  [28-29]. The binding energies of all XPS spectra were referenced using the C1s peak \n(284.6eV). Spin-orbit splitting is observed (2p 3/2 and 2p 1/2) are observed for Ni 2p, Zn 2p, and \nFe 2p. To probe the effect substrate temperature on the magnetic properties of NZFO  thin \nfilms, detailed analysis on XPS spectra of Ni and Fe at all four temperatures are shown in \nFig. 5. Gaussian -Lorentzian peak s were fitted to the experimental data of Ni  2p and Fe  2p. Ni \n2p3/2 and Fe  2p3/2 peaks were well fitted with double peak Gaussian -Lorentzian profile and 4 \n \nwe find that Ni and Fe exist within all films as +2 and +3 states for all four temperatures  (Fig. \n5). Ni 2p1/2 and Fe  2p1/2 peaks  are well fitted with single peak of Gaussian -Lorentzian profile \nfor 500  oC, 600  oC and a double peak Gaussian -Lorentzian profile for 700oC, 800oC, con-\nsistent with + 2 valence in the thin films grown at 500 and 600 oC, and a mixture of  +2 and +3 \nstates in the thin films grown at  700oC, and 800  oC. The contribution of Ni3+ and Fe3+ are \nhigher in th e NZFO thin films which were annealed at higher temperatures  i,e 700oC, and \n800oC. A single peak Gaussian -Lorentzian profile for Zn 2p was obtained for all four annea l-\ning temperatures which confirm that zinc exists within all films as Zn2+. Since the amounts of \noxidized, and hig her moment  Fe 3+ and Ni  3+ are greater in  the thin films grown and annealed \nat 700oC, and 800oC, higher magnetization is expected in these thin films, and this is o b-\nserved ( Fig. 6(a)). The O  1s can be deconvoluted  into two peaks for all four annealing te m-\nperatures, where higher binding energy peak belonging to the lattice oxygen and the lower \nbinding energy attributed to the surface adsorbed oxygen. We find  no shifting of binding e n-\nergy, however the full width at h alf maximum increases with the increase of temperature .  \n \n3.4 Magnetic properties  \n     We investigated  the effects of substrate temperature on magnetic properties by  measuring \nthe dc -magnetization as a function of the magnetic field at different temperature s (Fig. 6(a)). \nWe carried out M (H) measurements for the in-plane (magnetic field parallel  to the thin film \nsurface) configuration. The diamagnetic response of the substrates , which is significant at \nhigher field s, was subtracted . All of the thi n films grown at different growth temperature e x-\nhibit perfect saturated magnetic hysteretic behavior above ± 6 kOe , which is a typical sign a-\nture of ferro/ferrimagnetic behavior. We have estimated saturation magnetization (M S) as a \nfunction of annealing temperature  (Fig. 6 (a), inset) . We find a  monotonic increase of M S \nwith increase of annealing temperature ; the observed value at 800oC is around 363 emu/cc. \nThe observed increase in M S as a function of annealing temperature indicates possible cry s-\ntallite growth and smooth er surface of the films with an increase in temperature. Crystalli ne \ngrowth and  decreased  surface roughness could also explain the increase in saturation magne t-\nization as a function of substrate/deposition temperature .  \n     We find that the temperature -dependent H C increase s initially with annealing  temperature, \nand reaches a maximum value of 4 60 Oe at 700  oC, and then drops to ~  330 Oe with further \nincrease of substrate temperature. This feature can be attributed as expected  crossover at \n7000C from a single domain to multi -domain behavior with increasing crystallite size. Note \nthat comparatively high M S and low H C values of the NZFO thin films indicate the soft ma g-\nnetic nature.  \n     To probe the magnetic Tc, the temperature dependence of magnetization measurements of \nNZFO thin films grown at 800 C have been carried out in the in -plane (magnetic field para l-\nlel to the thin film surface) mode. We performed z ero-field-cooled (ZFC) and fi eld-cooled \n(FC) magnetization under a magnetic field of 1000 Oe in a wide temperature range of 300 - \n800 K (Fig. 6 (b)).  We observe no bifurcation or hysteresis in  M(T ) curve throughout the \ninvestigated temperature range . The disappearance of bifurcation in ZFC -FC magnetization \ncurve might  be due to the application of field higher than the coercive field  [30-31]. We find \nthat m agnetization of NZFO thin film decreases slowly up  to 690 K and then decreases su d-\ndenly with the increase of temperature and vanishes above ∼740 K . The estimated ferrima g-\nnetic –paramagnetic transition temperature  (TC ) is found around 716 (±10) K with a broad \ntransition up to 740 K, which reveals the existence of a short -range spin interaction in the \nsample above Tc.   \n \n 5 \n \n4. Discus sion \n     The crystalline quality  and magnetic moments of NZFO thin films are observed to i n-\ncrease with an increase of substrate  temperature. The increase of saturation magnetization \nwith the increase of the substrate temperature  is due to better  of crystallinity  at higher growth \ntemperatures . During the growth, t he kinetic energies of ablated particles from the target are \nnot affected by temperature directly, but the density of gas near the substrate decreases as the \nsubstrate/growth temperature increases. As a result, the mean free path of the ablated part i-\ncles might increase. So higher temperatures contribute to the synthesis of high quality crysta l-\nline thin films by increasing surface mobility of adatoms  [21-23],[27] . The increases of cry s-\ntalline quality with increase of the substrate temperature of the films are evident from the \nincrease of the XRD peak intensities  as shown in Fig.1..  The thin films grown at lower tem-\nperature  have lower crystalline quality  and thus higher surface to volume  ratio and as t he \ngrain surface is known to be poorly magnetized  and results in lower magnetization . In low \ntemperature grown thin films , very small nano -grains are present, which lead to large no n-\nmagnetic grain boundary volumes hence low magnetization value. High temperature growth \nleads to a reduction of the relative grain boundary volume, which causes the increase in the \nmagnetization  [21-23]. Liu et al. have reported that NZFO thin films can be fabricated under \nlow temperature, but the magnetic prope rties of Ni ckel Zinc  ferrite films grown under low \ntemperature were found to be amorphous with high coercivity (Hc) and very low satura tion \nmagnetization (Ms) [ 22-23]. They also concluded that high -temperature post -heating trea t-\nment or in -situ heating is n eeded to obtain a better spinel structure and soft magnetic property \nwith enhanced saturation magnetization. Rajagiri et al. also observed the increase of Ms with \nan increase of substrate temperature in polycrystalline MnFe 2O4 thin films and the increase of \nMs had been attributed to the increase of crystalline quality ( crystallite/grain size ) with the \nincrease of substrat e temperature during the growth [ 21]. The magnetic materials should e x-\nhibit low coercivity, high Ms for spin tonics and some other specific application as with very \nlow field magnetization can be switched. Based on our results, NZFO is a potential candidate \nin this regard. The coexistence of soft magnetic behavior with high Ms and Tc well above \nroom temperature i n NZFO thin films makes them suitable for applications in multifunctional \nmagnetic switchab le and other spintronic devices [7-8],[10],[13 -15],[33]. \n \n5. Conclusions  \n     Highly c - axis oriented NZFO thin films were grown on LNO buffered LSAT single cry s-\ntal substrate at different temperature by PLD technique. The crystalline qualities of these thin \nfilms were  found to increase with an increase of substrate temperature. The surface topogr a-\nphy of all the thin films was found very smooth and homogeneous with out any microcracks, \npores or holes. Observation of photoelectron characteristic peaks of Ni, Zn  Fe, and O in the \nhigh-resolution XPS spectra on the surface of all NZFO thin films confirm the presence of all \nindividual elements. The contribution of Fe 3+ and Ni3+ are found to be larger in the thin films \ngrown and annealed at 700  oC and 800  oC compared to the other thin films. These thin films \nexhibit well saturated hysteretic behavior with high Ms and low Hc at room temperature. The \nMs is found to be incr eased with an increase of substrate temperature, which can be attributed \nto the in crease of crystalline quality with the increase of substrate temperature. The ferrima g-\nnetic - paramagnetic phase transition of the thin films grown at 800oC has been found nea r to \n716  (±10) K well above room temperature. These nanostru ctures might be useful for n a-\nnoscale multifunctional and spintronics device applications as they exhibit well saturated \nhigh Ms, soft magnetic behavior, with magnetic Tc well above the room tempe rature.  6 \n \n \n \nAcknowledgments  \nThis work is partly supported by U.S. Office of Naval Research Grants No. N00014 -12-1-\n1038 and No. N00014 - 14-1-0561. D. K. P and R. 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Krishnan, Substrate \nTemperature Dependent Anomalous Magnetic Behavior in CoFe 2O4 Thin Film,  IEEE  Trans. \nMagn . 47 (2011) 337 -340. \n[33] R. Sharma, P. Thakur, P. Sharma, V. Sharma, Ferrimagnetic Ni2+ doped Mg -Zn spinel \nferrite nanoparticles for high density inform ation stor age, J. Alloys Compd.  704 (2017) 7 -17. \n \n \n \n 8 \n \n \n \nFigure Captions  \n \nFig. 1.  (Color online) XRD patterns of NZFO thin film grown at different temperatures . The \ndiffraction peaks with symbols * and # corresponds to the peaks of LSAT/LNO (overlapped) \nand NZFO diffraction patterns, respectively.  \nFig. 2.  (Color online) Atomic force mi crographs of NZFO thin films grown at different te m-\nperatures.  Average roughness  (Ra) and root mean square roughness  (Rq) of each thin film are \nlisted in their respective growth temperatures.  \nFig. 3.  (Color online) One dimensional AFM line profile  which shows the surface height  (Z) \nof NZFO thin films grown at different temperatures  within an area of 3×3 μm2. Rq of all thin \nfilms are mentioned in their respective growth temperatures.  \nFig. 4. (Color online) Fitted XPS spectra of Ni  (2p 1/2, 2p 3/2), Zn (2p 1/2, 2p 3/2), Fe (2p 1/2, 2p 3/2), \nO (1s) in NZFO thin film grown at 600 oC which confirms the presence of all elements. The \nexperimental data are represented by black circles and the fitted pattern by the red solid line. \nThe wine colored curve represents  the Shirley background.  \nFig. 5. (Color online) F itted XPS spectra of Ni (2p 1/2, 2p 3/2) and Fe (2p 1/2, 2p 3/2) of NZFO thin \nfilms grown at 500, 600, 700, 800 oC. The contribution of Ni3+ and Fe3+ are observed to be \nhigher in the thin films which were grown at higher temperatures i,e 700oC, and 800oC. \nFig. 6.  (Color online) (a) Room temperature M-H loops ( inset : growth temperature depen d-\nence of Ms ) of NZFO thin film grown at different temperature (b) Temperature dependence \nof magnetization measured with zero field cooling (ZFC) and field cooling (FC) with applied \nstatic magnetic field of  1000 Oe (300 - 800 K) of NZFO thin film grown at 800 oC. The \nblack  solid line represents the ZFC curve and the red solid line represents the FC curve.  \n \n \n 9 \n \n \n \n \n \n20 30 40 50 60 70 80 90*#\n*\n*\n  \n* LSAT/LNO\n 500oC 800oC \n 700oC  750oC\n 650oC\n 600oC\n  \n(004)# NZFOIntensity (a.u.)\nBragg Angle (2)Figure 1. Pradhan et al.10 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n11 \n \n \n \n \n \nFigure 3. Pradhan et al.0246810\n0246810\n0 1 2 30246810\n0 1 2 3Rq = 2.12 nm \n \n  \n500 oCRq = 3.26 nm600 oC\n  \n  \nRq = 2.42 nm 650 oC\n  \n  \nRq = 2.05 nm 700 oC \n \n  \nRq = 1.29 nm 750 oC\n  Z (nm)\nDistance mRq = 0.28 nm 800 oC\n  \n  12 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4. Pradhan et al.880 870 860 850\n 600oC\nSatellite SatelliteNi 2P3/2\nNi 2P1/2\nBinding energy (eV)Intensity (a.u)  Ni+2\n Ni+3\n experimental data\n shirley background\n summation\n  \n740 730 720 710SatelliteFe 2P3/2\nFe 2P1/2\nSatellite600oC\nBinding energy (eV)Intensity (a.u) experimental data\n Fe+2\n Fe+3\n shirley background\n summation\n  1050 1040 1030 1020Zn 2P3/2\nZn 2P1/2600oC\nBinding energy (eV)Intensity (a.u)\n  \n534 532 530 528O 1S600oC\nadsorbed oxygen\nBinding energy (eV)Intensity (a.u)\n  13 \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 5. Pradhan et al.880 870 860 850Satellite500oC\nSatelliteNi 2P3/2\nNi 2P1/2\nBinding energy (eV)Intensity (a.u) Ni+3\n Ni+2\nexperimental data\n shirley background\n summation\n  \n880 870 860 850SatelliteSatelliteNi 2P3/2\nNi 2P1/2700oC\nBinding energy (eV)Intensity (a.u)  Ni+2\n Ni+3\n experimental data\n shirley background\n summation\n  \n880 870 860 850\n 600oC\nSatellite SatelliteNi 2P3/2\nNi 2P1/2\nBinding energy (eV)Intensity (a.u)  Ni+2\n Ni+3\n experimental data\n shirley background\n summation\n  \n880 870 860 850800oCNi 2P3/2\nNi 2P1/2 SatelliteSatellite\nBinding energy (eV)Intensity (a.u)  Ni+2\n Ni+3\nexperimental data\n shirley background\n summation\n  \n740 730 720 710 experimental data\n Fe+2\n Fe+3\n shirley background\n summationSatelliteSatellite500oCFe 2P3/2\nFe 2P1/2\nBinding energy (eV)Intensity (a.u)\n  \n740 730 720 710SatelliteFe 2P3/2\nFe 2P1/2\nSatellite600oC\nBinding energy (eV)Intensity (a.u) experimental data\n Fe+2\n Fe+3\n shirley background\n summation\n  \n740 730 720 710 700Satellite\nSatellite700oC\nFe 2P1/2Fe 2P3/2\nBinding energy (eV)Intensity (a.u) experimental data\n Fe+2\n Fe+3\n shirley background\n summation\n  \n740 730 720 710 700Fe 2P1/2Fe 2P3/2 800oC\nBinding energy (eV)Intensity (a.u)  experimental data\n Fe+2\n Fe+3\n shirley background\n summation\n  \nSatellite14 \n \n \n \n \n-20 -10 0 10 20-400-2000200400\n500 600 700 800100200300400 Ms (emu/cm3)\nTSub (oC) \n  \n  \n 500oC\n 600oC\n 650oC\n 700oC\n 750oC\n 800oC\n  \n  \nMagnetic Field (kOe)Magnetization (emu/cm3)(a)\nFigure 6. Pradhan et al.300 400 500 600 700 800050100 \n Magnetization (emu/cm3)\n@ 1000 Oe\nTemperature (K)  ZFC\n FC\n  \n(b)"    },    {        "title": "1705.08603v1.Control_of_single_mode_operation_in_a_circular_waveguide_filled_by_a_longitudinally_magnetized_gyroelectromagnetic_medium.pdf",        "content": "May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\nTo appear in the Journal of Electromagnetic Waves and Applications\nVol. 00, No. 00, Month 20XX, 1{12\n10.1080/0950034YYxxxxxxxx\nControl of single-mode operation in a circular waveguide \flled by\na longitudinally magnetized gyroelectromagnetic medium\nVolodymyr I. Fesenkoa;b;\u0003, Vladimir R. Tuza;b;c, Illia V. Fedorind, Hong-Bo Sunc,\nValeriy M. Shulgaa;b, and Wei Hana;e\naInternational Center of Future Science, Jilin University, Changchun, People's Republic of\nChina ;bInstitute of Radio Astronomy of National Academy of Sciences of Ukraine, Kharkiv,\nUkraine ;cState Key Laboratory on Integrated Optoelectronics, College of Electronic Science and\nEngineering, Jilin University, Changchun, People's Republic of China ;dNational Technical\nUniversity `Kharkiv Polytechnical Institute', Kharkiv, Ukraine ;eCollege of Physics, Jilin\nUniversity, Changchun, People's Republic of China\n(March 2017 )\nA substantial control of dispersion features of the hybrid EH 01and HE 11modes of a circular\nwaveguide which is completely \flled by a longitudinally magnetized composite \fnely-strati\fed\nferrite-semiconductor structure is discussed. A relation between the resonant conditions of\nsuch a composite gyroelectromagnetic \flling of the circular waveguide and dispersion fea-\ntures of the supported modes are studied. Three distinct frequency bands with the single-\nmode operation under normal as well as anomalous dispersion conditions of the EH 01mode\nare identi\fed by solving an optimization problem with respect to the \flling factors of the\ncomposite medium. The possibility of achieving isolated propagation of the HE 11mode is\nrevealed.\nKeywords: Microwave Propagation; Waveguide Theory; Gyrotropy; Guided Modes;\n1. Introduction\nToday the general theory of metallic waveguides designed to operate in the microwave\npart of spectrum is well developed and described in a number of subsequent papers and\ntextbooks [1{3]. This theory implies a solution of wave equations being under constraints\nof boundary conditions related to perfectly conducting (PEC) walls with accounting the\nguide geometry. The required solution appears as a set of waveguide (eigen)modes which\ndi\u000ber by their cuto\u000b frequencies and dispersion. Nevertheless, while the corresponding so-\nlutions obtained for hollow and isotropically \flled waveguides with coordinate boundaries\n(e.g. rectangular or circular waveguides) are well known and can be derived in an ex-\nplicit analytical form, those of the waveguides with an inhomogeneous and/or anisotropic\n(gyrotropic) \flling are much more complicated and require some particular considera-\ntions accompanied by cumbersome numerical calculations for each concrete design [4{12].\nMoreover, in the latter case the dispersion properties of such waveguides become to be\n\u0003Corresponding author. Email: volodymyr.i.fesenko@gmail.com\n1arXiv:1705.08603v1  [physics.class-ph]  24 May 2017May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\nsu\u000eciently intricate and e\u000bects of anomalous dispersion, complex wave propagation,\nmode coupling, splitting and conversion may occur. On the other hand, such a diversity\nin dispersion features of the waveguide modes gives an ability to design systems with\ndemanded characteristics. That is the reason why in recent years main research e\u000borts\nare directed towards the study of electromagnetic features of waveguides \flled by arti-\n\fcially created materials (metamaterials) in order to realize waveguide systems having\nunique operation conditions including those suitable for transmission in the range of\nterahertz waves [13{15]. In particular, in the present paper, extraordinary dispersion fea-\ntures of modes of a circular waveguide \flled by a composite medium possessing combined\ngyroelectromagnetic e\u000bect [16{18] are of a special interest.\nIndeed, conventional hollow and isotropically \flled circular waveguides support trans-\nverse electric (TE) and transverse magnetic (TM) modes from which TE 11mode is the\nfundamental one, whereas all waves supported by longitudinally magnetized gyrotropi-\ncally (ferrite or plasma) \flled waveguides are not pure TE and TM modes, since they\ninclude longitudinal components of both the electric and magnetic \felds. It is convenient\nto distinguish these waves as hybrid HE and EH modes, depending whether the longitu-\ndinal component of either magnetic or electric \feld is dominant, and, in general, these\nwaves are unable to degenerate into TE and TM modes under the axial symmetry of\nthe guide [19]. In such a gyrotropic waveguide HE 11mode is the fundamental one which\nis inherited from the TE 11mode of the hollow circular waveguide. Moreover, in such a\ngyrotropic waveguide, all non-symmetric HE nmand EHnmmodes with nonzero index n\nsplit into waves acquiring left-handed and right-handed circular polarizations [19, 20].\nIt is a standard engineering practice to choose parameters of a circular waveguide in\nsuch a way to ensure propagation of only TE 11and TM 01modes, all other higher order\nmodes are cut o\u000b and are non-propagating ones. Thus, operation of the microwave links\nin the single-mode conditions with supporting only the fundamental mode is particu-\nlarly useful in tasks of the power transmission over long distances and for broadcasting\nsystems [21]. Besides, nonreciprocal behaviors of the hybrid HE 11mode which propa-\ngates through a circular waveguide loaded by a longitudinally magnetized ferrite rod are\nutilized in a wide class of ferrite-based devices such as circulators, rotators, polarized\nabsorbers and duplexers [3, 19]. Nevertheless, some speci\fc applications require for a\nwaveguide to operate at the modes other than the fundamental one. For instance, the\nhigher order TE 0mmodes (especially, the TE 01mode) are very attractive for their use\nfor the low-loss power transmission over long distances and in resonant cavities with very\nhigh quality factor since these modes possess signi\fcantly smaller attenuation with fre-\nquency increasing compared to the fundamental mode. The systems in the particle-beam\nphysics operate on the hybrid EH 01mode in order to accelerate particles to relativis-\ntic speeds in the plasma \flled circular waveguides [12, 22, 23]. Interest in the circular\nwaveguides operating on EH 01/TM 01mode is also conditioned by the fact that some of\nhigh-power microwave sources (such as the relativistic backward-wave oscillators and the\nmagnetically insulating transmission-line oscillators) generate exactly the TM 01circular\nwaveguide mode.\nIn spite of prospects of using higher order modes instead of the fundamental one for\nspeci\fc applications, there are several problems observed during their excitation and re-\ntention. One of the problems is that these modes are not dominant in circular waveguides.\nThus, in order to provide their support, a waveguide should be oversized that inevitably\nresults in appearance of the undesirable propagation of a number of other modes having\nlower cuto\u000b frequencies. Another issue is that for the higher order modes the coupling be-\ntween the desired and undesired modes usually appears at the irregularities of the guide\n(e.g. bands, non-ideal inner cross-section, etc.). Moreover, the simultaneous presence of\nseveral modes in the waveguide causes not only higher power losses but also distortion\n2May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\nof the transmitted signal due to the inter-modal dispersion e\u000bect.\nTherefore, there is an interest in designs of waveguides which are able to support only a\nsingle desired higher order mode with avoiding its signi\fcant distortion and attenuation\nduring transmission. A possibility of such a single-mode operation for hybrid EH 01and\nHE11modes in a hollow circular waveguide has been recently reported in [12, 15]. In\nparticular, it has been demonstrated that a single-mode operation conditions can be\nachieved for the hybrid EH 01mode that possesses an anomalous dispersion in the circular\nwaveguide with either metamaterial-lined PEC wall or metamaterial-coated PEC rod.\nIt is shown that insertion of a thin metamaterial liner which exhibits both dispersive\nepsilon-negative and near-zero properties into a circular waveguide leads to formation\nof the passband with anomalous dispersion for the hybrid HE 11mode which arises far\nbelow its conventional cuto\u000b frequency.\nIn our previous study [18] it is revealed that in a circular waveguide completely \flled\nby a longitudinally magnetized composite \fnely-strati\fed ferrite-semiconductor structure\nthe combined geometrical and material dispersion appears in the hybrid modes which\ndi\u000bers drastically from that of conventional dielectric, ferrite or plasma \flled waveguides.\nMoreover, simultaneous presence of both gyromagnetic and gyroelectric e\u000bects in the\nwaveguide system allows to gain a substantial control over the dispersion characteristics\nand \feld distribution of the supported modes by utilizing an external static magnetic \feld\nas a driving agent. As a further elaboration of the results obtained in [18], in this paper\nwe demonstrate that bands of the single-mode operation for the hybrid EH 01and HE 11\nmodes can be controlled by a concisions choice of material and geometrical parameters\nof such a composite gyroelectromagnetic \flling of a circular waveguide.\n2. Outline of problem\nThereby, in this paper we study dispersion peculiarities of modes of an axial waveg-\nuide encircled by a PEC wall with radius R(Figure 1a). The waveguide inner region\nis considered to be completely \flled by a longitudinally magnetized composite medium\nwhich is constructed by combining together of gyromagnetic (ferrite with constitutive\nparameters \"f, ^\u0016f) and gyroelectric (semiconductor with constitutive parameters ^ \"s,\u0016s)\nlayers providing that these constitutive layers are arranged periodically along the guide,\ni.e., along the z-axis. Therefore, the alternating layers form a gyroelectromagnetic super-\nlattice which period is Land thicknesses of the constitutive ferrite and semiconductor\nlayers aredfandds, respectively. The \flling factors balance \u000ef+\u000es= 1 is additionally\nintroduced in order to de\fne the ratio between the ferrite \u000ef=df=Land semiconductor\n\u000es=ds=Lfractions of the superlattice. For further reference the dispersion characteris-\ntics of the tensor components of permeability ^ \u0016fof the ferrite layers and permittivity ^ \"s\nof the semiconductor layers are presented in panels (b) and (c) of Figure 1, respectively.\nThey are calculated using common expressions for description of constitutive parameters\nof normally biased ferrite [20, 24] and semiconductor [25] materials taking into account\nthe losses. In particular, it is supposed that the practical structure is made in the form\nof a barium-cobalt/doped-silicon superlattice [26].\nIn what follows we stipulate two restrictions. First, the strength of an external static\nmagnetic \feld ~Mwhich is aligned along the axis of the guide (i.e., along the z-axis)\nis high enough to produce a homogeneous saturated state of magnetic as well as semi-\nconductor layers. Second, the whole composite \flling of the guide is considered to be\na \fnely-strati\fed structure. It implies that all characteristic dimensions of such a mul-\ntilayered ferrite-semiconductor structure are much smaller than the wavelength in the\ncorresponding parts of the composite medium, i.e., df\u001c\u0015,ds\u001c\u0015, andL\u001c\u0015(the\n3May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\nlong-wavelength approximation). Under this approximation the homogenization proce-\ndures from the e\u000bective medium theory (is not presented here; for details see, [27, 28])\nare involved in order to describe the multilayered ferrite-semiconductor structure under\nstudy via an equivalent anisotropic uniform medium whose optical axis is directed along\nthez-axis. Note, the procedures are veri\fed by the transfer matrix technique [29{31],\nand they are repeatedly used in the solid state physics [32{38].\nFigure 1. (a) Schematic of a circular waveguide completely \flled by a longitudinally magnetized multilayered\nferrite-semiconductor structure and dispersion curves of the tensors components of (b) ferrite permeability ^ \u0016and\n(c) semiconductor permittivity ^ \"depicted on the frequency scale normalized on the radius Rof the guide; for ferrite,\nunder saturation magnetization of 2930 G, parameters are: f0=!0=2\u0019= 4:2 GHz,fm=!m=2\u0019= 8:2 GHz,\n\"f= 5:5,b= 0:02; for semiconductor, parameters are: fp=!p=2\u0019= 4:9 GHz,fc=!c=2\u0019= 4:7 GHz,\"l= 1:0,\n\u0017=2\u0019= 0:05 GHz,\u0016s= 1:0.\nThus, the resulting equivalent gyroelectromagnetic medium that \flls the guide is fur-\nther characterized by two tensors of relative e\u000bective permittivity ^ \"effand relative ef-\nfective permeability ^ \u0016effwritten in the form\n^\"eff=0\n@\" i\"a0\n\u0000i\"a\"0\n0 0\"k1\nA; ^\u0016eff=0\n@\u0016 i\u0016a0\n\u0000i\u0016a\u00160\n0 0\u0016k1\nA; (1)\nwhere the complete expressions for the tensors components derived via underlying con-\nstitutive parameters of magnetic and semiconductor layers one can \fnd in [36{38].\nThereby, the initial problem is reduced to the consideration of a circular waveguide\nwhich is completely \flled by a gyroelectromagnetic uniform medium. The essence of the\nproblem is to obtain and numerically solve the dispersion equation with a subsequent\n4May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\nclassi\fcation of the waveguide modes. The solution procedure comprises a deriving of\na pair of coupled Helmholtz wave equations with respect to longitudinal components\nof the electromagnetic \feld inside the gyroelectromagnetic \flling of the guide and then\nimposition of the boundary conditions on the PEC circular wall (the treatment is omitted\nhere and can be found in [17, 20]).\n3. Classi\fcation and passbands conditions of hybrid modes\nOur objective here is to discuss the principles of classi\fcation, determination of passbands\npositions and cuto\u000b frequencies of modes of a circular gyroelectromagnetic waveguide.\nIn order to classify the modes, it is a normal practice to consider the structure without\nlosses (i.e. to solve the eigenwaves problem), and then determine the bands where a high\nlevel of losses can make a signi\fcant impact on the propagating waves in a practical\nsystem.\nSince in the waveguide under study the waves possess a hybrid character and arise\nin both symmetric and asymmetric forms, they need to be classi\fed as hybrid HE nm\nand EHnmmodes depending on which longitudinal component of either magnetic (H z>\nEz, HE-modes) or electric (E z>Hz, EH-modes) \feld is dominant. Here the indexes n\nandmare introduced to describe the number of \feld variations in azimuth and radius,\nrespectively. For the asymmetric modes ( n6= 0), each \feld variation in azimuth ( \u0006n)\nis related to two independent solutions of the dispersion equation, which di\u000ber by the\npropagation constant \r. This di\u000berence in the propagation constant distinguishes two\northogonal polarization states with the left-handed and right-handed circular rotation\n[19, 39]. The handedness of the circular polarization is de\fned by the sign of the integer\nn. Whenn<0 the waves acquire the left-handed circular polarization (HE\u0000\nnmand EH\u0000\nnm\nmodes), whereas when n>0 they are right-handed circularly polarized ones (HE+\nnmand\nEH+\nnmmodes). For the symmetric modes ( n= 0), the dispersion equation has a single\nsolution, and in this case the sign `+' or ` \u0000' in the mode notations is absent.\nThe hybrid modes classi\fcation is made involving an auxiliary reference waveguide\nthat is \flled by an isotropic homogeneous medium since the modes of such a waveg-\nuide are de\fned de\fnitely. Thus, for the waveguide structure under study it is supposed\nto start with such an isotropic (non-gyrotropic lossless) case (i.e. considering the limit\nM!0) and classify the modes as those of either TE-type or TM-type beginning from\ntheir cuto\u000bs, and then gradually increase the gyrotropic parameters of the underlying\nconstituents (i.e. non-diagonal constitutive tensors elements \u0016aand\"a) of the gyroelec-\ntromagnetic medium inside the guide and trace the changing in the propagation constant\n\r[39, 40]. According to such scheme, hybrid modes of the HE-type and EH-type of the\ngyroelectromagnetic waveguide under study appear to be directly associated with corre-\nsponding modes of the TE-type and TM-type of the auxiliary reference waveguide.\nIn order to reveal passbands and cuto\u000b frequencies of the hybrid modes of a gyrotropic\nwaveguide it is convenient to introduce some generalized parameters [18], namely e\u000bective\ntransverse permeability \u0016?, e\u000bective transverse permittivity \"?, and e\u000bective refractive\nindex\u0011\u0006written in the form:\n\u0016?=\u0016\u0000\u00162\na=\u0016; \"?=\"\u0000\"2\na=\"; \u0011\u0006=p\n(\u0016\u0006\u0016a)(\"\u0006\"a); (2)\nwhere in the latter term the upper sign `+' and lower sign ` \u0000' are related to the right-\nhanded and left-handed circularly polarized waves propagating through an unbounded\ngyrotropic medium, respectively.\nThe passbands of hybrid modes in a circular waveguide \flled by a gyrotropic medium\n(i.e., it can be a gyroelectric medium described by a permittivity tensor, a gyromagnetic\n5May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\nmedium described by a permeability tensor, as well as a gyroelectromagnetic medium\ndescribed by both permittivity and permeability tensors) exist inside the areas of \r\u0000k0\nspace where in the lossless case the e\u000bective refractive index \u0011\u0006acquires real numbers\n[41]. It corresponds to the propagation conditions of the right-handed and left-handed\ncircularly polarized eigenwaves of a gyrotropic medium (see areas outlined by the red\ndash-dotted lines in Figure 2). Moreover, these passbands are restricted laterally by the\nlines where e\u000bective transverse permeability \u0016?or e\u000bective transverse permittivity \"?\nreaches some extrema (see vertical gray dash lines in Figure 2).\nIn order to highlight the main di\u000berences between dispersion characteristics of the\nwaveguide \flled by a gyroelectromagnetic medium compared to those \flled by ferrite or\nplasma ones, the passbands of hybrid modes supported by the circular waveguide \flled\nby a lossless gyromagnetic ( \u000ef= 1,\u000es= 0), gyroelectric ( \u000ef= 0,\u000es= 1), as well as\ngyroelectromagnetic ( \u000ef=\u000es= 0:5) medium are depicted in panels (a), (b) and (c) of\nFigure 2, respectively. One can conclude that while in the waveguides \flled by either\ngyromagnetic or gyroelectric medium there are two passbands, in the waveguide \flled\nby the gyroelectromagnetic medium four passbands appear where supported modes can\nexist. For all types of the waveguide \flling every passband starts at the frequency where\n\u0016?= 0 or\"?= 0, and it ends at the frequency where the asymptotic line \u0016?!1 or\n\"?!1 arises.\nFurthermore, in each identi\fed passband the low-index modes are properly classi\fed\nassuming the blue and green curves in the \fgure express dispersion characteristics of\nthe asymmetric modes with the right-handed and left-handed circular polarization, re-\nspectively, whereas the black curves represent those of the symmetric HE 0mand EH 0m\nmodes. It is revealed that in the lossless case for the waveguide modes to exist both e\u000bec-\ntive transverse permeability \u0016?and e\u000bective transverse permittivity \"?must be positive\nquantities, whereas their underlying constitutive parameters \u0016and\"can possess di\u000ber-\nent signs. Remarkable, in those bands where parameters \u0016and\"have di\u000berent signs the\nmodes exhibit an anomalous dispersion line.\nFinally, taking into account the above discussed scheme of the hybrid modes clas-\nsi\fcation with involving an isotropically \flled auxiliary reference waveguide, the cuto\u000b\nfrequency of a particular supported mode can be estimated from the following expressions\n[39]:\nHE-modes: fHE\nnm=\u001f0\nnmc\u0010\n2\u0019Rp\"?\u0016jj\u0011\u00001\n;\nEH-modes: fEH\nnm=\u001fnmc\u0010\n2\u0019Rp\"jj\u0016?\u0011\u00001\n;(3)\nwhere\u001fnmand\u001f0\nnmare zeros of the Bessel function Jnof the \frst kind of order n\n(n= 0;1;2;:::) and its \frst derivative J0\nn, respectively.\nFrom this estimation it is revealed that in the circular waveguide \flled by a gyrotropic\nmedia an extraordinary dispersion feature may appear consisting in arising the EH 01\nmode below the HE 11mode cuto\u000b (see, Figure 2a). We also distinguish a particular\nmode as an isolated one if it stands alone within a certain passband of the circularly\npolarized eigenwaves of the unbounded gyrotropic medium.\n4. Dispersion control of the EH 01and HE 11modes\nHereinafter it is our goal to demonstrate an ability to realize a single-mode operation\nfor either EH 01mode or isolated EH 11mode in the waveguide system under study. In\norder to reach conditions for such an operation both the \flling factors of the composite\n6May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\nFigure 2. Dispersion curves of the hybrid HE\u0006\nnmand EH\u0006\nnmmodes of a circular waveguide completely \flled by\nlossless (b= 0,\u0017= 0) (a) gyromagnetic, (b) gyroelectric, and (c) gyroelectromagnetic medium. The blue curves\nand upper sign `+' correspond to the asymmetric modes with the right-handed circular polarization, whereas the\ngreen curves and upper sign ` \u0000' correspond to the asymmetric modes with the left-handed circular polarization.\nBlack curves represent dispersion of the symmetric HE 0mand EH 0mmodes. Extreme states \u0016?= 0,\u0016?!1\nand\"?= 0,\"?!1 are presented by the dashed vertical lines. All other constitutive parameters of ferrite and\nsemiconductor are \fxed as in Figure 1.\n7May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\nmedium and radius of the guide need to be carefully adjusted to distinct cuto\u000b of the\nhigher order modes. In particular, compared to the results of our reference paper [18],\nhere the waveguide radius is halved and \fxed, and then an optimization problem is solved\nwith respect to the \flling factors \u000efand\u000esto \fnd the single-mode operation conditions.\nFigure 3. Frequency bands versus \flling factor \u000efof the gyroelectromagnetic \flling of the waveguide corresponding\nto a single-mode operation of the EH 01mode (blue areas) and isolated HE\u0000\n11mode (green area). Color map\nand intensity bar represent the imaginary parts of e\u000bective transverse permeability \u001600\n?and e\u000bective transverse\npermittivity \"00\n?. All other constitutive parameters of ferrite and semiconductor are \fxed as in Figure 1.\nA graphical solution of the designated optimization problem is depicted in Figure 3,\nwhere the frequency bands allowable for the single-mode operation of the EH 01mode as\nwell as isolated HE\u0000\n11mode are identi\fed depending on the \flling factor \u000ef. One can con-\nclude that in the waveguide \flled by a gyroelectromagnetic medium with a predominant\nimpact of the ferrite subsystem ( \u000ef>\u000es) the EH 01mode appears to be the fundamental\none within the range of \flling factor 0 :64\u0014\u000ef\u00141. In particular, the conditions of the\nsingle-mode operation of the EH 01mode are found may occur in three particular pass-\nbands. In Figure 3 these passbands are colored in blue and denoted by Roman numerals\nI, II, and III. On the other hand, the isolated HE\u0000\n11mode arises in the waveguide \flled by\na gyroelectromagnetic medium with a predominant impact of the semiconductor subsys-\ntem (\u000es>\u000ef). Within the frequency band of interest, only one passband is found where\nsuch isolated single-mode operation conditions for the HE 11mode can be achieved. It\nappears in the range of \flling factor 0 :12\u0014\u000es\u00140:43. In Figure 3 this distinct passband\nis colored in green and denoted by Roman numeral IV.\nFrom estimations (3) it can be determined which hybrid mode (either HE 11mode or\nEH01mode) is the fundamental one in the waveguide under study comparing values of\nrelations\u001f0\n11=p\"?\u0016jjand\u001f01=p\"jj\u0016?obtained for the HE 11mode and EH 01mode, re-\nspectively. Since \u001f0\n11and\u001f01are constant quantities related to the Bessel function roots,\nthe cuto\u000bs of corresponding modes depend on the multipliers \"?\u0016jjand\"jj\u0016?in the de-\nnominators only. Thus, in the case when \u001f0\n11=p\"?\u0016jj<\u001f 01=p\"jj\u0016?the HE 11mode is the\nfundamental one, otherwise it is substituted by the EH 01mode. This remarkable feature\ndistinguishes a gyrotropic waveguide from conventional circular hollow or isotropically\n\flled waveguides where the EH 01mode cannot be the fundamental one.\nIn order to reveal propagation conditions for the identi\fed modes in the frequency\nband of interest the imaginary parts of e\u000bective transverse permeability \u001600\n?and e\u000bective\n8May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\nFigure 4. Dispersion curves of the hybrid EH 01and HE\u0000\n11modes, and corresponding magnitude patterns of the z-\ncomponent of (a-c) magnetic and (d) electric \feld in a circular waveguide completely \flled by a gyroelectromagnetic\nmedium. The medium \flling factors are: (a, b) \u000ef= 0:8,\u000es= 0:2; (c)\u000ef=\u000es= 0:5; (d)\u000ef= 0:15,\u000es= 0:85. All\nconstitutive parameters of ferrite and semiconductor are \fxed as in Figure 1.\ntransverse permittivity \"00\n?are plotted as color maps in the \u000ef\u0000k0space where the level\nof losses is denoted in the color bar on the right side of Figure 3. One can conclude that\npassbands I, III and IV lay far from the characteristic frequencies of the ferromagnetic and\nplasma resonances of the underlying materials, therefore, a low level of losses is expected\nto be in these passbands. Unfortunately, passband II is deprived of this advantage, and\nthus the wave propagation within this band may be signi\fcantly suppressed.\nIn order to verify the scheme of the hybrid modes classi\fcation, the \feld patterns of\nparticular modes in the identi\fed passbands I{IV are plotted in Figure 4. These pattern\nare obtained for certain values in the \r\u0000k0space which are selected in the middle\nof the corresponding dispersion curve for di\u000berent ratio of the \flling factors \u000efand\u000es.\nOne can see that despite of the fact that the di\u000berence in the ratio of material fractions\nsigni\fcantly in\ruences the dispersion curves appearance of both hybrid EH 01and HE 11\nmodes, their \feld patterns acquire distributions which are very similar to those of a\nconvenient hollow or isotropically \flled circular waveguide, and, thus, the hybrid modes\ncan be identi\fed de\fnitely.\nMoreover, from the appearance of the dispersion curves in Figure 4 it is revealed,\nthat the modes can possess either normal or anomalous dispersion line depending on\nthe characteristics of the gyroelectromagnetic \flling medium. For instance, since in the\nparticular region II the conditions \u0016>0 and\"<0 hold, the isolated EH 01mode acquires\n9May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\nan anomalous dispersion within the narrow frequency band (see, Figure 4b).\n5. Conclusion\nDispersion characteristics of a circular waveguide completely \flled by a longitudinally\nmagnetized gyroelectromagnetic medium are studied. It is shown that in such a waveguide\nthe single-mode passbands for the EH 01and HE 11modes can be achieved far below their\nnatural cuto\u000b frequencies without signi\fcant distortion of the modes \feld patterns.\nBy solving an optimization problem, it is revealed that cuto\u000b frequencies of both the\nHE11and EH 01modes can be e\u000bectively controlled by adjusting the ratio between the\nferrite and semiconductor components of the constitutive layers within the period of the\ncomposite medium for a \fxed radius of the guide. Two remarkable results are distin-\nguished. First, the EH 01mode is obtained to be the fundamental one in the structure\nwith a predominant impact of the ferrite subsystem, and, for this mode the single-mode\npropagation conditions accompanied by an anomalous dispersion line are achieved in\nthe frequency band where constitutive parameters of the underlaying materials possess\ndi\u000berent signs. Second, for the fundamental HE 11mode the single-mode operation is\nobtained in the structure with a predominant impact of the semiconductor subsystem.\nDisclosure statement\nNo potential con\rict of interest was reported by the authors.\nORCID\nVolodymyr I. Fesenko http://orcid.org/0000-0001-9106-0858\nReferences\n[1] Baden Fuller AJ. Microwaves, 1st ed. Oxford: Pergamon Press; 1969.\n[2] Collin RE. Field theory of guided waves. Piscataway (NJ): IEEE; 1991.\n[3] Pozar DM. Microwave engineering, 4th ed. Hoboken (NJ): John Wiley & Sons, Inc.; 2012.\n[4] Kales ML. 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J Appl Phys. 2008;103(2):023302.\n[23] Cook AM, Tikhoplav R, Tochitsky SY, Travish G, Williams OB, Rosenzweig JB. Observation\nof narrow{band terahertz coherent Cherenkov radiation from a cylindrical dielectric-lined\nwaveguide. Phys Rev Lett. 2009 Aug;103:095003; Available from: http://link.aps.org/\ndoi/10.1103/PhysRevLett.103.095003 .\n[24] Collin RE. Foundation for microwave engineering. New Jersey: WileyInterscience; 1992.\n[25] Bass FG, Bulgakov AA. Kinetic and electrodynamic phenomena in classical and quantum\nsemiconductor superlattices. New York: Nova Science; 1997.\n[26] Wu RX, Zhao T, Xiao JQ. Periodic ferrite{semiconductor layered composite with negative\nindex of refraction. J Phys Condens Matter. 2007;19(2):026211; Available from: http://\nstacks.iop.org/0953-8984/19/i=2/a=026211 .\n[27] Agranovich V. Dielectric permeability and in\ruence of external \felds on optical properties\nof superlattices. Solid State Commun. 1991;78(8):747{750; Available from: http://dx.doi.\norg/10.1016/0038-1098(91)90856-Q .\n[28] Elmzughi FG, Constantinou NC, Tilley DR. The e\u000bective-medium theory of magnetoplasma\nsuperlattices. J Phys: Condens Matter. 1995;7(2):315{326; Available from: http://stacks.\niop.org/0953-8984/7/i=2/a=009 .\n[29] Tuz VR, Batrakov OD, Zheng Y. Gyrotropic-nihility in ferrite-semiconductor composite\nin Faraday geometry. Prog Electromagn Res B. 2012;41:397{417; Available from: http:\n//jpier.org/pierb/pier.php?paper=12042603 .\n[30] Tuz VR. Gyrotropic-nihility state in a composite ferrite-semiconductor structure. J\nOpt. 2015;17(3):035611; Available from: http://stacks.iop.org/2040-8986/17/i=3/a=\n11May 25, 2017 Journal of Electromagnetic Waves and Applications GW-modes\n035611 .\n[31] Tuz VR, Fesenko VI. Gaussian beam tunneling through a gyrotropic-nihility \fnely-strati\fed\nstructure. In: Shulika O, Sukhoivanov I, editors. Contemporary optoelectronics. Vol. 199 of\nSpringer Series in Optical Sciences; Springer Netherlands; 2016. p. 99{113; Available from:\nhttp://dx.doi.org/10.1007/978-94-017-7315-7 .\n[32] Abraha K, Tilley DR. Theory of far infrared properties of magnetic surfaces, \flms and su-\nperlattices. Surf Sci Rep. 1996;24(5):129{222; Available from: http://www.sciencedirect.\ncom/science/article/pii/0167572996000039 .\n[33] Aronov IE, Beletskii NN, Berman GP, Bishop AR. Collective electromagnetic excitations in\na double-layer two-dimensional electron system in a high magnetic \feld. Phys Rev B. 1997\nOct;56:10392{10403; Available from: https://link.aps.org/doi/10.1103/PhysRevB.56.\n10392 .\n[34] Kushwaha MS. Plasmons and magnetoplasmons in semiconductor heterostructures. Surf Sci\nRep. 2001;41(1):1{416.\n[35] Farjami Shayesteh S, Rahmani A, Parker TJ. Investigation of optical phonons and electronic\nproperties of group III-V nitride multi quantum wells by using far infrared magnetoplasmon\nspectroscopy. Phys Status Solidi (c). 2006;3(6):2241{2245; Available from: http://dx.doi.\norg/10.1002/pssc.200565457 .\n[36] Tuz VR. Polaritons dispersion in a composite ferrite-semiconductor structure near\ngyrotropic-nihility state. J Magn Magn Mater. 2016;419:559{565; Available from: http:\n//www.sciencedirect.com/science/article/pii/S0304885316312355 .\n[37] Tuz VR, Fesenko VI, Fedorin IV, Sun HB, Shulga VM. Crossing and anti-crossing e\u000bects\nof polaritons in a magnetic-semiconductor superlattice in\ruenced by an external magnetic\n\feld. Superlattice Microst. 2017;103:285{294; Available from: http://www.sciencedirect.\ncom/science/article/pii/S0749603617301970 .\n[38] Tuz VR, Fesenko VI, Fedorin IV, Sun HB, Han W. Coexistence of bulk and surface polaritons\nin a magnetic-semiconductor superlattice in\ruenced by a transverse magnetic \feld. J Appl\nPhys. 2017;121(10):103102; Available from: http://dx.doi.org/10.1063/1.4977956 .\n[39] Veselov G, Raevskiy S. Layered metal-dielectric waveguides (in Russian). Radio i Sviaz',\nMoscow; 1988.\n[40] Whites KW. Electromagnetic wave propagation through circular waveguides containing ra-\ndially inhomogeneous lossy media. Washington (DC): DTIC Document; 1989. Report No.:\n(AD-A213 062).\n[41] Tuz VR, Vidil MY, Prosvirnin SL. Polarization transformations by a magneto-photonic\nlayered structure in the vicinity of a ferromagnetic resonance. J Opt. 2010;12(9):095102;\nAvailable from: http://stacks.iop.org/2040-8986/12/i=9/a=095102 .\n12"    },    {        "title": "2106.11283v2.A_low_loss_ferrite_circulator_as_a_tunable_chiral_quantum_system.pdf",        "content": "A low-loss ferrite circulator as a tunable chiral quantum system\nYing-Ying Wang,1Sean van Geldern,1Thomas Connolly,1,\u0003Yu-Xin Wang,2Alexander\nShilcusky,1Alexander McDonald,2, 3Aashish A. Clerk,2and Chen Wang1,y\n1Department of Physics, University of Massachusetts-Amherst, Amherst, MA, USA\n2Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL, USA\n3Department of Physics, University of Chicago, Chicago, IL, USA\n(Dated: November 5, 2021)\nFerrite microwave circulators allow one to control the directional flow of microwave signals and noise, and\nthus play a crucial role in present-day superconducting quantum technology. They are typically viewed as a\nblack-box, with their internal structure neither specified nor used as a quantum resource. In this work, we show\na low-loss waveguide circulator constructed with single-crystalline yttrium iron garnet (YIG) in a 3D cavity, and\nanalyze it as a multi-mode hybrid quantum system with coupled photonic and magnonic excitations. We show\nthe coherent coupling of its chiral internal modes with integrated superconducting niobium cavities, and how\nthis enables tunable non-reciprocal interactions between the intra-cavity photons. We also probe experimentally\nthe effective non-Hermitian dynamics of this system and its effective non-reciprocal eigenmodes. The device\nplatform provides a test bed for implementing non-reciprocal interactions in open-system circuit QED.\nI. INTRODUCTION\nMicrowave circulators, typically composed of a transmission\nline Y-junction with ferrite materials [1], are ubiquitous in su-\nperconducting circuit QED experiments [2]. They provide a\ncrucial link in the readout chain of superconducting quantum\nprocessors, by directing the signal traffic while protecting the\nqubits and resonators from thermal noise [3]. They also enable\nthe interactions between distinct quantum circuit modules to be\nnon-reciprocal [4, 5], a feature which is important for eliminat-\ning long-distance cross-talks in modular quantum computation\narchitectures. Despite their importance, microwave circulators\nare generally treated as broadband black-box devices in experi-\nments. Formulating a more microscopic quantum description is\noften challenging, as their internal modes involving the magnetic\nspin excitations (magnons) are generally too lossy and complex\nto be analyzed using canonical circuit quantization [6].\nOn the other hand, there has been growing interest in\nstudying and manipulating magnon excitations of ferromag-\nnetic/ferrimagnetic materials in the quantum regime [7, 8]. In\nparticular, the ferromagnetic resonance (FMR) mode of yttrium\niron garnet (YIG), a ferrimagnetic insulator with usage in com-\nmercial circulators, has shown sufficiently high quality factor\nand coupling cooperativity with microwave cavities to func-\ntion as a quantum oscillator mode in strong-coupling circuit\nQED [9–11]. Notably, coherent coupling of magnons with a\nsuperconducting qubit [12] and single-shot detection of a sin-\ngle magnon [13] have been demonstrated using a millimeter-\nsized single-crystalline YIG sphere in a 3D cavity. Furthermore,\nthere is a plausible pathway towards planar superconducting-\nmagnonic devices [14, 15] to connect circuit QED with spin-\ntronics technologies by advancing fabrication techniques of low-\ndamping YIG films [16].\nIt would be interesting to harness these recent advances in\nthe study of quantum magnonics to revisit the design of mi-\ncrowave circulators, potentially leading to new kinds of non-\nreciprocal devices in circuit QED. Our work here describes\n\u0003Present address: Department of Applied Physics, Yale University, New\nHaven, CT, USA\nywangc@umass.edua first step in this direction. Here we demonstrate a tun-\nable non-reciprocal device based on the waveguide circulator\nloaded with single-crystalline YIG, which explicitly makes use\nof well-characterized hybrid polariton modes. Such modes\nare the normal modes of coupled magnon-photon systems [9–\n11, 17, 18], and have an intrinsic chirality that is set by the\nmagnetic field [19–21]. While our device follows the same ba-\nsic working principles underpinning textbook circulators [1, 22],\ndetailed understanding of the internal modes allows us to incor-\nporate the physical source of non-reciprocity in the full descrip-\ntion of a larger system including two external superconducting\ncavities, using a non-Hermitian effective Hamiltonian.\nWhile our device can be configured to operate as a traditional\ncirculator for its non-reciprocal transmission of travelling waves,\nthe main focus of our study is to use the device for mediating\ntunable non-reciprocal interaction between localized long-lived\nquantum modes. Such non-reciprocal mode-mode couplings re-\nsult in distinct signatures in the eigenvalues and eigenvectors of\nthe non-Hermitian system Hamiltonian, which is relevant to the\nmore general study of non-Hermitian dynamics in contexts rang-\ning from classical optics to quantum condensed matter. Anoma-\nlous properties of the eigenvalues and eigenvectors of a non-\nHermitian Hamiltonian have given rise to a number of striking\nphenomena such as the existence of exceptional points [23, 24]\nand the non-Hermitian skin effect [25–27], but direct experimen-\ntal access to the underlying eigenmodes is often difficult. In this\nstudy, we provide comprehensive characterization of the eigen-\nmode structure, which is a step towards effective Hamiltonian\nengineering of non-reciprocal non-Hermitian systems.\nThe most tantalizing usage of non-reciprocity in quantum sys-\ntems (such as entanglement stabilization using directional in-\nteractions in chiral quantum optics setups [28, 29]) require ex-\ntremely high quality devices. In particular, they must approach\nthe pristine limit where undesirable internal loss rates are neg-\nligible compared to the non-reciprocal coupling rates. While\nmany experiments have focused on new avenues of achieving\nnon-reciprocity [30–36], this loss-to-coupling ratio, which can\nbe understood as the quantum efficiency of the non-reciprocal\ninteractions, has been typically limited to approximately 10%\n(\u00180.5 dB) or more, which is comparable to the linear insertion\nloss of typical commercial circulators as measured in modular\ncircuit QED experiments [4, 5]. This performance lags far be-\nhind the quality of unitary operations between reciprocally cou-arXiv:2106.11283v2  [quant-ph]  4 Nov 20212\n(a) VNA (b)\nYXZ\nIMT WCP\nFIG. 1. Device and measurement setup. (a) A YIG cylinder (black)\nis placed at the center of the intersection of three rounded-rectangular\nwaveguides placed 120 degrees away from each other. The light grey re-\ngion is vacuum inside an oxygen-free copper enclosure. The device can\nbe assembled in two different configurations: First, a drum-head shaped\ntransition pin can be attached at the end of each waveguide section to\nform an impedance matched waveguide-to-SMA transition (IMT). Al-\nternatively, a short weakly-coupled probe (WCP) can be attached to\neach waveguide section to explore the internal modes of the device. (b)\nThe device is mounted to a mezzanine plate that is thermalized to the\nmixing chamber of a dilution refrigerator, and is positioned at the center\nof a superconducting solenoid magnet which operates at 4K. The device\nis connected to three input cables (with attenuators as marked) and two\noutput amplifier lines (with directional couplers splitting signals) for\nS-parameter measurements using a vector network analyzer (VNA).\npled quantum components (i.e. two-qubit gate infidelity <1%).\nOur approach provides a route for transcending this limitation\non the quantum efficiency of non-reciprocal interactions.\nThe results of our study have implications in several areas: (1)\nIn the context of quantum magnonics, we present the first study\nof polariton modes with a partially magnetized ferrite material,\nwhich features a high quality factor and low operating field, both\nof which are crucial for constructing superconducting-magnonic\ndevices. (2) In the context of modular superconducting quan-\ntum computing, we demonstrate the first circulator with internal\nloss well below 1% of the coupling bandwidth, which would\nenable high-fidelity directional quantum state transfer. (3) For\nthe general non-Hermitian physics, we demonstrate an experi-\nmental probe of the non-reciprocal eigenvector composition of a\nnon-Hermitian system. Combining these advances, we have es-\ntablished an experimental platform that meets the conditions for\nfuture study of nonlinear non-reciprocal interactions with super-\nconducting qubits.\nII. EXPERIMENTAL SETUP\nOur experimental setup is shown in Fig. 1(a). Three rounded-\nrectangular waveguides, each with a cross section of 21.0 mm \u0002\n4.0 mm, placed 120 degrees away from each other, intersect to\nform the body of the circulator. A \u001e-5.58 mm\u00025.0 mm single-\ncrystalline YIG cylinder is placed at the center of the Y-junction,\nwith external magnetic fields applied along its height (the zaxis\nand the [111] orientation of the YIG crystal). At the end of\nthe three waveguide sections, we can either attach impedance-\nmatched waveguide-to-SMA transitions (IMT) to perform stan-dard characterization of the circulator (as in Section IV), or at-\ntach weakly-coupled probe pins (WCP) to explore the internal\nmodes of this YIG-loaded Y-shaped cavity (as in Section III).\nThe use of reconfigurable probes in the same waveguide package\nallows us to infer the operation condition and the performance\nof the circulator from the properties of the internal modes. Fur-\nthermore, the copper waveguide sections can be replaced by su-\nperconducting niobium cavities, with details to be described in\nSection V and Fig. 5. This modular substitution introduces ad-\nditional external high Q modes to the system, and understanding\nthe resulting Hamiltonian and the hybridized mode structure of\nthe full system will be a first step towards the study of pristine\nnon-reciprocal interactions in circuit QED.\nThe device package is thermalized to the mixing chamber\nplate (\u001820 mK) of a Bluefors LD-250 dilution refrigerator inside\nthe\u001e-100 mm bore of a 1 T superconducting magnet that applies\nmagnetic field along the zaxis [Fig. 1(b)]. A vector network an-\nalyzer is used to measure the complex microwave transmission\ncoefficients Sij(from Portjto Porti, wherei;j= 1;2;3) of\nthe device in series with a chain of attenuators, filters and ampli-\nfiers as in typical circuit QED experiments. A magnetic shield\nmade of a steel sheet is placed outside the bottom half of the re-\nfrigerator, and all data is acquired under the persistent mode of\nthe superconducting magnet to minimize magnetic-field fluctua-\ntions.\nIII. INTERNAL MODE STRUCTURE\nWe begin by discussing the internal mode structure of the de-\nvice, as probed by S21as a function of applied magnetic field\nBwhen the device is installed with WCP [Fig. 2(a)]. A se-\nries of electromagnetic modes (relatively field-independent) are\nobserved to undergo large avoided crossings with a cluster of\nmagnon modes of the YIG crystal, forming photon-magnon po-\nlariton modes. The magnon mode most strongly coupled to pho-\ntons is known to correspond to near-uniform precession of YIG\nspins, or the Kittel mode of FMR, whose frequency increases\nlinearly with magnetic field: !m=\r[B+\u00160(Nx;y\u0000Nz)Ms]\u0019\n\rB, as marked by the dashed line in Fig. 2(a). Here \ris the\ngyromagnetic ratio, and the (volume-averaged) demagnetizing\nfactorsNx;y;z in magnetic saturated state are very close to 1=3\nfor the aspect ratio of our YIG cylinder [37]. These avoided\ncrossings are similar to previous experiments showing strong\nphoton-magnon coupling [10, 11], but due to the much larger\nsize of the YIG in our experiment, a large cluster of higher-\norder magnetostatic modes, most of which have slightly higher\nfrequency than the Kittel mode [38, 39] also coherently interact\nwith the microwave photons, contributing to the complex trans-\nmission spectra in the vicinity of the crossings. Nevertheless, to\nhave a coarse estimate of the photon-magnon coupling strength,\nit is convenient to model each observed spectral line far away\nfrom the crossing region as a bare electromagnetic mode with\nfrequency!c=2\u0019hybridized with a single combined magnon\nmode. The implied coupling strengths g=2\u0019(in the cavity QED\nconvention) are about 1.2 GHz and 2.1 GHz for the two modes of\nparticular interest to this study [blue and red in Fig. 2(a)], plac-\ning the mode hybridization in the ultrastrong coupling regime\n(see e.g. [40]) with g=(!c+!m)\u001810%. Even atB= 0, with\na photon-magnon detuning of \u0001 =!c\u0000!m\u00192\u0019\u000110GHz, the\nparticipation of magnon excitations in the photon-branch of the3\n(a) (c)\nField ( T) Field ( T) Field ( T)Frequenc y (GHz)\nFrequenc y (GHz)\nLinewidth (MH z)\n(dB) (b)\nlower freq. modehigher freq. mode\nFIG. 2. Internal mode spectrum of the device. (a) VNA transmission measurement S21of multi-mode photon-magnon hybrid system formed\nin the waveguide circulator package with WCP. The blue (red) dashed line plots the frequency of the clockwise(counterclockwise) mode from a\nsimplified two-mode model of photon-magnon avoided crossing with g=2\u0019= 1.3 GHz (2.1 GHz) to compare with an observed spectral line. (b)\nThe right panel shows a finer sweep of S21in the low-field regime. The mode frequencies differ slightly from (a) since the data was acquired\nafter some modifications to the device packaging (a piece of Teflon spacer at the top of the YIG cylinder was removed). The left panel shows\nthe electromagnetic mode structures of the eigenmode solutions from our HFSS simulation for the WCP with good frequency agreement to the\nexperimental data (see Fig. 8 in Appendix). The color scale from red to blue represents electric field strength from high to low in log scale. The pair\nof modes around 11 GHz are connected to the circulating modes of the loaded circulator and (c) shows their linewidths.\npolariton modes remains quite substantial.\nUsing finite-element simulations (Ansys HFSS, Appendix A),\nwe identify that the five polariton modes in the frequency range\nof 8-12 GHz at B= 0include two nearly-degenerate mode pairs\nwith two-fold symmetry and another mode with three-fold sym-\nmetry. Electric field distributions of each of the modes are il-\nlustrated in Fig. 2(b). Each degenerate mode pair can be under-\nstood using a basis of standing-wave modes polarized along the\nxorydirection. The application of a magnetic field lifts this x-y\ndegeneracy, as the mode pair forms clockwise and counterclock-\nwise rotating eigenmodes with a frequency splitting [19–21].\nPrior use of these chiral polariton mode pairs have been in\nthe magnetically saturated regime [19–21]. Here we focus on\nthe low-field regime ( jBj<0:05mT) where the approximately\nlinear increase of frequency splitting between the mode pair re-\nflects increasing magnetization of YIG under increasing applied\nmagnetic field. After implementing demagnetization training\ncycles to suppress a relatively small hysteretic effect through-\nout our experiments, we expect an approximately linear mag-\nnetization curve ( M-H) for YIG until it approaches magnetic\nsaturation. In the limit of high permeability \u0016\u001d\u00160(with\u00160\nbeing the vacuum permeability), we have M=B=\u0016 0Nz(note\nthatBis the applied magnetic field strength) and Nz\u00190:285\nis thez-direction demagnetizing factor when the YIG is signif-\nicantly below magnetic saturation [37]. Saturation magnetiza-\ntionMs= 2440 Oe [41] of YIG is approached on the scale of\nB\u0018\u00160NzMs\u001970mT, which agrees with the changing curva-\nture of the mode-splitting spectra.\nOn the other hand, in the completely demagnetized state\n(M= 0) at zero field, the system is expected to satisfy macro-\nscopic time-reversal symmetry. As supported by numerical sim-\nulations, the x-ymode pairs should be in principle exactly de-\ngenerate since both the Y-junction geometry and the [111] YIG\ncrystal has 3-fold rotational symmetry around the zaxis. How-\never, appreciable zero-field splitting is observed experimentally.\nWe attribute this splitting to some anisotropy in the x-y plane\nbreaking this symmetry and allowing a preferred magnetization\naxis of the YIG at 0 field. Some possible explanations for this\nanisotropy are a small visible damage to our YIG crystal on\none edge or possible imperfections in eccentricity and align-\nment. If the magnetic domains of unsaturated YIG preferen-\ntially align with one in-plane axis compared to its orthogonal\naxis within the x-yplane, this anisotropy would result in a rela-tive frequency shift between the standing-wave modes along the\nin-plane easy and hard axes. This anisotropy-induced frequency\nshift\u0006\ffor thexandymodes can be modeled in numerical\nsimulations employing a permeability tensor of unsaturated fer-\nromagnets [42, 43] with certain anisotropic assumption, which\ncan plausibly explain the data (Appendix A). As Bincreases, we\nexpect\fto decay towards 0 when the magnetic domains are in-\ncreasingly aligned towards the zdirection, thus making any x-y\nplane energetic preference of negligible effect. We model this\ndecay with a thermodynamic toy model (Appendix B) whose\ndetails do not affect the conclusions of this study.\nFor the rest of this article, we will focus on the pair of polari-\nton modes near 11 GHz in Fig. 2(b), and refer to them as “the\ncirculator modes” for reasons that will become apparent. We\ncan model their frequencies in the partially magnetized regime\n(jBj<50mT) using a phenomenological model accounting for\nthe degeneracy-lifting anisotropy and the field-dependent mag-\nnetization of YIG. Let the zero-field frequencies of the xandy\nmodes be!x=!yif the device had perfect 3-fold symmetry,\n\fand\u0012=2be the magnitude of anisotropy caused degeneracy-\nlifting and the direction of the in-plane anisotropy axis (rela-\ntive to thexaxis), and off-diagonal imaginary coupling term\n\u0006ikB be the magnetic field induced degeneracy-lifting, linearly\nincreasing with a real coefficient k. We use the following Hamil-\ntonian to characterize the pair of circulator modes in the basis of\nxandymode amplitudes:\nH=~=\u0012\n!x+\fcos\u0012+mB2\fsin\u0012+ikB\n\fsin\u0012{ikB ! y\u0000\fcos\u0012+mB2\u0013\n(1)\nThis effective model of the polariton modes has absorbed the\nmagnon contributions in the regime where they have been adi-\nabatically eliminated. The formation of clockwise and counter-\nclockwise eigenmodes is due to magnon-mediated interactions\nmodeled by\u0006ikB. The level repulsion from the far-detuned\nmagnon modes is approximated by a small quadratic shift in\nfrequencymB2. The quadratic dependence was empirically\nchosen because the sum of the mode frequencies over field dis-\nplayed a roughly quadratic relationship with Bover the plotted\nfield range. By fitting the mode spectrum in Fig. 2(b), we ob-\ntain!x=2\u0019=!y=2\u0019= 11:054GHz,k=2\u0019= 9:82GHz/T,\nm=2\u0019= 50 GHz/T2,\f=2\u0019= 139 MHz.\nIt is well-known that the FMR modes of partially magnetized\nferrimagnetic insulators, where the magnetic domains are not4\nFrequency (GHz)Frequency (GHz)\ndBdBS12S23\nS21S13(a) (b)(d)\nS12\nS21(e)(f)\n(c)(f)\nπ/6\n−π/6\n−π/6Isolated\nOutput1 -1 0Isolated\nInputOutputωt = 0 ωt = π/4\nInputAmplitude\nPhase\nFrequency Frequencyωdriveωdrive\n(i)(g)\n(h)\nField (T) Field (T)S21\nS12S32\nS31\nFIG. 3. Illustration of the circulator working principle and low-temperature characterization of the non-reciprocity. In the circulator package\nwith IMT, the frequency splitting of clockwise and counterclockwise rotating modes as shown in (a) can be tuned such that the phase of the modes\nare\u0019=6and\u0000\u0019=6as shown in (b). This then produces a node at the upper port, thereby preventing any signal from leaving there at all times where\n!t= 0and!t=\u0019=4are shown pictorially in (c). (d, e) Measured microwave transmission (d) jS12jand (e)jS21jspectra as a function of magnetic\nfield B. (f-i) The isolation performance, (f) I12=jS12=S21j,(h)I21=jS21=S12j, (g)I23=jS23=S32j, (i)I13=jS13=S31j.S21is obtained by\nmeasuring the S12at –B, which provides a self-calibrated way to determine the isolation of the circulator.\naligned in equilibrium, have large damping. Therefore, one may\nexpect broad linewidths for photon-magnon polariton modes be-\nlow magnetic saturation. Indeed, we have observed linewidths\nexceeding 100 MHz for another polariton mode at 5 GHz at B <\n50 mT (not shown). Surprisingly, the polariton modes at higher\nfrequency display narrow linewidths, \u0014i\u00192MHz for the pair of\ncirculator modes [Fig. 2(c)], which corresponds to quality fac-\ntors on par with some circuit QED elements such as the read-\nout resonators. The narrow internal linewidth of the circulator\nmodes is crucial for constructing a low-loss circulator and even-\ntually achieving high quantum efficiency of non-reciprocal inter-\nactions in circuit QED. It is primarily aided by the use of single\ncrystalline YIG and the relatively low magnon participation in\nthe circulator modes compared to commercial circulators. The\nobserved\u0014imay be limited by either the spin relaxation in YIG\nor the Ohmic loss in copper. The former remains to be inves-\ntigated in this partially magnetized regime, and the latter may\nbe further reduced through better surface treatment or the use of\nsuperconducting materials in low-field regions of the waveguide\npackage.\nIV . CIRCULATOR CHARACTERIZATION\nThe device acts as a circulator when the end of each waveg-\nuide section is in IMT rather than WCP with an applied magnetic\nfield in the ^zdirection. In this configuration, the linewidths of\nall internal modes are substantially broadened forming a trans-\nmission continuum in the measurement, as shown in Fig. 3(d,e)\nforjS12jandjS21j. Nevertheless, the operating condition of the\ncirculator can be conceptually understood as having a pair of\ncounter-propagating internal modes with their magnetic-field-\ninduced splitting ( \u000e) satisfying the relationship \u000e= 2\u0014c=p\n3ver-\nsus their half linewidths ( \u0014c) [22]. As illustrated in Fig. 3(a-c),\nwhen driven at a frequency in the middle of the two resonances,\nthe two circulator modes are excited with equal amplitude and\na phase shift of\u0006\u0019\n6relative to the drive. The resultant standing\nwave pattern forms a node at the isolation port of the circulator.\nThis condition can be satisfied by choosing the correct combina-\ntion of frequency and magnetic field.We characterize the non-reciprocity of the circulator by the\nisolation ratioI12=jS12=S21j, which may be computed from\nFig. 3(d,e). However, since S12andS21are measured through\ndifferent cables and amplifier chains [Fig. 1(b)], it is challenging\nto calibrate their absolute values precisely. A much better self-\ncalibrated technique to extract the isolation ratio in our system is\nto use the Onsager-Casimir relation [44], S21(B) =S12(\u0000B),\nresulting from the microscopic time reversal symmetry. There-\nfore, we useI12=jS12(B)=S12(\u0000B)jto determine the isola-\ntion ratio of the circulator as shown in Fig. 3(f), with the (field-\nindependent) contribution from same transmission chain can-\ncelled out. The result indicates the circulator working condition\nis met for the pair of counter-propagating modes at \u001811.2 GHz\nwith external field \u00180.022 T. We see\u001520 dB of isolation over\na bandwidth of about 250 MHz, with maximum isolation of at\nleast 35 dB.\nThe same analysis on S21data yields the same isolation prop-\nerty [Fig. 3(h)] as expected. Similarly, I23andI13are measured\nas in Fig. 3(g) and (i), each showing a slightly different working\nfield and frequency (possibly due to imperfections of the device\ngeometry) but similar isolation magnitude and bandwidth. These\ndata are measured at an estimated circulating photon number on\nthe order of 10’s, but when we lower the power to below sin-\ngle photon level, the isolation property does not show notable\nchanges.\nAn important motivation of our work is to ultimately imple-\nment pristine non-reciprocal interactions between superconduct-\ning qubits or cavities. It is crucial to minimize the ratio between\nthe undesirable internal dissipation ( \u0014i) and the external bath\ncoupling (\u0014c) that enables non-reciprocity. In the case of a cir-\nculator, this ratio sets the limit for the circulator’s microwave\ninsertion lossL21[1, 22]:\nL21= 1\u0000jS21j2\u00151\u0000jS21j2\u0000jS11j2\u0000jS31j2\u0019\u0014i\n\u0014c(2)\nThis lower limit is obtained in principle when the circulator\nhas perfect impedance matching ( S11= 0) and isolation ratio\n(S31= 0). Typical commercial ferrite circulators used in circuit\nQED experiments have shown insertion loss around 10% [4, 5],\nwhich is dominated by internal loss. Experimental Josephson5\nFrequenc y (GHz)dB\ndB(b)\nLoss (%)\nField ( T)Linewidth (MH z)(a)\nlower freq. modehigher freq. mode\nFIG. 4. Characterization of the internal loss of the circulator at\nroom temperature. (a) Linewidths of the pair of circulator modes\nmeasured at room temperature. (b) Transmission S21and reflection\nS11near the maximum isolation regime of the circulator, measured at\nB= 24:8mT (top panel) and the internal loss of the circulator calcu-\nlated from it (bottom panel) at room temperature.\ncirculators so far have also reported insertion loss of -0.5 dB\n(11%) or higher [30, 31]. The lowest quoted insertion loss for\na commercially-listed waveguide circulator is -0.1 dB (or 2.2%)\nbut that is untested in the quantum regime. In order for the quan-\ntum efficiency of a non-reciprocal two-qubit interaction chan-\nnel to match the fidelity of state-of-the-art two-qubit operations,\nthe insertion loss would need to be improved to the sub-percent\nlevel.\nTo the best of our knowledge, it is an open challenge to cal-\nibrate the insertion loss of a microwave component in a dilu-\ntion refrigerator with a precision better than 1%. Even using\nspecialized Thru-Reflect-Line calibration components and well-\ncharacterized cryogenic switches, the resultant precision would\nstill be limited to about 0.1dB (or 2.3%) [45]. In order to infer\nthe loss of our circulator at 20 mK, we measure its Sparam-\neters at room temperature after a careful calibration procedure\nthat uses attenuators in series to suppress standing waves. We\nfind a conservative upper bound for room-temperature internal\nloss of\u00141{jS21j2{jS11j2\u00192%, as shown in Fig. 4(b). As-\nsuming\u0014cdoes not change as a function of temperature, com-\nparing the intrinsic linewidth of the circulator mode pair at room\ntemperature versus 20 mK would inform the internal loss at 20\nmK. The intrinsic linewidths, measured in WCP, are 4.1 and 6.3\nMHz at room temperature [Fig. 4(a)] and 1.8 and 2.2 MHz at\nlow temperature [Fig. 2(c)], indicating that \u0014c&260MHz and\n\u0014i=\u0014c.0:8%. If we instead use the relation of \u0014c=p\n3\u000e=2,\nwhich yields \u0014cin the range of 430 MHz to 550 MHz (and data\nin Section V would further suggest \u0014cat the high end of this\nrange), or\u0014i=\u0014c\u00190:4%. Further improvement of the circulator\nbandwidth and the coupling ratio can be achieved by applying\nimpedance transformation techniques to increase \u0014c[46].\nTranslating this small internal loss ratio to a sub-percent inser-\ntion loss for a circulator as a peripheral transmission-line device\nwould further require excellent impedance matching. However,\nwe emphasize that this requirement is not fundamental if the cir-\nculator is modeled as part of the quantum system itself mediating\ninteractions between other quantum resonance modes. Unlike\nmost ferrite circulators, our device operates in the regime of par-\ntial magnetization for YIG. It only requires a moderate external\nmagnetic field that is significantly below the critical field of a\nvariety of superconducting materials. This allows for 3D inte-gration of superconducting niobium cavities and shielded trans-\nmon qubits for studying circuit QED with non-reciprocal inter-\nactions. In the following section, we demonstrate direct cou-\npling of two external superconducting cavity modes with the\ncirculator modes and analyze the resultant non-reciprocal hybrid\nsystem as a whole.\nV . TUNING NON-RECIPROCITY OF EIGENMODE\nSTRUCTURE\nWe integrate superconducting cavities with the ferrite device\nby replacing the rectangular waveguide extensions with super-\nconducting 3D cavities made of niobium [Fig. 5(a)]. Two cav-\nities, attached at Port 1 and 2, are tuned to have resonance fre-\nquencies close to each other, !1\u0019!2\u001810:8GHz, both of\nwhich are within the bandwidth of the circulator. Each cavity is\ncoupled to the central Y-junction via a coupling aperture. As a\nresult, the circulator modes will mediate an interaction between\nthese two external cavities. Crucially, this circulator-mediated\ninteraction can have both coherent and dissipative aspects, and\ncan be non-reciprocal. The degree and the direction of non-\nreciprocity of the coupling can be tuned via the external mag-\nnetic field. Note that Port 3 remains impedance-matched to a\ntransmission line. This is also essential: it serves as the dominant\ndissipative bath that is necessary for achieving non-reciprocal\ninter-mode interactions [47].\nTo probe the hybridized mode structure of the composite sys-\ntem, we measure S31from a weakly-coupled drive port on Cav-\nity 1 to Port 3. The measured amplitude of S31as a function\nof magnetic field and frequency is shown in Fig. 6(a). There\nare a total of four bare oscillator modes in the vicinity (within\n0.5 GHz) of the frequency range of interest: two supercon-\nducting cavity modes and two internal circulator modes. Since\nthe loaded circulator modes with very broad linewidths ( >100\nMHz) are difficult to observe in the presence of the standing-\nwave background of the coaxial cables, this spectroscopy mea-\nsurement primarily reveals the eigenmodes that are localized in\nthe external superconducting cavities. Indeed, at jBj>0:03T,\nthe spectrum shows two sharp resonances which we identify as\nthe bare cavity modes to a good approximation. At lower fields,\nthe cavities appear to more strongly hybridize with each other\nand with the lossy circulator modes, but the spectrum can non-\ntheless be captured relatively well by the sum of two Lorentzian\nmodesaandb:\nS31=Aaei\u001ea\n\u0000i(!\u0000!a)\u0000\u0014a=2+Abei\u001eb\n\u0000i(!\u0000!b)\u0000\u0014b=2(3)\nBy fitting the spectrum to Eq. (3), we can extract the linewidth\n(\u0014i), frequency ( !i) and amplitude ( Ai) of the two Lorentzians\nat each magnetic field, as plotted in Fig. 6(c-e).\nThe magnetic field dependence of the two prominent\nLorentzians can be connected to the eigenmode solutions of an\neffective Hamiltonian model of system. We describe the sys-\ntem using the following 4 \u00024 non-Hermitian matrix, written in\nthe basis of the amplitudes of the two cavity modes and the two\ncirculator modes:6\nCopper Waveguide\nOutput P ort 3\nNiobium Cavity 1Niobium C\navity 2\nInput por t 1 Input por t 2YIG C ylinderCircula tion\nDirection a t\nPositiv e Field(b)\n+ -+ +(c)\n (a)\nx\nFIG. 5. Waveguide circulator-cavity integration. (a) The photo image, (b) a schematic top-down view, and (c) a diagrammatic illustration of the\neffective Hamiltonian (see Eq. (4), for clarity the \fandmB2terms have been neglected in the illustration) of our integrated non-reciprocal device.\nIt is composed of a Cu waveguide Y-junction loaded with a YIG cylinder, two Nb cavity segments with weakly-coupled drive ports (Port 1 and 2),\nand an output port with IMT (Port 3). For each cavity, the sidewall closest to the copper Y-junction is formed by a standalone niobium plate in the\nassembly [enclosed in blue in (a)], which contains a 5 mm-diameter aperture to create an evanescent coupling between the superconducting cavity\nmode and the circulator modes. One of the cavities is loaded with a transmon qubit [marked as \u0002in (b)] which stays unused in its ground state in\nthis study.\nHe\u000b=~=0\nB@!1\u0000i\u00141\n20 gy gx\n0!2\u0000i\u00142\n2gy \u0000gx\ngygy!y\u0000\fcos\u0012+mB2\u0000i\u00143\n2\fsin\u0012\u0000ikB\ngx\u0000gx \fsin\u0012+ikB ! x+\fcos\u0012+mB21\nCA (4)\nThe two niobium cavities have bare frequencies !1,!2, and\ninput coupling rates of \u00141and\u00142. The bottom right block of\nEq. (4) describes the two circulator modes, with their anisotropy\ndependence and imaginary coupling due to magnon hybridiza-\ntion following the same description as in Eq. (1). The zero-field\nfrequencies of the two circulator modes !x,!yare no longer\nequal since the device is no longer 3-fold symmetric. The y-\nmode with frequency !yis symmetric with respect to the yaxis,\nand therefore has an equal and in-phase coupling rate gywith\nthe two cavities. It rapidly leaks to the waveguide output Port 3,\nwith a decay rate \u00143\u001d\u00141;\u00142;gx;gy.\u00143is related to \u0014cof the\nloaded circulator as in Section IV by \u00143= 4\u0014c=3. Thex-mode\nis anti-symmetric with respect to the yaxis, preventing it from\ncoupling to the output port. This also leads to a 180\u000ephase dif-\nference in cavity coupling as accounted for by the negative sign\non two of the gxparameters\nThis effective non-Hermitian Hamiltonian can be diagonal-\nized as:\nHe\u000b=X\nn~!njnRihnLj (5)\nwheren=a;b;c;d are the eigenmode indices of the system,\n!nthe complex eigen-frequencies, and jnRiandjnLithe right\nand left eigenvectors of the non-Hermitian Hamiltonian, de-\nfined as:He\u000bjnRi=~!njnRiandHy\ne\u000bjnLi=~!\u0003\nnjnLi.\nThe scattering matrix element Sijfrom Portjto Portican be\ngenerally drived from the input-output theory relation: Sij=\n\u000eij\u0000ip\u0014i\u0014jGR\nij(!), where the 4\u00024retarded matrix Green’s\nfunction is defined as: GR(!) = (!\u0000He\u000b)\u00001, and\u0014iand\u0014j\nare the output and input coupling rates, respectively. Applying\nthis formalism to the S31measurement of our device, we arrive\nat the following Lorentzian spectral decomposition to describe\nthe spectrum:\nS31(!) =X\nn\u0000ip\u00141\u00143hyjnRihnLj1i\n!\u0000!n(6)where the real and imaginary parts of the eigen-frequency !n\ncorrespond to the observed Lorentzian frequencies and half\nlinewidths, respectively. The amplitudes of the Lorentzians are\nproportional to the product of the left eigenvector overlap with\nthe bare cavity mode j1iand the right eigenvector overlap with\nthe output circulator mode jyi.\nBy fitting the extracted Lorentzian parameters of the two\nprominent eigenmodes in Fig. 6(c-e) to the predictions of the\n4\u00024Hamiltonian model across all fields (Eq. 4), we can deter-\nmine all the free Hamiltonian parameters in this model. This\nincludes\u00143= 730 MHz, implying \u0014c= 550 MHz for the\nloaded circulator, consistent with (and at the high end of) the es-\ntimates in Section IV . Somewhat surprisingly, the experimental\ndata strongly suggests that the cavity-circulator coupling rates gx\nandgymust be magnetic field dependent. (For example, it heav-\nily constrains that gx=2\u0019>16MHz nearB= 0 andgx=2\u0019<12\nMHz atjBj>30mT.) We attribute this varying coupling to\nthe change in electromagnetic field distribution of the x- andy-\nmodes around the coupling aperture due to the x-yanisotropy of\nYIG. Assuming gxandgycontains a contribution proportional to\n\f(B)with the same decay shape over applied field, the effective\nHamiltonian model fits the Lorentzian parameters quite well and\nalso reproduces the overall transmission spectrum [Fig. 6(b)].\nThe eigenmode features of the system can be understood in-\ntuitively by considering first the inter-mixing of the x,ycircula-\ntor modes (i.e. diagonalization of the lower right block of He\u000b)\nand then their mixing with the two cavity modes. At B= 0,\nthe circulator modes are relatively close in frequency to the bare\ncavities, resulting in strong four-mode hybridization and sub-\nstantial linewidth-broadening and frequency shift to Mode a. As\nBincreases, the block-diagonalized circulator modes split fur-\nther in frequency in response to increasing magnetization of YIG\n(analogous to the unloaded internal mode spectrum in Fig. 2b),\nand become more detuned from the bare cavities, so the cavity-\ncirculator hybridization is continuously reduced. This is re-\nflected in the eventual flattening of the frequency and linewidth7\nExper imen t |S   |  31\nTheor y |S   |  31\nField ( T)Field ( T)\nLinewidth (MH z)Frequenc y (GHz)(c)\n(d)(e)\n(a)\n(b)Frequenc y (GHz) Frequenc y (GHz)\nAmplitude (MH z)1.0\n0.6\n0.4\n0.2\n0.0\n-0.05 -0.025 0.00 0.025 0.050102030\n10.80510.81010.81510.82010.8250.8\nField ( T)-0.05 -0.025 0.00 0.025 0.05A\nField ( T)0.00 0.02 0.040.000.501.001.50 Amplitude r atioωκ\n(f)A(B)\nA(-B)-0.05 -0.025 0.00 0.025 0.05\nmode bmode a\nmode bmode a\nmode bmode a\n3 x mode bmode a\n-30\n-40\n-50\n-60\n-70\n-80\n-30\n-40\n-50\n-60\n-70\n-80\nFIG. 6. Spectroscopy of the hybridized non-reciprocal modes of a circulator-cavity system. (a) VNA transmission measurement and (b)\nmodel prediction of jS31jfrequency spectrum over external magnetic field B. Remaining panels show magnetic field dependence of the system’s\neigenmodes and wavefunctions: (c) eigenmode linewidths \u0014n=2\u0019, (d) eigenmode frequencies !n=2\u0019, (e) amplitude parameter An(c.f. Eq.(3)),\nand (f) amplitude ratio (c.f. Eq.(8)) of experimental data (dots) from two-mode Lorentzian fit (Eq.(3)) and theory predictions (dashed lines). The\nsymmetry of \u0014nand!n(i:e:complex eigen-energy of the hybrid system) with respect to Bexemplifies the microscopic time-reversal symmetry of\nthe non-Hermitian system. The non-reciprocity is reflected in the difference in Anat\u0006B, which reveals the asymmetry in the left/right eigen-vector\nstructure (c.f. Eq.(8))). The effective Hamiltonian parameters from the fit are: !1=2\u0019= 10:8104 GHz,!2=2\u0019= 10:8040 GHz,!x=2\u0019= 10:707\nGHz,!y=2\u0019= 10:813GHz,\u0012= 37:7\u000e,\u00143=2\u0019= 730 MHz,gx=2\u0019= (9:0 + 0:011\f)MHz,gy=2\u0019= (5:0 + 0:006\f)MHz with\f=2\u0019= 139\nMHz atB= 0and decays withjBj.\nof the observed Lorentzians at high fields.\nIn our device, opposite magnetic fields produce opposite di-\nrections of non-reciprocity, hence the transmission spectra ob-\nserved at\u0006Bin Fig. 6(a) are markedly different. Interest-\ningly, the extracted data in Fig. 6(c,d) shows that the underly-\ning eigenmode frequency and linewidths at \u0006Bare equal, un-\nchanged under the mapping PofB:7!\u0000B. This is no co-\nincidence, but is rather the direct consequence of microscopic\nsymmetry requirements. Recall again that the Onsager-Casimir\nrelation [44] requires that the full scattering matrix Ssatisfy\nS(\u0000B) =ST(B). AsSis however directly determined by\nour non-Hermitian Hamiltonian, this necessarily implies that\nHe\u000b(\u0000B) =HT\ne\u000b(B). This in turn implies that the complex\neigenvalues of He\u000bare unchanged under P. Note that the op-\nerationPis not just a simple time-reversal operation, as it does\nnot involve transforming loss to gain (and vice versa). This prop-\nerty ofHe\u000bcan be easily seen to hold for our specific model in\nEq. (4). Nonetheless, we emphasize that our experimental ob-\nservation here of eigenvalue invariance under the mapping Pis\na demonstration of a general physical property; it is by no means\ncontingent on the specifics of our model.\nWhile the eigenvalues of He\u000bdo not directly reflect the non-\nreciprocal physics of our system, the same is not true of its\neigenvectors. As it involves matrix transposition, the opera-\ntionPexchanges the left and right eigenvectors of the effective\nHamiltonian:jnR(B)i=jnL(\u0000B)i\u0003. A defining feature of a\nnon-reciprocal Hamiltonian is that the left and right eigenvec-\ntors generally differ in their spatial structures (i.e. they look very\ndifferent when expressed in a basis of bare modes):\nRi;n=jhnLjiij\njhijnRij6= 1 (7)\nAs has been discussed elsewhere [48, 49], the Ri;ncharacter-izes a fundamental asymmetry in the response of our system.\nThe numerator characterizes the susceptibility of the eigenmode\nnto a perturbation or excitation entering from bare mode i. In\ncontrast, the denominator tells us the amplitude on bare mode i\nthat would result given that the system eigenmode nis excited.\nIn a Hermitian system these quantities are necessarily identical,\nexpressing a fundamental kind of reciprocity between suscepti-\nbility and response. In our non-Hermitian system, the non-unity\nratio here reflects the effective non-reciprocity of the inter-mode\ninteractions.\nThis non-reciprocal eigenvector structure is experimentally\nverified by the asymmetry of the Lorentzian amplitudes with re-\nspect toBin Fig. 6(e),\nAn(B)\nAn(\u0000B)=\f\f\f\fhnLj1i\nh1jnRihyjnRi\nhnLjyi\f\f\f\f=R1;n\nRy;n(8)\nWe plot this ratio in Fig. 6(f). For our device, a calculation based\non Eq. (4) shows that Ry;n\u00191for most of the field range (near\nzero field andjBj&15mT), allowing Fig. 6(f) to be under-\nstood as a measurement of the non-reciprocity ratio R1;n, in this\nfield range, showing the role of Cavity 1 in the two prominent\neigenmodes of the system. In particular, the most pronounced\nasymmetry is observed near the optimal working point of the\ncirculator (B=\u000628mT) for Mode b, which can leak through\ncavity 1 but cannot be excited from Cavity 1 or vice versa, as\nexpected for a mode dominated by photons in Cavity 2.\nIt is also interesting to discuss our system in the context of\ngeneral systems exhibiting non-reciprocal interactions between\nconstituent parts. Such systems are commonly described by phe-\nnomenological non-Hermitian Hamiltonian matrices H, whose\nmatrix elements in a local basis encode interactions with direc-\ntionality:jHijj6=jHjij. A prominent example is the Hatano-\nNelson model [25] of asymmetric tunneling on a lattice. In our8\n0246810101\n100\n10-1\n-0.04 -0.02 0.00 0.02 0.04MHz\nField (T)\nFIG. 7. Mediated non-reciprocal coupling rates between the ex-\nternal superconducting cavities. Red curves show the off-diagonal\ncoupling terms in the effective two-mode Hamiltonian (c.f. Eq.(9)),\njH12jandjH21j, as a function of magnetic field, and blue shows\nr=p\njH21j=jH12j.\ncase, we have a microscopically-motivated model that is fully\nconsistent with the requirements of microscopic reversability,\nbut which encodes non-reciprocity. As shown in appendix C,\none can adiabatically eliminate the internal circulator modes\nfrom our system to obtain an effective two-mode non-Hermitian\nHamiltonian that describes the external cavity modes and their\ncirculator-mediated interaction:\nH0\ne\u000b=~=\u0012\n!1;e\u000b\u0000i\u00141;eff\n2H12\nH21!2;e\u000b\u0000i\u00142;eff\n2\u0013\n(9)\nThe field-tunable non-reciprocity can be seen in the asymmetry\nof the off-diagonal coupling values H12andH21based on the\nmodel, as plotted in Fig. 7. We note that the scale of H12and\nH21of a few MHz (which can be increased by using a larger\ncoupling hole) is much larger than the achievable internal loss\nof the superconducting cavities , making the non-reciprocal cou-\npling the dominant interaction.\nFurthermore, this reduced Hamiltonian and its eigenvec-\ntors can be mapped to a reciprocal Hamiltonian and associ-\nated eigenvectors using a similarity transformation S(r), wherep\njH21j=jH12j\u0011r, as outlined in Appendix C. The similarity\ntransform effectively localizes the mode participation on the lat-\ntice site in the direction of stronger coupling more than would\nbe expected in the reciprocal case, causing the amplitude ratios\n(Ri;n) to deviate from 1, which can be viewed as a consequence\nof the non-Hermitian skin effect on a two site Hatano-Nelson lat-\ntice [26]. Furthermore,the similarity transformation can be used\nto explain the qualitative behavior of the disparate amplitude ra-\ntios seen in Fig. 6(f). In particular, for B&20mT, we have\nR1;a\u00191andR1;b\u0019r2(see Appendix C for details).\nVI. OUTLOOK\nIn this work, we have revisited the working principles of a\nY-junction ferrite circulator [22], a microwave engineering clas-\nsic from the 1960’s, in an entirely new context of hybrid quan-\ntum systems and non-Hermitian Hamiltonian. The use of re-\nconfigurable probes and single-crystalline YIG in a low-loss\nwaveguide package allows us to connect the properties of the\nphoton-magnon polaritons to the circulator performance. We\nhave further leveraged our direct access to the internal modesand our ability to tune their coupling in situ to construct a multi-\nmode chiral system and unambiguously reveal its non-reciprocal\neigenvector structure. An understanding of the circulator modes\nand the non-reciprocal eigen-vector structure of the multi-mode\nchiral system provides a foundation for future engineering of any\ntarget non-Hermitian Hamiltonian. This is achieved by our cre-\nation of a template model that one can use to couple any circuit\nQED element to in order to understand how it would integrate\ninto the non-reciprocal dynamics, as was done here with the su-\nperconducting cavities.\nLooking forward, our device architecture provides a versatile\ntestbed for studying non-reciprocal interactions in circuit QED\nby integration of superconducting qubits. This is enabled by two\nof its highlighted properties: the low internal loss of the circula-\ntor modes (<1% of the demonstrated coupling rates, compatible\nwith potential high-fidelity operations), and the relatively low-\nfield operation of the circulator ( \u001825 mT, below ferrimagnetic\nsaturation). The latter allows niobium waveguides or cavities to\nconveniently act as magnetic shields for superconducting qubits.\nWe have preliminarily tested that the coherence times of a trans-\nmon qubit housed in one of the niobium cavities are unaffected\nby in-situ application of a global magnetic field up to at least 0.1\nT. We expect a transmon housed in a niobium waveguide should\nreceive a similar level of protection from magnetic field.\nDirect non-reciprocal coupling of superconducting qubits\nwould open a new frontier in the study of non-reciprocal dynam-\nics currently dominated by linear systems [33, 34, 36, 50]. The\nphysics of a N-mode linear non-reciprocal system can always\nbe described efficiently by a N\u0002Nnon-Hermitian Hamiltonian\nmatrix (exemplified by our application of such a model) and its\ndynamics are always in the classical correspondence limit. Di-\nrect participation of multiple nonlinear modes (such as super-\nconducting qubits) in non-reciprocal coupling, as envisioned in\nchiral quantum optics [29], would lead to novel forms of entan-\nglement stabilization and many-body phases [28, 51]. Our sys-\ntem presents another potential platform to implement this regime\nin circuit QED in additional to those proposed using dynamic\ncontrol [52, 53]. Strong coupling of Josephson circuits with\nlow-loss non-reciprocal elements can even produce degenerate\nand protected ground states for robust encoding of qubits [54].\nACKNOWLEDGMENTS\nWe thank Juliang Li and Dario Rosenstock for experimental\nassistance. This research was supported by U.S. Army Research\nOffice under grants W911-NF-17-1-0469 and W911-NF-19-1-\n0380.\nAppendix A: Numerical simulation of the ferrite device\nFinite element analysis software that supports magnetody-\nnamic simulations, such as Ansys HFSS, can be used to sim-\nulate our ciculator system with both driven mode and eigen-\nmode solutions. Eigenmode analysis can solve for the frequency\nand field distributions of our device’s eigenmodes, while driven\nmode analysis reports the S-parameters over frequency. Here we\ndiscuss eigenmode simulations, but driven mode analysis can be\ncarried out similarly.9\nIt is well known that when the applied field is large and mag-\nnetization is saturated along zaxis, one can generalize to the\nwhole ferrite the equations of motion derived from the torque\nexperienced by an electron dipole moment under the presence\nof an applied field. This approach, augmented by the small sig-\nnal approximation of the Landau-Lifshitz equation of motion,\nyields the textbook Polder (relative) permeability tensor:\n[\u0016]z=0\n@\u0016ri\u00140\n\u0000i\u0014 \u0016r0\n0 0 11\nA (A1)\nwhere\u0016r= 1 +!0!m\n!2\n0\u0000!2,\u0014=!!m\n!2\n0\u0000!2, with!0=\r\u00160H0and\n!m=\r\u00160Msbeing the internal field strength and saturation\nmagnetization converted to frequencies, respectively.\nThe Polder permeability tensor is implimented in HFSS by\ndefault to solve for the interaction of a saturated ferrite with an\nAC microwave field. However, in our experiment we operate\nthe circulator at a low bias field, where the ferrite is not fully\nsaturated. We adopt a permeability tensor model proposed by\nSandy and Green [43] for a partially-magnetized ferrite:\n[\u0016]z=0\n@\u0016pi\u0014p0\n\u0000i\u0014p\u0016p0\n0 0\u0016z1\nA (A2)\nwhere\n\u0016p=\u0016d+ (1\u0000\u0016d)\u0012jMpj\nMs\u00133=2\n(A3)\n\u0014p=\u0014\u0012Mp\nMs\u0013\n(A4)\n\u0016z=\u0016\u0000\n1\u0000jMpj\nMs\u00015=2\nd(A5)\n\u0016d=1\n3+2\n3s\n1\u0000\u0012!m\n!\u00132\n(A6)\nwithMpbeing the net magnetization of the partially magnetized\nferrite. This model contains functional forms for \u0016pand\u0016zthat\nare purely empirical. However, the expressions for \u0014p, which\ndictates the chiral splitting of the circulator modes, and \u0016d,\nwhich represents the permeability in fully demagnetized state,\nare well motivated [42].\nThis model is implemented in simulations by defining mate-\nrials with the customized permeability tensor as given above.\nWhereas HFSS does not by default support eigenmode simula-\ntions for a ferrite under a DC bias field, manually defining the\npermeability tensor components allows us to simulate the circu-\nlator’s eigenmode structure at any magnetization (bias field) as\nshown in Fig. 8.\nThe simulation results agree semi-quantitatively with the ex-\nperimental data in Fig. 2(a) with a linear relationship between\napplied magnetic field and magnetization M=\u00160B=Nzmen-\ntioned earlier, including a dielectric resonance mode with steep\nmagnetic field dependence that is visible in the experimental\ndata.\nRelating to the anisotropy mentioned earlier in section III, the\nsimulation is treating the YIG cylinder as completely isotropic,\nleading to degenerate Mode x and y at 0 field. To account for the\nanisotropy, we introduce a general energetic preference along\nthexaxis, thus making the domains of the unsaturated YIG\nFIG. 8. Eigenfrequencies of the device from finite-element simula-\ntions. For the circulator device with WCP, HFSS eigenmode simulation\ngives the mode frequency over different magnetic fields(dots) and agree\nsemi-quantitatively with experimental data.\npreferentially align along the xaxis and breaking the rotational\nsymmetry.\nWhen all domains are oriented along the zaxis with net mag-\nnetization of zero, the permeability is calculated to be\n[\u0016]z=0\n@\u0016e\u000b0 0\n0\u0016e\u000b0\n0 0 11\nA (A7)\nwhere\u0016e\u000b=q\n!2\u0000!2m\n!2. To get the permeability matrix for do-\nmains align along the xandyaxes ( [\u0016]x,[\u0016]y), one can apply\na change of coordinates to Eq. (A7). The matrix for completely\nrandom domain orientations would be an equal average of the\nthree permeability matrices, [\u0016]x;[\u0016]y;[\u0016]z[42]. Applying a\nweighted average to the matrices will then allow for representa-\ntion of an energetic preference, as shown for a preference along\nthexaxis:\n[\u0016] = (1\n3+\u000e)[\u0016]x+ (1\n3\u0000\u000e)[\u0016]y+1\n3[\u0016]z (A8)\nUsing\u000e= 0:1in Eq. (A8) gives 260 MHz of splitting be-\ntween Mode x and Mode y, which is in good agreement with\nexperimental results from Fig. 2(b).\nAppendix B: Modeling YIG anisotropy in system Hamiltonian\nAs mentioned earlier, there is a clear broken rotational sym-\nmetry in the x-yplane apparent from the splitting in Fig. 2(b).\nSince the exact origin of the anisotropy is unknown, we will\ntreat it as a general energetic favoring in the x-y plane. As the ^z\nbias field is increased, the magnetization will align more along\n^z, makingx-yplane preferences less impactful. Based off this\nunderstanding, we wanted a simple functional form to describe\nhow the effect of this anisotropy decays with an increase in bias\nfield strength that we could use to describe the decay of \f. Since\nwe just want the general form of how the effect of an energetic\npreference decays over field, the actual form of the energetic10\npreference in the x-yplane is not important. We chose to use\na toy model of a magnetic domain with a simple Hamiltonian\n(Han) with a simple energetic preference given by Kalong the\nxaxis and a total net magnetic moment M:\nHan=\u0000BMcos(\u0012)\u0000Ksin2(\u0012) cos2(\u001e) (B1)\nTo see how the effect of this anisotropy changes as we vary the\nmagnetic field B, we utilized classical Boltzmann statistics. We\ndefine a partition function:\nZ=Z\u0019\n0Z2\u0019\n0e\u0000Han(\u0012;\u001e)=(kbT)sin(\u0012)d\u0012d\u001e (B2)\nso we can calculate the expectation of the magnetic moment\ndirection using Eq. (B3) for A=Mx;My;Mz;Mx=\nMsin(\u0012) cos(\u001e);My=Msin(\u0012) sin(\u001e);Mz=Mcos(\u0012).\nhA2i=Z\u0019\n0Z2\u0019\n0A2e\u0000Han(\u0012;\u001e)=(kbT)sin(\u0012)d\u0012d\u001e=Z (B3)\nWe calculate these expectation values numerically, and find\nthat the difference of hM2\nxi\u0000hM2\nyifollows approximately a\nsech(B)function. This motivates us to use this simple func-\ntional form to model the anisotropy-induced term \f:\n\f(B) =\f0sech(B=B 0) (B4)\nThe scaling factor B0was fit to the S31spectrum giving a\nvalue of 18.5 mT. While this is a rather crude phenomenological\ntreatment of the anisotropy, since the detuning of the circula-\ntor modes becomes large enough that there is little hybridization\nwith the cavities at relatively small magnetic fields ( \u001820 mT),\nthe exact dependence on magnetic field becomes less important\nto understand the non reciprocal dynamics of the cavities.\nAppendix C: Two-mode Hamiltonian and gauge symmetry\nWe aim to elucidate the non-reciprocity from the Hamiltonian\ngiven in Eq. (4) by reducing it to the form written in Eq. (9). In\norder to do this, we adiabatically integrate out the two circula-\ntor modes to reduce the Hamiltonian to a simple 2\u00022matrix\n(H0\ne\u000b)involving only the two cavity modes. The adiabatic elim-\nination is justified due to the large loss rate on the hybridized\ncirculator modes, making their relevant time scales much faster\nthan the time scale set by the coupling parameters to the cav-\nities. The form of the of the effective Hamiltonian is written\nout in Eq. (9). Due to the complicated dependence on the four\nmode model parameters, we have written simple frequency and\nloss terms on the diagonal entries and simple non-reciprocal\ncouplings on the off diagonal entries where their explicit val-\nues change as a function of magnetic field. The coupling terms\n(H12;H21) along with r=p\njH21j=jH12jare plotted in Fig. 7.\nThe non-reciprocal nature of the system then becomes immedi-\nately apparent as the H21andH12Hamiltonian terms are dif-\nferent outside of 0 field, showing a clear directionality in the\ninteraction. We can map this Hamiltonian to a reciprocal one\nusing the similarity transformation outlined in Eq. (C1) with thetransformation matrix written in Eq. (C2). This new recipro-\ncal Hamiltonain is now symmetric under flipping the sign of the\nmagnetic field Hrec(B) =Hrec(\u0000B).\nH0\ne\u000b!SH0\ne\u000bS\u00001\u0011Hrec (C1)\nS=\u0012\nr1=20\n0r\u00001=2\u0013\n(C2)\nThis means that plotting the ratio of Ri;n;recfromHrecwill al-\nways yield 1 for all B values. One can also map the eigenvectors\nof the original system ( j ii) to the reciprocal system ( j i;reci)\nbyj Ri;reci=Sj Rii;j Li;reci=S\u00001j Lii. Starting from the\nratioRi;n;rec= 1 using the eigenvectors of Hrec, it is then ap-\nparent that transforming the eigenvectors back to those of H0\ne\u000b\nwill allow one to simply caluclate Ri;n. To illustrate this, we\nstart with the explicit change in components from the transfor-\nmation of the right and left eigenvectors as written in Eqns. (C3,\nC4), we can then substitute these in to the earlier expression for\nthe ratioRi;nand see how the ratio deviates from the reciprocal\ncase of 1, as done in Eq. (C5) with i= 1as an example.\nj R;reci=\u0012\nx\ny\u0013\nsimilarity\u0000\u0000\u0000\u0000\u0000!\ntransformj Ri=1q\njxj2\nr+jyj2r\u00121prxpry\u0013\n(C3)\nj L;reci=\u0012\nx\u0003\ny\u0003\u0013\nsimilarity\u0000\u0000\u0000\u0000\u0000!\ntransformj Li=1q\njxj2r+jyj2\nr\u0012prx\u0003\n1pry\u0003\u0013\n(C4)\nR1;n=jhnLj1ij\njh1jnRij=jr1=2xp\njxj2r\u00001+jyj2rj\njr\u00001=2xp\njxj2r+jyj2r\u00001j(C5)\nIt is important to note two simplifying limits for Eq. (C5) that\nthe reader may verify themselves, for x=y\u001dr,R1;n\u00191and\nfory=x\u001dr,R1;n\u0019r2.\nOne can use this similarity transformation to understand the\nqualitative behavior of the disparate amplitude ratios seen in\nFig. 6(f). As mentioned earlier, the amplitude ratio in this case\ncan be roughly approximated as R1;natjBj&15 mT so we\ncan focus primarily on this ratio to understand the behavior in\nthis ���eld range. At larger fields ( B&20mT) Modes a and\nb are dominated by participation in the bare cavity modes so\nwe can approximate these modes by using the eigenmode val-\nues fromH0\ne\u000bfor the cavity mode components and zeros for\nthe circulator mode components. Under this approximation we\ncan look at the inner products in R1;njust from the components\nin the eigenmodes of H0\ne\u000b. Mode a is largely dominated by\nthe cavity 1 component with little circulator participation with\njharecj1ij=jharecj2ij\u001drfor all values of r, thus we can in-\nvoke the limit of Eq.(C5) previously mentioned to find the ra-\ntioR1;b\u00191which is what is seen in Fig. 6(f). The same ar-\ngument can be made for mode b, but in the opposite limit of\njhbrecj2ij=jhbrecj1ij\u001dr, leading to the other limit of Eq.(C5),\nmaking the ratio R1;b\u0019r2which can be seen by comparing\nFig. 6(f) with Fig. 7.11\n[1] A. Kord, D. L. Sounas, and A. Al `u, Proceedings of the IEEE 108,\n1728 (2020).\n[2] M. H. 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Shavit   \n \nMicrowave Magnetic Laboratory,  \nDepartment of Electrical and Computer Engineering, \nBen Gurion University of the Negev, Beer Sheva, Israel \n \nJuly 17, 2014 \nAbstract Unidirectional multiresonant tunneling of the ma gnetoelectric (ME)-field excitations through a \nsubwavelength (regarding the scales of regular el ectromagnetic radiation) vacuum or isotropic-\ndielectric regions has been observed in two-port microwave structures having a quasi-2D \nferrite disk with magnetic-d ipolar-mode (MDM) oscillations . The excitations manifest \nthemselves as the Fano-resonance peaks in the scattering-matrix parameters at the stationary states of the MDM spectrum. The ME near-field ex citations are quasimagne tostatic fields with \nnon-zero helicity parameter. Topological-phase prope rties of ME fields are determined by edge \nchiral currents of MDM oscillations. We show that while for a given direction of a bias \nmagnetic field (in other words, for a given direction of time), the ME-field excitations are \nconsidered as “forward” tunneling processes, in the opposite direction of a bias magnetic field \n(the opposite direction of time), there are “backward” tunneling processes. Unidirectional ME-\nfield resonant tunneling is observed due to distinguishable topology of the “forward” and \n“backward” ME-field excitations. We establish a close connection between the Fano-resonance \nunidirectional tunneling and topology of ME fields in different microwave structures.   \n PACS number(s) 41.20.Jb, 42.25.Fx, 76.50.+g \n I. INTRODUCTION  \n Topological phases have been at tracting much attention in various fields in physics. In \ncondensed matters, this leads to the foundati on of topological isolators. As one of the \ninteresting examples of such structures there ar e photonic crystals with chiral edge states. It \nwas proposed that due to these edge states there should be unideractional propagation of \nelectromagnetic energy [1 – 3]. Recently, unidirectional (chiral) edge modes of magnetostatic \n(MS) [or magnetic-dipolar-mode (MDM)] waves were found in a magnonic crystal [4, 5]. It was discussed that in such structures, the magnetic dipolar interaction joins the relative \nrotational angle between the spin space and orbital space. It was shown that the chiral edge MS \nmodes can break both the time-reversal symmet ry and reflection symmetry and can propagate \nin a direction opposite to the Damon-Eschbach surface MS modes [6].  \n    Recent studies reveal that unidirectional (chiral) edge states of magnetization can also be \nexhibited in another type of a ferrite structur e – a single ferrite-disk particle with a MDM \nspectrum [7 – 13]. At the MDM resonances in a quasi-2D ferrite disk, together with spinning \nrotation of elementary magnetic dipoles also orb ital rotation of the entire-structure topological \nmagnetic dipoles (multipoles) occurs. Such an orbital rotation of the entire-structure \ntopological magnetic dipoles (multipoles) app ears due to geometrical phases on a lateral \nsurface of a ferrite disk at the MDM resonances. It was shown [9, 10] that such topological \n(geometrical) phases arise from the chiral edge states (chiral Majorana edge states) on a lateral \nsurface of a ferrite disk. The persistent edge currents for magnetization arise because of winding properties essential for the motion of the magnetization in a confined cylindrical \ngeometry. The chiral transport means one-direc tion propagation of excitations at a given  2direction of time. In such a structure, a spatial version of the causality principle emerges. This \ncan lead to a situation in which earlier events affect only those future events that occur \n”downstream”.  So, in addition to the requirement that future events do not affect the past, one \nalso expects that the downstream events do not affect upstream events even in the future. [14].  \n    MDM oscillations in a quasi-2D ferrite di sk can conserve energy and angular momentum [7 \n– 10]. Because of these properties, MDMs are st rongly coupled to microwave fields and enable \nto confine microwave radiation energy in subw avelength scales. MDM chiral currents strongly \nmodify microwave radiation acting on a ferrite disk. In a vacuum subwavelength region \nabutting to a MDM ferrite disk, one can observe the qu antized-state power-fl ow vortices [11, \n12]. In such a region, a coupling between the ti me-varying electric and magnetic fields is \ndifferent from such a coupling in regular electromagnetic (EM) fields. These specific near fields, originated from MDM oscillations, we te rm magnetoelectric (ME) fields [13]. The ME-\nfield solutions give evidence for spontaneous symmetry breakings at the resonant states of \nMDM oscillations. In Ref. [15] it was shown that the ME field properties are related to the \nspace-time curvature. Because of rotations of lo calized field configurations in a fixed observer \ninertial frame, the linking between the EM and ME fields cause violation of the Lorentz \nsymmetry of spacetime. In such a sense, ME fi elds can be considered as Lorentz-violating \nextension of the Maxwell equations [16, 17]. The ME fields have helical (chiral) structure. \nThere are the right-hand (RH) and left-hand (L H) helices [15]. Topological pictures of \ninteraction of ME fields with EM fields well illu strate these helices [15]. If we go through such \na chiral structure, we can “worked through” back geometrical phases when changing a direction of a bias magnetic field. \n    In this paper, we show that due to a topological structure of ME fields one obtains \nunidirectional multiresonant tunneling for elec tromagnetic waves propagating in microwave \nstructures with embedded quasi-2D ferrite disks. The resonances manifest themselves as peaks \nin the scattering-matrix parameters at the stationary states of MDM oscillations in a ferrite \ndisk. The effect of unidirectional tunneling is exhibited as the Fano-resonance interferences, \nwhich are different for oppositely directed bias magnetic fields. In 1961, Fano proposed [18] \nthat in a system where a discrete energy level is embedded in a continuum energy state and \nthere is coupling between these two states, a spec ific resonant state arises around the discrete \nlevel. This quantum mechanical interference yields a characteristic asymmetric line shape in the transition probability. Fano-resonance tunneling is a well known effect in semiconductor \nquantum-well and quantum-dot structures [19, 20]. In Ref. [21] it was shown that interaction of \na MDM ferrite particle with its microwave-structure environment has a deep analogy with the Fano-resonance interference in natural and artificial atomic structures. It was stated that MDM \nparticles have the energy-eigenstate spectra an d tunneling of microwave radiation into a MDM \nferrite disk is due to twisting excitation. All these effects allow us to establish, in the present \nstudies, a close connection between the the obser ved Fano-resonance tunneling, geometry of a \nmicrowave structure, and topology of ME fields. Our observations support the model of a special role of the MDM chiral edge states in the unidirectional ME-field multiresonant \ntunneling. \n    The paper is organized as follows. In S ection II, we show how such general notions as \nnonreciprocity and unidectionality are related to gyromagnetic properties of media. We analyze \nthe effects of nonreciprocity and unidirectionality  for oscillating modes in resonant structures. \nIn Section III, we analyze chiral edge states of  MDMs in a quasi-2D ferri te disk. In Section IV, \nwe represent our numerical and experimental results on the unidirectional ME-field \nmultiresonant tunneling. In Section V we provi de some discussions and summarize our studies. \n II. NONRECIPROCITY AND UNIDIRECTIONALITY  \n  3The Fano-resonance tunnelling is observed as unidirectional phenomena in microwave \nstructures with MDM ferrite resonators. For deeper understanding these effects, we should \ndwell initially on such fundamental notions as nonreciprocity and  unidectionality in gyrotropic \nstructures. As a general consideration, we analyze the scattering-matrix ( S-matrix) properties of \ntwo-port (input-output) structures with such field behaviors. We also analyze the \nnonreciprocity and unidirectional ity properties for oscillating modes in resonant structures. \n A. Field nonreciprocity and unidirectio nality in gyrotropic structures  \n \nThe effect of nonreciprocity in the electromagnetic wave propagation is observed in gyrotropic \nmedia. In classical electromagnetism, Lorentz reciprocity is considered as the most common and general theorem for time-invariant linear media. It involves the interchange of time-\nharmonic sources and the resulting electromagnetic fields. In isotropic media, reciprocity can \nbe recognized and exploited even in the presence  of absorption, whilst time-reversal symmetry \nprecludes absorption. Gyrotropic (gyromagnetic or gyroelectric) media with nonsymmetrical \nconstitutive tensors caused by an applied dc magnetic field have been called nonreciprocal  \nmedia because the usual reciprocity theorem [22] does not apply to them. Rumsey has \nintroduced a quantity called the “reaction” and in terpreted it as a “phy sical observable” [23]. \nThis made it possible to obtain a modified reciprocity theorem based on the property of \ngyrotropic media that nonsymmetrical constitutive tensors of permittivity or permeability are \ntransposed by reversing the dc magnetic field \n0H\n[24]. Based on this aspect, a widely used \nformulation is the following: devices that are no nreciprocal in their electromagnetic properties \nare so because of asymmetry of the magnetic or dielectric tensors of the linear media they \ncontain. This concerns both microwave ferrite- based devices and magneto-optic-based devices. \nFor the scattering-matrix parameters in nonreciprocal devices, one has \n00() ( ) ,  ij jiSH S H i j \n. For a lossless structure with a gyrotropic medium, the scattering \nmatrix is unitary. On the microscopic level, the applicability of the reciprocity theorem for \ngyrotropic media is based on the time-reversal invariance, which is described by the Onsager \nprinciple [25], [26].  \n    The notion of unidirectionality in gyrotropic structures is different from the notion of \nnonreciprocity. To suppress backscattering, the structure should be lossy for backscattered \npropagating waves. So, a scattering matrix should be non-unitary. The S-matrix for an ideal \ntwo-port isolator (the structure with complete  suppression of backscattered energy) has the \nform 00\n10S, indicating that both ports are matched, but transmission occurs only in \ndirection from post 1 to port 2. While the S matrix is not unitary, it is also not symmetric. \nWorking of an ideal isolator with a gyrotropic medium is based on the nonreciprocal-\nabsorption  properties. In a general case, we have:  00() ()ij jiSH SH \n, but \n 00() ( )ij jiSH S H\n [27]. This analysis on unidirectionality makes questionable the \nstatement in [1 – 3] that in lossless  systems with broken time-reversal symmetry, one-way \npropagation of electromagnetic-wave energy  leads to suppression of backscattering. \n \nB. Nonreciprocity and unidirectionality for oscillating modes in resonant structures \n \nSome general aspects of nonreciprocity and unidirectionality in gyrotropic structures are well \nillustrated in consideration of oscillating modes in resonant structures. This consideration will  4allow better understanding of our analysis of th e properties of MDM osc illations in a ferrite-\ndisk resonator.   \n    For the electric and magnetic fields in isotropic media, represented in the wave-number \nspace as 3 ikr\nkEE e d k\n\n and  3 ikr\nkHH e d k\n\n, the reality of the fields means that  \n \n                                                     *\nkkEE\n      and    *\nkkHH \n .                                             (1) \n   \nThis makes it possible to obtain the orthonormality conditions for electromagnetic waves of \noscillating modes in a lossless distributed-parameter resonator with an isotropic medium:  \n*\nmn kk kkmn mn\nVVEE d V E E d V     \n,  \n                                         *\nmn kk kkmn mn\nVVHH d V H H d V     \n,                                (2) \n \nwhere mn is the Kronecker delta. Eqs. (2) show that in such a resonator one can normalize the \nfield of a given mode or to the field of a corresponding counter propagating mode, or to the \ncomplex conjugated field of the same mode. Let us dwell now on a lossless running-wave ring \nresonator containing an isotropic medium. In th is case, the resonance occurs due to a wave \nrunning only in one direction along a circle. Ne vertheless, also in such a resonator the \northonormality conditions, expressed by Eq. (2),  take place since for every clockwise running \nmode one may have the same-type counterclockwise running mode, and vice versa. It means \nthat for rotary motion of energy in one direction there exists the counterpart – the rotary motion \nof energy in the opposite direction. When an input monochromatic signal is not exactly at the \nresonance frequency, one has a reflected wave. Now the following question arises: Can we \nrealize a lossless running-wave ring res onator based on a one-way waveguide with \nunidirectional propagation of electromagnetic energy? More generally speaking, can one create \nan electromagnetic-wave resonator with the fields nonsymmetrical in the k\nspace? Suppose \nthat based on any of the structures described in Refs. [1 – 3] we realized a large in-plane loop. \nA radius of this loop is much bigger than the wavelength of electromagnetic waves. So, locally, \none has a one-way waveguide with the properties shown in Refs. [1 – 3]. It is evident that in a \ncase of one-wave propagation of energy, a small fluctuation of the field energy in a certain \npoint of the loop will lead to infinite accumulation of energy in an entire lossless structure \nduring multi-cyclic processes of the wave propaga tion. Such a rotary motion of energy in only \none direction, while preventing motion in the oppos ite direction (a ratchet device), is beyond \nthe laws of thermodynamics. Energy of a signal cannot propagate only in one direction in such \na travelling-wave loop. The proposed structure mu st be lossy. In othe r words, to realize a \nwaveguide with one-wave propagation of energy, one has to create certain channels for energy \nlosses. In such a case, the fields are not real and so the orthonormality conditions, expressed by Eq. (2), cannot be fulfilled.   \n    While realization of a lossless resonator with unidirectional propagation of electromagnetic \nenergy is, physically, a meaningless problem, th e question is about possibility to create a \nlossless gyromagnetic electromagnetic-wave re sonator with nonreciprocal wave propagation \nand real fields. When a lossless resonator contains a gyromagnetic medium characterized by a \npermeability tensor \n, the orthonormality conditions for electromagnetic waves are expressed \nas [28] \n  5                          *\nmn kkmn\nVEE d V  \n,      *\nmn kkmn\nVHH d V    .                              (3)  \n \nThe tensor  is a Hermitian tensor: \n \n                                                             * T.                                                         (4) \n \nMoreover, from the Onsager relations for kinetic  coefficients, one has for the components of \nthe tensor  [25, 26, 28]: \n \n                                                        00,,ij ji HH   \n,                                                 (5)                          \n \nwhere 0H\n is an external bias magnetic field.  \n    The orthonormality conditions (3) are different from the orthonormality conditions (2). A \ngeneral analysis shows that an electromagnetic-wave resonator containing a gyromagnetic-\nmedium sample has real eigenfrequencies, but co mplex eigenfunctions [28]. It means that there \nis no a standing-wave behavior of the resonator eigenfunctions. Let us consider, for example, a \nferrite cylinder of the length l longitudinally magnetized in direction of axis z. If one uses \nseparation of variables, at the resonance frequency there will be a standing wave along z-axis \nand, because of the Faraday rotation, an azimuthally running wave in the plane perpendicular \nto z-axis. At resonance frequency, the phases of  the waves along z-axis and in the plane \nperpendicular to z-axis should be correlated. But the phase of the azimuthally running wave is \nnot identified in the cross-sectional plane of a ferrite cylinder. So, one should conclude that \nsuch a representation of the eidenfunction as a standing-wave plus running-wave behavior is \nincorrect. In a lossless gyromagnetic electroma gnetic-wave resonator with nonreciprocal wave \npropagation one cannot identify a certain phas e difference between two given points at a \nresonance frequency.     The main reason for this is that microwave resonators with ferrite inclusions are \nnonintegrable systems because of the time-reversal symmetry (TRS) breaking effects. The \nconcept of nonintegrable, i.e. path-dependent, ph ase factors is one of the fundamental concepts \nof electromagnetism. A key aspect of the behavior of the ferrite-resonator configuration \nconcerns reflection and refraction of electromagnetic waves at ferrite-vacuum interfaces. In a \ngeneral case of oblique incidence of a wave on a single ferrite-vacuum interface, apparently \ndifferent situations arise by changing the direc tions of incident waves and bias, and incident \nside of the interface [29 – 31]. In a system of a cavity and a ferrite sample one obtains a TRS-\nbreaking microwave billiard. Due to the TRS breaking, the ferrite-vacuum boundary conditions \nentail the existence of the solutions to the differential equations which are topologically \ndistinct. This leads to creation of topological defects – th e Poynting-vector vortices [30 – 33].              \n    Can one create a microwave ferrite resonato r with real eigenstates and real eigenfunctions? \nYes. But there will be the resonator with magnetostatic-wave (MS-wave) [or the magnetic-dipolar-mode (MDM)] oscillations. It is well know n that in a case of small (compared to the \nfree-space electromagnetic-wave wavelength) sa mples made of magnetic media with strong \ntemporal dispersion, the role of an electric displacement current in Maxwell equations can be \nnegligibly small, so oscillating fields are the quasistationary fields [26]. A magnetic field \nH\n is \na quasimagnetostatic field ( 0H \n), which is expressed by a magnetostatic potential: \nH\n.  The spectral properties of oscillations in such a small ferrite sample are analyzed \nbased on the Walker equation for MS-potential wave function (,)rt [34]:  6                                                                   \n                                                                      0 .                                                  (6) \n \nOutside a ferrite this equation becomes the Laplace equation. The MDM oscillations appear \nbecause of a prevailing role of long-range dipole- dipole interactions in a small ferrite sample. \nImportantly, excitation of the real-eigenstate mu ltiresonance MDM oscillations in a quasi-2D \nferrite disk by microwave radiation, observed, for the first time, in Ref. [35], is possible due to \npresence of surface chiral currents. In the near-field region, these chiral currents create specific \ntopologically distinctive structures – the ME-fields – resulting in observation of the \nunidirectional multiresonant tunneling, discussed in this paper. \n \nIII. MDM CHIRAL CURRENTS AND THEIR INTERACTION WITH MICROWAVE \nRADIATION \n \nA. MDM eigenvalue problems, chiral st ates, and quantized electric fluxes \n \n    MDM oscillations in a quasi-2D ferrite disk  are mesoscopically quantized states. Long range \ndipole-dipole correlation in position of electron sp ins in a ferrimagnetic sample with saturation \nmagnetization can be treated in terms of collective excitations of the system as a whole. If the \nsample is sufficiently small so that the dephasing length phL of the magnetic dipole-dipole \ninteraction exceeds the sample size, this  interaction is non-local on the scale of phL. This is a \nfeature of a mesoscopic ferrite sample, i.e., a sample with linear dimensions smaller than phL \nbut still much larger than the exchange-interaction scales. In a case of a quasi-2D ferrite disk, \nthe quantized forms of these collective matter  oscillations – magnetostatic magnons – were \nfound to be quasiparticles with both wave-like and particle-like behavior, as expected for \nquantum excitations. The magnon motion in this system is quantized in the direction \nperpendicular to the plane of a ferrite disk. The MDM oscillations in a quasi-2D ferrite disk, \nanalyzed as spectral solutions for the MS-potential wave function ( , ) rt, has evident \nquantum-like attributes [7 – 10]. \n    In microwave experiments with a normally magnetized quasi-2D ferrite disk, regular \nmultiresonance MDM spectra have been observed [21, 35 – 39]. It was shown that in such a ferrite-disk resonator, MDM oscillations can be characterized by real eigenstates and real \neigenfunctions. Formulation of quasi-Hermitian eigenvalue problem and analytical spectral \nsolutions for MDMs in a normally  magnetized ferrite disk were obtained in Ref. [7, 8]. For the \ndisk geometry, the energy-eigenstate oscillati ons are described by a two-dimensional (with \nrespect to in-plane coordinates of a disk) differential operator ˆG: \n \n                                                                    2 ˆ\n16qgG  ,                                                     (7)                          \n \nwhere 2\n is the two-dimensional Laplace operator,  is a diagonal component of the \npermeability tensor, and qg is a dimensional normalization coefficient for mode q. Operator \nˆG is positive definite for negative quantities . The normalized average (on the RF period) \ndensity of accumulated magnetic energy of mode q is determined as \n  7                                                                    2\n16  qq\nqzgE  ,                                                  (8) \n \nwhere \nqz is the propagation constant of mode q along the disk axis z. The energy eigenvalue \nproblem is defined by the differential equation: \n \n                                                                    ˆ\nqq qGE ,                                                         (9) \n \nwhere q is a dimensionless membrane (“in-plane”) MS-potential wave function. In the \nenergetic representation, a square of a modulus of the wave function defines probability to find \na system with a certain energy value.  The scalar-wave membrane function  can be \nrepresented as  \n \n                                                                      qq\nqa                                                          (11) \n \nand the probability to find a system in a certain state q is defined as \n \n                                                                  2\n2* qq\nSad S .                                                 (12) \n  \nMDM oscillations in a ferrite disk are described by real egenfunctions: *\n . The \northonormality conditions are expressed as \n \n                                                      qq q qqq\nSSdS dS  \n      ,                                   (13) \n \nwhere S is a cylindrical cross section of a ferrite disk. \n    The above spectral solutions, based on differential operator G, we conventionally call G-\nmode solutions. In solving the energy-eigenstate spectral problem for the G-mode states, the \nboundary condition on a lateral surface of a ferrite disk, is expressed as [6 – 8] \n \n                                                             0\nrrrr\n      ,                                      (14) \n \nwhere  is a radius of a ferrite disk. There is a homogeneous boundary condition for a \ndifferential operator ˆG[see Eq. (7)]. For the magnetic field components, Eq. (14) is written as \n \n                                                                () () 0rrrrHH  ,                                       (15) \n \nwhere ()rrH and ()rrH are the radial components of a magnetic field on a border circle \nof a ferrite disk.  \n    The G-mode ferrite disk, which is not connected to the surrounding, returns to the original \nsituation after 2 rotation. The G-mode object connected to the microwave surrounding \nbehaves differently. It appears that the G-mode boundary conditions are different from  8standard electromagnetic boundary conditions. The G-mode spectrum is obtained based on \nsolution of the Walker equation (6). This equation is, in fact, the magnetostatic-description \nrepresentation of a differential equation 0B\n. It is evident that the boundary condition (15) \nmanifests itself in contradictions with the boundary condition for continuity of a radial \ncomponent of magnetic flux density B\n on a lateral surface of a ferrite-disk resonator. Such a \nboundary condition should be written as  \n \n                                                  ( ) ( ) ( )rr a rrrHH i H       ,                                   (16) \n \nwhere ( )rH is an annular magnetic field on a borde r circle. In the MS description, this \nequation appears as  \n \n                                                  a r\nrrrr  \n\n      ,                               (17) \n \nwhere  is the MS-potential membrane wave function,  is an azimuth wave number, and a \nis a off-diagonal component of the permeability tensor. Contrary to real wave functions , \nfunctions  are complex wave functions. The term in the right-hand side of Eq. (17) has the off-\ndiagonal component of the permeability tensor, a, in the first degree. There is also the first-\norder derivative of function  with respect to the azimuth coordinate. It means that for the MS-\nwave solutions one can distinguish the time direction (given by the direction of the \nmagnetization precession and correlated with a sign of a) and the azimuth rotation direction \n(given by a sign of  ). For a given sign of a parameter a, there are different MS-potential \nwave functions, () and (), corresponding to the positive and negative directions of the phase \nvariations with respect to a given direction of azimuth coordinates, when 02 . There is an \nevidence for the path dependence in the problem solutions. \n    The G-mode solutions obtained based on the boundary condition (15) are the stationary-state \nsolutions with singlevalued MS-potential wave functions . Contrary, the spectral solutions \nobtained based on the boundary condition (16) cannot be considered as stationary-state solutions \nwith singlevalued MS-potential wave functions. A singular border term in the right-hand-side of \nEq. (16), which expresses the discontinuity of a radial component of magnetic flux density for \nthe G-mode solutions, we represent as effective surface magnetic charge density: \n \n                                                             ()4m\nas riH  .                                                 (18) \n \nThis equation relates an azimuthal component of the magnetic field with surface magnetic \ncharge density. In fact, the charges ()m\ns are topological magnetic charges. \n    We can consider surface magnetic charge density ()m\ns as a certain fluctuation. Such \nmagnetic-charge edge states of MDMs contribute to appearance of a surface magnetic current \naround the border ring. For the time varying G-mode fields, the surface magnetic charge density \n()m\ns should be related to the surface magnetic current density ()m\nsj by a continuity equation. For \nmonochromatic wave process (~ite), we have: \n \n                                                                  () ()mm\ns s ji  .                                                    (19)  9 \nBoth quantities, ()m\nsand ()m\nsj, have time- and space-dependent phases. Evidently, magnetic \ncharges ()m\ns appear in a form of the magnetic-dipole (magnetic-multipole, in general) structure \non a lateral surface of a fe rrite disk. Magnetic currents ()m\nsj are not linear, but circulating \ncurrents.  So, Eq. (19) can take place only when magnetic charges ()m\ns are the charges moving \n(clockwise or counterclockwise) on a lateral surface of a ferrite disk.  This gives evidence for the \nfact that the G-mode picture should rotate in the laboratory frame when the magnetic charges \n()m\ns exist. Definitely, surface magnetic charges ()m\ns are not “free magnetic charges”. There are \ntopological charges determined by orientation of the magnetization vectors on a lateral surface \nof a ferrite disk. Also, surface magnetic currents ()m\nsjare not caused by motion of “free magnetic \ncharges”. In fact, there are chiral-rotation surface magnetostatic spin waves.    \n    In Fig. 1, we show surface magnetic charges and edge chiral magnetic currents. This is a view \non the upper plane of a ferrite disk. Suppose that there exists a magnetic-dipole fluctuation on a \nlateral surface of a ferrite disk and a surface magnetic current is a clockwise rotating surface \nwave. Let at a given time moment, th ere be a positive magnetic charge ()()m  at point P on a \ndisk lateral surface and a negative magnetic charge ()()m at a diametrically opposite point Q. Let \nthere be a magnetic current ()m\nsAj “departing”, with a certain phase A, from a point P. \nBecause of conservation of “magnetic neutrality”, another current wave ()m\nsBj with the same \nphase B A should “depart” to a point Q. Since, however, no real magnetic charges physically \nexist, these magnetic currents on a lateral surface  of a ferrite disk can be only the topological \ncurrents. Topologically, a circulating current ()m\nsAj is not the same as a circulating current \n()m\nsBj. Every of these separate currents gets around the orbital trajectory 2  during a \nhalf of period of microwave radiation. The current ()m\nsAj “arrives” to a point P with the phase \nA when the G-mode rotates at the angle of 2. The similar situation is for the current \n()m\nsBj. So, we can state that for the G-mode regular-coordinate angle 2 , a topological \nsurface magnetic current acquires the phase of . Because of a magnetic-dipole fluctuation \non a lateral surface of a ferrite disk the domain of the G-mode azimuthal angle , in a laboratory \nframe, is no more [0,2 ] but [0,4 ]. \n    Circulating currents ()m\nsAj and ()m\nsBj appear as topologically distinctive currents due to \nfinite thickness of the disk. Fig. 2 illustra tes a two-layer-ring model for surface magnetic \ncurrents. When a magnetic current of the upper (lower) layer is arriving to terminal P (where a \ntopological magnetic charge is supposed to be lo calized), it must continue its propagation at the \nlower (upper) layer and this is only one choice . The similar situation takes place at terminal Q. \nRegions of terminals P and Q are the regions of singularity. At the same time, we have to note \nthat topological magnetic charges are not point charges. They are distributed on a lateral surface \nof the disk. So, sharp “kinks” of the current lines are impossible.  \n    Due to the special topology of the two-laye r ring, orbital angular momenta are allowed to be a \nhalf-integer. In a quasi-2D ferrite disk, the two layers are very close to each other and the above two currents look like a ring magnetic current on a lateral surface of a ferrite disk. So, the continuity equation (19) has a form: \n  10                                                             () ()mm\ns s ji   ,                                                  (20)                        \n \nwhere ()\n() 1m\nsm\nsj\nj\n \n \n. The closed-loop surface magnetic current ()m\nsj\n are \nclockwise and counterclockwise edge chiral curre nts depending on a direction of a bias magnetic \nfield (in other words, depending on a direction of time).  \n    In a quasi-2D ferrite disk, the edge chiral currents ()m\nsj\n can be described by 1D scalar wave \nfunctions, which are double-valued functions. Su ch ring magnetic currents can create electric-\nfield fluxes. This effect was studied in details in Ref. [9, 10, 13]. On a lateral border of a ferrite \ndisk, the correspondence between a double-valued membrane wave function  and a \nsinglevalued membrane function ~ is expressed as: rr     , where iqfe\n  is \na double-valued edge wave function on contour 2 . The azimuth number q is equal to \n1\n2l , where l is an odd quantity ( l = 1, 3, 5, …). For amplitudes we have  f f  and f = 1. \nFunction  changes its sign when  is rotated by 2 so that 21iqe. As a result, one has \nthe eigenstate spectrum of MDM oscillations with topological phases accumulated by the edge \nwave function . On a lateral surface of a quasi-2D ferrite disk, one can distinguish two \ndifferent functions , which are the counterclockwise and clockwise rotating-wave edge \nfunctions with respect to a membrane function ~. A line integral around a singular contour : \n2\n**\n01()  ()\nrid i d\n \n\n         \n  is an observable quantity. It follows from the \nfact that because of such a quantity one can restore singlevaluedness of the spectral problem. \nBecause of the existing the geometrical phase fa ctor on a lateral boundary of a ferrite disk, \nMDMs are characterized by a pseudo-electric field (the gauge field) [9, 13]. We will denote here \nthis pseudo-electric field by the letter €\n. The geometrical phase factor in the G-mode solution is \nnot single-valued under continuation around a contour  and can be correlated with a certain \nvector potential ()m\n€\n. We define a geometrical phase for a MDM as [9, 13] \n \n                                        2\n*( )\n0[( )( ) ] 2m\nr€ id K d q\n       \n .                        (21) \n \nwhere 1\nre \n\n  and eis a unit vector along an azimuth coordinate. In Eq. (21), K is \na normalization coefficient. The physical meaning of coefficient K was discussed in Refs [9, 13, \n40]. In Eq. (21) we inserted a connection which is  an analogue of the Berry phase. In our case, \nthe Berry's phase is generated from the broken dyn amical symmetry. The confinement effect for \nmagnetic-dipolar oscillations requires proper phase  relationships to guarantee single-valuedness \nof the wave functions. To compensate for sign ambiguities and thus to make wave functions \nsingle-valued we added a vector-potential-type term ()m\n€\n (the Berry connection) to the MS-\npotential Hamiltonian. On a singular contour 2 , the vector potential ()m\n€\n is related to  11double-valued functions. It can be observable only via the circulation integral over contour , \nnot pointwise. The pseudo-electric field €\n can be found as  \n \n                                                                      ()m\n€ € \n.                                               (22) \n \nThe field €\n is the Berry curvature. In contrast to the Berry connection ()m\n€\n, which is physical \nonly after integrating around a closed 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 ,                          (23) \n \nwhere ()e\n  are quantized fluxes of pseudo-electric  fields. Each MDM is quantized to a \nquantum of an emergent electric flux. There are the positive and negative eigenfluxes. These \ndifferent-sign fluxes should be nonequivalent to avoi d the cancellation. It is evident that while \nintegration of the Berry curvature over the regular-coordinate angle  is quantized in units of \n2, integration over the spin-coordinate angle  1\n2 is quantized in units of . The \nphysical meaning of coefficient K in Eqs. (21), (23) concerns the property of a flux of a \npseudo-electric field.  \n   The edge chiral current ()m\nsj is a persistent magnetic current in an Aharonov-Bohm-like \ngeometry. On an edge ring region, we have the magnetization motion pierced by an electric \nflux e\n . The edge magnetic current can be observable only via its circulation integrals, not \npointwise. This results in the moment oriented along a disk normal. It was shown \nexperimentally [37] that such a moment has a response in an external RF electric field and so \ncan be classified as an electric moment. There is a so called anapole moment ea [9]. \n \nB. Interaction of MDMs with external microwave radiation \n \nBecause of the edge chiral currents, interaction of MDMs with external microwave radiation of \nMDM oscillations is manifested by unique topological properties.  \n    In numerous microwave experiments with a quasi-2D ferrite disk, multiresonant MDM \noscillations were observed both at a constant signal frequency with scanning of a bias magnetic \nfield and at a constant bias magnetic field with scanning of a signal frequency [21, 35 – 39]. In initial experimental studies [35, 36], it was shown that in a microwave structure with an \nembedded quasi-2D ferrite disk, multiresonance MDM oscillations are excited by RF magnetic \nfields lying in the disk plane. Later, it was shown experimentally that MDMs can also be \neffectively excited by RF electric field of a microwave structure, which is oriented along a disk \naxis [37]. One of the main conclusions, we can made from all these experiments, is the fact that positions of the resonance peaks are not dependent on the microwave-structure environment \nand are exceptionally determined by the disk parameters. It means that in all the experiments \nwe observe the energy-eigenstate spectra. As  we discussed above, the domain of the \nG-mode \nazimuthal angle  is not [0,2 ] but [0,4 ]. The G-mode object, being connected to the  12microwave surrounding, returns to the original situation after 4 rotation. It means that for a \ngiven frequency  of microwave radiation, the G-mode rotation frequency is 2.  \n     When (due to interaction with external microwave radiation) a macroscopic MS-potential \nwave function ~ is set into rotation, quantized vortex lines appears. At the vortex center (the \ncenter of a ferrite disk) the function ~ is zero. In different physical systems, there are many \nother examples of such vortices with rotating macroscopic wave functions. In particular, there \nare vortices in Bose-Einstein-condensa te systems [41 – 44]. The operator ˆGin Eq. (7) is a \ntwo-dimensional differential operator in the absence of rotation. The eigenstate of this operator \nis the energy of the ground state with no vortices. In a rotating frame, this differential operator \nhas a form:  \n                                                                    ˆˆ ˆ\nz GG L .                                                     (24)  \n \nThe last term in the right-hand side of this equation favors of states with non-zero angular \nmomentum. ˆ\nzLi y xx y is the orbital angular momentum along the rotation axis z – \nthe disk axis. A rotation frequency is 2 . For a certain MDM q, the energy of a vortex \nstate in a frame rotating with angular frequency q is ˆ\nq qqq zEE L  . \n    Microwave radiation can potentially couple to  MDM oscillations if the ferrite sample shows \na confined structure to satisfy conservation of energy and angular momentum. Tunneling of microwave radiation into a MDM ferrite disk is due to twisting excitations. Our studies give \nevidence for such near-field twisting excitations.  There are subwavelength field structures with \nquantized energy and angular momentums. It is  known that photons, like other particles, carry \nenergy and angular momentum. A circularly polarized photon carries a spin angular \nmomentum [45]. Also, photons can carry additi onal angular momentum, called orbital angular \nmomentum. Such photons, carrying both spin and orbital angular momentums are called \ntwisting photons [46]. Twisting photons are pr opagating-wave behaviors. These are \"real \nphotons\". In the near-field phenomena, which have subwavelength-range effects and do not \nradiate through space with the same range-prope rties as do electromagnetic wave photons, the \nenergy is carried by virtual photons, not actual photons. Virtual particles should also conserve energy and momentum. The question whether virtual photons can behave as twisting \nexcitations, is a subject of numerous discussions [47 – 50]. In particular, in Refs.  [47, 49] it \nwas discussed that vacuum can induce a torque between two uniaxial birefringent dielectric \nplates. In this case, the fluctuating electromagn etic fields have boundary conditions that depend \non the relative orientation of the optical axes of  the materials. Hence, the zero-point energy \narising from these fields also has an angular de pendence. This leads to a Casimir torque that \ntends to align two of the principal axes of the material in order to minimize the system’s \nenergy.  A torque occurs only if symmetry be tween the right-hand and left-hand circularly \npolarized light is broken (when the media are birefringent). In our case, a quantum vacuum \nfield takes energy from the MDM ferrite disk. The electromagnetic mode with frequency \n \ninteracts with the MDM of frequency 2 (because the ac part of the ponderomotive force has \nfrequency 2). This is the parametric pumping of the energy from magnetomechanical \n(magnetization-precession) oscillations into electromagnetic oscillations.  Energy taken is \nconverted into real photons. This is the dynamical Casimir effect. The dynamical Casimir \neffect is the generation of photons out of the quantum vacuum induced by an accelerated body. \nRotating MDMs in a ferrite disk cannot rule out a dynamical Casimir torque even in the case of \nuniform angular velocity. This raises the question of the angular-momentum coupling with the  13quantum vacuum field. Quantized vortices ar e sensitive probes of the angular-momentum \ncoupling of MDMs with the vacuum field.  \n    The properties of rotating G modes and edge chiral states in a ferrite disk essentially \ndetermine a character of interaction of MDMs wi th external microwave radiation. Suppose that \na small ferrite disk is placed in a region of a li nearly polarized (in a plane of a ferrite disk) RF \nmagnetic field. When no MDM oscillations are assumed, we can state that such a RF magnetic field excites homogeneous-ferromagnetic-resonance  oscillations of magnetization in a small \nferrite sample [28]. One observes an induced magnetic dipole lying in the disk plane. The \nquantity of this dipole \n()mP\n is determined both by the incident-wave magnetic field ()iH\n and \nby the scattered-wave magnetic field ()sH\n: \n \n                                                               () ( ) ( ) ( )me i sPH H\n,                                            (25) \n \nwhere ()e is the ellipsoid external susceptibility [28]. Formally, one can describe the magnetic-\ncharge distribution on a lateral surface of a ferrite  disk as an azimuth magnetic-charge standing \nwave. There are two equal-amplitude azimuth waves of surface magnetic charges. Also, on a \nlateral surface of a ferrite disk, the azimuth component of a magnetic field () ( ) ( )ext i sHH H  \n \ncan be represented as an azimuth magnetic-field standing wave. It means that there are two \nequal-amplitude azimuth waves of a magnetic field: () () () 1\n2ext ext extHH H   \n. \n    At certain quantization conditions, long range dipole-dipole correlation in position of \nelectron spins in a ferrite disk results in appearance of collective excitations of the system. \nThese quantized excitations are MDMs. Surface magnetic charges induced by a linearly \npolarized (in a plane of a ferrite disk) RF magnet ic field can be considered as the sources for \nMDM oscillations. In this case, however, the ma gnetic-charge distribution on a lateral surface \nof a ferrite disk is not an azimuth magnetic-charge standing wave. There exists only a \nclockwise or only counterclockwise (depending on a direction of a bias magnetic field) azimuth wave of surface magnetic charges. For excitation of G-modes by external RF magnetic \nfields, the right-hand-side term in Eq. (16)  appears as an external source. Based on a \nperturbation-theory analysis, we can assume that the azimuth component of a time-varying \nincident magnetic field \n()i\nrH or  ()i\nrH induces surface magnetic charges: \n \n                                                              () () ()4i i m\nas riH                                      (26)                             \n \nOn the other hand, with the energy eigenstate description based on the singlevalued G-mode \nMS-potential wave functions , one has to show that there exists a certain internal mechanism \nwhich removes the perturbation term in a right-hand side of Eq. (16). From the above analysis, \nwe can state that the rotating surface magnetic charge ()()im\ns\n creates, on a lateral surface of a \nferrite disk, the surface magnetic current ()()im\nsj\nand this edge chiral current, in its turn, \noriginates an incident-wave electric flux ()()ie . The G-mode stationary-state solution will take \nplace when a certain electric flux appears to compensate  the incident-wave electric flux ()()ie . \nWe call this flux as the scattered-wave electric flux ()()se . We can characterize this effect as  14the electric self-inductance effect. The flux ()()se  will induce, on a lateral surface of a ferrite \ndisk, the edge magnetic current ()()sm\nsj\n and this chiral current, in its turn, will create surface \nmagnetic charge ()()sm\ns compensating the incident-wave magnetic charge ()()im\ns. Every of \nthese magnetic currents, ()()im\nsj\n and ()()sm\nsj\n, is composed by two topologically \ndistinctive current components discussed above. Both currents, ()()im\nsj\n and ()()sm\nsj\n, \nhave the same direction along the azimuth coordinate and are mutually time-phase shifted at \n180. Fig. 3 shows the edge magnetic currents ()()im\nsj\n and ()()sm\nsj\n on a lateral surface \nof a ferrite disk and their possible correlation with the G-mode MS-potential wave functions  \nof a certain MDM.    \n    What will happen when a ferrite disk is placed in the RF electric field oriented normally to a \nferrite disk? Following the above model of the elec tric self-inductance effect, one can state that \nin this case the incident-wave electric flux ()()ie should be compensated by the scattered-wave \nelectric flux ()()se . A numerical analysis of the electric field structure clearly proves this \nstatement. In Refs. [11, 50] it was shown that near a MDM-resonant ferrite disk a normal \ncomponent of the RF electric field is zero durin g all the time period of microwave radiation.  \n    Physically, the above formal introduction of the quantized fluxes of pseudo-electric fields \n()e\n can be well justifies based on the Thomas precession effect. Rotating surface-charge \nmagnetic dipoles and edge chiral magnetic current s on a lateral surface of a ferrite disk interact \nwith the G-mode oscillations. The underlying physical mechanism is the spin-orbit interaction, \nwhich couples the magnetization spin degree of freedom to their orbital dynamics. In Fig. 3, \nsuch spin-orbit interaction is illustrated by highlighted parts of in the graphs of the edge \nmagnetic currents ()()im\nsj\n and ()()sm\nsj\nand shaded areas on the graph of the G-mode \nMS-potential wave functions . Because of the spin-orbit interaction, the spin is directly related \nto the winding of trajectory. Such degrees of freedom as geometric (Berry) phases are winding \ndependent. The Thomas precession talks us a bout rotation of a spinning particle under the \ncoordinate transformations. The circular spin current along a ring will inevitably produce the Thompson precession. Due to this precession one ha s an effective electric field. There is the \ngauge field, no gradient of an electrical potenti al takes place in this case [51]. Importantly, the \nelectric field due to Thomas precession may exis t in any magnetic particle which confines \nmagnetic moments in motion [51].  \n    Rotating surface-charge magnetic dipole s and edge chiral magnetic currents strongly \ntransform both magnetization dynamics inside a ferrite-disk particle and topology of the near-\nfield microwave radiation. As a result of the spin-orbit interaction, so called \nL-type MDMs \nappear. There are helical MS waves in a quasi-2D ferrite disk [9, 10]. The L modes are \nmicrowave MDM polaritons [12]. These polariton stat es create near fields with specific topology \n– the magnetoelectric (ME) fields [13]. There ar e topologically distinctive virtual photons. As \nwe well know, in a subwavelength region of regu lar microwave radiation the near-field structure \nis exhibited as a quasi-static electric or quasi-static magnetic field. These quasi-static electric \nand magnetic fields are mutually uncoupled [45] . The near-field structure of a MDM particle is \nessentially different. One can observe strong subw avelength localization of both the electric and \nmagnetic fields [12, 13, 15, 40].   15    Topological properties of ME fields (non-zer o helicity factor) arise from the presence of \ngeometric phases on a border circle of a MDM ferrite disk. Due to geometrical phases, the in-\nplane power-flow vortices appear. It is evident that, in ac cordance with the thermodynamics \nlaws, power cannot flow along a closed circle. So we have flat (Archimedes) spirals of the \npower-flow distributions. The in cident-wave power flow goes fro m peripheral regions to the \nvortex center. The scattered-wave power flow  goes from the vortex center to peripheral \nregions. There are or right-handed, or left-handed flat spirals. When, for example, the incident-\nwave power flow is a right-handed spiral, the scattered-wave power flow is a left-handed \nspiral. The L-mode solutions, being not the energy eigenstate solutions, are well observable in \nthe HFSS numerical simulation. In a “dynamical” HFSS program, one can observe such \ngeometric phases indirectly – by topological transformations of the wave fronts, subwavelength-scaled power flow vortices, a nd the helicity factors [11 – 13, 15, 40].   \n    ME fields are the topological-defect solutions, which are distinct from the Maxwellian \nvacuum solutions. It can be proven to exis t because the boundary conditions entail the \nexistence of homotopically distinct solutions; the solutions to the differential equations are then \ntopologically distinct. Experiments with a thin-film ferrite disk embedded in a microwave structure show that quantization of a microwave energy takes place due to the microwave-\nphoton angular momentum. The discrete topological states of the microwave cavity fields are \ncaused by discrete variation of energy of a ferrite disk appearing because of an external source \nof energy – a bias magnetic field [21]. The modes observed in a microwave cavity with an \nembedded MDM ferrite disk are quantum vacuum fluctuations [21].   \nIV. EXPERIMENTAL AND NUMERICAL RESULTS \n \nA. Chiral edge magnetic currents  \n \nWhile the L-mode resonances are energetically unstable, there are topologically stable \nresonances of rotating fields [11 – 13].  These so lutions are characterized by both the linear and \ncircular magnetic currents. The two magnetic currents, being coupled at the MDM resonance, \nform helical-structure (or chiral) magnetic currents. It means that all excitations in a ferrite disk \ncan propagate only clockwise or only counterclockw ise. When, for a given direction of a bias \nmagnetic field, we call the chiral-current process as a “forward” process in the formalism of \nfluctuation relations, in an opposite direction of a bias magnetic field we have a “backward” \nprocess. The backward process can be described as a forward process in the time-reversed twin system with the opposite chirality. Certainly, it should be assumed that the material \ncharacteristics and temperatures are the same in the forward and backward processes. In \nquantum-Hall structures, one can observe chiral ex citations on a 2D surface of a 3D system. In \nsuch systems, electric charge can propagate in bo th directions along one of the coordinate axes \nbut only in one direction along the second axis [14]. Similar situation, but with “magnetic \ncharges”, takes place in our case.   \n    While positions of the MDM resonance peaks are exceptionally determined by the disk \nparameters, the amplitudes and forms of these peaks can be strongly dependent on the \nmicrowave-structure environment. The observed effect of multiresonant unidirectional \ntunneling (UDT) is due to such a dependence of the MDM spectrum on the microwave-structure properties. Based on the experimental and numerical studies, we show that breaking \nof symmetry in geometry of a microwave structure strongly influence on the UDT \ncharacteristics. The main physical aspect concerns  the presence of chiral magnetic currents in a \nmicrowave structure. At the MDM resonances, these chiral magnetic currents result in \nunidirectional transfer of quantized angular  momenta through subwavelength vacuum or \nisotropic-dielectric regions.   16    We start with a microstrip structure with an embedded thin-film ferrite disk. Such a \nmicrostrip structure is shown in Fig. 4. For experimental studies, we use a ferrite disk with the \nfollowing parameters. The yttrium iron ga rnet (YIG) disk has a diameter of 3D mm and a \nthickness of 05.0t mm. The saturation magnetization of a ferrite is  1880  4sM G. The \nlinewidth of a ferrite is 0.8 OeH . The disk is normally magnetized by a bias magnetic field \n04210 H Oe. An experimental microstrip structure is realized on a dielectric substrate \n(Taconic RF-35, 3.52r , thickness of 1.52 mm). Characteristic impedance of a microstrip \nline is 50 Ohm. The S-matrix parameters were measured by a network analyzer. With use of a \ncurrent supply we established a quantity of a normal bias magnetic field 0H\n, necessary to get \nthe MDM spectrum in a required frequency range. For numerical studies, we use a ferrite disk \nwith the same parameters as pointed above. The only difference is that, for better \nunderstanding the field structures, in numerical anal yses we consider a ferrite disk with very \nsmall losses: the linewidth of a ferrite is 0.1 OeH . In the absence of MDM resonance \npeaks, the subwavelength coupling between two microstrip lines in these structures is \nextremely small. At the resonance peaks, one has strong subwavelength coupling. Taking into \naccount in-plane geometry, the presence of a ground metallic plane in a microwave structure, \nand direction of rotation (at a given direction of a bias magnetic field) of a power-flow density \nin a ferrite disk, one finds that the ways el ectromagnetic waves propagating from port 1 to port \n2 and, oppositely, from port 2 to port 1, are geometrically different.  \n    Fig. 5 represents the experimental frequency characteristics of modules of the reflection \ncoefficient (the  11S scattering-matrix parameter) and the transmission coefficients (the 21S and \n12S scattering-matrix parameters) for two opposite orientations of a normal bias magnetic field \n0H\n. Classification of the resonances shown in Fig. 5 is based on analytical studies in Ref. [8]. \nThere are resonances corresponding to MDMs w ith radial and azimuth variations of the \nmagnetostatic-potential wave functions in a fe rrite disk. The azimuth-variation resonances \nappear because of the azimuth nonhomogeneity of a microstrip structure. In the spectra, these \nresonances are observed between the peaks of radial-variation resonances. In the mode designation, the first number characterizes a number of radial variations for the MDM spectral solution. The second number is a number of azimuth variations [8, 21].      In Fig. 5, one can see that the reflection-coefficient spectrum is characterized by the Lorentz-\nresonance peaks. Contrarily, the transmission-coefficient excitations manifest themselves as \nthe Fano-resonance peaks at the stationary states of the MDM spectrum. It is worth noting that \na character of the Fano interference is differe nt for radial-variation and azimuth-variation \nMDMs [21, 40]. For the spectra in Fig. 5, we can state that while the reflection-coefficient \nspectrum is the same at two opposite orient ations of a normal bias magnetic field \n0H\n, for the \ntransmission-coefficient excitations there is strong sensitivity of the peak sizes on the direction \nof a bias magnetic field. At a very small tran smission level for non-resonant frequencies (about \n21 25 Sd B ), one clearly observes resonance peaks of the UDT. For better illustration of this \neffect, in Fig. 6 we show the reflection and transmission spectra normalized to the background \n(when a bias magnetic field is zero) level of the microwave structure. An analysis of the transmission spectra shows that the observed UDT e ffect is due to the field chirality in an \nentire microwave structure. When, for a given direction of a bias magnetic field, one has a \n“forward” process, the “backward” process is exhibited as a “forward” process in the time-\nreversed twin system with the opposite chirality. As the sources of the fields with different \nchirality, there are edge magnetic currents \n()()im\nsj\n and ()()sm\nsj\n  considered in the \nabove model. The field chirality results in unidirectional transfer of angular momenta through a  17subwavelength vacuum and isotr opic-dielectric regions. Simult aneous exchange between the \n“forward” and “backward” processes together with change of the time direction remains the \nsystem symmetry unbroken. This symmetry properties  of the chiral states are well illustrated in \nFig. 7. One can see complete coincidence between the spectra of the 21S and 12S scattering-\nmatrix parameters for oppositely directed bias magnetic fields.   \n    The shown above experimental results of the UDT effect are well verified numerically. In Fig. 8, one can see the numerically obtained transmission characteristics for the first MDM \nresonance at two opposite orientatio ns of a normal bias magnetic field \n0H\n. It is necessary to \nnote here that instead of a bias magnetic field used in experiments (04210 H Oe), in the \nnumerical studies we applied a higher-quantity bias magnetic field: 04434 H Oe. Use of such \na higher quantity (giving us th e same position of the resonance peak in the experiments and \nnumerical studies and) is necessary because of non-homogeneity of an internal DC magnetic \nfield in a real ferrite disk. A more detailed discussion on a role of non-homogeneity of an \ninternal DC magnetic field in the MDM spectr al characteristics can be found in Ref. [8]. \n    To explain the UDT effect in this microstrip structure, we can use the following model. At the MDM resonance frequency, for a given direction of a bias magnetic field the incident and \nscattered waves near a ferrite disk have different types of flat spirals. While, for example, the \nincident wave is a right-handed spiral, the scatte red wave should be a left-handed spiral. In an \nopposite direction of a bias magnetic field, we have a left-handed spiral for the incident wave \nand a right-handed spiral for the scattered wave. The UDT effect appears for the reason that in a nonsymmetrical microwave structure the interplay between the right-handed and left-handed \nspirals of the incident and scattered waves is diffe rent for different directions of a normal bias \nmagnetic field. \n    Since the UDT appears due to distinguishable topology of the “forward” and “backward” \nexcitations, this effect should be enhanced for a microwave structure with increased breaking \nof symmetry in geometry. Such a microstrip structure is shown in Fig 9. The symmetry \nbreaking is increased by an inclined slot in one of conductive strip. Fig. 10 shows the \nexperimental \n21S and 12S scattering-matrix parameters of this structure for two opposite \norientations of a normal bias magnetic field 0H\n. There are the transmission spectra both non-\nnormalized and normalized to the background level of the microwave structure. One can find \nthat, compared to the previous results, the UDT effect is strongly enhanced in the structure with an inclined slot. These results are well verified numerically in Fig. 11. \n \nB. Chiral edge electric currents  \n \nAt MDM resonances, chiral electric currents can be induced on a thin metal wire placed on a \nsurface of a ferrite disk [52]. In a structure show n in Fig. 12, the electric field of a microstrip \nsystem causes a linear displacement of electric charges when interacting with a short piece of a \nwire. At the same time, the magnetic field of a MDM vortex causes a circulation of electric \ncharges. Being combined, these two motions (w hich include translation and rotation) cause \nhelical motion of electrons on a surface of a metal wire. Such helical waves result in observation of very peculiar field structures. Be cause of a chiral surface electric current, the \nelectric and magnetic fields at the butt end of a wire electrode become not mutually oriented in \nvacuum at the angle of \n90. While the mutually perpendicular components of the electric and \nmagnetic fields give the power-flow-density vorte x, the mutually paralle l components result in \nappearance of nonzero helicity density * 1Im16FE E  \n[52]. Fig. 13 shows  18numerically obtained distributions of the fields  and currents on a wire electrode and also the \nfield structures near a butt end of a wire electrode. \n    The field chirality in a MDM microwave structure with a wire electrode results in \nunidirectional transfer of a ngular momenta through a subwavelength vacuum region. Our \nexperimental results give evidence for such  a tunneling effect. To enhance experimental \nobservation of the UDT effect due to chiral electric currents, we use structures with breaking of symmetry in geometry. There are the right- or left-handed metallic helices. Fig. 14 shows two-\nport microwave structures with a MDM ferrite disk  and a wire electrode (port 1) and with the \nright- or left-handed metall ic helices (port 2). In these structures we used a wire electrode with \ndiameter of 100 um. Metallic helices are made with the same wire. Diameters of the helices are \n2mm. The pitch is equal to 0.4mm.  Every helix has ten turns. A wire concentrator is placed near a metallic helix without an electric contact with it.  \n    Fig. 15 shows experimental results of the \n21S scattering-matrix parameter for the right- and \nleft-handed metallic helices and two opposite directions of a bias magnetic field. The \ntransmission spectra is normalized to the background (when a bias magnetic field is zero) level \nof the microwave structure. This background level is about 21 30 Sd B . It is evident that \nthere is a specific chiral symmetry. Simultaneous change of the helix handedness and direction \nof bias magnetic field remains the system symmetry unbroken. Numerical results the \ntransmission spectra shown in Fig. 16 are in good correspondence with experimental results.       The observed effect of unidirectional tunneling  can be well explained by an analysis of the \npower-flow-density distributions in  a vacuum region near a wire concentrator and a metallic \nhelix. Such distributions are shown in Fig. 17 for two resonance peaks corresponding to the 1\nst \nMDM – the peaks A and B in the 21S frequency characteristics in Fig. 16. One can clearly see \nthat the power transmission in a two-port microwave structure is maximal when a direction of \nthe power-flow vortex at a butt end of a wire electrode corresponds to the handedness of a \nmetallic helix. There is an evidence for the pr esence of the orbital-angular-momentum twisting \nexcitations in a subwavelength region of microwave radiation at the MDM resonances.  Due to \nchiral properties of the fields near a wire elec trode, one has unidirectional transfer of quantized \nangular momenta through a subwavelength vacuum region. \n    In our previous studies [13, 15, 40, 52], it was shown that topology of the near fields \noriginated from a MDM ferrite disk – the ME fi elds – is characterized both by the power-flow \nvortices and the helicity parameters. While th e power transmission in a two-port microwave \nstructure is strongly related to the power-flow vort ices of the twisting ex citations, the helicity \ncharacteristics of these excitations (being local characteristics of the field geometry) are very \nslightly correlated with the power transmission effect. Fig. 18 shows distributions of the normalized helicity factor for the two resonance peaks corresponding to the 1\nst MDM – the peaks \nA and B in the 21S frequency characteristics in Fig. 16.  The normalized helicity parameter is \ndefined as [13, 15, 40, 52] \n \n                                                        *Im\ncosEE\nEE \n\n\n .                                                   (27) \n \nIn Fig. 18, one can see the helicity parameter dist ributions are mainly related to a direction of a \nbias magnetic field and periodicity of the metal-he lix turns. There is very small relation between \nthe helicity parameter distribution and the helix handedness. \n   \nV. DISCUSSION AND CONCLUSION  19 \n    The effects of quantum coherence involv ing macroscopic degrees of freedom and occurring \nin systems far larger than individual atoms are one of the topical fields in modern physics [53]. \nRecently, much progress has been made in de monstrating the macroscopic quantum behavior \nof superconductor systems, where particles form highly correlated electron system. The \nconcept of coherent mixtures of electrons  and holes, underlying the BdG-hamiltonian \nquasiclassical approximation, well descri bes the topological superconductors. \n    Macroscopic quantum coherence can also be observed in some ferrimagnetic structures. \nMDM oscillations in a quasi-2D ferrite disk are macroscopically (mesoscopically) quantized \nstates. Long range dipole-dipole correlation in position of electron spins in a ferrimagnetic \nsample with saturation magnetization can be treated in terms of collective excitations of the \nsystem as a whole. If the sample is sufficiently small so that the dephasing length phL of the \nmagnetic dipole-dipole interaction exceeds the sa mple size, this interaction is non-local on the \nscale of phL. This is a feature of a mesoscopic ferrite sample, i.e., a sample with linear \ndimensions smaller than phL but still much larger than the exchange-interaction scales. In a \ncase of a quasi-2D ferrite disk, the quantized fo rms of these collective matter oscillations – \nmagnetostatic magnons – were found to be quasip articles with both wave-like and particle-like \nbehavior, as expected for quantum excitations. The magnon motion in this system is quantized \nin the direction perpendicular to the plane of a ferrite disk. The MDM oscillations in a quasi-\n2D ferrite disk, analyzed as spectral solutions  for the magnetostatic-potential wave function \n(,)rt, has evident quantum-like attributes. The discrete energy eigenstates of the MDM \noscillations in a quasi-2D ferrite disk are well observed in the first microwave experiments [35, \n36]. Experimental studies of interaction of the MDM ferrite particles with its microwave \nenvironment give evidence for multiresonance Fano-type interference characteristic for \nquantum structures [21, 37]. It was also shown that MDMs in a ferrite disk are topologically \ndistinctive eigenstates [13, 15]. MDM oscillat ions in a quasi-2D ferrite disk can conserve \nenergy and angular momentum. Because of these properties, MDMs strongly confine \nmicrowave radiation energy in subwavelength scales. In a vacuum subwavelength region abutting to a MDM ferrite disk, one can observe  the quantized-state power-flow vortices. In \nsuch a region, a coupling between the time-varyi ng electric and magnetic fields is different \nfrom such a coupling in regular electromagnetic fields. These specific near fields – the ME fields – give evidence for spontaneous symmetr y breakings at the resonant states of MDM \noscillations. This symmetry breaking is character ized by the helicity parameter[13, 15, 52]. \n    The ME-field singularity is strongly related to  edge chiral currents. At the MDM resonances \none has chiral magnetic currents on a lateral surfa ce of a ferrite disk. Also, in a MDM ferrite \ndisk with a wire electrode, one can observe chir al electric currents on a surface of a metal wire. \nBecause of the presence edge chiral currents, interaction of MDMs with external microwave \nradiation is characterized by unique topological prope rties. In this paper, we showed that due to \na topological structure of ME fields one obtai ns unidirectional multiresonant tunneling for \nelectromagnetic waves propagating in microwave structures with embedded quasi-2D ferrite \ndisks. The resonances manifest themselves as peaks in the scattering-matrix parameters at the stationary states of MDM oscillations in a ferr ite disk. The effect of unidirectional tunneling is \nexhibited as the Fano-resonance interferences, which are different for oppositely directed bias \nmagnetic fields. The observed multiresonant tunneling is due to generation of quantized \nvortices. The resulting quantum vacuum torque  is strong enough to provide a contactless \ntransfer of angular momentum to the helical-form metallic sample. \n    The MDM twisting excitations in a subwav elength region of microwave radiation in two-\nport structures can be well described by the formula for the scattering-matrix parameters: \n  20                                                       00 00\n12 21 21 12 &HH HHSS SS   .                                           (28) \n \nThe observed chiral fields are near fields which are fundamentally different from the fields \nradiated by usual microwave antennas with ferrite inclusions. In the last case, the radiation \ncharacteristics do not depend on a di rection of a bias field. So the S-matrix parameters of the \ntwo-port (transmitter-receiver) stru cture do not depend on a directio n of a bias field. In usual \nmicrowave antennas with ferrite inclusions we have symmetry of the S matrix: , ij jiSS i j .   \n \nRefrerences \n \n[1] F. D. 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Because of a \nmagnetic-dipole fluctuation on a lateral surfa ce of a ferrite disk the domain of the G-mode \nazimuthal angle , in a laboratory frame, is no more [0,2 ] but [0,4 ]. The MS-potential \ndistribution for the G-mode eigenfunction  is schematically shown as color regions inside a \nferrite disk. In the figure, there is a correspondence between colors used for surface magnetic \ncurrents and colors used for topological magnetic charges.   \nFig. 2. A two-layer-ring model for edge chiral magnetic currents for diffe rent time phases. When \na magnetic current of the upper (lower) layer is arriving to terminal P (where a topological \nmagnetic charge is localized), it must continue its propagation at the lower (upper) layer and this \nis only one choice. The similar si tuation takes place at terminal Q. Regions of terminals P and Q \nare the regions of singularity (the regions of topological magnetic charges). Topological \nmagnetic charges are distributed on a lateral surface of a ferrite disk. In the figure, there is a \ncorrespondence between colors used for surface magnetic currents and colors used for \ntopological magnetic charges.  \n  22Fig. 3. Edge magnetic currents ()()im\nsj\n and ()()sm\nsj\n on a lateral surface of a ferrite disk \nand their correlation with the G-mode MS-potential wave functions  of a certain MDM. The \nspin-orbit interaction is illustrated by highli ghted parts of in the graphs of the currents \n()()im\nsj\n and ()()sm\nsj\nand shaded areas on the graph of the wave functions . Every of \nthe magnetic currents, ()()im\nsj\n and ()()sm\nsj\n, is composed by two topologically \ndistinctive current components shown in Figs. 1 and 2.  \n Fig. 4. A microstrip structure with an embedded thin-film ferrite disk.  \n \nFig. 5. Experimental evidence for unidirectional multiresonant tunneling. Frequency \ncharacteristics of modules of the scattering-matrix  parameters for two opposite orientations of a \nnormal bias magnetic field \n0H\n. (a) The reflection coefficient; ( b), (c) the transmission \ncoefficients.  The resonances are classified based on analytical studies in Ref. [8]. The first \nnumber characterizes a number of radial variations for the MDM spectral solution. The second \nnumber is a number of azimuth variations.  \nFig. 6. The reflection and transmission spectra the same as in Fig. 5, but normalized to the \nbackground (when a bias magnetic field is zero) level of the microwave structure. ( a) The \nreflection coefficient; ( b), (c) the transmission coefficients. \n Fig. 7. Symmetry properties of the chiral stat es in a microwave structure with a MDM ferrite \ndisk. There is complete coincidence between the spectra of the \n21S and 12S scattering-matrix \nparameters for oppositely directed bias magn etic fields. Microwave radiation in two-port \nstructure can be described by the formul a for the scattering-matrix parameters: \n00 00\n12 21 21 12 &HH HHSS SS  . \n \nFig. 8. The numerically obtained transmission ch aracteristics for the first MDM resonance at two \nopposite orientations of a normal bias magnetic field 0H\n. (a) The 21S scattering-matrix \nparameter; ( b) the 12S scattering-matrix parameter. \n \nFig 9. A microwave structure with increased breaking of symmetry in geometry. The symmetry \nbreaking is increased by an inclined slot in one of conductive strip.  \n \nFig. 10. The experimental 21S and 12S scattering-matrix parameters of the structure shown in \nFig. 9 for two opposite orientations of a normal bias magnetic field 0H\n. (a), (b) Non-\nnormalized transmission spectra; ( c), (d)  Transmission spectra normalized to the background \nlevel of the microwave structure.   \nFig. 11. The numerically obtained transmission characteristics for a microwave structure with \nincreased breaking of symmetry in geometry. ( a) The \n21S scattering-matrix parameter; ( b) the \n12S scattering-matrix parameter. \n  23Fig. 12.  (a) A microstrip structure with a MDM ferrite disk and a wire electrode. (b) A \nmagnified picture of a ferrite disk and a wire.  \n \nFig. 13. Distributions of the fields and currents on  a wire electrode and the field structures near \na butt end of a wire electrode. ( a) Electric field on a wire electrode; ( b) surface electric chiral \ncurrent; ( c) and ( d) electric and magnetic fields near a butt end of a wire electrode; ( e) and ( f) \npower-flow density and helicity density n ear a butt end of a wire electrode.  \n \nFig. 14. (a) Two-port microwave structures with a MDM ferrite disk and a wire electrode (port 1) \nand with the right- or left-handed metallic helices (port 2). (b) A magnified picture. A wire \nconcentrator is placed near a metallic helix without an electric contact with it.  \nFig. 15. Experimental evidence for unidirectiona l multiresonance tunneling due to chiral edge \nelectric currents. Frequency characteristics of the \n21S scattering-matrix parameter for two \nopposite orientations of a normal bias magnetic field 0H\n. (a) The right- handed metallic helix; \n(b) the left-handed metallic helix. The transmission spectra is normalized to the background \n(when a bias magnetic field is zero) level of the microwave structure. The background level is \nabout 21 30 Sd B . The system has chiral symmetry: simultaneous change of the helix \nhandedness and direction of bias magnetic field remains the system symmetry unbroken.  \n \nFig. 16. Numerical results of the 21S scattering-matrix parameter for two opposite orientations of \na normal bias magnetic field 0H\n. (a) The right-handed metallic helix; ( b) the left-handed \nmetallic helix. \n \nFig. 17. The power-flow-density distributions in a vacuum region near a wire concentrator and a \nmetallic helix shown for two resonance peaks corresponding to the 1st MDM – the peaks A and \nB in the 21S frequency characteristics in Fig. 16. ( a) and ( b) the right- and left-handed metallic \nhelices at a normal bias magnetic field 0H\n directed upwards; ( c) and ( d) the right- and left-\nhanded metallic helices at a normal bias magnetic field 0H\n directed downwards. The power \ntransmission in a two-port micr owave structure is maximal when  a direction of the power-flow \nvortex at a butt end of a wire electrode corresponds to the handedness of a metallic helix. There \nis an evidence for the presence of the orbital-angular-momentum twisting excitations in a \nsubwavelength region of microwave radiation at the MDM resonances. \n Fig. 18. Distributions of the normalized helicity factor for the two resonance peaks \ncorresponding to the 1\nst MDM – the peaks A and B in the 21S frequency characteristics in Fig. \n16. ( a) and ( b) the right- and left-handed metallic helices at a normal bias magnetic field 0H\n \ndirected upwards; ( c) and ( d) the right- and left-handed metallic helices at a normal bias \nmagnetic field 0H\n directed downwards. \n \n \n--------------------------------------------------------------------------------------------------------------------- \n  \n \n   24 \n \n \n \n \n  \n \n \n \n  \n \n \n \n \n \n                                             0t                   t                  2t  \n                                             0                                       2  \n                                             0                     2                   4  \n \nFig. 1. Surface magnetic charges and edge chiral magnetic currents (a view on the upper plane of \na ferrite disk). At the time phase variation from 0t to t, an edge magnetic current \nacquires the phase of  while the G-mode regular-coordinate angle is 2 . Because of a \nmagnetic-dipole fluctuation on a lateral surface of a ferrite disk the domain of the G-mode \nazimuthal angle , in a laboratory frame, is no more [0, 2 ]  but [0,4 ]. The MS-potential \ndistribution for the G-mode eigenfunction  is schematically shown as color regions inside a \nferrite disk. In the figure, there is a correspondence between colors used for surface magnetic \ncurrents and colors used for topological magnetic charges.  \n  25\n \n                               0t                                 t                          2t  \n \nFig. 2. A two-layer-ring model for edge chiral magnetic currents for diffe rent time phases. When \na magnetic current of the upper (lower) layer is arriving to terminal P (where a topological \nmagnetic charge is localized), it must continue its propagation at the lower (upper) layer and this \nis only one choice. The similar si tuation takes place at terminal Q. Regions of terminals P and Q \nare the regions of singularity (the regions of topological magnetic charges). Topological \nmagnetic charges are distributed on a lateral surf ace of a ferrite disk. In the figure, there is a \ncorrespondence between colors used for surface magnetic currents and colors used for \ntopological magnetic charges.  \n \n \nFig. 3. Edge magnetic currents ()()im\nsj\n\n and ()()sm\nsj\n\n on a lateral surface of a ferrite disk \nand their correlation with the G-mode MS-potential wave functions  of a certain MDM. The \nspin-orbit interaction is illustrated by highli ghted parts of in the graphs of the currents \n()()im\nsj\n and ()()sm\nsj\nand shaded areas on the graph of the wave functions . Every of \nthe magnetic currents, ()()im\nsj\n and ()()sm\nsj\n, is composed by two topologically \ndistinctive current components shown in Figs. 1 and 2.  \n \n  26\n \n \nFig. 4. A microstrip structure with an embedded thin-film ferrite disk.  \n \n \n(a) \n  \n  \n                                        ( b)                                                                       ( c) \n Fig. 5. Experimental evidence for unidirectional multiresonance tunneling. Frequency \ncharacteristics of modules of the scattering-matrix  parameters for two opposite orientations of a \nnormal bias magnetic field \n0H\n. (a) The reflection coefficient; ( b), (c) the transmission \ncoefficients.  The resonances are classified based on analytical studies in Ref. [8]. The first \nnumber characterizes a number of radial variations for the MDM spectral solution. The second number is a number of azimuth variations. \n  27\n \n(a) \n   \n                                               ( b)                                                       ( c) \n Fig. 6. The reflection and transmission spectra the same as in Fig. 5, but normalized to the background (when a bias magnetic field is zero) level of the microwave structure. ( a) The \nreflection coefficient; ( b), (c) the transmission coefficients. \n  \n    \n  \n                                          ( a)                                                                 ( b) \n Fig. 7. Symmetry properties of the chiral stat es in a microwave structure with a MDM ferrite \ndisk. There is complete coincidence between the spectra of the \n21S and 12S scattering-matrix \nparameters for oppositely directed bias magn etic fields. Microwave radiation in two-port \nstructure can be described by the formul a for the scattering-matrix parameters: \n00 00\n12 21 21 12 &HH HHSS SS  . \n  28\n     \n  \n                                             ( a)                                                                 ( b) \n \nFig. 8. The numerically obtained transmission ch aracteristics for the first MDM resonance at two \nopposite orientations of a normal bias magnetic field 0H\n. (a) The 21S scattering-matrix \nparameter; ( b) the 12S scattering-matrix parameter. \n \n \n \nFig 9. A microwave structure with increased breaking of symmetry in geometry. The symmetry \nbreaking is increased by an inclined slot in one of conductive strip.  \n \n         \n  \n                                             ( a)                                                              ( b) \n \n  29\n     \n  \n                                             ( c)                                                              ( d) \n \nFig. 10. The experimental 21S and 12S scattering-matrix parameters of the structure shown in \nFig. 9 for two opposite orientations of a normal bias magnetic field 0H\n. (a), (b) Non-\nnormalized transmission spectra; ( c), (d)  Transmission spectra normalized to the background \nlevel of the microwave structure.  \n \n   \n  \n                                             ( a)                                                              ( b) \n \nFig. 11. The numerically obtained transmission characteristics for a microwave structure with \nincreased breaking of symmetry in geometry. ( a) The 21S scattering-matrix parameter; ( b) the \n12S scattering-matrix parameter. \n   \n  \n      \n                                                 ( a)                                                                ( b) \n \nFig. 12.  ( a) A microstrip structure with a MDM ferrite disk and a wire electrode. ( b) A \nmagnified picture of a ferrite disk and a wire.   30\n      \n  \n                                                 ( a)                                               ( b) \n \n                 \n  \n                                                 ( c)                                               ( d) \n \n             \n  \n                                                 ( e)                                               ( f) \n \nFig. 13. Distributions of the fields and currents on  a wire electrode and the field structures near \na butt end of a wire electrode. ( a) Electric field on a wire electrode; ( b) surface electric chiral \ncurrent; ( c) and ( d) electric and magnetic fields near a butt end of a wire electrode; ( e) and ( f) \npower-flow density and helicity density n ear a butt end of a wire electrode.  \n \n                \n  \n                                                 ( a)                                                                ( b) \n  31Fig. 14. (a) Two-port microwave structures with a MDM ferrite disk and a wire electrode (port 1) \nand with the right- or left-handed metallic helices (port 2). (b) A magnified picture. A wire \nconcentrator is placed near a metallic helix without an electric contact with it. \n \n \n   \n  \n                                                 ( a)                                                         ( b) \n \nFig. 15. Experimental evidence for unidirectiona l multiresonance tunneling due to chiral edge \nelectric currents. Frequency characteristics of the 21S scattering-matrix parameter for two \nopposite orientations of a normal bias magnetic field 0H\n. (a) The right- handed metallic helix; \n(b) the left-handed metallic helix. The transmission spectra is normalized to the background \n(when a bias magnetic field is zero) level of the microwave structure. The background level is \nabout 21 30 Sd B . The system has chiral symmetry: simultaneous change of the helix \nhandedness and direction of bias magnetic field remains the system symmetry unbroken.  \n \n  \n  \n                                            ( a)                                                         ( b) \n \nFig. 16. Numerical results of the 21S scattering-matrix parameter for two opposite orientations of \na normal bias magnetic field 0H\n. (a) The right-handed metallic helix; ( b) the left-handed \nmetallic helix. \n  32\n      \n  \n                                            ( a)                                                                 ( b) \n \n     \n  \n                                            ( c)                                                                 ( d) \n \nFig. 17. The power-flow-density distributions in a vacuum region near a wire concentrator and a \nmetallic helix shown for two resonance peaks corresponding to the 1st MDM – the peaks A and \nB in the 21S frequency characteristics in Fig. 16. ( a) and ( b) the right- and left-handed metallic \nhelices at a normal bias magnetic field 0H\n directed upwards; ( c) and ( d) the right- and left-\nhanded metallic helices at a normal bias magnetic field 0H\n directed downwards. The power \ntransmission in a two-port micr owave structure is maximal when  a direction of the power-flow \nvortex at a butt end of a wire electrode corresponds to the handedness of a metallic helix. There is an evidence for the presence of the orbital-angular-momentum twisting excitations in a \nsubwavelength region of microwave radiation at the MDM resonances. \n \n   \n  \n                                              ( a)                                                       ( b)  33\n   \n  \n                                              ( c)                                                       ( d) \n \nFig. 18. Distributions of the normalized helicity factor for the two resonance peaks \ncorresponding to the 1st MDM – the peaks A and B in the 21S frequency characteristics in Fig. \n16. ( a) and ( b) the right- and left-handed metallic helices at a normal bias magnetic field 0H\n \ndirected upwards; ( c) and ( d) the right- and left-handed metallic helices at a normal bias \nmagnetic field 0H\n directed downwards. \n "    },    {        "title": "1903.01022v1.Structural_and_magnetic_properties_in_sputtered_iron_oxide_epitaxial_thin_films____Magnetite_Fe__3_O__4__and_epsilon_ferrite_e_Fe__2_O__3_.pdf",        "content": "1 \n Structural and magnetic properties in sputtered iron oxide epitaxial thin films  \n- Magnetite Fe 3O4 and epsilon ferrite \n -Fe2O3 - \n \nMasato Watanabe  \nResearch Institute for Electromagnetic Materials  \n9-5-1 Narita, Tomiya 981 -3341  Japan  \nPhone:   +81-22-341-6343  \nFax:  +81-22-347-3789  \nE-mail.:   m_watanabe@denjiken.ne.jp , masato33@innovamateria.jp  \n  2 \n Abstract  \nEpitaxial thin film  fabrication of i ron oxide s including magnetite Fe3O4 and epsilon -\nferrite  \n-Fe2O3 with the po tential  for advancing electromagnetic devices has been \ninvestigated , which led to the first ever \n -ferrite epitaxial layer be ing synthesized in the \nconventional sputtering  process. Concerning  Fe3O4 (100) / MgO  (100) films , a cube -on-\ncube epitaxial relationship  and sharp rocking curves with FWHM of 50  - 350 arcsec were \nconfirmed  regardless of the small amount of Ge additions . Sputte ring Ar gas pressure \nPAr heavily  influence d their  magnetic and transport properties. High P Ar = 15 mTorr \ncause d a high magnetization of 6.52 kG  for the Ge added sample and the clear Verwey \ntransition at 122  K for the no n Ge addition case. Conversion electron M össbauer \nspectroscopy ( CEMS ) measurements revealed that low P Ar < 10 mTorr causes Fe/O off -\nstoichiometry on the oxidizing side for the no n Ge addition case and the reductive side \nfor the Ge addition case, respectively.  Regarding the \n-Fe2O3 (001) / SrTiO 3(111) epilayer \nsynthesis , bilayer microstructure composed of an approximatel y 5nm thick  initially \ngrown  \n-Fe2O3 (001) epilayer and subsequently grown \n -Fe2O3 (001) epilayer was \nconfirmed from cross -sectional TEM observations. The c oexistence of magnetically hard \nand soft phases was confirmed  from the magnetization measurements. As a possible \napplication of the single nm thick \n -Fe2O3 layer, 4-resistive -state multiferroic tunnel \njunction (MFTJ) is considered.  \n  3 \n Introduction  \nMagnetic iron oxides , so-called  \"ferrites \", have been utilized  for various  electro magnetic \napplications due to their versatile  magnetic properties [1, 2] and large natural \nabundan ce of main constitu ent iron and oxygen , which meets recent socia l requirements \nfor materials  such as  rare-metal -free ubiquit y. Among the huge v ariety  of ferrites, we \nfocused on two iron oxides: magnetite  Fe3O4 and epsilon ferrite \n -Fe2O3, both of which \nshow unique magnetic and electronic  functionalities , and attempted to fabricate their \nepitaxially grown thin films by conventional sputtering, which is an advantageous  \nprocess for industrial device applications.  \nMagnetite  of the first iron oxide is an ubiquitous magnetic material naturally found as \nthe main component of iron sand  [3]. Its crystal structure is an inverse spinel that is \ncomposed of Fe3+ at tetra hedral  A site s, Fe2+ and Fe3+ at octahedral B site s and oxygen \nsites . It has a  room temperature saturation magnetization 4\nM of 6.25 kG, which is the \nhighest among iron oxides , and Curie temperature T C of 858 K [4]. Magnetite  also show s \ncharacteristic electronic properties of half -metallicity [ 5-8] and large  anomalous Hall \nresistivity H ~10 - 40 μΩcm [9], which is comparable with Co based full Heusler \ncompounds [ 10], leading to the possibility for various spintronic devices  such as magnetic \ntunnel junction s (MTJ) . Due to  its high biocompatibility, magnetite 's biomedical \napplications such as hyperthermia and drug delivery system (DDS) have also been \npursued [ 11]. To date, o ur research group  conducted  research on small -amount element \naddition s in polycrystalline magnetite  films and found that the addition s of some \nelements  up to several percent, especially Ge, raise d their thermal stability  and caused \nan increase in magnetization [12, 13]. Since the effects of such element addition s in \nmagnetite  epifilms ha ve not yet been confirmed , we studied  the structural and magnetic 4 \n effect s on small -amount  Ge addition to magnetite  epifilms in this research.  \nEpsilon ferrite of the second iron oxide is an emerging hard magnetic material , which  \nhas an orthorhombic crystal structure with four iron sites for which magnetic moments \nare ferrimagnetically aligned [ 14]. From its high anisotropy field H A that comes from the \nhigh magnetic anisotropy constant s of Ka = 7.7 x 106 erg/cc along the a axis and K b = 1.2 \nx 106 erg/cc along the b axis [15] and low magnetization value 4\nMS = 1.26  kG (= 100 \nemu/cm3) [16], \n-ferrite shows high resonance frequencies f r of 35 -222 GHz [ 17, 18] and \nhigh coercivit ies over 20 kOe  [17, 19] , leading to the possibility of milli -wave absorption \ndevice  and magnetic recording media  applications . It also shows a rare room -\ntemperature multiferroicity  that means simultaneous manifestation of ferromagnetism \nand ferroelectricity  [16], which may have memory or sen sor applications  as indicated in \nthe last part  of this paper . Its synthesis  is difficult due to the characteristic s of the \nmetastable phase.  Two preparation methods available have been proposed for \nmetastable \n -ferrite  so far. One is based on Gibb 's energy  minimum of \n-ferrite  \nnanoparticles at the particle diameters of 8-30 nm [17], leading to the availability of \nchemical synthes es of \n -ferrite  nanoparticles . The other  is based on the constraint of \ncrystal lattices acco mpanied by epigrowth on certain single -crystalline substrate s, which \nhas mainly been conducted  for \n-ferrite epifilms by pulsed laser deposition (PLD).  A few \nsingle -crystalline substrates including SrTiO 3(111)  [16, 20], YSZ(100)  [21] and \nGaN(0001)  [22, 23] have been reported for the syntheses of \n -ferrite epifilms by PLD. \nHowever, epifilm synthesis by conventional sputtering, which is importan t in industrial \napplications, has to the best of my knowledge not yet been reported . Therefore,  we \nattempted \n -ferrite epifilm synthesis by RF magnetron sputtering.  \n 5 \n Experimental  \nSample preparation s for both magnetite  and \n -ferrite epifilms  were  done by RF \nmagnetron sputtering . The s puttering target  for the magnetite  epilayer was a magnetite  \ncomposite target on which Ge chips were regularly -arrayed  and that for the \n -ferrite  \nepilayer was an \n-Fe2O3 target , respectively . Single -crystalline substrates for epigrowth \nwere polished MgO(100) (Tateho  Chemical Industries) for magnetite  epilayer s and \npolished SrTiO 3(111) (Furuuchi Chemical Corp.) for \n -ferrite  epilayer s, both of which \nwere heated up to  800°C during deposition after evacuati ng at less than  the vacuum \ndegree of 2 x 10-7 Torr. Sputtering g as species were pure Ar gas for magnetite  epigrowth \nand Ar + O2 mixed gas with the O2 / (Ar +O 2) flow ratio ranging from 25% to 50%  for \n-\nferrite  epigrowth , respectively . Sputtering gas pressures , which strongly  affect film \ncharacteristics,  were varied from 2 to 15 mTorr for magnetite  epigrowth , and from 1 to 5  \nmTorr for \n -ferrite  epigrowth , respectively . The film thicknesses for magnetite  epilayers \nrang ed from 0.481  m to 0.548  m, which were measured by DekTak 150 (Bruker ). Ge \ncompositions in magnetite  epilayers were evaluated by wavelength dispersive X -ray \nspectroscopy WDS (INCA  WAVE , Oxford Instruments). Structural analyses were \nconducted using a high-precision X-ray diffractometer with a CuK  radiation , LYNXEYE \n1-dimension al detector and Ge(220) monochromator  (D8 D ISCOVER , Bruker). The film \nthicknesses of \n /\n-Fe2O3 bilayer s were evaluated by X -ray reflectometry. Magnetization \nmeasurements were performed  by a superconducting quantum interference device \nSQUID  (MPMS3, Quantum Design) and vibrating sample magnetometer VSM \n(Tamakawa) with maximum applied fields of 70 kOe and 15 kOe, respectively. \nDiamagnetic components that come from substrate s and sample holder s were subt racted \nfrom the raw magnetization data.  Conversion electron Mössbauer spectroscopy (CEMS ) 6 \n measurement was performed with a 57Co radiation embedded in a Rh matrix, constant -\nacceleration spectrometer and He -1% (CH 3)3CH gas -flow counter. Velocity correction was \nconducted  using \n -Fe. No 57Fe was doped in the sample s for CEMS . All the \nmeasurements were performed  at room temperature.  Resistivity measurements were \nperformed by the 4-point method using PPMS (Quantum Design)  in the temperature \nrange of 10  - 300K. Cross-sectional high-angle annular dark -field scanning transmission \nelectron microscopy (HAADF -STEM ) observation (Z-contrast image) and nano beam \nelectron diffraction (NBD) for the \n/\n-Fe2O3 bilayer  were conducted  by JEM -ARM200F  \n(200 kV, JEOL) . \n \nResults and discussions for Fe 3O4 epigrowth  \nA typical θ/2θ scan XRD profile for the 0.6 at%Ge -Fe3O4 / MgO(100) epifilm is shown in \nFig. 1. Sharp Fe3O4 (400) and Fe 3O4 (800) peaks were clearly observed close to the \nMgO(200) and MgO(400) peaks and no other phase was confirmed, which shows c -axis \nalignment of the magnetite  phase in the direction normal to the film plane.  In order to \nstudy the effect of sputtering gas pressure P Ar on crystallinity of the magnetite  epilayers, \nfull-widths at half maximum (FWHM) of Fe3O4 (400) rocking curves with varying P Ar are \nshown in Fig.  2 in comparison with FWHM of MgO(200). FWHM of the magnetite  \nepilayers was relatively low rang ing from 50 to 3 50 arcsec, which is the same order of \nmagnitude as those of compound semiconductor epi films  reported so far . The FWHM of \nmagnetite  epilayers for P Ar ≤ 7 mTorr w as lower than those for P Ar ≥ 10 mTorr , indicating \nthat magnetite  epilayers with lower P Ar ≤ 7 mTorr have higher crystallinity than the \nhigher PAr ≥ 10 mTorr  cases,  and shows almost the same crystallinity as those of \nMgO(200). Conventional θ/2θ scan XRD gives only structural information along the out-7 \n of-plane direction,  and therefore we conducted  Fe3O4(311) \n  scan XRD that contains its \nin-plane information , as shown in Fig s.3 (a)  and (b) . Different from Fe 3O4 (400), \nFe3O4(311) is appropriate for \n  scan XRD since it is sufficiently  separated from the MgO \nsubstrate spots in its reciprocal lattice space. Figure 3 (c) shows a simulated Fe3O4(311) \npole figure with the \n  = 0° direction along Fe 3O4 [100], of which the inner four spots and \nouter eight spots correspond to the four peaks in Fig. 3 (a) and the eight peaks in Fig.3 \n(b), respectively. Reproduction of the simulated pole figure is considered to show the \nfollowing c ube-on-cube orientation  relationships: Fe3O4[100] // MgO[100] , Fe3O4[010] // \nMgO[010]  and Fe3O4[001] // MgO[001] . \nWe have studied room -temperature magnetic properties for the Fe 3O4(100)  / MgO(100) \nepifilms  varying with Ar sputter gas pressure P Ar. Magnetization curves for P Ar = 15 \nmTorr and P Ar = 2 mTorr with the maximum field of 7 0 kOe  are shown in Fig s. 4 (a) and \n(b). Out -of-plane magnetizations for both PAr = 15 mTorr and 2  mTorr exceed the in-plane \nmagnetizations  at high er fields  over 10 kOe , which suggests  the existence of some \nperpendicular magnetic anisotropy. This large perpendicular anisotropy has been \nreported for magnetite  epifilms and polycrystalline films [ 24, 25] so far and its possible \norigin is attributed to out -of-plane moment alignment around antiphase boundaries \n(APB), which was confirmed by off-axis electron holography [ 26]. Figure 4 (c) summarizes \nout-of-plane magnetization s 4\nM with the maximum applied fields of 15 kOe and 70 kOe \nfor the Ge -doped and non -doped magnetite  epifilms  as a function of P Ar. For both the Ge-\ndoped and non -doped cases, magnetizations 4\n M increase with the increas e in P Ar. \nMoreover, the out -of-plane 4 M for PAr = 15 mTorr is 6.52 kG, which exceed s the 6.25 kG  \nfor single crystal  magnetite,  at the applied field of 70 kOe a nd still does not saturate at \n70 kOe  as shown in Fig s.4 (a) and (b) . Magnetization enhancement  over 12 kG, which is 8 \n much larger than the bulk value of 6 .25 kG, has bee n reported for 3nm thick  magnetite  \nepifilm , the possible origin of which has been attributed to  spin flip from ferrimagnetic \nmoment alignment to ferromagnetic type [4]. The same type of enhancement might occur  \npartially for t he epifilms as well. In this research, we grew magnetite  epifilms at a high \nsubstrate temperature of 800°C and magnesium atoms easily diffuse into the magnetite  \nlayer [ 27, 28 ], leading to the possible formation of an inter diffused Mg-Fe3O4 layer. Thus, \nwe conducted cross -secti onal TEM observation and composition analysis along the out-\nof-plane direction , which confirmed the existence of a ~100  nm thick Mg-Fe3O4 \ninter diffused layer in the magnetite epifilm with  a total thickness of ~ 500 nm. Mg ferrite \nMgFe 2O4 has 4\n M of 1.149 kG [29], which is much lower than that of the 6.25 kG for \nmagnetite , and so the interdiffused layer is considered  to cause a magnetization  \nreduction . Since  a slight magnetization enhancement was observed for the magnetite  \nepilayer with P Ar = 15 mTorr as de scribed above  despite the existence of the interdiffused \nlayer , the magnetization enhancement at the undiffused part of magnetite  epilayer is \nconsidered to be large enough to compensate  the magnetization reduction at the Mg \ninterdiffused part.  \nIn order to study the Verwey transition , which  is sensitive to the film quality  and \nstoichiometry [ 30], we performed  resistance measurements of the (a) Ge-doped and (b) \nnon-doped magnetite  epifilms with P Ar = 10 and 15  mTorr as a function of the inverse \nnumber of temperature 1000/T as shown in Fig.  5. The non -doped magnetite  epifilm s \nshow the clear resistivity change accompanied with the Verwey transition. The \ntransition temperature T V for PAr = 15 mTorr is 122 K, almost the same as the previously \nreported value of  bulk magnetite . On the contrary, resistivity changes for the Ge -doped \nsamples are diminish ed and their T V's decrease until <  100 K. Since the Ge addition also 9 \n increases the resistivity, the added Ge might be incorporated into B site s with two types  \nof valency and inhibit the hopping conduction.  Regardless of the Ge concentrations, w e \ncould not observe clear resistivity change s of the Verwey transition for the samples with \nPAr < 10 mTorr, the magnetizations of which are low as shown in Fig. 4(c).  \nFor further investigation of magnetism in the magnetite  epifilms, we performed  CEMS \nmeasurements. CEMS spectra for the Ge -added magnetite  epifilms are shown in Fig.6 \n(a) P Ar = 15 mTorr (0.4 at%Ge) and (b) P Ar = 2 mTorr  (1.6 at%Ge) . The observed spectra \ncan be  decomposed into  two sharp ferromagnetic sextet s that come from tetra hedral  A \n(Fe3+) and octahedral B (Fe2.5+) sites. Hyperfine structur e parameters , BHf, \nMB integral \nintensities for A and B sites and sextet intensity ratios  x for the Ge -added magnetite  \nepifilms are summarized in Table 1 in comparison with those of previously reported \nsputtered magnetite  epifilms and natural bulk magnetite  [31, 32 ]. The parame ters for \nthe PAr = 15 mTorr sample are in good agree ment  with  those of the previously reported \nfilm and bulk  magnetite . BHf and integral intensities for the PAr = 2 mTorr sample , which \nhas lower magnetization 4\n M than the PAr = 15  mTorr sample, deviate from the \npreviously reported values . Sextet intensity ratio  x contain s information about moment \ndirection according to the following Eq.: 𝑥= 4sin2𝜃(1+cos2𝜃) ⁄ , where θ stands for the \nangle between 57Fe magnetic moment and the incident direction of γ-ray for CEMS. \nStates for x = 0 and x  = 4 mean moment alignments along the out-of-plane and the in-\nplane directions , respectively.  State  for x = 2 means  randomly -oriented or <111> oriented \nmoments . From x < 2 for both the PAr = 15 mTorr and 2  mTorr samples, the samples show \nsome perpendicular magnetic anisotropy, which coincides  with the feature of \nmagnetization curves in Fig. 4 (a) and (b).  Integral intensit y ratios  for A and B sites in \nCEMS contain informatio n about iron vacancy in magnetite . Vacancy parameter \n  in 10 \n 𝐹𝑒3−𝛿𝑂4 has the following relation with the ratio of integral intensity for A site to that \nfor B site  β : 𝛽= 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙  𝐼𝑛𝑡.(𝐴 𝑠𝑖𝑡𝑒 )𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙  𝐼𝑛𝑡.(𝐵 𝑠𝑖𝑡𝑒 ) ⁄ = (1+5𝛿)(2−6𝛿) ⁄  [33]. \n \n= 0 and \n  = 1/3 correspond to stoichiometric magnetite  and stoichiometric γ-Fe2O3, \nrespectively. If we take account of the effect of the recoilless fraction , a slight correction \nto the oxidizing side from β = 0.5 (\n = 0) to β = 0.52 (\n = 4.9 x 10-3) is required for \nstoichiometric magnetite  [34]. Figure s 7 (a) and (b) show \n  and 4\nM at 15  kOe for Ge -\ndoped  and non -doped MNT epifilms as a function of P Ar. With the increas e in PAr, \n's \napproach zero for both the Ge-doped and non -doped samples , which correspond s to the \nincrease in 4\nM. Therefore,  the 4\nM reduction with the decreas e in P Ar is considered to \nbe due to  the composition deviation from the stoichiometry  of magnetite . The v anish ing \nof resistivity change at the Verwey transition with the decreas e in P Ar described in the \nprevious part of Fig. 5 is considered to be the same  as well , and so Fe/O composition \ndeviation is considered to significantly influence the magnetic and transport properties \nof magnetite . \n \nResults and discussions for \n /\n-Fe2O3 bilayer epigrowth  \nResearch on epigrowth of \n -ferrite so far has been mainly focused on  the PLD process \nand the conditions necessary for its epigrowth  have been reported to be  a high substrate \ntemperature and high oxygen content atmosphere during deposition [ 16, 20-23]. Thus, \nwe conducted sputtered ep igrowth deposition at the high substrate temperature of 800ºC \nwith a varying oxygen flow rate ratio Q(O2)/Q(Ar+O 2). For the no oxygen atmospheric \ncondition, the Fe3O4(111) epilayer was grown despite the use of the Fe2O3 sputter target  \nand found that \n -ferrite epi layer s can be grown in a high oxygen content atmosphere \nsuch as the PLD cases  as shown below . This study is the first to report of \n -ferrite epilayer 11 \n growth by conventional sputtering process . XRD profiles for the Fe 2O3 epifilms with \nQ(O 2)/Q(Ar+ O2) = 25  % and different s putter gas pressure s are  shown in Figs.8 (a), (b) \nand (c). A c-axis oriented \n -ferrite epilayer was confirmed only f or the lowest gas pressure \nof a 1 mTorr sample  and the higher gas pressure conditions of 2 and 5  mTorr  create \n -\nphase epilayers. Low sputter gas pressure cause s long mean free paths of sputter \nparticles, leading to high kinetic energies , less collisions and low sputter particle \nincorporation into the films , which may influence  the different phase formation  in the \nepilayers . The prepared film samples on 10  mm square SrTiO 3(111) substrates were \ncomposed of two different parts of the dark -colored and light -colored areas as shown in \nthe optical images of Fig. 9. From XRD  with narrowed X -ray beam s, the dark -colored \nand light -colored areas are composed of the c-axis oriented \n -phase and \n -phase, \nrespectively. X-ray reflect ometry  with collimated beams show s that t he thicknesses of \nthe dark -colored and light -colored areas are  29 nm and 4  nm, respectively.  \nIn order to clarify the microstructure of  the epilaye r, we conducted cross -sectional TEM \nobservations on the dark -colored area of the epifilm with Q(O 2)/Q(Ar+O 2) = 40  % and \nsputter gas pressure P(Ar+O 2)= 1  mTorr . Figure 10 shows cross -sectional TEM \nobservation and NBD  in the epifilm. It was revealed that the epifilm was composed of \ntwo phase construction where one is \n  phase of the initial epilayer on the substrate with \n~ 5 nm thick and the other  is \n-phase of the capped  epilayer . From the NBD  of Fig . 10 \n(b), both the \n - and \n -phase  epilayers were aligned with [001] along the out -of-plane \ndirection and [ -210] along the in -plane direction . The initial \n -phase epilayer is \ncontinuous in this image, however, discontinuous areas were also found to exi st from \nlarger area TEM observations.  Since a clear lattice image was confirmed in Fig.10, we \nalso conducted HAADF -STEM observation with higher magnification as shown in Fig.  12 \n 11. Within the scope of our TEM observation, no  crystalline boundaries were confirmed \nin the \n -phase epilayer though so me disorder of atomic arrangement was observed, which \nis different from the case of the previously reported -phase epilayer prepared by PLD. \nIt has been reported that PLD can create a thicker c -axis oriented \n -phase epilayer up to \nabout 100nm thick and the PLD epilayer is composed of nano -sized crystal domains with \nthe in-plane crystal orientation of six -fold symmetry [ 20]. \n-ferrite has been mainly \nstudied within the form of nanoparticles and its phase formati on is explained by free \nenergy minimum at the diameters of 8-30nm [ 17]. However, other forms of epitaxially \ngrown \n -ferrite have also been reported such as nanowire or nanobelt several \nmicrometer s in length by PLD [ 35, 36] and peculiar dendritic microstruc tures of mullite \non the scale of several micrometer s in Japanese stoneware, Bizen -yaki [ 37, 38]. Therefore,  \nfurther study is considered to be necessary regarding the generation mechanism of \n -\nferrite.  \nRoom-temperature magnetic properties of the two phase \n /\n-Fe2O3 epilayer with \ndifferent total thicknesses were investigated as shown in Fig.  11. The hysteresis cu rves \nshow that the sample was composed of decoupled magnetically soft and hard phases. The \nhard phase comes from original  hard  \n-ferrite and the soft phase  comes from \n -ferrite in \nwhich magnetic anisotropy was diminished by some imperfections of crystallin ity since \nno other magnetic phase could be confirmed.  The magnetization values obtained by \nassuming 5 nm thick \n -ferrite epilayer s had mostly good agreement  with the reported \nvalue of 1.26  kG in \n-ferrite by the same order of magnitude , which is consistent with the \nabove TEM observation result s. \nAs a possible application of such single nm-scale \n -ferrite epilayer, it may be applied to \nthe multiferroic barrier in multiferroic tunnel junction (MFTJ) . Spin filter MFTJ with 13 \n half-metallic an d multiferroic perovskite oxides was firstly reported by a French group \nin 2007 as a prototype of the 4-resistive -state memory that utilizes both tunnel \nmagnetoresistance (TMR) and tunnel electroresistance (TER) effects [ 39, 40]. 4-resistive \nstates in MFTJ  correspond to the configurations of magnetic moment M and electric \npolarization P as shown in Fig. 12. Because the high-temperature phase of magnetite  \nand \n -ferrite show s half-metallicity and multiferroicity  at room temperature , respectively, \nwe anticipate d that all ferrite MFTJ would work at room -temperature if it were  \ncomposed of a magnetite  electrode and \n -ferrite barrier as shown in Fig. 12. In order to  \nsecure the independent switching function of 4 -state memory, ME coupling  between M \nand P in the multiferroic barrier should  not be strong  and so the clarification of  \nmagnitude  of ME coupling in \n-ferrite is required . A study on optical ME coupling in \n-\nferrite utilizing  its second harmonic generation SHG effect showed that it has a strong \nME coupling  from the magnetic component of SHG  [15], thus it might be necessary to \ncontrol the magnitude of coupling by measures such as some element addition s in order \nto realize 4-resistiv e-state magnetite  / \n-ferrite MFTJ.  \n \nSummary  \nEpitaxial growth of magnetite  and \n -ferrite was conducted  by conventional RF \nmagnetron sputtering  on MgO(100) and SrTiO 3(111), respectively . Fe3O4 / MgO(100) \nepifilms  have a cube -on-cube epita xial relationship and a high crystallinity was \nconfirmed from the sharp rocking curves with FWHM of 50 - 350 arcsec  for both the Ge-\nadded and no n Ge addition cases . Sputtering Ar gas pressure P Ar significantly influence d \ntheir mag netic and transport properties.  The magnetite epifilm for  PAr = 15 mTorr had a \nhigh magnetization of 6.52 kG for the Ge added case and the clear Verwey transition at 14 \n 122 K for the no n Ge addition case. CEMS  measurements revealed that low P Ar < 10 \nmTorr causes Fe/O off -stoichiometry on the oxidizing side for the no n Ge addition case \nand the reductive side for the Ge addition case, respectively.  \nRegarding \n -Fe2O3 epilayer synthesis, bilayer microstructure comp osed of an \napproximately  5nm thick \n -Fe2O3 (001) epilayer initially grown on SrTiO 3(111) and a \nsubsequently grown \n -Fe2O3 (001) epilayer was confirmed from cross -sectional TEM \nobservations. The c oexistence of magnetically hard and soft phases was confirmed  from \nthe magnetization measurements. As a possible application of the single nm thick \n -\nFe2O3 layer, 4 -resistive -state multiferroic tunnel junction (MFTJ) could be considered.  \n  15 \n Acknowledgement s \nThis article was supported by JSPS KAKENHI  Grant number 17K06806, \"Magnetic \nanisotropy control by coherency strain in element -added magnetite thin films\"  and \npartially presented at 2nd International Conference on Magnetism and Magnetic \nMaterials, Budapest, Hungary in September 24 -26, 2018 (Allied Academies) . I thank  Dr. \nS. Abe, President K.  Arai of Research Institute for Electromagnetic Materials, Prof. S.  \nSugimoto and Prof. K.  Takanashi of Tohoku Univ. for their beneficial discussions, and \nalso thank Dr. M.  Ikeda of Quantum Design  Japan , Ms. A.  Tsutsui of Foundation for \nPromotion of Material Science and Technology of Japan  (MST), Dr. T.  Segi of KOBELCO \nResearch Institute and Mr. H.  Sato of Research Institute for Electromagnetic Materials \nfor their assistance  with and discussion s on the characteriza tions.  \n  16 \n Table 1. Hyperfine structur e parameters  obtained from CEMS for Fe3O4/MgO(100) \nepifilms with P Ar = 15  mTorr (0.4  at%Ge) and P Ar = 2 mTorr (1.6  at%Ge ). Previou sly \nreported data for sputtered Fe 3O4 epifilm s [31] and natural bulk Fe 3O4 [32] are also \nshown for comparison.  \n  \n17 \n  \n \nFig. 1 . θ/2θ scan XRD profiles for the 0.6 at%Ge added Fe 3O4 epifilms grown on MgO \n(100) . Magnifi ed profile around MgO(200) and Fe 3O4(400)  peaks is shown in the inset . \n \n \nFig. 2 . Full widths at half maximum (FWHM) of Fe 3O4(400) and MgO(200) rocking \ncurves as a function of sputtering Ar pressure P Ar for the Fe 3O4 / MgO(100) epifilms. The \ninset shows MgO(200) and Fe 3O4(400) rocking curve profiles for the epifilm at P Ar = \n5mTorr.  \n \n18 \n  \n \nFig. 3 . Fe3O4 (311) φ scan profiles for the Fe 3O4/MgO(100) epifilms at (a) ψ = 25.2393  deg \nand (b) ψ = 72.4512  deg, which correspond to the four inner and eight outer spots in (c) \nsimulated Fe 3O4 (311) pole figure.  \n \n  \n19 \n  \n \nFig. 4. Room -temperature magnetization hysteresis curves for the Fe 3O4(100)/MgO(100) \nepifilms with no n Ge addi tion at (a) P Ar = 15 mTorr and (b) P Ar = 2 mTorr.  The r ed and \nblack lines stand for out -of-plane and in -plane curves, respectively. (c) Out -of-plane \nmagnetization 4 πM at 15  kOe and 70 kOe  as a function of sputtering Ar gas pressure  \nPAr. Red blank square s and black blank circle s stand for Ge added and no n Ge ad dition \ncases  at 15 kOe . Solid marks are 4 πM at 70 kOe.  \n \n  \n20 \n  \n \nFig. 5. Resistance  as a function of 1000/T for (a) the Ge -added and (b) no n Ge added \nFe3O4(100)/MgO(100) epifilms with sputtering Ar gas pressure PAr = 15 and 10 mTorr.  \n \n  \n21 \n  \n \nFig.6. Converson electron Mossbauer spectroscopy (CEMS ) spectra for the \nFe3O4(100)/MgO(100) epifilms with (a) P Ar = 15 mTorr (0.4  at%Ge) and (b) P Ar = 2 mTorr \n(1.6 at%Ge). Measured CEMS data (black dots  and fitted red lines ) are decomposed into \nA site (green lines ) and B site (blue lines) components.  \n \n  \n22 \n  \n \nFig. 7. (a) Vacancy parameter δ (solid red circles for Ge added and blue circles for no n Ge \naddition cases) and Ge concentration (blank red circles) , and (b) room -temperature \nmagnetization 4 πM at 15  kOe for the Fe 3O4(100)/MgO(100) epifilms as a function of \nsputtering Ar pressure P Ar. The ideal δ = +4.9 x 10-3 for stoichiometric Fe3O4 taking its \nrecoilless fraction into account is indicated with a red dotted  line. \n \n  \n23 \n  \n \nFig. 8. θ/2θ scan XRD profiles for the α- and ε-Fe2O3 epifilms grown on SrTiO 3 (111) with \nsputtering Ar+25%O 2 mixed gas pressure P(Ar+25%O 2) = (a) 1  mTorr, (b) 2  mTorr and \n(c) 5 mTorr.  \n \n \n  \n24 \n  \n \n \n \nFig. 9. θ/2θ scan XRD profiles for the α- and ε-Fe2O3 / SrTiO 3 (111)  epifilm  (P(Ar+ 40%O 2) \n= 1mTorr ) in (a) the light -colored area (4nm  thick ) and (b) the dark -colored area (29nm  \nthick ) in the right side  sample images ( areas surrounded by red lines ). \n  \n25 \n  \n \nFig. 10 . Cross-sectional TEM image and nano beam electron diffraction (NBD, point A \nand B in the TEM image) for the α- and ε-Fe2O3 / SrTiO 3 (111)  epifilm ( P(Ar+ 40%O 2) = \n1mTorr ) in the dark -colored area shown in Fig.  9. \n \n  \n26 \n  \n \nFig. 11. Cross -sectional HAADF -STEM (Z-contrast) image for the α/ε-Fe2O3 / SrTiO 3 (111)  \nepifilm ( P(Ar+ 40%O 2) = 1mTorr ) in the dark -colored area shown in Fig.  9. \n \n \n  \n27 \n  \n \nFig. 12 . Room-temperature in-plane magnetization curves for the α- and ε-Fe2O3 / SrTiO 3 \n(111)  epifilm ( P(Ar+ 40%O 2) = 1mTorr ) with total thicknesses = (a) 121  nm and (b) 29  nm. \nMagnetizations 4πM were evalua ted assum ing 5nm thickness of the ε-Fe2O3 initial \nepilayer.  \n  \n28 \n  \nFig.13 . Schem a of 4-resistive -state multiferroic tunnel junction ( MFTJ)  utilizing the spin \nfilter effect in multiferroic barrier such as ε-Fe2O3 thin layer and tunnel electroresistance \n(TER) effect.  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Magnetostatic \noscillations in a thin ferrite disk are characterized by unique \ntopological properties: the Poynting -vector vortices and the field \nhelicity.  The chiral -topology near fields allow obtaining unique \nphase structure distribution  for far -field microwave radiation.   \nKeywords —antennas ; ferrites;  magnetostatic resonances; \nsubwavelength energy localization, chiral fields   \nI. INTRODUCTION  \nIn the search of novel communication systems, a strong \ninterest arises in subwavelength structures with quasistatic \noscillations as basi c building blocks for controlling far-field \nelectromagnetic radiation.  Quasistatic resonances in small \nobjects allow engineering of novel near-field structures with \nstrong subwavelength localization of electromagnetic energy. \nThese fields are distinguished  by topological characteristics. \nAmong them, there is a very unique property of the field \nchirality.  In optics, the near field chirality was observed in \nmetallic structures with plasmonic (electrostatic) resonances \n[1]. Recently, chiral near fields were ob tained in microwaves \nbased on small ferrite particles with magnetostatic resonances \n[2 – 6].  \nIn this paper we show that microwave chiral fields \noriginated from a small ferrite disk with magnetostatic \noscillations can f orm far -field radiation pattern with a strong \nand controllable squint. Such  an effect of  a spatial mode \ndivision  by a single radiation element is  unique. It is  very \nattractive for development of novel microwave antennas with \ncontrollable  far-field phase structure distributions. It is well \nknown that electromagnetic  fields can carry not only energy \nbut also angular momentum . The angular  momentum  is \ncomposed  of spin angular momentum (SAM) and orbital \nangular  momentum (OAM) describing its polarization state and \nthe phase structure distribution, r espectively.  The research on \nOAM  was not attractive  until Allen et al . investigated the \nmechanism of OAM in 1992  [7]. Henceforth, more and more \nattention has been paid to  OAM in both optical and radio \ndomains.  In contrast to SAM, which has only two possibl e \nstates  of left -handed and right -handed circular polarizations, the theoretical states of OAM are unlimited owing to its unique  \ncharacteristics of spiral f low of electromagnetic energy [8 ]. \nTherefore, OAM has the potential to  tremendously increase the \nspectral efficiency and capacity of  communication systems  [9]. \nNumerous experiments on OAM,  originally in optical \nfrequency and then in radio frequency,  have been carried out.   \nThe concept of OAM in radio frequency is relatively novel. \nFor generating OAM in r adio frequency a circular antenna \narray is used [ 10 – 12]. In this paper, we show, for the first \ntime, that the far-field phase structure distribution  for \nmicrowave radiation can be obtained by a single radiation \nelement with chiral -topology near fields.  \nII. MAGNETOSTATIC OSCILLA TIONS IN SMALL FERRI TE \nDISKS  \nPhysical justification of multiresonance magnetostatic  \n(MS) oscillations in microwave structures [] is based on the \nfact that in a small sample of a medium with strong temporal \ndispersion of the magnetic su sceptibility, variation of electric \nenergy is negligibly small and so the electric displacement \ncurrent is negligibly small as well. In an analysis of such \nstructures we should use three differential equations instead of \nthe four -Maxwell -equation analysis of electromagnetic fields \n[13]:  \n0B \n   (1) \n \n0 H \n  (2) \n \n1BE\nct  \n\n                (3) \nTaking into account a constitutive relation:  \n \n4 B H m \n  (4) \n \nwhere m is the magnetization, one obtai ns from ( 1): \n \n4 Hm    \n (5) \n \nThis presumes an introduction of MS -potential wave functions \n(r, t) for description of a magnetic field:   \nH \n (6) \n \nThe spectral problem is formulated for MS -potential wave \nfunctions (r, t), where a magnetization field is expressed as  \n \n    \nm  \n  \nHere \n\nis the susceptibility tensor. Formally, in a system \nof (1) – (3), a potential magnetic field and a curl electric field \nshould be considered as completely uncoupled fields. It turns \nout, however, that the magnetic and electric fields in ( 1) – (3) \ncan be united . It was found that in a case of a quasi -2D ferrite \ndisk, the spectral -problem solution for MS oscillations shows \nthe presence of the unified (electric and magnetic) field \nstructure which is  different from the Maxwell -electrodynamics \nunified -field structure . We term the fields originated from the \nMS oscillations as magnetoelectric (ME)  fields to distinguish \nthem from regular electromagnetic  (EM)  fields [ 5]. MS \noscillations in a quasi -2D ferrite disk are mesoscopically \nquantized states. Long range dipole -dipole correlation in \nposition of electron spins in a ferromagnetic sample can be \ntreated in terms of collective excitations of the system as a \nwhole.  The spectral solutions for the MS -potential wave \nfunction  \n( , )rt\n , has evident quantum -like attributes  [2]. \nThe incident EM wave has strong coupling with MS \nresonances of the ferrite disk and enable  us to confine \nmicrowave radiation energy in subwavelength scales.  In a \nvacuum subwavelength region abutting to a MS ferrite disk  one \ncan observe the quantized -state power -flow vortices [ 3 – 6]. \nThe ME-field solutions give evidence for spontaneous \nsymmetry breakings at the resonant states of MS oscillations. \nBecause of rotations of localized field configurations in a fixed \nobserver inertial frame, the linking between the EM and ME \nfields cause violation of the Lorentz  symmetry of spacetime. In \nsuch a  sense, ME fields can be considered as Lorentz -violating \nextension of the Maxwell equations  [5, 6].  \nTo characterize the ME -field singul arities, the helicity \n(chirality) parameter was introduced. A time average helicity \nparameter for the near fields of a ferrite disk with MDM \noscillations is defined as [ 5, 6, 14]: \n* 1Im\n16F E E\n  \n            (1) \n \nOne can also introduce a normali zed helicity parameter, \nwhich shows a time -averaged space angle between rotating \nvectors  \nE\n and \nE\n :    \n                    \n*\nIm\ncosEE\nEE \n\n\n                    (2) \nIn the regions where this paramet er is not equal to zero, a \nspace angle between the vectors \nE\n   and \nE\n   is not equal to \nπ/2. This breaks the field structure of Maxwell \nelectrodynamics.   III. MICROWAVE RADIATION S TRUCTURE WITH A MS -MODE \nFERRITE DISK  \nAs a radiation structure, we use a \n10TE -mode rectangular \nX-band waveguide with a small hole in a wide wall. A hole \ndiameter is 8 mm. A MS -mode ferrite disk is placed inside a \nwaveguide symmetrically to its walls so that the disk axi s is \nperpendicular to a wide wall of a waveguide. The disk axis \ncoincide with a hole axis. The structure is shown in Fig. 1. The \nyttrium iron garnet (YIG) disk has a diameter of 3 mm and the \ndisk thickness  is 0.05 mm; t he di sk is normally magnetized by \na bias magnetic field \n04900 H Oe; the saturation \nmagnetization of the ferrite is \n4  1880 sM G. The sizes of a \nferrite sample are extremely small compared to the EM \nwavelength in a waveguide.  \n \nFig.1 . Waveguide radiation structure with a M S-mode  ferrite disk and a hole in \na wide wall.  \nIn the shown microwave structure, a thin -film ferrite disk is \nthe only resonant element. The multiresonanc e spectrum of MS \noscillation in a closed waveguide structure with an embedded \nferrite disk is we ll studied in our previous works [ 3 – 6, 14 ]. At \nthe MS -mode -resonance frequencies  one observes strong \nlocalization of microwave energy by a ferrite particle. For \nevery resonance, the localized fields are topologically \ndistinctive.  One can observe power -flow vortices and non -zero \nhelicity (chirality) parameters. Inside a waveguide, the power -\nflow vortices near a ferrite disk are non -symmetric with respect \nto waveguide walls. This non -symmetry, being different for \ndifferent directions of the wave propagation  in a waveguide, \ndoes not affect practically , however, on nonreciprocity in the \nmicrowave propagation from one port to another. Fig s. 2 (a) \nand (b)  show  numerical results of the Poynting -vector \ndistributions inside a waveguide near a ferrite  disk at t he MS -\nresonance frequency of 8.146GHz . One can see strong energy \nlocalization and non-symmetrically distingu ishable pictures of \nthe power -flow vortices for different directions of the wave \npropagation in a waveguide.  At the same time, at the frequency \nbeyond th e MS resonance, the disk slightly perturbs the \nwaveguide field [see Fig 2 (c)].  \n \n \n(a)           (b)                (c) Fig.2. The Poynting -vector  distributions inside a waveguide near a ferrite  disk. \n(a) w ave propagation from port 1 to port  2 at a MS res onance; (b) wave \npropagation from port 2 to port 1 at a MS resonance;  (c) wave propagation at \nfrequency beyond a MS resonance.  The MS -resonance  frequency  is 8.146GHz . \nWhen a waveguide has a hole (see Fig. 1), nonsymmetrical \npictures of the power -flow distr ibution near a hole are evident \nat the MS -mode resonances. For different directions of the \nwave propagation in a waveguide, one has different pictures of \nthe power -flow distribution near a hole. Fig. 3 shows numerical \nresults of the Poynting -vector distrib ution in a small vacuum \ncylinder above a waveguide hall. Figs. 3 (a) and (b) correspond \nto two opposite directions of the wave propagation in a \nwaveguide at a MS -mode resonance frequency of 8.146GHz . \nEvidently, there are non -symmetrical pictures with respe ct to \nthe waveguide axes. When the frequency is beyond the MS -\nmode resonances, one has symmetry with respect to the \nwaveguide axes.  Fig. 3 (c) shows such a symmetrical picture \nof the power -flow distribution near a hole.   \n \n \n(a) (b)  (c) \nFig.3. The Poynti ng-vector distributions above a hole. (a) wave propagation \nfrom port 1 to port 2 at a MS resonance; (b) wave propagation from port 2 to \nport 1 at a MS resonance; (c) wave propagation at frequency beyond a MS \nresonance.  The MS -resonance  frequency  is 8.146GH z. \n \nThe symmetry -breaking  properties of the fields originated \nfrom a ferrite disk  at the res onant states of MS oscillations are \nexhibited not only by peculiar power -flow distributions. As \nvery important characteristics, there are also distributions of the \nfield helicity (chirality) parameter. Fig. 4 shows t he normalized \nhelicity parameter of the field above a hole  in the XY plane \n(see Fig. 1) . Figs. 4 (a) and (b) correspond to two opposite \ndirections of the wave propagation in a waveguide at a MS -\nmode reson ance frequency of 8.146GHz . Fig. 4 (c) shows that \nat a frequenciy beyond the MS resonance the fields are without \nany symmetry breakings and so no helicity properties are \nobserved.  \n \n \n(a) (b)           (c) \nFig.4. The normalized helicity parameter of the field a bove a hole. (a) wave \npropagation from port 1 to port 2 at a MS resonance; (b) wave propagation \nfrom port 2 to port 1 at a MS resonance; (c) wave propagation at frequency \nbeyond a MS resonance.  The MS -resonance  frequency  is 8.146GHz . \n \nAs an additional characterization of the symmetry breaking \nproperty  of the ME fields, we show in Fig. 5 t he normalized \nhelicity parameter of the field in the YZ -plane cross section of the structure in Fig. 1 in the regions both inside and outside a \nwaveguide.  \n \n \n \nFig.5. The n ormalized helicity parameter of the field in the YZ -plane cross  \nsection of the structure in the regions both inside and outside a waveguide  at \nthe MS -resonance  frequency 8.146GHz . The wave propagates from port 1 to \nport 2.  \nIV. RADIATION PATTERNS  \nWhile the show n non -symmetry in the Poynting -vector \ndistributions does not affect on nonrecipocity between the \nwaveguide ports, this gives strong nonreciprocity in the \nradiation patterns. It is well known that  a standard single \nradiation element does not exhibit  a radia tion pattern with a \nsquint. To obtain a squint in the radiation patter n, an array of \nradiation elements has to be used. In our single -element \nstructure we obtain a radiation pattern with a squint due to \nchiral microwave near fields originated from a MS -mode \nferrite disk. Fig. 6 shows the single -element radiation pattern s \nfor two cut plane s,\n0\n  and \n90\n , at frequency \n8.146GHz.   \n \n     \n(a)  (b) \nFig.6. Single -element radiation patterns for two cut planes,\n0\n  and \n90\n, at the MS -resonance  frequency 8.146GHz.  The wave in a \nwaveguide propagates from port 1 to port 2.  (a) 2-D pattern; (b) 3 -D pattern.  \n \nIn Fig. 7 , we show the r adiation patterns (\n0\n ) for two \ndifferent directions of the wave propagation in a waveguide at \nthe MS -resonance frequency. We compare these patterns with \na radiation pattern of a structure at a frequency beyond the MS -\nmode resonance.    \n \nFig.7. Radiation patterns (\n0\n ) for different directions of the wave \npropagation in a waveguide . (a) wave propagation from port 1 to port 2 at a MS \nresonance; (b) wave propagation from port 2 to port 1 at a MS resonance; (c) \nwave propagation at frequency beyond a MS resona nce. The MS -resonance  \nfrequency  is 8.146GHz . \n \nOur preliminary results show also that the desired squint \nangles can be obtained by variation of a hole diameter and also \nby use of an elliptical -form hole.   \nV. CONCLUSION  \nIn this paper we showed that microwave c hiral fields \noriginated from a small ferrite disk with magnetostatic \noscillations can form far -field radiation pattern with a strong \nand controllable squint. To the best of our knowledge, this is \nthe first time demonstration of a radiation pattern with a s quint \nobtained by a single radiation element. Such an effect of a \nspatial mode division  by a single radiation element is unique could be attractive for development of novel microwave \nantennas with controllable phase structure distributions.  \nREFERENCES  \n[1] T. J. Davis and E. Hendry, \"Superchiral electromagnetic fields created \nby surface plasmons in nonchiral metallic nanostructures\", Phys. Rev. B \n87, 085405 (2013) . \n[2] E O Kamenetskii, “Vortices and chirality of magnetostatic modes in \nquasi -2D ferrite disc particles ”, J. Phys.  A: Math . Theor . 40, 6539 \n(2007) .  \n[3] E.O. Kamenetskii, M. Sigalov, R. Shavit, \"Manipulating microwaves \nwith magnetic -dipolar -mode vortices\", Phys. Rev. A. 81, 053823 (2010).  \n[4] E.O. Kamenetskii, R. Joffe, and R. Shavit, \"Coupled states of \nelectromagn etic fields with magnetic -dipolar -mode vortices: magnetic -\ndipolar -mode vortex polaritons\", Phys. Rev. A 84 (2011).  \n[5] E.O. Kamenetskii, R. Joffe, and R. Shavit,  \"Microwave magnetoelectric \nfields and their role in the matter -field interaction\",  Phys. Rev. E 87, \n023201 (2013).  \n[6] M. Berezin, E. O. Kamenetskii, and R. Shavit, \"Topological properties \nof microwave magnetoelectric fields\",  Phys. Rev. E (in press) (2014).  \n[7] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, \n“Orbital angular momentum of  light and the  transformation of Laguerre -\nGaussian laser modes ”, Phys. Rev. A 45, 8185  (1992 ). \n[8] L. Allen and M. J. Padgett, “Poynting vector in Laguerre - Gaussian \nbeams and the interpretation of their angularmomentum  density,” Opt. \nCommun . 184, 67 (2000) . \n[9] R. Celechovsky, Z. Bouchal, “Optical implementation of the vortex \ninformation channel” New J. Phys. 9, 328 (2007).  \n[10]  B. Thide, H. Then, J. Sj oholm, K. Palmer, J.  Bergman, T.D. Carozzi, \nYa. N. Istomin, N.  H. Ibragimov and R.  Khamitova  “Utilization of \nphoton orbital angular momentum in the low -frequency radio domain” , \nPhys. Rev. Lett . 99, 087701 (2007).  \n[11] C. Deng,  W. Chen, Z . Zhang, Y . Li, and Z . Feng , “Generation of OAM \nradio waves u sing circular vivaldi antenna a rray”. Int. J. Antenn . Propag . \n2013 , 847859 (201 3). \n[12] O. Edfors, A.  J. Johansson, “Is orbital angular m omentum (OAM)  based \nradio communication an unexploited a rea?”, IEEE T rans.  Antenn . \nPropag . 60, 1126  (2012 ). \n[13] A.G. Gurevich and G.A. Melkov , Magnetic Oscillations and Waves  \n(CRC Press, New  York, 1996).  \n[14] R. Joffe, E. O. Kamenetskii, and R. Shavit, J. Appl. Phys. 113, 063912 \n(2013).  \n \n \n "    },    {        "title": "1812.06370v2.Yttrium_substituted_Mg_Zn_ferrites__correlation_of_physical_properties_with_Yttrium_content.pdf",        "content": "*Corresponding author: mohi@cuet.ac.bd  \n Yttrium -substituted Mg -Zn ferrites: correlation of physical properties  with \nYttrium content  \nM.A. Ali1, M.N. I. Khan2, F.-U.-Z. Chowdhury1, M.M. Hossain1, A.K. M. Akhter Hossain3, \nR. Rashid2, A. Nahar2, S.M. Hoque2, M.A. Matin4 and M.M. Uddin1* \n1Department of Physics, Chittagong University of Engineering and Technology (CUET), Chattogram  4349, \nBangladesh  \n2Materials Science Division, Atomic Energy Center, Dhaka 1000, Bangladesh . \n3Department of Physics, Bangladesh University of Engineering and Techn ology (BUET), Dhaka 1000, Bangladesh.  \n4Department of Glass and Ceramic Engineering , Bangladesh University of Engineering and Technology (BUET), \nDhaka 1000, Bangladesh.  \n \nABSTRACT  \n \nYttrium - (Y) substituted Mg -Zn ferrites with the compositions  of Mg 0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ \n0.05) have been synthesized by conventional standard ceramic technique.  The effect of Y3+ \nsubstitution on the structural, electrical , dielectric  and magnetic properties of Mg -Zn ferrites has \nbeen  studied.  The s ingle phase of spinel structure with a very tiny secondary phase of YFeO 3 for \nhigher Y contents has been  detected.  The theoretically estimated lattice constant  has been \ncompared with measured experimental lattice constant.  The bulk density, X-ray density a nd \nporosity  have been calculated. The Energy Dispersive X -ray Spectroscopy ( EDS ) study  confirm s \nthe presence of Mg, Zn, Y,  Fe and O ions in the prepared sample s. Frequency dependence of \nconductivity has been  studied and an increase in resistivity (an order) has been  observed. \nFrequency de pendence of dielectric constant  (ԑʹ), dielectric loss tangent (tanδ) has been  studied \nand the lowering of ԑʹwith the increase of Y content  was noted. Dielectric relaxation time was \nfound to vary between  15 to 31 nano seconds. The saturation magnetization ( Ms), coercive field \n(Hc), remanent magnetization ( Mr) and Bohr magneton ( µB) have been  calculated. The variation \nof Ms has been  successfully explained with  the variation of A –B interaction strength due to Y \nsubstitution. The soft ferromagnetic nature also confir med from the value s of Hc. The complex \npermeability has been  studied and the initial permeability was found to increase  with Y up to  x = \n0.01, thereafter it decrease s. The values of electrical resi stivity and dielectric constant  with \nproper  magnetic properties suggest the suitability  of Y-substituted Mg -Zn ferrites in microwave \ndevice application s. \nKeywords:  Mg-Zn ferrites,  structural properties, cation distribution, electrical and dielectric \nproperties , magnetic properties and permeability . 2 \n 1 Introduction  \nThe field of ferrites has become a possible subfield of materials science because of unique \ncombination of their electrical and magnetic properties. The physics involved in ferrites have \nalso drawn much attention to the scientific community and interests in  ferrites are still now \ngrowing, even after many years since  their discovery. The scientists, researchers, technologist \nand engineers are continuously trying to open the door for commercial application of ferrites and \nvarious types of ferrites with excelle nt properties are known in bulk, thin film and nano  particle \nform [1 ]. Among the large family of ferrites, the Mg -Zn ferrites with spinel structure are widely \nknown due to their significant properties which makes them suitable for application in computer \nmemory and logic devices, cores of transformers, recording heads, antenna rods, loading coils \nand microwave frequency devices (as a core of coils), and so forth [2, 3]. Like other soft ferrites, \nthey are also considered as a suitable  choice owing to their h igh Curie temperature (T c), high \nelectrical resistivity, low eddy current losses, low dielectric loss, low cost, high mechanical \nhardness and superior environmental stability [4, 5]. Even their application extends in every \nsector: electronic communication to medical industry, military to space technology [3].  \nThe synthesis  of new ferrites with different  composition s with better performance in practical \napplication by the modification of existing materials is always motivating the materials \nresearchers [6]. The properties of ferrites strongly depend on the chemical composi tion, cation \ndistribution on A -site and B -site, methods of preparation, sintering temperature and time, types \nof impurity ions and levels etc. [7, 8]. Therefore, there is a way of tuning the  physical properties \nof ferrites by changing the chemical composition. The cation distribution at A -site and B -site can \nbe changed  by different ions substitution for either divalent cation (M2+ of MFe 2O4) or Fe3+ ions.  \nThe aim of this research work is to enhance the electrical resistivity of Mg -Zn ferrites by \nsubstituting Fe3+ with rare -earth (Y3+) ions with an intension to achieve desired characteristic s of \nferrites for applications in the high frequency devices . The electrical conductivity in spinel \nferrites is mainly due to the electron hopping between Fe2+ (which are form during  sintering \nprocess [9, 10]) and Fe3+ions at octahedral (B -site) site [11]. It is well known that the rare -earth \nions occupy the octahedral (B) sites that  reduced the electron  hopping by limiting the motion of \nFe2+, results an increase in resistivity [12]. Therefore, increase of electrical resistivity is expected \nby Y3+substitution for Fe3+in Mg -Zn ferrites due to its (Y3+) tendency  to occupy the B -site due to  \nlarger ionic radius [13]. It is noted that the Yttrium (Y) has already been substituted in Ni -Zn 3 \n ferrites, Mg ferrites and Co ferrites [1 3-20]. Enhancement of electrical resistivity in Y substituted \nNi-Zn and Mg ferrites have also been reported [16 -18]. \nThe physical properties of different ions substituted Mg -Zn have been investigated  by many \nresearchers: Mn [2, 3 , 21], Sm [11], Ti [22], Nd [23, 24], Zr [25], Cu [26 -29], Cr [30 -32], Co \n[33, 34], Pr [35], Gd [36], Ni [37, 38]  of which some of them were dealt  with electrical and \ndielectric properties  [3, 11 , 21, 22, 26, 29, 32 , 38] and some of them were dealt with magnetic \nproperties  [2, 25, 28, 31 , 34-37] of substituted Mg -Zn ferrites . Few of them were also dealt with \nboth of dielectric and magnetic properties [27, 30, 33]. Improvement of electrical properties of \nMg-Zn ferrites has also been reported by Nd [24, 25] and Sm [ 11] substitutions that have been \nexplained by their larger ionic rad ius and B -site occupation. However , to the best of our \nknowledge, study of Y substituted Mg -Zn ferrites is not reported yet.  \nTherefore,  the study of the structural, electrical, dielectric and magnetic properties of Mg -Zn \nferrites as a function of Y contents has been done for the first time . \n \n2 Experimental techniques  \nConventional ceramic technique was used to prepare  Y-substituted Mg -Zn ferrite \n[Mg0.5Zn0.5YxFe2-xO4(0 ≤ x ≤ 0.05)] . The raw materials in nano form  were used ( US \nResearch Nanomaterials, Inc. ) to facilitate the homogeneous mixing  with purity > 99.5% and the \nparticle size are of 20, 10-30, 30 and  20 -40 nm for MgO, ZnO, Fe 2O3 and Y 2O3, respectively.  \nWe have already completed some projects using nano powder s as raw materials and the results \nare available elsewhere  [39-42]. The raw powders were weighed according to the stoichiometric \nratio for corresponding composition. The powders were th en mixed and milled for 6hrs using an \nagate mortar and  pestle. After completing the homogeneous mixing of powders, they were \nloosely pressed to make biscuit like shape and then were calcined at 850 °C for 4 hrs in a muffle \nfurnace . The calcined powders were then milled  again for 2.5 hrs. A 5% polyvinyl  alcohol \nsolution was added as a binder and desired shape of dimension 8.4 mm diameter and 2.4 mm \nthickness  of samples were prepared using a suitable die with a  hydraulic press by applying 10  kN \npressure. T he samples were finally sintered at 1250 °C  for 4 hrs in air at atmospheric pressure  \nwith the temperature step  of 5 °C per minute  and cooled naturally. The characterization of \nsamples was done by taking X -ray diffraction (XRD) using Philips X'pert  PRO X -ray 4 \n diffractometer (PW3040) with Cu -Kα radiation (λ=1.5405 Å), microstructure images with the \nEDS by a high resolution FESEM (JEOL JSM -7600F) , dielectric and permeability measurements \nby a Wayne Kerr precision impedance analyzer (6500B) in the frequency range of 10 –120 MHz \nwith a drive voltage of 0.5 V at room temperature . The room temperature  magnetic properties \nwere obtained by a physical properties measurement system (PP MS) from quantum design .  \n \n3 Results and discussion  \n3.1 Structural properties  \nThe X -ray diffraction (XRD) patterns of Y -substituted Mg -Zn ferrites with the chemical \ncomposition,  Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x≤ 0.05)  are shown  in Fig. 1.  The sharp and well \ndefined peaks in the XRD patterns  [ICDD PDF 22 -1012 ] confirmed  the spinel structure in which \npeaks are assigned for Miller indices. The single phase of spinel structure is identified up to \nx=0.03, there after a secondary extra phase is observed at 2  = 33.2°. The secondary phase \nappears at the g rain boundaries  (marked as red filled circle s in Fig. 1 ) results from YFeO 3 [ICDD \nPDF # 39 -1489]  due to high reactivity of Fe3+ ions with Y3+ ions [43].   \n \n \n \n \n \n \n \n \n \n \n \nFig. 1  The XRD patterns of Mg0.5Zn0.5YxFe2-xO4(0 ≤ x ≤ 0.05).  \n \nThe simi lar phase is also reported in Y -substituted Ni -Zn ferrites [16] and Mg ferrites [18].  The \nXRD data has also been used to determine the lattice constant s of the samples.  \n20 30 40 50 60 70(311)\n(222)(400)\n(422)(511)(440)\nx=0.05\nx=0.04\nx=0.03\nx=0.02\nx=0.01 \n Intensity (arb. u.)\n2 (degree)x=0.00(220)5 \n To calculate the lattice constant ( a), the formula has been used for all peaks of the samples is \ngiven by: 𝑎=𝑑 2+𝑘2+𝑙2, where h, k and l are the Miller indices of the crystal planes. The \nNelson -Riley (N -R) extrapolation method has been used to evaluate lattice constants for the \nsamples, the N -R function, F(θ), is given as  𝐹 𝜃 =1\n2 𝑐𝑜𝑠2𝜃\n𝑠𝑖𝑛𝜃+𝑐𝑜𝑠2𝜃\n𝜃  [44]. The values a for \neach peaks are  plotted against F() and the exact lattice constant has  been obtained from the \npoint where the least square fit straight line cut the y -axis. \nThe theoretical density also known as X-ray density has been calculated using following \nexpression:  𝜌𝑥=8𝑀\n𝑁𝐴𝑎03𝑔/𝑐𝑚3, where  NA is Avogadro’ s number (6.02× 1023 mol-1), M is the \nmolecular weight.  The bulk density is measured by the formula: 𝜌𝑏=𝑀\n𝑉𝑔/𝑐𝑚3where V (=πr2h) \nis the volume of the samples , r and h are radius and height of the samples.  Porosity  (P) of the \nsamples is calculated using the following equation:  𝑃= 𝜌𝑥−𝜌b\n𝜌𝑥 ×100 %.where,  ρb is the bulk \ndensity of the samples and ρx is the X -ray density. The calculated lattice constant, bulk density, \nX-ray density, and porosity are presented in Table 1.  \n \n \n \n \n \n \n \n \n \nFig. 2  (a) The obtained lattice constants (experimental and theoretical) and (b) bulk density and \nporosity of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05).  \n \nFig. 2 (a) illustrate s the c alculated lattice constant s (experimental and theoretical ) as a function \nof Y contents. Increase in lattice constants up to x = 0.03 (expt.) are due to substitution of  larger \nionic radius of Y ion (0.9 Å) compared to Fe ion  (0.67  Å). On the other hand, the values of a are \nnoted to be decreased for higher Y content s. An extra impurity phase is also observed  (Fig. 1) for \n0.00 0.01 0.02 0.03 0.04 0.054.04.14.2 \n Bulk density (g/cm3)\nY contents (x)1415161718\nPorosity (%)(b)\n0.00 0.01 0.02 0.03 0.04 0.058.4278.4308.4338.436\n  \nY contents (x)\natheo. (Å)aexpt. (Å)(a)\n8.478.488.498.50\n 6 \n x = 0.04 and  0.05, indicating diffusion of some of Y ions into the grain boundaries consequently \nthe YFeO 3 is formed. The spinel lattice is compressed due to the differences in the thermal \nexpansion coefficient  in presence of extra inter -granular phase, results a decreasing trend in a \n[12] that has been rep orted by other s [16-18]. Fig. 2  (a) also depicts the theoretical lattice \nconstant ( atheo), calculated using the equation which relates lattice constant with the ionic radii of \nA and B crystallographic lattice sites [45] : 𝑎𝑡=8\n3 3  𝑟𝐴+𝑅0 + 3 𝑟𝐵+𝑅0  , where R 0 is the \nradius of the oxygen ion (1.32 Å) [46]. The ionic radii of r A and r B have been calculated by \nassuming possible cation distribution. The cation distribution has been proposed based on the \nfollowing assumptions: the Mg2+ ions occupy both the A-sites (10 % of total distribution) and B-\nsites (90% of total distribution) [47] wh ile Zn2+ prefers to occupy the tetrahedral sites ( A-sites) \n[48]. The Fe3+ ions have preference for  both tetrahedral and octahedral sites [49].  The mean ionic \nradii r A of tetrahedral sites (A) and r B of octahedral sites  (B) have been calculated using relati ons \n[45]; \n𝑟𝐴=𝐶𝐴𝐹𝑒𝑟 𝐹𝑒3+ +𝐶𝐴𝑀𝑔𝑟 𝑀𝑔2+ +𝐶𝐴𝑍𝑛𝑟 𝑍𝑛2+ and𝑟𝐵=𝐶𝐵𝐹𝑒𝑟 𝐹𝑒3+ +𝐶𝐵𝑀𝑔𝑟 𝑀𝑔2+ +\n𝐶𝐵𝑌𝑟 𝑌3+ . A good agreement is to be noted between theoretical and experimental lattice \nconstant up to x = 0.03. The theoretical lattice constant  (atheo) is found to be higher than the \nexperimental lattice constant (aexpt). In order to calculate the theoretical lattice constant t he ideal \ncrystal unit cell is considered. The ideal unit cell  is perfectly filled in a close packed spinel \nstructure manner with regular cation  and anions  distribution . Therefore , a little deviation between \naexpt and atheo is expected  [50].  The aexpt values are decreasing trend for x = 0.04 and 0.05 \nfollowing disagreement between aexpt and atheo since the secondary phase is not considered in \ncalculation of atheo, however it is obvious for x = 0.04, 0.05 and seems to be started at 0.03 (Fig. \n1).  \nFor different Y contents , the values of X-ray density ( ρx) and bulk density ( ρb) are presented in \nTable 1. Pores formed during sintering process are not considered in calculation of ρx that \nproduces higher values of ρx than that of  ρb. Fig. 2(b) represents the variation of ρb and porosity \n(P) with Y contents. The value of ρb increases up to x = 0.02 and  beyond this  it decreases . The \nappearance of impurity phase (YFeO 3) in Fig. 1 can be  explained in a way that the inter -granular \nvoids might not be filled for higher Y contents.  It should be noted here that the values of ρb for \nsubstituted compositions are higher than that of the un -substituted one due to the difference in 7 \n atomic weight of Y (88.90585 amu) and Fe (55.845 amu) . The variation of density with Y \ncontents can also be more cleared in the subsection 3.2 . \nThe A-sites are small in volume due to movement of  oxygen ions  to adjust the metal ions \nconsequently A -sites expand to an extent corresponding to the reduction of B -sites [49]. The \noxygen positional parameter ( u), along with the bond lengths  at tetrahedral site s (R A) at \noctahedral sites (R B), the tetrahedral edge  length R, shared octahedral edge length Rʹ and \nunshared octahedral edge length Rʹʹ  have been calculated (Table 1)  using following  equations:  \n𝑢= 1\n𝑎𝑡 3 𝑟𝐴+𝑅0 +1\n4 [45], 𝑅𝐴=𝑎 3 𝛿+1\n8 , 𝑅𝐵=𝑎 1\n16−𝛿\n2+3𝛿2 12 \n[51],  \n𝑅=𝑎 2 2𝑢−0.5 , 𝑅′=𝑎 2 1−2𝑢 and𝑅′′=𝑎 4𝑢2−3𝑢+1116  [45], where δ (= u-\nuideal), is the inversion parameter which signifies departure from ideal oxygen parameter \n(uideal=0.375 Å) and a is the experimental lattice constant.  \nThe bond length R A and non-linear behavior of the bond length R B with Y substitution are  \nobserved and shown  in Table 1. The cation distribution on A-sites and B-sites results the \nvariation of the tetrahedral edge length R, shared octahedral edge length Rʹ and unshared \noctahedral  edge length Rʹʹ [52]. The R is found to decrease with Y substitution. On the other \nhand , the Rʹ and Rʹʹ are found  to increase with Y contents. The calculated results are in good \nagreement with that of the Ni substituted Mg -Zn ferrites  [37].  8 \n Table 1  Cation distribution (A - and B-sites), ionic radii ( rA and rB), theoretical ( atheo.) and experimental (aexpt.) lattice constant s, X-ray \ndensity ( ρx), bulk density ( ρb), porosity ( P), bond length RA and RB, tetrahedral edge length ( R), shared octahedral edge length ( Rʹ) and \nunshared octahedral edge length ( Rʹʹ) of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05 ) with Y contents . \nY contents  \n(x) A-site B-site r\nA \n(Å) r\nB \n(Å) a\ntheo \n.(Å) aexpt.  \n(Å) ρx \n(g/cm3) ρb \n(g/cm3) P \n(%) u \n(Å) RA \n(Å) RB \n(Å) R \n(Å) R' \n(Å) R'' \n(Å) \n0.00 Fe\n0.45 Mg\n0.05Zn\n0.5 [Fe\n1.55Mg\n0.45]O\n42-\n 0.744  0.665  8.4689  8.4309  4.89 4.006  18.08  0.3907  2.0547 1.9841 3.3554 2.6178 3.0060 \n0.01 Fe\n0.45 Mg\n0.05Zn\n0.5 [Fe\n1.54 Mg\n 0.45Y\n0.01]O\n42-\n 0.744  0.666  8.4720  8.4327  4.894  4.172  14.76  0.3906  2.0544 1.9849 3.3549 2.6200 3.0070 \n0.02 Fe\n0.45 Mg\n0.05Zn\n0.5 [Fe\n1.53 Mg\n 0.45Y\n0.02]O\n42-\n 0.744  0.668  8.4750  8.4366  4.895  4.197  14.25  0.3905  2.0546 1.9862 3.3553 2.6221 3.0080 \n0.03 Fe\n0.45 Mg\n0.05Zn\n0.5 [Fe\n1.52 Mg\n 0.45Y\n0.03]O\n42-\n 0.744  0.669  8.4781  8.4350  4.905  4.166  15.06  0.3904  2.0534 1.9861 3.3534 2.6243 3.0090 \n0.04 Fe\n0.45 Mg\n0.05Zn\n0.5 [Fe\n1.51 Mg\n 0.45Y\n0.04]O\n42-\n 0.744  0.670  8.4812  8.4269  4.927  4.119  16.38  0.3903  2.0507 1.9846 3.3490 2.6265 3.0100 \n0.05 Fe\n0.45 Mg\n0.05Zn\n0.5 [Fe\n1.50 Mg\n 0.45Y\n0.05]O\n42-\n 0.744  0.671  8.4842  8.4288  4.931  4.084  17.17  0.3902  2.0505 1.9854 3.3485 2.6286 3.0110 \n  \n \nTable 2  Cations -anions atomic % and cations to anions ratio of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05).  \nY contents  \n(x) Grain size \n(µm)  cations (At %)  anions (At %)  Ratio of cation : \nanion  \n0.00 1.22 40.03  59.97  2.802: 4.197  \n2.847: 4.152  \n2.820: 4.179  \n2.946: 4.053  \n3.355: 3.775  \n2.688: 4.312  0.01 3.65 40.68  59.32  \n0.02 3.75 40.29  59.71  \n0.03 1.28 42.09  57.91  \n0.04 1.07 47.94  53.94  \n0.05 1.87 38.40  61.60  9 \n 3.2 Microstructure study  \nThe role of microstructure in materials designing is very important to obtain the desired \nproperties for their useful applications. The physical properties can significantly be tuned by \nchanging the microstructure of materials. Fig. 3 demonstrates the micr ostructure of \nMg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05). A significant effect of Y ions substitution on the \nmicrostructure is clearly identified  from the images of Fig. 3 . \nSignificant change in shape and size of grains and grain boundaries is  observed. Moreover, the \ndensification of Y substituted samples can also be observed from the figure . An inverse relation \nis found between the variation of bulk density and  porosi ty [Fig. 2  (b)] and grains size \ndistribution as function of Y contents. The i ncrease in grains size with Y contents up to x = 0.02, \nindicating the appreciable extent of diffusion of Y ions into the Mg -Zn ferrites grains and most \ndenser composition. After x = 0.02, the extent of diffusion decreases due to larger ionic radius of \nY, residual stress is created, which results smaller grains and lowered the density. The variation \nof grain  sizes is also associated with the variation of lattice constants with Y contents.  \nThe elements of prepared samples can be detected using EDS. It is also  possible to detect the \nexist ence of unwanted elements. The p eaks and height of peaks in EDS spectra correspond to the \nelements present and concentrations of element. Each peak is unique for every element. The EDS \nspectra (not shown here)  of the samples were obtained and the presence of Mg, Zn, Fe, Y and O \nelements are confirmed.  The standard ratio of metal cations to anion in ferrites is 3:4 [ 37]. The \nobtained  atomic % of metal cations and anions are presented in Table 2. A fairly good agreement  \nwith standard ratio is observed an d deviation from standard ratio  is also observed for Ni -\nsubstitute d Mg-Zn ferrites [ 37]. The results  obtained from XRD and FESEM measurements \nrevealed the successful synthesis of Y -substituted Mg -Zn ferrites. Now, to expl ore the suitability \nthe prepared samples the electrical, dielectric and magnetic properties of Mg 0.5Zn0.5YxFe2-xO4 (0 \n≤ x ≤ 0.05) ferrites  have been studied which are presented in the following sections.  \n \n \n \n \n \n 10 \n  \n \n \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. 3  FESEM images of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) . \n(a) x=0.00  (b) x=0.01  \n \n(c)x=0.02  \n (d)x=0.03  \n \n(e)x=0.04  \n (f)x=0.05  \n 11 \n 3.3 Frequency dependence of ac conductivity  \nFig. 4 (a) shows the frequency dependence of ac conductivity ( ζac) of the samples. The ζac \ngradually increases with the increase in  frequency . The frequency dependent total ac \nconductivity follows  Jonscher’s  power law : [53] \n𝜍𝑎𝑐,𝑡𝑜𝑡𝑎𝑙 𝜔 =𝜍 0 + 𝜍𝑎𝑐(𝜔)= 𝜍𝑑𝑐+ 𝐴𝜔𝑛 \n \nwhere, the first part ( σdc) indicates the frequency -independent conductivity or dc conductivity. \nThe pre -exponential factor ‘A’ is constant and dependent on temperature, ‘ n’ is an exponent, \ndependent on both frequency and temperature in the range 0 to 1 and ω represents the angular \nfrequency. Frequency dependent behavior of these samples can be explained by employing \nKoop ’s heterogeneous and Maxwell –Wanger double layer model [5 4-56]. The ferrites comprise s \nof two layers: the layer consists of grains which are well conducting, separated by poorl y \nconducting thin layer, forming grain boundary. The frequency response of these two layers is \ndifferent resulting  different conductivity at low and high frequency regions. At the lower \nfrequency region, the dc conductivity is attributed due to the grain b oundaries which are more \nactive and the  exchange of electrons between Fe2+and Fe3+ions is less.  The grain s activity \nincreases with increasing the frequency  by promoting the electrons hopping between  Fe2+ and \nFe3+ ions consequently  an increase in hopping fr equency hence ac electrical conductivity  rises.  \n \n \n \n \n \n \n \n \n \n \n \nFig. 4  (a) The frequency dependence of ac conductivity for Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ \n0.05)  and (b) ac conductivity for different Y contents  at 20 Hz.  \n \n0.00 0.01 0.02 0.03 0.04 0.050.51.01.52.02.5(b)\n  ac (-1cm-110-6)\nY contents (x)\n10210310410510610710810-610-510-410-3\n x=0.02\n x=0.03\n x=0.04\n x=0.05\n  x=0.00\n x=0.01\nFrequency, f (Hz)\n  Logac (-1cm-1)(a)12 \n Fig. 4  (b) shows t he variation of ac conductivity  with Y contents  at 20 Hz . The conductivity of \nMg-Zn ferrites decreases with increasing Y contents unlike for x = 0.01  that is expected and \nconsistent with reported Ho substituted Ni -Zn ferrites for x = 0.015 [ 57] and x = 0.01 [58]. When  \nY ions are added into Mg -Zn ferrites, the Y ions substitute the Fe3+ ions at the B -site in the \ncompositions [13]. At the same time, some of Zn ions loss compensate d by pushing some Fe3+ at \nthe A-site. The electronic valence is higher for Fe ions than Zn ions; hence to balance the \nelectrical charge, metallic vacancies which are charge carriers have been increased resulting  \nincrease in conductivity. The resistivity of the compositions rises with increasing Y contents that \ncan be explained as:  the Fe3+ ions concentration at B -sites decrease due to increasing Y3+ ions at \nB-sites that accelerates the conduction process between Fe2+ and Fe3+at the B -site [59, 60] where \nFe2+ ions are produced during sintering process [30].  Therefore, the decrease in Fe3+ \nconcentration at B -site reduces the probability of electrons exchange between  Fe2+and Fe3+ and \nhence the resistivity is increased  that consistent with reported results [18]. Another point could  \nalso be noted here that for x = 0.04 and 0.05, ac conductivity was noted to be increased but still \nlower than parent one. The excess of Y contents and Fe2+ ions at the B -site might facilitate the \nconduction process [6 1]. The increase in the probability of electrons hopping is due to  larger \nionic radius of Y which causes the oxygen ions to close each other in the crystal lattice could be \nanother probable reason. The similar result for La substit uted Ni ferrites has also been reported \n[62]. \n \nTable 3 The variation of resistivity and dielectric parameters with Y contents . \nY contents  \n(x) ρac (105) (Ω-cm) ԑʹ (104)  ԑʹ ρac (106) tanδ fmax(tanδ)  \n(103) ηMʹʹ \n(nano sec) \n at 20 \nHz at 1 \nkHz at 20 Hz  at 1 \nkHz at 20 Hz  at 1 \nkHz at \n20 Hz    \n0.00 4.54 2.24  5.34 0.68  37.5  3.2 3.41 254.4  24.1 \n0.01 3.45 1.79  7.89 0.31  48.9  1.3 2.96 1.751  22.3 \n0.02 10.4 1.92  2.88 0.65  31.6  2.8 2.59 47.11  26.1 \n0.03 11.4 2.46  2.59 0.20  29.2  1.0 2.63 88.22  15.1 \n0.04 11.0 4.22  2.55 0.15  33.4  0.9 2.82 1.747  30.6 \n0.05 7.75 3.79  2.99 0.96  24.5  5.9 3.35 1.494  20.6 \n \n \n \n 13 \n 3.4 Dielectric properties  \nFrequency dependence of dielectric constant  \nThe dielectric constant has been calculated using equation: ԑʹ= Cd/ԑ0A, where C is the \ncapacitance, d is the thickness, ԑ 0 is the permittivity in free space and A is the surface  area of \npellet. The measured (at room temperature) frequency dependent dielectric constants ԑʹ of \nMg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05)  are presented in Fig. 5 (a), showing a  dispersion in ԑʹ, \nto a different extent, in the low frequency range ( <10 kHz) due to interfacial polarization.  \n \n \n \n \n \n \n \n \n \n \nFig. 5 Frequency dependence of (a) dielectric constant and (b) dielectric loss tangent of  \nMg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05).  Inset of (a) shows the variation εʹ with frequency \nrange  from 1  kHz to 120 MHz.  \n \nThe curves exhibiting three regions of frequencies: a sharp decrease up to 100 Hz, a slow \ndecrease up to 10  kHz and finally become almost zero and frequency independent at high \nfrequency region, which is common for spinel ferrites [6 3]. Similar results als o reported earlier \nfor Mg –Zn [6 4], Sm, Cr and Co substituted Mg -Zn ferrites [11, 32, 33] and Ni –Zn [40, 41].  The \nfrequency dependent behavior of the compositions can be explained using of Koops’  theory [5 4], \nassumes that the dielectric materials contain tw o layers of the Maxwell –Wagner type [5 5, 56]. \nThe conduction mechanism and dielectric polarization is similar in ferrites [6 5] and a significant \ncorrelation between these two has also been reported [6 6]. The electron hopping between Fe2+ \nand Fe3+ has been taken in consideration for  the polarization mechanism. In the low frequency \nregion, the electron exchange between Fe2+ and Fe3+ is able to follow the electric field up to \n10210310410510610710801234\n  (b)\nFrequency, f (Hz)tan x=0.03\n x=0.04\n x=0.05 x=0.00\n x=0.01\n x=0.02\n  \n102103104105106107108036\n103105107048 103\n \n \n  \n Frequency, f (Hz) x=0.00\n x=0.01\n x=0.02\n x=0.03\n x=0.04\n x=0.05\n  104(a)14 \n certain frequency (hopping frequency) results the local displacement of char ges between sites in \nthe applied field direction, which determine the polarization of the system. At high frequencies \n(after hopping frequency) the electron exchange between Fe2+ and Fe3+ cannot follow the applied \nelectric field and hence attained a consta nt value [ 67].  It can also be noted here that the \ndielectric constant for x = 0.01 (Fig. 5 (a)) is lower than that of parent one but higher for other \nconcentrations. The variation of dielectric constant of the compositions with Y contents can also \nbe expl ained by employing same formalism as stated in subsection 3. 3. \nFrequency dependence of dielectric loss  \nAn abnormal behavior is observed in the plots of the dielectric loss tangent (tan ) against \nfrequency for different Y contents, shown in Fig. 5 (b).  The tanδ vs frequency curves for the \nsample showing typical maximum of dielectric loss at a certain frequency and the peak shift to \nlower frequency for substituted compositions. This type of behavior in  Mg-Zn ferrites [ 68] as \nwell as in Cu substituted Mg -Zn ferrites [27]  has also been reported.  This abnormal behavior is \nreported even in other ferrites such as Ni–Zn [41], Cu–Cd [69], Li–Mg–Ti [7 0] and  Ni–Mg [7 1] \nferrite systems.  The condition for  showing a peak in tanδ  for a dielectric material can be \nexpressed  by the following relaxation relation 𝜔𝜏=1 [72], where ω = 2πf max and η is the \nrelaxation time, related to the jumping probability per unit time p by an  equation η   =1/2p; or \n𝑓𝑚𝑎𝑥∝𝑝 [27]. Therefore, the tanδ vs frequency curves exhibit a maximum when the jumps \nfrequency between Fe2+ and Fe3+ions at adjacent B -sites approximately equal to the applied \nelectric field [7 3]. Thus  the frequencies  can be calculated  (presented in T able 3) for  samples at \nwhich the maximum in tanδ is observed.  \nCorrelation of resistivity and dielectric constants as a function of Y contents  \nThe conduction mechanism and dielectric polarization is similar in ferrites [ 65] and a significant \ncorrelation between these two has been reported [ 66]. In order to correlate the dielectric constant \nwith ac resistivity, we have calculated the product of ԑʹ and  𝜌𝑎𝑐 (at 20 Hz and 1000 Hz)  and \npresented in Table 3 from which an approximate inverse proportionality of ԑʹ to  𝜌𝑎𝑐 can be \nobserved. Similar results have been reported and can be concluded that in case of dielectric \nmaterials where dielectric polarization is strongly dependent  on conduction mechanism, the 15 \n dielectric constant is approximately inversely proportional to t he square root of resistivity [18 , \n74]. \nStudy of e lectric modulus  \nFig. 6 (a) shows the variation of real part of electric modulus of different  compositions of \nMg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) . The value of Mʹ is very low (almost zero) in the \nfrequency region around < 104 Hz a nd increases with increase in frequency  showing a dispersion \nat frequency region  > 104 Hz, exhibiting a maximum at very high frequency region and beyond it \nis down to a certain values. The characteristic  feature might  be demonstrated in  the short range \nmobility of charge carriers due to  the conduction process.  \n \n \n \n \n \n \n \n \n \n \nFig. 6 Frequency dependence of (a) real and (b) imaginary of electric modulus of  \nMg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05).  \n \nFig. 6  (b) shows the variation of imaginary electric modulus ( M′′ (ω)) of the samples with \nfrequency, where the curves are characterized by (i) a clear peak in the pattern at different \nfrequencies, (ii) The peaks are lying  in the dispersion region of Mʹ and (iii) the peak shift \ntowards lower frequency for  x = 0.02  and 0.04 while the peak sh ifts towards high frequency side \nfor x = 0.01, 0.03 and  0.05. The characteristic  peaks distinguish the frequency range into two \nregions.  The low frequency side that determines frequency range where charge carriers are able \nto move over a long distance i.e., hopping of charge carriers is possible between two adjacent \nsites.  The high frequency region that determine frequency range where charge carriers are able to \nmove within short range i.e., the motion of charge carriers are localized within their poten tial \n1021031041051061071080246\n M x=0.00\n x=0.01\n x=0.02\n x=0.03\n x=0.04\n x=0.05 \nFrequency, f (Hz)\n  \n(a)\n1021031041051061071080.00.51.01.52.02.5\n  (b)\nFrequency, f (Hz)M x=0.00\n x=0.01\n x=0.02\n x=0.03\n x=0.04\n x=0.05\n  16 \n well. The peak also gives  information about the transition from long range to short range \nmobility with increase  in frequency [7 5]. Moreover, the frequency at which Mʹʹ is maximum is \nalso known as dielectric relaxation frequency and the relaxation tim e η is calculated using \nequation 𝜏𝑀′′=12𝜋𝑓𝑀′′ .The calculated relaxation time for different Y contents of the samples \nis presented in Table 3.  \n \nStudy of i mpedance spectroscopy  \nComplex impedance is a useful tool to demonstrate the transport properties taking place between \nthe grain and grain  boundaries, resulting  changes in electrical  conductivity  (ac and dc), dielectric \npermittivity, and dielectric  losses. Especially , the dominant resistance from grains or grain \nboundary in the polycrystalline ceramics can easily be resolved [7 6]. Fig. 7 (a) and (b) represent \nfrequency dependence of real and imaginary parts of the impedance for different Y contents. In \nFig. 7 (a) , the values of Z ʹ are found  to decrease with increase  in frequency, suggesting an \nincrease in ac  conductivity with frequency.  The coincidence of the values Zʹ at high frequency \nsuggests the possible release of space charge [ 77] that is  consistent with reported results [ 40, 41 ]. \nFor particular frequency (e.g. at 20 Hz) the variation of Zʹ with Y contents is same as observed \nand successfully discussed in case of conductivity and dielectric constants at same frequency.  \nFig. 7  (b) illustrates the imaginary part of the impedance Zʹʹ as a function of frequency for \ndifferent Y contents, showing a relaxation peak at low frequency side; all the curves coalesce \nwith each other at high frequency.  \nThe impedance plane plots of  Mg 0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) compositions are  presented in \nFig. 8  (a). Usually, the impedance spectrum of the polycrystalline materials is characterized by \nthe presence of one or more semicircles. The resistance of bulk grains or grain boundary can be  \ncalculated from the diameter of the semicircle at Zʹ axis and the total resistance (R T = Rg + Rgb) \nof the material is equal to intercept of the curve on the Zʹ axis at low frequencies [ 78]. In case of \nceramic materials , the contribution from grain boundar y is much more due to existence of \ndefects compared to the contribution from grain resistance. In Fig. 8  (a), two semicircles for each \ncomposition are observed but the semicircle at low frequency (right) remains incomplete within \nstudied frequency range. Fig. 8 (b) shows the equivalent circuit model used to calculate  the grain \nresistance (R 1), capacitance (C 1), grain boundary resistance (R 2) and capacitance (C 2); and the 17 \n values are shown in Table 4 . It is also clear  from Table 4 that t he grain resistance are too small  \ncompared to the grain boundary resistance  for the compositions with x = 0.00, 0.02, 0.03 and 05 . \nOn the other hand, the grain res istance are comparatively high er than grain boundary resistance \nin the composition of  x = 0.01and 0.04 . Similar r esults are also available for Gd substituted Ni \nferrites  [79]. It is assume d (for x = 0.01  and 0.04)  that the  size of the grain boundaries is much \nthinner than bulk grains resulting lower values of grain boundary resistance compared to that of \ngrain resistance  [79]. To make a clear picture of the grains and grain boundaries , TEM /AFM  or \nother  very high resolution  microscopy is necessary to produce the three dimensional image.  \n \n \n \n \n \n \n \n \n \n \nFig. 7  Frequency dependence of (a) real part and (b) imaginary part of impedance of \nMg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05).  \n \n \n \n \n \n \n \n \nFig. 8  (a) The complex impedance  spectra  of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05)  and (b) \nThe equivalent model  circuit  for two semicircle Cole -Cole plot . The inset of (a) shows the first \nsemicircle for x = 0.00, x = 0.02 and  x = 0.03.   (b) \n \n1021031041051061071080123456\n  x=0.00\n x=0.01\n x=0.02\n x=0.03\n x=0.04\n x=0.05\nFrequency, f (Hz)\n  Z ()(a)\n1021031041051061071080510152025\n  (b)\n x=0.00\n x=0.01\n x=0.02\n x=0.03\n x=0.04\n x=0.05\nFrequency, f (Hz)Z ()\n  \n0 1 2 3 4 5012345\n0.0 0.1 0.2 0.3 0.40.00.10.2  x=0.00\n x=0.01\n x=0.02\n x=0.03\n x=0.04\n x=0.05\nZ ()Z ()\n  \n(a)\n  18 \n  \nTable 4  The grain resistance (R 1) & capacitance (C 1) and grain boundary resistance (R 2) & \ncapacitance (C 2). \nY contents  (x) R1(Ω) C1(F) R2(Ω) C2(F) \n0.00 1.1E4 8.1E-12 1.3E5 1.4E-9 \n0.01 6.7E4 7.8E-10 2.8E4 1.4E-11 \n0.02 2.3E4 1.3E-11 2.1E5 3.4E-9 \n0.03 1.3E4 2.1E-11 4.6E5 2.7E-9 \n0.04 2.7E5 1.0E-9 6.0E4 1.2E-11 \n0.05 6.4E4 1.1E-11 1.4E5 4.6E-10 \n \n3.5 Magneti c properties  \nThe factors affecting the shape and the width of the hysteresis are the chemical composition of \nthe compound, porosity, grain size, etc. The ferrimagnet ic nature of all the samples has been \nconfirmed from the field dependence of magnetization curve. Fig. 9 (a) shows narrow hysteresis \nloops confirming soft magnetic nature of all samples [ 80]. The prepared samples with very low \ncoercivity (18 -35 Oe) makes them suitable for use in high frequency devices and as core \nmaterials [ 81-84]. The saturation magnetization, coercive field , remanent magnetization and \nBohr magneton have been calculated from the hysteresis curve  and presented in Table 5.  \n \n \n \n \n \n  \n  \n \nFig. 9  The (a) M–H loops of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05)  and (b) the variation \nof saturation magnetization with Y contents ( x). The inset [Fig. 9 (a)] shows the \nhysteresis loops for very low applied field.  \n \n-40 -20 0 20 40-60-40-200204060\n-100 -50 0 50 100-15015M (emu/g) x=0.00\n x=0.01\n x=0.02\n x=0.03\n x=0.04\n x=0.05(a)\nApplied field (kOe)\nB\nOe\n0.00 0.01 0.02 0.03 0.04 0.05404550556065\n(b)Ms (emu/g)\nY contents19 \n Table 5  Saturation magnetization ( Ms), coercive field ( Hc), remanent magnetization ( Mr) and \nBohr magneton ( µB) of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) . \nY contents ( x) Ms (emu/g)  Mr (emu/g)  Hc (Oe) µB \n0.00 63.7 2.55 26 2.51 \n0.01 53.0 2.38 30 2.08 \n0.02 50.1 2.84 18 1.99 \n0.03 58.3 2.43 31 2.31 \n0.04 40.1 5.04 26 1.59 \n0.05 46.8 2.84 35 1.86 \n \nFrom Fig. 9 (b) it is clear that the saturation magnetization  for Y -substituted compositions  are \nfound to be less than the parent one ( x = 0.00) due to the non -magnetic nature of Y ions. I n this \npresent case , non-magnetic Y3+ is substituted for Fe3+. But the decrease in  Ms is non -linear. The \nMs decreases  up to  x = 0.0 2 and then increase for x = 0.03 , thereafter  it decreases  for x = 0.04 and \nfinally slightly increases  for x = 0.05. The variation of Ms with Y contents can be explained by \nthe exchange interactions of cations. Although AB interactions are strongest and dominant in the \nferrites, but the AA and BB interactions also play an important role in determining the variation \nof magnetization. In Mg -Zn ferrites, the Fe3+ ions occupy both A -site and B -site but the Y has a \ntendency to occupy the B -site only. When Y is substituted for Fe3+ ions, it replaces some Fe3+ \nfrom B -site only, results a decrease of magnetization at B sub -lattice. According to Nee l two \nsub-lattice model the magnetization of ferrimagnetic is defined by the equation M=MB-MA, \nwhere MB and MA are magnetic moment of B sub -lattice and A sub -lattice.  Therefore, the \nmagnetization is  expected  to decrease  for Y3+ substituted compositions. The non -linear variation \ncan also be explained based on the migration of Mg2+ ions from B -sites to A -sites and cation \nredistribution . Generally, Mg ions occupy both the A -sites and B -sites.  Some of Mg ions from \nB-sites can also migrate to A -sites during h eat treatment process [ 84] and this migration may \nvary composition to composition . As a result , some of Fe3+ ions mig rate from A -site to B -site \nlead to an increase in ma gnetic moment of B -sites resulting a higher saturation  magnetization for \nx = 0.03 in comparison to  x = 0.01  & 0.02 and saturation magnetization is higher for x = 0.04 \nthan that of  x = 0.0 5.  \n \n 20 \n 3.6 Study of p ermeability  \nThe frequency dependence of real part ( 𝜇𝑖′) and imaginary part  𝜇𝑖′′  of complex permeability of \nMg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) in the frequency range from 20 Hz to 120 MHz is \npresented in Fig. 10 (a and b) . The 𝜇𝑖′ measures the ability of the materials to store energy, used \nto express the in phase relationship between the component of magnetic induction ( B) and the \nalternating magnetic field ( H) while the 𝜇𝑖′′ measures the dissipation of energy in the materials \nand also used to express the component of B 90° out of phase with H. The formalism used to \ncalculate the  𝜇𝑖′ and 𝜇𝑖′′of the𝜇𝑖∗are the following rel ations:  𝜇𝑖′=Ls/L0 and 𝜇𝑖′′𝜇𝑖′tanδ, where Ls is \nthe self -inductance of the sample core and 𝐿0=𝜇0𝑁2𝑆\n𝜋𝑑  is derived geometrically. Here , Lo is the \ninductance of the winding coil without the sample core, N is the number of turns of the coil ( N = \n5), S is the area of cross section of the toroidal sample as given below:  𝑆=𝑑× and 𝑑=𝑑2−𝑑1\n2, \nwhere d 1 = inner diameter, d 2 = outer diameter and h= height and also 𝑑  is the mean diameter of \nthe toroidal sample as given below: 𝑑 =𝑑2+𝑑1\n2. The 𝜇𝑖′ remains fairly constant up to certain \nfrequency, after which it increases to a maximum and after then sharply decreased with \nfrequency. The frequency region in which the real part remains constant is defined as the utility \nzone of ferrites and demonstrate d the suitability of the prepared ferrites in the wide range of \nfrequency such as broadband pulse transformer and wide band read -write heads for video \nrecording [39, 85]. On the other hand, the 𝜇𝑖′′decreases rapidly at lower frequency side and \ndisplays a  characteristics peak at a higher frequency side where the 𝜇𝑖′ starts to decrease. This \nphenomenon is termed as the ferrimagnetic resonance [ 86]. Fig. 10 (d) shows the variation of \npermeability with Y contents ( x). It is cleared from Fig.  10 [(a) and (c) ] that the compositions \nexhibit the resonance at lower frequency due to their higher permeability and vice versa in \naccordance with the Snoek’s limit 𝑓𝑟𝜇𝑖′=constant  [87].  \nThe value s of 𝜇𝑖′ at 20 Hz frequency for different Y contents are shown in Fig. 10 (d).  The \nvariation of initial permeability of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05 ) can be explained by \nthe following relation  [88]: 𝜇𝑖′∝𝑀𝑠2𝐷\n 𝐾1where μ i is the initial permeability, Ms the saturation \nmagnetization, D the average grain size and K 1 the magneto -crystalline anisotropy constant. The \nvalue of 𝜇𝑖′for x = 0.01  and 0.02 is greater than pure ( x = 0.00) Mg -Zn ferrite. As 𝜇𝑖′ proportional \nto the square of the Ms and directly pr oportional to the D, the Ms is lowered for x = 0.01 and 0.02 21 \n but the value of D is increased from 1.22 µm to 3.65 µm  and 3.75 µm  for 0.01 and 0.02, \nrespectively . The combine effect of both Ms and D may remain the  higher  𝜇𝑖′ values  for x = 0.01 \nand 0.02 than for x = 0.00. After that , the values of 𝜇𝑖′ are found to decrease gradually  for x ≥ \n0.03. Although, The Ms is slightly increased for x = 0.0 3 than x = 0.02 , but the grains size \ndecrease s to 1.28 µm , resulting  a decrease in 𝜇𝑖′ for x = 0.03 .  \n \n \n \n \n \n  \n   \n  \n \n \n   \n \n \n \n \n  \n \nFig. 10  The frequency dependence of permeability (a) real part, (b) imaginary part , (c) relative \nquality factor and  (d) the variation of μi with Y contents ( x) of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ \n0.05)  at 20 Hz. \n \n1021031041051061071080246 (a)\n x=0.03\n x=0.04\n x=0.05 x=0.00\n x=0.01\n x=0.02i \n \nFequency, f (Hz)\n10210310410510610710803691215\n1051061071080510152025\n  \n   x=0.00\n x=0.01\n x=0.02\n x=0.03\n x=0.04\n x=0.05\nFequency, f (Hz)i\n (b)\n0.00 0.01 0.02 0.03 0.04 0.053456(d)\n   \nY contents (x)\n10210310410510610710805101520\n  x=0.00\n x=0.01\n x=0.02\n x=0.03\n x=0.04\n x=0.05\nFequency, f (Hz)\n  RQF 103(c)22 \n The frequency dependence of relative quality factor (RQF) of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ \n0.05) ferrites is shown in Fig. 10 (c). The quality factor measures the performance of materials \nfor used in filter application. The RQF  is very low at very low frequency, found to increase with \nfrequency, showing a characteristics peak, and decrease to low values at high frequency. The \nvalues of RQF go down beyond 106 Hz where the loss factor [Fig. 10 (b)] starts to increase \nrapidly. The loss is resultant from the lagging of domain wall motion regarding the applied \nalternating magnetic field which is due to various domain defects [ 89] such as  non -uniform and \nnon-repetitive motion of domain -wall, bending over of domain -wall, localized vari ation of flux \ndensity and nucleation and annihilation of domain walls.  The frequency for maximum in RQF \nshifts to higher frequency with Y contents except for x = 0.01, which is shifted slightly to lower \nfrequency. The highest value of RQF  is obtained for x = 0.02.  \n4. Conclusions  \nMg 0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) ferrites have bee n successfully synthesized by  standard \nceramic technique. The single phase of spinel structure up to x = 0.03 and a secondary phase of \nYFeO 3 for x ≥ 0.04 has been confirmed. A n increase in lattice constant (both aexpt. and atheo.) is \nfound (≤ x = 0.03  for experimental  and thereafter decreases for x ≥ 0.04, due to  the secondary \nphase ) for the substitution of Y ions with larger radius than that of Fe . The bulk density increases \nx ≤ 0.02 from 4.006 g/cm3 to 4.197 g/cm3, there after it decreases with Y contents  to 4.084 g/cm3 \nfor x = 0.05 . On the contrary , the porosity decreases with Y contents  x ≤ 0.02, then increases  \nwith Y contents. FESEM images confirmed t he homogeneou s grain distribution and clear g rain \nboundaries . Absence of any unwanted element is also confirmed from the EDS data. Y \nsubstitution has significantly influenced the electrical resi stivity and dielectric constant . The ac \nelectrical resistivity increase s from 105 to 106 Ω-cm (at 20 Hz) for x ≥ 0.02-0.04. A fairly good \ninverse relation between electrical resi stivity and dielectric constant  is observed for all the \ncompositions. Existence of dielectric relaxation behavior is confirmed and the relaxation time is \nin th e range between 15 -31 nano seconds. The hysteresis loops confirm the soft ferromagnetic \nnature of the samples. The saturat ion magnetization is found to decrease  (non-linearly) due to Y \nsubstitution. The Ms is found to decrease  up to x = 0.02  from 63.7 emu/g to 50.1 emu/g , \nthereafter  it increase s for x = 0.03  (58.3 emu/g) and decreases for x = 0.04 (40.1 emu/g)  and \nfinally increased  again for x = 0.05  (46.8 emu/g) . The initial permeability  with wide stability \nregions ( ≤ 10 MHz)  is found to increase  for x = 0.01  and decreases  gradually  for higher Y 23 \n contents.  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Magn. 17, 2698 (1981).  \n  \n \n "    },    {        "title": "2106.01844v1.Macrospin_model_of_an_assembly_of_magnetically_coupled_core_shell_nanoparticles.pdf",        "content": "Macrospin model of an assembly of magnetically coupled core-shell nanoparticles\nNikolaos Ntallis,1Corisa Kons,2Hariharan Srikanth,2Manh-Huong Phan,2D.A. Arena,2and Manuel Pereiro1,\u0003\n1Department of Physics and Astronomy, Uppsala University, Uppsala 751 20, Sweden\n2Department of Physics, University of South Florida, Tampa, Florida 33620, USA\nHighly sophisticated synthesis methods and experimental techniques allow for precise measure-\nments of magnetic properties of nanoparticles that can be reliably reproduced using theoretical\nmodels. Here, we investigate the magnetic properties of ferrite nanoparticles by using theoretical\ntechniques based on Monte Carlo methods. We introduce three stages of sophistication in the macro-\nmagnetic model. First, by using tailor-made hamiltonians we study single nanoparticles. In a second\nstage, the internal structure of the nanoparticle is taken into consideration by de\fning an internal\n(core) and external (shell) region, respectively. In the last stage, an assembly of core/shell NPs are\nconsidered. All internal magnetic couplings such as inter and intra-atomic exchange interactions or\nmagnetocrystalline anisotropies have been estimated. Moreover, the hysteresis loops of the afore-\nmentioned three cases have been calculated and compared with recent experimental measurements.\nIn the case of the assembly of nanoparticles, the hysteresis loops together with the zero-\feld cooling\nand \feld cooling curves are shown to be in a very good agreement with the experimental data. The\ncurrent model provides an important tool to understand the internal structure of the nanoparticles\ntogether with the complex internal spin interactions of the core-shell ferrite nanoparticles.\nI. Introduction\nVarious technological \felds have motivated the design\nand fabrication of nanostructures with the ability of tai-\nloring the magnetic properties. The category of magnetic\nnanoparticles (NPs) are interesting from both fundamen-\ntal and technological points of view1{6. Thus, for ex-\nample, ferrite nanoparticles with the general formula of\nMFe 2O4(M = Fe, Co, Ni and Mn) have attracted great\nattention of researchers due to their potential applica-\ntions in biomedicine and industry2. In particular, spinel\nferrites are an attractive class of ferrimagnets o\u000bering\nboth soft and hard phases as well as a common crystal\nstructure that allow for a high quality crystal interface\nbetween core and shell layers7. The metallic ions are lo-\ncated at either octahedral coordinated sites|also called\nB-sites|or forming a tetrahedral geometry|denoted as\nA-sites. The ratio of the occupation of these sites pro-\nduces di\u000berent spinel structures de\fned by the inversion\ndegree number X. The normal and inverse spinel struc-\ntures represent the two limiting cases with X= 0 and\nX= 1, respectively. Between these two limits, a mixed\nspinel exists where the divalent transition metal M is dis-\ntributed between sites A and B8,10. A value of X= 2=3\nrepresents a random distribution of metallic ions between\nsites A and B. The degree of inversion can a\u000bect the\nmagnetic properties of ferrites as, for example, satura-\ntion magnetization and coercive \feld11,12. Ferrites are\nfound to have a ferrimagnetic magnetic ordering, which is\ndue to the dominant antiferromagnetic exchange between\nsites A and B8,10.\nThe inverse spinel structure is of paramount impor-\ntance for the current work where Fe3+cations are equally\ndistributed at both A and B sites while the divalent M\nions are found only at octahedral sites. In this work,\nCoFe 2O4(CFO) was selected due to its high anisotropy\nthat should limit the degree of canting as spins will be\nmore tightly bound to the crystal lattice compared toFe3O4(FO), which has a relatively high moment but\nmuch lower anisotropy. A common crystal structure and\nnegligible di\u000berences in lattice constant between the two\nmaterials (8.40 \u0017A for FO, 8.39 \u0017A for CFO8) enables syn-\nthesis of high quality core/shell NPs and, consequently,\na bi-magnetic structure of these two compounds can be\nconstructed without introducing large lattice mismatch\ndistortions. Moreover, as CFO is in a hard magnetic\nphase while FO is in a soft magnetic phase at normal\nconditions, the variants of these compounds can possess\ninteresting exchanged coupled related properties.\nThus, two variants of core/shell NP assemblies have al-\nready been synthesized and magnetically characterized9.\nMore speci\fcally, a conventional assembly of NPs is ex-\nperimentally synthesised by adding CFO to the core\nof the NP and capping the core with a shell of FO\n(CFO@FO). The inverted assembly is achieved by plac-\ning FO in the core and adding CFO in the shell\n(FO@CFO). In order to give a better understanding of\ntheir underlying reversal mechanisms, we employ Monte\nCarlo (MC) atomistic simulations for both type of as-\nsemblies based on tailor-made hamiltonians. Thus, we\nintroduce progressively di\u000berent degrees of sophistication\nof the model. Initially, we performed MC simulations\nfor single nanoparticles by varying several of their inter-\nnal degrees of freedom in order to examine how interface\nand surface e\u000bects a\u000bect their magnetic response. In a\nsecond stage, the hamiltonian is improved to describe a\nsingle nanoparticle composed of a central region denoted\nas core and an external shell region. The hamiltonian\nalso considers the interface e\u000bects between the core and\nshell regions. Finally, the hamiltonian, and consequently,\nthe macrospin model is recasted in a more sophisticated\nshape so that the model now is capable to describe not\nonly single nanoparticles with internal structure but also\nan assembly of core-shell nanoparticles. A depiction of\nthe three NP systems modeled is shown in Fig. 1. More-\nover, we also indicate in several colors the regions ofarXiv:2106.01844v1  [cond-mat.mes-hall]  3 Jun 20212\nthe nanoparticles that are essential to understand the\ndi\u000berent terms considered in the hamiltonians given by\nEqs. (1-3).\nII. Model and Results\nInitally, we employed atomistic Monte Carlo simula-\ntions with the implementation of the Metropolis algo-\nrithm of isolated nanoparticles composed of CoFe 2O4and\nFe3O4. The Hamiltonian used for the calculations is\nH=\u0000X\ni;jJijSiSj\u0000X\niKicos2\u0012i\u0000g\u0016BX\niBSi(1)\nwherei;jdenote \frst-neighbour atomic positions and\nSiis the atomic magnetic moment. The \frst term de-\nscribes the Heisenberg exchange interaction, Jij, between\natomic sites iandj. The ansatz given in Eq. (1) implies\nthat a ferromagnetic (antiferromagnetic) interaction is\ndescribed by a positive (negative) exchange coupling. In\nthe second term, a uniaxial atomic magnetic anisotropy\nof strength Kiis introduced to collect the spin-orbit ef-\nfects. The angle \u0012ide\fnes the deviation of the atomic\nmagnetic moment ( Si) with respect to the atomic easy\naxis anisotropy. The last term represents the Zeeman\nterm under the in\ruence of a global external magnetic\n\feld. For the very small sizes in diameter of the nanopar-\nticles considered here ( \u00186 nm for CFO and \u00187 nm for\nFO), the surface e\u000bects become important, and thus, for\nall the simulated systems, unless stated otherwise, we de-\n\fne a surface shell thickness of width d(cf. Fig. 1 a)).\nThe widthdis chosen to be equal to the unit cell length\nof the associated material. For the core spins, the easy\naxis anisotropy is set along zdirection whereas for the\nsurface atoms the anisotropy axis direction is randomly\nselected for each spin site in agreement with experimen-\ntal evidence9. BulkJijvalues are collected in Table I for\nCFO and FO nanoparticles13. Likewise, the bulk values\nof the magnetic anisotropy for the core region (red color\nin Fig. 1 a)) of the nanoparticles are ki= 0:0036 mRy\nandki= 0:000112 mRy per atom for CFO and FO, re-\nspectively8,10. Both CFO and FO have an inverse spinel\nstructure, i.e., the divalent ion (Co+2for CFO and Fe+2\nfor FO) is found only in B sites. The Co+2ion has a\nmoment of 3 :0\u0016B=atom, whereas Fe+2presents a mo-\nment of 4:0\u0016B=atom. In both samples, Fe+3ions have a\nmoment of 5.0 \u0016B=atom. The spherical particles with ra-\ndiusrare constructed by replication of the bulk inverse\nspinel structure. The spherical shape is reproduced by\nremoving the unit cells that are a distance bigger than r\naway from center of the nanoparticle.\nFigure 2 shows hysteresis loops for CFO and FO single\nnanoparticles at T= 5Kby varying the strength of the\nsurface anisotropy ( Ks\ni) with respect to the one of the\ncore (Kc\ni). As surface atoms interact with a number of\nneighbouring oxygen atoms smaller than the number of\noxygen atoms interacting with bulk atoms, the exchange\ncouplings of the surface atoms are expected to be smaller\nFIG. 1. Since the studied nanoparticles are very small in size,\nthe surface e\u000bects become very important. Thus, we sketch\nin a) an spherical single nanoparticle with the region in gold\nrepresenting the portion of the nanoparticle that belongs to\nthe surface. The width of the surface is denoted by d. In b),\na core-shell nanoparticle is shown with the di\u000berent regions\nindicated by using di\u000berent colors. Finally, in c) is represented\nan assembly of core-shell nanoparticles, while on the right is\ndrawn the internal structure of a single nanoparticle taken\nfrom the assembly. The arrows represent the macrospins used\nin the model.\nTABLE I. Bulk exchange coupling constants for CFO and FO\nwith di\u000berent interaction transition-metal (TM) atomic sites.\nAll values are given in energy units of mRy.\nCFO TMi TMj i j J ij\nFe Fe A A \u00000:094\nFe Co A B \u00000:143\nFe Fe A B \u00000:164\nCo Co B B 0 :296\nFe Co B B \u00000:117\nFe Fe B B \u00000:047\nFO Fe Fe A A \u00000:132\nFe Fe A B \u00000:150\nFe Fe A B \u00000:177\nFe Fe B B 0 :308\nFe Fe B B \u00000:08\nFe Fe B B \u00000:06\nthan the exchange couplings for bulk atoms. Thus, we set\nJs\nij=Jc\nij= 0:5. This value is chosen on a perfect spherical\nsurface formed by a cubic unit cell with the mean value\nof the coordination number drooping to half in the sur-\nface. Thus, by setting the aforementioned ratio to 0 :5 we\nassume that, in mean average, the coordination number3\nFIG. 2. Calculated hysteresis loops for CFO (left panel) and\nFO (right panel) single nanoparticles for di\u000berent ratios of the\nsurface anisotropy with respect to the core anisotropy\u0010\nKs\ni\nKc\ni\u0011\nat a temperature of 5 K.\nin the surface is half of the one for the atoms in the bulk.\nFO single nanoparticles clearly shows a larger saturation\nmagnetization with a value of 1 :31\u0016B=atom with respect\nto CFO which has a value of 0 :97\u0016B=atom. On the con-\ntrary, CFO possesses a larger coercive \feld reaching a\nvalue\u00182:5 T. It is worthwhile to mention here that the\nvariation of the coercive \feld with respect to the ratio\nKs\ni=Kc\niis almost negligible, in particular, for FO single\nnanoparticles. This is a direct consequence of the the\nrandomly easy axis distribution of the surface atoms.\nBy increasing the level of sophistication of the model,\nwe proceed to the calculation of single core-shell nanopar-\nticles for the two aforementioned variants, i.e., CFO@FO\nand FO@CFO. In order to study the core-shell morphol-\nogy, we introduce the following hamiltonian:\nH=\u0000bulkX\ni;jJijSiSj\u0000interf :X\ni;jaiJijSiSj\u0000surf:X\ni;jasJijSiSj\n\u0000X\niKicos2\u0012i\u0000g\u0016BX\niBSi (2)\nIn Eq. (2), we have introduced two additional terms\nwhich separate the interface and surface e\u000bects from the\nbulk of the nanoparticle. We assume that both the width\nof the surface (d s) and interface (d i) are equal to the\nCFO unit cell constant, i.e. d s=di=0:835 nm. As both\nCFO and FO possess the same inverse spinel structure\nand very similar lattice constant, we would expect an al-\nmost perfect match in the interface. However, due to the\nspherical shape of the nanoparticle, and consequenly, the\ncurvature of the interface, there is still a slight mismatch\nbetween the boundaries in the interface which produces a\nnonzero surface tension. For the aforementioned reasons,\nin the current model we assume that the spinel structure\nis preserved but it is slightly distorted on the interface\nand surface. The distortion is introduced by rescaling\nthe exchange constants Jijon the interface by the factorTABLE II. Structural parameters of CFO@FO and FO@CFO\nnanoparticles. All values are given in distance units of nm.\nCore Radius Shell Thickness Diameter\nCFO@FO 2.9 1.7 9.2\nFO@CFO 3.5 1.3 9.6\naiand on the surface by as. The initial value of Jijon\nthe interface is assumed to be the mean average of the\nsum ofJijof the associated bulk values of the di\u000berent\nstructures. Unless stated elsewhere, ashas a value of 0 :5,\ni.e., on the surface we assume that interaction strengths\ndrop to the half value than the one found in the bulk.\nAs a consequence, we assume the anisotropy becomes\nrandomized on the surface but uniaxial on the interface.\nThe structural parameters of the core-shell nanoparticles\nstudied here are given in Table II.\nIt has to be noted here that the size of both types of\ncore-shell nanoparticles are far below the coherent radius\nlimit of 3:6lex, wherelexis the exchange length14. Typ-\nical values of the exchange length are 4.9 nm and 5.2\nnm for Fe 3O4and CoFe 2O4bulk systems, respectively8.\nThus, domain formation is very unlikely. In consequence,\nthe magnetization reversal process can only be attributed\nto incoherent states on the interface or surface. Figures\n3 and 6 show hysteresis loops at 5K for CFO@FO and\nFO@CFO core-shell nanoparticles, respectively. The hys-\ntersis loops are plotted for di\u000berent scaling factors, i.e.,\nai,asandak. The anisotropy scaling parameter per atom\nis represented by ak. Both types of nanoparticles possess\nsimilar properties as single particles. Thus, under the\nabsence of magnetocrystalline anisotropy (black dotted\ncurves,ak= 0) both types of core-shell nanoparticles\nshow a knee-like behaviour close to saturation indicat-\ning the frustration produced by Jijin ferrites. Also in\nthe range from -3 T to 3 T in the applied external mag-\nnetic \feld, none of the CFO@FO and FO@CFO core-\nshell nanoparticles possess a full closed loop. However,\nby scaling (reducing) the exchange constants on the in-\nterface without magnetocrystalline anisotropy, the afore-\nmentioned e\u000bect is highly reduced although it is worth-\nwhile to mention here that the core-shell nanoparticles\nstill have a non-zero coercive \feld (cf. red curves in\nFigs. 3 and 6). The latter arises from the fact that the\nmismatch between the core and shell introduces canting\nin the interface spins and thus induces an increase of the\nexchange anisotropy of the system. Introduction of mag-\nnetocrystalline anisotropy into the nanoparticles led to\nan increase in the coercive \feld. For example, for a scal-\ning factorak= 2:2, the coercive \feld is calculated to\nbe 2:3 T and 1:9 T for FO@CFO and CFO@FO, respec-\ntively.\nFor exchange coupled systems, the exchange interac-\ntion on the interface is a critical parameter regardless\nof the soft or hard phase of the nanoparticle. Figure\n4 shows the variation of the coercive \feld with respect4\nto the interface scaling factor for CFO@FO nanopar-\nticle at 5K for two di\u000berent values of the anisotropy\nscaling factor. In the limit of no anisotropy coupling\n(ai!0), the coercive \feld approximates to that of the\nhard phase. For both anisotropy scaling parameters, the\ncoercive \feld possess a non-monotonic behaviour. When\nthe anisotropy constant approaches the bulk values, i.e.,\nak!1, the coercive \feld continuously decreases but the\ndecay is broken in the interval [0 :3;0:5]. In this region,\nthe coercive \feld curve shows a plateau with a very tiny\nincrease. After this value the decrease rate reduces, i.e.,\nthe coercive \feld decreases but with a smaller slope. On\nthe other hand, for ak= 2:2 the coercive \feld shows a\npeak forai\u00180:2, so that, the interface e\u000bects induce\nmagnetic \ructuations for such coupled systems. Notably\nthe hardness of the bi-magnetic systems is heavily depen-\ndent on the interplay between the anisotropy barrier and\nthe exchange interactions on the interface. As already\nmentioned above, the disorder induced by the interface\ne\u000bects, even under the in\ruence of strong anisotropy, re-\nsults in a non-zero coercive \feld. By increasing of the\nanisotropic scaling factor, a high coercive \feld can be\nreproduced with a small interface coupling factor. On\nthe other hand, for the anisotropy bulk value case, a\nvery strong interface coupling must be considered in or-\nder to achieve the maximum coercive \feld. Therefore, a\nnanoparticle system requires an almost perfect interface\nwith negligible imperfections or dislocations. From these\nfacts, we can conclude that for bi-magnetic nanoparticle\nsystems, the interface shows an enhanced anisotropy.\nApart form the coercive \feld, the remanent magneti-\nzation is a critical parameter for magnetic applications.\nFigure 5 shows the evolution of the normalized rema-\nnent magnetization with respect to the interface scaling\nfactor. For both ak= 1:0 andak= 2:2, the remanent\nmagnetization shows an increasing trend up to a critical\nvalue of the interface scaling factor ai= 0:3 and then\nclearly reduces down to a value of Mr=Ms= 0:2. For\nboth anistropy scaling factors, the Mr=Msratio show a\nmaximum close to 0 :8.\nIndependently from the interface coupling, the surface\nis free to start a reversal process due to reduced coordina-\ntion of the atomic spins. From our simulations it is found\nthat, in order to reach the maximum Mr=Msa larger\nvalue of the interface coupling constant is needed, with\nrespect to the one for the maximum coercive \feld. As\nthe surface and interface are already disordered an even\nstronger coupling between the phases is needed in order\nfor the soft phase to overrule and rise the remament state.\nInterestingly though when the Mr=Msis at its maximum\nvalue still the nanoparticle possess a considerable coer-\ncive \feld as the disorder of the interface does not allow\nfor a full reversal process to be completed. Despite that,\nit is clear that in ferrites the exchange interactions are\nextremely strong and play a major role in the magnetic\nbehaviour. As seen in Figs. 4 and 5, even with a scaling\nofak= 2:2 for a value of ai= 0:3, a clearly exchanged\ncoupled behaviour can been achieved. Moreover, the size\nFIG. 3. Hysteresis loops for CFO@FO core-shell nanoparticle\nat 5 K. Loops are plotted for di\u000berent scaling factors ai,as\nandak.\nFIG. 4. Coercive \feld as a function of the interface scaling\nfactor aifor CFO@FO nanoparticles.\nof the particle is also extremely important as it does not\nallow for any kind of domain formation14.\nExperimental measurements have been performed on\ndense assemblies of CFO@FO and FO@CFO core-shell\nnanoparticles9. Under these experimental conditions, the\ndipolar interaction can a\u000bect the magnetic response, es-\npecially for NPs composed of CFO and FO due to their\nconsiderable magnetic moments. As the size of the parti-\ncles is below the coherent radius limit, in order to study\nthe assembly in a computationally e\u000ecient form, we de-\nvelop a coarse-grain macrospin model for each nanopar-\nticle. To be more speci\fc, we used 3 to 6 macrospins\nper particle. Under this level of theory, we recast the\nHamiltonian in the following form:5\nFIG. 5. Variance of the normalized remanent magnetization\nfor CFO@FO nanoparticle with respect to interface scaling\nfactor ai.\nFIG. 6. Hysteresis loops for FO@CFO single nanoparticle at\n5 K. Loops are plotted for di\u000berent scaling factors ai,asand\nak.\nH=\u0000X\ni;jJijSiSj\u0000X\nm;nJinterMmMn+X\nm;nMm[D]Mn\n\u0000X\nik(cos)2\u0012i\u0000g\u0016BX\niSiB (3)\nwherefSigrepresent the moment of macro spins within\na nanoparticle while fMngis the net moment of\nparticlen, so that, indices fi;jgdenote summation\nwithin a nanoparticle (intra-nanoparticle interaction)\nwhilefm;ngdenote summation between nanoparticles\n(inter-nanoparticles interaction). The parameter Jijis\nthe Heisenberg interaction coupling between macrospins\niandjwithin the nanoparticle while Jinter describes the\ninterparticle interactions of the Heisenberg form. The\nthird term in Eq. (3) represents dipolar interactions be-\ntween nanoparticles with [ D] being the dipolar tensor andit is calculated as an interaction between the net moment\nof each particle Mn. The magnetic anisotropy constant is\nrepresented by kand\u0012iis the angle between Siand easy\naxis direction while the last term represent the Zeeman\ncontribution with Bbeing the applied external magnetic\n\feld.\nFor each nanoparticle, the total magnetic moment is\nsubdivided into Nmacrospins, N1for the core and N2\nfor shell, and the sum of the macrospins for the core and\nshell equals the total moment per nanoparticle, as deter-\nmined from bulk magnetometry. Thus, the total moment\nper nanoparticle is divided between the core and shell\ncontribution, separately. For the CFO@FO core-shell\nnanoparticle, the core and shell moments are estimated to\nbe 4:14\u000210\u000020Am2and 12:38\u000210\u000020Am2, respectively;\nwhile for FO@CFO nanoparticle, the core and shell mo-\nments are 6 :80\u000210\u000020Am2and 10:74\u000210\u000020Am2,\nrespectively. The multiple macrospins per core and shell\ncan simulate the e\u000bects of spin canting at the core-shell\nand shell-vacuum interface. The ensemble of nanopar-\nticles is simulated by placing the macrospins for each\nnanoparticle on a 12 \u000212\u000212 grid ( i.e.,123nanoparti-\ncles) with a mean spacing of 10 nm between nanoparticles\nand periodic boundary conditions.\nIn order to de\fne the interaction parameters, we\nperform a \ftting procedure of the calculated single-\nnanoparticles energy. The atomistic energy of the sys-\ntem is calculated for di\u000berent magnetic con\fgurations\nand mapped back into the macrospin model. The toler-\nance of the \ftting procedure ensures that all \ftted pa-\nrameters have an error smaller than 10\u00004meV/atom. In\nTables III and IV, we present the parameters used for\nthe macrospin model. For both compounds, the indices\nf1\u00003grefer to the core and f4\u00006gto the shell. In\nall cases, the anisotropy was assumed to be uniaxial for\neach nanoparticle, and its axis was randomly selected per\nparticle in the assembly of nanoparticles. The interpar-\nticle interaction, Jinter, was set to 0 :1 mRy. With the\naim to reduce the number of free \ftting parameters, the\nmacrospins labelled as f1gfor the core and f4gfor the\nshell, are imposed to interact with the same exchange\ninteraction strength (see Table III).\nIn Table IV, the anisotropy contributions are distin-\nguished between core and shell. FO region shows a con-\nsiderable enhancement when it is interfaced with CFO,\nthus reducing the volume of the soft phase in each\nnanoparticle. The CFO region, being the harder mag-\nnetic phase, is the most probable nucleation region which\nsparks the magnetization reversal process. According to\nthis model, the nucleation process is initiated from the\nsurface of the CFO@FO or the core of the FO@CFO.\nThe application of at least three macrospins per material\nallows us to implicity take into account by a mean mag-\nnetization state the incoherent states on the interface or\nsurface.\nFigure 7 shows the calculated hysteresis loops from\nthe current macro spin model. The agreement with ex-\nperimental data9is remarkably good, especially in low6\nTABLE III. Heisenberg interaction constant Jijfor the\nmacrospin model. All Jijare given in units of mRy .\nCompound J12J13J23J24J34J25J35J45J46J56\nCFO@FO 3.4 3.4 -2.1 2.3 1.9 2.2 1.8 2.4 2.4 -1.8\nFO@CFO 3.1 3.1 -1.8 2.1 2.2 2.3 1.7 2.8 2.8 -1.9\ntemperature regime. The small value used for Jinter in-\ndicates that even with a dense ensemble of particles, the\nmagnetic behaviour is dominated by the intra-particle\ncharacteristics. For low temperatures, both samples are\ncharacterized by a knee behaviour appearing right after\nthe remanent state. Even though we have assumed a non-\nperfect interface, this behaviour is not so pronounced for\nthe single nanoparticles. Thus, for the case where the\nnanoparticles are grouped forming an assembly, it is pos-\nsible to attribute a random anisotropy axis distribution\ncreating now a range of energy barriers to be bypassed in\nthe cycle to complete the magnetization reversal. The in-\ntroduction of dipolar interactions has an e\u000bect, that the\ncoercive \feld drops down to values of about 1 :1T\u00001:3T.\nThis value is clearly lower than the one calculated for\nsingle nanoparticles. Even though the assembly is dense,\ndipolar interactions do not dramatically decrease the co-\nercive \feld due to the moderate magnetic moment of the\nnanoparticles. It has to be noted here that even though\nthe assembly is quite dense, we managed to reproduce the\nexperimental data by using a macrospin model. This fact\nmeans that we do not need to take into account the dis-\ntribution of the dipolar \felds because the intra-particle\ninteractions are quite strong and, as already explained\nabove, there cannot be any kind of magnetic domain for-\nmation in the nanoparticles. The magnetization reversal\nprocess depends strongly on the incoherent modes on the\ninterface and surface arising from the exchange strength\nvariations.\nThe accuracy of the current model decreases as the\ntemperature of the system is increased (cf. Fig. 7). For\nthis reason, we also simulate the the ZFC/FC curves of\nthe assemblies as shown in Fig. 8. Notably the pro\fle of\nthe ZFC/FC curves is reproduced quite well with the cur-\nrent macrospin model however, in both cases, the block-\ning temperature TBis overestimated. The latter e\u000bect\nis more pronounced in the FO@CFO variant where the\nTBis overestimated by 30 K. This overestimation is also\nfound in the loop simulations as the coercive \feld is over-\nestimated in all cases for temperatures larger than 5 K.\nThe latter is a clear indication that there is a tempera-\nture dependence of the magnetic parameters, disregarded\nin the current model.\nIII. Summary and Conclusions\nIn conclusion, using progressively more complex mod-\nels beginning with the case of isolated single nanoparti-TABLE IV. Anisotropy constant kfor the macrospin model.\nThe units of the uniaxial magnetocrystalline anisotropy are\ngiven in mRy .\nCore Shell\nCompound k1 k2 k3 k4 k5 k6\nCFO@FO 7.1 13.1 13.1 8.1 7.6 0.1\nFO@CFO 0.8 6.4 6.4 9.2 8.1 8.0\nFIG. 7. Hysteresis loops for CFO@FO (left column) and\nFO@CFO (right column) assembly nanoparticles at several\ntemperatures (5 K, 75 K and 100 K). In blue is shown the ex-\nperimental hysteresis loop taken from Ref. [9] for comparison\nwith the prediction of the theoretical model.\ncles, we have investigated the magnetic properties of fer-\nrite nanoparticles. The proposed macrospin model is able\nto describe substantially well the experimental measure-\nments of an assembly of such core-shell nanoparticle sys-\ntems. By taking the bulk values of the exchange interac-\ntions and magnetocrytalline anisotropy, we discussed the\ncalculated hysteresis loops in terms of the ratio between\nthe surface anisotropy with respect to the core anistropy\nfor CFO and FO nanoparticles (cf. Fig. 1 a)). By gaining\nsome information about the internal parameters of iso-\nlated nanoparticles, we improved the model by consider-\ning regions with di\u000berent magnetic texture, so that, the\nparticles now have a core, interface, shell and surface (cf.\nFig. 1 b)). This improvement requires consideration of\nmore types of exchange coupling between the di\u000berent re-\ngions as well as magnetocrystalline anistropies per region.\nBy scaling exchange interactions and anisotropies in the\ninterface and surface, we can get information about the\ndetails of the hysteresis loops so that we have full control\nover features seen in the experimental hysteresis loops,\nsuch as, the knee present in the vicinity of the remanent\n\feld that are measured in experiments for an assembly of\nnanoparticles (cf. Fig. 7). In the last stage of the macro-7\nFIG. 8. Zero-\feld cooling (ZFC) and \feld cooling (FC) curves\nfor the assembly nanoparticles CFO@FO (left panel) and\nFO@CFO (right panel). The predictions of the theoretical\nmodel (circle symbol) are compared with the experimental\nmeasurements taken from Ref.[9] (solid line).\nmagnetic model, we brought the core-shell nanoparticles,\nwith their internal structure, all together forming a cubic\ncrystal structure (see Fig. 1 c)). This system is describe\nby the hamiltonian of Eq. (3). In doing so, the model is\ncapable of describing with high \fdelity the experimental\nhysteresis loops, ZFC and FC curves. Overall, experi-\nmental and theoretical results are in close agreement al-\nthough discrepancies in the hysteresis loops increase at\nhigher temperatures. It is well-known in literature that\nthe Heisenberg exchange interaction is a function of the\ntemperature15. We speculate here that the small devi-\nation of the calculated hysteresis loops with respect to\nthe experimental ones at 100 K is due to the fact we donot include a temperature dependence in the estimated\nHeisenberg exchange interactions.\nOverall, and based on the good description of the ex-\nperimental results, the current model underpins the pro-\nposed internal structure of the nanoparticles, not only\nwith the core and shell, but also with two thin layers\nof interface and surface, so that, the interface plays an\nimportant role in describing the observed knee in the ex-\nperimental hysteresis loops.\nAcknowledgments\nThe computations were enabled by resources provided\nby the Swedish National Infrastructure for Computing\n(SNIC) at Chalmers Center for Computational Science\nand Engineering (C3SE), High Performance Comput-\ning Center North (HPCN), and the National Supercom-\nputer Center (NSC) partially funded by the Swedish Re-\nsearch Council through grant agreement no. 2016-07213.\nNN and MP acknowledge \fnancial support from the\nKnut and Alice Wallenberg Foundation through grant\nno. 2018.0060. HS and MHP acknowledge support (syn-\nthesis and magnetic measurements) from the US Depart-\nment of Energy, O\u000ece of Basic Energy Sciences, Divi-\nsion of Materials Sciences and Engineering under Award\nNo. DE-FG02-07ER46438. This material is based upon\nwork supported by the National Science Foundation un-\nder Grant No. ECCS-1952957. DAA acknowledges the\nsupport of the USF Nexus Initiative and the Swedish\nFulbright Commission. We also thank Joshua Robles for\nassistance with the nanoparticle synthesis.\n\u0003Corresponding author: manuel.pereiro@physics.uu.se\n1I. Shari\f, H. Shokrollahi and S. Amiri, Ferrite-based mag-\nnetic nano\ruids used in hyperthermia applications, J.\nMagn. Magn. Mater. 324, 903 (2012).\n2S. Y. Srinivasan, K. M. Paknikar, D. Bodas and V.\nGajbhiye, Applications of cobalt ferrite nanoparticles\nin biomedical nanotechnology, Nanomedicine 13, 1221\n(2018).\n3M. W. Mushtaq, F. Kanwal, M. Imran, N. Ameen, M.\nBatool, A. Batool, S. Bashir, S. M. Abbas, A. Ur Rehman,\nS. Riaz, S. Naseem and Z. Ullah, Synthesis of surfactant-\ncoated cobalt ferrite nanoparticles for adsorptive removal\nof acid blue 45 dye, Mater. Res. Express. 5, (2018).\n4H. Zhu, S. Zhang, Y.-X. Huang, L. Wu and S. Sun,\nMonodisperse MxFe3\u0000xO4(M = Fe, Cu, Co, Mn)\nNanoparticles and their electrocatalysis for oxygen reduc-\ntion reaction, Nano Lett. 13, 2947 (2013).\n5A. Quesada, C. M. Granados, O. Lopez, A. Erokhin, S.\nLottini, E. Pedrosa, J. Bollero, A. Aragon, Ana M., F.\nStingaciu, M. Bertoni, C. de Juli\u0013 an Fern\u0013 andez, C. Sangre-\ngorio, J. F. Fernandez, D. Berkov, and M. Christensen,\nEnergy Product Enhancement in Imperfectly Exchange-\nCoupled Nanocomposite Magnets, Adv. Elect. Mater. 2,\n1500365 (2016).6Ferrite nanoparticles: Synthesis, characterisation and ap-\nplications in electronic device\", Materials Science and En-\ngineering: B 215, 37 (2017).\n7M. A. G. Soler and L. G. Paterno Nanostructures (William\nAndrew Publishing, 2017)\n8J. M. D. Coey, Magnetism and Magnetic Materials (Cam-\nbridge University Press, 2012).\n9C. Kons, K.L. Krycka, J. Robles, N. Ntallis, M. Pereiro,\nM.-H. Phan, H. Srikanth, J.A. Borchers and D.A. Arena,\nSpin Canting in Exchange Coupled Bi-Magnetic Nanopar-\nticles: Interfacial E\u000bects and Hard/Soft Layer Ordering,\narXiv:2105.11501 [cond-mat.mes-hall].\n10B. D. Cullity and C. D. Graham Introduction to magnetic\nMaterials (Willey, 2008)\n11Y.H. Hou, Y.J. Zhao, Z.W. Liu, H.Y. Yu, X.C. Zhong,\nW.Q. Qiu, D.C. Zeng and L.S. Wen, Structural, elec-\ntronic and magnetic properties of partially inverse spinel\nCoFe2O4: A \frst-principles study, J. Phys. D.: Appl.\nPhys. 43, 445003 (2010).\n12N. Da\u000be, F. Choueikani, S. Neveu, M. Arrio, A. Juhin, P.\nOhresser, V. Dupuis, P. Sainctavit, N. Da\u000be,F. Choueikani,\nS. Neveu, M. Arrio, A. Juhin, V. Dupuis, P. Sainc-\ntavit, Magnetic anisotropies and cationic distribution in\nCoFe2O4 nanoparticles prepared by co-precipitation route:8\nIn\ruence of particle size and stoichiometry, J. Magn.\nMagn. Mater. 460, 243 (2018).\n13C.M. Srivastava, G. Srinivasan and N.G. Nanadikar, Ex-\nchange constants in spinel ferrites, Phys. Rev. B 19, 1\n(1979).\n14N. Ntallis and K.G. Efthimiadis, Size dependence of\nthe magnetization reversal in a ferromagnetic particle, J.\nMagn. Magn. Mater. 99, 373 (2015).\n15A. Szilva, M. Costa, A. Bergman, L. Szunyogh, L. Nord-\nstr om and O. Eriksson, Interatomic Exchange Interac-\ntions for Finite-Temperature Magnetism and Nonequilib-\nrium Spin Dynamics, Phys. Rev. Lett. 111, 127204 (2013)."    },    {        "title": "1109.1915v1.Modélisation_multidomaine_du_comportement_magnéto_mécanique_des_aciers_dual_phases.pdf",        "content": "arXiv:1109.1915v1  [physics.class-ph]  9 Sep 201120èmeCongrès Français de Mécanique Besançon, 29 août au 2 septemb re 2011\nModélisation multidomaine du comportement\nmagnéto-mécanique des aciers dual-phases\nF.S. Mballa-Mballaa,b, O. Huberta, S. Lazrega, P. Meillandb\na.LMT-Cachan (ENS-Cachan/UMR CNRS 8535/UPMC/Pres Universu d Paris) 61, avenue du\nprésident Wilson 94235 Cachan Cedex\nb.ArcelorMittal Maizières Research BP 30320 - Voie Romaine F- 57283 Maizières-lès-Metz Cedex\nRésumé :\nLa microstructure des aciers dual-phases et leur comportem ent mécanique sont fortement sensibles\naux variations de procédé (traitements thermiques). Un con trôle en ligne par méthode magnétique est\nenvisagé, la mesure s’appliquant à un matériau soumis à un ét at de contrainte. Le dual-phase est\nun milieu biphasé (ferrite/martensite), où chacune des pha ses est considérée comme étant une sphère\nimmergée dans un milieu homogène équivalent. La modélisati on de chaque phase s’appuie sur un modèle\nmagnéto-mécanique couplé. Il s’agit d’un modèle monocrist allin explicite représentatif du polycristal\nisotrope de la phase considérée. La mise en place de règles de localisation permet la simulation du\nmilieu biphasé. Expériences et modèle sont comparés.\nAbstract :\nThe microstructure and mechanical behavior of dual-phase s teels are highly sensitive to the variation\nof the process (heat treatments). Online control by magneti c method is relevant. A measurement under\napplied stress must be considered. The dual-phase is a two-p hase medium (ferrite / martensite). Each\nphase can be considered as a sphere embedded in a homogeneous equivalent medium. The model used\nfor each phase is based on a magneto-mechanical coupled mode l. This is an explicit single crystalline\nmodel representative of the behavior of the corresponding p hase. Localization rules allow the simulation\nof the two-phases medium. Experiments and modeling are comp ared.\nMots clefs : comportement magnéto-mécanique, microstructure biphasé e, localisation.\n1 Introduction\nLes dernières années ont vues un intérêt grandissant des ind ustries automobiles pour l’utilisation\nd’aciers à haute performance tels que les aciers dual-phase s (DP). La production de ces aciers implique\nplusieurs procédés de fabrication : métallurgie primaire e t métallurgie secondaire, coulée, laminage (à\nchaud/froid) et traitements thermiques. Ces procédés cond uisent à une microstructure biphasée prin-\ncipalement composée d’îlots de martensite dure dispersés d ans une matrice ferritique ductile (figure 1)\nen proportion variable selon l’histoire thermo-mécanique du matériau. Leur microstructure (fraction\net composition des phases) étant fortement sensible aux var iations du procédé de fabrication (traite-\nment thermique, laminage), Les industriels cherchent à met tre en place une méthode de contrôle non\ndestructive en ligne. La méthode employée exploite les prop riétés magnétiques de ces matériaux. La\nmise en œuvre de ce procédé de contrôle rentre dans le cadre du projet ANR DPS-MMOD [1].\nLe travail présenté dans cette communication a été réalisé d ans le cadre de ce projet : il s’agit d’une\ncontribution à la modélisation du comportement magnéto-mé canique des aciers DP. La méthode de\nmodélisation retenue s’appuie sur le modèle multidomaine, développé au LMT-Cachan. Ce modèle\nest basé sur la partition en domaines magnétiques des matéri aux magnétiques. Il s’agit d’un modèle\nmonocristallin. On montre sous certaines hypothèses qu’il existe une direction de sollicitation pour\n120èmeCongrès Français de Mécanique Besançon, 29 août au 2 septemb re 2011\nFigure 1 – Microstructure d’un acier dual-phase : martensite en bla nc, ferrite en sombre.\nlaquelle le comportement obtenu est représentatif du polyc ristal isotrope du matériau concerné. Il\ns’agit, dans le cadre de l’application recherchée, de prend re en compte :\n– l’orientation du chargement magnétique dans le repère du m onocristal représentatif.\n– un chargement mécanique statique imposé et ses effets sur la mesure.\n– la nature biphasée de la microstructure de l’acier DP, ce qu i suppose de mettre en place des règles\nde localisation mécanique et magnétique adaptées.\n2 Modélisation multidomaine [2, 3]\nLes deux phases en présence (ferrite f/ martensite m) sont ferromagnétiques : elles s’aimantent et\nse déforment en présence d’un champ magnétique. Les deux pha ses sont modélisées séparément à\nl’aide d’un modèle de comportement proposé récemment : le mo dèle multidomaine [2]. Ce modèle\nprend son origine dans la configuration en domaines magnétiq ues des matériaux magnétiques. On\nconsidère un monocristal de fer de symétrie cubique divisé e n 6 familles de domaines notés α( axes\n<100>du cristal) correspondant aux 6 directions de facile aimant ation (figure 2a). On note /vector γα=\n(γα\n1γα\n2γα\n2)Tle vecteur directeur de chacun des domaines (1). Il est colin éaire au vecteur aimantation\n/vectorMα=Ms./vector γα, oùMsdésigne l’aimantation à saturation du matériau. Le vecteur aimantation est\ninitialement confondu avec les axes du cristal. Sous l’effet d’un chargement magnétique ou mécanique,\nles fractions de domaines ainsi que les directions d’aimant ation changent.\n/vector γα=/parenleftbig\ncosφαsinθαsinθαsinφαcosθα/parenrightbigT(1)\n(1)\n(2)(3)\n(4)(5)\n(6)y\nx zφcθc\n<100> <110><111>\nnc<001>\n<010>\nπ/4asin(3-1/2)acos(3-1/2)(a) (b)\nH\nσ\nφcθcnc(c)\nxyz\nFigure 2 – (a) représentation simplifiée du monocristal; (b-c) tria ngle standard et paramétrage du\nchargement.\nSi on utilise des hypothèses de déformation et de champ homog ène sur le cristal [4], l’énergie libre\ntotale par domaine Wαest la somme de trois contributions énergétiques :\n220èmeCongrès Français de Mécanique Besançon, 29 août au 2 septemb re 2011\n– L’énergie de champ : Wα\nH=−µ0/vectorH./vectorMα.\n– L’énergie magnéto-mécanique : Wα\nσ=−σ:ǫα\nµ.\n– L’energie magnétocristalline : Wα\nK=K1/parenleftbig\nγ2\n1γ2\n2+γ2\n2γ2\n3+γ2\n1γ2\n3/parenrightbig\nµ0est la perméabilité du vide ( 4π.10−7H/m),/vectorHetσsont le champ magnétique et le tenseur des\ncontraintes appliqués au cristal, ǫα\nµest le tenseur de magnétostriction du domaine considéré (2) ,K1\nest la constante d’anisotropie principale du matériau.\nǫα\nµ=3\n2\nλ100(γ2\n1−1\n3)λ111(γ1γ2)λ111(γ1γ3)\nλ100(γ2\n2−1\n3)λ111(γ2γ3)\nsym λ 100(γ2\n3−1\n3)\n (2)\n/vector nc=/parenleftbig\ncosφcsinθcsinθcsinφccosθc/parenrightbigT(3)\nCompte tenu de la symétrie cubique du cristal, une direction de chargement ( champ magnétique ou\ntraction uniaxiale) trouve son équivalent dans le triangle standard défini par les axes [100],[110]et\n[111]du cristal (figure 2b). On considère une direction de chargem ent/vector ncdéfinie par les deux angles\nsphériques φcetθc(3). Les directions de chacun des domaines minimisent par dé finition l’énergie libre\ntotale. Compte tenu de la restriction au triangle standard, il est possible de réaliser une minimisation\nanalytique. On aboutit par exemple aux expressions suivant es pour le domaine (1) de la figure 2a :\nφ1(H,σ,/vector n c) =µoMsHφc+arctan/parenleftbig3\n2λ111σsin(2φc)/parenrightbig\nµoMsH+2K1+3λ100σcos(2φc)\nθ1(H,σ,/vector n c) =π\n2−µoMsH(π\n2−θc)+arctan/parenleftbig3\n2λ111σsin(2(π\n2−θc)/parenrightbig\nµoMsH+2K1+3λ100σcos(2(π\n2−θc))(4)\navec/vectorH=H./vector ncet/vector σ=σ./vector ncles vecteurs champ magnétique et contrainte. Les fractions de chaque\ndomaine fαsont ensuite obtenues en utilisant une relation explicite i nspirée d’une fonction statistique\nde Boltzmann [4] :\nfα=exp(−As.Wα)/summationtext\nαexp(−As.Wα)(5)\nAsest un paramètre proportionnel à la susceptibilité initial eχ0du matériau tel que As=3χ0\nµ0M2\ns. Le\ncomportement étant considéré homogène dans le monocristal , les règles d’homogénéisation s’appliquent\npour le calcul des valeurs moyennes de l’aimantation et la dé formation :\n/braceleftbigg/vectorM=/summationtext\nαfα/vectorMα\nǫµ=/summationtext\nαfαǫα\nµ(6)\nOn aboutit à un modèle capable de rendre compte du comporteme nt magnéto-mécanique d’un mono-\ncristal. Or, puisque toutes les directions de chargement po ssibles peuvent être restreintes au triangle\nstandard, le comportement d’un polycristal isotrope est né cessairement donné par un chargement le\nlong d’une direction appartenant à ce triangle. La non linéa rité des phénomènes en présence implique\nque cette direction n’est pas une direction moyenne, et est s usceptible de changer d’orientation avec la\ncontrainte ou le niveau du champ magnétique imposé. On fait l ’hypothèse que ce changement est assez\npetit pour être négligé [3]. La direction moyenne théorique est obtenue pour φc= 39oetθc= 78o[5] .\n3 Modélisation d’un acier DP, localisation des champs\nLa présence de plusieurs phases de nature différente crée une perturbation locale du champ (magnétique\nou mécanique) imposé à chacune des phases. Le champ local ne v aut ainsi généralement pas le champ\n320èmeCongrès Français de Mécanique Besançon, 29 août au 2 septemb re 2011\nmoyen. On parle de localisation du champ macroscopique cons idéré. On considère le milieu biphasé\n(f,m). On note Iune inclusion, jouée alternativement par la phase fou la phase m.\nLe champ magnétique local /vectorHIvu par la phase Iest une fonction complexe du champ macroscopique\n/vectorHet des propriétés du milieu moyen. Dans le cas d’un problème d ’inclusion sphérique [4], et en utilisant\nune hypothèse de champ homogène par phase et de comportement linéaire, la relation de localisation\nprend la forme :\n/vectorHI=/vectorH+1\n3+2χo/parenleftBig\n/vectorM−/vectorMI/parenrightBig\n(7)\nχ0et/vectorMreprésentent la susceptibilité et l’aimantation du milieu moyen, /vectorMIétant l’aimantation de\nl’inclusion. L’extension au comportement non linéaire sup pose d’utiliser une susceptibilité sécante pour\nla définition de χ0, soitχ0=/bardbl/vectorM/bardbl//bardbl/vectorH/bardbl. Cette approche est appliquée à la microstructure biphasée\n(f,m). On obtient :\n/vectorHf=/vectorH+1\n3+2χo/parenleftBig\n/vectorM−/vectorMf/parenrightBig\n;/vectorHm=/vectorH+1\n3+2χo/parenleftBig\n/vectorM−/vectorMm/parenrightBig\n(8)\navec :\n/vectorH=ff/vectorHf+fm/vectorHm;/vectorM=ff/vectorMf+fm/vectorMm (9)\nLes champs /vectorHfet/vectorHmsont introduits en entrée de deux calculs multidomaines. Ce s calculs fournissent\nune aimantation par phase, puis une moyenne permettant une n ouvelle évaluation des champs locaux.\nCe processus est itéré jusqu’à stabilité des grandeurs magn étiques (illustré figure 4a).\nLa solution du problème d’inclusion d’Eshelby constitue la base de modélisation du comportement\ndes milieux hétérogènes en mécanique. La déformation de mag nétostriction ǫµ\nIest la déformation que\nsubirait l’inclusion considérée en l’absence de la résista nce exercée par la matrice. Aux contraintes\nd’incompatibilité s’ajoute une contrainte associée au cha rgement extérieur σ. Si de plus on décompose\nles déformations en une somme de déformation élastique et de déformation de magnétostriction et qu’on\nconsidère les constantes d’élasticité homogènes (même con stantes élastiques pour les deux phases), on\naboutit à la formulation suivante de la contrainte σIdans l’inclusion I:\nσI=σ+C(I−SE) : (ǫµ−ǫµ\nI) (10)\nCest le tenseur d’élasticité du milieu; SEest le tenseur d’Eshelby, ne dépendant que des paramètres\nmatériau de la matrice et de la forme de l’inclusion. ǫµest le tenseur de magnétostriction moyen. Cette\napproche est appliquée à la microstructure biphasée ( f,m) en considérant des modules et des formes\nidentiques. On obtient :\nσf=σ+C(I−SE) : (ǫµ−ǫµ\nf)σm=σ+C(I−SE) : (ǫµ−ǫµ\nm) (11)\navec :\nσ=ffσf+fmσm;ǫµ=ffǫµ\nf+fmǫµ\nm (12)\nLes champs σfetσmsont introduits en entrée de deux calculs multidomaines au m ême titre que\nles champs magnétiques. Ces calculs fournissent une déform ation et une aimantation par phase, puis\ndes moyennes permettant une nouvelle évaluation des champs mécaniques et magnétiques locaux. Ce\nprocessus est itéré jusqu’à stabilité des grandeurs. Resta nt dans un cadre mécanique linéaire, la mise\nen place numérique de cette deuxième localisation ne pose pa s de problème majeur.\n420èmeCongrès Français de Mécanique Besançon, 29 août au 2 septemb re 2011\n4 Résultats expérimentaux et modélisation\nUne première étape a consisté à identifier les paramètres du m odèle multidomaine pour les deux phases.\nNous disposons d’échantillons de fer pur et d’un acier XC100 trempé à l’eau dont la structure est 100%\nmartensitique. La figure 3 regroupe les résultats expérimen taux obtenus pour les deux matériaux (com-\nportements magnétique et magnétostrictif). On se reporter a à [5] pour les procédures expérimentales.\nLes simulations à l’aide du modèle multidomaine monophasé s ont présentées sur les mêmes figures. Le\ntableau 1 regroupe les paramètres utilisés. Ceux du fer sur s ont issus de la littérature. Les paramètres\nde la martensite sont mal documentés. Nous avons procédé à un e optimisation pour leur évaluation.\nLe coude de saturation est relativement mal décrit par le mod èle pour les deux matériaux. C’est un\ndéfaut associé à l’hypothèse de champ homogène intrinsèque au modèle multidomaine.\nFigure 3 – Comportements magnétique (a,c) et magnétostrictif (b,d ) du fer pur (a,b) et de la mar-\ntensite pure (c,d) - comparaisons modèle/expérience.\nθc(◦);φc(◦)λ100 λ111K1(J.m−3)Ms(A.m−1)As(m3.J−1)\nferrite 88;41 21 .10−6−21.10−64,8.1041,71.1063,5.10−3\nmartensite 90;36 3 .10−63.10−610.1041,05.1064.10−4\nTable 1 – Paramètres utilisés pour la modélisation.\nUne fois les paramètres du comportement de la ferrite et de la martensite identifiés, il est possible de\nsimuler le milieu biphasé. On utilise une fraction de 42% de m artensite (fraction identifiée à partir\ndes micrographies). La figure 4 permet d’apprécier l’effet de la procédure de localisation en champ (le\ncalcul se fait en contrainte homogène). La figure 4a illustre la convergence des champs locaux après\nquelques itérations : la phase magnétiquement dure (marten site) voit un champ plus élevé que le champ\nmoyen; le champ perçu par la ferrite est plus faible que le cha mp moyen. La figure 4b montre l’effet de\ncette localisation sur le comportement magnétique. On cons tate que l’aimantation moyenne prévue est\nplus faible avec localisation. La figure 5 nous montre le résu ltat des simulations comparé aux résultats\nexpérimentaux. Comme on peut le voir, le comportement magné tique est bien décrit par le modèle,\n520èmeCongrès Français de Mécanique Besançon, 29 août au 2 septemb re 2011\npassé600A.m−1, on voit que la procédure de localisation n’a plus qu’un effet marginal. On constate que\nla déformation de magnétostriction est surestimée à aimant ation élevée. La procédure de localisation\nen contrainte devrait permettre d’améliorer ce résultat.\n010203040506070809010011.11.21.31.41.51.6x 104Hlocf (A/m) ferrite\niterationmartensite\nferrite(a)\n101102103104024681012x 105\nH (A/m)M (A/m)\n  \nExpérimental\nLocalisé\nNon localisé(b)\nFigure 4 – Illustration de la localisation ( fm=42%) - (a) convergence des champs locaux pour\nH= 1,25.104A.m−1-(b) Comportement magnétique : expérience et simulation.\n020004000600080001000012000024681012x 105\nH (A/m)M (A/m)\n  \nExpérimental\nLocalisé\nNon localisé(a)\n024681012\nx 10500.511.522.533.5x 10−6\nM (A/m)εµ\n  \nExpérimental\nLocalisé\nNon localisé(b)\nFigure 5 – Comparaison modèle/expérience pour le comportement mag nétique (a) et magnétostrictif\n(b) du dual-phase ( fm=42%)\n5 Conclusion\nLe modèle proposé permet un estimation rapide du comporteme nt magnéto-mécanique d’une micro-\nstructure biphasée. Cette estimation requiert de connaîtr e le comportement de chacune des phases.\nOn constate une surestimation du comportement magnétostri ctif, imputable à la non prise en compte\nde la localisation en contrainte ou à une modélisation trop s impliste (magnétostriction isotrope) de la\nmartensite. La technique développée servira à faire dialog uer des méso-modèles micromagnétiques [1].\nRéférences\n[1] ANR DPS-MMOD 2008 Dual-Phases Steel Modeling - MATETPRO 08 - 322447\n[2] S. Lazreg, O. Hubert 2009 Modèle multidomaines pour l’id entification inverse d’états mécaniques par méthode ma-\ngnétique Colloque national MECAMAT pp. 306-309\n[3] S. Lazreg, O. Hubert 2011 A multidomain modeling of the ma gneto-elastic behavior for non-destructive evaluation\nJ. Appl. Phys. 109doi :10.1063/1.3540416\n[4] L. Daniel, O. Hubert, N. Buiron, R. Billardon 2008 Revers ible magneto-elastic behavior : A multiscale approach. J.\nof the Mechanics & Physics of Solids 56pp.1018-1042\n[5] S. Lazreg 2011 Identification inverse d’états multiaxia ux élasto-plastiques par méthode magnétique Thèse de doctorat\n, ENS-Cachan\n60 2 4 6 8 10 12\nx 105−0.500.511.522.533.544.5x 10−6\nM (A/m)εµ\n  \nExpérimental\nLocalisé\nNon localiséb"    },    {        "title": "1408.1571v1.Deviations_from_cooperative_growth_mode_during_eutectoid_transformation__insights_from_phase_field_approach.pdf",        "content": "Deviations from cooperative growth mode during eutectoid transformation: insights from\nphase-field approach\nAccepted in Acta Materialia on 6thAugust, 2014\nKumar Ankita,b,\u0003, Rajdip Mukherjeea,b, Tobias Mittnachtb, Britta Nestlera,b\naInstitute of Materials and Processes, Karlsruhe University of Applied Sciences, Moltkestr. 30, 76133 Karlsruhe, Germany\nbInstitute of Applied Materials - Reliability of Components and Systems, Karlsruhe Institute of Technology, Haid-und-Neu-Str. 7, 76131 Karlsruhe, Germany\nAbstract\nThe non-cooperative eutectoid transformation relies on the presence of pre-existing cementite particles in the parent austenitic phase\nand yields a product, popularly known as the divorced eutectoid. In isothermal conditions, two of the important parameters, which\ninfluence the transformation mechanism and determine the final morphology are undercooling (below A 1temperature) and inter-\nparticle spacing. Although, the criteria which governs the morphological transition from lamellar to divorced is experimentally\nwell established, numerical studies that give a detailed exposition of the non-cooperative transformation mechanism, have not been\nreported extensively. In the present work, we employ a multiphase-field model, that uses the thermodynamic information from the\nCALPHAD database, to numerically simulate the pulling-away of the advancing ferrite-austenite interface from cementite, which\nresults in a transition from lamellar to divorced eutectoid morphology in Fe-C alloy. We also identify the onset of a concurrent\ngrowth and coarsening regime at small inter-particle spacing and low undercooling. We analyze the simulation results to unravel\nthe essential physics behind this complex spacial and temporal evolution pathway and amend the existing criteria by constructing a\nLamellar-Divorced-Coarsening (LDC) map.\nKeywords: Non-cooperative growth, Divorced eutectoid transformation, Phase-field method, Coarsening\n1. Introduction\nTimeTemperatureA1(a)Simulated !transformationTimeTemperatureA1ΔT(b)\nFigure 1: Typical spherodizing annealing heat treatment cycles [9]. (a) Sub-\ncritical annealing is carried out slightly below the A 1temperature and does not\ninvolve the formation of austenite. (b) Inter-critical annealing involves heating\nthe hypereutectoid steel to fully austenise it, with a small amount of cementite\nremaining undissolved and then, holding it just below A 1temperature. The\nfinal transformation product is known as the divorced eutectoid. The divorced\neutectoid transformation, that is numerically simulated in the present work (for\nthree di \u000berent undercoolings, \u0001T), is shown by the colored (thick) line.\nThe eutectoid transformation in steel involves the decom-\nposition of the parent austenite ( \r) into two product phases,\nferrite (\u000b-Fe) and cementite ( \u0012-Fe 3C). When both the prod-\nuct phases, evolve cooperatively, sharing a common growth\nfront with austenite, the morphology of the resulting product\nis lamellar, popularly known as pearlite [1, 2, 3, 4]. On the\n\u0003Corresponding author. Tel.: +49 721 608-45022.\nEmail address: ankit.kumar2@hs-karlsruhe.de (Kumar Ankit)\n(a)\n(b)Figure 2: Cooperative and non-cooperative growth regimes are observed during\neutectoid transformation in Fe-0.92C-0.66Si-1.58Mn-1.58Cr-0.12Ni-0.05Mo-\n0.178Cu (wt. %) alloy. Samples are austenized at 870\u000eC for 2.5 hours and held\nfor\u001865 minutes below the eutectoid temperature (at T 1and T 2). These are\nfinally quenched to ambient temperature. (a) Lamellar and (b) divorced eutec-\ntoid morphologies are obtained for T 1=710\u000eC and T 2=705\u000eC, respectively\n(private communication with Z.X. Yin and H.K.D.H. Bhadeshia).\ncontrary, under a given set of conditions (low undercooling\nand small inter-particle spacing of pre-exisiting cementite), the\n\u000b=\r advancing transformation front begins to pull-away from\ncementite, leading to the formation of a divorced eutectoid.\nHillert et al. [3] establish that a pearlitic colony comprises\nof inter-penetrating bi-crystals of ferrite and cementite phases.\nSteels with a fully pearlitic microstructure (0.8 wt.% C), find\nextensive application in the manufacture of ropes, where high\ntensile strength is desirable.\nManufacture of a significant proportion of engineering com-\nPreprint submitted to Acta Materialia August 13, 2018arXiv:1408.1571v1  [cond-mat.mtrl-sci]  7 Aug 2014ponents, obligate the use of steels with low hardness and good\nmachinability (for e.g. in ball-bearings [5]). Two, well-known\nspherodizing annealing heat treatment cycles, that are adopted\nto soften the pearlite, prior to machining, are shown in Fig. 1.\nThe sub-critical annealing involve the spherodization of the fine\npearlite, by holding the hypoeutectoid steel isothermally, just\nbelow the A 1temperature, as shown in Fig. 1(a). The driv-\ning force for morphological transition is the reduction in \u0012=\u000b\ninterfacial area. For softening hypereutectoid steels, intercrit-\nical annealing [Fig. 1(b)] is a more economical method (see\n[5] and references therein). The steel is fully austenised, such\nthat a small amount of cementite particles remain undissolved,\nand then held below A 1to generate a spherodized transforma-\ntion product (cementite particles embedded in ferritic phase),\npopularly known as divorced eutectoid microstructure , which\nis much softer than the lamellar counterpart i.e. pearlite [Fig.\n2(a)]. Experimental studies indicate that the presence of pre-\nexisting cementite particles in the parent austenitic matrix re-\nsults in the non-cooperation between the ferrite and cementite\nphases [6, 7, 8] and yields divorced eutectoid as the final trans-\nformation product [Fig. 2(b)].\nThe history of divorced eutectoid dates back to the time\nof Honda and Saito [10], who report the morphological de-\npendence of the final-transformed product (lamellar to com-\npletely spehrodized) on the austenising temperature. Oyama\net al. [6] describe a heat treatment schedule, that is adopted\nfor spherodizing a microstructure, comprising of a mixture of\npearlite and proeutectoid cementite. Verhoeven and Gibson\n[11] develop a theoretical framework (for binary Fe-C alloy)\nto establish the criteria, that governs the transition from lamel-\nlar to divorced eutectoid morphology. Luzginova et. al [12]\nstudy the influence of chromium concentration on the forma-\ntion of divorced pearlite in a hypereutectoid steel. Pandit and\nBhadeshia [13] amend the earlier theory of lamellar to divorced\neutectoid transition, by accounting for the di \u000busion of carbon\nalong the transformation front.\nIt is apparent from the brief literature survey, that much of the\ninvestigation of divorced eutectoid transformation is primarily\nlimited to experimental and theoretical studies. They delineate\nthe basic concept of the evolution mechanism, but unable to\nprovide the finer details required for tailoring the mictrostruc-\nture to achieve the desired properties (e.g. better machinabil-\nity). Therefore, a theoretical understanding of the complex evo-\nlution pathways during the divorced eutectoid transformation is\nparamount to comprehend the final microstructure, which is in-\ndispensable from a technological point of view.\nIn view of establishing a synergy between theoretical and ex-\nperimental studies concerning the eutectoid transformation, the\nphase-field method holds great promise in terms of the ability to\ndescribe the interface evolution in the di \u000busion length scale. In\nthe present article, we use a multiphase-field model [14] to scru-\ntinize (and amend) the existing theory by providing an in-depth\nunderstanding of the carbon redistribution mechanism, which\nhas profound implications in eventual optimization of the pro-\ncess control parameters related to heat treatment of steel. Based\non the insights from numerical simulations, our further inten-\ntion is to depict the interplay between two important parameters– (a) spacing between the pre-existing cementite particles and\n(b) undercooling, which can result in di \u000berent eutectoid mor-\nphologies.\nIn the following section, the phase-field model, used for the\npresent numerical simulations, is briefly outlined. The simula-\ntion results concerning the lamellar to divorced eutectoid transi-\ntion and the concurrent growth and coarsening regime are dis-\ncussed in subsections 3.1 and 3.2 respectively. In subsection\n3.3, we summarize the presented simulation results by con-\nstructing a lamellar-divorced-coarsening (LDC) map. Section\n4 concludes the article.\n2. Phase-field model\nThe multiphase-field model is a common di \u000buse-interface\napproach for studying microstructural evolution accompany-\ning phase transformations. The primary advantage of such a\ndi\u000bused-interface approach lies in the elegance with which it\ntreats moving boundary problems by obviating the necessity to\nexplicitly track the position of interfaces. In the present work,\nwe use this approach for numerical simulations, which is cou-\npled with CALPHAD database to study a binary Fe-C alloy sys-\ntem. The multiphase-field model equations, that are used in the\npresent study, is briefly outlined in this section. The reader is re-\nferred to the previous studies [14, 15, 4, 16] for a more detailed\ndescription of the model equations and numerical methods.\nThe evolution of phases is governed by the phenomenologi-\ncal minimization of the grand potential functional \n,\n\n(T;\u0016;\u001e)=\nZ\nV\"\n\t(T;\u0016;\u001e)+ \n\u000fa(\u001e;r\u001e)+1\n\u000fw(\u001e)!#\ndV; (1)\nwhere Tis the temperature, \u0016is the chemical potential vec-\ntor comprising of K\u00001 independent chemical potentials, \u001eis\nthe phase-field vector containing the volume fractions of the N-\nphases and\u000fis the length scale related to the interface. a(\u001e;r\u001e)\nandw(\u001e)represent the gradient and obstacle potential type en-\nergy density, respectively and Vrepresents the domain volume.\nThe grand potential density \t(T;\u0016;\u001e), which is the Legendre\ntransform of the free energy density of the system f(T;c;\u001e)is\nwritten as an interpolation of individual grand potential densi-\nties\n\t(T;\u0016;\u001e)=NX\n\u000b=1\t\u000b(T;\u0016)h\u000b(\u001e)\n\t\u000b(T;\u0016)=f\u000b(c\u000b(T;\u0016);T)\u0000K\u00001X\ni=1\u0016ic\u000b\ni(T;\u0016); (2)\nwhere h\u000b(\u001e)is an interpolation function of the form h\u000b(\u001e)=\n\u001e2\n\u000b(3\u00002\u001e\u000b). The evolution equation for the N phase-field vari-\nables can be written as,\n\u001c\u000f@\u001e\u000b\n@t=\u000f \nr\u0001@a(\u001e;r\u001e)\n@r\u001e\u000b\u0000@a(\u001e;r\u001e)\n@\u001e\u000b!\n\u00001\n\u000f@w(\u001e)\n@\u001e\u000b\u0000@\t(T;\u0016;\u001e)\n@\u001e\u000b\u0000\u0003; (3)\n2where \u0003is the Lagrange parameter to maintain the constraintPN\n\u000b=1\u001e\u000b=1. The concentration fields are obtained by a mass\nconservation equation for each of the K\u00001 independent con-\ncentration variables ci. The evolution equation for the concen-\ntration fields can be derived as,\n@ci\n@t=r\u00010BBBBBB@K\u00001X\nj=1Mi j(\u001e)r\u0016j1CCCCCCA(4)\nMi j(\u001e)=NX\n\u000b=1M\u000b\ni jg\u000b(\u001e); (5)\nwhere each M\u000b\ni jrepresents the mobility matrix of the phase \u000b\n(related to the di \u000busivity). The function g\u000b(\u001e)is in general\nnot the same as h\u000b(\u001e)which interpolates the grand potentials,\nhowever, in the present description, we utilize the same. The\nthermodynamic data-fitting procedure to approximate the varia-\ntion of the grand-potential of the respective phases as a function\nof chemical potential and the relation of the numerical simula-\ntion parameters with the corresponding quantities in the sharp-\ninterface limit, are explained in the previous work [4].\n3. Results and Discussion\n3.1. Lamellar to Divorced transition\nAs we are primarily interested in amending the criteria which\ndetermines whether the eutectoid transformation front evolves\nby a cooperative (lamellar growth, which leads to the forma-\ntion of pearlite) or a non-cooperative mechanism (resulting in\ndivorced eutectoid), we use the same input parameters (volume\ndi\u000busion constants and surface energies) for the present phase-\nfield simulations that was used earlier by Ankit et al. [4] to\nsimulate a pearlitic morphology. In order to account for the role\nof di\u000busion of carbon along the transformation front simultane-\nously, the interface di \u000busion constant is assumed to be 1000\ntimes greater than volume di \u000busion constant in ferrite. The\ninterface relaxation coe \u000ecient is derived from a thin-interface\nanalysis which is described elsewhere [17, 14].\nWe study the temporal evolution of austenite, ferrite and ce-\nmentite phases which is governed by the initial particle spacing\nat intercritical temperature and the undercooling below the eu-\ntectoid temperature ( A1). The simulation domain width in the\ntransverse direction directly controls the spacing (represented\nby\u0015) while the radius of the particles is kept same for consis-\ntency of the numerical results. In order to compare the present\nphase-field results with the classical theories, which introduce a\ncriteria for lamellar to divorced transition based on experimen-\ntal findings [11, 5], we limit the present discussion to a sym-\nmetric arrangement of pre-existing cementite particles which\nare embedded in an austenite matrix. The undercooling below\nthe eutectoid temperature ( \u0001T) as well the particle spacing ( \u0015)\nis varied to study their e \u000bect on the resulting microstructure.\nFig. 3(a) shows the dependence of undercooling and parti-\ncle spacing in stimulating a transition from lamellar to divorced\nmorphology. It is noteworthy, that the numerical results ac-\ncentuate the experimental findings which emphasize a greater\nPeriodic boundary condition\nλλ\nPeriodic boundary conditionIsolated boundary condition\nIsolated boundary condition\nInitial phase configurationΔT = 10 K\nLarger spacing\nSmaller spacingλ = 1.1 µ m λ = 0.28 µ m \nΔT = 5 KHigher  \nundercooling\nLower  \nundercooling\nλ = 1.1 µ m Same spacingDifferent undercooling\nDifferent undercooling and spacing Divorced eutectoid microstructure(a)\n 45 54 63 72 81 90 99\n 4340 4375 4410 4445 4480 4515\n 0  25  50  75  100  125  150  175Chemical Potential\nDistanceDistance(b)Figure 3: (a) Numerically simulated microstructures at two di \u000berent undercool-\nings below the eutectoid temperature ( \u0001T=5 and 10 K) and particle spacing\n(\u0015=0:28 and 1:1\u0016m) starting from the same initial arrangement of the phases.\nThe diagram shows that a cooperative growth regime is favored at higher un-\ndercooling and spacing leading to the formation of pearlitic lamellae. At lower\nundercoolings and smaller particle spacings, a non-cooperative mechanism pre-\ndominates which results in the formation of a divorced eutectoid microstructure.\n(b) 1-D chemical potential profile for \u0001T=5K and\u0015=0:58\u0016m plotted along\nthe dashed line connecting the center of both the cementite particles. The profile\nshows that the carbon partitioned at the \u000b=\rtransformation front is incorporated\nby both the particles which results in non-cooperative eutectoid transformation.\ntendency of the ferrite-austenite interface to pull away from ce-\nmentite particles at low undercooling and small spacing. On the\ncontrary, at larger spacing and higher undercooling, a cooper-\native growth regime is favored which results in the formation\nof pearlitic lamellae. On analyzing the simulated chemical po-\ntential profile in 1-D as shown in fig. 3(b), it is apparent that\na divorced morphology forms due to the incorporation of parti-\ntioned carbon (at the advancing \u000b=\rtransformation front) into\nthe existing cementite particles. Thus, a near overlap of the\npresent simulation results with the existing theory demonstrate\nthe general capability of phase-field method in capturing the\ntopological changes during eutectoid transformation.\nFig. 4 compares the temporal evolution of the numerically\nsimulated isolines corresponding to interphase interfaces for a\nlamellar growth [Fig. 4(a)] and divorced eutectoid [Fig. 4(b)],\nstarting from a symmetric arrangement of cementite particles.\n3 0 50 100 150 200 250 300 350 400 450 500 550 600\n 0 50 100 150 200 250 300 350 400 0 15 30 45 60 75 90 105 120 135 150\n 0 10 20 30 40 50 60 70 80 90 100t = 0 st = 1.8x10-4 st = 5.4x10-4 st = 1.45x10-3 st = 2.25x10-3 st = 0 st = 3x10-5 st = 1.6x10-4 st = 2.5x10-4 st = 2.7x10-4 s(a)(b)\nDistance (X-grid points)Distance (X-grid points)Distance (Y-grid points)\n1 grid-point = 0.00279 μmFigure 4: Temporal evolution of the isolevels \u001e\u000b=0:5 (dashed lines representing \u000b=\rinterface) and \u001e\u0012=0:5 (solid lines representing \u0012=\rand\u0012=\u000binterfaces) for the\n(a) cooperative (resulting in pearlitic lamella) and (b) non-cooperative (resulting in divorced eutectoid) regimes. The pulling-away of the advancing ferrite-austenite\ninterface is evident from the numerically simulated isolevels shown in (b). A comparison of the temporal evolution of the isolines in (a) and (b) indicate that the\ninitial particle spacing ‘ \u0015’ (number of grid-points along x-axis) governs the switch between both the evolution regimes (for constant undercooling \u0001T=10K).\nAs the undercooling is constant for both the cases ( \u0001T=10K),\nthe evolution mode (cooperative or non-cooperative) is deter-\nmined by the initial particle spacing ‘ \u0015’ (represented by X-axes\nin Fig. 4). At a lower value of ‘ \u0015’ (0:27\u0016m), the\u000b=\r inter-\nface pulls-away from \u0012, more commonly known as, the non-\ncooperative growth. However, at a larger value of ‘ \u0015’ (1:11\u0016m),\nthe growing phases, \u000band\u0012maintain a common transformation\nfront, by evolving cooperatively.\n3.2. Concurrent growth and coarsening\nThe most phenomenal finding of the present numerical stud-\nies is the isolation of concurrent growth and coarsening regime\nduring eutectoid transformation. Fig. 5 shows the temporal\nevolution of phase contours which are overlaid on the chemical\npotential map, when the initial cementite spacing is reduced to\n0.294\u0016m at an undercooling of 5 K below the A1(eutectoid)\ntemperature. An intermittent coarsening regime sets in before\nthe pulling away of the \u000b=\rinterface from cementite particles.\nIn order to provide a detailed exposition of this newly identi-\nfied regime (which is not clearly visible in Fig. 5), we plot both\nthe phase contours separately [in Fig. 6(a)] as well as the 1-D\nchemical potential profile along the dashed-line [in Fig. 6(b)]\nfor di \u000berent simulation time-steps. Depending on the initial\ndistance from the ferrite-austenite transformation front, the ce-\nmentite particles are labeled as 1 and 2. On comparing the 1-D\nchemical potential profiles for two di \u000berent simulation time-\nsteps ( t1andt2), we find that a change in the carbon redistri-\nbution mechanism is stimulated which leads to coarsening of\nparticles prior to the divorce from the growth front.\nTo begin with, the \u000b=\r transformation front advances and\nforms an interface with the adjacent cementite particle 1. As\na result of this interaction, particle 1 starts to grow due to the\nincorporation of partitioned carbon primarily via the interface(transformation front) di \u000busion flux. It is noteworthy, that the\nparticle 1 which shares a common interface with ferrite expe-\nriences a greater influx of partitioned carbon as compared to\nparticle 2, since the interface di \u000busivity is assumed to be 1000\ntimes faster than the di \u000busion in austenite in all the present\ncases. As the di \u000busion fields of both the cementite particles\noverlap, particle 1 grows while the particle 2 shrinks, as shown\nin Figs. 6(a) and 6(c). At this stage, the driving force for coars-\nening predominates over the growth. The same is also reflected\n[Fig. 6(a)] by a temporal increase in the curvature of \u0012=\rinter-\nface of particle 1 which slowly approaches infinity and subse-\nquently curves inwards. An advancement of \u000b=\rtransformation\nfront towards particle 2 causes a shift in the carbon redistribu-\ntion mechanism again; the driving force for cementite growth\nexceeds coarsening. We attribute a reduction in the distance\nbetween\u000b=\r transformation front and particle 2 which makes\nthe incorporation of partitioned carbon feasible at smaller dis-\ntances via bulk di \u000busion flux. As a result, the chemical poten-\ntial near the advancing \u000b=\rfront ascends leading to the growth\nof particle 2. This change in the carbon redistribution mecha-\nnism which results in predominance of growth over a coarsen-\ning regime is evident from the 1-D plot shown in Fig. 6(b).\nIt is worth clarifying that the “concurrent growth and coars-\nening” regime (denoted by ‘C’) reported in the present work\nprincipally di \u000bers from the particle coarsening in alloys which\nhas been extensively reported in the literature [18, 19, 20, 21,\n22]. Although, the reported regime ‘C’ does involve curva-\nture driven coarsening of particles, the primary di \u000berence with\nthe phenomena of conventional coarsening is attributed to the\nenergetics of \u000b=\u0012=\r phase triple-junction which determines if\nthe transformation proceeds by a cooperative (to yield lamel-\nlar pearlite) or by a non-cooperative regime (yielding divorced\neutectoid). Further, the accompanying eutectoid transformation\n44331.284422.85\n 4432\n4432\nt = 0 sec t = 1.11x10-4 secλλ\n4337.064416.73\n4327.374565.36\nt = 5.76x10-4 sec t = 8.13x10-4 secFigure 5: Temporal evolution of the phase contours which are plotted over the corresponding chemical potential maps during concurrent growth and coarsening\nregime ( \u0001T=7:5 K and\u0015=0:294\u0016m). Coarsening can be observed clearly in Fig. 6.\n 0 9 18 27 36 45 54 63\nInitial\nCoarsening\nNo coarsening\n 4270 4305 4340 4375 4410\n 0  25  50  75  100Coarsening\nNo coarsening\nDistanceChemical potential Distance\n 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2\n 0.0005  0.0006  0.0007  0.0008R/R0\nTimeTime (s)R/R 0Particle 1Particle 2t0t2t1\nt2t1\nt2t1(a)\n(b)\n(c)\nFigure 6: (a) Phase contours showing the subsequent shrinkage and growth of\nparticle 2 as the \u000b=\rfront advances. (b) 1-D chemical potential profiles plotted\nalong the dashed line in (a) which shows a deviation in the carbon redistribution\nmechanism during temporal evolution (as seen at t1=6:44\u000210\u00004seconds and\nt2=7:79\u000210\u00004seconds). (c) Temporal evolution of the scaled radius ( R=R0) of\nparticle 2 illustrating sharp deviations in the trend. The corresponding contours\nof particle 1 are also plotted along side at di \u000berent time-steps which explains\nhow the temporal change in curvature of \u0012=\rinterface of particle 1 results in\nthe onset of growth and coarsening regimes respectively.modifies the e \u000bective curvature of \u0012=\rinterface of particle ‘A’\nwhich increases the rate of coarsening as depicted by a decline\nin normalized radius of particle ’B’ shown in Fig. 6(b). It can\nbe argued that the reported regime holds a close resemblance\nwith the discontinuous coarsening of grain boundary precipi-\ntates which could result in the formation of precipitate free zone\n(PFZ) along prior austenite grain boundaries [23]. However, on\na careful examination, it is apparent that the physics of tem-\nporally evolving interphase interfaces which is reported in the\npresent study is not only di \u000berent, but also more complex when\ncompared to the grain boundary interfaces involved in discon-\ntinuous coarsening.\n3.3. Lamellar-Divorced-Coarsening map (LDC)\n 1 2 3 4 5 6\n 0  5  10  15  20  25  30Spacing ( µm)\nUndercooling (K)L\nL\nD\nDD\nCC CL : Lamellar\nD: Divorced\nC : Coarsening\n 0.2 0.4 0.6 0.8\n 5  7.5  10D\nDD\nCCC\nFigure 7: A morphological transition map showing the predominance of lamel-\nlar (L), divorced (D) and concurrent growth and coarsening (C) modes during\nthe eutectoid transformation in Fe-C alloy system. The initial spacing between\nthe cementite particles (at the intercritical temperature) as well as the under-\ncooling below the A1temperature govern the switching among the three nu-\nmerically simulated modes.\nHaving numerically simulated and comprehended the ipse-\nity of the concurrent growth and coarsening regime which pre-\n5cedes the non-cooperative eutectoid transformation, we con-\nstruct a Lamellar-Divorced-Coarsening (LDC) transition map\nas shown in Fig. 7 to summarize the parametric study. The LDC\ntransition map generated by conducting phase-field simulations\nfor three di \u000berent undercoolings (5, 7.5 and 10 K) below the\neutectoid temperature and initial particle spacings predicts the\nmorphology that is favored for a given set of initial conditions\nduring an isothermal transformation. In a nut-shell, the most\nsignificant contribution of the work presented in the current let-\nter is the addition of an alphabet ‘C’ (acronym for concurrent\ngrowth and coarsening regime which is favored at smaller spac-\ning and lower undercooling) to the classical Lamellar-Divorced\n(LD) map [11, 13]. Further, the present numerical findings are\nalso in complete agreement with the existing theory for the di-\nvorced to lamellar morphological transition; lamellar morphol-\nogy being more favorable at large spacings and high undercool-\ning.\n4. Conclusions\nIn conclusion, the spacing of the cementite particles embed-\nded in the austenite matrix as well as undercooling below the\neutectoid temperature entirely determines the final microtruc-\nture. An in-depth phase-field study of the isothermal transfor-\nmation presented in this article, aids in selection of parameters\nto tailor the eutectoid microstructure appropriately. The present\napproach also captures the important transition between lamel-\nlar and divorced morphologies and sheds light on the change in\ncarbon redistribution mechanism which is primarily governed\nby initial configuration of the phases. The concurrent growth\nand coarsening regime is identified for the first time which may\nbe fundamentally di \u000ecult to isolate in experiments. Thus, the\npresent numerical studies provide new insights into the transfor-\nmation mechanism and amend the classical model of eutectoid\ntransformation.\nIn future, it will be interesting to study the influence of asym-\nmetrical arrangement of cementite particles on the final eutec-\ntoid morphologies. Large-scale numerical studies of the di-\nvorced eutectoid transformation for a random distribution of\nparticles needs to be conducted to facilitate a direct compari-\nson with the experimental microstructures.\nAcknowledgements\nThe authors thank Z.X. Yin and Prof. H.K.D.H. Bhadeshia\nfor the contribution of experimental microstructure and Prof.\nA. Choudhury for preliminary discussions. KA, RM and BN\nacknowledge the financial support of DFG in the framework of\nGraduate School-1483.\nReferences\n[1] Zener C. Kinetics of the Decomposition of Austenite. Wiley, NY , 1947.\n[2] Hillert M. Jernkont Ann 1957;147:757.\n[3] Hillert M. Decomposition of Austenite by Di \u000busional Processes. Inter-\nscience Publishers, New York, 1962.[4] Ankit K, Choudhury A, Qin C, Schulz S, McDaniel M, Nestler B. Acta\nMater 2013;61:4245.\n[5] Bhadeshia H. Prog Mater Sci 2012;57:268 .\n[6] Oyama T, Sherby O, Wadsworth J, Walser B. Scripta Metall Mater 1984;\n18:799 .\n[7] Syn C, Lesuer D, Sherby O. Metall Mater Trans A 1994;25:1481.\n[8] Tale \u000bEM, Syn CK, Lesuer DR, Sherby OD. Metall Mater Trans A 1996;\n27A:111.\n[9] O’Brien J, Hosford W. J Mater Eng Perform 1997;6:69.\n[10] Honda K, Saito S. J Iron Steel Inst 1920;102:261.\n[11] Verhoeven J, Gibson E. Metall Mater Trans A 1998;29:1181.\n[12] Luzginova N, Zhao L, Sietsma J. Metall Mater Trans A 2008;39 A:513.\n[13] Pandit A, Bhadeshia H. P Roy Soc A-Math Phy 2012;468:2767.\n[14] Choudhury A, Nestler B. Phys Rev E 2012;85:021602.\n[15] Molnar D, Mukherjee R, Choudhury A, Mora A, Binkele P, Selzer M,\nNestler B, Schmauder S. Acta Mater 2012;60:6961 .\n[16] Mukherjee R, Choudhury A, Nestler B. Model Simul Mater Sc 2013;\n21:075012.\n[17] Karma A, Rappel WJ. Phys Rev E 1996;53:R3017.\n[18] Mendoza R, Savin I, Thornton K, V oorhees P. Nat Mater 2004;3:385.\n[19] Wu D. Nat Mater 2004;3:353.\n[20] Ardell A, Ozolins V . Nat Mater 2005;4:309.\n[21] Clouet E, La ´e L, ´Epicier T, Lefebvre W, Nastar M, Deschamps A. Nat\nMater 2006;5:482.\n[22] Hoyt J. Nat Mater 2011;10:652.\n[23] Ahmadabadi M, Shirazi H, Ghasemi-Nanesa H, Nedjad S, Poorganji B,\nFuruhara T. Mater Design 2011;32:3526.\n6"    },    {        "title": "1811.04696v1.Current_direction_anisotropy_of_the_spin_Hall_magnetoresistance_in_nickel_ferrite_thin_films_with_bulk_like_magnetic_properties.pdf",        "content": "Current direction anisotropy of the spin Hall magnetoresistance in nickel\nferrite thin films with bulk-like magnetic properties\nMatthias Althammer,1, 2, a)Amit Vikam Singh,3Tobias Wimmer,1Zbigniew Galazka,4Hans Huebl,1, 2, 5\nMatthias Opel,1Rudolf Gross,1, 2, 5and Arunava Gupta3\n1)Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching,\nGermany\n2)Physik-Department, Technische Universit ¨at M¨unchen, 85748 Garching, Germany\n3)University of Alabama, Center for Materials for Information Technology MINT and Department of Chemistry, Tuscaloosa,\nAL 35487 USA\n4)Leibniz-Institut f ¨ur Kristallz ¨uchtung, 12489 Berlin, Germany\n5)Nanosystems Initiative Munich (NIM), 80799 M ¨unchen, Germany\n(Dated: 13 November 2018)\nWe utilize spin Hall magnetoresistance (SMR) measurements to experimentally investigate the pure spin current trans-\nport and magnetic properties of nickel ferrite (NiFe 2O4,NFO)/normal metal (NM) thin film heterostructures. We use\n(001)-oriented NFO thin films grown on lattice-matched magnesium gallate substrates by pulsed laser deposition, which\nsignificantly improves the magnetic and structural properties of the ferrimagnetic insulator. The NM in our experiments\nis either Pt or Ta. A comparison of the obtained SMR magnitude for charge currents applied in the [100]- and [110]-\ndirection of NFO yields a change of 50% for Pt at room temperature. We also investigated the temperature dependence\nof this current direction anisotropy and find that it is qualitatively different for the conductivity and the SMR magni-\ntude. From our results we conclude that the observed current direction anisotropy may originate from an anisotropy\nof the spin mixing conductance or of the spin Hall effect in these Pt and Ta layers, and/or additional spin-galvanic\ncontributions from the NFO/NM interface.\nThe advent of (spin) angular momentum transport without\nan accompanying charge current, i.e. the flow of pure spin\ncurrents, has led to the discovery of several remarkable ef-\nfects that are relevant for next generation spin electronic de-\nvices1–3. Amongst these effects is the spin Hall magnetore-\nsistance3–9in magnetically ordered insulator (MOI)/ normal\nmetal (NM) heterostructures, which enables the detection of\nnovel magnetic phases in MOIs10–12. While initial investiga-\ntions of the SMR heavily relied on yttrium iron garnet (YIG),\nthe report on the observation of the SMR in many other MOIs\nranging from ferrimagnetic5,13,14to antiferromagnetic15–19or-\nder confirms the universality of this effect. The magnitude of\nthe SMR effect crucially depends on the transparency of the\nMOI/NM interface as well as the spin Hall effect (SHE) and\nthe spin diffusion length of the NM. Nevertheless, the impact\nof the current direction with respect to the crystalline orien-\ntations on the SMR remains up to now unexplored. In this\npublication we experimentally show that the SMR amplitude\nin nickel ferrite thin films with bulk-like magnetic properties20\ninterfaced with Ta or Pt depends upon the relative orientation\nof the charge current jcompared to the NFO crystal axes.\nThe heterostructures investigated in this study are NFO/NM\nbilayers, where the NM is Pt and Ta. The ferrimagnetic NFO\nthin films (\u0019100 nm) are grown on (001)-oriented MgGa 2O4\n(MgGO) substrates via pulsed laser deposition. The bulk\nMgGO single crystals were obtained by the Czochralski\nmethod at the Leibniz-Institut f ¨ur Kristallz ¨uchtung, Berlin,\nGermany21, and substrates were then prepared by CrysTec\nGmbH, Berlin, Germany. During growth the substrate was\nkept at 700\u000eC in an oxygen atmosphere with 10 mTorr. For\na)Electronic mail: matthias.althammer@wmi.badw.dethe magnetotransport experiments we then defined NM Hall\nbar structures on top of the NFO with a width of 80 mm and\na length of 800 mm via optical lithography, sputter deposition\nof the NM and lift-off. For the NMs we used Ta and Pt layers,\nthat were deposited ex-situ in an ultra-high vacuum sputtering\nsystem with a base pressure of 2 \u000210\u00009mbar. The deposition\nwas carried out in an argon atmosphere at 5 \u000210\u00003mbar and a\ngrowth rate of 2 ˚A=s for both materials. The magnetotransport\nexperiments were carried out in two superconducting magnet\ncryostats at temperatures Tranging from 5 K to 300 K. One\ntransport setup is based on a 2D-vector magnet with magnetic\nfields limited to m0H=7 T and a second has full 3D magnetic\nfield vector control ( m0H\u00142:5 T). For the resistance mea-\nsurements we applied a DC current of 10 mA to the Hall bar\nand measure the longitudinal DC voltage drop. To rule out\nany spurious thermal voltages, we utilized the current reversal\ntechnique as detailed in Refs. 22 and 23.\n[010]\n[100] NFO\nMgGO[001],n\nh\n38° 40° 42° 44°100101102103104105\nMgGO(004)NFO(004)I(cps)\n2θPt(111)(b) (a)\nFIG. 1. (a) Obtained x-ray diffraction results from a 2 q-wscan on\na 100 nm thick NFO layer grown on a MgGO substrate and covered\nwith a 10 nm thick Pt layer. Reflections from the (001)-oriented NFO\nlayer, the MgGO substrate and the (111)-textured Pt are visible. (b)\nIllustration of the sample geometry used for transport experiments.\nOn top of the NFO layer two NM Hall bar structures are deposited\nwithjk[100]andjk[110]crystal orientations of the NFO thin film.arXiv:1811.04696v1  [cond-mat.mes-hall]  12 Nov 20182\nAs a first step we investigated the orientation of the sputter\ndeposited Pt layer on top of the NFO layer on (001)-oriented\nMgGO substrates by x-ray diffraction using a reference sam-\nple with a blanket 10 nm thick Pt film. The obtained results\nfor the 2 q-ware shown in Fig. 1(a). The sample exhibits re-\nflections from the (001)-oriented NFO thin film and the low\nintensity (111)-reflection of Pt suggests that Pt grows (111)-\ntextured on top of the NFO layer. Due to the low intensity of\nthe Pt reflection, we were unable to investigate any in-plane\nepitaxial relationship between Pt and NFO. For a Ta refer-\nence sample no reflections originating from the Ta layer were\nfound, thus we do not have information on the growth of Ta\non our NFO thin films.\nFor the magnetotransport experiments we utilized two dif-\nferently oriented NM Hall bars with respect to the crystalline\norientation of the NFO layer as illustrated in Fig. 1(b). This\nallows us to investigate the SMR for two different current di-\nrections: along the [100]-direction and the [110]-direction of\nNFO. For the study of the SMR in these samples, we used\nangle-dependent magnetoresistance (ADMR) experiments24.\nIn ADMR experiments an external magnetic field with fixed\nmagnitude m0His applied to the sample, while measuring the\nlongitudinal resistivity rlongof the Hall bar as a function of the\norientation of the magnetic field direction h=H=H. In our\nexperiments we rotated the external magnetic field in several\nplanes to investigate the observed magnetoresistance (MR).\nThe first plane is the in-plane (ip) rotation of the external mag-\nnetic field, where ais defined as the angle between the charge\ncurrent direction and h(See inset in Fig.2(a)). Four more ro-\ntation planes have been used in these experiments. Two rota-\ntion planes perpendicular to the two charge current directions\n(oopj, bas defined in the inset of Fig. 2(b)), and two rotation\nplanes residing in the plane defined by each charge current di-\nrection and the surface normal (oopt, gas defined in the inset\nof Fig. 2(c)). We determined the minimum value of rlongfor\neach ADMR measurement and calculated the relative MR am-\nplitude as the difference with respect to this minimum value\ndivided by this minimum value.\nIn Fig. 2 we show the ADMR results obtained for Pt and\nTa Hall bars on NFO thin films at 300 K and m0H=2:5 T.\nWe first look into the MR response of the Pt Hall bars for\nthe in-plane rotation plane (see Fig. 2(a)). Clearly, for both\nHall bars we observe two maxima and two minima over the\nfull 360\u000erotation and the MR follows a sin2-dependence. For\nthe both current directions we observe maxima in the MR for\nh?tand minima for hktin agreement with SMR theory3,8,9.\nHowever, the extracted maximum MR for the two current di-\nrections is different: For jalong the NFO [100]-direction we\nfind a maximum MR of 8 :1\u000210\u00004, while for jalong the NFO\n[110]-direction we obtain 1 :2\u000210\u00003. This is a 50% change\nin maximum MR for these two current directions and does not\noriginate from a difference in Pt resistivity as discussed below.\nIn addition, the MR for the [110]-direction is comparable to\nSMR values obtained for YIG/Pt heterostructures, where the\ncurrent was oriented along the [1¯10]- and the [1¯21]-direction\nof the YIG film3,5.\nTo further investigate wether the increase in maximum MR\nfor the in-plane rotation for jalong the NFO [110]-directionis only due to the SMR, we also conducted ADMR experi-\nments in the oopj- and oopt-configuration for both charge cur-\nrent directions. These results are shown in Fig. 2(b) for the\noopj-configuration and in (c) for the oopt-configuration. For\nthe oopj-configuration we see that the MR does not follow a\ntypical cos2-dependence, which can be explained by the large\nuniaxial anisotropy with the hard axis along the surface nor-\nmal for NFO20. Thus, for the applied field of 2 :5 T it is not\npossible to fully align the magnetization direction malong\nthe out-of-plane direction. Nevertheless, we observe distinct\nmaxima for h?tand minima for hkt. Again we find that\nin the oopj-configuration the maximum MR for jalong the\nNFO [110]-direction is larger than for jalong the NFO [100]-\ndirection. In the oopt-configuration we only observe a negli-\ngible angle-dependence of the MR signal (due to the fact that\nthe magnetic anisotropy in NFO still plays a role at the inves-\ntigated magnetic field magnitude) for both current directions,\nin agreement with SMR theory5,8. Thus, for both current di-\nrections we observe the typical SMR fingerprint in ADMR\nexperiments and can conclude that the observed current di-\nrection anisotropy of the MR originates from the SMR. We\nnote that similar results have been obtained in the investigated\ntemperature range from 5 K to 300 K for all three rotation\nplanes. Moreover, we conducted the same ADMR experi-\nments on several different NFO/Pt samples and always found\nthis charge current direction anisotropy of the SMR response,\nsuch that we can rule out sample thickness variations as well\nas changes in the resistivity of Pt as the cause for the observed\ncharge current direction dependence.\nFor comparison, we also investigated the current direction\nanisotropy in Ta/NFO Hall bar structures. The extracted MR\nis shown in Fig. 2(d) for the ip-, (e) for the oopj-, and (f) for\nthe oopt-configuration, respectively. Also for Ta we find an\nangle-dependence of the MR for the ip and oopj-configuration\nfor both current directions and negligible angle-dependence\nfor the oopt-configuration. Thus also for the Ta layer the sole\ncause for the observed MR is the SMR. In contrast to the Pt\nHall bars, the maximum MR is now larger for jalong the\nNFO [100]-direction (8 :4\u000210\u00004for the [100]-direction and\n7:7\u000210\u00004for the [110]-direction). The difference in the max-\nimum MR for Ta is small and thus also the current direction\nanisotropy of the SMR.\nIn order to further investigate this current direction\nanisotropy of the SMR we conducted ADMR experiments in\nall three orthogonal rotation planes for temperatures 5 K \u0014\nT\u0014300 K and a maximum external magnetic field m0H\u00147 T.\nTo extract the SMR amplitude from these measurements we\nsimulated the SMR response of rlongusing3,8,9:\nrlong=r0+r1(1\u0000mt)2; (1)\nwhere r0is the resistivity of the NM layer, when mis collinear\nto the spin polarization of the spin accumulation in the NM\nlayer induced by the SHE. The SMR amplitude is described by\nr1andmtis the projection of monto the t-direction ( t=n\u0002j).\nFor our simulations, we assumed r0to be field dependent,\nwhile r1is field-independent. For the determination of mag-\nnetization direction for each field direction we globally opti-\nmized the free enthalpy density normalized to the saturation3\n04812\nMR(x10-4)\nγh\njtn\nβh\njtn\n0° 90° 180° 270° 360°048\nMR(x10-4)\nγ(a)h||j h||t h||-n h||j h||-n\n(b) (c)\n(d)(e) (f)300 K, 2.5 T\n300 K, 2.5 Th||-t\nPt: Pt: Pt:\nTa:Ta: Ta:\nhjt\nα\nFIG. 2. ADMR data of a NFO(100)/Pt(3.5) bilayer (a)-(c) and a NFO(100)/Ta(5) bilayer (d)-(f) sample grown on a MgGO (001) substrate. The\ndata has been recorded at 300 K and an external magnetic field of 2 :5 T. In the plot, black squares and red diamonds represent the experimental\ndata for Pt and black up-triangles and red down-triangles represent the experimental data for Ta for the charge current direction along the\n[100]- and [110]-direction, respectively. For both materials and both charge current directions, we observe an angle-dependence of the MR\nin the in-plane and the perpendicular to current direction plane rotations, while negligible angle-dependence is visible in the third orthogonal\nplane. Thus the observed MR exhibits the symmetry fingerprint of the SMR.\nmagnetization of the NFO5,24:\nGM(m) =\u0000m0H(m\u0001h)+B001m2\n001+Bc(m4\n100+m4\n010+m4\n001);\n(2)\nwith B001the uniaxial out-of-plane anisotropy field, Bcthe\ncubic anisotropy field and mhklthe projection of monto the\n[hkl]-direction of NFO. For each temperature we then opti-\nmized a set of riandBiparameters until excellent agreement\n(reduced c2\u00141\u000210\u00006) between simulation and experimen-\ntal data was obtained in all rotation planes and for all m0H. For\nthe cubic magnetic anisotropy of the NFO thin film we found a\ntemperature independent value of Bc=10 mT, corresponding\nto in-plane magnetic easy axes along the [110]-direction and\n[1¯10]-direction, which agrees with ferromagnetic resonance\nstudies on samples grown under the same conditions20.\nTo better analyze the temperature dependence of the\nother parameters we first plot the SMR magnitude SMR =\nr1=r0(m0H=7 T)for Ta and Pt as a function of Tfor the\ntwo different charge current directions in Fig. 3(a). As evident\nfrom this plot, the current direction anisotropy persists for all\ninvestigated temperatures. For Pt, SMR is larger for jalong\nthe NFO [110] direction over the whole temperature range.\nAt low temperatures ( T\u001425 K), the difference in SMR mag-\nnitude for the current directions in Pt is smaller than at higher\ntemperatures. For the two Ta Hall bars we find that the SMR\nis larger for jalong the NFO [100] direction, albeit the differ-\nence is less pronounced than for Pt. Moreover, for T\u001475 K\ntheSMR in Ta is larger than in Pt, suggesting that Ta might be\nthe better choice for SMR investigations at low temperatures.\nFor the magnetic anisotropy determined from these ADMR\nexperiments, we find that B001monotonically increases with\ndecreasing temperature as illustrated in Fig. 3(b). Such a be-\nhavior could be either explained by the increase in saturation\n6912SMR (x10-4)\n0 50 100 150 200 250 30012B001(T)\nT(K)(a)\nj||[100]j||[100]\n(b)Pt:Ta:\nj||[110] Ta:j||[110]Pt:FIG. 3. Extracted simulation parameters from the temperature and\nfield-dependent ADMR experiments. (a) SMR as a function of\ntemperature for Pt with jalong the NFO [100]-direction (black\nsquares), [110]-direction (red diamonds), and for Ta with jalong\nthe NFO [100]-direction (black triangles), [110]-direction (red tri-\nangles). Over the whole temperature range the current direction\nanisotropy of the SMR amplitude persists for Pt and Ta. (b) Evo-\nlution of the uniaxial magnetic anisotropy parameter B001with tem-\nperature.\nmagnetization or due to strain effects caused by the differ-\nence in thermal expansion of the MgGO substrate and the\nNFO layer. As we do not observe any saturation behavior\nat low temperatures, which one would expect for the shape\nanisotropy contribution of a thin film, we conclude that B001\nis dominated by the strain in the NFO layer . This finding\nagrees well with the previous analysis of the uniaxial out-of-\nplane magnetic anisotropy in bulk-like NFO thin films20.\nIn order to further investigate the origin of the observed cur-4\n(a) (b)\nFIG. 4. Extracted temperature dependence of the ratios of the SMR\nmagnitude SMRNM[100]=SMRNM[110](black symbols) and conduc-\ntivities sNM[100]=sNM[110](blue circles) for (a) Pt and (b) Ta. The\nanisotropy ratios of the SMR and the conductivities exhibit different\ntemperature dependence ruling out any simple correlation between\nthese two anisotropic quantities.\nrent direction anisotropy of the SMR, we compared the tem-\nperature evolution of the ratios in SMR to the temperature de-\npendence of the ratio of the conductivities sfor the two differ-\nent current directions for Pt and Ta. The result of this analysis\nis shown in Fig. 4. As evident from Fig. 4(a), Pt also exhibits a\ncurrent direction anisotropy of the conductivity. However, this\nanisotropy in conductivity cannot be the only reason for the\nobserved SMR current direction anisotropy as the conductivity\nratio and the SMR ratio have a quite different dependence on\ntemperature. For sPt[100]=sPt[110]we observe a decrease with\ndecreasing temperature. In contrast, SMR Pt[100]=SMR Pt[110]\nshows a more complex non-monotonic temperature depen-\ndence. At high temperatures, the SMR ratio remains rather\nconstant at 0 :69, then starts to increase for T\u0014200 K and\nreaches a maximum value of 0 :76 for 10 K\u0014T\u001425 K. From\nthis we conclude that it is not possible to simply correlate\nthe observed current direction anisotropy of the SMR to the\nanisotropy of the conductivity for Pt.\nFor Ta as illustrated in Fig. 4(b), sTa[100]=sTa[110]remains\nabout constant over the whole temperature range with a value\nof 1:01. For SMR Ta[100]=SMR Ta[110], we find a slight decrease\nwith decreasing temperature from 1 :09 at room temperature\ndown to 1 :07 at T=5 K. Nevertheless, the evolution of these\ntwo ratios with temperature is rather different and thus again\nwe can not find an universal relation between the two for Ta.\nEven though we carried out temperature- and field-\ndependent ADMR experiments on the NFO/NM sample, the\norigin of the current direction anisotropy of the SMR in these\nsamples is difficult to determine. From our experiments we\nconclude that a texturing of the NM layer seems to increase\nthis anisotropy. In our opinion there are three possible con-\ntributions to the SMR that are responsible for the current di-\nrection anisotropy. One possibility could be that the spin mix-\ning conductance relevant for the spin current flow across the\nNFO/NM interface exhibits an anisotropy for the alignment\nof the spin orientation with respect to the NFO crystal. As\nthe direction of the spin orientation at the NFO/NM inter-\nface in the SMR experiments depends on the charge current\ndirection, the transparency of the interface could be differ-\nent for the two investigated charge current directions and thus\nthe SMR magnitude might also be different. However, to ourknowledge such an anisotropy of the spin mixing conductance\nhas never been experimentally observed or theoretically pos-\ntulated. A second possible mechanism would be an anisotropy\nof the spin Hall effect in the NM with respect to the charge\ncurrent direction. Thus, the amount of spin current gener-\nated depends on the charge current direction and leads to an\nanisotropy of the SMR amplitude. Such an anisotropy of the\nSHE has been theoretically predicted based on ab-initio cal-\nculations, but only for materials with hexagonal symmetry25.\nLast but not least, the bulk spin Hall effect is not the only\ncause for the conversion of a charge current into a pure spin\ncurrent in a NM. It is quite possible that contributions from\nthe spin-galvanic effect26–32arising at the NFO/NM interface\nalso give rise to additional contributions to the SMR. As the\nspin-galvanic effect is caused by spin-orbit fields as a result of\nthe broken inversion symmetry at the NFO/NM interface, it is\nquite possible that such a contribution depends on the charge\ncurrent direction. In such a scenario the combined action of\nbulk SHE and interfacial spin-galvanic effect will determine\nthe SMR magnitude and cause the observed current direction\nanisotropy, as previously observed in epitaxial Fe/GaAs het-\nerostructures33. Clearly, determining the origin of the cur-\nrent direction anisotropy requires more sophisticated experi-\nments, which is beyond the scope of this publication. How-\never, our first promising results suggest that understanding this\nanisotropy in the SMR may provide an additional pathway to\nengineer the spin current transport across MOI/NM interfaces.\nIn summary, we showed that the SMR from NFO thin\nfilms with bulk-like magnetic properties grown on MgGO\nsubstrates interfaced with Ta and Pt is comparable to results\nobtained on the prototype ferrimagnetic insulator YIG, such\nthat these NFO thin films are well suited for pure spin current\nexperiments in agreement with already published results34.\nOur results further illustrate that one can change the SMR\namplitude and thus also the amount of spin current across the\nMOI/NM interface by changing the charge current direction.\nWhile at the current stage of our investigations we can not\npinpoint the physical origin of the observed charge current di-\nrection anisotropy, we have to consider at least three possible\nreasons. First, it may originate from an anisotropy of the SHE\nin textured or even epitaxial NM layers. Second, spin-galvanic\ncontributions originating from the inversion symmetry break-\ning at the MOI/NM interface have to be taken into account.\nThird, an anisotropy of the spin mixing conductance in NFO\nmay explain the observed behavior. From this perspective,\nfurther experiments on fully epitaxial MOI/NM systems are\nexpected to allow for a further clarification of possible origins.\nMoreover, a more systematic investigation of the charge cur-\nrent direction anisotropy in these NFO/Pt bilayers may allow\nto find a clue to the underlying symmetry of the charge cur-\nrent direction anisotropy of the SMR. Our results presented\nhere open up a new avenue for engineering the charge current\nto spin current conversion in MOI/NM heterostructures.\nWe thank Timo Kuschel, Juan Shan and Jutta Schwarzkopf\nfor fruitful discussions. 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Asokan4\n1Department of Physics, Manonmaniam Sundaranar University, Tiru nelveli,\nTamilnadu, India\n2Nano Analysis Center, Korea Institute of Science and Technology, Seoul 136-791,\nRepublic of Korea\n3Pohang Accelerator Laboratory, Pohang University of Science an d Technology,\nPohang -790 784, Republic of Korea\n4Inter-University Accelerator Centre, Aruna Asaf Ali Marg, New D elhi-110067 India.\nE-mail:sgautam71@kist.re.kr(S.Gautam)\nAbstract. Magnetic nanomaterials (23-43 nm) of Cu xCo1−xFe2O4(x = 0.0, 0.5\nand 1.0) were synthesized by auto combustion method. The crysta llite sizes of\nthese materials were calculated from X-ray diffraction peaks. The b and observed\nin Fourier transform infrared spectrum near 575 cm−1in these samples confirm the\npresence of ferrite phase. Conductivity measurement shows the thermal hysteresis and\ndemonstrates the knee points at 475oC, 525oC and 500oC for copper ferrite, cobalt\nferrite and copper-cobalt mixed ferrite respectively. The hyster sis M-H loops for these\nmaterials were traced using the Vibrating Sample Magnetometer (VS M) and indicate\na significant increase in the saturation magnetization (M s) and remanence (M r) due\nto the substitution of Cu2+ions in cobalt ferrite, while the intrinsic coercivity (H c)\nwas decreasing. Among these ferrites, copper ferrite exhibits hig hest sensitivity for\nhumidity.\nPACS numbers: 61.05.cp, 61.46.-w, 75.60.Ej, 75.75.+a, 07.07.Df\nKeywords : X-ray diffraction, nanomaterials, magnetic properties, mixed fer rites, sensors\nSubmitted to: NanotechnologyMagnetic and humidity sensing properties of Cu xCo1−xFe2O4 2\n1. Introduction\nThe increased concern about environmental protection led to the development insensors\nfield. Apart from the technological importance ferrite materials ha ve shown advantages\nin the field of sensors due to its mechanical strength, resistance t o chemical attack and\nstability. Ferrites have spinel structure, which is mainly used in gas [1 , 2, 3], stress [4]\nand humidity [5] sensors. Humidity sensors are potentially in demand in industries like\ncloth driers, air coolers, broiler forming, cereal stocking and medic al field [6]. Humidity\nsensors based on the metal oxide materials have advantages such as low cost, simple\nconstruction and ease of placing the sensor in the operating enviro nment. The ability\nof a metal oxide to sense the presence of water molecules depends on the interaction\nbetween water molecules and surface of the metal oxide i.e. the rea ctivity of its surface.\nThe reactivity depends on the composition and morphological struc ture, which depends\non the preparation procedure. Ferrites can be prepared by sol-g el method [7], co-\nprecipitation method [8], hydrothermal method [1], milling [9], and self co mbustion\nmethod [10]. A review on the different humidity sensing mechanism and o perating\nprinciple for ceramics is reported in the literature [6]. Kamila Suri et alreported the\nhumidity sensing properties of α-Fe2O3and polypyrrole nanocomposites [5]. Tulliani et\nalalsoreportedthehumiditysensing propertiesof α-Fe2O3anddopingeffects[11]. Most\nofthehumidity sensors reportedinliteratureworks areatelevate d temperatures. Inthis\npaper a potential ceramic humidity sensor working at room tempera ture is investigated.\nThe structural, electrical and magnetic properties of copper, co balt and its mixed ferrite\nmaterials(Cu xCo1−xFe2O4withx=0 .0, 0.5and1.0)preparedbyself combustion method\nare reported. These nanoceramics have been used as humidity sen sors due to its porous\nnature created during the combustion process.\n2. Experimental Details\nCopper ferrite has been prepared by mixing copper nitrate, ferro us nitrate aqueous\nsolutions with citric acid in 1 : 1 stoichiometric ratio. The pH of the solut ion is adjusted\nto 7 using liquor ammonia. The obtained sol was then allowed to evapor ate in a beaker\nby keeping the solution temperature at 80-90oC and it results into high viscous gel. The\nresultant gel has been kept inside a preheated oven at 300oC. Within 5 to 10 minutes,\na large amount of gas is evolved according to the equation [12] and th e self combustion\nreaction has completed.\nC6H8O7+6NO3→6CO2+H2O+6OH−+6NO\nIn this citric acid acts as a fuel to produce the necessary bonding w ith metal ions and\nprevents the metal to precipitate as metal hydroxides [13].\nX-ray diffraction pattern have been taken in X’Pert PRO diffractome ter, using Cu-K α\nradiationofwavelength1.54 ˚Aandmicrostructureanalysiswascarriedoutonascanning\nelectron microsocope (SEM). The Fourier transform infrared (FT IR) spectrum for theMagnetic and humidity sensing properties of Cu xCo1−xFe2O4 3\nferrite samples and gel (copper ferrite) samples are recorded us ing Bruker Tensor 27 in\nthe region of 4000 cm−1to 400 cm−1. Conductivity measurements have been carried\nout using two probe method from room temperature to 700oC using Keithley source\nmeasure meter model 2400 on pellets having 13 mm diameter and 1 mm t hickness.\nThe temperature is varied from 30oC to 700oC in steps of 25oC for both heating\nand cooling cycles. The magnetic properties are investigated using E G & G PARC\n4500(USA) vibrating sample magnetometer(VSM). The humidity sen sing behaviour of\nthe material was measured with an indigenous set-up made of glass c hamber in which\nrelative humidity can be varied. The compressed air dehydrated ove r silica gel and\ncalcium chloride was directed into the chamber. The humidity level is va ried from 38 to\n95% by bubbling air through water and mixing it with dry air. These ferr ite samples of\n13 mm diameter and 1 mm thickness are placed in between two silver elec trodes in the\nchamber which are connected to the Keithley source meter model 2 400 to measure the\nchange in resistance with respect to relative humidity (RH).\n3. Results and discussions\n3.1. XRD Analysis\nFigure 1 shows the indexed x-ray diffraction (XRD) patterns of Cu xCo1−xFe2O4(x=0.0,\n0.5 and 1.0). Different peaks were identified by using the JCPDS datab ase (for copper\nferrite, JCPDS No. 25 0283). XRD patterns show the formation of single phase cubic\nstructure with dominant peak corresponding to (311) reflection in dicating that the\ncrystallites are preferentially oriented along (311) plane. The brea dth of the Bragg peak\nis a combination of both instrument and sample dependent effects. T o decouple these\ncontributions, it is necessary to collect a diffraction pattern from t he line broadening\nof a standard material such as silicon to determine the instrumenta l broadening.\nThe instrumented corrected broadening βhklcorresponding to the diffraction peak of\nCuxCo1−xFe2O4(x=0.0,0.5 and 1.0) was estimated by using the relation\nβhkl=/bracketleftbig\n(βhkl)2\nmeasured−(βhkl)2\ninstrumental/bracketrightbig2(1)\nUsing the βhklof XRD peaks the crystallite size is calculated by Scherrer’s formula\nDhkl=0.9λ\nβhklcosθhkl(2)\nWhereDhkl= volume weighed crystallite size, λ=wavelength of CuK α(1.54˚A ) and\nβhkl= instrumental corrected full width at half maximun (FWHM) of peak in radian.\nThe XRD pattern of Cu xCo1−xFe2O4was refined by Rietveld method using FullProf\nsuite [14], within FD3M space group and shown in Figure 2 for Cu 0.5Co0.5Fe2O4. The\naverage crystallite size is calculated for the three high intense refle ctions using Debye-\nScherrer’s formula for cobalt ferrite, copper cobalt ferrite and c opper ferrite are found\nto be 35 nm, 33 nm and 27 nm respectively. The lattice parameters ar e refined using\nPowderX [15] calculations for cobalt ferrite, mixed copper cobalt fe rrite and copper\nferrite are found to be 8.381 ˚A 8.372 ˚A and 8.37 ˚A respectively. The reflections (400)Magnetic and humidity sensing properties of Cu xCo1−xFe2O4 4\n10 20 30 40 50 60 70 80 90 100 110 120 XRD pattern           Cu xCo 1-x Fe 2O4 Intensity \n 2 θ (deg.) CuFe 2O4\nCo 0.5 Cu 0.5 Fe 2O4(533) (422) (111) \n(731) (440) \n(511) (400) (220) \n(311) \nCoFe2O4 \nFigure 1. X-ray diffraction patterns (Cu K α) for Cu xCo1−xFe2O4(x=0.0, 0.5 and\n1.0) nanoparticles.\nand (731) are excluded in refinement due to large residual values. I t is observed that\nwith increase in Cu content the lattice constant and unit cell volume d ecreases. The\ndecrease in lattice constant and unit cell volume is due to the smaller io nic radii of the\ndoped cation i.e. Cu2+(0.730˚A ) than that of Co2+(0.745˚A ). The increase in the\nX-ray density ( ρx−ray) is due to the increase in the molar masses of the doped sample\ni.e. Cu2+(63.55 g mol−1) as compared to Co2+(58.93 g mol−1).\nAfter the addition of Cu to cobalt ferrite a shift in most intense (311 ) peak is observed.\nIf the diffraction peak shift to the lower angles, a tensile stress can be realized, where as\na shift towards higher angles indicates a compressive stress [16]. Th e compressive strain\nalong [311] direction has been calculated using the following relation\n∆d/dundoped= (ddoped−dundoped)/dundoped (3)\nWhere ∆ dis the change in the d-spacing w.r.t. undoped sample (pure cobalt fer rite).\nThe strain is due to the substitution of Cu ion in place of Co ion as the ion ic radii of Cu\n(0.73˚A ) is less than Co (0.745 ˚A ). The strain calculated for (311) direction is given\nin Table 1. Since all the XRD patterns are recorded under the same e xperimental\nconditions therefore the crystalline nature of these materials can be compared by\ncalculating the degree of crystallinity (Nc) by using the relation\nNc= (Idoped−Iundoped)/Iundoped (4)\nWhereIdopedis the integrated intensity when x=0.5 and 1.0, Iundopedis the integrated\nintensity when x=0 (i.e. pure cobalt ferrite). A positive value of N cindicates the\nimprovement inthecrystallinity comparedwiththeundopedandnega tivevalueindicate\nthe decrease in crystallinity (Table 2).Magnetic and humidity sensing properties of Cu xCo1−xFe2O4 5\n15 30 45 60 75 90 105 120 Cu 0.5 Co 0.5 Fe 2O4\n Observed \n Calculated \n Obs.-Cal. \n Bragg's Positions  Intensity (arb. units) \n 2 θ (deg.)  XYN (X_1) \nFigure 2. Reitveld’s fitting for XRD pattern of Cu 0.5Co0.5Fe2O4nanoparticles with\n“goodness of fit” χ2= 0.218, Bragg’s R-factor=0.442 and RF-factor=0.351. The\ngraphs is plotted for observed points and calculated points on the u pper line. Below is\nthe difference between the two. Middle line points shows the Bragg’s p ositions for the\nFD3M space group, which calculates the values of lattice constants as a=b=c=8.372\nand angle=90o.\nTable 1. Hysteresisloopparametersforcopperferrite, cobaltferritean dcopper-cobalt\nmixed ferrite.\nMaterial Ms (emu/g) Hc (Oe) Mr (emu/g)\nCuFe2O4 20.00 838 11.34\nCoFe2O4 6.31 1951 3.42\nCu0.5Co0.5Fe2O4 9.06 1047 4.07\nTable 2. X-ray powder diffraction data of Cu xCo1−xFe2O4for most intense (311)\nreflection.\nConc.(x) 2 θ(deg.) d hkl˚A FWHM(deg.) D hkl(nm) Lattice Strain Crystallinity\nx=0.0 36.0221 2.49332 0.2480 33.686 — —–\nx=0.5 35.5226 2.52723 0.3306 25.234 0.01360 -0.713\nx=1.0 35.5275 2.52689 0.3306 25.234 0.01346 -0.475Magnetic and humidity sensing properties of Cu xCo1−xFe2O4 6\nFigure 3. SEM micrographs for Cu xCo1−xFe2O4(x=0.0, 0.5 and 1.0).(a) CoFe 2O4\n(b) Cu 0.5Co0.5Fe2O4(c) CuFe 2O4, each at 600nm and 10 µm scale respectively.\nSEM micrographs were used to see the grain micro-structure of th e nanoparticles,\nwhich would provide a better view of the grain development and grain s izes. SEM\nmicrographs are shown in figure 3(a-c) at different resolution scale s. From the\nmicrographs, it is clear that grains have also different morphologies t han spherical only.\nThe grain sizes measured by ImageJ (1.42q) for Cu xCo1−xFe2O4(x=0, 0.5 and 1.0) are\n8.913, 7.095 and 10.203 µm respectively.\n3.2. FT-IR spectroscopy\nThe FT-IR spectra of copper ferrite gel, copper ferrite, cobalt f errite and mixed cobalt-\ncopper ferrite samples are recorded in the range of 400-4000 cm−1and shown in\nfigure 4(a-d). In the spectrum of gel, peak at 1320 cm−1is due to NO 3vibration\n[17] indicating the presence of nitrate ions in the gel. This peak is not p resent in the\nferrite materials as seen in figure 4(b-d). The peaks at (1573 - 158 5 cm−1) exhibit the\npresence of citrate ions, chemically bounded to the metal atoms [1 8]. The intense bands\nobserved at 575 cm−1, 571 cm−1and 564 cm−1in copper ferrite, cobalt ferrite and mixed\nferrite respectively. The change in band position on going from one c oncentration to\nother may be due to change in the inter-nuclear distance of Fe3+-O2−in the equivalent\nlattice sites. These bands areattributed to the stretching vibrat ion of Fe3+-O2−andthis\nis the characteristic peak of ferrites [10]. This peak is not present in the copper ferrite\ngel and it reveals that ferrite phase is produced only after the com bustion reaction.Magnetic and humidity sensing properties of Cu xCo1−xFe2O4 7\n4000 3000 2000 1000 -50 050 100 \n  \nWavelength (cm -1 )050 100 \n571 \n(d) (c) % Transmittance 050 100 \n564 575  050 100 \n1585 1573 1585 \n1320  \n(b) (a) \nFigure 4. (color online) FTIR spectra for (a) copper ferrite gel (b) copper ferrite (c)\ncobalt ferrite and (d) copper-cobalt ferrite. The band near 575 c m−1in (b) to (d)\nsamples confirm the presence of ferrite phase.\n3.3. Conductivity studies\nFigure5(a-c)shows, theconductivity changeswithtemperature forcopperferrite, cobalt\nferrite and Cu 0.5Co0.5Fe2O4respectively. During the heating cycle, the conductivity of\nall the three samples increases as the temperature increases. Du ring the cooling cycle,\nthe conductivity decreases with the fall in temperature but follows a new path that\nleads to a thermal hysteresis. The observed step change in condu ctivity on cooling can\nbe attributed to the defects present in the pristine material which gets smoothened out\nduring the heating cycle. The conduction mechanism in ferrites is exp lained on the\nbasis of the Verwey de Boar mechanism [19] that involves exchange of electrons between\nthe ions of the same element having more than one valence state. At low temperature\nlow conductivity is observed which may be the result of large voids and less cohesion.\nBut the high conductivity at high temperature may be due to polaron hopping. It is\nreported that copper ferrite acts both as n- and p-type semicon ductors [20, 21]. The\ntwo competing mechanism may be due to the hopping of electrons bet ween Fe2+and\nFe3+ions and jumping of holes between Co2+and Co3+, and Cu2+and Cu1+as shown\nin the following redox reaction:\nFe2+→Fe3++e−, Co3+→Co2++e+(hole),\nFe2++Co3+→Fe3++Co2+Cu2+→Cu1++e+,\nFe2++Cu2+→Fe3++Cu1+(5)\nAt high temperature the fractions of Fe2+and availability of electrons will be much\nmore than at low temperature. Therefore the conduction at lower temperature is due\nto extrinsic type while at higher temperature is due to polaron hoppin g. Moreover weMagnetic and humidity sensing properties of Cu xCo1−xFe2O4 8\nhave observed a sharp change in the Arrhenius plot (not shown her e) for all samples,\nwhich is due to the change in the conduction mechanism. The activatio n energy for\nthe all sample is lower at low temperature as compared to high temper ature. In copper\n0.0 0.5 1.0 1.5 0 200 400 600 800 \n04812 16 \n0 200 400 600 800 04812 16  \n Conductivity (mho cm -1 ) Heating \n Cooling (a) \n(b) \n   Temperature oC(c) \nFigure 5. (color online) Variation of conductivity with temperature for (a) co pper\nferrite (b) cobalt ferrite and (c) copper-cobalt ferrite.\nferrite, figure 5(a), the heating and cooling cycle retraces the sa me path up to 475oC,\nbut gets separated in the region 475oC to 700oC. Curie temperature of copper ferrite\nis reported as 455oC in the literature [22]. Chao Liu et al, has reported that in cobalt\nferrite the phase transition from ferromagnetic to paramagnetic is observed at 517oC\n[23] whereas in the present work the knee point is observed at 525oC. The observed\nheating-cooling transition temperatures (Knee points) are 475oC, 525oC, 500oC for\ncopper ferrite, cobalt ferrite and copper-cobalt mixed ferrite re spectively.\n3.4. Magnetic Properties\nFigure 6 shows the hysteresis loop for copper ferrite, cobalt ferr ite and mixed copper-\ncobalt ferrite materials recorded using vibrating sample magnetome ter. Various\nmagnetic properties such as saturation magnetization (Ms) reman ence (Mr) and\ncorecivity (Hc) are calculated from the hysteresis loop and given in T able 2. Hysterisis\nloop for copper ferrite and cobalt ferrite shows that these two ma terials have net\nmagnetization even before applying the magnetic field. But the copp er-cobalt mixed\nferrite loses this property. Copper ferrite is known to be magnetic ally soft, with the low\ncoercive values at room temperature. Pure Cu2+ions are diamagnetic in nature and\nhence the formation of copper ferrite gives the low coercive values . In copper ferrite,Magnetic and humidity sensing properties of Cu xCo1−xFe2O4 9\nthe effect of variation of crystallite size (50 - 220 nm) on saturation magnetization\n(39 to 47 emu/g) is reported [24]. The observed Ms value of 20 emu/ g in the present\nwork can be linked to the decrease in crystallite size as evident from X RD analysis\nand it is in agreement with the work of Ahemed A. Farghali et al[24]. However the\nmeasured coercivity value of 838 Oe is much larger and it can be attrib uted to the\nstrength of magnetic moments formeddue to the self-combustion preparation technique.\nThe hysteresis loop for cobalt ferrite has a high coercive field of 195 1 Oe with small\nsaturation magnetization of 6.31 emu/g. Pure Co2+and Fe3+ions are ferromagnetic in\nnature. So in cobalt ferrite the bonding between A (tetrahedral) a nd B (Octahedral)\nsites lead to higher coercivity multidomain structure. Cannas et al[10] reported that\ncobalt ferrite prepared by self combustion method has 65.9 emu/g m agnetization and\n1550.8 Oe coercive field. Cobalt ferrite prepared by sol-gel method (800oC annealing\ntemperature) has 2020 Oe coercivity and 76.5 emu/g Ms as reporte d by Jae Gwang Lee\net al[7]. Yu Qu et alreported that the value varies from 3.3 emu/g to 29.5 emu/g\nfor different annealing temperature and a maximum coercive field of 1 180 Oe [25]. The\nabove discussion shows that the magnetization and coercive field va lues strongly depend\nonthe preparationtechnique andtemperature. The mixed ferrite , Cu0.5Co0.5Fe2O4, has\n-8000 -4000 0 4000 8000 -20 -10 010 20 Magnetization (emu/g) \nApplied field (Oe)  CuFe 2O4\n CoFe 2O4\n CuCoFe 2O4Temp =300 K \nFigure 6. (color online) B-H hysteresis loop for copper ferrite, cobalt ferrit e and\ncopper-cobalt ferrite at room temperature.\nintermediate values in both magnetization and coercivity. This shows that mixing of\ncobalt with copper ferrite increases the H cvalues and decreases the M svalues. The\naddition of high coercive Co2+with copper ferrite, leads to the A-A (Co2+& Cu2+)\ninteraction in the tetrahedral site and create the single domain sys tem, which requires\ngreater energy rather thanthe movement of walls (multi domain). Thus the involvement\nof Cu2+ions essentially decreases the net coercive values in mixed ferrites. The presence\nof Co2+and Cu2+ions at the same site has also been discussed through x-ray absorp tionMagnetic and humidity sensing properties of Cu xCo1−xFe2O4 10\nspectroscopy elsewhere [26].\n3.5. Humidity sensor\nThe ferrite materials are porous in nature and have surface oxyge n atoms which\nessentially arise due to the sample preparation technique. When the material adsorbs\nthe humidity, its resistivity decreases due to the increase of charg e carriers, protons, in\nthe ferrite and water system [27]. The adsorption of water on the s urface of the material\nleads to the dissociation of hydrogen ions. These hydrogen ions bon ded with the surface\nlattice oxygen atom, forms the hydroxyl groups [5] as shown in the equation\nH++Oo↔[OH]−\nwhere O ocorresponds to oxygen at lattice sites. The hydroxyl groups thu s produced\nare bonded with the lattice iron atoms and liberate the free electron s [28].\n[OH]−+Fe↔[OH−Fe]+e−\nThus conductivity increases with increase in humidity because of the production of free\nelectrons.\nFigure 7(a-c) shows the response magnitude of the copper ferrit e, cobalt ferrite and\ncopper-cobalt ferrite respectively, for various humidity ranges. Response magnitude is\ndefined as,\nResponse magnitude = ∆ σ/σ\nwhere ∆σis the change in conductivity at particular RH and σis the conductivity at\nlow RH. In copper ferrite, figure 5(a), two linear regions are notice d. From 38 to 58\n% RH, the increase in sensitivity is slow and it is fast in the region 65 to 84 % RH.\nThe total conductivity increases by 17 times in the humidity range of 38 % to 84 % of\nRH and it explains its potential use in humidity sensing. In cobalt ferrit e, figure 7(b),\nthe sensitivity varies linearly as two regions from 37 to 48 % RH and 53 t o 84 % RH.\nCopper-cobalt mixed ferrite, figure 5(c) has also two linear regions , one in the range of\n38 to 60 % RH and the other in 67 to 81 % RH. In comparison, at 80 % RH, the copper\nferrite, cobalt ferrite and copper-cobalt mixed ferrite materials s how the response of\n17.05, 15.12 and 11.70 respectively.\n4. Conclusions\nCopper ferrite, cobalt ferrite and copper-cobalt mixed ferrite na nomaterials were\nprepared by self combustion method. X-ray diffraction pattern sh ows the crystalline\nnatureofthematerialsandthese nano-crystallites arepreferen tially oriented along(311)\nplane. FTIR spectrum of bulk material shows the characteristic pe ak of ferrites. The\ntemperature variation of the electrical conductivity of all the sam ples shows a thermal\nhysteresis and definite break in conductivity, which corresponds t o ferrimagnetic-\nparamagnetic transition. The activation energy in the paramagnet ic region is higherMagnetic and humidity sensing properties of Cu xCo1−xFe2O4 11\n0510 15 20 25 \n0612 18 \n30 40 50 60 70 80 04812 \nResponse magnitude (arb. units)  \n  Response magnitude (arb. units) (a) \n   \n(b)  (points) experimental \n (line) linear fit \n  \n Relative Humidity (RH) (%)  (c) \nFigure 7. (coloronline)VariationofresponsemagnitudewithRHfor(a)coppe rferrite\n(b) cobalt ferrite and (c) copper-cobalt ferrite.\nthan in the ferrimagnetic region. VSM studies revealed that substit ution of cobalt\nwith copper ferrite increases the corecivity and decreases the sa turation magnetization.\nHumidity sensing properties are studied and response magnitude ind icates that copper\nferrite has maximum sensitivity.\nAcknowledgments\nThe authors acknowledge the Inter-University Accelerator Cent re, New Delhi for\nfinancial support (IUAC - UFUP 41334) and Korea Institute of Scie nce and Technology\n(KIST- 2V01450).\nReferences\n[1] Chu X F, Jiang D, Guo Y and Zheng C 2006 Sens. Actuators B: Chem 120177.\n[2] Gopal Reddy C V, Manorama S V and Rao V J 2000 J. Mater. Sc. Lett. 19775.\n[3] Tao S, Gao F, Liu X Q and Sorensen O T 2000 Mater. Sc. Eng. B 77172.\n[4] Paulsen J A , Ring A P , Lo C C H , Snyder J E and Jiles D C 2005 J. Appl. Phys. 971044502.\n[5] Suri K, Annapoorni S, Sarkar A K, Tandon R P 2002 Sens. Actuators B: Chem 81277-282.\n[6] Traversa E 1995 Sens. Actuators B: Chem 23135.\n[7] Lee J G, Park J Y and Kim C S 1998 J. Mater. Sc. 333965.\n[8] Li X, Chen G, Lock Y P and Kutal C 2002 J. Mater. Sc. Lett. 211881.\n[9] Vasambekar P N, Kolekar C B, Vaingankar A S, 1999 J. Mater. Sc.: Mater. in Elec. 10667.\n[10] Cannas C, Falqui A, Musinu A, Peddis D and Piccaluga G. 2006 J. Nanop. Research 8255.\n[11] Tulliani J M and Bonville P 2005 Ceramics Internationals 31507.Magnetic and humidity sensing properties of Cu xCo1−xFe2O4 12\n[12] Marin ˇSek M, Zupan K and MaCek J, 2003 J. Mater. Res. 181551.\n[13] Cannas C, Musinu A, Peddis D and Piccaluga G 2004 J. Nanop. Research 6223.\n[14] Rodriguez J C, 1993 Physica B 19255.\n[15] Dong C, 1999 J. Appl. Cryst. bf 32 838.\n[16] Panigrahi M R and Panigrahi S 2010 Physica B 4051787.\n[17] Rey J F Q, Plivelic T S, Rocha R A, Tadokoro S K, Torriani I and Mucc illo E N S 2005 J. Nanop.\nResearch 7203.\n[18] Nakamoto K Infrared and Raman Spectra of Inorganic and coordination co mpounds 1986 (fourth\ned., Wiley, New York).\n[19] Devan R S, Kolekar Y D and Chougule B K 2006 J. Phys.: Condens. Matter 189809.\n[20] Patankar K K, Mathe V L, Shiva Kumar K V, 2001 Mater. Chem. Phys. Vol. 72 (2001) p. 23.\n[21]Rosenberg M, Nicoloau P, Bunget I, 1996 Phys. Status Solidi 15721.\n[22] Kittel C Introduction to Solid State Physics 2003 (seventh ed., John wiley & sons, Singapore).\n[23] Liu C, Rondinone A J and Zhang Z J 2000 Pure Appl. Chem. 7237.\n[24] Farghali A A, Khedr M H and Abdel Khalek A A 2007 J. Mater. Processing Tech. 18181.\n[25] Qu Y, Yang H, Yang H, Fan Y, Zhu H and Zou G 2006 Mater. Lett. 603548.\n[26] Gautam S, Muthurani S, Balaji M, Thakur P, Padiyan D P, Chae K H, Kim S S and Asokan K,\nJ. Nano. Nanotech. (in press).\n[27] Liu X Q, Tao S W and Shen Y S 1997 Sens. Actuators B: Chem 40161.\n[28] Arshaka K, Twomey K and Egan D 2002 Sensors 250."    },    {        "title": "1606.05783v1.Enhanced_diffusion_and_anomalous_transport_of_magnetic_colloids_driven_above_a_two_state_flashing_potential.pdf",        "content": "arXiv:1606.05783v1  [cond-mat.soft]  18 Jun 2016Enhanced diffusion and anomalous transport of magnetic coll oids driven above a\ntwo-state flashing potential\nPietro Tierno1,2∗and M. Reza Shaebani3†\n1Departament d’Estructura i Constituents de la Mat` eria, Un iversitat de Barcelona, 08028 Barcelona\n1Institut de Nanoci` encia i Nanotecnologia, IN2UB, Universitat de Barcelona, Barcelona, Spain\n1Department of Theoretical Physics, Saarland University, D -66041 Saarbr¨ ucken, Germany\nWe combine experiment and theory to investigate the diffusiv e and subdiffusive dynamics of para-\nmagnetic colloids driven above a two-state flashing potenti al. The magnetic potential was realized\nby periodically modulating the stray field of a magnetic bubb le lattice in a uniaxial ferrite garnet\nfilm. At large amplitudes H0of the driving field, the dynamics of particles resembles an o rdinary\nrandom walk with a frequency-dependent diffusion coefficient . However, subdiffusive and oscilla-\ntory dynamics at short time scales is observed when decreasi ngH0. We present a persistent random\nwalk model to elucidate the underlying mechanism of motion, and perform numerical simulations\nto demonstrate that the anomalous motion originates from th e dynamic disorder in the structure of\nthe magnetic lattice, induced by slightly irregular shape o f bubbles.\nI. INTRODUCTION\nTransport and diffusion of microscopic particles\nthrough periodic potentials is a rich field ofresearchfrom\nboth fundamental and technological points of view [1, 2].\nInvestigation of the particle motion along ordered [1] or\ndisordered [3, 4] energy landscapes helps to better un-\nderstand the dynamics in more complex situations, such\nas Abrikosov [5, 6] and Josephson vortices in supercon-\nductors [7, 8], cell migration [9], or transport of molec-\nular motors [10]. Moreover, a periodic potential can be\nused to perform precise particle sorting and fractiona-\ntion [11–15], thus, being of significant impact in diverse\nfields in analytical science and engineering which make\nuse of microfluidic devices. Colloidal systems provide\nan ideal opportunity to investigate different transport\nscenarios, because of having particle sizes in the visi-\nble wavelength range and dynamical time scales which\nare experimentally accessible. In order to force colloidal\nparticles to move along periodic or random trajectories,\nstatic potentials can be readily realized by using opti-\ncal [16], magnetic [17], or electric fields [18]. Dynamic\nlandscapes (obtained by periodically or randomly mod-\nulating the potential) are a subject of growing interest\nsince a rich dynamics can be induced due to the presence\nof competing time scales. Moreover, flashing potentials\nwhere static landscapes are modulated in time, are usu-\nally employed to study molecular systems [19–21] or as\nan efficient way to transport and fractionate Brownian\nspecies [22, 23].\nHere we present a combined experimental and theoret-\nicalstudyfocusedonthe dynamicsofmicroscaleparticles\ndriven above a flashing magnetic potential. This poten-\ntial is generated by periodically modulating the energy\nlandscape created by an array of magnetic bubbles. As\nit was previously reported [24], this experimental system\n∗ptiernos@ub.edu\n†shaebani@lusi.uni-sb.deis able to generate different dynamical states depending\non the applied field parameters, such as frequency or am-\nplitude of the external field. In particular we report the\nobservation of enhanced diffusive dynamics at high field\nstrength H0, while the motion at lower H0is subdiffusive\nwith acrossoverto normaldiffusion atlongtimes. At low\nvalues of H0, the lattice structure is slightly disordered\nduring the switching of the magnetic field direction. The\nresultingrandomnessisdynamic, i.e. not reproducibleaf-\nter a cycle of external drive, and enhances with decreas-\ningH0. By means of numerical simulations we verify\nthat in the presence of lattice disorder, the particles fre-\nquently experience oscillatory motion in local traps. The\nback and forth motion of particle in local traps happens\nmore frequently as the dynamic disorder in the structure\nof the magnetic lattice increases. The trapping events\nchange the statistics of the turning angles of the particle\nfrom an isotropic distribution (limited to the directions\nallowed by the lattice structure) to an anisotropic one\nwith a tendency towards backward directions. Using a\npersistent random walk model, we show that anomalous\ndiffusion arises when the turning-angle distribution of a\nrandom walker is asymmetric along the arrival direction.\nWhen the walker has the tendency towards backward\ndirections, the resulting antipersistent motion is subd-\niffusive or even strongly oscillatory at short time scales.\nHowever, the walker has a finite range memory of the\nsuccessive step orientations, i.e. the direction gets ran-\ndomized after long times and the asymptotic behavior is\nordinary diffusion with a smaller long-term diffusion co-\nefficient compared to an ordinary random walk. We ob-\ntain good agreement between the analytical predictions,\nsimulations, and experiments.\nThe paper is organized in the following manner: First\nwe introduce the setup in Sec. II. Section III contains\nthe experimental results obtained at different field pa-\nrameters. In Sec. IV, the results of numerical simula-\ntions for transport in the presence of dynamic disorder\nare discussed and compared with the corresponding ex-\nperimental data. The motion of particles is modeled at2\n(c)(a)\n(d)(b)\nFIG. 1. (a) Schematic showing a paramagnetic particle drive n above a magnetic bubble lattice by a square wave magnetic fie ld\nH. One Wigner-Seitz cell is shaded in blue. (b) Microscope ima ge of a bubble lattice with a superimposed trajectory of one\nparticle (blue lines) for ω=12.6rads−1andH0=0.14Ms. (c) Contour plots of the normalized magnetostatic energy E/kBTof\none particle above the bubble lattice at two different times s eparated by half a period. Energy maxima (minima) are colore d in\nwhite (blue). The blue arrows indicate the possible paths th e particle can undertake in the next jump. The red arrows show the\npath undertaken by the particle to reach the energy minimum. (d) Time evolution of the x−coordinate of one paramagnetic\nparticle subjected to an external square wave field with H0=0.14Msandω=15.7rads−1. The red inset zooms in one piece of\nthe trajectory.\nthe level of individual steps via an antipersistent random\nwalk approach in Sec. V, and finally Sec. VI concludes\nthe paper.\nII. EXPERIMENTAL SETUP\nA schematic illustrating the experimental system is\nshown in Fig. 1(a). The colloidal particles used are\npolystyrene paramagnetic microspheres (Dynabeads M-\n270, Invitrogen) having diameter d=2.8µm and mag-\nnetic volume susceptibility χ∼0.4. The particles are di-\nluted in high-deionized water and let sediment above the\nperiodic potential generated by a bismuth-substituted\nferrite garnet film (FGF). The FGF has composition\nY2.5Bi0.5Fe5−qGaqO12(q∈[0.5,1]) and was previously\ngrown by dipping liquid phase epithaxy on a 0 .5mm\nthick gadolinium gallium garnet substrate [24]. The\nfilm has thickness ∼4µm and saturation magnetizationMs=1.7×104Am−1. In the absence of external field, this\nFGF is characterized by a labyrinth of stripe domains\nwith alternating magnetization and a spatial periodicity\nofλ= 9.8µm. This pattern is converted into a periodic\nlattice of cylindrical magnetic domains by using high fre-\nquencymagneticfieldpulsesappliedperpendiculartothe\nfilm, with amplitude H0and oscillating at angular fre-\nquencyω. As shown in Fig. 1, the cylindrical domains,\nalsoknownas“magneticbubbles” [25], areferromagnetic\ndomains with radius R=4.2µm, having the same magne-\ntization direction and arranged into a triangular lattice\nwith lattice constant a=11.6µm. We can visualize both\nthe magnetic domains in the film and the particles using\npolarization microscopy, due to the polar Faraday effect.\nThe external oscillating field is obtained by connect-\ning a coil perpendicular to the film plane ( z-direction)\nwith a wave generator (TT i- TGA1244) feeding a power\namplifier (IMG STA-800). The custom-made coil was\nmounted on the stage of a polarization light microscope\n(Nikon, E400) equipped with a 100 ×, 1.3NA objective3\n(a) (b)\nFIG. 2. (a) Mean squared displacement versus time for a param agnetic colloid above a glass substrate (open squares) and a bove\na two-state flashing potential (solid squares), under a field with amplitude H0=0.37Msand angular frequency ω=6.3 rads−1.\n(b) Effective diffusion coefficient Dvs angular frequency ωfor a colloidal particle driven by an applied field with H0=0.37Ms.\nThe solid line denotes the relationship D=a2ω\n16π. The inset shows DvsH0, at a constant angular frequency of ω=6.3rads−1.\nand a 0.45 TV lens. The movements of the particles are\nrecorded at 60 fps for ∼30 min with a b /w CCD cam-\nera inside an observation area of 179 ×231µm2. We then\nuse particle tracking routines [26] to extract the particle\ntrajectories and later calculate correlation functions.\nIII. PARTICLE DYNAMICS IN A FLASHING\nPOTENTIAL\nOnce deposited above the garnet film, the paramag-\nnetic particles pin to the Bloch walls, which are located\nat the boundary of the magnetic bubbles. To generate a\ntwo-state flashing potential we apply to the FGF film a\nsquare-wave modulation of type\nH=H0sgn/parenleftbig\nsin(ωt)/parenrightbig\nez, (1)\nwhereH0is the field amplitude, ωthe angular frequency,\nand sgn( x) denotes the sign function. The applied field\nperiodically changes the radii of the bubbles in the FGF,\nincreasing (decreasing) the size of the bubbles when it is\nparallel (resp. antiparallel) to their magnetization direc-\ntion, thus alternating between two distinct states. One\ncan understand the effect of the applied field on the en-\nergy landscape by calculating the magnetostatic poten-\ntial at the particle elevation [27] see Fig. 1(c). In partic-\nular, when the field expands the bubbles ( state 1), the\nenergy displays a paraboloid-like minimum within the\nmagnetic domains. Thus, the magnetic colloids are at-\ntracted towards the center of the bubble domains. When\nthe field has opposite polarity, it shrinks the size of the\nbubbles and enlarges the interstitial region ( state 2). In\nthis situation, the magnetostatic potential features sixregions of energy minima with triangular shape at the\nvertices of the Wigner-Seitz cell around each bubble.\nAs a consequence, during the transition 1→2, a parti-\ncle can jump to 6 possible places [Fig. 1(c), left], while in\nthe transition 2→1the possibilities reduce to 3 [Fig. 1(c),\nright]. Since we apply a field perpendicular to the sam-\nple plane (no tilt), the potential preserves its spatial\nsymmetry, i.e. there is no net drift motion as it was\ninduced in Ref. [17] by using a precessing field. We\nanalyze the particle dynamics by measuring the mean\nsquared displacement via a temporal moving average\nMSD(t)=/angbracketleft/parenleftbig\nx(t+t′)−x(t′)/parenrightbig2/angbracketright∼tα. Here,xdenotes the po-\nsition of the particle projected along one of the crystallo-\ngraphic axes and αthe exponent of the power-law which\nis used to distinguish the diffusive ( α=1) from subdif-\nfusive (α<1) dynamics. In our system we observe both\ntypes of dynamics, which are discussed in the following\nsubsections.\nA. Diffusive dynamics\nIn Fig. 2(a) we compare the MSD for a paramagnetic\ncolloid freely diffusing on a glass plane in the absence of\nFGF film and the one which is strongly driven by the\nflashing potential. Both cases exhibit a normal diffusion\nbut with different diffusion coefficients. From the experi-\nmental dataofthe MSD, the effective diffusion coefficient\nof the particle can be estimated as D= lim\nt→∞MSD(t)/2dt,\nwithdbeing the spatial dimension (Here d=1 since the\ndata is projected along one of the crystallographic axes).\nWe find that Din the presence of the flashing poten-\ntial (D=14.6µm2s−1) is enhanced by nearly two orders4\nof magnitude with respect to the one on a glass plate\n(D=0.14µm2s−1). In this regime of motion, which is\nobserved for field amplitudes H0∈[0.28,0.42]Msand\nfrequency ω<35rads−1, the particle moves from one do-\nmain to the next without performing continuous oscilla-\ntion around a single site, thus, it performs an ordinary\nrandom walk on a triangular lattice. The step length l\nof the walker can be estimated to be given by one side\nof the Wigner-Seitz cell, i.e. ℓ=a\n2=5.8µm, and the com-\nponent of the MSD along a crystallographic axis equals\n/angbracketleftx2/angbracketright=(ℓ√\n2)2=ℓ2\n2. Each step takes a half-period of the\nmagnetic field pulse, thus, the duration of each step is\ngiven by ts=T\n2=π\nω. One thus obtain the diffusion coeffi-\ncient as [28, 29] D=/angbracketleftx2/angbracketright\n2ts=l2\n4ts=a2ω\n16π. We confirm this rela-\ntioninFig.2(b)bymeasuring Dversusthefrequencyand\namplitude ofthe applied field. While Dincreaseslinearly\nwith frequency, it decrease rapidly beyond ω∼30rads−1\nsince the overdamped particle is unable to follow the fast\nvibrations of the potential, and reduces significantly its\ndiffusive dynamics. In contrast to frequency, Dis almost\nindependent of the amplitude of the applied field, since\nthe lattice constant of the magnetic bubble array does\nnot change significantly for amplitude H0<0.5Ms[17].\nBeyond this value, however, the magnetic bubble lattice\nstarts melting and transport of the particles is not pos-\nsible anymore.\nB. Subdiffusive dynamics\nOur experimental setup allows us to independently\nvary both the amplitude H0and the frequency ωof the\ndriving field. At strong fields ( H0>0.28Ms), the parti-\ncle jumps between nearest bubbles with an equal prob-\nability to choose any of the possible movements, thus,\nperforms a normal random walk on a triangular lattice\n(α≃1 on all time scales). In contrast, when the field is\nweak (H0<0.09Ms), the landscape deformations induced\nby the applied field are small and the particle is unable\nto leave the magnetic bubble. The corresponding parti-\ncle trajectory is composed of simple oscillations between\nthe center of one bubble and one of its six surrounding\nenergy minima. In such a pure localization the MSD\nsaturates rapidly, leading to an exponent α≃0. In the in-\ntermediate regime of amplitudes, i.e. H0∈[0.09,0.28]Ms,\nwe observe a subdiffusive dynamics at short time scales\nwith a crossover to normal diffusion at long times. Fig-\nure 3 shows several examples of the temporal evolution\nof MSD for different values of H0andω. The initial\nanomalous exponent varies between 0 and 1 depending\non the strength H0of the applied magnetic field. While\nin a previous work [24] it was found a stable subdiffu-\nsive exponent α=0.5 forH0= 0.13Msand at different\nfrequencies, the value of αmay vary by changing the am-\nplitude of the switching field. To see this more clearly,\nwe rescale both axes in such a way that the asymptotic\ndiffusive regimes of the curves collapse on a master curve10-11 1011101102103104105\n15.7\n18.8\n15.7\n15.70.37\n0.17\n0.13\n0.09\n10-11 101102102\n10-21\nFIG. 3. Mean squared displacement versus time for a param-\nagnetic colloid subjected to a two-state flashing potential for\ndifferent values of amplitude H0and frequency ωof the ap-\nplied magnetic field. Inset: The dimensionless mean squared\ndisplacement /tildewideMSD = MSD /(4ℓ\nvDasymp)versusthescaled time\ntω/π.\n(see inset). This can be achieved if the MSD is scaled\nby4ℓ\nvDasymp(Dasympbeing the long-term diffusion coef-\nficient,ℓthe step size, and vthe velocity of the particle),\nand the time axis is scaled byπ\nωto synchronize the os-\ncillations. Only the samples which reach the long-term\ndiffusion limit within our experimental time window are\nconsidered. With increasing H0, the curves are initially\nmore steep and converge faster to the asymptotic limit.\nIndeed, the lattice structure is slightly disordered at\nweak fields, because of the non-uniform deformations\nof the magnetic bubbles during their periodic expan-\nsion/contraction. These deformations arise from the\npresence of pinning sites in the film and other inho-\nmogeneities (such as dislocations or magnetic and non-\nmagnetic inclusions present in the FGF crystal). These\ndefects exert an influence on the Bloch walls analogous\nto the action of a frictional force against the motion of\nthe walls[30]. Furthermore, when the field isapplied, the\nmagnetic bubbles interact via long range dipolar forces\n[31] and small deviations from the 2D projected circular\nshape induce a slight distortion on the triangular lattice.\nFor the fields used in our experiments, these distortions\nare not strong enough to create permanent defects in the\nfilm (such as disclinations or dislocations), but slightly\nvary the lattice spacing from place to place. Thus, the\nspatial structure of the lattice is not perfectly symmetric\nin practice. In the presence of disorder, the particle may\noscillate around each site before it moves to the next do-\nmain (see Fig. 1(d) for an example of such movements),\nwhichleads tosubdiffusive dynamicsat shorttime scales.\nHowever, this bias decays with time and the directions of\nthe particle motion become asymptotically randomized,\nleading to diffusive dynamics at long times. With de-\ncreasing H0, the lattice disorder and thus the frequency5\nof trapping events enhances, which further decreases the\ninitial anomalous exponent from one (ordinary random\nwalk) towards zero (pure localization).\nThe frequency ωof the driving field does not affect the\nMDS behavior; it only rescales the time because the time\nstep is given by ts=π\nω. Thus, it can be understood why\nthe crossover time to asymptotic diffusion was found to\nfollow a power-law τc∼ω−1withω[24].\nIV. SIMULATION RESULTS\nIn the previous section we explained the origin of the\ndisorder in the lattice structure during switching of the\nmagnetic field. As a result, the particle spends part of\nthe time in local traps, where it experiences an oscilla-\ntory movement. In a back and forth motion, the particle\nchoosesnearlybackwarddirectionswhen it turns. There-\nfore, the turning-angle distribution f(φ) changes from\nan isotropic one for ordinary random walk on the lattice\n(limited tothe alloweddirectionsbythelatticestructure)\nto an anisotropic one with a tendency towards backward\ndirectionswith respecttothecurrentdirectionofmotion.\nIn the extreme limit of pure localization (remaining per-\nmanently in a trap), f(φ) is a delta function at φ=π.\nSuch anisotropic turning-angle distributions slow down\nthe spread of colloids on the lattice and cause subdiffu-\nsion at short times. The structural irregularities in the\nmagnetic bubble lattice are more pronounced at smaller\nvalues of H0. Hence, with decreasing H0the chance of\ntrapping events increases and the resulting f(φ) becomes\nmore anisotropic, which decreases the initial anomalous\nexponent.\nIn order to ensure that the lattice disorder causes\nasymmetric turning-angle distribution f(φ) and subdif-\nfusive dynamics, we perform Molecular Dynamics sim-\nulations of motion on a dynamic disordered triangular\nlattice of synchronousflashing magnetic poles. To obtain\nsmoothparticletrajectoriesandadetailedtimeevolution\nof the MSD, in particular to monitor the oscillatory dy-\nnamics at short times, MD simulations are advantageous\ncomparedto other possible methods such as Monte Carlo\nsimulations. The simulation cell consists of nearly 3000\nmagnetic poles and a colloidal particle which is initially\nreleased near the center of the system to avoid boundary\neffects. We impose periodic boundary conditions, and\nconsider in-plane magnetic interactions between the im-\nmobile poles and the magnetic colloid. The time step of\nour in-house code is chosen to be ∆ t=1×10−4s, so that\nthe time ts=π\nωbetween two successive switching of the\nfield direction is resolved into more than 1000 time steps.\nAn explicit Euler update scheme is used for integration\nand the simulations run until the crossoverto asymptotic\ndiffusion occursorthe total number oftime steps exceeds\n5×105.\nThestructuraldisorderiseffectivelyintroducedbyran-\ndom displacements of magnetic poles from their ordered\nlattice positions. The strength of disorder can be quanti-0.20.40.60.81.0\n0.20.40.60.81.0\n0.20.40.60.81.0\nFIG. 4. (left) Schematic drawing of the disordered lattice a nd\n(right) a polar representation of the corresponding turnin g-\nangle distribution f(φ) after 10000 steps in simulations for\ndifferent values of the lattice disorder δ.\nfied by the parameter δ, denoting the maximum possible\ndisplacement of each magnetic pole from its ordered lat-\ntice position. We randomly choose the position of each\nnode within a circle of radius L·δaround the correspond-\ning node of the ordered triangular lattice, where Lde-\nnotes the size of the Wigner-Seitz cell. The dynamic dis-\norder is generated by instantaneous random rearrange-\nments of the poles at each switching event. Thus, the\npoles move to new positions where they stay immobile\nuntil the next switching event. The new random position\nof each pole remains within the allowed range around\nthe original position of the corresponding ordered lat-\ntice node. This way we keep the deviations dynamic but\nsmaller than δ·Lin all cases.\nWe analyze the particle trajectory for different values\nofδ. As shown in Fig. 4, with increasing δthe distortion\nof the lattice is more pronounced, thus, the particle ex-\nperiences oscillatory movements in traps more frequently\nand for longer times, which increases the asymmetry of\nthe turning-angle distribution f(φ). By tunning δ, as a\nsingle free parameter in our simulations, a remarkable\nagreement with the experimental MSD data is achieved\n(see Fig. 5). Moreover, by smoothening the MSD curves\nobtained from simulations, we fit the initial slope of the\ncurves to MSD( t)∼tαto get the anomalous exponent at6\n10-11101\n0102103\n10-210-11 101102103\nFIG. 5. MSD versus time, obtained from experiments (green\nsymbols), simulations (red dashed lines), and the persiste nt\nrandom walk model via Eq. (7) (black solid lines). The ex-\nperimental data at ω=15.7rads−1and various amplitudes H0\nare compared to the best fits obtained by tunning a single fit-\nting parameter (i.e. the lattice disorder δin simulations, or\nthe turning-angle anisotropy ηin the analytical model). The\nbest-fitting parameters for each value of H0are given.\nshort times. The results shown in Fig. 6 reveal that\nwith increasing the amount of structural disorder, the\nslope gradually decreases from α=1 for normal diffusion\ntowards α=0 corresponding to pure localization. The\ngrowth of MSD is extremely slow for δ≥25%, in accord\nwith the experimental data at weak fields H0<0.09Ms.\nV. PERSISTENT RANDOM WALK MODEL\nSo far, we have shown that for small amplitudes of\nthe external field the turning-angle distribution f(φ) is\nasymmetric, which is accompanied by a subdiffusive dy-\nnamics at short times. In this section, we theoretically\nconsider the particle motion at the level of individual\nsteps and show how the asymmetry of f(φ) with a ten-\ndency towards backward directions leads to subdiffusion.\nIt was demonstrated in a previous work [24] that the\nstatistical properties of the subdiffusive motion could be\ncaptured by the “ random walk on random walk ” model\n[32] (RWRW), which is a simple example of stochastic\nmotion in a complex environment. In the RWRW model,\nthe walker performs an ordinary random walk on an en-\nvironment which is constructed by an ordinary random\nwalk process as well. While this model showed a quan-\ntitative agreement with the experiments for certain field\nparameters [24], the randomness of the environment in-\ndeed varies depending on the choice of H0. Thus a more\ngeneral theoretical framework is needed in order to cap-\nture the particle dynamics in the subdiffusive regime for\nall field parameters. Here, we lookat the individual steps0.00.20.40.60.8\n0 5 10 15 20 25 30 35 401.0\nFIG. 6. The anomalous exponent αat short times (after\nsmoothening the MSD curves) in terms of the structural dis-\norderδof the lattice.\noftherandomwalkerandpresentanantipersistentmodel\nwhich enables us to reproduce fine details of motion such\nas the oscillatory dynamics observed at smaller values of\nH0. Persistent random walk models [33–36] have been\nused to describe e.g. stochastic transport on cytoskeletal\nfilaments [37, 38], self-propulsion subject to fluctuations\n[39, 40], or diffusive transport of light in foams and gran-\nular media [41, 42]. In the following, the asymmetry of\nf(φ) is quantified with η≡/angbracketleftcosφ/angbracketright, which varies from 0 for\nan isotropic distribution f(φ)=1\n2πto−1 for an extremely\nasymmetric distribution f(φ)=δ(φ−π). We obtain ana-\nlytical expressionsfor the time evolution of the MSD, the\ncrossovertime to asymptotic diffusion, and the long-term\ndiffusion coefficient in terms of the control parameter η\nand compare the analytical predictions with the experi-\nmental data.\nWe consider a random walk in 2D, with uncorrelated\nstep sizes ℓwhich are obtained from an arbitrary distri-\nbutiong(ℓ). The successive step orientations arehowever\ncorrelated such that a new orientation is obtained from\na turning-angle distribution f(φ) along the previous di-\nrection of motion (see Fig. 7). While an isotropic f(φ)\nleads to a normal diffusion, introducing anisotropy along\nthe arrival direction induces persistency and results in an\nanomalous transport on short time scales. The choices\noff(φ) which encourage the walker towards backward\n(forward)directionsleadtosubdiffusion (superdiffusion).\nSuchanapproachhadbeenused e.g.inthe contextofcell\nmigration along surfaces [43], animal movement [44], and\ndynamics of polymer chains [45, 46]. Starting from the\norigin, let us assume that the first direction of motion is\nchosen randomly, i.e. the initial condition is P(α1)=1\n2π.\nThex-coordinate of walker after nsteps can be obtained\nby projecting each step along the x-axis\nx=n/summationdisplay\ni=1xi=n/summationdisplay\ni=1ℓicosαi, (2)\nwhereαi=α1+φ2+···+φi. Forsimplicity, inthefollowing\nwe consider a constant step length g(ℓ)=δ(ℓ−/angbracketleftℓ/angbracketright) and7\n0.41.2\n0.8\n0=-0.95\n=-0.8\n=0(b)(a)\nFIG. 7. (a) Trajectory of the walker during a few successive\nsteps. (b) Examples of the turning-angle distribution f(φ) for\ndifferent values of the asymmetry measure η.\nf(φ) distributions with left-right symmetry. The first\nmoment of displacement /angbracketleftx/angbracketrightthen reads\n/angbracketleftx/angbracketright=/angbracketleftℓ/angbracketright/angbracketleftn/summationdisplay\ni=1cosαi/angbracketright=/angbracketleftℓ/angbracketrightn/summationdisplay\ni=1/integraldisplayπ\n−πdφif(φi)···/integraldisplayπ\n−πdφ2f(φ2)\n×/integraldisplayπ\n−πdα1P(α1)cos(α1+φ2+···φi) = 0,\n(3)\nsince the integral over α1vanishes. Here /angbracketleft···/angbracketrightdenotes an\nensemble average. The second moment of displacement\nis given by [43–46]\n/angbracketleftx2/angbracketright=/angbracketleftn/summationdisplay\ni=1n/summationdisplay\nj=1xixj/angbracketright=/angbracketleftn/summationdisplay\ni=1x2\ni+2/summationdisplay\ni>jxixj/angbracketright\n=n/summationdisplay\ni=1/angbracketleftx2\ni/angbracketright+2n/summationdisplay\ni=1i−1/summationdisplay\nj=1/angbracketleftxixj/angbracketright.(4)\nSimilar to Eq. (3), the first term can be obtained as\n/angbracketleftx2\ni/angbracketright=/angbracketleftℓ2/angbracketright/integraldisplayπ\n−πdφif(φi)···/integraldisplayπ\n−πdφ2f(φ2)×\n/integraldisplayπ\n−πdα1P(α1)cos2(α1+φ2+···φi) =/angbracketleftℓ2/angbracketright\n2.(5)\nthe second term on the right hand side of Eq. (4) can beFIG. 8. The normalized auto-correlation function /angbracketleftφiφi+τ/angbracketright\nversus time for different values of η. Positive (negative) ηcor-\nresponds to persistent (antipersistent) motion. η=1 denotes\na ballistic motion without any turning.\nevaluated in the following way\n/angbracketleftxixj/angbracketright=/angbracketleftℓ/angbracketright2/integraldisplayπ\n−πdφif(φi)···/integraldisplayπ\n−πdφ2f(φ2)×\n/integraldisplayπ\n−πdα1P(α1)cos(α1+φ2+···φi)cos(α1+φ2+···φj)\n=/angbracketleftℓ/angbracketright2\n2/integraldisplayπ\n−πdφif(φi)···dφ2f(φ2)cos(φj+1+···φi)\n=/angbracketleftℓ/angbracketright2\n2/parenleftBigg/integraldisplayπ\n−πdφf(φ)Re[eiφ]/parenrightBiggi−j\n=/angbracketleftℓ/angbracketright2\n2ηi−j,\n(6)\nwhereη=/integraltextπ\n−πdφf(φ)cosφ=/angbracketleftcosφ/angbracketright. A few examples of\nthe turning-angle distribution f(φ) and the correspond-\ning values of the asymmetry parameter ηare shown in\nFig. 7(b). From Eqs. (4) to (6) one can obtain the sec-\nondmoment /angbracketleftx2/angbracketright. Similarconclusionsforthemomentsof\nthey-coordinate can be drawn due to symmetry. Thus,\none gets the following expression for the mean squared\ndisplacement\n/angbracketleftr2/angbracketright=/parenleftBig\n/angbracketleftℓ2/angbracketright+/angbracketleftℓ/angbracketright22η\n1−η/parenrightBig\nn+/angbracketleftℓ/angbracketright22η\n(1−η)2(ηn−1).(7)\nThe case η=0 corresponds to an isotropic distribution\nf(φ), for which Eq. (7) reduces to /angbracketleftr2/angbracketright=/angbracketleftℓ2/angbracketrightn, i.e. a nor-\nmal diffusion. Negative (positive) values of ηdenote an\nincreased probability for motion in the near backward\n(forward) directions, thus, leading to antipersistent (per-\nsistent) motion. For η>0 one obtains a superdiffusive\nshort-time dynamics, while η<0 leads to subdiffusion or\noscillatory dynamics. In Fig. 5, the theoretical predic-\ntions for the MSD via Eq. (7) are compared with the\nexperimental and simulation results. It can be seen that\nour theoretical approach remarkably reproduces the ob-\nserved behavior by fitting the single free parameter of\nthe model, i.e. η. Notably, the overall behavior of the\nMSD, the frequency of oscillations, the crossover time,8\nand even the asymptotic diffusion coefficient are all cap-\ntured by the theory.\nThe persistent random walker has a finite-range mem-\nory,beyondwhichthedirectionofmotiongetscompletely\nrandomized. According to Eq. (6), the auto-correlation\nbetween step orientations is given by\n/angbracketleftφiφi+τ/angbracketright=/angbracketleftcosφ/angbracketrightτ, (8)\nwhich vanishes in the limit of τ→∞since|/angbracketleftcosφ/angbracketright|≤1.\nSome examples are shown in Fig. 8 for various val-\nues of η. One can estimate the crossover time\nncto asymptotic diffusion by numerically solving/parenleftBig\n/angbracketleftℓ2/angbracketright+/angbracketleftℓ/angbracketright22η\n1−η/parenrightBig\nnc∼/angbracketleftℓ/angbracketright22η\n(1−η)2ηnc.Moreover, it can be\nseen from Eq. (7) that the asymptotic diffusion coeffi-\ncient also depends on ηas\nDasymp=1\n4v/parenleftBig/angbracketleftℓ2/angbracketright\n/angbracketleftℓ/angbracketright+/angbracketleftℓ/angbracketright2η\n1−η/parenrightBig\n, (9)\nwithvbeing the averageparticle velocity. For a constant\nstep size, Dasympranges from 0 for η=−1 to1\n4v/angbracketleftℓ/angbracketrightfor\nη=0.VI. CONCLUSIONS\nWe combined experiment and theory to investigate the\ndynamics of paramagnetic colloids driven above a two-\nstate flashing potential. This potential was realized by\nperiodically modulating the stray field generated at the\nsurface of a magnetic bubble lattice in an uniaxial garnet\nfilm. The particles experience either enhanced diffusive\nor anomalous sub-diffusive dynamics, depending on the\nstrength of the external drive. Applying a strong field\nleadstoanordinaryrandomwalkonthemagneticbubble\nlattice, while weaker fields result in structural disorder in\nthe lattice which slows the particle dynamics. By means\nofa persistent randomwalkapproachand numericalsim-\nulationsweverifiedthat increasinglatticedisordersharp-\nens the turning-angle distribution of the particle towards\nbackward directions, which decreases the anomalous ex-\nponent, postpones the crossover time to asymptotic dif-\nfusion, and modifies the long-term diffusion coefficient.\nAcknowledgment\nP. T. acknowledges support from the European Research\nCouncil Project No. 335040, from Mineco (Grant No.\nFIS2013-41144-P and RYC-2011-07605), and AGAUR\n(Grant No. 2014SGR878). 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Sabirov1 \n \n1 IMDEA Materials Institute, Calle Eric Kandel 2, Getafe 28906, Madrid, Spain  \n2 Department of Materials Science and Engineering, Delft University of Technology, \nMekelweg 2, 2628CD Delft, The Netherlands  \n3 Department of Electrical Energy, Metals, Mechanical constructions & Systems , Ghent \nUniversity, Technologiepark 46, 9052 Ghent, Belgium  \n \n \nAbstract  \n \nIn this work, we investigate t he sensitivity of the microstructure and mechanical \nproperties of an ultrafast heat treated low carbon -steel to the peak temperature. In all \nstudied cases , the steel was heated within the intercritical temperature range (i.e. between \nthe AC1 and AC3 temperatures). Both the peak temperature and soaking time were varied,  \nand their effect on the size , the fraction of individual micro structural constitu ents and \ntheir tensile mechanical response  were investigated . It is sho wn that the increasing peak \ntemperature and soaking time promote austenite formation  and recrystallizat ion processes \nin the ferritic matrix. The highest nanohardness is shown by martensitic grains, while \nrecovered ferrite demonstrated slightly higher nanohardness compared to recrystallized \nferrite. The applied heat treatment parameters have strong effect on  the nanohardness of \nmartensite , whereas nanohardness of ferrite microconstituents is not sensitive to variation \nof the  peak temperature and soaking time.  The non -recrystallized ferrite is harder than its \nrecrystallized counterpart due to the higher dislocation density of the former.  Increasing \n                                                 \nCorresponding author : Miguel Angel Valdés Tabernero.  \nPostal address: IMDEA Materials Institute, Calle Eric Kandel  2, Getafe 28906, Madrid, Spain. \nPhone: +34 91 5493422 . E-mail:  miguelvaldestabernero@gmail.com  \n 2 peak temperatures  promote strengthening in the material at the expense of its ductility \nmainly due to increased martensite fraction . The steel demonstrates  enhanced strain \nhardening ability independently of the peak temperature. Analysis of the experimental \nresults show ed that the industrial processing window of ± 10 ºC may lead to some \nheterogeneity of the local micr ostructure in the ultrafast heat treated sheets . However, the \nlatter should not have any negative effect on the overall mechanical behavior of the \nultrafast heat treated steel sheets on the macro -scale.  \n \nKeywords:  metals and alloys ; ultrafast heating ; phase transitions, scanning electron \nmicroscopy, SEM ; nanoindentation  \n \n1. Introduction  \nSteel sheets manufacturing is a multistage process , where the  steel is subjected to several \nrolling operations and finally to a heat treatment, which determines its final \nmicrostructure and, therefore, its  properties. The standard approach for processing \nadvanced high strength steels (AHSS) is based on the homogenization of the \nmicrostructure at elevated temperatures followed by cooling with well controlled rates  \n[1]. The typical route used to manufacture components for the automotive industry, where \nsteel is the primarily used  material  [2], is a relati vely long process  resulting in very high  \nenergy consumption and carbon dioxide emission  [3]. A potential solution to  decrease the \nCO 2 emissions produce d by vehicles is to reduce  the total weight of the car , without \ncompromising the passengers’  safety. In order to do so, the mechanical properties of the \ncar components should be improved. Therefore, the steel industry is continuously looking \nfor new solutions to fulfill the current societal demands  by processing the steel in the most \nenvironmentally -frien dly manner. Hence, in the last decades , new processing routes were \ndeveloped , such as the rapid o r “flash processing” treatment [4,5] . In literature, this \nprocess is also referred to as “ultra -fast heating” (UFH) [6,7]  or “ultra -rapid annealing” \n(URA) [8,9] . It is based on heating  the material to intercritical or fully austenitic \ntemperature with heating rates well above 100 ºC/s, which is at least one order of \nmagnitude higher than the conventional heating rates ( ≤ 10 ºC/s), followed by a short \nsoaking at peak temperature  and imme diate  quench ing to the room temperature. Thus , the \ntreatment time and the energy consumed for  the process are significantly reduced [10]. 3 Complex multiphase microstruct ure consisting of ferrite and martensite and some \nretained austenite  is typically formed after UFH treatment . The resulting micro structure \nand properties of the UFH -treated material are greatly affected by the initial \nmicrostructure as well as  the heat treatment parameters , such as heating rate, soaking time \nand peak temperature [11]. It has been reported that high heating rates increase the \naustenite start and finish temperatures [12,13] . Moreover, i n low carbon steels , \nrecrystallization temperature tends to increase with increasing heating rate and may even \nexceed  the austenite start temperature ( AC1) [14,15] . Hence, ferrite to austenite \ntransformation takes place in a non -recrystallized matrix, which leads to the formation of \nnumerous austenite nuclei , while  the ferritic matrix undergoes  recrystallization and \nrecovery  simultaneously . Thus , the complex microstruc ture with finer grain size is \ndeveloped [16]. In order to promote grain refinement, the i sothermal soaking time  is \ntypically kept as short as possible   in the 0.1-0.2 s  range  [7,11,15,17] . However, such \nsoaking times constitute a challenge for the steel sector due to its d ifficult  implement ation  \nin the current industrial lines,  impeding the expansion of this processing route. Slightly \nlonger soaking times (1 – 3 s) can be a feasible solutio n to maintain  the reduced  grain size \nbrought about  by ultrafast heating [18], but only  if the effect of other processing \nparameters is well understood . One key parameter is the peak temperature , as it influences  \nthe mechanism of the austenite formation and growth . At conventional heating rates the \naustenite formation  kinetics are determined by carbon diffusion, whereas at ultrafast \nheating rates formation of austenite starts by carbon diffusion control, which is later \novertaken by a massive mechanism  [7,9,19] . The martensite formed after quenching gets \nsofter due to higher volume fraction of intercritical austenite and reduced carbon content \ntherein  [20]. Additionally, t he recrystallized ferrite volume fractio n in the heat treated \nsteel tends to increase with increasing peak temperature [21]. Despite significant effect \nof peak temperature on  the final microstructure  of the UFH  treated steels , there are no \nsystematic studies focused on the microstructure sensitivity to the peak temperature \nvariations.  In addition , it is important to simulate real  industrial  conditions, as the  typical \nindustrial  processing temperature window  of ± 10 ºC is by an order of magnitude higher \ncompared to that maintained in a lab oratory  dilatometry or thermo -mechanical simulator  \n(± 1 ºC) [7,11] . Therefore, it is essential  to understand the effect of peak temperature  \nranges  during UFH on the microstructure and properties in order to select the optimum \nconditions.  4 The reported studies on the UFH treatment of steels have mainly focused on \nmicrostructure and basic mechanical p roperties  (hardness, tensile strength , ductility) . In \nreferences  [17,22] , it was reported that the UFH treatment leads to an improvement on \nthe material strength compare d to the  conventional heat treat ment , without a reduction \nin ductility. However, although it has been shown in many studies  that mechanical \nbehavior  of  multi -phase materials on macroscale depends on the morphology, \narchitecture and properties of the individual microconstituents [1,23,24] , there are no such \nin-depth  studies on the  UFH treated steels . Hence, u nderstanding the heat treatment \nparameters -microstructure -properties  relationship both at macro - and micro -scales , is \nnecessary to develop specific microstructures and properties and to  design  and optimize  \nprecise  UFH  treatments depending on the requirements and specifications of the final \nproduct.  Therefore, t he objective of this work is to investigate  the influence  of peak \ntemperature and short soaking times ( ≤ 1.5 s) on the microstructure and properties of the \nindividual micro structural constit uents, as well as to relate both to the macro -mechanical \nresponse  of the material .  \n \n2. Material and experimental procedures  \n2.1. Material  \nThe chemical composition of the  low carbon steel selected for this investigation  was 0.19 \n% C, 1.61 % Mn, 1.06 % Al, 0.5 % Si  (in wt. %) . The as-received material was 1 mm \nthick sheet (50% cold rolled) with a microstructure of 76 % of ferrite and 24 % of pearlite \n(Figure 1). This material was subjected to two kinds of heating experiments: a) \ndilatometry measurements to determine the formation of austenite at different intercritical \ntemperatures, and b) annealin g tests to different intercritical temperatures with varying \nsoaking time followed by quenching. Both types of experiments are described in detail \nbelow.  5  \nFigure 1: Initial ferritic -pearlitic microstructure of the material after 5 0 % cold reduction.  \n \n2.2. Dilatometry experiments  \nDilatometry measurements were carried out to analyze the austenitization kinetics at \ndifferent temperatures for the same heating rate. For these experiments, specimens with \ndimensions of 10x5x1 mm3 were machined from the as -received material. Tests were \ncarried out in a Bähr DIL805A/D dilatometer (Bähr -Thermoanalyse GmbH, Hüll -Horst, \nGermany). A K-type thermocouple was welded to the midsection of each specimen to \ncontrol their temperature during the  experiment. Specimens were heated from room \ntemperature to different temperatures in the intercritical region (860 ºC, 880 ºC and 900 \nºC) at 200 ºC/s and soaked  for 600 s. Then , specimens were heated to a maximum \ntemperature of 1100 ºC at 200 ºC/s and soa ked for 0.2 s  (to ensure full austenitization) \nfollowed by quenching  to room temperature at -300 ºC/s. The volume fraction of the \naustenite phase  formed during isothermal holding  was obtained via analysis of the \ndilatometry data applying  the lever rule  to the dilatation -time curve  [25]. \n \n2.3. Intercritical heat treatment  \nStrips having a  length of  100 mm and width of 10 mm were cut along the rolling direction  \nfrom  the cold rolled sheet . A K-type thermocouple was spot -welded to the midsection of \n6 each strip. A thermo -mechanical simulator Gleeble 3800 was used to perform heat \ntreatments. At the first stage  of heat treatments , samples were heated at  10 ºC/s to 300 ºC  \nand kept at this temperature for  30 s to simulate  a preheating in some industrial continuous \nannealing lines to minimize  the thermal stresses during heating. At the second stage, part \nof samples was heated from 300 ºC at 800 ºC/s (which corresponds to the ultrafast heating \nrate) to the intercritical peak temperature of 860 ºC followed by soaking for 0.2 s or 1.5 s \nand quenching to room temperature  with cooling rate of ~160 oC/s. Hereafter, t hese \nspecimens will be referred to as UFH860 -0.2s and UFH860 -1.5s, respectively.  \nTo investigate the influence  of peak temperature, additional heat treatments with \nmaximum temperatures equal to 880 ºC and 900 ºC and same soaki ng times (0.2 s and \n1.5 s) were performed. These conditions are referred to as UFH880 -0.2s and UFH880 -\n1.5s for the 880 ºC, and UFH900 -0.2s and UFH900 -1.5s for the 900 ºC  heat treatment. \nMicrostructural analysis and hardness mea surements along the axis of t he heat treated \nstrips showed a homogeneously heat treated zone having a length of 10 mm.  The \nspecimens processed by the  Gleeble thermo -mechanical simulator were  then subjected to \na thorough microstructural and mechanical characterization . \n \n2.4. Microstructural  characterization  \nScanning electron microscopy (SEM) and electron backscatter diffraction (EBSD) \nanalyses were carried out for a thorough m icrostructural characterization of the heat \ntreated samples. Specimens were ground and polished to a mirror -like surface applying \nstandard metallographic techniques with final polishing using OP -U (colloidal silica). For \nSEM characterization, polished spec imens were etched with 2 vol.% nital solution for \n~10 s. The EBSD studies were performed using a FEI Quanta™ Helios NanoLab 600i \nequipped with a NordlysNano detector controlled by the AZtec Oxford Instruments \nNanoanalysis (version 2.4) software. The data w ere acquired at an accelerating voltage \nof 18 kV, a working distance of 8 mm, a tilt angle of 70º, and a step size of 65 nm in a \nhexagonal scan grid. The orientation data were post -processed using HKL Post -\nprocessing Oxford Instruments Nanotechnology (vers ion 5.1©) software and TSL Data \nanalysis version 7.3 software. Grains were defined as a minimum of 4 pixels with a \nmisorientation ≥ 5º. Grain boundaries having a misorientation ≥ 15º were defined as high -\nangle grain boundaries (HAGBs), whereas low -angle gr ain boundaries (LAGBs) had a 7 misorientation < 15º. The volume fractions of transformed/untransformed grains and \nrecrystallized/recovered ferritic grains were determined by a two -step partitioning \nprocedure described in [17,26] . In this procedure, grains with hig h (> 70º) and low ( ≤ 70º) \ngrain average image qualities are separated in a first step, allowing to distinguish between \nuntransformed (ferrite) and transformed (martensite) fractions, respectively . In the second \nstep, recrystallized and non -recrystallized f erritic grains are separated using the grain \norientation spread (GOS) criterion: Grains with GOS  below 1º are defined as the \nrecrystallized grains, while grains with GOS  above 1º are defined as the non -\nrecrystallized ones [27]. Microstructure was observed on the plane perpendicular to the \nsample transverse direction (the RD –ND plane).  \n \n2.5. Mechanical characterization  \nHysitronTI950 Triboindenter with a Berkovich tip was employed for nanoindentation \ntesting. First, s quare areas having a size of ~10 x 10 µm2 were analyzed by EBSD , and \nindividual micro structural constituents  were determined . At least ten areas were tested for \neach material’s condition. In order to target specific phases/grains, these square areas \nwere scanned, using the scanning probe microscopy (SPM) mode of the instrument  prior \nto the nanoindentation . In SP M mode, the nanoindenter tip  is in contact with the surface \nof the tested material scanning it, giving the topography of the sample. Nanoindentation \ntests were carried out in displacement control mode at a constant strain rate ( 𝜀̇=ℎ̇/h) of \n0.07 s-1, where h  is the penetration depth and ℎ̇ the penetration rate of the indenter. At \nleast 20 indents were performed on each phase at an imposed maximum depth of 150 nm. \nThe nanohardness was determined from the analysis of the load –displacement curves \nusing the Oliver  and Pharr method [28]. \nVickers hardness tests of all heat treated samples were carried out using Shimadzu HMV \nhardness tester according to the ASTM E92 – 17 Standard . The  RD–ND plane  of samples \nwas ground and polished using 1 µm diamond paste at the final stage. A  load of 4.9 N  \nwas appl ied for 15 s . \nA Kammrath&Weiss module was used for t ensile testing of d og bone sub -size samples \nat room temperature at a constant cross head speed corresponding to an initial strain rate \nof 10-3 s-1. These samples had a gauge length of 4 mm, a gauge width of 1 mm and a 8 thickness of 0.9 mm. They were machined from the homogeneously heat treated zone of \nthe heat treated  strips, so their  tensile axis was parallel to the RD. All samples  were \ncarefully grou nd and mechanically polished using OP-U (colloidal silica) at the final \nstage . At least three specimens were tested for each condition, and the results were found \nto be reproducible.  \n \n \n3. Results  \n3.1. Microstructural characterization  \n3.1.1.  Dilatometry  \nFigure 2 represents the evolution of austenite fraction during isothermal holding at \ndifferent intercritical temperatures. It is seen that the higher the peak temperature, the \nhigher the initial austen ite fraction, as the material is closer to the A C3 temperature . For \ninstance, at 860 ºC the austenite volume fraction is 19 %, and increases to 47 % with \nincreasing peak temperature to 90 0 ºC. It is also seen that  the peak temperature strongly \naffects the kinetics of austenite formation  and growth. A ustenite rapidly grows at the early \nstages of annealing at 900 ºC compared to annealing at lower peak temperatures of 860 \nºC and 8 80 ºC, which present a similar behavior. Soaking at 900 ºC for 600 s is sufficien t \nfor full austenitization , which i s reached after 522 s. At 880 ºC, the austenite fraction \nachieved at the end of the intercritical holding is 99 % , whereas  at 860 ºC  is 94 % . \nNevertheless, taking into account the positive  slope of the curve, it is clear that the \nmaterial will reach the complete austenitization after soaking for longer time . 9  \nFigure 2: a) Dilatation -time curves for material heated to 1100 ºC at 200 ºC/s with soaking for \n600 s at dif ferent temperatures (860 ºC, 880 ºC and 900 ºC); b) Effect of the peak temperature \non the austenite volume fraction during isothermal holding.  \n \n3.1.2.  SEM analysis  \nSEM analysis of the Gleeble  processed samples was performed to qualitatively \ncharacterize the influence  of both peak temperature and short soaking times on the \nmicrostructure. Figure 3 displays the microstructure variation at the different peak \ntemper atures stud ied for holding time s of 0.2 s ( Figure 3 a-c) and 1.5 s ( Figure 3 d-f). \nThe resultant microstructure is heterogeneous after all heat treatments b eing mainly \nformed by ferrit ic matrix and martensite  (marked by a white dashed arrow in Figure 3 b). \nIn turn, t he matrix  is formed by recrystallized (Rx) and non -recrystallized  (non-Rx) \nferrite, as it is demonstrated in Section 3.1.3 . A qualitative analysis shows that, \nindepend ently on the heat treatment parameters, all the conditions present a ferritic matrix \n10 consisting  of coarse and fine grains  due to  combination of diffe rent processes , which take \nplace during UFH (recovery, incomplete recrystal lization and grain growth at early \nstage s). Images on Figure 3 a-f demonstrate  that the increasing  peak temperature leads \nto grain growth  for both micro structural constituents (ferrite and martensite ) even after a \nholding time of 0.2 s. Moreover, it is poss ible to observe that the martensite fraction \nsubstantially grows  with increasing soaking time  independently of the peak temperature  \nand also with the peak temperature for a specific soaking time . As described  in Section \n3.1.1 , increasing peak te mperature and soaking time lead  to a higher fraction of \nintercritical austenite , which  is transformed into martensite upon quenching. The higher \nthe intercritical austenite fraction, the lower its carbon content  due to the C redistribution \nin its interior . A small amount of retained austenite grains was also identified  by EBSD \nanalysis in all conditions (see Section 3.1.3 ). In addition , very small amount of \nspheroidized cementite was observed  in the microstructure , which remains in the material \nfrom the initial cold rolled state  (marked by red arrow s on Figure 3 d). Its presence is \nrelated to  the short time given for its dissolution, hence, it is more commonly observed in \nthe samples annealed for the shorter soaking time of 0.2 s.  11  \nFigure 3: SEM photos illustrating the influence  of peak temperature  (860, 880 and 900 ºC ) and \nsoaking time (0.2 and 1.5 s) on the microstructure: a), b) and c) are for  0.2 s at 860, 880 and 900 \nºC, respectively; d)  & g), e) and f) are for  1.5 s at 860, 880 and 900 ºC , respectively.  \nSC: spheroidized cementite; M: martensite; F: ferrite; RA: retained austenite.  \n \n3.1.3.  EBSD analysis  \nEBSD analysis was performed i n order to identify and quantitatively characterize the  \ndifferent phases present in the microstructure of the heat treated samples . Figure 4a \nillustrates a typical EBSD phase map measured on the UFH860 -0.2s sample. Fine \nretained austenite grains (in white color) and martensite grains (in black color) are \nembedded into the ferrit e matrix composed of recrystallized ( Rx) ferrite (in orange color) \nand non-recrystallized ( non-Rx) ferrite (in blue color).  LAGBs are seen mainly in the \ninterior of the non -Rx ferrit e grains, whereas majority of the Rx ferrit e grains are free of \nLAGBs. The morphology of the microstructure and the individual micro constituents are \nvery similar in all  studied conditions,  \n12  \nFigure 4: a) Representative EBSD phase maps for the UFH860 -0.2s sample.  Rx ferrite is shown \nin orange; non -Rx ferrite in blue; martensite  in black and austenite is shown in white. HAGBs are \nrepresented in black and LAGB s in white. b) Histogram of grain orientation spread distribution \nin the ferrite  matrix for the EBSD phase map presented in (a). Rx ferrite  grains have GOS<1o \n(orange bars) , whereas non -Rx ferrite  grains have GOS>1o (blue bars), as described in Section \n2.4.  \n \nwhereas size and volume fraction of individual microconstituents depend on the heat \ntreatment  parameters . The results of the quantitative analysis are summarized in Table 1. \na \nb 13 Table 1: Data on the volume fraction of microstructural constituents as a function of heat \ntreatment parameters.   \nPeak \ntemperature (ºC)  860 880 900 \nSoaking time (s)  0.2 1.5 0.2 1.5 0.2 1.5 \nMartensite (%)  6.9 ± 3.2  12.6 ± 3.1  11.6 ± 2.1  20.2 ± 2.5  16.3 ± 3.8  27.8  ± 4.6  \nRetained \naustenite (%)  2.2 ± 0.4  2.1 ± 0.3  1.2 ± 0.3  2.5 ± 0.7  1.7 ± 0.3  1.7 ± 1.0  \nFerrite (%)  90.9 ± 4.0 85.3 ± 2.8  87.2 ± 1.9  77.3 ± 2.6  82.0 ± 3.6  70.5 ± 3.9  \nRx ferrite  48.4 ± 9.8  61.8 ± \n13.0 67.4 ± 3.8  73.5 ± 3.9  83.3 ± 3.7  80.3± 3.6  \nNon-Rx ferrite  51.6 ± 9.8  38.2 ± \n13.0 32.6 ± 3.8  26.5 ± 3.9  16.7 ± 3.7  19.7 ± 3.6  \n \nAnalysis of  the effect of soaking time for each temperature  shows  that at 860 ºC, the \nmartensite fraction increases from 6.9  % after 0.2 s to 12.6 % after 1.5 s. When \ntemperature raises up to 880 ºC, the volume fraction of martensite formed after 0.2 s is \n11.6 % , which increases to  20.2 % after 1.5 s. Finally, the 900 ºC treatment leads to the \nhighest increment  in martensite fraction during soaking , from 16.3 % after 0.2 s to 27.8 \n% after 1.5 s . For the shortest soaking time , the martensite  volume fraction increases by \nthe same amount  (~4.7 %)  when peak temperature is increased from 860 ºC to 880 ºC and \nthen from 880 ºC to 900 ºC . Similar dependence can also be noted after soaking for 1.5 s.  \nThe ferrite volume fraction presents a revers e trend, as both phases are formed in the \nintercritical temperature range . The portion of RA is minor in all heat treated conditions , \nbeing between 1.2 to 2.5 %.  \nThe morphology of the ferritic  matrix is greatly affected by both parameters, temperature \nand soak ing time. While at 860 ºC for 0.2 s the matrix is to larger extent formed by non-\nRx ferrite (~ 52 % ), whereas Rx ferrite prevails in the matrix after 1.5 s, reducing the \nvolume fraction of the non-Rx grains to ~ 38 %  (Figure 5). With increasing holding time 14 at 880 ºC, the non -Rx fraction is reduced to a lesser extent from 33 % to 27 %. Finally, \nsoaking at 900 ºC significantly reduces the volume fraction o f non-Rx ferrite : Average \nvolume fraction values are in the range of 17 -20 % and are not affected by  soaking time. \nTherefore, the most pronounced effect of soaking time in the studied temperature -time \nrange occurs at the lowe r peak temperature s of 860 – 880 ºC, as seen from  Figure 5. \n \nFigure 5: Volume fraction of non -Rx ferrite obtained from EBSD analysis for different \ntemperatures (860, 880 and 900 ºC) and soaking times (0.2 and 1.5 s). In red for 0.2 s and in \nblue for 1.5 s.  \nThe effect of the holding time (0.2 and 1.5 s) on the Rx and non -Rx ferrite grain size  is \nshown in Figure 6. The effect of peak temperature after soaking for 0.2 s is shown in \nFigure 6a. First, t he fraction of grains having size below 1 µm tends to decrease with \nincreasing peak temperature.  Second, a fter UFH to 860 ºC, the majority of  grains have  a \nsize between 1 and 2 µm, although it tends to shift to higher values with peak temperature \nand reaching the range of  2 - 3 µm at 900 ºC . Third, t here are some grains having  a size \nabove  6 µm even after heating to the lowest peak temperature of 860 ºC and their area \nfraction  increases with peak temperature. The  first two  observations are even more \npronounced when the holding time increase s to 1.5 s  (Figure 6b). It is shown that  at 860 \nºC, the fraction of grains below 1 µm has increased with respect to the 0.2 s counterpart, \nand it is considerably reduced at higher peak temperatures (880 ºC and 900  ºC). Moreover, \nthe main fraction of grains present s a larger  size when temperature is raised, although the \n15 fraction of grains larger than 6 µm has decreased for all temperatures at 1.5 s compared \nto the 0.2 s condition.  On the other hand, the average grai n size for the non-Rx ferrite at \n0.2 s is higher compared to the Rx ferrite at all studied temperatures ( Figure 6 c), as the \ngrains retained the initial  cold rolled micro structure. In addition, the distribution is \nnarrower compared to the Rx  grains , as there are almost no grains below 1 µm. At the \nlowest temperature the distribution seem s to be wider than at higher temperatures (880 \nand 900 ºC), although this is better seen  after 1.5 s ( Figure 6 d). Coarse grains (> 6 µm) \nare prone to disappear with temperature.  \n \nFigure 6: a) & b) representation of the area fraction for recrystallized (RX) ferrite grain size \nversus the equivalent circle diameter (ECD) vs. after 0.2 and 1.5 s ho lding time respectively for \nthe different temperatures studied; c) & d) non -recrystallized ( non-RX) ferrite grain size after \n0.2 and 1.5 s holding time , respectively. Data are obtai ned from the EBSD measurements.  \nFigure 7 represents the area fraction for martensite grains plotted versus the equivalent \ncircle diameter (ECD) after soaking for 0.2 s and 1.5 s at the studied peak  temperatures . \nFor 860 and 880  ºC after 0.2 s ( Figure 7 a), the vast majority of martensite grains present \nan ultrafine size  (below 1 µm). Contrary to the result seen in ferrite , the area fraction of \ngrains < 0.5 µm increases when temperature is raised up to 880 ºC. However, w hen \ntemperature further increase s to 900 ºC, the curve is shifted to the larger grain size values, \n16 with the peak  above 1 µm and  showing a wider size dis tribution than at 860 or 880 ºC . \nSimilar to  ferrite, the effects are more pronounced with increasing soaking time  to 1.5 s. \nAt 860 ºC the austenite formed at intercritical temperatures grows, while at 880 ºC  the \nfraction of grains below 1 µm increases resulting in a wider size distribution. Finally, at \n900 ºC the increased  intercritical austenite grain size shows  a wide r distribution  than the \n880 ºC and 860 º C . \n \nFigure 7: Martensite area fraction vs  ECD  after soaking for 0.2 s (a) and 1.5 s (b) at the peak \ntemperatures of 860 ºC, 880 ºC and 900 ºC.  \n \n3.2. Mechanical characterization  \n3.2.1.  Properties  of the individual microconstituents  \nNanoindentation  tests were performed on selected grains to investigate  the influence  of \nmaximum temperature and soaking time on the mechanical response  of the \nmicro structural constituents . Figure 8a shows an EBSD map where dark areas c orrespond \nto martensite and white areas to ferrite,  which were later identified by SPM imaging prior \nnanoindentation  testing  (Figure 8b). An SPM image  of the area with the nanoindentation \nimprints  was also recorded immediately after the test as observed in Figure 8c. Finally, \nthe microstructure was e tched with nital 2 vol % to take SEM images  (as shown in Figure \n8d) that corroborate the differentiation made by EBSD. Typical load – depth curves for \nthe main two micro structural constituents are shown in Figure 8e, where  the red and blue \ncurve s correspond to martensite and Rx ferrite , respectively .  \n17 Continuous l oad-depth curves were obtained from nanoindentaion on  martensitic grains, \nwhile majority  of the ferritic grains exhibited p op-in events, particularly the softer Rx -\nferrite grains. They are caused by sudden penetration bursts during the loading  process. \nThis effect has been related to the transition from an elastic to an elasto -plastic contact. \nThe probability of pop -in events and the pop -in load increase as the  dislocation density  \ndecreases , as discussed in o ur previous work [14]. \nThe measured nanohardness values for the main micro structural constituents: Rx ferrite, \nnon-Rx ferrite  and martensite are summarized in Table 2 as a function of peak \ntemperature and soaking time.  It is clearly seen that neither the soaking time nor the \ntemperature affects the nanohardness of Rx ferrite, which has average values  withi n the \nrange of 2.5–2.6 GPa . Nevertheless, there is a significant difference between Rx and non-\nRx ferrite , as the lat ter presents significant ly higher nanohardness values  (3.1–3.2 GPa ), \nbeing also similar for all studied conditions. The martensite  phase exhibits the highest \nnanohardness  for all  condition s, with the average values showing greater variation for  \neach condition . For instance, a fter heating to  860 ºC  and soaking  for 0.2 s , the martensite \naverage nanohardness value is 7.6 GPa, which redu ces slightly  to 7.4 GPa , when soaking \nincreases to 1.5 s, but still within measured standard deviation . The softening effect with \nsoaking time is  much  more evident  at higher peak temperatures : at 880 ºC, the \nnanohardness  drops by 10.3 %,  varying from 6.8 G Pa (after 0.2 s) to 6.1 GPa (after 1.5 \ns), while at 900 ºC , the drop is 16.7 %, as nanohardness  varies from  6.6 GPa and 5.5 GPa \nfor holding times of 0.2 and 1.5 s . Taking into consideration only the raise in temperature \nfrom 860 to 900 ºC, the average nanohardness drops by 13.1 % at 0.2 s ( from 7.6 GPa to \n6.6 GPa ), while  a more pronounced  drop of 25.7 % was observed at 1.5 s  (from 7.4  GPa \nto 5.5 GP a). 18  \nFigure 8: a) Band slope EBSD map with marked grains where  the nanoindentatio n tests were  \nperformed; b) SPM  map of the same area  without indentation imprints; c) SPM map of the area \nafter indentation; d) SEM image of the area after testing, etched with nital 2 vol.%; e) Typical  \nindentation load – penetration depth curves from nanoindentation measurements on  marte nsitic \ngrain (in red color) and ferritic grain (in blue color).  \n \n19 Table 2: Data on nanohardness (in GPa) of the individual microconstituents.   \nPeak \ntemperature \n(ºC) 860 880 900 \nSoaking time \n(s) 0.2 1.5 0.2 1.5 0.2 1.5 \nMartensite  7.6 ± 2.4  7.4 ± 1.1  6.8 ± 0.8  6.1 ± 0.7  6.6 ± 1.0  5.5 ± 0.6  \nRecrystallized \nferrite  2.6 ± 0.1  2.6 ± 0.1  2.6 ± 0.1  2.5 ± 0.1  2.6 ± 0.2  2.6 ± 0.1  \nNon-\nrecrystallized \nferrite  3.2 ± 0.2  3.1 ± 0.2  3.2 ± 0.2  3.2 ± 0.3  3.1 ± 0.2 3.2 ± 0.2  \n \n3.2.2.  Macro -mechanical characterization  \nThe effect of peak temperature and short soaking times on the macro -mechanical behavior \nof the AHSS was studied through hardness testing. The r esults are presented in Table 3. \nIt is seen that an increment on the peak temperature produces a rise in the hardness values \nindependently of the given soaking time. For 0.2 s , the increase in hardness between 860 \nºC and 88 0 ºC is insignificant , going from 252 to 255 HV0.5 respectively , while at 900 \nºC, the hardness increases to 264 HV0.5. The increase in hardness with peak temperature \nis much more evident for a soaking time  of 1.5 s , being 245 HV0.5 at 860 ºC , and \nincreasin g to 272 and 293 HV0.5 at 880 and 900 ºC , respectively.   20 Table 3: Data on h ardness of the heat treated strips . \nPeak \ntemperature \n(ºC) 860 880 900 \nSoaking time \n(s) 0.2 1.5 0.2 1.5 0.2 1.5 \nHardness  \n(HV0.5)  252 ± 4 245 ± 5  255 ± 5 272 ± 7 264 ± 5 293 ± 8 \n \nAdditionally , tensile testing was carried out for all conditions using miniaturize d dog \nbone samples . Figure  9 illustrates the typical engineering stress - engineering strain \ncurves . Data on  the mechanical properties determined from the curves (0.2% proof \nstrength σ0.2, ultimate tensile strength σUTS, uniform elongation εu and elongation to \nfailure  εf) are given  in Table 4. One can see  that at 860 ºC , the yield strength slightly \nvaries with soakin g time being 444 MPa and 441 MPa for 0.2 and 1.5  s, respectively. \nHowever, for higher peak temperature s and holding times, the yield point is enhanced . \nFor instance, at  880 ºC, the σ0.2 -values increase  by more than 20 MPa to 468 and 473 \nMPa for 0.2 and 1.5 s , respectively . For the maximum peak temperature analyzed (900 \nºC), the yield strength for both soaking times shows the maximum values, being 479 MPa \nafter 0.2 s and 493 MPa in the 1.5 s UFH treatment. Ultimate tensile strength values show \na similar t endency tha n the yield point. While at 860 ºC, the σUTS is independent of the \nsoaking time, with a value of  ~925 MPa, higher peak temperature s and soaking times  \nenhanced the strength of the material. For example, at 880 ºC and after 0.2 s, the material \npresents a σUTS of 933 MP a which increases up to 959 MPa at 900 ºC. The increment in \nstrength is more pronounced after longer soaking times, being  1017 and 1053 MPa for \n880 ºC and 900 ºC respectively . Nevertheless, the uniform elongation shows  the opposite \ntrend, being reduced from 24  % to 15 % when the temperature is increased  from 860 to \n900 ºC after 0.2 s. This reduction  in elongation with temperature is less significant after \n1.5 s, decreasing from 18 % at 860 ºC to 15 % at 900 ºC.  21  \nFigure  9: Typical  engineering stress – engineering strain curves from tensile testing of \nspecimens heated to  different peak temperatures and soaked  for: a) 0.2 s; b) 1.5 s. \n \n22 Table 4: Basic  mechanical properties determined by tensile testing of the heat treated samples . \nPeak \ntemperature \n(ºC) 860 880 900 \nSoaking \ntime (s)  0.2 1.5 0.2 1.5 0.2 1.5 \nσ0.2 444 ± 12 441 ± 8 468 ± 5 473 + 12  479 + 12  492 + 24  \nσUTS 926 ± 20 925 ± 10 933 ± 5 1017 ± 28 959 ± 21 1053 ± 42 \nεu (%) 24 ± 1 18 ± 2  18 ± 1  16 ± 2  15 ± 1 15 ± 1 \nεf (%)  33 ± 1 30 ± 2 27 ± 1 24 ± 1 23 ± 1 23 ± 3 \n \n \n4. Discussion  \n \n4.1. Influence of maximum temperature on the m icrostructure – properties \nrelationship in the UFH steel  \nDilatometry tests  (Figure 2) demonstrate that the higher the peak temperature , the higher \nthe initial volume fraction of austenite, as it was reported elsewhere [29,30] . These results \nare in a good agreement with the outcomes of the EBSD analysis  (Table 1), where  the \naustenite/martensite fraction increases with temperature for the same soaking time. The \nvariations between the initial austenite fraction measured for the studied temperatures \nthrough dilatometry and EBSD can be rationalized by the difference in  the applied heating \nrate. While during dilatometry the maximum heating rate employed was 200 ºC/s, the \nsamples analyzed by EBSD were processed at 800 ºC/s. It is known that high heating \nrates shift transformations temperatures (A C1 and A C3) to higher values [31,32] . \nTherefore, less amount of austenite is formed for the same peak temperature , when high er \nheating rates are applied. Moreover, heating at 800 ºC/s  implies that  the entire thermal \ntreatment is faster than heating at 200 ºC/s, giving less time for the austenite nucleation 23 process to be accomplished, resulting in the lower initial fraction of austenite observed in \nthe samples heated at 800 ºC/s. Furthermore, both characterization techniques, \ndilatometry and EBSD, confirm that the increase in peak tempera ture favors the formation \nof austenite nuclei, as nucleation highly depends on temperature [15,33] . For instance, \nduring the dilatometry test, intercricitical holding at 900 ºC results in a faster formation \nof austenite, and similar observations are made from the  EBSD results analyzing the \nmartensite grain size ( Figure 7). In the latter case, the fraction of grains having size below \n0.5 µm increases , when temperature is raised from 860 to 880 ºC for the shortest soaking \ntime ( Figure 7a). This means that the austenite nucleation is favored  at 880 ºC , whereas \nat 860 ºC the already formed austenite tends to grow as the area fraction of grains above \n1 µm is enlarged, being more  evident after soaking for 1.5 s ( Figure 7b). Hence, it is \npossible to state that at 860 ºC , the austenite gro wth is more significant than  nucleation, \nwhereas at 880 ºC, this behavior is inversed. At 900 ºC, both effects, nucleation a nd \ngrowth, are promoted, as the area fraction of grains below 0.5 µm and above 1 µm \nincrease with respect to the 860 ºC case, due to the high internal energy. The rapid grain \ngrowth during ultrafast heating to high peak temperatures has been reported by \nMassardier et al.  [8]. In addition to the grain size, the peak temperature also affects the \ninterior structure of the formed austenite, which transforms into martensite  after \nquenching. While at 860 ºC martensitic grains are chemically homogeneous, increasing \nthe temperature and soaking time results in the appearance of non -homogeneous \nmartensite regions, as it is shown in Figure 3e,f and Figure 6d which correspond to the \nUFH900 -1.5s sample. Similar observations were reported by Castro et al.  [17]. This effect \ncan be rationalized based on the existence of carbon gradients in the grain interior, due to \nthe high fraction of austenite formed, and the lack of time for carbon to diffuse through \nthe grains [34]. \nRegarding the ferrite phase, it is possible to observe that UFH  delays the recrystallization  \n[7,11,14,15,17,35] , leading to a matrix formed primarily  by non-Rx ferrite , when the \nmaterial is heated to  860 ºC for 0.2 s  (Figure 5). The fraction of non -Rx ferrite \nsignificantly decreases  with time and peak tempe rature , although it is not affected by the \nsoaking time at the maximum peak temperature of 900 ºC  and saturates at  17 – 20 % . The \nferrite tends to transform into austenite at high temperatures, thus the driving force for \nrecrystallization is reduced. Moreover, non-Rx grain size decreases with time and \ntemperature ( Figure 6 c & d), favored by recrystallization and by the consumption of 24 grains due to austenite  forma tion [32]. Nevertheless, Rx ferritic grains  show just the \nopposite behavior (Figure 6) growing with temperature and time . In addition, f or low \ntemperatures (860 ºC), ferri te tends to nucleate whereas,  the temperature  increment  favors \nthe ferrite grain growth , as the first nucle i formed rapidly grow s [17]. This is related to \nthe high stored energy from both, deformation induced via cold rolling and heat treatment.  \n \n4.2. Influence  of the peak temperature on the mechanical behavior  of the individual \nmicroconstituents  \nDifferent micro structural constituents formed during heat treatment  show dissimilar \nresponse  during nanoindentation testing. For instance , Rx ferrite present s a lower \nnanohardness compared to the non-Rx ferrite  independently o f the heating rate and \nsoaking time  (Table 2). The lat ter exhibits large  orientation gradients, as reported in our \nprevious work [36], mainly associated to the high geometrically necessary dislocat ions \n(GND ) density and residual stresses [37,38]. Moreover,  it should be noted that  the \naustenite to martensite transformation during quenching generates a volume expansion, \nwhich needs to be accommodated by the surrounding ferrite, introducing new dislocations  \nin both , Rx ferrite and non -Rx ferr ite [39]. The increment of the dislocation density can \naffect the ferrite mechanical behavior, as reported  in [40]. The nanohardness of any of the \nferrite micro structural constituents is not altered by  the processing parameters  (Table 2). \nOn the other hand, the nanomechanical response of the martensit ic grains  is greatly \naffected by the processing parameters. At the lowest peak  temperature (860 ºC) and \nshort est holding time  (0.2 s) , the martensite fraction is low ( Table 1) due to the short time \ngiven to the austenite nuclei to form and grow. Thus, the carbon concentration within  the \naustenite  grains increases , due to the high fraction of ferrite  [41]. As a consequence, the \nmartensite grains formed  at 860 ºC are the hardest (Table 2) in comparison to those \nformed at  higher peak temperatures, as the hardness strongly depends on the carbon \ncontent  [20,42] . Hence, the martensite strength is reduced with both, peak t emperature \nand soaking time, due to the carbon homogenization  in the auste nite grains form ed during \nthe heat treatment. Similar results on the effect of holding time in DP steels and peak \ntemperature during Quenching & Partitioning  processing were reported by Mazaheri et \nal. [43] and Hidalgo et al.  [44], respectively.  The softening effect of increasing soaking \ntimes on the martensitic grains is more pronounced at higher peak temperatures  due to 25 the high er austenite fraction ( Figure 2) and the more intensive  grain growth ( Figure 7). \nThe latter also increase s the diffusion distance of carbon [17] causing it s redistribution \ninside the grains, reflected in the lower standard deviations of the measured nanohardness  \n(Table 2).  \n \n4.3. Relation of the peak temperature with the macro -mechanical behavior of the \nmaterial  \nThe slight decrease in average hardness observed at 860 ºC when soaking time is \nincreased from 0.2 s to 1.5 s (Table 3) is due to  the high er fraction of Rx ferrite present \nat 1.5 s  (Table 1), as Rx ferrite exhibits lower nanohardness compared to the non-Rx \ncounterpart ( Table 2). In addition, the presence of coarser grains at 1.5  s than at 0.2 s also \nleads to reduction in average hardness , obeying the Hall – Petch law [45]. However, \nraising the  temperature to 880  or 900  ºC for 0.2 s  leads to higher  hardness due to the \nincreased fraction of  martensite [46]. On the other hand, holding time s of 1.5 s  at 880 and \n900 ºC produce a notable increase in hardness compared to their 0.2 s counterparts, as a \nconsequence  of the  considerable reduction of the ferrite volume fraction ( Table 1). \nSimilar results are observed for both, the yielding point and the ultimate tensile strength \nduring tensile testing. At 860 ºC, when holding time is increased from 0.2 to 1.5 s, there \nis no variation o f σ0.2 or σUTS (Table 4), although martensite fraction is increased with \ntime ( Table 1). This observation can b e associated with the reduction of  the non -\nrecrystallized ferrite fraction , which presents a higher resistance to deformation compared \nto its recrystallized counterpart [47]. In addition , the grain size of ferrite also increase s \nwith soaking time  resulting in a lower strength  [45]. Nevertheless, the increment of \ntemperature (880 ºC and 900 ºC) for a fixed soaking time  strengthens the material (yield \nand ultimate tensile strength), due to a significant increase in the martensite volume \nfraction [48]. Our observations are consistent with the previous work  published elsewhere \n[49,50] . Nevertheless, the increases in strength  with both , temperature  and soaking  time, \nresults in a significant loss in the  ductility of the material. This is associated to the drop \nin the ferrite fraction, which is softer and more ductile than the martensite  [51], which is \nalso confirmed via nanoind entation ( Table 2). In addition  to the ductility, the strain \nhardening coefficient was analyzed for  the different conditions, following t he common \npower -law relationship in Eq. (1) [52] 26 𝜎=𝑘𝜀𝑛 (1) \nThe strain hardening rate (n) in Eq. (1) was obtained using the following Eq.(2) \n𝑛= ln𝜎𝑎\n𝜎𝑎−1\nln𝜀𝑎\n𝜀𝑎−1 (2) \nwhere 𝜎𝑎 and 𝜀𝑎 represent the true stress and true strain in the point a, respectively.  \nFigure 10 shows  the variation of the  strain hardening rate with true strain for 0.2 and 1.5 \ns samples. For both holding times, at 860 ºC the material present s a higher strain \nhardening rate  than its counterparts  for any strain , decaying  at a lower rate. This effect is \nassociated with the higher fraction of non-Rx ferrite present in the microstructure, as \ncompared to the ferrite formed at higher temperatures . In the n on-Rx ferrite, the onset of \nplastic deformation requires higher stress ( Table 2), due to its high er dislocation density. \nTherefore, when temperature or soaking time are increased, the strain hardening \ndecreases, as a consequence  of the reduction in the non-Rx ferrite fraction. When the \nmicrostructure shows a high martensite volume fraction, the differen ce in strain hardening \nrate is reduced. Several authors have associated this behavior to the martensite islands \nsurrounded by ferrite. The volume expansion caused by the austenite to martensite \ntransformation needs to be accommodate d by the surrounding fer rite grains, resulting in \nstrain hardening of the matrix [53,54] . 27  \nFigure 10: Representative strain hardening rate - true strain curves determined for  samples \ntreated at 860 ºC, 880 ºC and 900 ºC after soaking for 0.2 s ( a) and 1.5 s (b). \n \n4.4. Potential microstructural and property gradients in the UFH processed sheets  \nAnalysis of the experimental results from the microstructural (Section 3.1) and \nmechanical (Section 3.2) characterization shows , that t he industrial processing window \nof 20 ºC (i.e. ±10 ºC) should lead to some heterogeneity of the microstructure on the \n28 meso -scale (i.e. 0.1…1 mm). It will show up as some deviations in the size and local \nvolume fraction of individual micro structural constituents (martensite, Rx ferrite and non -\nRx ferrite), as well as the hardness of martensite. Nevertheless, such local heterogeneities  \nof the microstructure should no t degrade the overall mechanical behavior of the processed \nsheets on macro -scale. First, there are no significant differences in basic mechanical \nproperties of the UFH processed steel within processing  window of 20 ºC (Table 4). \nSecond, the UFH processed steel shows high strain hardening ability independently of \nthe peak temperature  (see Figure  9, Figure 10). The latter should eliminate  any minor \nnegative effects from the microstructur al heterogeneity in the UFH processed sheets  \nappeared  due to the deviation of the local peak temperature within ±10 ºC window . \n \n5. Conclusions  \nWe have studied t he effect of the UFH parameters, peak temperature  (860 ºC, 880 ºC and \n900 ºC ) and soaking  times  (0.2 s – 1.5), on the  microstructure and mechanical response  \nof a Fe-0.19C -1.61Mn -1.06Al -0.5Si steel at different scales . The main conclusions of our \nstudy include : \n1) The increase in p eak temperature  promotes aus tenite formation. Independently of the \npeak t emperature, mainly nucleation of small austenite grains with their limited growth \noccurs  at the short est soaking time (0.2  s), whereas both formation of nuclei and their \ngrowth proceed after soaking for  longer time (1.5 s) .  \n2) Morphology of the ferritic  matrix is significantly altered by both peak temperature and \nsoaking time. The lower peak temperature and shorter soaking time promote \nnucleation of the recrystallized ferritic grains, while the fraction of the non -\nrecrystallized ferritic matrix undergoin g recovery process remain s high. With \nincreasing both parameters, the average grain size of the recrystallized  ferritic grains  \nand their volume fraction tend  to increase. These processes are  accompanied by \ndecrease  of the ferrite volume fraction due to the increasing volume fraction of the \nintercritical austenite (i.e. martensite after quenching).  \n3) Independently on the applied heat treatment parameters , the highest nanohardness is \nmeasured on martensitic grains followed by non-recrystallize d ferrite and \nrecrystallized ferrite. Peak temperature and soaking time strongly affect t he \nnanohardness of the martensitic grains . The higher the temperature the larger the 29 grains, reducing the carbon concentration therein . On the contrary, nanohardness of  \nthe ferritic microconstituents is affected neither by temperature nor soaking time. The \nnon-recrystallized ferrite is harder than its recrystallized counterpart due to the high er \ndislocation density of the former.  \n4) At the peak temperature of 860 ºC, the in crease in soaking time within studied range  \ndoes not produce an improvement in the mechanical properties despite higher  \nmartensite volume fraction, due to the decrease  of the non-recrystallized ferrite  \nfraction  and the grain growth. Nevertheless, increasin g peak temperature s to 880 ºC \nand 900 ºC favors the strengthening of the material, as the effect of martensite becomes  \nthe dominant factor . However, the ductility is considerably reduced with both, \ntemperature and soaking time, due the lower fraction  of the ductile ferrit ic phase.  \n5) Analysis of the experimental results from the microstructural and mechanical \ncharacterization shows that  the industrial processing window of 20 ºC may lead to \nsome heterogeneity in the microstructure of the UFH processed sheets . 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Scott1,2 *  \n \n \n1School of Physics and Astronomy, St. Andrews University, St. Andrews KY16 9SS, UK  \n2School of Chemistry, St. Andrews University, St. Andrews KY16 9ST, UK \n3Dept. Physics, SPECLAB, Univ. Puerto Rico, San Juan, PR 90021 USA \n§Present Address: Paul Scherrer Institut, 5232 Villigen, PSI, Switzerland.  \n \nAbstract \n \nThere have been a large number of papers on bismuth ferrite (BiFeO 3) over the past few \nyears, trying to exploit its room-temperatu re magnetoelectric multi ferroic properties.  \nAlthough these are attractive, BiFeO 3 is not the ideal multiferroic, due to weak magnetization \nand the difficulty in limiting leakage curre nts.  Thus there is an ongoing search for \nalternatives, including such materials as gallium ferrite (GaFeO 3).  In the present work we \nreport a comprehensive study of the perovskite PbTi 1-xPdxO3 with 0 < x < 0.3.  Our study \nincludes dielectric, impedance and magnetizat ion measurements, conductivity analysis and \nstudy of crystallographic phases present in the samples with special attention paid to minor phases, identified as PdO, PbPdO\n2, and Pd 3Pb.  The work is remarkable in two ways: Pd is \ndifficult to substitute into ABO 3 perovskite oxides (where it might be useful for catalysis), \nand Pd is magnetic under only unusual conditions  (under strain or internal electric fields). \nThe new material, as a PZT derivative, is expected to have much stronger piezoelectric properties than BiFeO\n3. \n \n*Author to whom correspondence to be addressed. Electronic mail: jfs4@st-andrews.ac.uk  (J. \nF. Scott)  \n  \n Introduction \n1.1. Ferroelectric Memories \n \nBeginning around 1984 random access memories utiliz ing ferroelectric thin films (FRAMs or \nFeRAMs) were developed.  In comparison with other non-volatile computer memories these were fast (60 ns in early embodiments), low- power (voltage-driven as opposed to current-\ndriven magnetic memories), radiation-hard (no single-event upset SEU), and cheap.[1,2]  \n \nWithin the past few years they have become a major disruptive industry at > $100 \nmillion/year, replacing magnetic-stripe cards and used for train and subway fares, cash-points \n(“e-money”), vending machines, employee identification cards, storage lockers, and convenience stores.  They are contact-free card s, which can be updated (money added), and \nthey are manufactured up to 8 Mb capacity by both Fujitsu and Toshiba in Japan and Texas \nInstruments in the USA (the latter for Washington subways).  They are sold under the brand names FeliCa and Suica in Asia; Samsung has published 64 Mb FRAMs (the state of the art) \nbut as far as we know is not marketing them commercially.[3]\n \n \n1.2. Multiferroic magnetoelectrics \n \nIn research laboratories worldwide one major effort has been to combine the advantages of FRAMs (speed, low power) with the advantages of magnetic devices (non-destructive read-out).  This involves the use of materials that are simultaneously ferromagnetic and ferroelectric (multiferroic) and which have a linear (magnetoelectric) coupling between those \ntwo phenomena.  BiFeO\n3 has been the leading contender,[4,5,6,7] but other materials such as GaFeO 3 have also been explored.[8] In the present paper we consider a new material, PbTi 1-\nxPdxO3, that appears potentially superior to BiFeO 3 for some devices. \n \n1.3. Pd in Perovskite Oxides \n \nPure perovskite-phase lead palladate PbPdO 3 is not stable (PbPdO 2 with Pd2+ is favored), and \nindeed other investigators trying to substitute Pd into ABO 3 perovskite oxides report that the \nsubstitution is reversible and complicated due to various Pd oxidation states adopted.[9,10] \nHowever, we have found that up to 30% can be substituted into PbTiO 3, mostly at the Ti4+ B-\nsite, where it is a good match for valence and i onic size.  We find that typically 30-40% of \nthe Pd goes in as Pd2+, however, a point to which we shall return.   \n \nIn addition, Pd4+ ([Kr] 4d6, low-spin configuration) is normally not ferromagnetic; [11] \nhowever, it has an instability towards magnetism and can be ferromagnetic under internal \nelectric fields or strain, [12,13] such as might be encountered if it is located not at the Ti4+ B-\nsite but at the oversized Pb2+ perovskite A-site. \n \nThus in examining lead palladium titanate as a multiferroic, we are rejecting a popular view \nthat it will not substitute in perovskite oxides and that it would not be ferromagnetic if it did.  In this sense the present work leads to a radical new direction in multiferroics. We emphasize that although Pd is expensive (comparable to gold), the intended devices are thin films, typically 30-100 nm thick, for which the use of Pd is a negligible additional cost, and remind \nreaders that Ag/Pd was the most common electrode material in ordinary multilayer capacitors \n(Kyocrera, TDX, AVX, etc.) until its relatively recent replacement by Ni. \n  \n  \n1.4. Initial Experiments \n \nIt had been proposed that doping ferroelectric lead titanate (PbTiO\n3) with palladium (Pd) \nresults in a magnetoelectric multiferroic material at room temperature. This is remarkable because Pd atom itself is non-magnetic, but when  it replaces lead (Pb) and titanium (Ti) ions, \nmagnetism is observed as predicted from the density functional theory (DFT) studies[14] (discussed in section 1.5).\n \n \nThere is some experimental evidence for multiferroicity and giant magnetoelectric coupling \nat room temperature too. Kumari et al. [14] showed that 30% Pd doped lead titanate zirconate \nPb(Zr,Ti)O 3 (PZT) displays room-temperature weak fe rromagnetism, strong ferroelectricity, \nand strong ME coupling.  However their work was not as conclusive as the present study, due \nto multiple unresolved secondary phases present in the samples and the rather complicated \nelectronic structure of PZT. \n The seminal work of Kumari et al.  [14]\n is important but differs from the present in several \nimportant ways: (1) The main impurity phase PbPdO 2 could not be seen in XRD because its \nstrong Bragg reflections are coincidentally un der the main PZT peaks; (2) The magnetic \nproperties measured did not reveal the magnetic PbPdO 2 contribution; (3) The SEM studies \ndid not isolate any minor phases, including PbPdO 2; (4) Activation energies for conduction, \nimportant for device applications were unmeasured; and (5) No change in T c with Pd% could \nbe measured due to the dependence of Curie temperatures on Ti/Zr ratio. \n  \n  \n1.5. Predictive Theory \n \nFerromagnetism arising from doping PbTiO\n3 with Pd was predicted by DFT calculations of \nPaudel and Tsymbal (Kumari et al. [14]) and more generally for Pd in earlier work by Sun, \nBurton, and Tsymbal [12].  \n \nThree different types of Pd doping were modeled computationally. When Pd4+ substitutes Ti4+ \nion, the induced defect states lie in the conduc tion and valence bands with no occupied defect \nstates in the band gap of PbTiO 3: isoelectric substitution does not produce magnetism. The \nsame applies for Pd2+ substituting for Pb2+ ions. However, a magnetic moment of 2µ B is \nobserved when Pd4+ replaces both Ti4+ and Pb2+ cations that are first or second nearest \nneighbors. Exchange-split peaks are observed in the band gap of the system at 2.0 eV, related to spin polarization of both Ti and Pb density of states (DOS). These defect states hold two \nelectrons that can be thought of as donated by a Pd\n4+ ion residing nominally on the Pb2+ atom. \nMost of the magnetic moment (1µ B) is residing on the Pb site; Pd on the Ti site produces a \nsmall moment (0 .1µB), and the rest comes from spin polarization of the DOS of oxygen \natoms bonded to Pd replacing Pb. \n \nExperimental Details \nPbTiO 3 preparation : stoichiometric amounts of PbO and TiO 2 were calcined in PbTiO 3 \nsacrificial powder at 600°C and 800°C for 2h and 4h, respectively. Optimal sintering \nconditions were found to be 900°C for 16h. Pd-doped PbTiO 3 preparation : stoichiometric \namounts of PbO, TiO 2 and PdO were calcined in sacrificial PbTiO 3 powder. B-site substitution only was assumed for stoichiometric calculations. Calcination temperatures were \nchosen between 600°C and 800°C for 1 to 10 hours. Pellets for electrical measurements were sintered at 700°C to 900°C for 4 to 16 hours to minimize the formation of Pd-containing side \nproducts. PbPdO\n2 preparation: PbPdO 2 was prepared from stoichiometric amounts of PbO \nand PdO. Pellets were sintered in PbO sacrificial powder for 72 hours at 700°C.  PXRD measurements : PANalytical EMPYREAN powder X-ray diffractometer in reflection \nmode was used to collect XRD data. XRD patterns were collected for 2 θ values of 10° to 90° \nwith the measurement duration set to 1-2 hours. Rietveld refinement : All powder diffraction \ndata were analyzed by the Rietveld method using the General Structure Analysis System \n(GSAS).[15]\n The cif model (crystallographic informat ion file) of room temperature PbTiO 3 \n(P4mm symmetry)[16] modified by incorporation of  Pd atoms as well as the cif models of \nPbPdO 2,[17] PdO[18] and Pd 3Pb[19] were used for refinements. Background and peak \nprofiles were refined in order to determine para meters of the unit cell and weight fractions of \ncrystallographic phases. \nSEM and EDX : Micrographs were obtained using a Joel JSM 5600 scanning electron \nmicroscope set to a 5 kV accelerating voltage . All micrographs presented were obtained from \nthe fracture surfaces of pellets. Oxford Inca EDX system was used for compositional \nanalysis. \nXPS:  Pellets were mounted to the ultra-high vacuum (UHV) system using conductive tape. \nDuring spectra acquisition, flood gun was used at 2.0 eV. To remove surface layers of the pellets, Ar\n+ ion bombardment was performed in the UHV system for 20 minutes at 44µA/ \n3kV.  \nDielectric and impedance studies :  The pellets were electroded by applying conductive \nsilver paint on both faces of the circular pellets. In some cases gold electrodes were applied using an Emitech K550x sputter-coater. A coatin g current of 30 mA was applied until the resistance measured across the electrode wa s under 50 Ohms. Dielectric measurements \nwere performed using a Wayne Kerr 6500B impedance analyser: capacitance and \ndielectric loss ( tanδ) values as were collected in the temperature range of 25 °C to 550 °C, \nwith an applied AC excitation of 500 mV  (frequencies ranging from 25 Hz to 2 MHz). \nElectroded pellets placed in a compression jig within a Carbolite MTF 10/25/130 benchtop \ntube furnace. The cooling/ heating rates were 2 K min-1. For the low temperature \nmeasurements (50 K - 300 K), an Agilent 429A impedance analyzer was used with the sample mounted in a closed cycle cryocooler.  Magnetization measurements : The SQUID measurements were made in Quantum Design \nMPMS SQUID magnetometer. \n \n \nResults and Discussion  \n \n1.6. Scanning Electron Microscopy (SEM) and Energy Dispersive X-ray \nSpectroscopy (EDX) Measurements \n \nSEM and EDX were combined to determine the ceramic microstructure and perform \nelemental analysis of the samples, particularly focusing on Pd content in different phases. SEM micrographs of various undoped and Pd doped PbTiO\n3 samples reveal relatively \nsmall grains and a high-degree of porosity, which was expected as PbTiO 3 synthesis is \nknown to be challenging. This is due to the vol atilization of PbO at elevated temperatures, \nwhich often causes deviation from stoichiometry, inhomogeneous microstructure and porosity\n [20,21] and a high tetragonality (c/a = 1.064 with c = 0.4156 nm and a = 0.3902 \nnm[22]) that develops during cooling across the Curie point, which induces large internal stresses in the unit cell that can easily re sult in flaws and cracks. Sintering at low temperatures (600 °C-800 °C) results in the grain size of 0.7±0.2 µm, which increases to \n2.7±0.6 µm when sintered at high temperature (900 °C), which is illustrated in Figure 1a .  \n \nPd does not easily substitute for cations in perovskite oxides. Even though it is used with \nperovskite oxides for catalysis, the perovskite  acts as a supporting framework that allows \ndispersion and activation of metallic Pd nanoparticles at its surface. Substitution of B-sites \nis very rarely observed and normally is under 10%.[9,23,24] Therefore it was expected that \nduring PbTi 1-xPdxO3 preparation some other Pd-containing phases might. Powder X-ray \ndiffraction (PXRD) patterns reve al impurity phases of PbPdO 2 and PdO in calcined \npowders of 10% and 30% Pd-doped lead titana te (see Figure S1, Figure S2 and Figure S3, \nSupplemental Material [25]). Pd 3Pb phase was detected after sintering at 800 °C or higher, \ntherefore samples used for further measurements  were prepared below this temperature.  \n These secondary phases were difficult to identify in the samples sintered at low temperatures, \npossibly due to their low concentrations and very small grain size (as shown in Figure 1a , \nleft). They were successfully resolved only for samples with considerable amounts of impurity phases (PbPdO\n2 and Pd 3Pb), prepared at high temperatures. Figure 1b  shows the \nmicrograph of different phases present in 30 % Pd doped sample, which was sintered at \n900°C. EDX data were used to assign phases to their chemical composition by considering \nelemental fractions of Pb, Ti and Pd. There appears to be a clear separation between Pd-\ncontaining PbTiO 3 phase (small circular grains, 1-2 µm) and undoped PbTiO 3 (large cuboids, \n10 µm). The PbPdO 2 phase was identified too, as a spherical cluster of very small grains. \n Samples analyzed can be grouped into undoped and doped samples, which were further \ngrouped depending on their sintering properties.  The variations in particle size and effective Pd substitution for each group are shown in Table 1 . Increase in particle size was \nobserved not only at higher sintering temperatures but also after sintering in oxygen atmosphere. EDX studies helped to determine that Pd substitutes better at low \ntemperatures. Synthesis under oxygen flow at low temperatures is optimal as it maximizes \nboth ferroelectric (related to grai n size) and ferromagnetic prope rties (related to higher Pd \nfraction), the only drawback being inhomogeneity of the ceramics due to an oxygen vacancy gradient. \n1.7. X-ray Photoelectron Spectroscopy (XPS)  \n \nXPS data were collected from the surface of 30% Pd-doped pellet. Pd 3d\n5/2 and Pd 3d 3/2 \nregions were detected at binding energies of 336.95 eV and 342.07 eV, respectively ( Figure \n2a). The peaks are symmetric and coincide with the region expected for Pd2+. Therefore it \nwas concluded that only Pd2+ cations are present on the surface of the pellet. The observation \ncan be explained by oxygen loss from the surface, which turns Pd4+ into Pd2+. \n To investigate bulk of the pellet, Ar\n+ bombardment in the UHV setup was performed to \nremove the surface of 10% Pd doped PbTiO 3. The Pd 3d 5/2 and Pd 3d 3/2 regions were no \nlonger symmetric, revealing presence of both Pd4+ and Pd2+ states, required for magnetism \n(Figure 2b ). The Pd 3d 5/2 and Pd 3d 3/2 doublets were deconvoluted into Pd2+ components at \n336.82 eV, 342.15 eV, and Pd4+ at 337.72 eV, 343.05 eV. \n A shift observed in Pd\n2+ binding energy can be attributed to differential charging effects \ninduced by ion beam. The ratio of Pd4+ to Pd2+ states was determined to be 1.47:1.00. \nAccording to the DFT calculations, Pd4+ is required to be present in PbTiO 3 in order for \nmagnetism to be observed - this condition is fulfilled.  \n 1.8. Dielectric Studies  \n \nThe dielectric spectroscopy data collected for undoped PbTiO 3, as well as 10% and 30%  Pd \ndoped pellets are shown in Figure 3a . The profile of the dielectric curves reveal one peak \ncorresponding to ferroelectric to paraelectric transition. All three samples were also found \nto exhibit frequency independent (normal ferroelectric) behavior ( Figure 3b ), which is \nuseful for device applications.  \n \nThe magnitude of relative permittivity was strongly influenced by the quality of ceramics \nanalyzed: Pd doping improved densification of pe llets (density increased up to 40%); dense \nceramics with fewer pores were observed to have larger relative permittivity values \nthroughout the temperature sweep. Th is is normally explained by 90 ° domain size effects, \nwhich contribute for a very large part of relative permittivity.[26] The number of \nferroelectric 90◦ domains per volume is maximized for well-sintered ceramics. Another \nexplanation could possibly be the strain induced by Pd doping. Due to the mismatch in radii of Ti and Pd (0.01Å [27]), the volume of unit cell slightly increases upon doping. \nTherefore undoped nanoregions expand under the stress of the matrix,[28]\n which creates \nmore space between Ti atoms and the oxyge n octahedra allowing larger off-center \ndisplacements directly related to polarization strength. [28,29] \n Upon Pd doping, the measured ferroelectric T\nc values were lower. This can be rationalized \nby changes in the unit cell induced by doping with Pd4+ ion, whose radius (0 .615 Å [27]) is \nslightly larger than that of Ti4+ (0.605Å [27]). The phenomenon of transition temperature \ndecreasing with the substitution of larger cations for smaller cations was observed in various perovskites and related structures and the effect can be as large as a 100 °C shift \n[30,31].  \nThe relative permittivity profile in 30% Pd doped samples is rather different from that \nobserved for undoped and 10% Pd doped PbTiO 3. Even though in the measured \ntemperature range there is only one transition, above T c the relative permittivity continues \nto increase. This is anomaly observed due to conduction processes in the ceramic, when \nconducting electrons and/or ions are measured as displacement current related to surface \npolarization charge density. Conduction is  commonly observed in lead-containing \nferroelectrics and is attributed to oxygen vacancies [32,33]. \n \nOxygen vacancies become mobile at elevated temperatures ( > 150°C [34]) and ionic \nconductivity becomes appreciable. However, oxygen vacancies cannot be the only source \nof conductivity in Pd-doped PbTiO 3. The 10% Pd doped sample sintered at 900 °C for 16h \n(in which lead loss should be maximal) does  not show anomalous increase in dielectric \nconstant, whereas 30% Pd doped pellets sinter ed at much lower temperatures and for \nshorter durations exhibit considerable cond uctivity. This observation reveals that \nconductivity is mainly created by PbPdO 2 impurity phase (larger weight-fraction in 30% \ndoped sample, see Figure S3, Supplemental Mate rial [25]). This is not surprising as \nPbPdO 2 is reported to have a metallic-like conductivity at 90 K - 300 K. [17]  \n \nKumari et al. [14] have shown that PZT (PbZr 1-xTixO3) is ferroelectric and magnetoelectric \nmultiferroic at room temperature over a wide range of Zr/Ti ratios, using the positive-up negative-down (PUND) method and quantitative  measurements of magnetoelectric tensor \ncomponents, so we do not show such data here. We do find that leakage current is a problem for 30% Pd at T = 292K and suggest that such studies will be much more precise with thin films rather than our bulk ceramic specimens. Such films have not yet been made \nbut should be possible with spin-on techniques and a liquid Pd source, or via sputtering.  \n \n1.9. Impedance Spectroscopy \n \nImpedance spectroscopy measurements were performed on undoped, 10% and 30% Pd \ndoped PbTiO 3. All plots show only one feature: semicircles in the complex plane modulus \nand impedance plots are symmetric, while Debye peaks in Z” and M” spectroscopic plots \nhave nearly the same peak frequency, indicating they originate from the same single electroactive region (impedance plots collected for 10% Pd doped PbTiO\n3 are shown in \nFigure 4) .  \n \nA single electroactive region observed in impedance spectra normally corresponds to a \nvery well sintered sample with very larg e grains and thin and well-defined grain \nboundaries. On the contrary, SEM studies of both undoped and Pd doped PbTiO 3 samples \nshow very small grains. Therefore it is very unlikely that grain boundary contribution is negligible – electroactive regions probably have similar time constants τ = RC.  \n \nNyquist semicircles in M* plot are distorted in the high frequency range. This can be explained by inductance of the measuring l eads at high frequencies. Furthermore, Pd-\ndoped PbTiO\n3 samples are magnetic which also amplifies this effect. The parasitic \ninductance also affects semicircles in Z* plot s: it causes a tail at high frequencies, which \ncrosses the real impedance axis, but this is not obvious in the data collected.[35]  \n \n \n1.10. Conductivity Analysis \n The control of conductivity is essential in ferroelectric ceramics - they should be insulating \nto minimize dielectric loss. The bulk conductivity in the three pellets and their variation with temperature was extracted from complex impedance and electric modulus \nmeasurements ( M”, M*, Z” and Z* values). Ferroelectric mate rials are normally wide-band \ngap semiconductors and their temperature dependent conductivity processes generally \nfollow the Arrhenius law: ߪൌ  ߪ\n exp ቀെாೌ\n்ቁ, where σ is the conductivity, ߪ is a pre-\nexponential factor and Ea is the activation energy. \n \nArrhenius plots and corresponding parameters ( Figure 4 ) reveal different features of \nelectrical transport in undoped, 10% and 30% Pd doped PbTiO 3 pellets. PbTiO 3 exhibits \ntwo resolved thermally activated conductivity re gions - in ferroelectric and in paraelectric \nregions. The activation energy is similar for both (0.58 eV and 0.46 eV, respectively), \nwhich suggests the same type of conduction mechanism.  For the 30% Pd samples, the \n0.74±0.02 eV activation energy most likely arises oxygen vacancy hopping; it agrees with the literature value of 0.74 eV in the paraelectric phase of PZT.[34]\n Migration barriers for \noxygen vacancies are known to be phase dependent,[36] which explains the slight \ndifference in activation energy values. The 10% doped pellet exhibits similarly behaved \nconductivity in paraelectric region, however, the conductivity in ferroelectric region is no \nlonger resolved. This can be explained by PbPdO 2 phase, conductivity of which decreases \nwith temperature and is dominant at low temperatures,[17] hindering Arrhenius behavior of \noxygen vacancies, which becomes dominant at high temperatures. The conductivity in the \n30% Pd doped sample is more complex. Th e sample is highly conducting even at room \ntemperature and therefore little change is observed at low temperatures. A thermally activated conductivity region is resolved below T\nc with E a=0.73 eV, which is not too \ndifferent from oxygen vacancy conduction observed previously. Conductivity in the paraelectric phase has a very low E a of only 0.335 eV, which is characteristic of electronic \nconduction or mixed ionic-electronic conduction. \n \n1.11. Effect of Oxygen Annealing \n \nTo improve dielectric properties of Pd-doped PbTiO 3, pellets were sintered in a flowing \noxygen atmosphere. This oxygen annealing was expected to decrease oxygen vacancy concentration and the amount of PbPdO\n2 present in the samples. The relative permittivity \nprofile of the 30% Pd-doped PbTiO 3 is significantly improved by oxygen annealing \n(Figure 5a ). Comparison of relative permittivitie s shows that oxygen annealing suppresses \nconductivity and also promotes Pd substitutio n into perovskite structure as previously \nsupported by EDX studies (decrease in T c). Comparison of the XRD patterns ( Figure 5b ) \nshows that sintering in oxygen atmosphere decreases the amount of PbPdO 2, with a minor \nincrease in PdO concentration.  \n \n 1.12. Magnetization Studies \n \nMagnetization measurements were performed as a function of magnetic field on samples of \nboth 10% and 30% Pd doped PbTiO\n3 samples, as well as on a separately prepared sample \nof the main magnetic impurity phase PbPdO 2  (Figure 8a ). The data on all samples were \nfound to exhibit a significant diamagnetic cont ribution to the signal, clearly evident over \nmost of the temperature range.  In the plots presented, this contribution has been subtracted by performing a linear fit to the raw data in  order to better highlight the ferromagnetic \ncontributions of interest. \n The 10% and 30% Pd doped samples are found to be weakly ferromagnetic even at room \ntemperature ( Figure 8b ), with 30% Pd doped sample possessing a significantly larger \nmagnetic moment at saturation (0 .2·10−4 and 2 .8 · 10−4 emu g-1 respectively). Both samples \nalso exhibited an appreciable hysteresis, wi th the highest coercivity being found for 10% \ndoped sample (~260 Oe at 300K).   For device a pplications a high coercivity is an essential \nrequirement, as it determines the stability of the memory by making it more robust to the influence of external magnetic fields. Further measurements made at 400K on 10% Pd doped PbTiO\n3 still indicated strong ferromagnetic behavior. \n Since both 10% and 30% Pd doped PbTiO\n3 samples contain measurable fractions of \nPbPdO 2, it is imperative to ascertain the possible contributions of this impurity phase to the \nmagnetic signals. PbPdO 2 is known to be diamagnetic at temperatures higher than 90 K, \n[17, 37, 38]  but the sample measured here was ferro magnetic at 300 K (inset Fig 10a). The \ncharacter of this phase is however very different from that observed in the perovskite samples, with a very small coercivity (~ 50 Oe) and a much higher saturated moment (Figure S4, Supplemental Material [25]). It is known that when doped with magnetic metal ions, such as iron,\n [39] copper or cobalt [38], that PbPdO 2 can become ferromagnetic even \nat room temperature, and it is likely that this is the origin of the ferromagnetic signal in the \nPbPdO 2 measured here. While the significantly higher moment of PbPdO 2 means that only \na small amount of this impurity phase, if similarly doped, might make measureable ferromagnetic contributions to the signals observed for 10% and 30% Pd doped samples, \nthe very different character of the ferromagnetic signal in both cases confirms that the \nferromagnetism observed  in the Pd-doped PT  samples is intrinsic to those materials.  To gain some insight into the evolution of the magnetic order with temperature, a \nsaturation field was applied at 2K and then the field set to zero so that the remanent magnetization could be recorded. The sample was then heated in zero field and the value of \nthe magnetization was recorded as a function of  temperature (Figure 11a). In this way any \nchanges in magnetic order are likely to manifest as rapid changes in magnetisation. The general trend for both 10% and 30% Pd doped PbTiO\n3 is a slow reduction of magnetization \nwith increasing temperature, as would be expected for an ordered system. There is however a small feature around 90K. This does not appear to correlate with any features \nfound in relative permittivity measurements, and in fact occurs at the known metal-\ninsulator transition (90K) for undoped PbPdO\n2.[17] This feature thus seems most likely to \narise from a change in conductivity of the impurity phase and not from any changes to the perovskite phase.  It is thus reasonable to  conclude that the magnetic state of the \nmultiferroic samples remains essentially unchanged from 2K up to the highest temperature \nmeasured (400 K). \n \nConclusion \n \nPd doped PbTiO\n3 perovskite was found to be with Pd substitutional at both Pb- and Ti-sites \nand was shown to exhibit ferromagnetic propertie s at temperatures as high as 400 K. High \nlevels of Pd substitution (30%) were observed  in the perovskite structure along with the \nformation of PbPdO 2 and Pd 3Pb impurity phases. Pd4+ and Pd2+ states were identified, \nfulfilling the condition for ferromagnetism determined from the density functional theory studies by Kumari et al.[14]\n  Lowering of ferroelectric Curie point (to 740 K and 730 K for \n10% and 30% doping, respectively) and increase in relative permittivity (by 1.5 times and \n7.0 times for 10% and 30% doping, respective ly) were observed. The samples exhibited relatively low saturation magnetisation at room temperature (0 .2 · 10−4 and 2 .8 · 10−4 emu \ng-1 for 10% and 30% doping, respectively), but high coercive field (258 Oe). Even though \na ferromagnetic signal was detected for a minor phase present in the sample (PbPdO 2), it \nwas shown that it is a small contribution to the total signal. Pd-doped samples showed \nincrease in conductivity, which was explained by metallic-like conductivity of PbPdO 2 in \nthe lower temperature range and oxygen vacancy , as well as electronic conductivity, at \nhigh temperatures. Oxygen annealing was shown to be effective in decreasing this lossy behavior.   \nIn order to produce prototype room-temperature devices it is still necessary to eliminate \nminor phases, especially PbPdO\n2 and Pd 3Pb, which occur in our ceramic specimens and \nexacerbate the leakage currents.  New efforts are suggested for thin-film fabrication, \nincluding sol-gel spin-on and sputtering techniques.  However, the present work serves as \nan existence proof for compounds that will combine the excellent piezoelectric and pyroelectric properties of PZT with ferromagnetism.  This has obvious commercial applications for nonvolatile memories as well as for transducers and actuators. In \ncomparison with two-component multiferroic sandwich devices, which are limited by \nstrain coupling and hence the speed of sound, they offer improved response time. \n \nAcknowledgements \nA part of the work was carried out at the University of Puerto Rico (UPR) with financial \nsupport provided by the DoD-AFOSR Grant # FA95501610295. J.F.S. acknowledges his visit expenses to UPR from IFN-NSF Grant # 1002410. Work at St. Andrews was supported \nby EPSRC grant EP/P024637/1. \n \n \n \n \nFigure 1.  a) SEM micrographs showing the effect of sintering at higher temperature: \nmicrographs show samples sintered at 750°C (l eft) and 900°C (right). b) SEM micrograph of \n30% Pd doped sample, sintered at 900°C. Elemental composition of different phases and \nparticle sizes are shown. \n  \n \n \n \n  \n \n \n \n \n \n \n \n \n  \n \n \n \nTable 1.  Particle size and effective Pd substitution to perovskite phase: variation with \ndifferent synthesis conditions. \n \nGroup Sintering \nT (◦C) Pd doped \n(%) Particle \nSize (µm) Detected \nPd (%) \nUndoped \nPbTiO 3 700 \n900 \n 0 \n0 0.7 ± 0.2 \n0.6 ± 0.2 \n nil \nSintered at \nlow T 750 \n750 \n 10 \n30 \n 0.8 ± 0.3 \n0.7 ± 0.1 \n 10 \n26 \n \nAnnealed in \nO2        750 \n750 \n 10 \n10 2.4 ± 1.7  \n 1.6 ± 0.3 \n 25 and 9 \n10 and 3 \n \nSintered at \nhigh T        900 \n       900  10 \n30 \n 2.7 ± 0.6 \n1.5 ± 0.5 \n 8 \n10 \n \n \n \n \n  \n \n \n \n  \n \n  \n \n \n \n  \n \n \n \n   \nFigure 2. XPS spectra of the Pd 3d peaks from the surface of 10% Pd doped PbTiO 3 pellet: \na) before Ar+ bombardment; b) after Ar+ bombardment. \n \n \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. a) Relative permittivity versus temper ature for undoped, 10% and 30% Pd doped \nPbTiO 3 pellets. T c values are labeled. b) Relative permittivity of 10% Pd doped PbTiO 3 as a \nfunction of temperature and frequency (feature at 325°C corresponds to the curing of silver \nelectrodes). \n  \n \n \n  \n \n  \n \n  \n \n \n \n \nFigure 4. High-temperature impedance spectroscopy data collected for 10% Pd doped \nPbTiO 3 sample: a) M’’ spectroscopic plot; b) Z ’’ spectroscopic plot; c) complex plane M* \nplot and d) complex plane Z* plot. \n \n \n  \n \n  \n \n \n \nFigure 5. Plots showing temperature dependence of conductivity for (a) undoped , (b) 10% \nPd-doped and (c) 30% Pd-doped PbTiO 3. Data points were extracted from values measured \nfor M”, M*, Z” and Z*. \n \n  \n \n  \n \n  \n \n \n \n  \n \n \nFigure 6. The oxygen annealing effect on 30% Pd doped PbTiO 3: (a) difference in relative \npermittivity and (b) XRD patterns when compared with sintering in ambient atmosphere.  \n \n \n \n \n \n  \n \n  \n \n \n \n  \n \n \n \n  \n \n  \n \n \n \n \n \nFigure 7. Comparison of magnetisation M(H) for 10%, 30% Pd doped PbTiO 3 and the \nPbPdO 2 secondary phase at 2K (a) and 300K (b). Insets show M(H) for PbPdO 2 (a,b) and \n10% doped PbTiO 3 (b). (c) Evolution with temperature of the remanent magnetisation \ncreated at 2 K, for 10% and 30% Pd doped PbTiO 3. The inset shows variation in \nmagnetization derivative with temperature. \n \n \n      \n \n  \nReferences  \n[1]  J. F. Scott, Ferroelectric Memories , Springer, Heidelberg, Germany, (2000).   \n[2]  J. F. Scott, C. A. De Araujo, Science, 246, 1400, (1989). \n[3]  J. F. Scott, International Symposium on Nonvolatile Memory. The Technology Driver of \nthe Digital Age, Proceedings edited by S. M. Sze, 75, (2017). \n[4]  J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu, D. Viehland, V. \nVaithyanathan, D. G. Schlom, U. V Waghmare, N.  A. Spaldin, K. M. Rabe, M. Wuttig, R. \nRamesh, Science, 299, 1719, (2003). \n[5]  T. Zhao, A. Scholl, F. Zavaliche, K. Lee, M. Barry, A. 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Int. 42, 15762, (2016). \n \n "    },    {        "title": "1304.7255v1.Ferroelectricity_from_iron_valence_ordering_in_rare_earth_ferrites_.pdf",        "content": "arXiv:1304.7255v1  [cond-mat.str-el]  26 Apr 2013physica statussolidi, 30 August2018\nFerroelectricity from iron valence\norderinginrareearth ferrites?\nManuelAngst*,1,2\n1PeterGr¨ unbergInstitutPGIandJ¨ ulichCentreforNeutron ScienceJCNS,JARA-FIT,ForschungszentrumJ¨ ulichGmbH,5 2425J¨ ulich,\nGermany\n2Experimental Physics IVC,RWTHAachen University, 52056 Aa chen, Germany\nReceivedXXXX, revisedXXXX,accepted XXXX\nPublishedonline 19April2013 - Phys.Status SolidiRRL,DOI :10.1002/pssr.201307103 (2013)\nKeywords: Multiferroics,Ferroelectrics, Charge Order,Spin-Charg e Coupling, LuFe2O4, RareEarthFerrites\n∗Corresponding author: e-mail m.angst@fz-juelich.de , Phone: +49-2461-612479\nThepossibilityofmultiferroicityarisingfromchargeor-\nderinginLuFe 2O4andstructurallyrelatedrareearthfer-\nrites is reviewed. Recent experimental work on macro-\nscopicindicationsofferroelectricityandmicroscopicde -\ntermination of coupled spin and charge order indicates\nthat this scenario does not hold. Understanding the ori-\ngin of the experimentally observed charge and spin or-\nder will require further theoretical work. Other aspects\nofrecentresearchinthesematerials,suchasgeometrical\nfrustration effects, possible electric-field-inducedtra nsi-\ntions,ororbitalorderarealsobrieflytreated.\nBilayer subunit in RFe2O4compounds with proposed charge\norder rendering itpolar.\nCopyrightlinewillbe provided by the publisher\n1 Introduction The interplay of magnetic and elec-\ntric degrees of freedom in correlated electron materials is\noneofthemajortopicsofcontemporarycondensedmatter\nphysics, with a rapidly rising number of publications per\nyear [1]. Materials combining magnetism with ferroelec-\ntricity, termed multiferroics, offer the prospect of a larg e\nmagnetoelectric coupling, which has a high potential for\napplicationsinfutureinformationtechnology[2].Becaus e\nthe traditional mechanism of ferroelectricity is incompat i-\nble with magnetism [3], much interest has focused on un-\nconventionalroutes to ferroelectricity (see [4] for a rece nt\nextensivereview).Particularlyintriguingisthemechani sm\nof ferroelectricity originating from charge ordering (CO)\n(reviewed in [5,6]), the ordered arrangement of different\nvalence states of an ion, typically a transition metal. Be-\ncause any ferroelectric polarization is built from electri c\ndipolemoments,i.e.non-centrosymmetricchargedistribu -\ntions, any CO breaking inversion symmetry automatically\ninduces a polarization, which may be very large [5]. Thepresenceofdifferentvalencestatesofatransitionmetali on\nimplies an active spin degree of freedom on the same ion,\nandthereforeastrongmagnetoelectriccouplingmaybeex-\npected as well. This mechanism of multiferroicity is thus\nvery attractive from the point of view of prospective ap-\nplications.However,whiletheconceptisstraight-forwar d,\nexamples of oxides where this mechanism is experimen-\ntally indicated to occur are exceedingly rare and none is\nreallywell understood.\nTheoftencited(e.g.[4,5,6])prototypicalexamplema-\nterial providing “proof of principle” for this mechanism\nis LuFe 2O4, a charge- (and spin-) frustrated system. In\nthe following, I will review recent research on this and\nisostructural materials, which may serve also as an illumi-\nnatingcase studyfor(muchless investigated)otherpoten-\ntial examples of CO-based ferroelectricity. LuFe 2O4had\nbeenproposedin2005tobeamultiferroicduetoFe2+/3+\nCO[17],basedonbothmacroscopicindicationsbydielec-\ntric spectroscopy and pyroelectric current measurements,\nCopyrightlinewillbe provided by the publisher2 M. Angst: Rareearthferrites\nFigure 1 a)R3mCrystal structure of\nRFe2O4anddescriptionwithdifferentcells\n(after [7]). Apart from the primitive rhom-\nbohedral and the R-centered hexagonal\ncells the lattice could also be described by\none of three C-centered monoclinic cells,\nwhich are rotated by 120◦with respect to\neach other. Cells describing the charge and\nspinorder inthe threedomains areobtained\nfrom these by tripling the monoclinic band\ndoubling themonoclinic caxes (omittedfor\nclarity). b) R3+ion size effect on cell vol-\nume and ratio of intralayer Fe −Fe distance\nahexto bilayer thickness db(c.f. c), com-\npiled from [8,9,10,11,12,13,14]. c) Iron\nbilayers with one of the monoclinic cells,\nand the corresponding supercell. Blue tri-\nangle illustrates geometrical frustration ef-\nfects (see text). Red lines denote a minimal\nset of interactions [15,16] necessary for 3D\ncharge order.\nand on a plausible microscopicmodel of a polar CO. This\nproposed “ferroelectricity from CO” is the main focus of\nthe review, in particular recent findings by structure re-\nfinementof a non-polarCO [18] and the suggestionof the\nabsence of polar order by dielectric spectroscopy [19,20].\nOtheraspectsthathavebeenofrecentfocusinthesemate-\nrials, such as geometrical frustration expressed in CO and\nmagnetic order, strong spin-charge coupling, or potential\nelectric-field-inducedphasetransitions,aretreatedasw ell.\nThe main experimentalfindingsreviewed are summarized\nin Sec. 7, which concludes with a brief outlook on future\nresearch directions both in this family of compounds and\nfor“ferroelectricityfromCO” in general.\n2 TheRFe2O4familyRare earth ferrites RFe2O4\nwithRa3+ionwithoutpartiallyoccupied dlevels(Y,Ho,\nEr, Tm, Yb, Lu, or In), known since the 1970s [21], crys-\ntallize in a rhombohedral( R3m) structure [8,9,10,11,12,\n13,14] (see Fig. 1a) featuring characteristic triangular b i-\nlayers of trigonal-bipyramidal-coordinatedFe as the elec -\ntronicallyactivesubunit(c.f.Fig.1c).MagnetismoftheF e\nspinsattractedaninitialwaveofresearchintothesemater i-\nals, with unusual features such as anomalous thermomag-\nnetization [22] or giant coercivity [23] discovered. In all\ninvestigatedcompounds,averylargeIsinganisotropywith\nspins pointing perpendicular to the layers was observed\n[23,24]. The average valence of the Fe ions is 2.5+, and\ntherefore Fe2+/3+valence order or CO may also be ex-\npected, first investigatedby M¨ ossbauer spectroscopy[25] .\nHence, there are two binary degrees of freedom at the Fe\nsites: Isingspin ↑/↓andvalence 2+/3+.\nGeometrical Frustration The arrangement of Fe\nions leads to “geometrical frustration” [26] hampering\nboth spin order (SO) and CO, illustrated in Fig. 1c) for/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53\n/s32/s61/s32/s48/s46/s48/s48/s48/s32/s61/s32/s48/s46/s48/s52/s48/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s84 /s32/s40/s75/s41/s89/s70/s101\n/s50/s79\n/s52/s45\n/s32/s61/s32/s48/s46/s48/s53/s53Figure 2 Magnetization\nM(T)of polycrystalline\nYFe2O4−δwith differ-\nentδ, measured in 0.4T\nupon heating, after the\nsample has been cooled\nin the same field ( ◦) or in\nzero-field ( •). Two tran-\nsitions in the most sto-\nichiometric sample are\nmarked. Data from [27].\nthe Ising spins by a blue triangle: repulsive or antiferro-\nmagnetic (AF) nearest-neighbor (NN) interactions within\nthe layers can be satisfied only for two of the spin pairs\nof the triangle. For an isolated layer, this would lead to\na macroscopic degeneracy of ground states. The proxim-\nity of another triangular layer changes the topology of\nthe problem, particularly since the inter-layer Fe-Fe NN-\ndistance (marked U2in Fig. 1c) is about 8%shorter than\nthe intra-layer one (marked U1). However, there is poten-\ntial for frustration in the inter-layer interaction as well ,\nbecause each Fe ion has three NN in the other layer (see\nalsoFig.8).Furthermore,therhombohedralstackingleads\ntofrustrationoftheinteractionsbetweendifferentbilay ers,\nhindering full 3D order. Even when macroscopic ground-\nstate degeneracy is broken, geometrical frustration often\n[26] leads to a competitionof differentphases with nearly\nthe same energy, facile creation of defects, and complex\nunusualgroundstates,suchastheproposed[17]ferroelec-\ntric CO inthesematerials,discussedin Sec.4.\nStrong impact of oxygen stoichiometry A soon\nobserved experimental difficulty in these compounds was\na strong variation of physical propertiesfor different sam -\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 3\nFigure3 Magnetization M(T)ofthreedifferentLuFe 2O4\ncrystalsmeasuredin 100Oe/bardblc,uponcoolingexceptwhere\nnoted.Panela)after[28], panelb)after[29].\nples. For YFe 2O4−δthis was clearly linked to tiny vari-\nations in oxygen-stoichiometry [27], as shown in Fig. 2:\nmagnetization M(T)for the most off-stoichiometricsam-\npleshowsalargedifferencebetweenzero-field-cooledand\nfield-cooled Msuggesting “glassy” magnetism without\nestablishment of long-range SO, whereas M(T)for the\nstoichiometric sample indicates two distinct well-ordere d\nantiferromagnetic phases. Single-crystal neutron diffra c-\ntion confirmed a tendency towards 3D long-range order\nforsamplesclosertostoichiometry[30],butduetotheab-\nsence of sufficiently stoichiometric crystals until recent ly\n[31], the magnetic structures remain unsolved. Such a\nsample-dependencewas observed also for Er [32] and Lu\n(see Fig. 3), although the link to O-stoichiometry [33] is\nless clear, mostly due to the range of stoichiometries for\nwhich long-range order is observable shrinking with de-\ncreasingR3+size (c.f. [32] and Fig. 1b). In a wider range\nof stoichiometries, links to the remanent magnetization\nandresistivityare clear,however[34].\nIon-size effects and structural modifications\nChangingthe R3+size leads to an expectedchange of the\ncell volume ( ∼5%between In and Y), but more relevant\nfor CO and SO is a 15%change in the ratio between the\nlayer-separation within a bilayer, db, and the intra-layer\nFe-Fe distance ahex(Fig. 1b), which can be expected to\nmodify the relative importance of interactions U1andU2.\nPartial substitution, e.g. by Co, Cu, or Mn, on the Fe site\ngenerally leads to decreased conductivity, but also sup-\npressed magnetic ordering (see e.g. [35,36]). However,\neffects due to substitution can be difficult to disentangle\nfromeffectsduetoa potentiallychangedO-stoichiometry.\nFinally, a larger structural modification can be done by\nintercalating RFe2O4with one or more blocks of RFeO3,\neach adding an additional R−and a single-Fe layer be-\ntween the bilayers [37]. There have been relatively few\nstudies on these intercalated compounds (e.g. [38]). Fi-\nnally, a high-pressure orthorhombic modification with a\nlarge supercell, possibly CO, has recently been reported\nFigure 4 Real part of the dielectric permittivity ε′vs\nT(lines) in LuFe 2O4for frequencies between 1Hzand\n1GHzequally spaced with two frequencies per decade.\nThe data additionallymarkedwith symbols( ◦,+)areboth\ntaken at100kHz, but with electrodesfromdifferentmate-\nrial: Graphite ( ◦) and silver ( +and all other data). Figure\nfrom[19], c/circleco√yrt2012AmericanPhysicalSociety.\n[39], but the details of this structure have yet to be eluci-\ndated.\n3 Anomalousdielectricdispersionandpyroelec-\ntriccurrents Manyrareearthferrites,e.g.ErFe 2O4[40],\nhave been studied extensively by dielectric spectroscopy,\nwithanomalouslylargerealpartsofthedielectricconstan t\nε′in the low-frequency limit generally observed. A typi-\ncal (recent) example of such a measurement is shown in\nFig. 4. Fromthe frequency-dependenceofthetemperature\nof maximum rise of ε′(T)a connection of the dispersion\nto electrons hopping between Fe2+and Fe3+has been\nconcluded and an origin of the large dielectric constants\nin motion of ferroelectric domain boundaries suggested\n[17,40]. Within this interpretation, giant ( >20%) room-\ntemperature magneto-dielectric response [41] would then\nsuggest multiferroicity with large magnetoelectric cou-\npling, potentially useful for applications. Research on\npossible ferroelectricity in rare earth ferrites focused o n\nLuFe2O4, because in thiscompoundsuperstructurereflec-\ntions were found, which for example by resonant x-ray\ndiffraction at the Fe K−edge could be clearly associated\nwith Fe2+/3+charge order [17,42], allowing the discus-\nsionofpotentiallypolarCO phases(seeSec. 4).\nMacroscopic proof of ferroelectricity requires demon-\nstration of a remanent electric polarization, which should\nbe switchable by application of an electric field. A rema-\nnent polarizationlargerthan in the traditionalferroelec tric\nBaTiO 3was suggested by pyroelectric current measure-\nments [17], shown in Fig. 5a). These measurements were\nperformed by first cooling the single-crystal sample in an\nelectric field of ±10kV/cm, then switching off the elec-\ntric field, and finally measuring, upon warming in E= 0,\nthe pyroelectric current associated with the change of the\nCopyrightlinewillbe provided by the publisher4 M. Angst: Rareearthferrites\nFigure 5 a) Electric polarization Pvs temperature T\nof LuFe 2O4as deduced from pyroelectric current mea-\nsurements conducted on warming after the sample had\nbeen cooled in electric fields E/bardblcof±10kV/cm. Panel\nfrom [17], c/circleco√yrt2005 Nature Publishing Group. b) Pvs\nEhysteresis loops of LuFe 2O4at140Kfor two differ-\nentfrequencies,from[20], c/circleco√yrtEDPSciences,Societ` aItal-\niana di Fisica, Springer-Verlag 2012. With kind permis-\nsion of The European Physical Journal (EPJ). c) P(E)of\nLu2Fe2.14Mn0.86O7at2kHz, from[38], c/circleco√yrt2009 Ameri-\ncanInstituteofPhysics.\npolarization. The dependence of the pyroelectric current\nand thus the polarization on the direction of the cooling\nfieldisconsistentwithamacroscopicpolarization,andthe\ndecay to zero around the charge ordering temperature of\nTCO∼320−330K[43,44] supports a connection of P\nwith the charge ordering,althoughthe lack of a saturation\nregion with no pyrocurrent flowing anymore above TCO\nrenders this last conclusion tentative. It was noted, how-\never, that similar behavior can also be observed in (non-\nferroelectric)leakydielectrics,whereitoriginatesfro mlo-\ncalizationoffreecarriersatinterfaces[45].\nSubsequentmeasurementsbyvariousgroupspartlyre-\nproduced the pyrocurrent results of Fig. 5a), as noted e.g.\nin [42], or tried to establish P(E)hysteresis loops at low\ntemperaturesto moredirectlyshowferroelectricity.Outo f\nmany attempts at the latter, only few were published [20,\n46,47], some showing hysteresis, but not the saturation\nnecessary[48]toestablishpolarizationswitching.Fig.5 b)\nshowsP(E)measuredat 140Kandtwo frequencies[20]:\nthelow-frequencyloopshowssubstantialhysteresis,butn o\nindications of saturation, whereas the loop measured with\na frequency of 80kHzexhibits practically linear behavior\nwithoutappreciablehysteresis.\nBecause the polarization is linked with the dielectric\nconstant, P= (ε−1)ε0E, detailed modeling of the di-\nelectric dispersion can show the origin of observed polar-\nization behaviors. This was done for the results obtained\non LuFe 2O4single crystals over 9 decades of frequencies\n[19] and shown in Fig. 4. The qualitative behavior with\nlargeε′at low frequencyandhigh Tisthe same as in pre-\nvious observations [17], but a strong impact of the elec-\ntrodematerialasshownfortheexampleof 100kHz(differ-\nent symbols) already suggests significant contact effects,e.g.duetoSchottky-typedepletionlayersatthecontactin -\nterfaces, which act as an additional capacitance [19]. The\nwhole data-set of ε′(f,T)andε′′(f,T)could be well de-\nscribedwithanequivalent-circuitmodelfeaturingcontac t-\ncapacitance and -conductance, and sample hopping- and\ndc-conductivity, in addition to the intrinsic dielectric c on-\nstantεiof the material, all parametersindependent of fre-\nquency. The fitted εi(T)is only around 30and does not\nshow any indications of ferroelectric or antiferroelectri c\ntransitions.TworecentstudiesonpolycrystallineLuFe 2O4\nreached the same conclusion[20,49]. For low frequencies\norhighT,thecircuitreducestoaleakycontactcapacitance\nin series with the sample resistivity, resulting in large ε′.\nBecause the step in ε′mainly depends on contact capaci-\ntanceandsampleresistance[19]amagneto-resistancewill\nlead to an apparent magnetodielectricresponse, according\nto the analysis in [49] the latter can be fully accountedfor\nby the former. A very recent study [50] of dielectric spec-\ntroscopy using various contact materials suggests a small\n(∼1%) sample magnetodielectric effect at least at 220K\n(at the AFM-fM transition, c.f. Fig. 11), but the simulta-\nneouslyobserved18timeslargerchangeinresistivitymay\ninfluencethe fittedsamplecapacity.\nP(E)loops can only show the intrinsic behavior if\nmeasured at frequencies and temperatures where ε′(f,T)\nisclosetoεi(T).ForthemeasurementsshowninFig.5b),\nthis is the case for P(E)measured at 80kHz, but clearly\nnot forP(E)measuredat 0.2Hz.Thereis one P(E)loop\nin the literature [38] that doesseem to indicate ferroelec-\ntricity given the expected tendency to saturate in high E\n(Fig. 5c), measured on intercalated Lu 2Fe3O7, in which\nconductivityhasbeensuppressedby ∼29%MnforFesub-\nstitution. The occurrenceof at least piezoelectricityin t his\ncompoundissupportedbypiezoresponseforcemicroscopy\n[51]. Although the switchable polarization is very weak\n(comparepanelscanda),smallereventhanin“spin-spiral\nmultiferroics”likeTbMnO 3,thissuggeststhatintercalated\ncompoundsdeservemoreattentionthantheycurrentlyget.\n4 Charge Order It was the combination of macro-\nscopic indications of ferroelectricity, discussed in Sec. 3,\nwith a likely microscopicmodelof chargeorder(CO) that\nis electrically polar, which made a convincing case for\nLuFe2O4exhibiting ferroelectricity from CO [17]. This\nCO-model was deduced by scattering methods, in which\nsignaturesoftheorderinreciprocalspaceareinvestigate d.\nFor manyRFe2O4, at least diffuse scatteringof x-rays\nat the(1\n31\n3ℓ)lineinreciprocalspace(inhexagonal-cellno-\ntation) suggested a tendency towards a CO that could be\ndescribed,forexample,withaso-called√\n3×√\n3cell.Al-\nternatively, a propagation vector with (1\n31\n3) in-plane com-\nponent can be conveniently described by expressing the\ncrystal structure without CO in a C−centered monoclinic\ncell, which is then enlarged three times along its b−axis.\nBecause such a CO breaks the 3-fold rotation symmetry\nof the crystal structure, domains of CO with symmetry-\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 5\nFigure 6 Sketch of the 4symmetry-allowed Fe2+(black)\n/ Fe3+(white) charge configurations following (1\n31\n30) or\n(1\n31\n33\n2) propagation (with additional contribution by ( 000)\nor (003\n2), respectively). Arrows indicate bilayer polariza-\ntion.\nequivalent (2\n31\n3) and (1\n32\n3) propagations and 120◦rotated\nmonocliniccells, assketchedinFig. 1, alsoexist[44].\nThe simplest arrangement of Fe2+and Fe3+ions in\na bilayer leading to such propagations and maintaining\ncharge neutrality is the one proposed in [17] and shown\nin Fig. 6 top-left (two cells of the CO are shown). Here,\nforeachbilayer,theupperlayerhasasurplusofFe3+ions\nand the lower layer a surplus of Fe2+ions, which implies\nanelectricdipolemomentbetweenthelayers(indicatedby\narrows), i.e. this CO is indeed ferroelectric, rendering th e\nbilayerspolar.\n4.1 Long-range CO from diffraction In the case of\nLuFe2O4, actual superstructurereflections were observed,\nbelowTCO∼320K, by several groups using various\ndiffraction techniques (Fig. 7). Most studies resolving\n3D peaks found the principal superstructure reflections\nto be near positions (1\n3,1\n3,halfinteger) and symmetry-\nequivalent [17,18,15,42,43,44,52,53,54,55,56,57,58,\n59],within-plane/out-of-planecorrelationlengthsgive nas\ne.g.30/6nm[53] or80/7nm[55].Resonantx-raydiffrac-\ntion at the Fe K-edge of these reflections support their\nassociation with Fe2+/3+CO [17,42,54,58]. Sharp re-\nflections at other positions, including near (1\n3,1\n3,integer)\n[60,61,62] are sometimes observed in electron diffraction\nstudies, generally on samples with higher oxygen content\n[59,62]. The additional reflections in [59] were attributed\ntoorderingofexcessoxygenions.High-resolutionx-raydiffraction[15,43,44] foundthe\nexact positions of the (1\n3,1\n3,halfinteger) reflections to be\nwell described by s+pA,s+pB, ands+pC, where\nsis a structural reflection following the rhombohedral\ncentering condition −h+k+ℓ= 3nwithninteger.\nThe three symmetry-equivalent propagation vectors, cor-\nresponding to CO domains, are pA= (1\n3+δ,1\n3+δ,3\n2),\npB= (2\n3−2δ,1\n3+δ,3\n2),andpC= (1\n3+δ,2\n3−2δ,3\n2)and\nδ∼0.003T−dependent [44]. A secondary set of reflec-\ntionsisdescribedby p′\nA= (τ,τ,3\n2),p′\nB= (2τ,τ,3\n2),and\np′\nC= (τ,2τ,3\n2),andτ∼9δfollowingthe T−dependence\nofδ. Although this seems to suggest an incommensu-\nrate modulation of the Fe valence state, there is no evi-\ndence of a “true” incommensuration, which would imply\na wide distribution of Fe valence states from 2+to3+,\nin conflict with spectroscopic measurements [25,63,64].\nMuch more likely is therefore a locally commensurate\nstate,interspersedwithdiscommensurationsoranti-phas e-\nboundaries (Fig. 8) of an average separation, estimated\nfromτ∼0.028[44], of about 12nm. It has been demon-\nstratedonothercharge-orderingoxidesthataproliferati on\nof such anti-phase-boundaries, which can cost very little\nenergyduetogeometricalfrustration,canleadtosharpsu-\nperstructurereflectionsat incommensuratepositions[65] .\nConsequently,thepossibleCOcorrespondingto(1\n31\n33\n2)\nand for completeness also (1\n31\n30) have to be examined\nthrough symmetry-analysis [66]. For each of these prop-\nagation vectors, there are two irreducible representation s,\ncorrespondingto the two Fe ions of the primitive unit cell\n(labeled1and2inFig.1a)havingthesame( Γ2)ordiffer-\nent (Γ1) valence. Each corresponding CO contains more\nthan than two valence states, in contradiction to the re-\nsults of M¨ ossbauer spectroscopy, which imply a bimodal\nvalence distribution [63]. Such a bimodal distribution is\nobtainedbycombining,inauniqueway[44],theCOfrom\n(1\n31\n33\n2)[resp.(1\n31\n30)]withtheonefrom( 003\n2)[resp.(000)].\nThe four resulting CO are shown in Fig. 6. The CO\nwithΓ2(lowerrow)havebilayerswithanetcharge,indeed\nfor (1\n31\n30)/(000) the whole structure would be charged.\n(1\n31\n30)/(000) withΓ1gives the ferroelectric CO with po-\nlar bilayersproposedin [17]. However,since the observed\n(alsoin[17])superstructurereflectionsfollow(1\n31\n33\n2)prop-\nagation, one of the two CO shown on the right of Fig.\n6 should be realized, either (top) the antiferroelectric CO\nwith polar bilayers but zero net polarization, as proposed\nin [44], or (bottom) the CO with stacking of oppositely\nchargedbilayers.Thelatterwasinitiallydismissedbecau se\nit involvesa transferofchargebetweenneighboringbilay-\nersseparatedby ∼6˚A,butanexperimentaldetermination\nof which CO is realized requires a full structural refine-\nment.\nThis was recently achieved [18], the key being the\nscreeningofalargequantityofsmallcrystalsofthehighes t\nquality, as judged by M(T)(Fig. 3), for one with almost\nonlyoneofthethreepossibleCO-domainspopulated.The\nstructure obtained at 210K, readily refined in the space\nCopyrightlinewillbe provided by the publisher6 M. Angst: Rareearthferrites\nFigure 7 Charge order CO superstructure in LuFe 2O4by different techniques. a) X-ray diffraction scattered in tensity at\n(hhℓ)at200and360K.From[44], c/circleco√yrt2008AmericanPhysicalSociety.b)Neutrondiffractionsca nsalong(1\n31\n3ℓ)atthree\ntemperatures.From[52], c/circleco√yrt2008AmericanPhysicalSociety.c)Neutrondiffractionint ensityinspin-flipchannelat( hhℓ)\nat245K. From [29], c/circleco√yrt2012 American Physical Society. d) Electron-diffraction d ark-field image taken at 92Kwith a\n(1\n31\n35\n2)superstructurespot,circledintheinset.TheCOdomainse xtend∼30nmintheab-planeand ∼6nmalongc.Inset:\nelectrondiffractionpatternwith ( 110)incidence.From[53], c/circleco√yrt2009AmericanPhysicalSociety.\ngroupC2/mwith a residual of 5.96%,containsfouriron-\nsiteswithvalences,determinedwiththebond-valence-sum\n(BVS) method [67], 1.9,2.1,2.8, and2.9[18]. Given the\nuncertaintiesinherent to the empirical BVS-method, these\nvalues can be considered to be remarkably close to the\nideal values of 2and3(for a comparison of other charge\norderingferritesseeFig.5in[68]).ThethusidentifiedCO\nis the one with charged bilayers, shown in Fig. 6 bottom\nright.Attemptstoforce-refinetheotherCOortorefineina\nlower-symmetryspacegroupconfirmedthisstructuresolu-\ntion,andindependentconfirmationwasalsoobtainedindi-\nrectlyviaspin-chargecoupling(c.f.Sec.5.3).Theground -\nstate CO ofLuFe 2O4isthusnotferroelectricanddoesnot\ncontaintheinitiallyproposedpolarbilayers.\n4.2 Short-range correlations and charge dynam-\nicsAboveTCO,thesharpCOsuperstructurereflectionsin\nLuFe2O4are replaced by diffuse scattering along (1\n31\n3ℓ),\nwith a characteristic zig-zag pattern, shown in Fig. 7a)\nlower graph, discernable up to about 550K[43,61]. A\ndetailed analysis of the scattered intensity [44] suggests\nthat it can be described as very broad and overlapping\npeaks at positions of structural reflections ±(1\n3−δ′,1\n3−\nδ′,0)andsymmetry-equivalent.Correspondingtothelarge\nFigure 8 Example of a possible anti-phase-boundary\n(dashed)perpendicularto the in-plane propagationof CO,\nwhich switches the majority valence. Top-view of bilayer\n(•=Fe2+,◦=Fe3+), Fe ions in the lower layer are drawn\nsmaller.peak-width along ℓ, at360Ka correlation extending only\nto2−3bilayerswasconcluded,whereastheremainingrel-\native sharpness along ( hh0) suggests that in each individ-\nual bilayer, medium-range CO is maintained much above\nTCO. Applying symmetry-analysis on the positions of the\ndiffuse peaks suggests a marginal tendency towards ferro-\nstacking of neighboring bilayers, one of the CO shown in\nFig. 6 left. In [44], representation Γ1and thus a tendency\ntowards ferroelectric CO, was assumed. However, as it is\nnow clarified that in the long-range ordered state, repre-\nsentationΓ2with charged rather than polar bilayers is re-\nalized,itseemsmorelikelythattheshort-rangeCOisalso\nfollowingΓ1, withoutpolarbilayerseverbeingpresent.\nThis seems at first impossible, as this corresponds to\nan overall net charge of the structure. However, given the\nsmall correlation-volume deduced from the broadness of\nthe diffuse peaks, a charge-transfer over a few bilayers\n(via a presently unknown pathway) is sufficient, and fea-\nsible considering that this is not a static, but a dynamic\nshort-range CO, i.e. short-range-correlated valence fluc-\ntuations. M¨ ossbauer spectroscopy can directly assess the\ndynamics of CO and charge fluctuations: spectra of sev-\neralRFe2O4were analyzed with a model featuring iso-\nmer shifts, quadrupole splittings, and linewidths for both\nvalence states and additionally a Fe2+→Fe3+hopping\nfrequency, showing still discernable natural hopping even\nsomewhatbelow TCO[25].\nFig. 9a) shows the hopping frequency νhoppingdeter-\nmined in a study on powdered crystals of optimal quality.\nThe hopping above TCOcan be described by an Arrhe-\nnius law with an activation energy of ∼0.16eV.νhopping\ndecreases continuously upon cooling through TCO, and\nbelow can again be described by an Arrhenius law, but\nwith a higher activation energy ∼0.36eV. The Arrhenius\nlaw with a change of slope at TCOis consistent with the\nT−dependenceof thedc conductivityextractedbydielec-\ntric spectroscopy on similar samples [19]. The substantial\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 7\nFigure 9 Charge dynamics in LuFe 2O4. From\n[63],c/circlecopyrt2008 American Physical Society. a)\nArrhenius-plot of the Fe2+→Fe3+hopping\nfrequency determined from M¨ ossbauer spec-\ntra (◦, bars are from [25]). b) Temperature-\ndependence ofthestrength neffofanopticalex-\ncitationassignedto Fe2+→Fe3+chargetrans-\nfer.\nFigure 10 Electric-field effects in LuFe 2O4. a) Current-voltage hysteresis loop at 300K, with the current limited to\n100mA. From [69], c/circleco√yrt2008 American Institute of Physics. b) Resistivity ρvsT−1/4. full symbols: low E-field d.c.\nmeasurements; open symbols: high-field pulsed measurement s. From [70], c/circleco√yrt2011 American Institute of Physics. c)\nNeutrondiffractionintensityof(2\n32\n37\n2)asafunctionofcalibratedtemperature,obtainedbycooli ngwith0.7mA(△)orno\ncurrent(•).From[56], c/circleco√yrt2010AmericanPhysicalSociety.\nelectron hopping present even below TCO, where diffrac-\ntion indicates long-range CO, likely involves fluctuations\nof anti-phase boundaries, a scenario shown to be at work\nin another CO ferrite, and connected to T−dependentap-\nparentincommensuration[65].InLuFe 2O4theincommen-\nsurationbecomesconstantonlybelow ∼170K,suggesting\nthat the charge dynamics become completely frozen only\nattemperaturessubstantiallybelow TCO.\nThe capacity of Fe2+→Fe3+charge transfer can be\nprobed not only through natural electron hopping, but\nalso by driving the process at optical frequencies. Fig.\n9b) shows the T−dependence of the strength of an op-\ntical excitation assigned to Fe2+→Fe3+charge transfer\n[63]. Upon cooling, it starts to decrease around 500K,\nreasonably close to the reported [43,61] onset of diffuse\nscattering,anditbottomsoutonlyat TLT∼170K,consis-\ntent with the freezing of charge dynamics at this temper-\nature. The nature of the phase below TLTwill be further\ndiscussedinSec.6.2.\n4.3 Electric-field and current effects on CO\nGiven that the pyroelectric current measurements shown\nin Fig. 5a) had been prepared by first cooling the sample\nin an electric field, and that diffraction analyses failed to\ndeduce a ferroelectric CO, it had been proposed [44] that\nsuch a ferroelectric CO might be stabilized by electric\nfieldsEor by cooling in E. However, at low Telectric\nfields up to 20kV/cmwere shown to have no effect on\nthe CO [56]. At higher T, intriguing electric-field effects\nhave been observed macroscopically in the form of non-linear current-voltage characteristics (Fig. 10a) [69,71 ]:\nRather than eventually reaching a state with lower resis-\ntance as might be expected for a ferroelectric phase, a\ncurrent-breakthrough is observed. The breakthrough de-\npends on the environment so that a possible application\nin gas-sensing has been proposed [72]. Effects were also\nreported on dielectric constants [73] and magnetism [57,\n74,75]. With in-situ electron diffraction experiments, a\ndisappearance of the CO superstructure reflections at this\nthreshold was observed [76,77], suggesting an electric-\nfield or current-inducedmelting of the CO. In contrast, an\nex-situ in-field cooling experimentsuggests improved CO\ncorrelations, without affecting the intensity-distribut ion\nbetweendifferentreflections[78].\nHowever, current-voltage measurements using single\nshort (∼1ms) current pulses [70] found Ohmic behavior\nin polycrystalline LuFe 2O4(see Fig. 10b) up to Emuch\nhigher than fields where the breakdown in dc measure-\nments occurs, with the time-dependence on longer pulses\nsuggesting self-heating effects [70]. Similar pulsed mea-\nsurements on YbFe 2O4[79] suggest non-linearity in the\nin-plane conductivity, attributed to sliding charge densi ty\nand possibly a break-down of CO, whereas large self-\nheating effectsforlongerpulsesare also found.An in-situ\nneutron diffraction experiment [56] demonstrates the ab-\nsence of any current-dependenceon either charge or mag-\nnetic order when the sample-temperature is properly re-\ncalibrated (Fig. 10c). A CO unaffected by static (melting\nof the CO by 1.55eVphotons was demonstrated by a re-\nCopyrightlinewillbe provided by the publisher8 M. Angst: Rareearthferrites\ncent pump-probe experiment [80]) electric fields or cur-\nrentswas also foundby in-situ infraredspectroscopy[81],\nby in-situ x-ray diffraction with the real sample tempera-\nture established from the thermal expansion of the silver\nelectrodes [82], and by resonant x-ray diffraction after in -\nfieldcooling[58].Althougharecentinfrared-spectroscop y\nstudy[83]suggestsverysubtlefield-effectsnotattributa ble\ntoself-heating,theconclusionthatadifferent,possibly fer-\nroelectric,CO cannot be stabilized by electric fields is un-\navoidable.\n4.4 Theoretical considerations The experimen-\ntally deduced CO with charged bilayers is surprising, be-\ncauseCO is usuallyassumedto be drivenby therepulsion\nof electrons between different sites (“Wigner crystalliza -\ntion”), which should result in minimizing the occurrence\nof neighbors of the same valence. It is clear that packing\nelectronsclosetogetherinachargedbilayershouldleadto\na significant energy penalty. Although the full description\nofthe3DCOrequiresatleastthefourinteractions U1−U4\nindicated in Fig. 1c) [15,16], the clear indications by the\ndiffuse scattering above TCO(Sec. 4.2) that correlations\nwithin the bilayers are much stronger than those between\ndifferent bilayers make a first analysis based on only U1\nandU2(intrabilayer interactions) useful. Concerning the\nintralayer interaction U1all the CO of Fig. 6, as well as\nmany others, such as a stripe-like CO, are exactly degen-\nerate due to geometricalfrustration(c.f. Sec. 2). However ,\nconcerning the three nearest neighbors (NN), which are\nin the other layer ( U2), the charged bilayer CO (Fig. 6\nbottom) have on average5\n3same-valence NN, whereas\nthe polar CO (Fig. 6 top) have only4\n3same-valence NN,\nand is therefore more stable for repulsive U2. However, a\nstripe-likeCO canhave 1same-valenceNN,whichshould\nmakeitevenmorefavorable,asfoundalsobyMonte-Carlo\nsimulations taking into account the long-range nature of\nthe Coulomb interaction [84]. This is an indication that\nCoulomb repulsion is not the sole relevant driving force\nfor CO inRFe2O4. It is not really surprising: a proof of\npurelyelectrostaticallydrivenCO hasbeenelusiveinbulk\nmaterials and strong influence of lattice distortionson CO\nare well known, e.g. in manganites [85]. In addition to\nlattice effects, indicated to be relevant in LuFe 2O4by in-\nfrared [7,86] and Raman [87] spectroscopy,CO drivenby\nmagneticexchangehasalso beendemonstrated[88].\nDensity-functional-theory (DFT) calculations per-\nformed for LuFe 2O4[84] suggest a lattice-contribution\nof about40%to the total energy gain by CO for the fer-\nroelectric ( 402meV/f.u.) and stripe ( 384meV/f.u.) CO\ninvestigated, estimated by separately stabilizing the CO\nwithfixedatomicpositionsandwithrelaxingthestructure.\nCharged bilayer or antiferroelectric CO could not be ob-\ntained,becauseacellwith 3bilayerswasused.Aferrielec-\ntric CO with a ↑↓↑stackingof bilayerpolarizations(and a\nnet polarizationclose to the oneindicatedin Fig. 5a, lend-\ning support to this model) and later using a larger cell an\nantiferroelectricCO [44] was foundto be yet more stable,by a total of 15meV/f.u.. These small energydifferences\nbetween antiferroelectric, ferroelectric, and stripe CO a re\nan indication of the relevance of geometrical frustration,\nas expected (Sec. 2). Spin-orbit coupling and magnetism\nwas considered in a later study [89] (see Sec. 5.3). In this\nstudy, a fixed ferroelectric CO was used, but the reported\nenergygainof 78meV/f.u.isconsiderablylargerthanthe\naboveenergy-differencesbetweendifferentCO,indicatin g\nthat magnetismmaybe relevantinstabilizingthe CO. The\nchargedbilayerCO hasyetto betestedbyDFT.\nThe CO in RFe2O4has also been studied by lattice\ngas models. Modeling a single bilayer [90,91,92,93,94]\nalready yields several competing CO phases, among them\nthe ferroelectric CO (Fig. 6 top left) and non-polar CO\nphaseswithin-planepropagationssuchas(1\n20),(1\n41\n4),(1\n61\n6)\nand (5\n125\n12), again pointing out the importance of geomet-\nrical frustration.Because in [90,91,92,93,94] chargeneu -\ntrality of this bilayer was implied, the charged bilayer CO\nwas not considered. A lattice gas model defined on the\nwhole 3D lattice of Fe has been considered in [15,16],\nthough without considering in detail the charged bilayer\nCO found experimentally. Starting point is the Hamilto-\nnian\nH=/summationdisplay\nUνν′\nij·σν\niσν′\nj, (1)\nwherei/jenumerate the primitive unit cells, ν/ν′= 1,2\ndenotetheFeionintheprimitivecell, σassumesthevalue\n1(−1) for the valence state 2+(3+) of the Fe ion, and\nUdenotes screened Coulomb-interactionsbetween differ-\nent Fe ions. In reciprocal space, the eigenvalues of the\nFourier-transformedHamiltonian (1) give the energies for\nchargepatternsdefinedbyacorrespondingeigenvectorand\napropagationvector,andthechargepatternwiththelowest\nenergy corresponds to the realized CO. Four interactions\n(U1−U4in Fig. 1c) are considered.It isclear that at least\ninteractions U1−U3are needed to yield a 3D CO: They\nleadtoinstabilitiesat(1\n31\n3ℓ),butthesearedegeneratealong\nℓ, yieldingonly2DCO [15]. Thisis dueto the rhombohe-\ndralstackingofthebilayers[16], ageometricalfrustrati on\neffect.\nWith a non-zero U4, dependingon the signs of U4and\nU2·U3possibleorderingwavevectors(1\n3±δ,1\n3±δ,3\n2)or\n(1\n3±δ,1\n3±δ,0)areobtained. U4<0(attractive,i.e.favor-\ning same valence) must be assumed to yield the observed\npropagationvector,whichwasproposedtobeduetoover-\nscreening [15]. The observed (c.f. Sec. 4.2) discrepancy\nbetween the dominant fluctuations above TCOwith peaks\nnear(1\n3,1\n3,integer)andthelong-rangeCObelow TCOwith\nreflections near (1\n3,1\n3,half-integer) is difficult to reconcile\nwithamean-fieldlatticegasmodel.In[16],thecouplingto\na non-criticalmode,specificallyaparticularRaman-activ e\nphonon-mode,was proposed. Raman scattering has so far\nbeen conducted only on polycrystalline samples, not find-\ninganyindicationsforthisscenario[87].However,aprob-\nlemofthemodelisthatwiththesimplequadraticHamilto-\nnian(1),aninstabilityatonlyasingle q-positioncanbeob-\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 9\ntained,i.e.theadditionallyobservedreflectionsnear( 003\n2)\ncannotbeexplained.\n4.5 Ion-size effects on CO Given the large distor-\ntions of the bilayer in RFe2O4as a function of Rion-\nsize (Sec. 2 and Fig. 1b), different CO for different R\nmay be expected. Yb is quite close in ion size to Lu, and\nwhereaslong-rangeCOwasnotyetfound,presumablydue\ntostoichiometry-problems,diffusescatteringnear(1\n31\n3ℓ)is\ngenerally observed [95], and an electron-diffractionstud y\n[96] even found maxima at ℓhalf-integer. Given similar\nspin-chargecoupling(seeSec.5.3),thesameCOasforLu\nis likely. For R=In, which has a significantly smaller ion\nsize,singlecrystalsarenotavailable,butsynchrotronx- ray\npowder diffraction suggests the appearance of long-range\nCO below ∼250K, with peaks indexed by (1\n31\n3τ), with\nτincommensurate [11]: the limit of thick bilayers seem\nto favor an incommensurationout-of-planerather than in-\nplane.\nFortheotherextremeofthinbilayers, R=Y,the same\ntype of diffuse scattering along (1\n31\n3ℓ) is observed at all\nT(e.g. [97]) for samples with significant oxygen defi-\nciency as in off-stoichiometric Lu or Yb samples. How-\never,the behaviorofsufficientlystoichiometricmaterial is\nmore complex. Abrupt changes of resistivity accompany-\ningthetwomagnetictransitionsmarkedbyarrowsinFig.2\n[98]suggestthesearealsostructuraltransitions,withmo n-\noclinicandfinallytriclinicdistortionsasindicatedbypo w-\nderx-raydiffraction[99].\nBecause weak superstructure peaks are difficult to ob-\nserve in powder diffraction, and sufficiently stoichiomet-\nric single crystals were absent until recently, subsequent\nwork on the CO of stoichiometric YFe 2O4focused on\nelectron-diffraction (e.g. [97,100,101]). At low T, these\nstudies found clear superstructure spots with very differ-\nent propagation vectors, such as (1\n142\n71\n14) between 85and\n130K,andconsistentwithtriclinicsymmetry.Athigher T\noften a (1\n41\n40)-type superstructure is observed, and some-\ntimes (1\n21\n20)- or more “conventional” (1\n31\n33\n2)-types, partly\nwithdifferentphasescoexistingorrelaxingfromonetothe\notheras a functionof time. The question of 3D CO in sto-\nichiometric YFe 2O4above the magnetic ordering temper-\nature, in particular at room temperature, is not clear, with\neitheronlydiffuse(1\n31\n3ℓ)linesreported[100]orwithcoex-\nistence of diffuse lines with broad (1\n31\n3n\n2) spots, the latter\nbydark-fieldimagingassessedtocorrespondto3DCOdo-\nmainscorrelatedoversome30bilayers[97].\nTheseobservationssuggestasubtlecompetitionofsev-\neral CO phases, that correspondto highly complex charge\npatterns, in YFe 2O4, reminiscent of some of the predic-\ntionsbasedonsingle-bilayermodelHamiltonians(seeSec.\n4.4). To elucidate the strong differencein CO to LuFe 2O4\nwill require detailed modeling, but first the real-space\nCO giving rise to the observed superstructure reflections\nwill have to be solved, presumably by single-crystal x-ray\ndiffractiononsufficientlystoichiometriccrystals[31]./s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s32/s73/s110/s116/s101/s103/s114/s97/s116/s101/s100/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s48/s72/s32/s40/s84/s41/s32/s40/s177/s49/s32/s48/s32 /s51/s46/s53/s41/s32/s78/s101/s117/s116/s114/s111/s110\n/s32/s77/s111/s109/s101/s110/s116/s32/s86/s83/s77/s32/s50/s50/s48/s75\n/s48/s46/s48/s48/s46/s51/s48/s46/s54/s32/s77 /s32/s40\n/s66/s32/s47/s32/s102/s46/s117/s46/s41Figure 11 Magnetic\nfieldHdependence\nof magnetization M\nand the intensity of the\n(102)+(003\n2) magnetic\nreflection at 220Kin\nLuFe2O4. After[29].\n4.6 Fe-site substitution and intercalation As\nmentioned in Sec. 2, possible changes in oxygen sto-\nichiometry hamper substitution studies on the Fe site.\nNevertheless, the expected quick depression of CO or\ncharge diffuse scattering seems to hold (see, e.g., [102]).\nFor intercalated materials, e.g. Lu 2Fe3O7, superstructure\nreflections have been observed by electron diffraction, in-\nterestingly up to 750K, and interpreted in terms of both\nthebilayersandtheintercalatedsingleFe-layersexhibit ing\nCO [38,103,104]. X-ray and neutrondiffraction on single\ncrystalswouldcertainlybe desirable.\n5 Magnetism and Spin-Charge coupling The\nRFe2O4family first attracted attention due to intriguing\nmagnetic propertiesof Ising spins (pointing perpendicula r\nto the layers) on a stronglyfrustrated lattice, and the mag-\nnetic properties depend strongly on oxygen-stoichiometry\n(c.f.Sec.2).\nOff-stoichiometric samples typically show signifi-\ncantly higher magnetization when cooled in a magnetic\nfield than when cooled in zero field (c.f. Figs. 2 and 3a),\nwith a remanent magnetic moment after cooling in strong\nmagnetic fields between 1.5(Y) and 2.8µB/f.u.(Lu)\n[105]. Frequency-dependence in ac magnetic susceptibil-\nity is observed below about 220−250K, with detailed\nanalyses suggesting either spin-glass- [106] or cluster-\nglass-like [107,108,109] freezing, with large clusters of\nsize∼100nmin-plane directly observed by magnetic-\nforce-microscopy in one example [107]. Neutron diffrac-\ntionstudiessuggest2Dmagneticcorrelations,withdiffus e\nmagnetic scattering typically observed along the (1\n31\n3ℓ)\nline[30,110]similartotheCOinnon-stoichiometricsam-\nples (c.f.Sec. 4). Thissuggeststhat mainlythe correlatio n\nbetweendifferentbilayersislost.Inlessoff-stoichiome tric\nsamples,atendencytowardsadevelopmentofpeaksalong\nthis line has been observed [30], but so far few neutron\ndiffraction studies of stoichiometric crystals has been\nperformed, and to date only the magnetic structure of\nLuFe2O4is known, which will therefore be the focus in\nthefollowing.\n5.1 3D magnetic correlations in LuFe 2O4Sharp\nmagneticBragg-reflectionshavebeenobservedaftercool-\ning belowTN∼240Kat (1\n3,1\n3,integer) and (1\n3,1\n3,half-\ninteger)positionsin several single crystalsof LuFe 2O4by\nneutron diffraction [29,52,55,56,57] and soft resonant x-\nray diffraction[29]. Magnetic correlationsbetween diffe r-\nCopyrightlinewillbe provided by the publisher10 M. Angst: Rareearthferrites\nFigure 12 H−TMagnetic\nphase diagram of LuFe 2O4from\ndata of [29,63]. Shown are a\nlow-Hantiferromagnetic (AFM)\nandahigh- Hferrimagnetic (fM)\nphase near the ordering temper-\natureTN∼240K, as well\nas a lower-temperature phase\n(LT)withre-entrantdisorder[52]\nand a subtle structural modifi-\ncation [63]. Hysteretic regions\nwhere more than one phase can\nbe stabilized are hatched. The\nLT→fMtransition line ( △)\nreaches7Tat50K(not shown)\n[63]. The spin structures of the\nAFM and fM phases [29] are\nalso shown, with large semi-\ntransparent arrows indicating the\nbilayer net magnetizations.\nent bilayers have also been deduced from powder neutron\ndiffraction [107,111]. An example of (1\n31\n3ℓ) scans at dif-\nferent temperatures is shown in Fig. 7b): the weak reflec-\ntions at (1\n3,1\n3,half-integer) at 280K(thick olive) are due\ntoCO,themuchlargerintensitymainlyat(1\n3,1\n3,integer)at\n220Kismagneticinorigin,asshownbypolarizedneutron\ndiffraction (see, e.g. Fig. 5 in [29]), with an out-of-plane\ncorrelation length of 15nm. At130K(red) a large dif-\nfuse background appears together with peak-broadening,\nsuggestinga re-entrantmagneticdisordering[52,56].Thi s\nlow-temperaturestate (LT phase)will be discussed in Sec.\n6.2.\nIn contrast to stoichiometric YFe 2O4, where the ab-\nsence of any magnetic-field induced phase transitions is\nreported [27,99], magnetization-measurements on sto-\nichiometric LuFe 2O4reveal a complex magnetic field\n(H/bardbl(001)hex)-temperature ( T) phase diagram [29,63],\nshown in Fig. 12. Apart from the already mentioned LT\nphase with reentrant magnetic disorder, and the param-\nagnetic state, two further phases are present. Isothermal\nmagnetization M(H)loops (e.g. Fig. 11 grey line) are\nconsistent with a hysteretic metamagnetic transition be-\ntween a low- Hphase that is antiferromagnetic (AFM)\nand a high-Hphase that is ferrimagnetic (fM), with a net\nmagnetic moment at low temperatures (the fM phase re-\nmains stable down to T= 0when cooled in µ0H >2T)\nof∼3µB/f.u., suggesting a ↑↑↓-like arrangement of Fe\nspins. Partially strong hysteresis of the transition betwe en\nAFM and fM phases (hatched in Fig. 12), also present\nbetween LT and either fM or AFM, is one of the unusual\nfeatures of the phase diagram. Magnetic Bragg reflections\nareobservedforboththeAFMandthefMphaseat (1\n31\n3ℓ),\nwith an intensity shift surprisingly from ℓinteger to half-\ninteger reflections going from AFM to fM [29,55]. As\nexpected from the presence of a net moment, in the fM\nphase magnetic intensity is also observed on structuralreflections. In contrast, magnetic intensity at ( 003\n2)-type\nreflectionsisa featureoftheAFM phase(see Fig.11).\n5.2 Competing spinstructures Given the corre-\nspondence of Ising spins ↑or↓with valence states 2+\nand3+,bothbeingbinaryorders,themoststraightforward\nway to analyze spinstructures is to neglect the different\nspin quantum numbers for the different valence states of\ntheFeionsandproceedfromthehexagonalcellanalogous\nto the CO. As the same propagation vectors are involved\nas considered for the CO in Sec. 4.1, there will again be\nthreedomainsandthesamefourpossibleorderingsshown\nin Fig.6 also thenapplytothe spinorder(SO), withblack\nand white coloring now indicating spins ↑and↓. If the\nweaker(1\n3,1\n3,half-integer)magneticreflectionsareconsid-\nered to be due to a decoration by the CO, then the proper\nSO should be the ordering shown in Fig. 6 bottom left,\nbecause for a Γ1representation there would be no inten-\nsity on (1\n31\n30), which is the strongest magnetic reflection\n(Fig. 7b). This ferrimagnetic SO was found to fit well\nall (1\n3,1\n3,integer)-type reflections and was thus initially\nproposedto apply[52].\nHowever, the later found [29] magnetic intensity at\n(107\n2)(Fig. 11) is inconsistent with this SO, showing that\nsymmetry-analysisbased on the hexagonalcell cannotde-\nscribe the SO of LuFe 2O4. In retrospective this is not sur-\nprising,becausetheSOtakesplaceat TN<TCOandifany\nspin-charge coupling is present (c.f. Sec. 5.3) the proper\ncell on which SO is to be considered is the CO supercell.\nBecause the magnetic cell is equal to the CO cell, sym-\nmetry analysis is then based on a ( 000) propagation, and\nwith four independent sets of Fe sites present in this cell,\nand four irreducible representations for each site, many\nSOhavetobeconsidered.AccordingtoLandau-theory,all\nsites should order according to the same irreducible rep-\nresentation if the ordering occurs in a second-order phase\ntransitionat a single temperature.Thisconditionyields 40\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 11\npossibleSO,whichhoweverwereallfoundtobeinconsis-\ntent with the experimental data [29]. In a brute-force ap-\nproach all 312= 531′441SO of Ising spins ( ↑,↓,0, al-\nlowingforsomeFesitesremainingdisordered)werecom-\npared with the data for both AFM and fM phases, which\nidentifiedtheSOshowninFig.12asthecorrectones,ver-\nified by refinement [29]. For these SO, all four sets of Fe\nsites order according to an irreducible representation, bu t\nwith differentonesfor differentsets, implyingthat the SO\ndoes not occur via a single second-order phase transition.\nIndeed,given that both AFM and fM, separated by a first-\norder transition, are present near TNandH= 0, a simple\nsecond-ordertransitionseemsunlikely.\nAFMandfMSOarecloselyrelated.Inbothcases,each\nbilayer(andeachindividuallayer)hasa 2 : 1(↑↑↓or↓↓↑)\nferrimagneticarrangementoftheFespins,withthebilayer\nnet moment indicated by thick semi-transparent arrows in\nFig. 12. For the ferrimagnetic fM phase, the net moments\nof the two bilayers of the cell are both ↑, yielding a net\nmoment,asobserved.FortheAFMphase,thespinsinone\nof the two bilayers are exactly as in the fM phase, and for\nthe otherbilayer theyare all reversed,yieldingthe antife r-\nromagneticbehavior indicated by magnetizationmeasure-\nments.Intriguingly,ascanbeseeninFig.12,theAFM-fM\ntransition extrapolates to H≈0forT→TN, implying\nthe near-degeneracyof these two SO phases at TNin zero\nfield, a hallmarkof geometricalfrustration.A competition\nof AFM and fM SO has also been observed in a study on\npolycrystalline samples [111]. Because AFM and fM SO\naredistinct only bythe flip of all spins in oneof the bilay-\ners,thissuggeststhatthespincorrelationswithinthebil ay-\nersaremuchstrongerthantheinter-bilayercorrelations. In\nanalogyto the CO (Sec.4.2), a randomstackingof the net\nmoments of still relatively well spin-ordered bilayers can\nthen be expected above TN, with diffuse magnetic scat-\ntering along (1\n31\n3ℓ). This has indeed been observed (Fig.\n7c), up to room temperature, and deviations of the mag-\nneticsusceptibilityfromCurie-Weissbehaviorsuggestth at\nspin-fluctuationscorrelatedwithineachbilayerremainre l-\nevant beyond even TCO[29]. Note that these correlations\naredynamic(fasterthan ∼MHz)ratherthanstatic,because\nno indications of magnetic order are visible in M¨ ossbauer\nspectraat 260K[63].\n5.3 Spin-charge coupling\nExperimental indications Experimental indica-\ntions of a spin-charge coupling by anomalies in the T-\ndependence of the intensity or incommensuration of the\nCO superstructure reflections at TNhave been observed\n[44,58], but they are weak. However, this might be ex-\npected if the SO within individual bilayers is formed al-\nready above TN. The SO refinements [29] yielded some\nhintsofastrictassignmentofSOandCO,butthedifferent\nmagnetic moments of Fe2+and Fe3+ions could not be\nreliably determined. However, spectroscopic techniques\nperformedonavarietyofsamplesprovidestrongevidence\nthat such spin-charge coupling is a common feature at\nFigure 13 X-ray absorption spectra XAS with right ( µ+)\nand left (µ−) circular polarized photons and X-ray Mag-\nnetic Circular Dichroism XMCD ∆µ=µ+−µ−of\nLuFe2O4across the Fe L2/3edges in4T(perpendicular\ntothelayers)at 120Kand260K(inset).Alsoshownisthe\nintegration of the XMCD, with the sum rules yielding an\norbitalmagneticmomentof ∼0.7µB/f.u.. From[18], c/circleco√yrt\n2012AmericanPhysicalSociety.\nleast in YbFe 2O4and LuFe 2O4[18,64,112,113,114]. In\nthe first studies, polarized M¨ ossbauer spectroscopy was\napplied to single crystals of YbFe 2O4[113] and LuFe 2O4\n[114] under varying H, it was concluded that all of the\nFe2+spins as well as1\n3of the Fe3+spins were aligned\n/bardblH, with the remaining2\n3of the Fe3+spins aligned an-\ntiparalleltoH.\nLater studies by x-ray magnetic circular dichroism\n(XMCD) confirmed this arrangement for LuFe 2O4[18,\n64,112]. Fig. 13 shows the absorptionspectra and XMCD\nmeasured (in the fM-phase) on a crystal of the same type\nas used to establish the magnetic phase diagram and SO\n(c.f. Fig. 12). The larger negative peak of the XMCD sig-\nnal at the Fe2+L3edge and the smaller positive peak at\nthe Fe3+L3edge directly implies a larger Fe2+net mo-\nment in magnetic field-direction, and a smaller Fe3+net\nmomentoppositetothis, whichgiventhe knownmagnetic\ncell-size and saturation magnetization leaves the above\nspin configuration as the only possible one. From the\nXMCD sum rules, a sizeable orbital magnetic moment of\n∼0.7−0.8µB/Fe2+ion can be concluded [29,64,112],\nrendering the Fe2+total magnetic moment to ∼4.75µB,\nquiteclosetothespin-onlyFe3+momentof 5µB.There-\nsulting saturation momentof 4.75−1\n3·5∼3.08µB/f.u.\nisin agreementwith magnetizationdata.\nIn Fig. 12, both spin and charge structures are shown,\nand it can be readily seen that for the fM-phase the align-\nmentofFe2+andFe3+momentsaccordingtoXMCDand\nM¨ ossbauerspectroscopyis indeedobserved,i.e.the previ -\nouslydiscussedCO andSO areconsistentwithstrict spin-\ncharge coupling as deduced spectroscopically. Moreover,\nthe SO shown in Fig. 12 combinedwith the spectroscopic\nCopyrightlinewillbe provided by the publisher12 M. Angst: Rareearthferrites\nFigure 14 Intensity of the (2\n32\n37\n2) CO superstructure peak\nmeasured by neutron diffractionat 300K, after cooling in\nmagneticfieldsof differentstrengths.From [55], c/circleco√yrt2009\nAmericanPhysicalSociety.\nresults is incompatible with any of the CO shown in Fig.\n6 except the chargedbilayer stacking deduced in Sec. 4.1,\nprovidingadditionalsupportforthismodelofCO [18].\nThe inset of Fig. 13 shows a XMCD spectrum mea-\nsuredin4TaboveTN, at260K.Thoughmuchreducedin\namplitude, the shape is still the same as in the fM-phase\nbelowTN. This is consistent with the model of randomly-\nstackedbutotherwisestillwellspin-orderedbilayersint he\nparamagneticstate:Polarizedbyasufficientlystrongmag-\nnetic field, the random-stacking of 50% ↑and 50% ↓bi-\nlayer net moments will be broken, with the overall spin\narrangementtendingtowardsthefMSO.Italsoshowsthat\nthespin-chargecouplingisveryrobust,notrequiringlong -\nrange order. The magnetic correlations could thus be im-\nportantnot onlyforthe magnetictransitionat TN, but also\nforthechargeorderingat TCO.\nInterestingly,a magnetic-fieldinfluenceonthe CO has\nbeen observedfor LuFe 2O4[55]: Fig. 14 shows the inten-\nsityofthe(2\n32\n37\n2)COsuperstructurereflection,measuredat\n300K,aftercoolingthecrystalindifferentmagneticfields\nfrom350K. A significant suppression of ∼15%already\nby rather small fields is visible. This is surprising, given\nthat at the AFM-fM metamagnetictransition no changeof\nthe CO was observed [18]. However, in contrast to these\nmeasurements with H/bardblc, in [55] the field was applied in\n[110]direction.Thisin-plane Hbreaksthethree-foldrota-\ntionsymmetryof the crystal,andlikely affectsthe relativ e\nstability of the three magnetic domains (based, as the CO\ndomains, on the three monoclinic cells shown in Fig. 1a).\nThestrictspin-chargecouplingimpliesthecorrespondenc e\nofmagneticandCO domains,andthereforeanin-plane H\nshouldaffectthe domainpopulation.The(2\n32\n37\n2) reflection\nstudied in [55] belongs to one of the CO domains, its in-\ntensitysuppressionby Hsuggeststhat populationofother\nCO domainsis favored, which could be tested by simulta-\nneously tracking different CO reflections belonging to all\ndomains.\nTheoretical considerations The single-bilayer lat-\ntice gas models [90,91,92,93,94] also consider differ-ent SO and spin-charge coupling, finding that the spin-\ncorrelations tend to stabilize CO with (1\n31\n3ℓ)-type propa-\ngation compared to alternatives (c.f. Sec. 4.4). However,\nbecause these models do not consider the charged bilay-\ners indicated by experiment, the focus below is on more\nphenomenologicalconsiderationsandtheDFT calculation\n[89].\nBecause the inter-bilayer spin-coupling is very weak\n(c.f. Sec. 5.2), it is a good first approximation to con-\nsider each bilayer, the one with Fe2+majority and the\nonewithFe3+majority,individually.Variousmagneticsu-\nperexchangeparametersbetweenFeionsofthesamelayer\nand between the two layers have then to be examined,\nwhich may be expected to strongly depend on the valence\nstate of the respective Fe ions. An experimental determi-\nnation of exchange parameters could be done by inelastic\nneutronscattering,whichsofarhavebeenreportedonlyin\na limited range on a crystal [78], and on a polycrystalline\nsample, with the observation of a spin-gap ∼7meVcon-\nfirmingtheIsingnatureofspins[111].\nThe exchangeparameterswere estimated theoretically\nby mappingthe energydifferencesbetween differentlyor-\ndered states obtained by DFT to the corresponding en-\nergy differences of an Ising Hamiltonian [89]. This study\nconsidered an individual bilayer, however, one assumed\nto exhibit the ferroelectric CO (Fig. 6 top left). The by\nfar largest interaction was found to be an antiferromag-\nnetic superexchangebetween Fe3+ions of the same Fe3+\nmajority layer. From this, an in-plane antiferromagnetic\narrangement of Fe3+moments is expected, which is in-\ndeedobservedexperimentally(seeFig.12).Acorrespond-\ningcalculationtakingintoaccountthechargedbilayerCO\nwould be highly desirable: as mentioned already in Sec.\n4.4, the large energy-differences between different spin-\nstates found in [89] imply that the CO and SO have to be\ncalculatedtogether,inacellofsufficientsize,whichmake s\nthecalculationschallenging.\n5.4 Off-stoichiometric samples and other rare\nearthsThe agreementbetweenXMCD experimentscon-\nducted on samples with [18] and without [64] long-range\nCO indicates that the strict spin-charge coupling with the\nFe2+↑↑↑/Fe3+↑↓↓arrangementis a robust and generic\nfeature of LuFe 2O4. Because the superexchange interac-\ntions leading to spin-charge coupling and SO depend on\nthe underlyingCO, it seems verylikely that in all samples\nindividual bilayers exhibit both CO and SO as discussed\nin Secs. 4.1, 4.2, and 5.2, with only the occurrence of\nlong-range 3D order with correlations between different\nbilayers being affected by stoichiometry. The generic ob-\nservation of at least diffuse charge(e.g. [107]) and diffus e\nmagnetic (e.g. [110]) scattering at the (1\n31\n3ℓ) line supports\nthisassessment.Because thechargedynamicsslowsdown\nupon cooling regardless of stoichiometry [25], a glassy\nfreezing of the essentially still charge-ordered bilayers\ncan be expected to replace the long-range CO when the\nlatter is prevented. Similarly, a glassy freezing of SO bi-\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 13\nlayers can be expected to replace the long-range AFM\nphase upon cooling, and such a cluster-glass-like behav-\nior involving frequency-dependence in ac susceptibility\nhas indeed been found for some samples, e.g. in [108].\nOn the other hand, when a magnetic field forces the net\nmoments of all bilayers to align, a stable fM-like phase\nwith the same saturation moment should be recovered, as\nisindeedsuggestedbyallmacroscopicmeasurements(see\ne.g. [106,108] for corresponding phase diagrams) and by\nelectron holography[115]. Note, however,that because of\ntheunderlyingrandomstackingofFe2+andFe3+majority\nbilayers, this fM phase is not expected to be 3D ordered,\nbut shouldratherexhibitdiffusemagneticscatteringalon g\n(1\n31\n3ℓ),whichcouldbecheckedbyin-fieldneutrondiffrac-\ntion on less-stoichiometric crystals. Thus, based on the\nabove, for off-stoichiometric samples, exactly the “spin-\nglass-like2D-ferrimagneticorder”proposedearlyon[110 ]\nisexpectedto occur.\nTheaboveconsiderationsshouldholdalsoforYbFe 2O4,\nwhich has a similar Rion size. Indeed, not only the same\ndiffuse scattering at (1\n31\n3ℓ) has been observed [95] as on\nnon-stoichiometric LuFe 2O4, but the same spin-charge\ncoupling is also deduced [113]. For R=In with consider-\nablysmallerionsize,magnetismandspin-chargecoupling\nappear to be rather different as indicated both by a much\nsmaller saturation moment of 1.5µB/f.u.and dissimilar\nM¨ ossbauer spectra [11]. An XMCD study on InFe 2O4\nmight be helpful to elucidate spin-charge coupling in this\nmaterial. For Rwith considerably larger ionic radii than\nLu or Yb, there are also clear differences. For example,\nin stoichiometric YFe 2O4there is no fM phase, and given\nthe wildly varying CO observed at low T(see Sec. 4.6)\na similar spin-charge coupling as in LuFe 2O4is rather\ndoubtful,despitesimilaritiesofdiffusescatteringobse rved\nin off-stoichiometric crystals. In agreement with substan -\ntial changes upon increasing the rare earth ionic size, an\nXMCD study on LuFe 2O4with Lu partially substituted\nby the largerY or Er ions [116], foundmarkedchangesin\nthe spectra. Whereas the negative peak at the Fe2+edge\nremained largely unaffected by the substitution, suggest-\ning that “all Fe2+spins aligned with H” still holds, the\npositivepeakat the Fe3+edge was completelysuppressed\nfor∼50%substitution, suggesting an antiferromagnetic\norderofFe3+spins.\n6 The ground state:orbital order?\n6.1 Orbital order, orbital magnetic momentum,\nand magnetic anisotropy Concomitant with CO and\nSO, orbital order (OO) typically occurs in other transition\nmetal oxides, strongly influencing the magnetism as well\n[117]. The possible presence of OO has been considered\nfor LuFe 2O4theoretically [84,90,92], for example the\nDFT calculations [84] suggest a clear OO. However, the\ndiscrepancy of the predicted CO with the experimentally\ndeducedonesuggeststhat thepredictionsof OO also have\ntobetakenwithcare.\nFigure 15 Sketch of the energy-splitting of the Fe 3dor-\nbitals (right) due to the trigonal-bipyramidal coordinati on\n(left). For Fe3+, each orbital is singly occupied,the extra-\nelectronofFe2+cangotoeitheroftwo e′′\ngorbitals.\nThenaturalstartingpointfortheconsiderationofOOis\nthe trigonal bipyramidalcrystal-field level diagram shown\ninFig.15.ForFe3+,allfiveorbitalsaresinglyoccupiedby\naspin↑electron,andthereisnoorbitaldegreeoffreedom.\nHowever, the extra ↓-electron at Fe2+ions can go to an\norbital of the doubly-degeneratelowest level. It should be\nmentioned that e′\ngande′′\nglevels are quite close in energy\nin LuFe 2O4: while band-structure calculations suggested\nthate′\ngis the lowest level [84], polarized x-ray absorption\nspectroscopyatthe O K−edgehaslater shownthat the e′′\ng\nlevelisactuallylower[64].\nOO would then imply a preferential occupation of ei-\nther theyzor thezxorbital−or a given real linear com-\nbination of these −for each site. An alternative to this\nis the preferential occupation of a complexlinear com-\nbination that could carry an orbital angular momentum\nL(/angbracketleftψ|L|ψ/angbracketright= 0forψreal) and thus an orbital mag-\nnetic moment. From a linear combination of yzandzx\norbitals, no eigenstates of LxorLycan be obtained, but/radicalBig\n1\n2(∓zx−i·yz)is an eigenstate of Lzwith eigenvalue\nm=±1, allowing an orbital magnetic moment of up to\n±1µB/Fe2+ion.Thefactthatanorbitalmagneticmoment\n∼0.7−0.8µB/Fe2+isobserved(seeSec.5.3)impliesthat\nin the fM state (and at low Tfor samples without long-\nrangeSO) OO cannotoccur.Consistent withthis, noaddi-\ntionalreflectionshaveeverbeenfound,andnoanisotropies\nin resonant diffraction suggestive of ferro-type OO have\nbeenfound[42].Forsampleswithlong-rangeSO,thiscon-\nclusion can also be extended to the AFM and paramag-\nnetic phases (c.f. Fig. 12), because high-resolution x-ray\ndiffraction indicates no structural changes that necessar -\nily would accompany an OO-transition at the fM-AFM or\nAFM-paramagnetic transitions [18] (though for the latter\nsmall intensityanomaliesaresuggestedin [44,58]).\nFromthepresenceofanorbitalmagneticmomentnec-\nessarily perpendicular to the layers, the large magnetic\nanisotropy making this a model Ising-system follows im-\nmediately, because spin-orbit coupling aligns the direc-\ntionsofspin-andorbitalmagneticmoments[64].Thelarge\nCopyrightlinewillbe provided by the publisher14 M. Angst: Rareearthferrites\nmagnetic anisotropy is to a good part responsible for the\ngiant coercivityof LuFe 2O4, with coercivefields reaching\naround10Tat lowT[23,107], and to “exchange-bias”-\nlikeproperties[118].\n6.2 The low-temperature phase The only phase in\ntheH−Tphase diagram (Fig. 12) consistent with OO is\nthe LT-phase, because both AFM↔LTandfM↔LT\ntransitions are also structural transitions as indicated b y\nBragg-peak splitting [63]. As discussed in Sec. 4.2, the\ntransition to this phase is accompanied by the complete\nfreezing of residual charge dynamics, which is consistent\nwith an OO scenario, given that OO often accompanies\nmetal-insulatortransitions[117]. Due to the impactofOO\non the magnetic exchange interactions, a change of the\nmagneticstructureuponenteringthe LT phase wouldthen\nbe expected,consistent with the observedre-entrantdisor -\nder(Fig.7b).\nThese indications that OO might be the driving force\nbehind the transition into the LT phase are, however, at\nmost tentative. As discussed in Sec. 6.1, a full OO would\nimply the quenching of the orbital magnetic moment and\nthe removal of the strong tendency of spins to be aligned\nperpendiculartothelayers.Therelativelygooddescripti on\noftheℓ−dependenceofthediffusecomponentofmagnetic\nscattering at 130Kassuming spin direction perpendicular\nto the layers [52] is therefore an argument against an OO\nscenario.\nStructurally, despite of the Bragg-peak splitting, there\nis no basic change of the CO across TLT: the interbilayer\ncorrelations of CO seem to improve slightly [44], but the\ncell size (no extra reflections), C2/msymmetry, and the\nvalence states of the Fe ions estimated by bond-valence-\nsum are unaffected within error bars [18]. Although small\nchanges in lattice parameters have been reported, the\nchanges of refined atom positions are much too subtle\nto conclude about a possible OO [18]. Further structural\nrefinement studies across TLTare certainly desirable. A\nfurtherpossibilitytodetectOOistoassessasymmetriesin\nresonant x-ray diffraction, ideally at the Fe Ledges at the\n(ττ3\n2) reflections [82,119], although the analysis is com-\nplicated by the possible emergence of magnetic intensity\natthese positionsin theLTphase[52,56].\nThenatureoftheLTphase,andwhetheritisconnected\nwithOOthereforeisanopenquestionleftforfuturework.\nIts experimentalinvestigationis also hamperedby its very\nfragile nature, as it is present in some samples showing\nlong-range CO and SO [18,29,52,56,63] (indications are\nalso visible in M(T)on the polycrystalline samples stud-\nied in [22,111]), but not in others [15,55]. Due to this, it\nis also not completely clear whether the LT phase repre-\nsents the true ground state of LuFe 2O4, though it appears\nlikely.IftheLT phase isthetruegroundstate, it willmake\ntheoretical investigations with zero-temperature method s,\nsuch as DFT, difficult because of the presence of signifi-\ncantmagneticdisorder.7 Summary and outlook In the following, I will\nsummarizethemainresultsreviewedand highlight critical\npoints for future investigations on rare earth ferrites, co n-\ncluding by answering the title question and by looking at\nthe larger picture of possible CO-based ferroelectricity i n\nothercompounds.\nForthemoststudiedrareearthferrite,LuFe 2O4,acon-\nsistent and rather comprehensive picture of strongly cou-\npled charge- and spinorder has emerged from experimen-\ntal work. Single-crystal x-ray and neutron diffraction re-\nvealedsharpchargeorder(CO)superstructurepeaksbelow\nTCO∼320Kandspinorder(SO)superstructurepeaksbe-\nlowTN∼240Kin stoichiometricsamples. Bothare unaf-\nfected by static electric fields or currents, except for self -\nheating effects. The corresponding orders have been de-\ntermined by refinements.The CO involvescharged,rather\nthanpolar,bilayers.AsrevealedbyM¨ ossbauerandoptical\nspectroscopies, substantial charge fluctuations occur eve n\nbelowTCO, down toTLT∼170K. BetweenTLTandTN\ntwo related SO phases compete, due to geometrical frus-\ntration.BelowTLT, a subtle structural distortion and re-\nentrantmagneticdisorderoccur.Whetherthisisconnected\nwithpossible orbital ordering is a major question left for\nfuture work . M¨ ossbauer spectroscopy and x-ray magnetic\ncirculardichroismdemonstrateastrictcouplingofSOand\nCO, present also above TNand likely even above TCO,\nwhere diffuse scattering experiments indicate that within\nindividualbilayersSOandCOcorrelationsarestillstrong .\nThepresenceofcoupledSOandCOcorrelationswithinin-\ndividualbilayersisarobustfeatureofthecompound.How-\never, oxygen off-stoichiometry quickly destroys the inter -\nbilayer spin- and charge-correlations, leading to “glassy ”\nbehavior.\nWhile charge and spin order have been established,\nthey are not really understood: this would imply elucidat-\ning what drives these orders, whereas currently no theo-\nretical works have considered the experimental charge or\nspin order. What is clear is that the charge order cannot\nbe driven entirely or even predominantly by electrostatic\nrepulsion, because the realized pattern is not the one min-\nimizing this repulsion. Lattice effects as well as magnetic\nexchange appear to be relevant, but the determination of\nthe main driving forces remains a challenge principally\nfor future theoretical work , thoughexperimental work on\nphonons and magnetic excitations could make an impor-\ntant contribution: this is the most critical issue for future\ninvestigations .\nRareearthferritesotherthanLuFe 2O4havebeenstud-\niedmuchless ,whichislargelyduetotheabsenceofsingle\ncrystals that are stoichiometric enough to reveal 3D CO\nand SO. With this hurdle being overcome [31], progress\ncanbeexpected.COandSOcorrelationsleadingtodiffuse\nscattering very similar as in off-stoichiometric LuFe 2O4\narecommonlyobserved,andforYbFe 2O4alsoM¨ ossbauer\nspectroscopy leads to the expectation of overall behav-\nior very similar to LuFe 2O4. However, for rare earths (or\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 15\nIn) with larger deviations of ionic size from Lu, experi-\nmentsonhigh-qualitypolycrystallinesamplesindicateth at\ncompletelydifferentCO and SO is established. For exam-\nple, electron diffraction on YFe2O4demonstrates a num-\nber ofcompeting very complex CO phases that have yet\nto be solved . A partial alternative to single-crystal work\nis to rely on experimental techniques not requiring long-\nrange CO and SO, such as XMCD, which has been suc-\ncessfullyusedtostudyion-sizeeffectsonspin-chargecou -\npling [116]. The intercalated compounds R2Fe3O7have\nbeen even less studied, but given indications of charge or-\nder stable up to much higher temperatures [103] deserve\nmore attention .A similar tuning as with intercalationmay\nbe possible with thin films [120,121,122,123], on which,\nhowever, 3Dchargeorderinghasyet tobedemonstrated .\nFerroelectricity from charge ordering? While the\ncoupled complex charge and spin orders on a geometri-\ncally frustrated lattice are also of high intrinsic interes t,\nit is not the property that has attracted most research to\nthe rare earth ferrites. The title question “Ferroelectric ity\nfrom iron valence ordering in rare earth ferrites?” accord-\ningto currentexperimentalknowledgehasto be answered\nin the negative, at least for the “prototype” LuFe 2O4. The\nchargeorderleadingtothe(1\n31\n3ℓ)-typereflectionsdoesnot\ninvolve polar bilayers, and according to recent dielectric\nspectroscopy studies, the large remanent polarization de-\nduced from pyroelectric current measurements may likely\nbe affected by contact effects. Future theoretical work\nleadingtoanunderstandingofwhatdrivestheexperimen-\ntallyindicatednon-polarchargeorderwouldbeimportant\nforafinalresolutionandtoprovidehintsforotherfamilies\nof compounds . In rare earth ferrites other than LuFe 2O4\nthe charge order can be different and has not been solved,\ntherefore in principle the possibility of ferroelectricit y\nfrom charge ordering remains within the family. How-\never, with dielectric behavior quite common among other\nRFe2O4andeven the most likely ferroelectricbehaviorin\na doped intercalated compound suggesting a polarization\nthree orders of magnitude weaker than what is expected\nfor ferroelectricityoriginatingfrom charge ordering,th ere\narenostrongindicationsforsuccess.\nLooking beyond rare earth ferrites, the findings re-\nviewedhereprovidesomegenerallessonsforCO-induced\nferroelectricity.First and most important,a non-negligi ble\nresidual conductivity is unavoidable in CO materials, par-\nticularly at and above TCO. Apart from being detrimental\nwith regardsto applications, this also rendersthe standar d\nmacroscopicprobesof ferroelectricitydoubtfulas pointe d\nout in [45] and for rare earth ferrites discussed in Sec.\n3. Therefore it is imperative to obtain also microscopic\nproof of at least a polar crystal structure, better yet of a\npolar crystal structure that can be switched by an elec-\ntric field−thoughfor such in-electric field measurements\nresidual conductivity and associated self-heating effect s\nare also problematic, see Sec. 4.3. Second, CO materials\nhave a propensity for variable stoichiometry, which canmake the elucidation of the CO crystal structure very dif-\nficult. Therefore, paying due respect to sample quality is\ncrucial.Finally,inCOmaterialsthereisoftenastrongcou -\npling of charge with orbital and spin degrees of freedom\nleading to complexitythat hamperstheoretical treatments .\nForrareearthferrites,theoreticalworksuggestsand/ora s-\nsumes ferroelectric CO (see Sec. 4.4), in contrast to the\nexperimentalfindings. For other materials, there are many\nmore examples of compounds proposed to exhibit ferro-\nelectricity from CO based on calculations than examples\nofexperimentalindicationsofthesame.\nWiththeprototypicalexampleofrareearthferritesap-\nparentlybeinganon-example,avalid“proof-of-principle ”\nexample of a material exhibiting multiferroicity driven by\nchargeorderinghasyettobedemonstratedexperimentally.\nAmong the few possibilities in oxides with at least some\nexperimental indications most promising seems classical\nmagnetite,which doesseemtohaveacharge-orderedcrys-\ntalstructurethatispolar[124],althoughitappearstobeo f\na relaxor-type with switching only possible at low Tand\nhighfrequency[125].Apartfromoxides,low-dimensional\nmolecularcompounds(see e.g. [126]) are increasinglybe-\ning scrutinized for multiferroicity originating from CO. A\nrecently proposed [127], but also disputed [128], example\nis a metal-organiccharge-transfersalt with apparentsmal l\npolarization but no demonstrated polar crystal structure.\nAlthoughthetemperaturescaleinalloftheaboveisfarbe-\nlowroomtemperature,anunambiguousproofofthemech-\nanism would be important,openingthe route to search for\nsuchmaterialscloserto applicability.\nAcknowledgements I thank J. de Groot, T. Mueller, R. P.\nHermann, and in particular Th. Br¨ uckel, for useful comment s on\ndraft versions of thisarticle,T. M¨ ulleralsofor digitizi ngthe data\nfrom[27]showninFig.2.Iamverygratefultomystudentsand to\nthemanycollaboratorsIhadwhileworkinginthisfield,inpa rtic-\nular the coauthors of Refs. [18,19,7,28,29,44,52,63,108] . Spe-\ncial thanks go to D. Mandrus for first suggesting this fieldto m e,\nto M. Subramanian for providing high-quality LuFe 2O4powder\nsamples veryearly-on,andtoJ.deGroot forhisimmense cont ri-\nbutions during his PhD studies. 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B86, 245103 (2012).\nCopyrightlinewillbe provided by the publisher"    },    {        "title": "2308.07735v1.Microstructural_and_material_property_changes_in_severely_deformed_Eurofer_97.pdf",        "content": "Microstructural and material property\nchanges in severely deformed Eurofer-97\nKay Songa,⋆, Guanze Heb, Abdallah Rezaa, Tamas Ung ´arc, Phani\nKaramchedb, David Yangd, Ivan Tolkacheva, Kenichiro Mizohata\ne, David E J Armstrongb, Felix Hofmanna,†\naDepartment of Engineering Science, University of Oxford, Parks Road, Ox-\nford, OX1 3PJ, UK\nbDepartment of Materials, University of Oxford, Parks Road, Oxford, OX1\n3PH, UK\ncDepartment of Materials Physics, E ¨otv¨os University, Budapest, P.O.Box 32,\nH-1518, Hungary\ndCondensed Matter Physics and Materials Science Department, Brookhaven\nNational Laboratory, P.O. Box 5000, Upton, NY 11973-5000\neUniversity of Helsinki, P.O. Box 64, 00560 Helsinki, Finland\n⋆Corresponding author email: kay.song@eng.ox.ac.uk\n†felix.hofmann@eng.ox.ac.uk\nORCID:\nKay Song: 0000-0001-8011-3862\nGuanze He: 0000-0002-3638-1445\nAbdallah Reza: 0000-0001-9594-2591\nTamas Ung ´ar: 0000-0003-4392-8856\nPhani Karamched: 0000-0002-5599-866XDavid Yang: 0000-0002-1062-7371\nIvan Tolkachev: 0000-0001-8985-726X\nKenichiro Mizohata: 0000-0003-1703-2247\nDavid E J Armstrong: 0000-0002-5067-5108\nFelix Hofmann: 0000-0001-6111-339X\n1arXiv:2308.07735v1  [cond-mat.mtrl-sci]  15 Aug 2023Abstract\nSevere plastic deformation changes the microstructure and properties of steels, which\nmay be favourable for their use in structural components of nuclear reactors. In this study,\nhigh-pressure torsion (HPT) was used to refine the grain structure of Eurofer-97, a fer-\nritic/martensitic steel. Electron microscopy and X-ray diffraction were used to characterise\nthe microstructural changes. Following HPT, the average grain size reduced by a factor\nof∼30, with a marked increase in high-angle grain boundaries. Dislocation density also\nincreased by more than one order of magnitude. The thermal stability of the deformed\nmaterial was investigated via in-situ annealing during synchrotron X-ray diffraction. This\nrevealed substantial recovery between 450 K - 800 K. Irradiation with 20 MeV Fe-ions\nto∼0.1 dpa caused a 20% reduction in dislocation density compared to the as-deformed\nmaterial. However, HPT deformation prior to irradiation did not have a significant effect\nin mitigating the irradiation-induced reductions in thermal diffusivity and surface acous-\ntic wave velocity of the material. These results provide a multi-faceted understanding of\nthe changes in ferritic/martensitic steels due to severe plastic deformation, and how these\nchanges can be used to alter material properties.\nKeywords: High-pressure torsion, Eurofer-97, nanocrystalline microstructure, annealing,\nion-irradiation\n2Graphical Abstract\n31 Introduction\nA key challenge for next-generation nuclear power is the development and processing of\nhighly radiation-resistant materials for structural components [1]. Reduced activation fer-\nritic/martensitic steels, such as Eurofer-97, are leading candidates due to their favourable me-\nchanical and thermal properties [2], as well as good resistance to void swelling [3]. While the\nexchange of Mo, Nb and Ni for W and Ta in Eurofer-97 helps to achieve low activation ca-\npability [4], the material still suffers from irradiation-induced changes such as hardening and\nembrittlement, which affect its performance below 600 K [5].\nRecently, nanostructuring and grain refinement via severe plastic deformation (SPD) has\nbeen proposed as a potential method for increasing the radiation tolerance of materials [6]. The\ngoal for a radiation-resistant material is for its structure and properties to remain stable, or un-\nchanged, under irradiation. It is proposed that SPD increases the density of grain boundaries\nin the material, which act as sinks for irradiation-induced defects such as vacancies and inter-\nstitials, removing them from the bulk material and reducing their overall effect on the material\nproperties [7].\nThe effects of SPD on the irradiation response of structural steels include reducing\nirradiation-induced segregation of alloyed solutes [8, 9], reducing the population of irradiation-\ninduced defects [10], and lowering irradiation hardening [11]. Some studies have shown that\ngrain refinement can suppress irradiation-induced void swelling by over an order of magni-\ntude [12–14]. However, Aydogan et al. have found higher swelling near the surface region (top\n1 µm) in the irradiated SPD material compared to the as-annealed material [15].\nThe suitability of grain refinement by SPD for increasing radiation resistance of structural\nsteels is also unclear due to the complexity of studying different steels with varying compo-\nsitions and crystal structures. For example, in face-centred cubic (FCC) materials, the grain\n4refinement mechanism depends greatly on the stacking fault energy and twinning. However for\nbody-centred cubic (BCC) materials, the results of SPD depend more on dislocation slip and\nthe plasticity of the individual material [16,17]. The range of conditions over which SPD can be\ncarried out (method of deformation, temperature, pre-/post-annealing treatments) also affect the\ngrain refinement and resultant material properties [6, 18]. Furthermore, the process of SPD it-\nself also changes the properties of steels by increasing yield strength but also reducing ductility,\nwhich will affect its operational performance [16, 19]. Therefore, more comprehensive studies\nof the combined effect of SPD and irradiation are warranted in light of its potential use to modify\nnuclear structural materials. In particular, there are comparatively fewer studies in the literature\non the effect of SPD on ferritic/martensitic steels compared to austenitic steels [16, 18]. As\nsuch, the study of Eurofer-97, a prototypical ferritic/martensitic steel is particularly interesting.\nHigh-pressure torsion (HPT) is a method which subjects materials to a compressive force\nand simultaneous torsional straining. It is an effective method of producing nanocrystalline\nmaterials with grain sizes less than 100 nm [20]. HPT produces a disc with radially increasing\nshear strain, γ, from 0 at the centre of the disc to:\nγ=2πNr\nh(1)\nwhere Nis the number of rotations, ris the distance from the centre of the disc, and his the\nthickness of the disc [20]. As such, a range of strains can be accessed within the same sample.\nHere we present a study on the effect of HPT, performed at room temperature, on the mi-\ncrostructure, thermal stability, and irradiation response of Eurofer-97. A range of experimental\ncharacterisation techniques including electron microscopy, X-ray diffraction and transient grat-\ning spectroscopy were carried out. The results allow us to study the evolution of grain size,\ndislocation density and character as a function of deformation, annealing temperature and irra-\ndiation. The thermal transport and elastic properties of the deformed and irradiated materials\n5were also probed. Considering the combination of these results, a multi-faceted picture of the\neffect of HPT on the structure and properties of Eurofer-97, as well as its thermal stability and\nirradiation response, emerges. This in turn allows us to examine the effectiveness of SPD as a\nmeans of enhancing the radiation resistance and material properties of Eurofer-97 as a nuclear\nstructural component.\n2 Materials and Methods\n2.1 Sample Preparation\nThe Eurofer-97 material was provided by the UK Atomic Energy Authority. The original 4 mm\nthick plate was produced by B ¨ohler Bleche GMBH (Heat 993394) [21]. After hot rolling, the\nmaterial was normalised at 1253 K for 10 minutes and then tempered at 1033 K for 1.5 hours.\nThe manufacturer-provided composition is shown in Table 1.\nElement Cr W V Ta C Mn N Fe\nwt% 9.08 1.07 0.24 0.13 0.11 0.56 0.04 Bal.\nTable 1: The chemical composition of Eurofer-97 with all values in wt% [21].\nSamples were cut into discs of 5 mm diameter and 1 mm thickness. HPT was performed\nusing a Zwick Roell Z100 Materials Testing machine in a quasi-constrained set-up [20] with\ncustom-made anvils made from M42 tool steel. The samples were compressed with 80 kN of\nforce (∼4 GPa pressure), and 9 complete rotations were performed while maintaining this force.\nThe final specimen thickness after HPT was 0.5 mm. Undeformed samples were also retained\nas reference specimens. Shear strains investigated for the specimens in this study range from 0\nat the centre of the disc to 280 at the edge of the disc.\nAll samples were mechanically ground with SiC paper, then polished with diamond sus-\npension and colloidal silica (0.04 µm). The final polishing step was electropolishing with 5%\n6perchloric acid and 15% ethylene glycol monobutyl ether in ethanol at 293 K, using a voltage\nof 30 V for 2-3 minutes using a Struers LectroPol-5.\n2.2 Ion Irradiation\nIrradiation with 20 MeV Fe4+ions at room temperature was performed using the tandem accel-\nerator at the Helsinki Accelerator Laboratory. Both the HPT-processed and undeformed sam-\nples were irradiated. The dose profile was calculated using the Quick K-P model in the SRIM\ncode [22], using 20 MeV Fe ions on a Fe target with 40 eV displacement energy at normal in-\ncidence (Figure 1). The calculated damaged layer extends to 3.5 µm below the sample surface.\nThe average dose in the first 2 µm below the surface, where the injected ion concentration is\nsmall, is 0.08 displacements-per-atom (dpa). The peak dose is 0.5 dpa at a depth of 3 µm and\nthe average dose across the whole implanted layer (from 0 to 3.5 µm depth) is 0.17 dpa. All\nthese doses fall into the transition regime before the onset of defect microstructure saturation in\nferritic materials [23–25].\nFigure 1: The calculated dose and injected ion profile from SRIM [22] for 20 MeV Fe-ion\nirradiation on Fe target.\n72.3 Electron Microscopy\nSEM-based characterisation was carried out using a Zeiss Merlin FEG-SEM at the David Cock-\nayne Centre for Electron Microscopy, University of Oxford. Backscattered electron (BSE)\nimaging was used for qualitative characterisation of grain morphology and carbide distribution\non the sample surface. For quantitative grain size analysis of the undeformed material, electron\nbackscatter diffraction (EBSD) was carried out at 30 kV with a beam current of 10 nA.\nFor the HPT-processed material, conventional EBSD did not provide sufficient spatial reso-\nlution for microstructural characterisation due to the refined grain size. Lift-out samples were\nmade using focused ion beam (FIB) milling (Zeiss Crossbeam 540 FIB-SEM and Zeiss NVision\n40 FIB-SEM). The samples were thinned with Ga ions to approximately 100 nm. Lift-outs were\ntaken at 2 different locations on each HPT sample (unirradiated and irradiated), corresponding\nto shear strains of γ= 110 andγ= 230 .\nTEM imaging was carried out with a Jeol 2100 TEM with a LaB 6source, operating at\n200 kV . EDX measurements of the TEM samples to characterise carbides in the material were\ncarried out on a JEOL ARM-200F TEM, operating at 200 kV . On-axis transmission Kikuchi\ndiffraction (TKD) measurements were carried out using a Zeiss Merlin FEG-SEM at 30 kV , for\nquantitative characterisation of grain size and dislocation density.\nGrain size analysis was carried out using the Oxford Instrument HKL Channel 5 Tango\nsoftware, with the grain boundary critical misorientation set to 10◦. Geometrically necessary\ndislocation (GND) analysis was carried out in the same software, which uses the kernel average\nmisorientation approach described in [26]. The data was first filtered with the Kuwahara method\nto remove some noise due to the small step sizes used for TKD [27]. The average misorientation\nat each point was calculated with respect to its second nearest neighbours as it was determined\nto be optimum for resolving the GND density distribution while not being too affected by step\nsize-related noise.\n82.4 Synchrotron X-ray Diffraction\n2.4.1 Grazing-incidence X-ray Diffraction\nX-ray diffraction (XRD) measurements were made at the Diamond Light Source I11 beamline\nwith high-resolution multi-analyser crystal (MAC) detectors. The detector set-up consists of 5\nMAC arms, which hold 9 analysing Si crystals each, mounted on a 2 θ-circle with 30◦separation\n[28,29]. Due to the narrow rocking curves of the Si crystal, a high instrument resolution can be\nachieved. A grazing-incidence geometry was used with a 10◦incidence angle, and X-ray energy\nof 15 keV ( λ= 0.82656 ˚A,∆E/E = 10−4). This ensured that the majority of the diffraction\nsignal was dominated by contributions from the top 3 - 4 µm thick layer of the samples, where\nthe implanted layer lies. The beam size was approximately 250 µm in height and 1 mm in width.\nMeasurements were taken at positions halfway between the centre and edge of the disc samples.\nAccounting for the size of the beam and its footstep on the sample surface due to the incident\nangle, the area measured corresponds to shear strains between 65 to 160.\n2.4.2 In-situ heating X-ray Diffraction\nTo study the thermal stability of the deformed material, in-situ heating during XRD was per-\nformed at the Diamond Light Source I12 beamline. Samples were mounted in a Linkam TS1500\nheating stage for diffraction in transmission geometry. The nominal heating rate was 20 K/min,\nthen holding for 5 minutes at the target temperature, followed by cooling to room temperature\nat a nominal rate of 30 K/min. The samples were held under an argon atmosphere throughout\nthe heating cycle to prevent oxidation. The highest nominal temperature reached during the\nannealing cycle was 1323 K.\nThe X-ray beam was monochromated to an energy of 80 keV ( λ= 0.15498 ˚A,∆E/E =\n10−4). The energy calibration and the instrument broadening function were measured using a\nceria standard sample. The beam size of 0.5 mm ×0.5 mm at the sample was defined using\n9slits. The position at which measurements were taken on the sample corresponded to a shear\nstrain of ∼110. Two-dimensional XRD patterns were collected on a Pilatus 2M CdTe detector\npositioned at a distance of 750 mm from the sample. Patterns were collected with 5 seconds of\nexposure for the entire duration of the heating and cooling cycle. Azimuthal integration of the\n2D raw data to obtain intensity vs. 2 θ(scattering angle) spectra was performed with the DAWN\nsoftware package [30].\nThough the Linkam heating stage provided a temperature reading for each diffraction pat-\ntern, a calibration was necessary to determine the actual temperature of the sample due to im-\nperfections in the thermal contact between the stage and the samples. The sample temperature\nwas estimated by fitting the lattice parameter of each diffraction pattern and using the ther-\nmal expansion coefficient of Eurofer-97 [31]. Further details are included in Appendix A. All\ntemperatures discussed from this point onwards refer to the calibrated sample temperature.\n2.4.3 Convolutional Multiple Whole Pattern (CMWP) Analysis\nAnalysis of the intensity vs. 2 θdiffraction spectra (from both I11 and I12 beamlines) was\nperformed using the convolutional multiple whole profile (CMWP) fitting method [32,33]. This\nallowed for the extraction of crystallite size distribution, dislocation density and dislocation\ncharacter. Briefly, CMWP analysis is based on fitting the sum of a background spline function\nand a profile function, which is a convolution of the theoretical size and strain profiles, as well\nas the measured instrument profile. The crystallite size profile is modelled as a log-normal\ndistribution. The strain profile considers the dislocation density, as well as the characteristic\nlengthscale of their strain fields, which depends on their physical ordering and arrangement\n[34, 35].\n102.5 Transient Grating Spectroscopy\nTransient grating spectroscopy (TGS) is a laser-based non-destructive technique to measure\nthermal diffusivity and surface acoustic wave velocity. The details of the technique are de-\nscribed elsewhere [36–38]. Briefly, two short pump laser pulses (0.5 ns duration at 532 nm,\nwith pulse energy ∼1.5 µJ and 1 kHz repetition rate) are overlapped on the sample surface at a\nfixed angle to form an interference pattern with wavelength λ. Partial absorption of the light and\nsubsequent thermal expansion occurs at positions of constructive interference, creating a spatial\ngrating on the surface. At the same time, this rapid thermal expansion launches two monochro-\nmatic, counter-propagating surface acoustic waves (SAW). Following excitation by the pump\nbeams, this ‘transient grating’ will decay due to the diffusion of thermal energy from maxima to\nminima and also into the material bulk. By measuring this decay via the diffraction of a set of\nprobe laser beams (continuous wave at 559.5 nm), the thermal diffusivity and the SAW velocity\nof the material can be determined. The key strength of TGS applied to ion-irradiated materials\nis that the thickness of the probed layer is approximately equal toλ\nπ, soλcan be adjusted to\nensure the signal is dominated by contributions from the irradiated layer.\nThis study uses λ= 5.116±0.002µm for most measurements. Extra measurements were\nmade for the HPT-deformed and irradiated sample with λ= 5.116±0.002µm. The resultant\nprobed layer thickness for all measurements was ∼1.6 µm, which ensures that the signal orig-\ninates from the 3.5 µm thick damaged layer in the irradiated samples. The pump and probe\nbeam sizes were respectively 140 µm and 90 µm (1\ne2width). In order to measure properties\ncorresponding to different shear strains, lines of measurement points spaced 150 µm apart were\ntaken across the diameter of the HPT-processed samples.\n113 Results\n3.1 Electron Microscopy\nFigure 2: The microstructure of the same area on the as-received Eurofer-97 characterised by\n(a) EBSD and (b) backscattered electron (BSE) imaging. (c) A higher magnification BSE image\nof the as-received material (same area enclosed by the yellow rectangle in (b)). Red and blue\narrows indicate the presence of large and small carbides, respectively. (d) A schematic of the\nsample coordinates with respect to the deformed material. The orientation shown applies to all\nmicrographs in this figure. BSE imaging of Eurofer-97 after deformation at (e) γ= 110 and (f)\nγ= 230.\nEBSD of the as-received Eurofer-97, before deformation via HPT, shows lath structures\ncharacteristic of the martensitic microstructure of the material (Figure 2(a)). The area-weighted\naverage grain size of the undeformed material is 5.3 ±3.2 µm, using an angular misorientation\nthreshold of 10◦. Following electropolishing, a dispersion of particles, each between a few\nnm to 200 nm in size, becomes evident on the sample surface (Figure 2(b) and (c)). Particles\nof similar size and spatial distribution have been observed in other studies of Eurofer-97 and\nsimilar steels, and have been identified as carbides [15, 39, 40]. The larger carbides mainly\nappear at the boundaries of prior austenite grains and are up to 200 nm in size (red arrows in\n12Figure 2(c)). There are also smaller carbides, <10 nm in size, distributed more homogeneously\nacross the sample (blue arrows in Figure 2(c)).\nIn this work, cylindrical coordinates will be used to describe directions in the HPT-deformed\nmaterial (Figure 2(d)). zis normal to the surface of the HPT sample (outwards positive), ris in\nthe radial direction, and θis the tangential direction along which torsion is applied. Following\ndeformation via HPT, a significant level of grain refinement is achieved (Figure 2(e) and (f)).\nQuantitative measurements of the grain size following HPT will be presented in the subsequent\nsections as EBSD was not able to achieve the required spatial resolution on the sample surface.\nFrom SE and BSE imaging, it can be seen that most of the refined grains are equiaxed, with\nno preferential alignment to the direction of torsion when observed along the −zdirection\n(in the r-θplane, Figure 2(e) and (f)). The carbides are still present following HPT but are\nmore homogeneously distributed. A mix of large and small carbides is present with similar size\ndistributions to the undeformed sample.\nTo further study the grain microstructures of the HPT material, FIB was used to prepare\nTEM lamella from different samples and radial positions. A full comparison of TEM micro-\ngraphs from all samples is provided in Appendix B. There is no discernible difference in mi-\ncrostructure between γ= 110 andγ= 230 , suggesting that strain saturation is reached before\nγ= 110 . TEM of both the unirradiated and irradiated HPT material also shows that there is\nno depth dependence of the microstructure down to 10 µm below the bulk sample surface. An\nexample is shown in Figure 3(a) for the top 6 µm of the deformed and irradiated sample.\nWhen viewed along the radial direction (on the θ-zplane), the grains appear elongated\n(Figure 3(a)-(c)). The axis of elongation is not completely aligned with the direction of torsion\nθ(parallel to the platinum layer on the top edge of the lift-out) but inclined approximately\n15◦. This has been observed previously in other studies of HPT materials, even at shear strains\nbeyond microstructural saturation [41–44].\n13Figure 3: TEM images of the deformed irradiated sample with the lift-out taken from a location\ncorresponding to γ= 230 . (a) Low magnification overview of the lift-out, showing the plat-\ninum (Pt) protective layer on the bulk sample surface, with a carbon layer underneath (black).\n(b)&(d) Magnified micrographs showing the grain shapes and microstructure in the irradiated\nand unirradiated regions, respectively. The figures have been rotated to align with the θ-zplane\nof the bulk sample. (c)&(e) Bright-field images of individual grains within the irradiated and\nunirradiated regions respectively.\nThe presence of large carbides is detected in the TEM images as indicated by the red arrows\nin Figure 3(b), similar to those indicated with red arrows in Figure 2(c). EDX was performed\nto confirm that these regions, which appear grey and without sharp edges, are indeed carbides\nenriched with chromium and tungsten (Appendix C).\nBright-field images of individual grains were recorded from both the irradiated (Figure 3(c))\nand unirradiated (Figure 3(e)) parts of the lift-out. Numerous dislocation lines are evident in\nboth grains. The black spots (blue arrows), most likely carbides, also appear along the length\nof the dislocation lines. No evidence of irradiation-induced microstructural changes (e.g. dislo-\n14cation loops) were observed.\nFigure 4: TKD maps taken from the (a)&(d) unirradiated samples and the irradiated samples in\nthe (b)&(e) unirradiated regions, and (c)&(f) irradiated regions, from respectively γ= 110 and\nγ= 230 . Note that the top and bottom edges of each sample are parallel to the surface of the\nbulk sample from which the lift-outs were taken. The grain boundaries (angular misorientation\n>10◦) are indicated in white. Non-indexed areas are indicated in black.\nComparisons of the microstructure of different deformed samples were done with TKD\nmeasurements (Figure 4). The average indexing rate was >80%. Regions of low confidence\nindexing occurs near the grain boundaries and around carbides (large compact black areas indi-\ncated by white arrows with red outlines in Figure 4(c) and (f)).\nComparing the effect of shear strain on the microstructure (top row compared with bottom\nrow in Figure 4), it can be seen that the aspect ratio of the grains appears slightly larger for\nγ= 230 compared to γ= 110 . The difference in strain level does not affect the indexing rate.\nComparing the unirradiated sample with the unirradiated regions of the irradiated sample\n15at each strain level (Figure 4(a) and (c), (b) and (d)), it can be seen that the microstructure at\nnominally the same deformation conditions is similar across different samples. This confirms\nthe reproducibility of the HPT method used for this study.\nThere appears to be no discernible irradiation-induced changes to the microstructure in the\nTKD maps (Figure 4(c) and (f)). The overall grain shapes and distribution of unindexed regions\nappear to be the same for the irradiated and unirradiated regions of the sample at each strain\nlevel.\n3.1.1 Grain Size\nFigure 5: The grain diameter distribution for the (a) undeformed and (b) deformed ( γ= 230)\nmaterial from TKD measurements. The grey regions indicate the range of inaccessible grain\nsizes due to the limited measurement step size. (c) The average grain diameter for the unim-\nplanted and implanted regions at the two strain levels where lift-outs were taken. The error\nbars represent ±1 standard deviation of the grain diameter. Note that all average and standard\ndeviation grain sizes reported are weighted by grain area.\nThe grain size distribution shape appears to be similar before and after deformation with\nHPT (Figure 5(a) and (b)). Due to the presence of grains much smaller than the step sizes of\nEBSD and TKD, it was difficult to probe the lower end of the distribution ( <250 nm for EBSD\nand<20 nm for TKD). The average grain size following HPT, to a shear strain of γ= 230, is\nreduced by over a factor of 35 compared to the undeformed reference. Comparing the grain sizes\n16following HPT, there does not appear to be much change with strain level within measurement\nerror (Figure 5(c)). This confirms that the material microstructure is in the saturated regime at\nor before γ= 110. There is also no significant change in grain size following irradiation.\n3.1.2 GND Density\nFigure 6: GND density map of the (a) undeformed and (b) deformed material ( γ= 230). Note\nthat the magnification and sample orientations are not the same for the two samples. (c) The\naverage GND density for the unimplanted and implanted regions at γ= 110 and γ= 230. The\nerror bars represent ±1 standard deviation of the GND density.\nThe distribution of geometrically necessary dislocations (GND) in the undeformed mate-\nrial shows extended swirling structures that are distributed evenly across and within the grains\n(Figure 6(a)). The average GND density is on the order of 1014-1015m−2as expected for a\nmaterial that has undergone cold working [10, 45]. The GND density increases by an order of\nmagnitude following HPT to 1015-1016m−2(Figure 6(b)). The regions of high dislocation\ndensity are mainly near grain boundaries. Some bands of high dislocation density are oriented\nperpendicular to the elongated axis of the grains.\nComparing dislocation density between the different lift-out samples of the deformed ma-\nterials reveals little difference in average GND density as a function of strain between γ= 110\nandγ= 230 (Figure 6(c)). Once again this confirms that strain saturation is achieved before\nγ= 110. Implantation also did not change the average GND density of the samples. This is\n17unsurprising as at this damage level, most irradiation defects likely exist as point defects or\nsmall loops [46] and thus do not contribute to the large-scale lattice curvature from which GND\ndensity is determined.\n3.1.3 Grain Boundaries and Texture\nFigure 7: The distribution of grain boundary misorientation angle for the (a) undeformed and\n(b) deformed material, along with (c)-(d) the corresponding {110}pole figures. (e) The theo-\nretical {110}pole figure of pure iron that has been processed by torsion [47].\nThe undeformed Eurofer-97 material has a high proportion of grain boundaries with <10◦\nand>50◦misorientation (Figure 7(a)). After torsion, this distribution changes to a large propor-\n18tion of grain boundaries between 30◦−60◦(Figure 7(b)). There is a significant peak around 30◦\nwhich was greatly enhanced following deformation with HPT. The true proportion of angular\nmisorientation below 2.5◦could not be accurately determined due to experimental limitations\n(grey regions in Figure 7(a) and (b)).\nThe{110}pole figure for the undeformed Eurofer-97 shows little preferred orientation or\ntexture (Figure 7(c)). After deformation with HPT, significant texture evolution occurs and\npreferred orientation is evident (Figure 7(d)). The experimental observations qualitatively agree\nwith the theoretically predicted preferred orientations for torsioned iron (Figure 7(e)). There is\na∼15◦rotation between the experimentally determined and the predicted texture. This is\nsimilar to the angular difference between the top surface of the bulk HPT sample and the axis\nof elongation for the grains in the θ-zplane (Figure 4(d)).\n3.2 X-Ray Diffraction\n3.2.1 Grazing-Incidence X-Ray Diffraction\nPeak broadening is clearly observed in the raw diffraction patterns following HPT to γ= 110\n(Figure 8(a)). This suggests an increase of intragranular strain and/or a decrease in grain size.\nThe diffraction patterns shown in Figure 8(a) are functions of K= 2 sin θ/λ, and the fittings\nfrom CMWP (overlaid black lines) appear to agree well with the measured data. Due to strain\nanisotropy, the peak broadening is not isotropic, i.e. it is not linear with K. Strain anisotropy\nin peak broadening can be rectified by taking into account the dislocation contrast factors, C,\nin the modified Williamson-Hall plot [48], where the FWHM or the integral breadth is plotted\nagainst K2C. The contrast factor depends on the relative orientation of the diffraction vector, the\nBurger’s vector and the line vector of dislocations, and the elastic constants of the material [35].\nFor polycrystalline samples, the average contrast factor of all the grains and dislocations is\ndetermined in the CMWP fitting procedure. The modified Williamson-Hall plots in Figure\n19Figure 8: (a) A section of the diffraction patterns, plotted as a function of K= 2 sin θ/λ, for\nthe samples investigated. The traces have been vertically offset for clarity. The scatter points\nare the experimental data points and the black lines are the fitted patterns from CMWP. (b) The\nmodified Williamson-Hall plot with Ccalculated from CMWP fitting. The dashed lines are\nincluded as a guide to the eye. (c) The area-weighted average crystallite size, (d) dislocation\ndensity, and (e) qparameter determined from CMWP. Further information about each parameter\nis given in the main text. The error bars represent the fitting errors calculated by considering\nthe variations in the weighted-sum-of-squares residuals [32].\n8(b) are produced following the procedures described in [35, 48]. They demonstrate that strain\nanisotropy in these samples can indeed be accounted for by considering the contrast factor fitted\nfrom CMWP, as the integral peak widths are a linear function of K2C[49].\nCMWP line-profile analysis fits the microstructural parameters including average crystallite\nsize, dislocation density and the contrast factors, which describes the average character of the\ndislocations (‘ qparameter’) [32,35]. The results of this analysis are shown in Figure 8(c)-(e). A\n20combined procedure using successive applications of a Monte Carlo (MC) statistical algorithm\nand Levenberg-Marquardt nonlinear least-squares algorithm fits the peaks and physical param-\neters of the diffraction patterns. The error bars presented describe the range for each parameter\nwhere the corresponding fit had a resultant weighted sum of square residuals (WSSR) that fall\nwithin p% confidence value in the MC procedure [50]. At least 2000 iterations were completed\nduring the MC fitting, and p= 3.5 was used as it has been previously shown to yield a good fit,\neven for complicated patterns. Further details of the fitting procedure can be found in [32, 33].\nThe crystallite size fitted from CMWP for the undeformed material is ∼300 nm (Figure\n8(c)), which is much smaller than that measured by EBSD (Figure 2(a) and 5(a)). This is to be\nexpected as X-rays are much more sensitive to smaller angular misorientations ( <1◦), hence\neffectively mapping out regions corresponding to ‘subgrains’ in the undeformed material [51].\nFor grains larger than about two microns, the size contribution to X-ray peak broadening is not\npossible to measure as its breadth becomes smaller than the instrument broadening contribu-\ntions. Therefore, XRD is unable to accurately determine the size of ‘grains’ in the as-received\nmaterial in this study. In contrast, SEM-based methods are useful for measuring angular mis-\norientations of 10 - 15◦, and on the lengthscale of microns, mostly resulting in the size mea-\nsurement of ‘grains’ (Figure 5(a)).\nAfter deformation via HPT to γ= 110 , the average crystallite size is 42 nm. This value is\ncomparable to the TKD grain size measurements (Figure 5(b)). Irradiation does not cause any\nsignificant change in crystallite size for the undeformed material. There is a slight irradiation-\ninduced increase in crystallite size for the HPT-deformed material.\nDislocation density increases by a factor of ∼10following HPT (Figure 8(d)). Subsequent\nirradiation of the deformed material causes a 20% reduction in dislocation density. In contrast,\nthe undeformed reference material shows a slight increase in dislocation density following irra-\ndiation, likely due to irradiation-induced defects. The reduction in strain following irradiation\n21of the deformed material can also be directly observed from the changes in peak broadening\nof the diffraction spectrum (Figure 8(a)) and the slope of the modified Williamson-Hall plot\n(Figure 8(b)). While the results of the deformed material shown in Figure 8 were measured for\nγ∼110, similar results (not shown here) were also observed for γ∼230. This demonstrates\nthat self-ion irradiation removes some effects of the dislocations introduced by severe plastic\ndeformation.\nTheqparameter describes the average character (screw vs. edge) of the dislocations present\nin the material. Its theoretical range depends on the elastic anisotropy ( Ai= 2c44/(c11−c12))\nand the ratio c12/c44) [35]. For Eurofer-97, ab initio calculations give Ai≈1.5andc12/c44≈1\n[52]. This leads to a corresponding qrange of between 1 (pure edge dislocations) to 2.5 (pure\nscrew dislocations). Our fitting of the qparameter suggests that the dislocations present in the\nundeformed material ( q∼2.3) are predominantly of a screw-type character, and after HPT\n(q∼1.4) they are of more edge-type character (Figure 8(e)). Irradiation appears to have little\neffect on the qparameter.\n3.2.2 In-Situ Annealing X-ray Diffraction Measurements\nIn the reference undeformed sample, the peak widths did not change as a function of temperature\nup to 1043 K, at which point a phase transition took place (Figure 9(b)). The temperature at\nwhich the phase transition from ferrite/martensite (body-centred cubic/tetragonal) to austenite\n(face-centred cubic) take place can depend on the heating rate. In other studies of Eurofer-\n97 in the literature, the transition begins to take place between 1025 K - 1133 K [31, 53, 54].\nIn this study, our measurements show that all ferrite/martensite peaks disappeared at 1093 K\nduring the heating ramp. The material contains only austenite phase during holding at 1153 K,\nand during the cooling stage until the material cooled to 606 K. The martensite phase returned\nwith cooling below 606 K. This is consistent with previous findings in Eurofer-97 annealing\n22Figure 9: (a) The calibrated temperature profile of the annealing cycle during the in-situ XRD\nmeasurements. The diffraction intensity profiles as a function of K= 2 sin θ/λand annealing\ncycle time are shown for (b) the reference undeformed sample, and (c) the HPT-deformed sam-\nple, respectively. The vertical axes representing time in (b) and (c) are aligned with (a), from\nwhich the annealing temperature can be read.\nexperiments [53]. However, some austenite phase is still present in the material with broader\ndiffraction peaks. Interestingly, the martensite peaks are also wider following the heating and\ncooling cycle than in the initial state. This could be due to residual strain in the martensite\nphase as it forms within the austenite phase upon cooling [55,56]. There is a volume expansion\nassociate with the formation and growth of martensite regions within the austenite grains. This\nwill impart a hydrostatic pressure within the austenite regions, which will also in turn cause\nresidual strains in the martensite grains [57, 58]. Appendix E shows a different set of in-situ\nannealing XRD measurements, in which a phase transition occurred for a shorter period of time,\nand no austenite phase was retained following cooling. The martensite peak widths following\ncooling remained as narrow as right before the phase change at 973 K. This is further evidence\nthat the post-annealing peak broadening is associated with the retained austenite phase.\nDuring heating of the HPT-deformed material, the peak widths are initially large, then grad-\n23ually reduce until around 900 K (Figure 9(c)). Similar to the undeformed sample, a phase\ntransition from ferrite/martensite to austenite occurs at 1043 K, then from austenite back to\nmartensite at 606 K during the cooling stage. Both austenite and martensite phases remain in\nthe material even at room temperature following the heating cycle. Furthermore, the peaks after\ncooling, for both the martensite and austenite phases, are broader than before and during the\nannealing cycle, respectively.\nFigure 10: The microstructural parameters of the in-situ annealing diffraction measurements\ndetermined from CMWP analysis. An explanation of each parameter is included in the main\ntext. The error bars are from the fitting algorithm discussed in Section 3.2.1.\nFrom the CMWP fitting, the area-weighted average crystallite size is unchanged between\n303 K and 367 K ( ∼75 nm) during the heating stage (Figure 10(a)). Further increase in tem-\nperature causes a corresponding increase in crystallite size until ∼725 K, where the average\ncrystallite size is around 280 nm. After this point, the crystallite size increases dramatically\nwith temperature, suggesting that the size-broadening contribution in the diffraction patterns\nis approaching zero. However, we note that recrystallisation has likely not progressed far at\nthis temperature as the XRD patterns still showed continuous Debye-Scherrer rings up until the\nphase transition (see supplementary files for raw patterns).\nThere appear to be three distinct stages of evolution for dislocation density (Figure 10(b)).\n24Between 303 K to 550 K, the dislocation density increases slightly and then decreases but the\naverage remains within 10% of the initial value ( ∼1.3×1016m−2). Between 550 K to 800\nK,ρdecreases linearly to ∼4×1015m−2. Further reduction in ρabove 800 K appears to\nbe at a greater rate than in the previous stage, but it is also likely that the strain-induced peak\nbroadening is too small to be quantitatively probed in this particular experiment.\nTheqparameter, which is indicative of the average nature of the dislocation character, also\nchanges during annealing (Figure 10(c)). At temperatures above 387 K, the qparameter slowly\nincreases from 1.75 to 2 at 680 K, after which the fitting from CMWP analysis appears to be less\nreliable. This suggests a general trend of the average dislocation character from more edge-like\nto more screw-like, recovering features of the undeformed material ( q∼2.3, Figure 8(e)).\n3.3 Material Property Changes\nThermal diffusivity of the reference Eurofer-97 measured by TGS in this study (Figure 11(a))\nis in agreement to within 10% of previous measurements using the laser flash technique ( 8.3×\n10−6m2s−1) [59]. Following irradiation, the thermal diffusivity of the undeformed material\ndecreases by 6.7%, which is comparable to the results of FeCr [24] and stainless steel 316\nL [60] exposed to similar damage doses. It is well-known that irradiation can cause a decrease\nin thermal diffusivity due to the introduction of defects to the crystal that act as scattering\nsites for electrons. Since thermal conductivity in metals is predominately electron-mediated, an\nincrease in electron scattering rate results in a reduction of thermal diffusivity [61].\nPlastic deformation via HPT also has the effect of reducing the thermal diffusivity of\nEurofer-97 (Figure 11(a)). It is interesting to note that at nominally zero shear strain (at the\ncentre of the HPT sample), the thermal diffusivity is not the same as the reference as-received\nmaterial. This difference shows that even though the centre experiences nominally only com-\npression, which does not cause significant grain refinement, there is still a significant reduction\n25Figure 11: (a) Thermal diffusivity and (b) surface acoustic wave (SAW) velocity of Eurofer-\n97, comparing the reference (dark blue triangle), irradiated (red circle), deformed (light blue\nasterisk), and deformed then irradiated samples (pink cross). The difference in (c) thermal\ndiffusivity and (d) SAW velocity following irradiation for the undeformed (blue circle) and\ndeformed (pink asterisk) set of materials. The values for the deformed material are obtained by\nsampling points along the diameter of the specimen to measure areas of different shear strains.\nin thermal diffusivity. This suggests that there is an appreciable increase in dislocation density\nin the sample centre during the initial compression.\nThermal diffusivity decreases with increasing shear strain (Figure 11(a)). At γ= 160 ,\nthe maximum strain level measured on the HPT samples due to the sample geometry, there\nis a decrease of 12% from the reference Eurofer-97 value. HPT deformation increases the\ndislocation density in the material and increases the ratio of grain boundaries to perfect crystal.\n26Both these factors lower the mobility of charge carriers in the material and thus cause a reduction\nin thermal diffusivity [62, 63]. The rate of thermal diffusivity decrease appears to reduce with\nincreasing shear strain, particularly at γ >70.\nIrradiation of the deformed material causes a further reduction in thermal diffusivity in\naddition to the decrease associated with the deformation. At γ= 160 in the irradiated sample,\nthere is a 19% reduction in thermal diffusivity compared to the reference unirradiated Eurofer-\n97 sample.\nFollowing irradiation, the material that has undergone prior HPT shows a smaller reduction\nin thermal diffusivity compared to the undeformed material (Figure 11(c)). This could be due\nto the irradiation-induced defects being absorbed by the greater proportion of grain boundaries\npresent in the HPT material [6], therefore causing less change to the thermal properties. There\ncould also be competing effects between the introduction of irradiation-induced defects and the\nirradiation process itself causing simultaneous annealing of the pre-existing dislocations from\nHPT (Section 3.2.1). As these two processes act in opposition to each other, the net change\nto thermal diffusivity after deformation and irradiation is smaller than for irradiation alone.\nHowever, we also note that the irradiation-induced difference in thermal diffusivity between the\nundeformed and deformed cases is small, and could be within measurement error.\nSurface acoustic wave (SAW) velocity is related to the elastic constants of the material.\nHowever, since Fe and Fe-based alloys have strong elastic anisotropy, the relationship be-\ntween SAW velocity and the elastic constants is dependent on the grain orientation or local\ntexture [64]. Since our grains are many times smaller than the measurement spot even in the\nundeformed case, we cannot quantitatively measure the changes to the elastic constants in this\nstudy. However, changes to the SAW velocities were still measured, indicating that the material\nstiffness has been altered by HPT and irradiation (Figure 11(b)).\nA 1% reduction in SAW velocity due to irradiation in the undeformed sample is observed.\n27This could be due to the creation of point defects from irradiation. Previous studies with He-ion\nirradiation in tungsten have shown a ∼2% reduction in SAW velocity at 0.2 - 0.3 dpa [65, 66].\nHowever, studies in copper and nickel (different crystal structures to Eurofer-97) have shown\nan increase in SAW velocity for irradiation up to 5 dpa, which the authors attributed to the\ninteraction of defect clusters with dislocations causing a pinning effect [67, 68].\nDeformation causes a significant reduction in SAW velocity (Figure 11(b)). Note that due\nto the measurement geometry, the SAW velocities reported here are in the radial direction r\n(perpendicular to the shear direction θ). At γ= 160 , the SAW velocity has decreased by 2.5%\ncompared to the undeformed material. There also does not appear to be any saturation effect\nuntil γ= 140 . This is a significantly greater level of strain than required for the saturation\nof thermal diffusivity changes (Figure 11(a)) and hardness [69], both of which approached\nsaturation at γ >70. Grain size and GND density, as observed by TKD, have both saturated by\nγ= 110 (Figure 5 and 6), which is also lower than the shear strain threshold for SAW velocity\nchange saturation.\nIrradiation of the deformed material causes a further reduction in SAW velocity, however\nthis could be within measurement error. The change to SAW velocity due to irradiation is\nless for the materials that have undergone prior HPT deformation compared to the undeformed\nspecimens (Figure 11(d)). This could again be due to a combination of the grain boundaries\nabsorbing irradiation defects and annealing of dislocations caused by HPT during the process\nof irradiation.\n4 Discussion\n4.1 Effect of HPT\nHPT causes significant grain refinement in Eurofer-97. From SEM-based analysis, the area-\nweighted average grain size decreases from 5.26 µm to 146 nm, which is a reduction by a\n28factor of ∼30 (Figure 5). From CMWP analysis of the grazing-incidence XRD data, the area-\nweighted average crystallite size decreases from 295 nm to 42 nm, which is a reduction by a\nfactor of ∼7 (Figure 8(c)). From CMWP analysis of the transmission XRD measurements,\nthe grain size of the HPT-deformed sample before annealing is 75 nm (Figure 10(a)). Though\nthe grain size reduction ratio from HPT is different between electron microscopy and XRD, all\nmeasurements of the grain sizes following deformation are in good agreement. This has been\nobserved from other studies of deformed materials that compare electron microscopy techniques\nwith XRD [70, 71].\nThe discrepancies between grain size measurements of the undeformed material from EBSD\nand XRD can be attributed to their respective sensitivity to misorientations in the lattice (Section\n3.2.1). As the XRD coherent scattering domains are small, the crystallite size measurements are\nmore indicative of ‘subgrain’ size. On the other hand, for EBSD, the misorientation threshold\nused 10◦, which allows the measurement of ‘grain’ size. However, the agreement between TKD\nand XRD measurements for the grain and crystallite size of the HPT-deformed material suggests\nthat HPT causes the formation of high-angle grain boundaries. The distinction between ‘grains’\nand ‘subgrains’ ceases to exist following grain refinement. This is also supported by the greater\ndistribution of grain boundaries with >30◦misorientation of the HPT material compared to the\nundeformed (Figure 7(a)-(b)).\nThe GND density measured by EBSD/TKD for the undeformed and HPT samples are re-\nspectively 9.5×1014m−2and7.5×1015m−2(Figure 6). In comparison, the total dislocation\ndensity values measured by grazing-incidence XRD is 2.5×1014m−2and3.8×1015m−2,\nrespectively for the undeformed and HPT-deformed samples (Figure 8(d)). From transmission\nXRD, the dislocation density of the deformed material is 1.3×1016m−2(Figure 10(b)). The\nfactor of dislocation density increase from HPT deformation is ∼8 from EBSD/TKD, and ∼15\nmeasured from XRD. These values are in good agreement despite the fact that EBSD/TKD only\n29probes a thin layer near the sample surface and only GNDs. In comparison, XRD measurements\naverage over a much larger sample volume and mainly measure statistically stored dislocations\n(SSDs) [72]. The comparison between TKD and XRD measurements of dislocation density\nin the HPT-deformed material suggests that the density of GNDs is higher than the density of\nSSDs. This can be rationalised by considering that in severely deformed materials, dislocations\nare mainly located in the vicinity of grain boundaries (also seen in Figure 6(b)) and in the form\nof GNDs [73].\nFor characterisation with EBSD/TKD, only the indexed points contribute to the GND den-\nsity calculation, which means that sampling is skewed towards the regions further away from\nthe grain boundaries. In contrast, XRD probes the full sample volume, including regions of\nstrong lattice distortions close to grain boundaries, which are not indexed with EBSD/TKD.\nThe detection of a greater increase in dislocation density following HPT from XRD measure-\nments compared to EBSD/TKD supports the theory that large amounts of dislocations are stored\nin the grain boundaries rather than the grain interiors [74, 75]. It is important to note that for\nEBSD/TKD, the GND density is related to the degree of misorientation between adjacent points,\nand hence dependent on the step size used in the measurement. A smaller step size leads to more\nmeasurement noise [76]. The step size used for the EBSD of the undeformed sample is 25 times\nlarger than the step size used for TKD in the deformed sample, due to the measurement con-\nstraints and the vastly different grain sizes in each sample. Therefore, the GND density in the\nHPT-deformed samples could be slightly overestimated.\nHPT processing also causes dislocation character to change from more screw-like to more\nedge-like (Figure 8(e)). At room temperature in coarse-grained BCC metals, the dislocation\npopulation is predominantly of screw type, which is the case for the undeformed Eurofer-97.\nThis is due to the high Peierl’s barrier and low mobility of screw dislocations [77], leading\nto a greater retention than edge dislocations [78]. Interestingly, it has been observed in other\n30nanocrystalline BCC metals produced by HPT including tungsten [79], molybdenum [80, 81]\nand tantalum [82], there is an increased population of dislocations with edge character com-\npared to the coarse-grain material. There are several explanations proposed to explain these\nobservations. One of which hypothesises that the smaller grain sizes, and hence a greater pro-\nportion of grain boundaries, means that fewer dislocations are retained in the crystal as they\nare annihilated at the grain boundaries [83]. This reduces the significance of the difference in\nmobility between edge and screw dislocations, leading to more comparable populations of each\ntype. Furthermore, HPT deformation causes a large increase in dislocation density (forest dis-\nlocations), which in turn impedes the motion of glide dislocations [84]. The mobility of glide\ndislocations becomes increasingly governed by the forest dislocation density. This again makes\nthe difference in mobility between screw and edge dislocations more insignificant. The other\nhypothesis is that hydrostatic compression increases the mobility of screw dislocations [83,85].\nThis is because higher stress reduces the activation volume for thermally-activated kink-pair\nformation. This in turn reduces the thermal energy barrier for dislocation motion and assists the\nmovement of screw dislocations at low temperatures. An increase in screw dislocation mobility\nwould lead to a more comparable probability of retention between screw and edge dislocations.\nFurther studies are needed to determine which mechanism, or combination of them, is at play\nhere.\n4.2 Thermal Stability of HPT Eurofer-97\nConsidering the changes in crystallite size and dislocation density, the HPT-processed material\nfirst undergoes a recovery process between 450 K to 800 K (Figure 10). During the start of the\nrecovery stage (450 K to 550 K), dislocation density is not expected to change significantly.\nThe growth in crystallite size in this temperature range can be attributed to the rearrangement of\ndislocations within grains, causing subgrain coarsening whilst total dislocation density remains\n31unchanged [45, 86]. Further annealing causes both the growth of these subgrains as well as the\nannihilation of dislocations, possibly at grain boundaries [87], reducing the overall dislocation\ndensity as crystallite sizes increase. At ∼800 K, the onset of recrystallisation causes rapid grain\ngrowth and a reduction in dislocation density. Annealing also causes the average character of\ndislocations in the HPT Eurofer-97 material to revert from more edge-like to more screw-like,\npartially recovering the features of a coarse-grain BCC material [77, 78].\nOther thermal stability studies of HPT-processed steel have been reported in the literature.\nFor stainless steel 316L with γup to 360, the dislocation density and crystallite size prior\nto annealing were comparable to this study, and these quantities stayed stable up to 560 K\nafter which dislocation density reduced significantly [88]. It is also interesting to note, that\nin that study, the steel structure was initially austenitic and transformed to martensitic during\nthe process of HPT. During annealing, it slowly transformed back to austenitic, beginning at\n650 K and up to 93% transformed following annealing at 1000 K. A study with differential\nscanning calorimetry of a Fe-10Ni-7Mn martensitic steel processed by HPT up to γ= 785\nshowed that dislocation annihilation with vacancies began around 504 K [89]. This is similar to\nthe temperature at which dislocation density started to reduce substantially in this study.\n4.3 Effect of Irradiation\nIon irradiation to ∼0.1dpa did not cause any microstructural changes that were observable by\nTEM or TKD. However, clear changes were measured with XRD and TGS. The reduction in\ndislocation density and increase in crystallite size following irradiation in the HPT-processed\nsamples (Figure 8(c)-(d)) suggest an irradiation-induced-annealing type process. Similar effects\nhave been observed following irradiation of HPT-processed austenitic stainless steel 316 [9],\nmartensitic Fe-Cr-W steel [10], EK-181 [15], and T91 steel [12]. However, these other studies\ntypically examined much higher doses ( >1 dpa) and elevated temperatures ( >673 K). Ob-\n32servations of crystallite growth and dislocation density reduction in our study suggest that the\nmovement of dislocations and subgrain coarsening aided by irradiation can happen at much\nlower irradiation dose and temperature.\nA previous nanoindentation study of this same set of materials found that HPT-processing\nhardened the material significantly, approaching saturation beyond γ= 70 [69]. Subsequent\nirradiation to ∼0.01 dpa caused slight softening, but further hardening is observed following\nirradiation to ∼0.1 dpa. The magnitude of hardening changes due to irradiation is much smaller\nthan from HPT deformation.\nIn light of the previous hardness results [69] and the findings from the present study, we can\ndraw a combined picture of the effect of HPT and irradiation. HPT introduces grain refinement\nand a large population of dislocations due to plastic deformation. Subsequent irradiation intro-\nduces point defects and dislocation loops into the material. This is supported by the fact that the\ndipole character parameter fitted from CMWP suggests that the dislocations are arranged in a\nstronger dipole formation following irradiation, a possible indication of dislocation loops (Ap-\npendix D). However, from grazing-incidence XRD measurements we see a reduction of total\ndislocation density following irradiation in the HPT material compared to the unirradiated HPT\nmaterial (Figure 8). This indicates a relaxation or annihilation of dislocations due to irradiation.\nHPT deformation and irradiation both individually cause an increase in hardness, and a de-\ncrease in SAW velocity and thermal diffusivity of Eurofer-97. The effect of HPT alone causes\nmore changes to the material properties, and total dislocation density, than irradiation alone.\nThe effects of HPT deformation and irradiation to ∼0.1 dpa on the material properties are ad-\nditive, whereas for dislocation density, subsequent irradiation removed some dislocations from\nprior deformation. Thus, we can conclude that changes to material properties are not simply\na function of the total dislocation density but on their type and lengthscale. The deformation-\ninduced extended dislocations annihilated in the HPT material during irradiation at ∼0.1 dpa do\n33not contribute as much change in material properties as the small defects introduced by irradia-\ntion. However, the irradiation softening at ∼0.01 dpa [69] could indicate that the initial removal\nof dislocations from HPT deformation is significant for material hardening. On the other hand,\nwhile the more dispersed but smaller irradiation-induced point defects and dislocation loops\ncontribute less to XRD peak broadening, they still have a significant effect on material proper-\nties. This is particularly noticeable for changes to thermal diffusivity. The additional reduction\nof thermal diffusivity following irradiation of the HPT-deformed material is comparable to the\nreduction from solely the HPT process. The effect of small point defects, such as vacancies, on\nthermal diffusivity in other irradiated materials has been previously investigated [90].\n5 Summary and Conclusion\nIn this study, grain refinement in Eurofer-97 has been successfully achieved via HPT at room\ntemperature. The thermal stability and irradiation response of the resultant material have been\ncharacterised. The key findings are as follows:\n• Grain refinement via the creation of high-angle grain boundaries was achieved with HPT.\nGrains smaller than 100 nm were created and the average grain size reduced from 5.26\nµm to 146 nm.\n• The average dislocation density in the material increased by over an order of magnitude\nfollowing HPT, up to 1016m−2. The majority of these dislocations are located at or near\ngrain boundaries.\n• Grain refinement from HPT increased the proportion of dislocations with edge-like char-\nacter. This is consistent with observations of other nanocrystalline BCC materials in\nliterature.\n34• Saturation of grain size refinement and dislocation density increase was achieved by γ=\n110. Thermal diffusivity and SAW velocity changes saturated at γ= 70 andγ= 140 ,\nrespectively. The saturation of microstructure and material properties at different shear\nstrains indicates that they are governed by different mechanisms.\n• Recovery was observed during annealing between 450 K to 800 K, with subgrain coarsen-\ning from the rearrangement and subsequent annihilation of dislocations. Recrystallisation\nand rapid grain growth begin around 800 K.\n• Irradiation of the undeformed Eurofer-97 material caused an increase in dislocation den-\nsity. However, for the HPT-deformed material, irradiation caused a reduction in disloca-\ntion density, suggesting an irradiation-induced annealing process.\n• HPT processing prior to irradiation did not prevent additional irradiation-induced reduc-\ntions in thermal diffusivity and SAW velocity, which also occurred in the undeformed\ncase. The irradiation-induced changes are slightly smaller in magnitude in the HPT ma-\nterials compared to the undeformed but the effect is not significant.\nBy considering the microstructure and material properties of Eurofer-97, a multi-faceted\nview of the effect of SPD, and its subsequent impact on thermal stability and radiation resistance\nfor ferritic/martensitic steels, emerges.\nDeclarations\nFunding\nThe authors acknowledge use of characterisation facilities within the David Cockayne Cen-\ntre for Electron Microscopy, Department of Materials, University of Oxford, alongside finan-\ncial support provided by the Henry Royce Institute (grant ref EP/R010145/1). KS acknowl-\n35edges funding from the General Sir John Monash Foundation and the University of Oxford\nDepartment of Engineering Science. FH, DY and AR acknowledge funding from the European\nResearch Council (ERC) under the European Union’s Horizon 2020 research and innovation\nprogramme (grant agreement No. 714697). DEJA acknowledges funding from EPSRC grant\nEP/P001645/1. This material is based upon work done at Brookhaven National Laboratory,\nsupported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, under\nContract No. DE SC0012704.\nConflicts of interest\nThe authors have no relevant financial or non-financial interests to disclose.\nData and code availability\nAll data, raw and processed, as well as the processing and plotting scripts are available at: A\nlink will be provided after the review process and before publication.\nAcknowledgements\nThe authors are grateful for the assistance provided by DLS beamline scientists Stephen Thomp-\nson and Eamonn Connolly (I11), Robert Atwood and Stefan Michalik (I12). The authors are\nalso grateful for the assistance from Andy Bateman and Simon Hills (Department of Engineer-\ning Science, University of Oxford) for sample cutting and preparation.\n36Appendix A\nDuring the in-situ annealing experiment at DLS I12 beamline, the temperature was initially\nrecorded as the temperature read-out ( TL) from the Linkam heating stage device. However,\ndue to imperfections in the thermal contact between the heater and the sample, this read-out\nvalue may not be exactly the same as the actual temperature of the sample ( Ts). In order to find\nthe relationship between TLandTs, the lattice parameter of the sample (calculated from the\nmeasured diffraction data) and the coefficient of thermal expansion were used. For simplicity,\nthis relationship was approximated as linear and the calculations only consider the patterns up\nto the point of ferrite/martensite-to-austenite phase transition.\nFigure A-1: The lattice constant and corresponding temperature calculated and measured from\nthe experimental diffraction patterns, and the Linkam temperature read-out. The method for\nobtaining the calibrated temperature curve is explained in the accompanying text.\nConsider the undeformed Eurofer-97 sample as an example. Figure A-1, considering only\nthe left-hand-side axis for now, shows the lattice constant fitted from the experimental diffrac-\ntion patterns ( as). It also shows the lattice constant predicted from the Linkam temperature\n37read-out ( aL) as a function of time. aLat a certain temperature TLis given by:\naL(T=TL)=as(T=303 K)+α×as(T=303)×(TL−303)\nwhere α= 12 .74×10−6is the average coefficient of thermal expansion of Eurofer-97\nbetween 293 K to 1073 K [91]. It is assumed at 303 K, the start of the experiment, that TL=\nTs= 303 K and thus aL=as= 2.875 ˚A (fitted from experiment).\nSince the lattice constant is linearly dependent on temperature, the value of Tsis at a given\ntime can be calculated from the lattice constant by replacing aLandTLof the above equation\nand rearranging:\nTs=as(T=Ts)−as(T=303 K)\nα×as(T=303 K)+ 303\nwhich gives the corresponding right-hand-side y-axis of Figure A-1.\nA linear fit was obtained for TsandTL. Then a proportionality constant was obtained,\nassuming the linear fits for TsandTLintersected at 303 K, to correlate TsandTL(as shown in\nthe black line of Figure A-1).\n38Appendix B\nThe full comparison of TEM micrographs taken for each lift-out examined is included in Figures\nB-1 and B-2. Low-magnification images have been rotated such that they are aligned with the\ncoordinates indicated.\nFigure B-1: TEM images of the HPT sample at γ= 110 andγ= 230 with no irradiation at\ndifferent levels of magnification. Dislocation lines are evident within certain grains. There is\nno significant depth dependence of microstructural features.\n39Figure B-2: TEM images of the HPT sample at γ= 110 with the irradiated region indicated,\nand imaged at different levels of magnification. Dislocation lines are evident within certain\ngrains. There is no depth dependence of features and irradiation defects could not be unam-\nbiguously identified in this study due to the complexity of the microstructure.\nAppendix C\nSecond-phase regions were observed in TEM and TKD as discussed in the main text. Energy\ndispersive X-ray spectroscopy (EDX) was performed on a TEM lift-out of the deformed ( γ=\n230) and irradiated sample (Figure C-1). Those regions are found to be enriched with Cr (up to\n30%), W (up to 10%), Mn and C. This is consistent with the carbides found in Eurofer-97 by\nanother study [40].\n40Figure C-1: EDX maps of the carbide regions (circled in red) found between individual grains\nafter HPT.\nAppendix D\nThe effective outer cut-off radius of the dislocations ( Re) gives an indication of the range of\nelastic strain from the dislocations. The ‘randomness’ of the dislocation arrangement can be\ndescribed by cylinders of radius Rethat contain equal numbers of screw dislocations of positive\nand negative sign, and these cylinders fill the whole crystal [34]. Since Reactually depends on\nthe dislocation density ρ, the dimensionless parameter Mis defined to give an indication of the\ndipole characteristic of the dislocation. It is calculated by M=Re√ρ. When M≫1, the\ndislocations are distributed randomly and in an uncorrelated arrangement. Smaller M, on the\nother hand, indicates that dislocations of opposite sign are positioned close to each other [33].\nFrom CMWP fitting, reductions in the MandRevalues is seen following irradiation in the\nHPT-deformed sample (Figure D-1). This suggests that the strain fields are becoming shorter\nrange and possibly indicates the formation of dislocation loops [92]. To characterise the exact\n41Figure D-1: The values of the Mparameter (blue bars) and the effective outer cut-off radius of\nthe dislocations (red bars, Re) fitted from CMWP analysis.\ndensity of dislocation loops and line dislocations separately, the contrast factor Cof each type\nneeds to be known. This has been calculated for hexagonal materials [93] but remains to be\ninvestigated for cubic materials.\nAppendix E\nThe purpose of this section is to demonstrate that the peak broadening in the martensite and\naustenite phases post-annealing is due to the presence of the retained austenite phase. We will\nre-present the data from Figure 9 here, with more analysis, for a clearer comparison with another\nset of in-situ annealing XRD data.\nThe fitted peak widths as a function of time is shown alongside the raw data in Figure E-1.\nKey features of the raw data was discussed in Section 3.2.2 of the main text. The peak tempera-\nture reached for this set of annealed samples is 1153 K. The phase transition (ferrite/martensite\n42Figure E-1: a) The temperature ramp and raw time vs K= 2 sin θ/λplots of the b) undeformed\nand c) HPT sample (as discussed in Section 3.2.2 of the main text). The peak widths (FWHM)\nof the d) undeformed and e) HPT sample as a function of time during the annealing cycle. The\ncalibrated temperature profile is shown as the black dashed line. The ferrite/martensite phase\nwas fitted to a body-centred cubic (BCC) structure, and the austenite was fitted to a face-centred\ncubic (FCC) structure.\nto austenite) occurred from around 1025 K onwards. The peak widths of both samples just\nbefore the onset of the phase transition are similar. After the annealing and cooling cycle, some\naustenite phase remained in the sample as evidenced in the diffraction pattern. The peak widths\nof both the martensite and austenite phases showed significant broadening after the marten-\nsite phase reappeared around 606 K during the subsequent cooling stage (Figure E-1(d) and\n43(e)). The peak width of the sample after annealing is similar in both the undeformed and HPT\nsamples.\nFigure E-2: The temperature ramp and time vs. K= 2 sin θ/λ for the (a) undeformed and\n(b) HPT samples annealed to a lower target temperature compared to Figure E-1. (c)-(d) Their\nrespective peak widths (FWHM) as a function of time as well as the temperature profile (black\ndashed line). The ferrite/martensite phase was fitted to a body-centred cubic (BCC) structure,\nand the austenite was fitted to a face-centred cubic (FCC) structure.\nAnother set of samples was annealed for this study but not discussed in the main text as the\ntemperature ramp was not well-controlled (Figure E-2). Specifically, for the annealing of the\nundeformed sample, the peak temperature only reached 883 K due to poor thermal contact. The\npeak widths of the undeformed sample did not change significantly during the whole annealing\n44cycle and no phase transitions occurred. For the deformed sample, the peak annealing tempera-\nture was 1000 K, and it underwent a phase transformation starting at 977 K. The peak widths of\nthe ferrite/martensite phase right before the phase transformation occurred are comparable to the\npeak widths of the undeformed sample. Upon cooling the austenite phase disappeared around\n913 K. Due to the different heating rate of these samples compared to the samples presented in\nFigure E-1, it is not surprising that the phase transition occurred at a different temperature.\nThe key difference compared to the first set of annealed samples (Figure E-1) is that the\npeak widths following cooling remained small and there is no retained austenite phase (Figure\nE-2). The continuity of the Debye-Scherrer rings (included in the supplementary files) after\ncooling is reduced compared to the initial state. This suggests that some grain growth has taken\nplace to produce crystals that are dislocation- and strain-free, as expected. 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Wills Physics Laboratory, University of Bristol,\nTyndall Avenue, Bristol BS8 1TL, United Kingdom\n3Materials Theory, ETH Z¨ urich, Wolfgang-Pauli-Strasse 27 , 8093 Z¨ urich, Switzerland\n(Dated: January 14, 2013)\nWe assess the potential of the ferrimagnetic spinel ferrite s CoFe 2O4and NiFe 2O4to act as spin\nfiltering barriers in magnetic tunnel junctions. Our study i s based on the electronic structure\ncalculated by means of first-principles density functional theory within different approximations\nfor the exchange correlation energy. We show that, in agreem ent with previous calculations, the\ndensities of states suggest a lower tunneling barrier for mi nority spin electrons, and thus a negative\nspin-filtereffect. However, amoredetailed analysis basedo nthecomplexband-structurereveals that\nboth signs for the spin-filtering efficiency are possible, dep ending on the band alignment between\nthe electrode and the barrier materials and depending on the specific wave-function symmetry of\nthe relevant bands within the electrode.\nI. INTRODUCTION\nThe ability to generate and detect spin-polarized cur-\nrents is a central requirement for any practical spintron-\nics device. A promising approach to achieve this goal\nis to use tunnel junctions containing ferro- or ferrimag-\nnetic barrier materials, thus presenting different tunnel-\ning probabilities for majority (spin-up, ↑) and minority\n(spin-down, ↓) electrons. Efficient spin-filtering has been\ndemonstrated for ferromagnetic insulators such as EuS,1\nEuO,2and BiMnO 3.3However the magnetic ordering\ntemperatures of these magnets are rather low. There-\nfore the identification of suitable barrier materials that\noperateat roomtemperatureoraboveisofgreatinterest.\nSpinel ferrites are insulating ferrimagnets with high\nCurie temperatures ( TC=790 K for CoFe 2O4and 865 K\nforNiFe 2O4),4andthereforearepromisingcandidatesfor\nefficient room temperature spin-filtering. A measure of\nthe ability of a material or a device to select a particular\nspin direction is the spin-filtering efficiency, Psf, which is\ndefined as\nPsf=I↑−I↓\nI↑+I↓,\nwhereIσis the spin- σcomponent of the current, which\nis assumed to be carried by the two spin species in paral-\nlel. Recent experiments on ferrimagnetic spinels appear\npromising, as a spin-filtering efficiency of +22% has been\nmeasured for NiFe 2O4at low temperatures.5The mea-\nsured positive sign of Psfis in apparent contradiction\nwith results of band-structure calculations, demonstrat-\ning that the bottom of the conduction band is lower for\nspin-down electrons than for spin-up,6which would lead\nto a lower tunneling barrier for minority spin electrons.\nIt was suggested that this apparent discrepancy could\nbe due to effects related to the wave-function symmetry\nof the tunneling states.5Furthermore, for CoFe 2O4both\npositive and negative Psfhave been reported in junctions\nmade of different electrode materials and where Psfwas\nmeasuredwithdifferentexperimentaltechniques. There-ported values of Psfrange from −44% to +26%.7–12Due\nto these large variations in experimental results (with\nboth signs occurring for the spin-filtering efficiency) a\nconclusive picture of spin-filtering in spinel ferrites has\nnot emerged, yet. As such, a first-principles investigation\nof the spin-filtering efficiency in these materials is highly\ndesirable, in order to provide a reference for future ex-\nperimental studies and to allow further optimization of\nthe corresponding devices.\nSo far, theoretical predictions for the spin-filter effect\nin CoFe 2O4and NiFe 2O4are almost exclusively based\non density of states (DOS) calculations within a self-\ninteraction corrected (SIC) local spin-density approxi-\nmation (LSDA).6The spin-splitting of the conduction\nband minimum (CBM) in these calculations suggests a\nlower tunnel barrier for minority spin electrons and thus\na negative sign for the spin-filtering efficiency. However,\nit is well known that in many cases this simple density\nof states argument can be misleading, and the tunnel\nprobability can be strongly dependent on the specific\nwave-function symmetry.13The implications of this were\nfirst noticed in a Fe/MgO/Fe heterostructure,14,15where\nsymmetry-dependent tunneling results in half-metallic\nbehaviour of the Fe/MgO(001) stack. Since then, the\nso-called complex band-structure , which determines the\ndecay length of Bloch states with different wave-function\nsymmetries inside an insulating barrier, has been used\nto account for many, otherwise unexplained, experimen-\ntal results in spin-dependent tunnel junctions. Further-\nmore, it is of interest to compare the SIC-LSDA result of\nRef. 6 to the electronic structure obtained by using alter-\nnative approaches such as LSDA+ U, hybrid functionals,\nor other SIC approaches.\nHerewe presenta detailed comparisonofthe electronic\nstructure of CoFe 2O4and NiFe 2O4calculated within dif-\nferent approximations for the exchange-correlation po-\ntential. This allows us to identify features of the DOS\nthatarefairlyrobustwith respecttothe specific choiceof\nexchange-correlationpotential and features that are very\nsensitivetothischoice. Inaddition,wecalculatethecom-2\nplex band-structure for both materials within the atomic\nSIC method (ASIC)16,17, which facilitates the identifica-\ntion of suitable electrode materials that can lead to high\nspin-filtering efficiency. We show that, for both CoFe 2O4\nand NiFe 2O4and the two transport directions [001] and\n[111], electrons tunnel with the highest probability at the\ncenter of the two-dimensional Brillouin zone in the plane\northogonal to the transport direction. Furthermore, de-\npending on the exact alignment of the electrode Fermi\nlevel relative to the CBM of the barrier, the tunneling\ncurrent may present either a predominant majority or a\npredominant minority contribution, i.e. Psfmay change\nsign depending on the level alignment.\nThe paper is organized as follows. After having briefly\npresented the computational method and the details of\nthe crystallographic unit cell used for this study, we pro-\nceed to describe the electronic structure of CoFe 2O4and\nNiFe2O4. In particular, we first discuss the DOS and\nreal band-structures, and then move on to present the\ncomplex ones. The final section summarizes our main\nconclusions.\nII. METHODS\nWe employ the vasp18andsiesta19density functional\ntheory (DFT) code packages for the calculation of DOS\nand real band-structures and the smeagol code20,21to\ncalculate the complex band-structure. The vaspcal-\nculations have been performed by using the projector-\naugmented wave (PAW) method22with standard PAW\npotentials supplied with the vaspdistribution, a 500 eV\nplane wave energy cutoff, and a Γ centered 6 ×6×6k-\npoint mesh for the Brillouin zone sampling. We employ\nthegeneralizedgradientapproximation(GGA) according\nto the Perdew-Burke-Ernzerhof formulation23together\nwith the Hubbard “+ U” correction,24whereU= 3 eV\nandJ= 0 eV is applied to the dstates of all transition\nmetal cations, as well as the hybrid functional approach\naccording to Heyd, Scuseria and Ernzerhof (HSE),25us-\ning the standard choice for the fraction of Hartree-Fock\nexchange ( α= 0.25) and a reduced plane wave energy\ncutoff of 400eV. When using the localised basis set code\nsiesta, structural relaxations were performed using the\nGGA while the atomic self-interaction correction (ASIC)\nscheme was used to determine the electronic structure,\nincluding the complex band-structure. A 6 ×6×6k-point\nMonkhorst-Pack mesh was used to converge the density\nmatrix to a tolerance of 10−5and a grid spacing equiva-\nlent to a plane-wave cutoff of 800eV was used.\nFor most of our calculations we use the smallest possi-\nble unit cell (containing 2 formula units) to describe the\ninverse spinel structure. The corresponding distribution\nofcationsonthespinel Bsitelowersthespacegroupsym-\nmetry from Fd¯3mtoImma.26We also present some re-\nsults obtained for a cation distribution with P4122 sym-\nmetry, which requires a doubling of the unit cell to 4 for-\nmulaunits (the k-pointsamplingisthen adjustedaccord-TABLE I. Band gap (E g) and spin-splitting of the CBM\n(∆CBM) for CoFe 2O4and NiFe 2O4calculated with different\nexchange-correlation functionals. All values are in eV.\nCoFe2O4 NiFe2O4\nEg ∆CBM Eg ∆CBM\nGGA+U 0.52 0.92 0.83 0.86\nHSE 1.60 1.09 2.32 1.00\nASIC 1.08 1.00 2.07 0.46\ningly). We have previously shown that both Immaand\nP4122arelowenergyconfigurationsfor the inversespinel\nstructure in CoFe 2O4and NiFe 2O4, and that the specific\ncation arrangement has only a minor influence on the\nglobal electronic structure of these systems.27We note\nthatexperimentallyadisordereddistributionofFe3+and\nCo2+/Ni2+cations over the spinel Bsite with effective\ncubicFd¯3msymmetry, i.e. with no long-range cation\norder, is generally observed, even though recently indica-\ntions for short range cation order in both NiFe 2O4bulk\nandthinfilmsampleshavebeen reported.28,29Foramore\ndetailed comparison between the different cation config-\nurations see Refs. 27 and 30.\nStructural relaxations have been performed at the\nGGA level, with all cations being fixed to their ideal\ncubic positions.26The relaxed bulk lattice constants a0\nobtained by using vasp(siesta) are 8.366 ˚A (8.360 ˚A)\nand 8.346 ˚A (8.356 ˚A) for CoFe 2O4and NiFe 2O4, respec-\ntively, and are in very good agreement with experimental\ndata (see Ref. 27 and references therein).\nIII. RESULTS AND DISCUSSION\nA. Electronic structure\nIt has been previously shown that GGA leads to a\nhalf-metallic solution for CoFe 2O4and results in only\na very small insulating gap in the case of NiFe 2O4(see\ne.g. Refs. 26 and 31 and references therein). The DOS of\nCoFe2O4and NiFe 2O4calculated by using a selection of\nbeyond-GGAfunctionals aredepicted in Fig.1. It can be\nseen that all the studied exchange-correlation potentials\nlead to an insulating state for CoFe 2O4and an enhanced\nband gap for NiFe 2O4. When compared to the GGA+ U\nband gaps, both the inclusion of Hartree-Fock exchange\nwithin the HSE calculation as well as the ASIC treat-\nment leads to a large increase in the band gap values for\nboth the Co and Ni based ferrite, with the largest band\ngaps obtained for HSE (see Table I). We also note that\nour results are consistent with recent HSE and LSDA+ U\ncalculations for NiFe 2O4.32\nGoing into more details we notice that, while the oc-\ncupied DOS are very similar for GGA+ Uand HSE, the\nASIC methodplacesthe localFespin-majoritystatessig-\nnificantly lower in energy. This results in a gap between\ntheseFe statesandthe higher-lyingCo(Ni) dandoxygen3\n-5.00.05.0DOS [eV-1]Co (Oh)t2g\negGGA+UCoFe2O4\nHSE ASIC\nNi (Oh)t2g\negGGA+UNiFe2O4\nHSE\n-5.00.05.0\nDOS [eV-1]ASIC\n-5.00.05.0DOS [eV-1]Fe (Oh)t2g\negFe (Oh)t2g\neg\n-5.00.05.0\nDOS [eV-1]\n-10-505\nE [eV]-5.00.05.0DOS [eV-1]Fe (Td)e\nt2\n-10-505\nE [eV]-10-505\nE [eV]-10-505\nE [eV]Fe (Td)e\nt2\n-10-505\nE [eV]-10-505\nE [eV]-5.00.05.0\nDOS [eV-1]\nFIG. 1. (Color online) Total and projected DOS per formula un it for CoFe 2O4(left panels) and NiFe 2O4(right panels)\ncalculated with different exchange-correlation potential s (from left to right: GGA+ U, HSE and ASIC). The t2gandegstates of\nFe, Co, and Ni on the Ohsites and the eandt2states of Fe on the Tdsites are shown as black (blue) and dark grey (red) lines,\nrespectively. The shaded grey area in all panels depicts the total DOS. Minority spin projections are shown using negati ve\nvalues. The zero energy is set to the middle of the band gap.\npvalence bands. Interestingly, for CoFe 2O4the valence\nband maximum in ASIC is made up of the majority spin\nCoegstates, whereasforboth GGA+ UandHSEthe cor-\nresponding minority spin t2gstates are slightly higher in\nenergy. We also note that the difference in the calculated\nGGA+Uband gap of CoFe 2O4(NiFe2O4) compared to\nthe previously obtained values of 0.9 eV (0.97 eV) for the\nImmastructure,26and 1.24 eV (1.26 eV) for the P4122\nstructure27, is due to the fact that in the present work all\ncalculations are performed at the GGA volume, whereas\nthe calculations in Refs. 26 and 27 have been performed\nat the larger GGA+ Uoptimized volume. In addition\nto the expected dependence on the exchange correlation\npotential, ourresults thus indicate a strongvolume sensi-\ntivity in particular of the calculated CoFe 2O4band gap.\nExperimental estimates for the band gaps of spinel fer-\nrites are sparse and vary over a broad range comprised\nbetween 0.11eV and 1.5eV for CoFe 2O4and between\n0.3eV and 3.7eV for NiFe 2O4.33,34A recent optical ab-\nsorption study of NiFe 2O4suggests an indirect gap of\n1.6eV in the minority spin channel,32which thus repre-\nsents an upper bound for the corresponding fundamental\nband gap.\nIn all cases, and for both CoFe 2O4and NiFe 2O4, the\nCBM is lower in energy for the spin-down states than for\nspin-up ones, in agreement with the SIC-LSDA calcula-tions of Ref. 6. In the case of CoFe 2O4, all the three ap-\nproaches used in our work predict a spin-splitting of the\nCBM (∆CBM in Table I) of around 1 eV. For NiFe 2O4\nhowever,GGA+ UandHSEyielda∆CBMofaround0.9-\n1.0 eV, while ASIC gives a somewhat smaller splitting of\nonly 0.46eV. In all the cases, the obtained spin-splittings\nof the CBM are smaller than those reported in Ref. 6,\n1.28 eV (1.21 eV) for CoFe 2O4(NiFe2O4). We note,\nhowever, that even smaller values, namely of 0.47 eV\nfor both CoFe 2O4and NiFe 2O4, have been obtained in\nprevious GGA+ Ucalculations at the relaxed GGA+ U\nvolume.27Recent experiments estimate the spin-splitting\nof the CBM in the tens of meV range for CoFe 2O4-\ncontaining junctions.9\nInordertoshedfurtherlightonthenatureofthebands\naround the gap, the calculated GGA+ Uand ASIC band-\nstructures for both CoFe 2O4and NiFe 2O4are shown in\nFig. 2. Apart from the larger band gaps obtained by the\nASIC approach, it can be seen that the relative energies\nof the minority and majority spin bands in the upper va-\nlence band region for CoFe 2O4differ between GGA+ U\nand ASIC. This is consistent with our previous discus-\nsion of the DOS. For the calculation that is performed\nwith GGA+ U, the top of the valence band is formed\nby a minority spin band with maximum at the X point,\ni.e. the minority spin band gap is indirect. In contrast4\n-2-1012E [eV]\nL Γ X-2-1012E [eV]-2-1012\nL Γ X-2-1012(a)  CoFe2O4 - GGA+U (b)  CoFe2O4 - ASIC\n(d) NiFe2O4 - ASIC (c)  NiFe2O4 - GGA+U\nFIG.2. (Coloronline)Bandstructuresforenergies aroundt he\nband gap of CoFe 2O4[upper panels (a) and (b)] and NiFe 2O4\n[lower panels (c) and (d)] calculated by using the GGA+ U\nexchange-correlation functional [left panels (a) and (c)] and\nthe ASIC scheme [right panels (b) and (d)]. Majority and\nminorityspinbandsare shownas full (black)anddashed(red )\nlines.\na direct gap with mixed spin character at Γ is obtained\nby ASIC. Since, as we will show in the following, the\ntunneling probabilities are dominated by states around\nthe Γ point, we do not expect that this qualitative dif-\nference between the two exchange-correlationfunctionals\nwill critically affect the transport properties.\nBased on our analysis of the DOS and the band-\nstructure in the vicinity the gap, we can conclude that\ndespite some differences, all computational methods con-\nsistently predict a lower tunnel barrier for the minority\nspin electrons and therefore a negative spin-filtering ef-\nficiency for both CoFe 2O4and NiFe 2O4. However, as\nshown in Fe/MgO/Fetunnel junctions,14,15in the case of\nhigh quality epitaxial interfaces between the electrodes\nand the barrier material such DOS considerations are\nonly of limited value for the description of actual trans-\nport properties. Instead, the specific symmetry of the\ndecaying wave-functions inside the barrier has to be con-\nsidered. This can be achieved through calculation of the\ncomplex band-structure.13\nB. Complex band structure\nThe complex band-structure along a particular crys-\ntalline direction is calculated with the DFT non-\nequilibrium Green’s function code smeagol .20,21Thecomplex band-structure is nothing but the solution of\nthe secular band equation extended to imaginary wave-\nvectors. Let us assume that the transport direction of a\ngiven tunnel junction is along the zdirection and that\nthe material composing the barrier has a particular crys-\ntalline axis aligned along that direction. For any given\nk-vector in the transverse x-yplane,k/bardbl= (kx,ky), and\nfor any energy, E, the band equation E=E(kx,ky,kz)\ncan be solved for kzif one admits imaginary solutions\nkz=q+iκ. This means that the wave-function of an\nelectron approaching the tunneling barrier with trans-\nverse wave-vector k/bardblexponentially decays into the bar-\nrier along the zdirection over a length-scale given by\n1/κ. Clearly such decay rate depends on the transverse\nk-vector and the energy, i.e. κ=κ(kx,ky;E). Here we\nconsider the situation of electron transport along both\nthe [001] and [111] directions of the cubic spinel struc-\nture.\nIn Fig. 3 we plot the minimal value of κas a function\nofkxandky(calculated on a 100 ×100 grid) at differ-\nent energies within the gap. We include data for both\nCoFe2O4and NiFe 2O4considering both transport direc-\ntions for the Immaconfiguration, and we also present\ndata for the P4122 configuration and transport along the\n[001] direction. The crucial result emerging from Fig. 3\nis that in all cases κis smallest at the Γ point of the\ntwo-dimensional Brillouin zone corresponding to the x-y\nplane. This means that, due to the exponential depen-\ndence of the wave-function on κ, electron tunneling away\nfrom the Γ point will contribute very little to the trans-\nport. Assuch,intheanalysisthatwillfollow,wewillonly\nconsider transport through the Γ-point. We note that Γ-\npoint filtering is a highly desirable property for both tun-\nnel junctions and spin injection. As has been shown for\nthe Fe/MgO barrier, as the thickness of the MgO layer\nincreases so does the selectivity of the Γ-point. This in\nturn increases the tunneling magneto-resistance (TMR).\nAlthough the Γ-point filtering is not strictly necessary\nfor a large TMR, it significantly reduces the importance\nof the material choice for the electrodes.\nHaving established that the transport predominantly\noccursattheΓ-point, furtherinsightcanbegainedbyex-\nploringthe energydependence of κ(0,0;E). In particular\nit is important to establish the spin and orbital symme-\ntry of the complex bands corresponding to the smallest\nvalue of κ(0,0;E) for each energy, since incident waves\nwith that particular symmetry will dominate the tun-\nneling current. In Fig. 4 we present the complex band-\nstructuresof CoFe 2O4and NiFe 2O4, calculated alongthe\n[001] and [111] directions at the Γ-point in the transverse\n2D Brillouin zone for the Immaconfiguration. One can\neasily recognize that, for both CoFe 2O4and NiFe 2O4,\nthe main features which we discuss in the following are\nsimilar for the two different transport directions. We\nnote that the transport calculation along [111] requires\na larger unit cell, in order to obtain lattice vectors that\nare either perpendicular or parallel to the transport di-\nrection, which leads to a larger number of complex bands5\n(a)Imma-CoFe 2O4, transport along [001]\n(b)Imma-CoFe 2O4, transport along [111]\n(c)P4122-CoFe 2O4, transport along [001]\n(d)Imma-NiFe2O4, transport along [001]\n(e)Imma-NiFe2O4, transport along [111]\n(f)P4122-NiFe 2O4, transport along [001]\nFIG. 3. Minimal value of κat different energies (indicated at\nthe top left in each graph) within the gap for CoFe 2O4(a-c)\nand NiFe 2O4(d-f) along different transport directions, calcu-\nlated within the ASIC approach. Zero energy corresponds to\nthe middle of the band gap.00.20.4κ (Å-1)\n[001]NiFe2O4\n00.20.4κ (Å-1)\n[001]CoFe2O4-1.5-1-0.500.511.5200.20.4κ (Å-1)\n[111]\n-1.5-1-0.500.511.52\nE [eV]00.20.4κ (Å-1)\n[111]\nFIG. 4. The complex band-structure corresponding to kx=\nky= 0 for NiFe 2O4(upper two panels) and CoFe 2O4(lower\ntwo panels) along [001] and [111], calculated within ASIC\nfor theImmaionic configuration. The up and down arrows\nindicate the spin-character of the lowest lying complex ban ds.\nThe vertical dashed lines indicate the energies that were us ed\nfor thekx-kyplots in Fig. 3.\ncompared to the [001] case. In both materials the slow-\nest decay rate close to the valence band maximum cor-\nresponds to electrons with majority spin character (in\nagreement with the real band-structure shown in Fig. 2).\nThis remains the case for energies up to around 0.5 eV\nfrom the top ofthe valenceband, althoughthe decayrate\nincreases quickly with energy. In contrast, the lowest de-\ncay rate for energies taken in the upper part of the band\ngap is dominated by states with minority spin symme-\ntry. ForNiFe 2O4this decayrateremainsalmostconstant\nfor a wide energy window of about 1.5 eV, whereas for\nCoFe2O4the gap region is divided more symmetrically\nbetween the majority and minority spin-dominated re-\ngions. ThesmallerASICcalculatedbandgapofCoFe 2O4\ncomparedto that ofNiFe 2O4resultsin slightlyslowerde-\ncayswithinthegapregionforbothmajorityandminority\nspins.\nIn Fig. 5 we also present the complex band struc-\nture of CoFe 2O4and NiFe 2O4in theP4122 configura-6\n-1.5-1-0.500.511.5200.20.4κ (Å-1)NiFe2O4\n-1.5-1-0.500.511.52\nE [eV]00.20.4κ (Å-1)CoFe2O4\nFIG. 5. The complex band-structure corresponding to kx=\nky= 0 for NiFe 2O4(upper panel) and CoFe 2O4(lower panel)\nalong [001] for the P4122 configuration, calculated within\nASIC. The up and down arrows indicate the spin-character\nfor some of the lowest lying complex bands.\ntion for transport along [001]. One can recognize the\nslightly larger band-gap compared to the Immaconfig-\nuration, but for NiFe 2O4the complex bands look very\nsimilar compared to the Immacase. For CoFe 2O4one\ncan see that the bands in the mid-gap region connect in a\nsomewhat different way than in the Immaconfiguration.\nHowever, the spin-characters of the lowest complex band\nin the upper and lower gap region remain unaffected by\nthe different cation distribution, even though the energy\nrange dominated by the minority spin complex bands is\nsomewhat more extended in the P4122 case.\nFrom the complex bands it becomes clear that pos-\nitive as well as negative values for Psfare possible for\nboth NiFe 2O4and CoFe 2O4, depending on whether the\nFermi level of the electrode lies in the upper or lower gap\nregion ofthe spinel tunnel barrier, and on the availability\nof majority or minority spin carriers in the metal. If the\nFermi level of the metallic electrode lies within ∼0.5 eV\nfrom the top of the valence band, the slowest decay rate\nin both CoFe 2O4and NiFe 2O4will be for electrons with\nmajority spin. In contrast, if the Fermi level of the elec-\ntrode is more than 0.5 eV above the valence band edge\nof the spinel barrier, then the slowest decaying state is\nin the minority spin channel. The exact position of the\nFermi level of the metal depends on the band alignment\nbetween the two materials. Thus, the spin filter effi-\nciency of the spinel ferrite barrier will depend strongly\non the band alignment and eventually also on the orbital\nsymmetry of the electrode states at the Fermi level. In\naddition, a good lattice match is of course required, oth-\nerwise translationalsymmetry is broken in the transverseplane and the complex band-structure argument breaks\ndown. Here, the possibility to grow good quality films\nof CoFe 2O4and NiFe 2O4with either [001] or [111] ori-\nentation (see e.g. Refs 35, 36, and 37) opens up a wide\nrange of possible electrode materials. In fact, high qual-\nity epitaxial junctions of CoFe 2O4or NiFe 2O4with var-\nious electrode materials, such as La 2/3Sr1/3MnO3, Au,\nFe3O4, Nb-doped SrTiO 3, Pt, Co, Al, and SrRuO 3, have\nalready been fabricated.5,7–12\nSo far we have only discussed the spin character of\nthe complex bands, whereas it is well known from the\nFe/MgO/Fesystem,thattheorbitalcharacteroftherele-\nvantbandscanalsohaveacrucialinfluenceonthetunnel-\ning properties. The determination of orbital character of\nthe complex bands in the inverse spinel ferrites CoFe 2O4\nand NiFe 2O4is complicated by the different symmetries\nof the specific cation configurations used in the calcula-\ntions. For example, the lowest lying state above the gap\nat Γ inImma-NiFe2O4, i.e. the one which connects to\nthe complex band with minority spin character that has\nthe smallest extinction coefficient over a rather large en-\nergy region within the gap, transforms according to the\nfully symmetric irreducible representation Agof the cor-\nrespondingorthorhombicpointgroup mmm. Thismeans\nthat, assuming an electrode with cubic bulk symmetry,\nthis state can in principle couple to ∆ 1and ∆ 2/∆′\n2bands\nfor transportalong the [001]direction (whether ∆ 2or ∆′\n2\ndepends on how exactly the electrode is oriented with re-\nspect to the spinel structure), or to Λ 1and Λ 3for trans-\nport along the [111] direction. However, these consider-\nation hold only for the case with Immasymmetry and\nit is unclear how different cation arrangement, in partic-\nular a completely disordered cation distribution, would\nchange these symmetry-based selection rules. Generally,\nthe lower symmetry of the various cation arrangements\nleads to fewer symmetry restrictions regarding the possi-\nble coupling with electrode bands. Since a full symmetry\nanalysisofall combinationsthat can possiblyoccur is be-\nyond the scope of this paper, we restrict our analysis to\nthe spin character of the decaying states within the bar-\nrier, which was discussed in the preceeding paragraphs.\nIV. SUMMARY AND CONCLUSIONS\nIn summary, we have calculated the electronic struc-\nture of both NiFe 2O4and CoFe 2O4using different ap-\nproaches to evaluate the exchange-correlation potential.\nTheseinclude GGA, GGA+ U, HSEand ASIC. We found\nthat, while there are certain characteristic differences in\nthe predicted band-structure, the densities ofstates ofall\nbeyond-GGA methods consistently suggest a lower tun-\nnel barrier for minority spin electrons. Due to the well-\nknown limitations of this simple density of states picture\noftunneling, wehavefurtheranalyzedthecomplexbands\nof the two materials at the ASIC level.\nWe have shown that the tunneling along the [001]\nand [111] directions is dominated by zone-center con-7\ntributions ( kx=ky= 0), and that for both NiFe 2O4\nand CoFe 2O4the spin character of the slowest decaying\nstate changes within the gap. Therefore, NiFe 2O4and\nCoFe2O4are both capable of acting as either positive or\nnegative spin filters, depending on the band alignment\nand wave-function symmetry of the electrodes. Given\nsuch a relatively sensitive dependence of the tunneling\ncurrent on the position of the electrode Fermi level, we\nenvision that gating may allow the spin filtering to be\nswitched from positive to negative.\nHowever, we also want to note that based on the com-\nplex band-structure of the barrier alone, it is not pos-\nsible to make a definite prediction about the transport\nproperties observed in a specific experiment. One may\nstill encounter a situation where incident wave-functions\nwith the desired symmetry, i.e. matching that of the\nsmallest κ(0,0;E) inside the barrier, are not available\nwithin the electrodes, simply because of the correspond-\ning real band-structure38,39. Furthermore, it has been\ndemonstrated recently for the case of an Fe-MgAl 2O4-Fe\ntunnel junction, i.e. containing a non-magnetic spinel\nas barrier material, that the different unit cell sizes of\nthe spinel barrier and the Fe electrodes can open up new\ntransport channels due to “backfolding” of bands from\nthe in-plane Brillouin zone boundary onto the Γ point.40This leads to a relatively low tunnel magneto-resistance\nfor the Fe-MgAl 2O4-Fe junction, even though the corre-\nsponding complex and real band structures would indi-\ncate a highly symmetry-selective barrier.40,41Therefore,\nin order to fully assess the spin-filter efficiency for a spe-\ncific combination of electrode and barrier materials, a\nfull transport calculation for the entire device needs to\nbe performed. Nevertheless, the analysis of the complex\nband-structure provides a powerful interpretative tool\nand offers a good indication on what are the dominant\ncontributions to the tunneling current. In the present\ncase, it allows the rationalization of both signs of the\nspin-filter efficiency occurring in NiFe 2O4and CoFe 2O4\ntunnel junction, depending on the band alignment with\nthe electrode.\nACKNOWLEDGMENTS\nThis work has been supported by Science Foun-\ndation Ireland under (Grants SFI-07/YI2/I1051 and\n07/IN.1/I945) and by the EU-FP7 (iFOX project).\nWe made use of computational facilities provided by\nthe Trinity Centre for High Performance Computing\n(TCHPC) and the Irish Centre for High-End Comput-\ning (ICHEC).\n∗caffreyn@tcd.ie; Current address: Institute of Theoretica l\nPhysics and Astrophysics, Christian-Albrechts-Universi t¨ at\nzu Kiel, Leibnizstrasse 15, 24098 Kiel, Germany\n†claude.ederer@mat.ethz.ch\n1J. S. Moodera, X. Hao, G. A. Gibson, and R. Meservey,\nPhys. Rev. Lett. 61, 637 (1988).\n2T. S. Santos and J. S. Moodera, Phys. Rev. B 69, 241203\n(2004).\n3M. Gajek, M. Bibes, A. Barth´ el´ emy, K. Bouzehouane,\nS. Fusil, M. Varela, J. Fontcuberta, and A. 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Lett.\n100, 222401 (2012)."    },    {        "title": "1310.4703v1.Insights_into_the_Phase_Diagram_of_Bismuth_Ferrite_from_Quasi_Harmonic_Free_Energy_Calculations.pdf",        "content": "arXiv:1310.4703v1  [cond-mat.mtrl-sci]  17 Oct 2013Insights into the Phase Diagram of Bismuth Ferrite from Quas i-Harmonic Free\nEnergy Calculations\nClaudio Cazorla and Jorge ´I˜ niguez\nInstitut de Ci` encia de Materials de Barcelona (ICMAB-CSIC ), Campus UAB, 08193 Bellaterra, Spain\nWe have used first-principles methods to investigate the pha se diagram of multiferroic bismuth\nferrite (BiFeO 3or BFO), revealing the energetic and vibrational features t hat control the occurrence\nof various relevant structures. More precisely, we have stu died the relative stability of four low-\nenergy BFO polymorphs by computing their free energies with in the quasi-harmonic approximation,\nintroducing a practical scheme that allows us to account for the main effects of spin disorder.\nAs expected, we find that the ferroelectric ground state of th e material (with R3cspace group)\ntransforms into an orthorhombic paraelectric phase ( Pnma ) upon heating. We show that this\ntransition is not significantly affected by magnetic disorde r, and that the occurrence of the Pnma\nstructure relies on its being vibrationally (although not e lastically) softer than the R3cphase.\nWe also investigate a representative member of the family of nano-twinned polymorphs recently\npredicted for BFO [Prosandeev et al., Adv. Funct. Mater. 23, 234 (2013)] and discuss their\npossible stabilization at the boundaries separating the R3candPnma regions in the corresponding\npressure-temperature phase diagram. Finally, we elucidat e the intriguing case of the so-called super-\ntetragonal phases of BFO: Our results explain why such structures have n ever been observed in the\nbulk material, despite their being stable polymorphs of ver y low energy. Quantitative comparison\nwith experiment is provided whenever possible, and the rela tive importance of various physical\neffects (zero-point motion, spin fluctuations, thermal expa nsion) and technical features (employed\nexchange-correlation energy density functional) is discu ssed. Our work attests the validity and\nusefulness of the quasi-harmonic scheme to investigate the phase diagram of this complex oxide,\nand prospective applications are discussed.\nPACS numbers: 77.84.-s, 75.85.+t, 71.15.Mb, 61.50.Ah\nI. INTRODUCTION\nMagnetoelectric multiferroics, a class of materials in\nwhich ferroelectric and (anti-)ferromagnetic orders coex-\nist, are generating a flurry of interest because of their\nfundamental complexity and potential for applications\nin electronics and data-storage devices, among others.\nIn particular, the magnetoelectric coupling between their\nmagnetic and electric degrees of freedom opens the pos-\nsibility for the control of the magnetization viathe appli-\ncationofabiasvoltagein advancedspintronicdevices.1–7\nPerovskite oxide bismuth ferrite (BiFeO 3or BFO) is\nthe archetypal single-phase multiferroic compound. This\nmaterial possesses unusually high anti-ferromagnetic\nN` eel and ferroelectric Curie temperatures ( TN∼650 K\nandTC∼1100 K, respectively8–10) and, remarkably,\nroom-temperaturemagnetoelectric coupling has been ex-\nperimentally demonstrated in BFO thin films and sin-\ngle crystals.11–13Under ambient conditions, bulk BFO\nhas a rhombohedrally distorted structure with the R3c\nspace group [see Fig. 1(a)]; such a structure can be de-\nrived from the standard cubic ABO3perovskite phase\nby simultaneously condensing ( i) a polar cation displace-\nment accompanied by an unit-cell elongation along the\n[111] pseudo-cubic direction, and ( ii) anti-phase rota-\ntions of neighboring oxygen octahedra about the same\naxis (this is the rotation pattern labeled by a−a−a−\nin Glazer’s notation14). The basic magnetic structure\nis anti-ferromagnetic G-type (G-AFM), so that first-\nnearest-neighboring iron spins are anti-aligned; superim-posed tothis G-AFM arrangement, in bulk samplesthere\nis an incommensurate cycloidal modulation.\nInterestingly, in spite of the extensive studies per-\nformed, there are still a few controversial aspects con-\ncerning the pressure-temperature ( p−T) phase diagram\nof BFO. Under ambient pressure BFO transforms from\ntheR3cphase to a paramagnetic β-phase at the Curie\ntemperature TC∼1100 K; upon a further temperature\nincrease of about 100 K, the compound transforms to\na cubicγ-phase that rapidly decomposes and melts at\nabout 1250 K. The exact symmetry of the paramagnetic\nβ-phase has been contentious for some time. Based on\nRaman measurements, Haumont et al.15suggested that\nthis was a cubic Pm3mstructure; however, subsequent\nthermal, spectroscopic, and diffraction studies by Palai\net al.16indexed it as orthorhombic P2mm. Next, Ko-\nrnevet al.17predicted the appearance of a tetragonal\nI4/mcmphase just above TCusing first-principles-based\natomistic models. However, further analysis and exper-\nimental XRPF measurements suggested that this phase\nis actually monoclinic P21/m.18Lastly, Arnold et al.19\nperformed detailed neutron diffraction investigationsand\narrived at the conclusion that the paramagnetic β-phase\nhas the orthorhombic Pnmastructure that is character-\nistic of GdFeO 3[a−a−c+rotation pattern in Glazer’s no-\ntation; see Fig.1(b)].20\nThe pressure-driven sequence of transitions that BFO\npresents at room temperature is not fully understood\neither. The first-principles study of Ravindran et al.\npredicted a pressure-induced structural transition of the2\nR3c→Pnmatype to occur at p∼13 GPa.21How-\never,alatersynchrotrondiffractionandfar-infraredspec-\ntroscopy study has suggested that BFO undergoes two\nphase transitions below 10 GPa: the first one at 3 .5 GPa\nfrom the rhombohedral R3cto a monoclinic C2/mstruc-\nture, and the second one at 10 GPa to an orthorhombic\nPnmaphase.22Most recently, Guennou et al.23reported\nX-raydiffractionandRamanmeasurementsshowingthat\nin the range between 4 GPa and 11 GPa (i.e., between\nthe stability regions of the R3candPnmaphases) there\nare three, as opposed to one, different stable structures\nof BFO. The authors describe such phases as possessing\nlarge unit cells and complex patterns of O 6-octahedra\nrotations and Bi-cation displacements.\nInterestingly, BFO’s phase diagram was recently re-\nexamined theoretically by Prosandeev et al.24, employ-\ning an atomistic model that captures correctly the first-\nprinciples prediction25that the R3candPnmastruc-\ntures are local energy minima. These authors found\nthat, at ambient pressure, the Pnmaphase is stable\nat high temperatures, while the R3cstructure is the\nground state. Additionally, they predicted an intermedi-\nateorthorhombicphasepresentingacomplexoctahedral-\ntilting pattern that can be seen as a bridgebetween the\na−a−a−anda−a−c+cases, with the sequence of O 6\nrotations along one direction displaying a longer repe-\ntition period. In fact, Prosandeev et al. found that\nthere is a whole family of metastable phases that are\ncompetitive in this temperature range and whose rota-\ntion pattern can be denoted as a−a−cq,26whereqis a\ngeneral wave vector characterizing the non-trivial mod-\nulation of the O 6tilts about the third axis. Figure 1(c)\nshows one such phase whose corresponding q-vector is\n2π/a(1/2,1/2,1/4), where ais the pseudo-cubic lattice\nconstant. There are reasons to believe that such com-\nplex phases can also appear under high- pconditions or\nupon appropriate chemical substitutions;24further, they\nseemtobethe keytounderstandthelowest-energystruc-\ntures predicted for the ferroelectric domain walls of this\nmaterial.27\nFinally, another family of novel phases was recently\ndiscovered in strongly-compressed BFO thin films.28,29\nThese so-called super-tetragonal structures can display\naspect ratios c/aapproaching 1.30, and are markedly\ndifferent from the BFO phases mentioned above. Var-\nious theoretical works have found that many of them\ncan occur,25,30all being metastable energyminima of the\nmaterial.25From the collection of structures reported by\nDi´ eguezet al.,25a monoclinic Ccphasewith a canted po-\nlarization of about 1.5 C/m2and anti-ferromagnetic or-\nder ofC-type (i.e., in-plane neighboring spins anti-align\nand out-of-plane neighboring spins align) emerges as a\nparticularly intriguing case [see Fig. 1(d)]. At T= 0 K\nthis monoclinic phase turns out to be energetically very\ncompetitive with the paraelectric Pnmastructure [see\nFig. 1(b)] that we believe becomes stable at high temper-\natures and high pressures. However, to the best of our\nknowledge, this super-tetragonal phase has never beenobserved in BFO bulk samples suggesting that both tem-\nperatureand pressuretendto destabilizeitin favorofthe\nPnmastructure.\nOne would like to use accuratefirst-principles methods\nto better understand what controls the relative stability\nof the different phases of BFO, and thus what determines\nits complex and still debated phase diagram. However,\na direct first-principles simulation of such a complex ma-\nterial at finite temperatures is computationally very de-\nmanding, and not yet feasible. Within the community\nworking on ferroelectrics like BaTiO 3, PbTiO 3and re-\nlated compounds, such a difficulty has been overcome\nby introducing mathematically simple effective models,\nwith parameterscomputed fromfirstprinciples, that per-\nmit statistical simulations and, thus, the investigation\nofT-driven phenomena.31–36In particular, as mentioned\nabove, the so-called effective-Hamiltonian approach has\nbeen alsoapplied to BFO,17and much efforthas been de-\nvoted to the construction of reliable models capturing its\nstructuraland magneticcomplexity.24,37Yet, asfaras we\nknow, we still do not have models capable of describing\nall the relevant BFO structures mentioned above. Fur-\nther, BFO has proved to be much more challenging than\nBaTiO 3or PbTiO 3for model-potential work; thus, a di-\nrect and accurate first-principles treatment is highly de-\nsirable.\nFortunately,BFOpresentsapeculiarfeaturethatenor-\nmously simplifies the investigation of its phase diagram.\nUnlike the usual ferroelectric materials, whose transi-\ntions are typically driven by the condensation of a soft\nphonon mode, BFO presents strongly first-order recon-\nstructive transformations between phases that are ro-\nbustly metastable. This makes it possible to apply to\nBFO tools that are well-known for the analysis of solid-\nsolid phase transitions in other research fields,38–44and\nwhich are based on the calculation of the free energy of\nthe individual phases as a function of temperature, pres-\nsure, etc. Thesimplestofsuchtechniques, which requires\nrelatively affordable first-principles simulations, is based\non aquasi-harmonic approximation to the calculation of\nthe free energy (QHF method in the following). This is\nthe scheme adopted in this work to investigate BFO’s\np−Tphase diagram.\nWe should stress, though, that application of the QHF\nscheme to BFO is not completely straightforward. In-\ndeed, the spin and vibrational degrees of freedom in\nmultiferroic materials can be expected to couple signif-\nicantly (i.e., spin-phonon coupling effects become non-\nnegligible,45–49see Fig. 2) implying that the free energies\nof ferromagnetic, anti-ferromagnetic, and paramagnetic\nphases belonging to a same crystal structure may differ\nsignificantly. The situation becomes especially compli-\ncated whenever we have structural transitions involving\nparamagnetic phases, as capturing the effect of disor-\ndered spin arrangements would in principle require the\nuse of very large simulation boxes.50–52In this work we\nhave introduced and applied an approximate scheme to\ncircumvent such a difficulty.3\nFIG. 1: (Color online) Sketch of the four crystal structures considered in this work as seen from two perpendicular direc tions.\nBi, Fe, and O atoms are represented with purple, brown, and re d spheres, respectively. Unit cells are depicted with thick solid\nlines and the O 6octahedra and O 5pyramids appear shadowed.\nTherefore, here we present a QHF investigation of the\np−Tphase diagram of BFO, monitoring the relative sta-\nbility of the four representative phases shown in Fig. 1:\nthe rhombohedral ground state (“ Rphase” with R3c\nspace group), the orthorhombicstructure that gets stabi-\nlizedathightemperaturesandpressures(“ Ophase”with\nPnmaspace group), a phase that is representative of the\nrecently predicted nano-twinned structures displaying\ncomplex O 6-rotation patterns (complex or “ Cphase”),\nand the most stable of the super-tetragonal polymorphs\nthat have been predicted to occur in strongly-compressed\nthin films (“ Tphase” with Ccspace group). Our calcu-\nlations take into account the fluctuations of spin ordering\nin an approximate way and reveal the subtle effects that\ncontrol the occurrence (or suppression) of all these struc-\ntures in BFO’s phase diagram.\nThe organization of this article is as follows. In Sec-\ntion II we provide the technical details of our energy and\nphonon calculations, and briefly review the fundamen-\ntals of the QFH approach. We also explain the strat-\negy that we have followed to effectively cast spin-phonon\ncoupling effects into QHF expressions. In Section III we\npresent and discuss our results. Finally, in Section IV we\nconclude the article by reviewing our main findings andcommenting on prospective work.\nII. METHODS\nA. First-principles methods\nIn most of our calculations we used the general-\nized gradient approximation to density functional the-\nory (DFT) proposed by Perdew, Burke, and Ernzerhof\n(GGA-PBE),53as implemented in the VASP package.54\nWe workedwith GGA-PBE becausethis is the DFT vari-\nant that renders a more accurate description of the rela-\ntive stability of the RandOphases of BFO, as discussed\nin Ref. 25. A “Hubbard-U” scheme with U= 4 eV was\nemployed for a better treatment of Fe’s 3 delectrons. We\nused the “projector augmented wave” method to repre-\nsent the ionic cores,55considering the following electrons\nas valence states: Fe’s 3 s, 3p, 3d, and 4s; Bi’s 5d, 6s,\nand 6p; and O’s 2 sand 2p. Wave functions were repre-\nsented in a plane-wave basis truncated at 500 eV, and\neach crystal structure was studied on its corresponding\nunit cell (see Fig. 1). For integrations within the Bril-\nlouin zone (BZ), we employed Γ-centered k-point grids4\nwhose densities were approximately equivalent to that of\na 10×10×10 mesh for the ideal cubic perovskite 5-\natom cell (e.g., 8 ×8×8 in the Rphase with Z= 2,\nand 6×6×6 in the Ophase with Z= 4). Using these\nparameters we obtained energies that were converged to\nwithin 0.5 meV per formula unit (f.u.). Geometry relax-\nations were performed using a conjugate-gradient algo-\nrithm that kept the volume of the unit cell fixed while\npermitting variationsof its shape, and the imposed toler-\nance on the atomic forces was 0 .01 eV·˚A−1. Equilibrium\nvolumes were subsequently determined by fitting the sets\nof calculated energy points to equations of state. Tech-\nnical details of our phonon calculations are provided in\nSecs. IID and IIE.\nB. Quasi-harmonic Free Energy Approach\nIn the quasi-harmonic approach, one assumes that the\npotential energy of the crystal can be captured by a\nquadratic expansion around the equilibrium configura-\ntion of the atoms, so that\nEharm=Eeq+1\n2/summationdisplay\nlκα,l′κ′α′Φlκα,l′κ′α′ulκαul′κ′α′,(1)\nwhereEeqisthe totalenergyoftheundistorted lattice, Φ\nthe force-constant matrix, and ulκαis the displacement\nalong Cartesian direction αof the atom κat lattice site\nl. In the usual way, we tackle the associated dynamical\nproblem by introducing\nulκα(t) =/summationdisplay\nquqκαexp[i(ωt−q·(l+τκ)],(2)\nwhereqis a wave vector in the first Brillouin zone (BZ)\ndefined by the equilibrium unit cell; l+τκis the vector\nthat locates the atom κat celllin the equilibrium struc-\nture. Then, the normalmodes arefound bydiagonalizing\nthe dynamical matrix\nDq;κα,κ′α′=\n1√mκmκ′/summationdisplay\nl′Φ0κα,l′κ′α′exp[iq·(τκ−l′−τκ′)],(3)\nand thus treat the material as a collection of non-\ninteracting harmonic oscillators with frequencies ωqs\n(positively defined and non-zero) and energy levels\nEn\nqs=/parenleftbigg1\n2+n/parenrightbigg\nωqs, (4)\nwhere 0 ≤n <∞. Within this approximation, the\nHelmholtz free energy at volume Vand temperature T\nis given by\nFharm(V,T) =1\nNqkBT/summationdisplay\nqsln/bracketleftbigg\n2sinh/parenleftbigg/planckover2pi1ωqs(V)\n2kBT/parenrightbigg/bracketrightbigg\n,\n(5)whereNqis the total number of wave vectors used in our\nBZ integration and the dependence of frequencies ωqson\nvolume is indicated. Finally, the total Helmholtz free\nenergy of the crystal can be written as\nFqh(V,T) =Eeq(V)+Fharm(V,T).(6)\nWe note that the greater contributions to Fharmcome\nfrom the lowest-frequency modes. This implies that,\nwhen analyzing the thermodynamic stability of different\ncrystal structures, those which are vibrationally softer\nin average will benefit more from the dynamical term in\nFqh.\nFinally, let us analyze the form that Fharmadopts in\nthe limits of low and high temperatures. In the first case,\none obtains\nFharm(V,T→0) =1\nNq/summationdisplay\nqs1\n2/planckover2pi1ωqs; (7)\nthis result is usually referred to as the zero-point energy\n(ZPE). As we will see in Sec. III, ZPE corrections may\nturn out to be decisive in the prediction of accurate tran-\nsitionpressuresinvolvingtwocrystalstructureswithsim-\nilar static energies. In the second limiting case, usually\ntermed as the classical limit (i.e., for /planckover2pi1ωqs≪kBT), one\narrives at the expression\nFharm(V,T→ ∞) = 3NuckBTln/bracketleftbigg/planckover2pi1¯ω\nkBT/bracketrightbigg\n.(8)\nHere,Nucis the number of atoms in the unit cell, and ¯ ω\nis the geometric average frequency defined as\n¯ω= exp(/an}bracketle{tlnω/an}bracketri}ht), (9)\nwhere/an}bracketle{t···/an}bracketri}htis the arithmetic mean performed over wave-\nvectorsqand phonon branches s. It is worth noting that\nlow-frequency modes are the ones contributing the most\nto ¯ω, and therefore to Fharm. As it will be shown in the\nnext section, Eq. (8) allows us to obtain compact and\nphysically insightful expressions for Fharmin which spin-\nphonon coupling effects are effectively accounted for.\nC. Spin-Phonon Couplings\nWe would like to identify a practical scheme to incor-\nporate the main effects of the spin fluctuations on the\ncalculation of QH Helmholtz free energies. To introduce\nour approach, let us begin by considering the following\ngeneral expression for the energy of the material, which\nis the generalization of Eq. (1) to the case of a compound\nwith localized magnetic moments whose interactions are\nwell captured by a Heisenberg Hamiltonian:\nEharm({um},{Si}) =E0+1\n2/summationdisplay\nmnΦ0\nmnumun\n+1\n2/summationdisplay\nijJij({um})SiSj,(10)5\n−25−20−15−10−5 0 5 10 15 20 25ωFM − ωG (cm−1) \n Γ L (q,q,q) Γ L (q,q,q)\nFIG. 2: Phonon frequency shifts among the ferromagnetic\n(FM) and G-type anti-ferromagnetic spin arrangements of\nBFO in the Rphase, calculated along one representative di-\nrection in the first BZ. Corresponding ωqsfrequency pairs\nwere identified by comparing the FM and G-AFM phonon\neigenmodes.\nwhere the Sivariables represent the magnetic mo-\nments associated with specific atoms and the Jij’s are\nthe distortion-dependent exchange interactions coupling\nthem. (For brevity, in the following we will talk about\nspinsinstead of magnetic moments ; nonetheless, note\nthat our arguments can be applied to cases involving or-\nbital magnetization.) To simplify the notation, we use\ncomplex indexes – mandnfor the atomic displacements\nandiandjforthe spins– that include informationabout\nthe cell, atom, and Cartesian component defining the\nstructural and magnetic variables. Finally, we write the\ndependence of the exchange constants on the atomic dis-\nplacements as:\nJij({um}) =J(0)\nij+/summationdisplay\nmJ(1)\nijmum+1\n2/summationdisplay\nmnJ(2)\nijmnumun,(11)\nwhere, for our purposes, it is sufficient to truncate the\nseries at the harmonic level. The J(0)\nijparameters de-\nscribe the magnetic interactions when the atoms remain\nattheirequilibriumpositions; typically, theseparameters\nwillcapturethebulkoftheexchangecouplings. The J(1)\nijm\ncoefficients describe the forces that may appear on the\natoms when we have certain spin arrangements, and the\nJ(2)\nijmnparameters capture the dependence of the phonon\nspectrum on the spin configuration.\nItisinterestingtonotethat, whiletheenergyinEq.(1)\ncanbeunambiguouslydescribedasaharmonicexpansion\naround an equilibrium state of the material, the inter-\npretation of Eq. (10) is much more subtle. Indeed, be-\ncause we work with spin variables that have a fixed norm\n(nominally, |Si|= 5µBin the case of the Fe3+cations in\nBiFeO 3), the reference structure of our spin-phonon QH\nenergy is defined formally as one in which the atomic\nspins are perfectly disordered and the atoms are locatedat the correspondingequilibrium positions. Such a struc-\nture cannot be easily considered in a first-principles cal-\nculation; hence, wehavetoobtainthe parameters E0and\nΦ0\nmnthat characterize it in an indirect way. In essence,\nthe fitting procedure would involve many different spin\nconfigurations, and parameters E0and Φ0\nmnwould cap-\nture the part of the energy and force-constant matrix\nthat isindependent of the spin order. Further, a thor-\nough calculation of the Jij({um}) constants would be a\nvery challenging task. Indeed, a detailed modeling of the\nspin-phonon couplings would require us to choose which\nspin pairs iandjare affected by which distortions pairs\nmandn, a problem that quickly grows in complexity\neven if we restrict ourselves to spin interactions between\nfirst nearest neighbors.\nIn this work we did not attempt to pursue such a de-\ntailed description, but adopted instead an approximate\napproach that provides the correct results in particular\nimportant cases. To illustrate our scheme, let us think of\nBFO’sRphase and consider two specific spin arrange-\nments that are obviously relevant: (1) the G-AFM struc-\nture (which is the ground state of the R,O, andCBFO\nphases mentioned above) and (2) a perfectly ferromag-\nnetic (FM) arrangement, which is the exactopposite case\nto G-AFM in the sense that all the interactions between\nfirst nearest-neighboring spins are reversed. Let us also\nrestrict ourselves to spin-spin interactions between first\nnearestneighborsand, forthe sakeofsimplicity, letusas-\nsume that all first-nearest-neighboring spins are coupled\nby the same J, so that we can drop the iandjindexes.\n(Thisisactuallythecaseforthe RphaseofBFO,andthe\ngeneralization to other lower-symmetry cases is straight-\nforward.) Then, for a given spin arrangement γ(where\nγcan be G-AFM or FM in this example), we can relax\nthe atomic structure of the material and construct the\nfollowing energy Eγ\nharm\nEγ\nharm({um}) =Eγ\neq+1\n2/summationdisplay\nmnΦγ\nmnumun,(12)\nwhich is the analogous of Eq. (1) above. Hence, we have\nstraightforward access to all the parameters in this ex-\npression from first principles. Now, we want our general\nspin-phonon energy in Eq. (10) to reproduce Eγ\nharmfor\ntheγ-orders of interest. If we are dealing with the G-\nAFM and FM cases, it is trivial to check that this can be\nachieved by making the following choices:\nE0=1\n2/parenleftbig\nEFM\neq+EG\neq/parenrightbig\n, (13)\nΦ0\nmn=1\n2/parenleftbig\nΦFM\nmn+ΦG\nmn/parenrightbig\n, (14)\nJ(0)=1\n6|S|2/parenleftbig\nEFM\neq−EG\neq/parenrightbig\n, (15)\nJ(1)\nm= 0, (16)\nJ(2)\nmn=1\n6|S|2/parenleftbig\nΦFM\nmn−ΦG\nmn/parenrightbig\n. (17)\nWhile these choices may seem very natural, there are6\nsubtle approximations and simplifications hiding behind\nthem. For example, the resulting model contains no ex-\nplicit information about the atomic rearrangements that\nmay accompany a particular spin configuration; never-\ntheless, the energies of the equilibrium structures are\nperfectly well reproduced for the G-AFM and FM cases.\nAnalogously, while the phonons of the G-AFM and FM\ncases will be exactly reproduced by this model, the spin-\nphonon interactions have been drastically simplified, and\nwe retain no information on how specific atomic motions\naffect specific exchange constants. Hence, the resulting\nmodel should not be viewed as an atomistic one; rather,\nit is closer to a phenomenological approach in which we\nretain information about the effect of magnetic order on\nthe whole phonon spectrum.\nFinally, we wouldlike to useour spin-phononenergyto\ninvestigate the properties of BFO at finite temperatures,\nespecially in situations in which the material is either\nparamagnetic (PM) or does not have a fully developed\nAFM order. To do so, we will assume that, for the caseof\nfluctuating spins, the energetics of the atomic distortions\nis approximately given by:\n˜Eharm({um};x) =E0+3x|S|2J(0)\n+1\n2/summationdisplay\nmn/parenleftBig\nΦ0\nmn+6x|S|2J(2)\nmn/parenrightBig\numun,\n(18)\nwherex=/an}bracketle{tSiSj/an}bracketri}ht/|S|2isthecorrelationfunctionbetween\ntwo neighboring spins, with /an}bracketle{t.../an}bracketri}htindicating a thermal av-\nerage. Note that in the limiting FM ( /an}bracketle{tSiSj/an}bracketri}ht=|S|2)\nand G-AFM ( /an}bracketle{tSiSj/an}bracketri}ht=−|S|2) cases, this equation re-\nduces to the expected EFM\nharmandEG\nharmenergies. Note\nalso that this model includes a spin-phonon contribu-\ntion to the energy even in the paramagnetic phase, as\nlong as there are significant correlations between neigh-\nboring spins. Indeed, for a non-zero value of x, the\nphonon spectrum is given by the force-constant matrix\nΞ(x)≡Φ0+6x|S|2J(2).\nIn this work, we evaluate xas a function of tempera-\nture by running Monte Carlo simulations of the Heisen-\nberg spin system described by the J(0)\nijcoupling con-\nstants, thus assuming frozen atomic distortions. Then,\nsince for a certain value of xEq. (18) is formally analo-\ngoustoEq.(1), wecanapplytheQHtreatmentdescribed\nabove to estimate the Helmholtz free energy ˜Fharmof the\ncoupled spin-phonon system.\nBefore concluding this section, let us discuss some ap-\nproximateexpressionsthatcanbeobtainedfor ˜Fharmand\nwhich are illustrative of how our approach captures the\neffectofspinfluctuationsandofthepeculiarnatureofthe\nparamagnetic state. We have usually observed that the\nnormal-mode frequencies ωγ\nqs, obtained by diagonalizing\nthe dynamical matrix associated to Φγ, depend signifi-\ncantly on the magnetic order. However, the correspond-\ning eigenvectors are largely independent from γ. As aresult we have the following approximate relations\n˜ωqs≈ωFM\nqs/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftbiggωGqs\nωFMqs/parenrightbigg2/parenleftbigg1−x\n2/parenrightbigg\n+/parenleftbigg1+x\n2/parenrightbigg\n=\nωG\nqs/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftbigg1−x\n2/parenrightbigg\n+/parenleftbiggωFMqs\nωGqs/parenrightbigg2/parenleftbigg1+x\n2/parenrightbigg\n,(19)\nwhere{˜ωqs}are the frequencies associated to Ξ(x). Us-\ning this result, we can write the Helmholtz free energy in\nthe high temperature limit as\n˜Fharm(V,T→ ∞,{S}) =/parenleftbiggFFM\nharm+FG\nharm\n2/parenrightbigg\n+\n3\n2NkBT/summationdisplay\nq,sln/bracketleftBigg/parenleftbig\nωFM\nqs/parenrightbig2(1+x)+/parenleftbig\nωG\nqs/parenrightbig2(1−x)\n2ωFMqs·ωGqs/bracketrightBigg\n(20)\nwhere terms FFM\nharmandFG\nharmare calculated in the clas-\nsical limit through Eq. (8) and correspond to perfect FM\nandG-AFM spinarrangements. Notethatin thelimiting\ncasesx= 1 and x=−1, Eq. (20) consistently reduces\ntoFFM\nharmandFG\nharm. Interestingly, in the ideal paramag-\nnetic case x= 0 we find that, since all ωqsare positive,\nthe inequality ˜Fharm>1\n2/parenleftbig\nFFM\nharm+FG\nharm/parenrightbig\nholds. This\nresult sets a lower bound for the error that we would be\nmaking if the spin disorder in ideal paramagnetic phases\nwas neglected. For example, in the case of the Rphase\nof BFO, if we used a frozen G-AFM spin structure in our\nQH calculations, the resulting free energy error would be\nof order1\n2/parenleftbig\nFFM\nharm−FG\nharm/parenrightbig\n.\nD. Phonon Calculations\nIn order to compute the QH free energy of a crystal,\nit is necessary to know its full phonon spectrum over the\nwhole BZ. There are essentially two methods which can\nbe usedforthe calculationofthe phononfrequencies: lin-\near response theory and the direct approach. The first\nmethod is generally exploited within the framework of\ndensity functional perturbation theory (DFPT);56–59the\nmain idea in DFPT is that a linear order variation in the\nelectrondensityuponapplicationofaperturbationtothe\ncrystal is the responsible for the variation in the energy\nup to third order in the perturbation. If the perturbation\nis a phonon wave with wave-vector q, the calculation of\nthe density change to linear order can thus be used to\ndetermine the dynamical matrix at wave-vector q. This\nprocedure can be repeated at any wave-vector and with-\nout the need to construct a supercell. In the direct ap-\nproach, in contrast, the force-constant matrix is directly\ncalculated in real-space by considering the proportional-\nity between the atomic displacements and forces when\nthe former are sufficiently small (see Sec. IIB).60,61In\nthis case, large supercells have to be constructed in order7\nto guarantee that the elements of the force-constant ma-\ntrix have all fallen off to negligible values at their bound-\naries, a condition that follows from the use of periodic\nboundaryconditions.62Once the force-constantmatrix is\nthus obtained, we can Fourier-transform it to obtain the\nphononspectrumatany q-point. Inthisworkwechoseto\nperform phonon frequencies calculations with the direct\nmethod. Nevertheless, convergence of the force-constant\nmatrix elements with respect to the size of the supercell\nin polar materials may be slow due to the appearance of\ncharge dipoles and macroscopic electric fields in the limit\nof zero wave-vector; in the next section we explain how\nwe have efficiently dealt with this issue.\nWe performed a series of initial tests in the Rphase\nto determine the value of the various calculation param-\neters that guarantee Fharmresults converged to within\n5 meV/f.u. (As it will be shown later, this accuracy\nthreshold translates into uncertainties of about 100 K\nin the predicted transition temperatures.) The quanti-\nties with respect to which our QH free energies need to\nbe converged are the size of the supercell, the size of\nthe atomic displacements, and the numerical accuracy\nin the calculation of the atomic forces and BZ sampling\n(see Eq. 5). We found the following settings to fulfill\nour convergence requirements: 2 ×2×2 supercells (i.e.,\n8 replications of the 10-atom unit cell of the Rphase),\natomic displacements of 0 .02˚A, and special Monkhorst-\nPack63grids of 12 ×12×12q-points, corresponding to\nthe BZ of the R-phase unit cell, to compute the sums in\nEq. (5). Regarding the calculation of the atomic forces\nwith VASP, we found that the density of k-points for\nBZ integrations had to be increased slightly with respect\nto the value used in the energy calculations (e.g., from\n8×8×8 to 10×10×10 for the BZ of the unit cell of the\nRphase) and that computation of the non-local parts of\nthe pseudopotential contributions had to be performed\nin reciprocal, rather than real, space. These technicali-\nties were adopted in all the phonon calculations, adapt-\ning in each crystal structure to the appropriate q- and\nk-point densities. The value of the phonon frequencies\nand quasi-harmonic free energies were obtained with the\nPHON code developed by Alf` e.62,64In using this code,\nwe exploited the translational invarianceof the system to\nimpose the three acoustic branches to be exactly zero at\nthe Γq-point, and used central differences in the atomic\nforces (i.e., we considered positive and negative atomic\ndisplacements). As an example of our phonon frequency\ncalculations, we show in Fig. 3 the full phonon spectrum\nobtained for the Rphase of BFO with a G-AFM spin\narrangement at zero pressure and when accounting for\nlong-range dipole-dipole interactions as described in the\nnext section.\nE. Treatment of long-range Coulomb forces\nAs noted in the previous section, the displacement of\natoms in an insulator like BFO creates electric dipolesand long-range dipole-dipole interactions; as a conse-\nquence, the interatomic force constants Φ mndecay typ-\nically with the third power of the interatomic distance.\nThese long-rangeinteractionsplay a criticalrolein deter-\nmining the spectrum of long-wavelength phonons. In the\ndirect approach, the phonon frequencies are exactly cal-\nculated at wave-vectors qthat are commensurable with\nthe supercell; thus, unaffordably large simulation boxes\nwould in principle be needed to accurately describe long-\nwavelength phonons.\nNevertheless, the long-range dipole-dipole interactions\ncan be modeled at the harmonic level from knowledge\nof the atomic Born effective charge tensors and the di-\nelectric tensor of the material.59,65Taking advantage of\nsuch a result, Wang et al.proposed a mixed-space ap-\nproachinwhichaccurateforceconstants Φarecalculated\nwith the direct approach in real space and long-range\ndipole-dipole interactions with linear response theory in\nreciprocal space.66Wang’s approach is based on the ad\nhocinclusion of a long-rangeforce-constantmatrix of the\nform\nϕlκα,l′κ′α′=4πe2\nNV/parenleftBig/summationtext\nβqβZ∗\nκβ,α/parenrightBig/parenleftBig/summationtext\nβqβZ∗\nκβ,α′/parenrightBig\n/summationtext\nββ′qβǫ∞\nββ′qβ′(21)\nwhereNis the number of primitive cells in the super-\ncell and Vits volume; eis the elemental charge, ǫ∞is\nthe electronic dielectric tensor, and Z∗\nκβ,αis the Born ef-\nfective charge quantifying the polarization created along\nCartesian direction αwhen atom κmoves along β. It\ncan be shown that, by Fourier-transforming the modified\nforce-constant matrix Ω=Φ+ϕ, one obtains the cor-\nrect behavior near the Γ point; further, for q/ne}ationslash= 0 wave\nvectors one obtains a smooth interpolation that recovers\nthe exact results at the q-points commensurate with the\nsupercell employed for the calculation of Φ.66\nIn Table I and Fig. 3, we report the phonon frequen-\ncies that we have obtained for the Rphase of BFO us-\ning Wang’s mixed-space approach, and compare them\nto previous experimental and theoretical data found in\nRefs. 67–69. As it may be appreciated there, the agree-\nment between our Γ-phonon results and the measure-\nments is very good, indeed comparable to that achieved\nwith DFPT calculations performed by other authors.\n(Actually, Wang’s method has already been applied with\nsuccess to the study of the phonon dispersion curves and\nthe heat capacity of BFO.70) After checking the numer-\nical accuracy of Wang’s technique, we performed a test\nin which we assessed the Fqhdifferences obtained by us-\ning the original and mixed-space versions of the direct\napproach. We found that the effect of considering long-\nrange dipole-dipole interactions in the QH energies was\nto varyFqhin less than 5 meV/f.u., which is our tar-\ngeted accuracy threshold. In view of the small size of\nthese corrections, and for the sake of computational af-\nfordability, we decided not to consider ϕterms in our\nsubsequent calculations, for which we just employed the\noriginalreal-spaceversionofthe direct approach. In fact,8\n0 20 40 60 80ωG−AFM (meV) \n Γ K L (q,q,q) (q,−q,0)\nFIG. 3: Phonon spectrum of the Rphase of BFO with G-\nAFM spin order, calculated with the direct approach and\nconsidering long-range dipole-dipole interactions. The c or-\nresponding equilibrium volume per formula unit is 64.35 ˚A3.\n−34−33.5−33−32.5−32−31.5−31\n 0 200  400  600  800  1000  1200Fqh (eV/f.u.)\nT (K)R\nO\nR (without ZPE)\nO (without ZPE)\n−0.06−0.04−0.02 0 0.02\n 200 500 800 1100 \n ΔFqh\n−0.06−0.04−0.02 0 0.02\n 200 500 800 1100 \n ΔFqh\nFIG. 4: Quasi-harmonic free energies of the RandOphases\nof BFO, calculated at p≈0 GPa (i.e., neglecting T-induced\nvolume expansion effects) and considering a perfect G-AFM\nspin order in both structures. The size of the ZPE correction s\nis shown in the vertical axis. Inset: Plot of the quasi-harmo nic\nfree energy difference ∆ Fqh≡Fqh(R)−Fqh(O) expressed as\na function of temperature.\nas it has already been pointed out by Alf` e,62in the typi-\ncal caseanincorrecttreatment ofthe longitudinal optical\nmodes near the Γ-point compromises only a small region\nof the BZ, and the resulting errors in the free energy are\nsmall and can in principle be neglected.III. RESULTS AND DISCUSSION\nA. Stability of the RandOphases at\nconstant-volume and frozen-spin conditions\nIn this section, we present our QH results for the R\nandOphases of BFO. For the sakeof clarity, we first dis-\ncuss the results obtained when spin-disorderand volume-\nexpansion effects are neglected.\nIn Fig. 4 we plot the Fqhenergy of the RandO\nphases calculated at p≈0 GPa as a function of tem-\nperature. Volumes were kept fixed at their equilibrium\nvaluesV0obtained at T= 0 K, which are equal to\n64.61˚A3/f.u. and 61.99 ˚A3/f.u., respectively. We con-\nsidered the perfect G-AFM spin order to be frozen in\nboth structures. We computed Fqhover sets of fourteen\ntemperature points taken at intervals of 100 K and fit-\nted them to third-order polynomial curves. ZPE cor-\nrections (see Eq. 7) were included in the fits and are\nequal to 0.263(5) eV/f.u. and 0.248(5) eV/f.u., respec-\ntively, for the RandOphases. (An estimate of the\nerror is given within parentheses.) We find that at\nT= 0 K the Rphase is energetically more favorable\nthan the Ophase by 0.046(5) eV/f.u. As the temper-\nature is raised, however, the Helmholtz free energy of\ntheOphase becomes lower than that of the Rphase\ndue to the increasingly more favorable Fharmcontribu-\ntions. For instance, at T= 300 K, Fharmamounts to\n0.048(5) eV/f.u. for the Ophase and 0.077(5) eV/f.u.\nfor theRphase, whereas at T= 1000 K the obtained\nvalues are −1.481(5) eV/f.u. and −1.414(5) eV/f.u., re-\nspectively. Consequently, a first-orderphase transitionof\ntheR → Otypeispredictedtooccurat Tt=900(100)K.\nWe show this in the inset of Fig. 4, where the energy\ndifference ∆ Fqh≡Fqh(R)−Fqh(O) is represented as a\nfunction of temperature; since quasi-equilibrium condi-\ntions are assumed, the corresponding transition temper-\nature coincides with the point at which ∆ Fqh= 0. We\nnotice that this estimation of Ttis reasonably close to\nthe experimental value of 1100 K.10,19\nLet us now discuss the origin of the obtained solid-\nsolid transformation in terms of the phonon eigenmodes\nand frequencies of each phase. In Fig. 5 we plot the\nphonon density of states (pDOS) calculated for the R\nandOphases at their equilibrium volumes. We find\nthat the value of the geometric frequency ¯ ω(see Eq. 9)\nis 27.16 meV in the Ophase and 28.58 meV in the R\nstructure (expressed in units of /planckover2pi1). Therefore, as it was\nalready expected from the results shown in Fig. 4, the\nOphase of BFO is, in average, vibrationally softer than\ntheRphase. In particular, the pDOS of the Ophase ac-\ncumulates a larger number of phonon modes within the\nlow-energy region of the spectrum, and extends over a\nsmaller range of frequencies.\nWe restrict our following analysis to the low-energy\nphonons (i.e., ωqs≤¯ω), which provide the dominant\ncontributions to Fharm. In the Rphase, we observe a\nsharp pDOS peak centered at /planckover2pi1ω∼10 meV followed9\nTO modes This work Expt . Calc. LO modes This work Expt . Calc.\nE(TO1) 78 74 102 E(LO1) 85 81 104\nE(TO2) 136 132 152 E(LO2) 161 175 175\nE(TO3) 238 240 237 E(LO3) 242 242 237\nE(TO4) 252 265 263 E(LO4) 258 276 264\nE(TO5) 265 278 274 E(LO5) 323 346 332\nE(TO6) 330 351 335 E(LO6) 352 368 377\nE(TO7) 361 374 378 E(LO7) 393 430 386\nE(TO8) 412 441 409 E(LO8) 445 468 436\nE(TO9) 488 523 509 E(LO9) 483 616 547\nA1(TO1) 151 149 167 A1(LO1) 172 178 180\nA1(TO2) 219 223 266 A1(LO2) 240 229 277\nA1(TO3) 285 310 318 A1(LO3) 461 502 428\nA1(TO4) 506 557 517 A1(LO4) 550 591 535\nA2(LO1) 101 109 109\nTABLE I: Γ-point phonon frequencies of the Rphase of BFO with G-AFM spin order, calculated using the dire ct approach\nand considering long-range dipole-dipole interactions. E xperimental values are taken from Refs. 67 and 68, and previo us\nLSDA-DFPT calculations from Ref. 69. Frequencies are expre ssed in units of cm−1.\n   \n 0 10 20 30 40 50 60 70 80 90Phonon DOS (arb. units)\nEnergy (meV)R (G−AFM)\nC (G−AFM)   \n 0 10 20 30 40 50 60 70 80 90Phonon DOS (arb. units)\nEnergy (meV)R (G−AFM)\nT (C−AFM)   \n 0 10 20 30 40 50 60 70 80 90Phonon DOS (arb. units)\nEnergy (meV)R (G−AFM)\nO (G−AFM)\n   \n 0 10 20 30 40 50 60 70 80 90Phonon DOS (arb. units)\nEnergy (meV)R (G−AFM)\nO (G−AFM)\nFIG. 5: (Color online) Phonon density of states of vari-\nous BFO phases obtained at p= 0 GPa. The correspond-\ning equilibrium volumes are 64.61 ˚A3/f.u., 61.99 ˚A3/f.u.,\n71.12 ˚A3/f.u., and 64.17 ˚A3/f.u. for the R,O,T, andC\nphases, respectively.\nby a deep valley. By inspecting the spectrum of phonon\neigenmodes obtained at Γ and the full phonon bands dis-\nplayed in Fig. 3, we identify that pDOS maximum withthe first optical transverse mode TO1 (see Fig. 6). This\nphonon mode involves opposed displacements of neigh-\nboring Bi atoms within the plane perpendicular to the\npseudo-cubic direction [111], and is polar in the [10 ¯1]\ndirection.71Figure 7 gives additional information on the\nthree lowest-lying phonons of the Rphase across the BZ.\nThere we can see that the softest phonons are acoustic in\ncharacterandcorrespondto q-pointsintheneighborhood\nof Γ. As we move away from Γ, the lowest-lying phonon\nmodes change of character and can be represented by the\noptical distortion shown in Fig. 6.\nThe situation for the Ophase is rather different. As it\ncan be appreciated from Fig. 5, the number of phonons\nin the very low-frequency region is much greater than\nin theRphase. Small frequency values are in general\nrelated to phonon modes of strong acoustic character,\nwhich are the responsible for the elastic response of ma-\nterials: the softer a crystal is, the smaller the slopes of\nits acoustic bands around the Γ q-point, and the larger\nthe number of low-energy phonons that result. By ap-\nplying this reasoning to the present case and consid-\nering our pDOS results, one would arrive at the con-\nclusion that BFO in the Ophase should be elastically\nsofter than in the Rphase. However, this is not the\ncase: we computed the equilibrium bulk modulus of BFO\n(i.e.,B≡ −V∂p\n∂V) atT= 0 K describing the response\nof the material to uniform deformations and found, re-\nspectively, 99(2) GPa and 158(2) GPa for the RandO\nstructures. Interestingly, this apparent contradiction is\nquickly resolvedby inspecting the behavior of the (three)\nlowest-lying phonons calculated at each BZ q-point (see\nFig. 8). As clearly observed in Fig. 8, the Ophase of\nBFO presents very low-lying optical bands with phonon10\nFIG. 6: (Color online) Sketch of the first optical transverse\nΓ-point phonon mode obtained in the Rphase of BFO at\nequilibrium. Bi displacements are represented with black a r-\nrows, and Bi, Fe, and O atoms with purple, brown and red\nspheres, respectively.\nfrequencies that can be below 2 meV. The correspond-\ning eigenmodes are dominated by the stretching of Bi–O\nbonds, with the Fe ions having a very minor contribution\n(see middle panel in Fig. 8). In fact, these soft optical\nphonons, with qvectorsfar away from Γ, are the ones re-\nsponsible for the stabilization of BFO’s Ophase at high\ntemperatures.\nB. Effect of spin disorder on the R → Otransition\nIn order to assess the effect of spin fluctuations on the\npredicted R → O phase transition, we put in practice\nthe ideas explained in Sec. IIC. As described there, our\npractical approach to capture the effects of spin disorder\nrequires the calculation of the QH energies for the G-\nAFM (FG\nqh; this is the case already considered in the pre-\nvious section) and FM ( FFM\nqh) spin arrangements, from\nwhich we derive the parameters describing (1) the spin-\nindependent part of the energy ( E0andΦ0), (2) the\nHeisenberg spin Hamiltonian for zero atomic distortions\n(J0), and (3) the effects of the spin arrangement on the\nphonon spectrum ( J(2)). Our DFT calculations render\nJ(0)values of 34.67meV and 32.67meV, respectively, for\ntheRandOphases, indicating a similar and strong ten-\ndency towards the G-AFM order. Further, Fig. 2 shows\nillustrative results of the shifts in phonon frequencies, for\ntheRphase of BFO, that occur as a function of the\nspin structure; these are the effects captured by the J(2)\nterms.\nFigure 9 reports the results of a series of Monte Carlo\n(MC) simulations performed with the Heisenberg model\ndefined by the J(0)coupling. We used a periodically-\nrepeated simulation box of 20 ×20×20 spins, and\ncomputed the thermal averages from runs of 50000 MC\nsweeps. The aim of these simulations was to determine\nthe value of the spin average /an}bracketle{tSi·Sj/an}bracketri}htthat has to be\nused in Eqs. (18)-(19) and which depends on T. Note 0 0.2 0.4 0.6 0.8 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Eigenmode Character\n|q| (2π/a)Acoustic Optical 0 0.2 0.4 0.6 0.8 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Eigenmode Contribution\n|q| (2π/a)Fe Bi O 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7E(1), E(2), E(3) (meV)\n|q| (2π/a) 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7E(1), E(2), E(3) (meV)\n|q| (2π/a)\nFIG. 7: (Color online) Analysis of the three lowest-energy\nphonon eigenmodes obtained at the q-points used for the sam-\npling of the BZ of BFO’s Rphase. We represent their eigenen-\nergies as a function of wave-vector module in the top panel,\nthe contribution of each atomic species to the mode eigenvec -\ntors in the middle panel, and a quantification of their acoust ic\nand optical characters in the bottom panel. For this quantifi -\ncation, we took advantage of the normalization and orthogo-\nnality relations satisfied by the eigenvectors of the dynami cal\nmatrix calculated at Γ and q/negationslash= 0 points.\n 0 0.2 0.4 0.6 0.8 1\n 0  0.2  0.4  0.6  0.8  1Eigenmode Character\n|q| (2π/a)Acoustic Optical 0 0.2 0.4 0.6 0.8 1\n 0  0.2  0.4  0.6  0.8  1Eigenmode Contribution\n|q| (2π/a)Fe Bi O 0 2 4 6 8 10 12\n 0  0.2  0.4  0.6  0.8  1E(1), E(2), E(3) (meV)\n|q| (2π/a) 0 2 4 6 8 10 12\n 0  0.2  0.4  0.6  0.8  1E(1), E(2), E(3) (meV)\n|q| (2π/a)\nFIG. 8: (Color online) Same as Fig. 7, but for BFO’s Ophase.11\n−1−0.8−0.6−0.4−0.2 0\n0 200 400 600 800 1000< SiSj>/S2\nT (K)R\nO\n 0 0.2 0.4 0.6 0.8 1\n0 200 400 600 800 1000Order Parameter (G−type)\nT (K)R\nO\n 0 0.2 0.4 0.6 0.8 1\n0 200 400 600 800 1000Order Parameter (G−type)\nT (K)R\nO\nFIG. 9: Monte Carlo results obtained for a simple Heisenberg\nmodel reproducing the spin magnetic order in the RandO\nphases of BFO. Top: Average value of the normalized spin\nproduct Si·Sj(withS≡5/2µB) as a function of tempera-\nture. Bottom: Calculated order parameter SG(see text) as a\nfunction of temperature.\n−0.06−0.05−0.04−0.03−0.02−0.01 0 0.01 0.02 0.03\n5025045065085010501250Fqh(R) − Fqh(O) (eV/f.u.)\nT (K)Frozen G−AFM\nUnfrozen spins\nFIG. 10: Quasi-harmonic free energy difference between the\nRandOphases of BFO. We show the results obtained in\ntwo different situations, i.e., frozen G-AFM and T-dependent\nspin orders (solid symbols and error bars correspond to the\nlast case). Lines are linear fits to the free energy results.\nthat here we have abandoned the compact notation of\nSection IIC, and Siαdenotes the αCartesian compo-\nnent of the spin at cell i. Additionally, these simulations\nallow us to monitor the occurrence of magnetic transi-\ntions through the computation of the G-AFM order pa-\nrameter SG=1\nN/summationtext\ni(−1)nix+niy+nizSiz. Here,nix,niy,\nandnizare the three integers locating the i-th lattice\ncell, and Nis the total number of spins in the simula-\ntion box; further, for the calculation of SG, we need to\nconsider only the zcomponent of the spins because of a\nsmall symmetry-breaking magnetic anisotropy that was\nincluded in our Hamiltonian to facilitate the analysis(see\nSupplemental MaterialofRef. 72). Ourresults showthatin theRphase the magnetic phase transition occurs at\nT∼600 K, a temperature that is rather close to the ex-\nperimental value TN∼650 K.8–10The results for the O\nphase are very similar.\nNow, let us assess the consequences of considering\nthese effects on the QH free energies of the RandO\nphases(see Eqs.18-19). Figure10reportsthe freeenergy\ndifference between the RandOphases, as obtained by\nconsidering (∆ ˜Fqh) or neglecting (∆ Fqh) the effect of the\nspin fluctuations. As one can appreciate, the two curves\narealmost identicaland providethe same transitiontem-\nperature. At T= 300 K, for instance, both ∆ FG\nqhand\n∆˜Fqhare about −0.032(5) eV/f.u., and at T= 1000 K\nwe get approximately 0.006(5) eV/f.u.; that is, the dif-\nferences fall within the accuracy threshold set in our free\nenergy calculations.\nHowever,the consequencesofconsidering T-dependent\nspin arrangements in the calculation of the QH free en-\nergy of an individual phase are actually quite sizable.\nIndeed, the error function defined as δ˜Fqh≡˜Fqh−FG\nqh\nmayamounttoseveraltenthsofeVathightemperatures.\nFor instance, for the Rphase,δ˜Fqhis 0.068(5) eV/f.u.\natT= 300 K and 0.102(5) eV/f.u. at T= 1000 K.\nHence, the reason behind the numerical equivalence be-\ntween functions ∆ FG\nqhand ∆˜Fqhis thatδ˜Fqherrors are\nessentiallythe same in both RandOstructures and thus\ntheycancel. Consequently, itispossibletoobtainreason-\nableTtpredictionsin BFOeven ifone neglects the strong\ndependence of spin magnetic order on temperature.\nIn view of this conclusion, and for the sake of compu-\ntational affordability, we will disregard spin-disorder ef-\nfects for the rest of phases considered in this work. Also,\nwe note that the inequality ˜Fharm>1\n2/parenleftbig\nFFM\nharm+FG\nharm/parenrightbig\n,\nmentioned in Sec. IIC, is fulfilled for both the Rand\nOstructures at all temperatures, even when x=/an}bracketle{tSi·\nSj/an}bracketri}ht/|S|2/ne}ationslash= 0 and /planckover2pi1¯ω≈kBT. Plausibly then, the lower\nbound set there for the QH errors caused by neglecting\nthe spin disorder can be tentatively generalized to any\nvalue ofx.\nC. Effect of volume expansion on the R → O\ntransition\nTo address the effect of volume expansion on Tt, we\nperformed additional energy, phonon, and Fqhcalcula-\ntions over a grid of five volumes spanning the interval\n0.95≤V/V0≤1.10 for both RandOphases. At\neach volume, first we computed the value of Fqhat a\nseries of temperatures in the range between 0 K and\n1600 K, taken at 100 K intervals. Then, at each Twe\nfitted the corresponding Fqh(V,T) points to third-order\nBirch-Murnaghan equations73,74and performed Maxwell\ndouble-tangent constructions over the resulting Rand\nOcurves to determine pt(T) (i.e., the pressure at which\nthe first-order R → O transition occurs at a given T).\nBy repeating this process several times we were able to\ndraw the R–Ophase boundary, pt(T), in the interval12\n 800 900 1000 1100 1200 1300 1400\n−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6T (K)\np (GPa)R O−31.8−31.7−31.6−31.5−31.4\n 60 62 64 66 68 70Eeq (eV/f.u.)\nV (Å3)R\nO\n−31.8−31.7−31.6−31.5−31.4\n 60 62 64 66 68 70Eeq (eV/f.u.)\nV (Å3)R\nO\nFIG. 11: Calculated two-phase boundary delimiting the R\nandOregions in the bulk phase diagram of BFO at ele-\nvated temperatures. The solid line is a guide to the eyes\nand the symbols represent explicitly calculated points. In set:\nEeqcurves obtained for the RandOphases as a function of\nvolume without considering ZPE corrections.\n−0.15 −0.10 −0.05 0.0 0.05\n5025045065085010501250Fqh(R) − Fqh(X) (eV/f.u.)\nT (K)O (G−AFM)\nC (G−AFM)\nT (C−AFM)\nFIG. 12: Quasi-harmonic free energy differences among the\nRphase and the rest of crystal structures analyzed in this\nwork. Perfect G-AFM spin order and constrained equilibrium\nvolumes were considered. Lines are guides to the eyes and the\nsymbols represent explicitly calculated points.\n−0.1 GPa ≤p≤0.6 GPa. Figure 11 reports these re-\nsults. As one can appreciate there, the calculated transi-\ntion temperature at equilibrium now is 1300(100)K. Vol-\nume expansion effects, therefore, shift upwards by 400 K\nour previous tentative Ttestimation. Also, we find that\nthe volume of the crystal varies from 66.51 ˚A3/f.u. to\n62.34˚A3/f.u. during the course of the R → Otransfor-\nmation. These values can be compared to recent experi-\nmental data obtained by Arnold et al.19which are TC≈\n1100K,V(R) = 64.15˚A3/f.u. and V(O) = 63.10˚A3/f.u.\nIn general, our agreement with respect to Arnold’s mea-\nsurements can be regarded as reasonably good, although\nourQHcalculationsoverestimatethe transitiontempera- 0 0.2 0.4 0.6 0.8 1\n 0  0.2  0.4  0.6  0.8  1Eigenmode Character\n|q| (2π/a)Acoustic Optical 0 0.2 0.4 0.6 0.8 1\n 0  0.2  0.4  0.6  0.8  1Eigenmode Contribution\n|q| (2π/a)Fe Bi O 0 2 4 6 8 10 12\n 0  0.2  0.4  0.6  0.8  1E(1), E(2), E(3) (meV)\n|q| (2π/a) 0 2 4 6 8 10 12\n 0  0.2  0.4  0.6  0.8  1E(1), E(2), E(3) (meV)\n|q| (2π/a)\nFIG. 13: (Color online) Same as Fig. 7, but for the complex\nCphase considered in this work.\nture and volume reduction ∆ V=V(O)−V(R) observed\nin experiments.\nMoreover,in the vicinityofthe transitionstate[0GPa,\n1300(100) K], we assumed the slope of the phase bound-\nary to be constant and numerically computed dT/dp≈\n−1100 K/GPa. By introducing this value and ∆ V=\n−4.17˚A3/f.u. in the Clausius-Clapeyron equation, we\nfound the latent heat of the ferroelectric phase transfor-\nmation to be about 0.71 Kcal/mol. Unfortunately, we do\nnot knowofany experimentaldata to comparethis result\nwith. Interestingly, if we assume the slope of the R–O\nphase boundary to be constant regardless of the p−T\nconditions, the extrapolated zero-temperature R → O\ntransition turns out to be pt(0)∼1.2 GPa. This re-\nsult differs greatly from the pt(0K) value obtained when\nstraightforwardly considering static Eeqcurves (see inset\nof Fig. 11) and enthalpies (i.e., Heq=Eeq+peqV), which\nis 4.8 GPa. This disagreement may indicate that assum-\ning global linear behavior in pt(T) is unrealistic and/or\nthat ZPE corrections in BFO are very important. We\nwill comment again on this point in Sec. IIIF, when ana-\nlyzing in detail the role of ZPE corrections in prediction\nofp-induced phase transformations at T= 0 K.\nD. The complex Cphase\nNovelnanoscale-twinned structures, denoted here as\ncomplex orCphases, havebeen recentlysuggestedto sta-\nbilize in bulk BFO under conditions of high- Tor high-p,\nand upon appropriate chemical substitutions.24From a13\nstructural point of view, these Cphases can be thought\nof abridgeappearing whenever we have RandOre-\ngions in the phase diagram of BFO. Thus, the energies of\nthe nanoscale-twinned structures lie very close to that of\nthe ground state. Interestingly, these Cphases have been\nlinked also to the structure of domain walls whose energy\nis essentially determined by antiferrodistortive modes in-\nvolving the rotation of O 6octahedra.27These intriguing\nfeatures motivated us to study the thermodynamic sta-\nbility of this new type of phases with the QH approach.\nWe note that, in the original paper by Prosandeev et\nal.,24several phases are proposed as members of the C\nfamily. For reasons of computational affordability, we re-\nstrict our analysis here to one particular structure (with\nPca21space group and Z= 8) that has been introduced\nabove and is depicted in Fig. 1(d).\nIn Fig. 12 we plot the QH free energy of our Cphase\nexpressed as a function of temperature, taking the result\nfor theRstructure as the zero of energy. As one may\nobserve there, at low Tthe ∆Fqhdifference is negative\nand quite small in absolute value. At T= 0 K, for in-\nstance, this quantity amounts to −0.025 (5) eV/f.u., and\nroughly lies between the values corresponding to the T\nandOphases. As Tis raised, however, ∆ Fqhincreases\nsteadilywith anapproximateslopeof2 ×10−5eV/K,and\natT≈1000 K it becomes positive within our numerical\nuncertainties. This change of sign marks the occurrence\nof a potential R → C transformation. However, such\na transition would be prevented by the onset of the O\nphase, which becomes the equilibrium state at a lower\ntemperature. Note that, according to our results, the\nprevalence of the Ophase occurs in spite of the fact that,\nat 0 K, this phase is energetically less favorable than the\nCstate by 0.021 eV/f.u.\nThe free energy competition between the OandC\nphases is very strong, as can be deduced from the pDOS\nplots enclosed in Fig. 5. In particular, the Cphase shares\ncommon pDOS features with both the RandOstruc-\ntures, which is hardly surprising given that its atomic ar-\nrangementcanbeviewedasamixturebetweenthe Rand\nOsolutions. For instance, in the ω→0 limit, the Cand\nOdistributions are practically identical, and the rangeof\nphonon frequencies over which they expand is very sim-\nilar. Moreover, the number of low-lying optical phonon\nmodes found in the Cphase is, as we calculated for the\nOstructure, very high (although we note that in the C\ncase the contribution of the Fe anions to the eigenmodes\nis not negligible, see Fig. 13). Then, for intermediate\nfrequencies the CpDOS presents a series of modulations\nwhich are more characteristic of the Rphase. Also, the\nenergy of the first CpDOS peak is closer to that of the\nRphase, and from an elastic point of view both Cand\nRphases are very similar (that is, the bulk modulus of\nthe two structures are coincident within our numerical\nuncertainties). A quantitative testimony of these pDOS\nsimilarities is given by the geometric frequencies ¯ ωcal-\nculated in the O,C, andRphases, which are 27.16 meV,\n28.00 meV, and 28.58 meV, respectively. Furthermore,ZPE corrections in the Cphase amount to 0.254 eV/f.u.,\na value that roughly coincides with the arithmetic av-\nerage obtained for the corresponding OandRresults.\nIn conclusion, we can state that BFO in the Cphase is\nin average vibrationally softer than in the Rphase, but\nmore rigid than in the Ophase.\nIt is worth noticing that, although we do not pre-\ndict here a temperature-induced phase transition of the\nR → Ctype, this can not be discarded to occur in prac-\nticegiventhatthe calculated∆ Fqhdifferencesamongthe\nR,OandCstructures are very small. Note that small\nvariations in the computed free energies – as for instance\ndue to the use of a different exchange-correlation func-\ntional in our DFT calculations, related to our QH ap-\nproximation, etc. – could very well change this delicate\nbalance of relative stability (see discussion in Sec. IIIG).\nFurther, the Cphase considered here is only one among\nthe many nanoscale-twinned structures that have been\npredictedto exist,24and itis reasonableto speculatethat\nsome of them might indeed be predicted to be the equi-\nlibrium solutions by the DFT scheme employed here. At\nany rate, our results do suggest that these Cstructures\nare, at the very least, very close to becoming stable in\nthe regions of the phase diagram in which R → Otran-\nsitions occur. Moreover, they are obvious candidates to\nmediate(i.e., to appear in the path of) the R → Otrans-\nformation. Hence, ourresultsareclearlycompatible with\nthe possibility that Cphases can be accessed experimen-\ntally, as robust meta-stable states, depending on kinetic\nfactors.\nE. The super-tetragonal Tphase\nUnder zero p−Tconditions, the energy of the Tstruc-\nture depicted in Fig. 1(d) differs from that of the R\nphase by only few hundredths of eV per formula unit.25\nThisTphase possesses a giant c/aratio, a large elec-\ntric polarization with a small in-plane component, and\nanti-ferromagneticspin order of type C (C-AFM); hence,\nin principle, this phase would be potentially relevant for\ntechnologicalapplications. Nevertheless, the Tphasehas\nnever been observed in bulk samples of BFO (although\nit is stabilized in thin films under high compressive and\ntensile epitaxial constraint28,29). Aiming at understand-\ning the causes behind the frustrated stabilization of a\nbulk-like Tphase in BFO, we studied it with the QH\napproach.\nIn Fig. 12 we plot the QH free energy of the Tphase\ntaken with respect to that of the Rstructure and ex-\npressed as a function of temperature. The Tphase\nis assumed to present frozen C-AFM spin order, and\na frozen G-AFM arrangement is considered for the R\nphase. As one may observe there, the free energy dif-\nference ∆ Fqh(T) is negative and very small at low tem-\nperatures (e.g., ∆ Fqh(0 K) = −0.012 (5) eV/f.u.) but\nprogressively increases in absolute value as Tis raised\n(e.g., ∆Fqh(1000 K) = −0.078 (5) eV/f.u.). This result14\n 0 0.2 0.4 0.6 0.8 1\n 0  0.2  0.4  0.6  0.8  1Eigenmode Character\n|q| (2π/a)Acoustic Optical 0 0.2 0.4 0.6 0.8 1\n 0  0.2  0.4  0.6  0.8  1Eigenmode Contribution\n|q| (2π/a)Fe Bi O 0 2 4 6 8 10 12\n 0  0.2  0.4  0.6  0.8  1E(1), E(2), E(3) (meV)\n|q| (2π/a) 0 2 4 6 8 10 12\n 0  0.2  0.4  0.6  0.8  1E(1), E(2), E(3) (meV)\n|q| (2π/a)\nFIG. 14: (Color online) Same as Fig. 7, but for the super-\ntetragonal Tphase considered in this work.\nimplies that vibrational thermal excitations energetically\ndestabilize the Tphase as compared to the RandO\nstructures, in agreement with observations.\nThis conclusion may not seem so obvious from inspec-\ntion ofthe pDOS resultsenclosedin Fig. 5. As wecan see\nthere, at frequencies below 5 meV, the Tphase presents\na larger phonon density than the Rphase, which would\nin principle suggest that the Tstructure is vibrationally\nsofter. However, the lowest-lying pDOS peak in the R\nphase is much higher than in the Tstructure, and this\nfeature turns out to be dominant. In particular, the cal-\nculated geometric frequency ¯ ωamounts to 33.07 meV in\ntheTphase and to 28.58 meV in the ground state. In-\nterestingly, ZPE corrections (see Eq. 7) in both RandT\nphases are practically identical ( ∼0.26 eV/f.u.).\nThe relatively high number of phonon modes that the\nTphase presents at very low frequencies is reminiscent\nof the results discussed above for the Ostructure. In-\ndeed, as can be seen in Fig. 14, in the Tphase we also\nfind low-lyingphonons of verylow energythroughout the\nBZ. Additionally, the Tphase also presents a relatively\nsmall bulk modulus and is elastically softer than the R\nstructure: we obtained 73(2) GPa in this case, while we\ncalculated 99(2) GPa for the Rphase. These bulk modu-\nlus results are consistent with what one would generally\nexpect from inspection of the pDOS plots enclosed in\nFig. 5; in this sense, the Tstructure behaves normally,\nincontrastwith thebehaviorofthe Ostructurediscussed\nabove. Finally, let us note that, as shown in Fig. 14, the\nlowest-energy phonons of the Tphase are largely domi-\nnated by the oxygen cations. This results is in contrast−0.2−0.15−0.1−0.05 0 0.05\n−2 0 2 4 6 8 10Hqh(R) − Hqh(X) (eV/f.u.)\np (GPa)O\nT\nC−0.2−0.15−0.1−0.05 0 0.05\n−2 0 2 4 6 8 10Hqh(R) − Hqh(X) (eV/f.u.)\np (GPa)with ZPE\nwithout ZPEO\nT\nC\n−0.2−0.15−0.1−0.05 0 0.05\n−2 0 2 4 6 8 10Hqh(R) − Hqh(X) (eV/f.u.)\np (GPa)with ZPE\nwithout ZPEO\nT\nC\nFIG. 15: (Color online) Calculated enthalpy energy differ-\nences among the four crystal structures analyzed in this wor k,\natT= 0 K and expressed as a function of p. Results ob-\ntained when considering (resp. neglecting) ZPE correction s\nare shown in the top (resp. bottom) panel. Blue dots in the\npressure axis mark the occurrence of first-order phase trans i-\ntions (i.e., ∆ Hqh(pt) = 0).\nwith our findings for the R,O, andCstructures. Such\na differentiated behavior is probably related to the fact\nthat, unlike to all the other phases, the basic building\nblock of the Tstructure are O 5pyramids [see Fig. 1(c)];\nhaving so many oxygen-dominated low-frequency modes\nsuggests that such pyramids are more easily deformable\nthan the rather rigid O 6octahedra characteristic of the\nother phases.\nF. Pressure-induced transitions at 0 K\nIn this section we analyze the thermodynamic stability\nof the four studied crystal structures under hydrostatic\npressure at T= 0 K. We take into account ZPE correc-\ntions and consider also negative pressures.\nIn Fig. 15 we plot the enthalpy energy (i.e., H=\nE+pV) of the O,T, andCphases as a function of\np, taking the result for the Rstructure as the pressure-15\nR3c−G a = 5.606˚Ab= 5.606˚Ac= 13.950˚A\n(P = 3.6 GPa) α= 90◦β= 90◦γ= 120◦\nAtom Wyc . x y z\nBi 6 a 0.0 0 .0 0 .4959\nFe 6 a 0.0 0 .0 0 .2734\nO 18 b 0.4186 −0.0174 0 .0402\nPnma−G a = 5.696˚Ab= 7.838˚Ac= 5.465˚A\n(P = 3.6 GPa) α= 90◦β= 90◦γ= 90◦\nAtom Wyc . x y z\nBi 4 c 0.0512 0 .25 0 .5098\nFe 4 a 0.0 0 .0 0 .0\nO 4 c−0.0285 0 .25 0 .0960\nO 8 d 0.1998 −0.0469 0 .3044\nTABLE II: Calculated structural data corresponding to the\np-induced R → O transition that is predicted when quan-\ntum ZPE corrections are considered. Wyckoff positions were\ngenerated with the ISOTROPY package.75\ndependent zero of enthalpy. A first-order transforma-\ntion between phases AandBoccurs at pressure ptwhen\nthe enthalpy energy difference ∆ H(pt)≡HA−HBbe-\ncomes zero. In all the cases we present the results ob-\ntained both when neglecting ZPE corrections (i.e., for\nE=Eeqandp=−∂Eeq/∂V) and when fully con-\nsidering them (i.e., for E=Eeq+Fharm(T→0) and\np=−∂[Eeq+Fharm(T→0)]/∂V). Additional phonon\nand static energy calculations were performed whenever\nrequired in order to compute accurate enthalpies in the\npressure interval −2 GPa≤p≤10 GPa.\nAs we increase the pressure, we find two phase tran-\nsitions of the T → R andR → O types. The T → R\ntransition occurs at −0.3(1) GPa and the associated vol-\nume change is ∆ V= 6.76˚A3/f.u.; at this transition pres-\nsure, the Tphase presents a volume of 71.94 ˚A3/f.u. and\na very large c/aratio of about 2. The R → O transi-\ntion occurs at 3.6(1) GPa, and the volume changes from\n63.04˚A3/f.u. to 61.13 ˚A3/f.u.; the corresponding struc-\ntural data is given in Table II. Interestingly, the pressure-\ndependence of the enthalpies shown in Fig. 15 resemble\nthe results reported in Fig. 12 for Fqhas a function of\ntemperature. In particular, under compression the T\nphase becomes higher in enthalpy than the rest, and the\nenthalpy of the Ophase turns out to be the smallest.\nAlso, the Cphase gets energetically favored over the R\nstructure upon increasing pressure, although it never be-\ncomes the most stable structure.\nThe bottom panel in Fig. 15showsthe enthalpyresultsobtained when ZPE corrections are neglected. Interest-\ningly,whilethemaintrendsareconserved,thepressureof\ntheR → Otransformation turns out to be shifted up to\n4.8(1) GPa. This result shows that atomic quantum de-\nlocalizationeffectsin perovskiteoxidesmaybe important\nfor accurate prediction of p-induced phase transitions.\nOur results for the R → O transformation are con-\nsistent with those of previous theoretical studies,21,25\nthe quantitative differences being related to the vary-\ning DFT flavors employed, consideration of typically-\nneglected ZPE corrections, and other technicalities. As\nregards the connection with experiment, it is worth not-\ning that we predict the R → Otransition to occur at a\npressure(3.6GPa)thatisratherclosetotheoneatwhich\ntheRphase has been observed to transform into a com-\nplex structureby Guennou et al.23(i.e.,∼4 GPaat room\ntemperature). It is therefore tempting to identify the ex-\nperimentally detected complex structure with the family\nofCphasesofwhichwehaveinvestigatedarepresentative\ncase; indeed, verifying a possible R → C → O transition\nsequence was one of our motivations to investigate the\neffects of pressure. However, our calculations render a\ndirectR → Otransition, which suggests that the exper-\nimentally observed complex structures might actually be\nvery long-lived meta-stable states, as opposed to actual\nequilibrium phases. On the other hand, as explained in\nSectionIIID,gettingaccuratepredictionsneartransition\npoints at which Fqh(R)≈Fqh(C)≈Fqh(O) is clearly a\nchallenging task, and many factors can come into play\nand affect the results. Hence, we cannot fully discard the\npossibility that, under pressure, the Rstructure trans-\nforms into a complex equilibrium phase.\nG. The role of the exchange-correlation energy\nfunctional\nIn previous sections we have highlighted that the dif-\nferences in the Helmholtz free energies and enthalpies of\ntheR,O, andCphases are calculated to be exceedingly\nsmall. In such conditions, our predictions for the equi-\nlibrium phase may depend, among other factors, on the\nemployed exchange-correlation DFT energy functional.\nIn this sense, Di´ eguez et al.already found25that, in\nBFO, energy differences between stable structures de-\npend strongly on the DFT energy functional used, with\nvariations in Eeqthat may be as large as 0.1 eV per for-\nmula unit.\nTo estimate the magnitude ofthis type ofuncertainties\nin our∆Fqhresults computed with a PBE+ Ufunctional,\nwe repeated our QH investigation of temperature-driven\ntransitions – at constant volume and frozen-spin condi-\ntions – using a LDA+ Uscheme. Our LDA+ Uresults\nshow, in accordance with the presented PBE+ Ustudy,\nthat the orthorhombic Ophase gets thermodynamically\nstabilized over the rest of structures at high tempera-\ntures, and that the Tphase goes steadily higher in free\nenergy. Further, the LDA+ Uresults indicate that the16\nR → Otransition occurs at approximately 500 K, which\nismuchlowerthantheexperimentalresult. Interestingly,\nmost of the the discrepancy between this LDA+ Uresult\nand our PBE+ Uprediction (900 K) can be traced back\nto the different equilibrium energies in the 0 K limit,\nwith the phonon contributions to the free energy play-\ning a secondary role. Indeed, from the PBE+ Ucalcula-\ntions we get Eeq(O)−Eeq(R) =−0.061eV/f.u., while the\nLDA+Uresult is−0.016eV/f.u. Obviously, the LDA+ U\nfunctional brings the RandOphases much closer in en-\nergy, which leads to the stabilization of the Ostructure\nat a much lower temperature. Additionally, the Fqhof\ntheCphase remains always about 50 meV/f.u. higher\nthan that of the Rphase, the difference being weakly\ndependent on temperature.\nHence, our calculations confirm that quantitative pre-\ndictions of transition temperatures are strongly depen-\ndent on the employed DFT functional. We can also con-\nclude that the LDA functional does not capture properly\nthe relative stability of the RandOphases of BFO,\nand that the PBE functional is a much better choice. In\nthis sense, our work ratifies the conclusions presented in\nRef. 25.\nIV. CONCLUSIONS\nWe have performed a first-principles study of the p−T\nphase diagramofbulk multiferroic BFO relyingon quasi-\nharmonic free energy calculations. We have analyzed the\nthermodynamic stability of four different crystal struc-\ntures that have been observed, or predicted to exist, at\nnormal and high porTconditions and/or in thin films\nunder epitaxial constraints. In order to incorporate the\neffects of spin-phonon coupling on the quasi-harmonic\ncalculation of the Helmholtz free energies, we have de-\nveloped an approximate and technically simple scheme\nthat allows us to model states with varying degrees of\nspin disorder.\nConsistent with observations, we find that the rhom-\nbohedral R3cferroelectric phase ( Rphase) is the ground\nstate of the material at ambient conditions of pres-\nsure. Then, an orthorhombic Pnmastructure( Ophase),\nwhich is the vibrationally-softest of all the considered\nstructures, is found to stabilize upon increasing Torp.\nMore precisely, two first-order phase transitions of the\nR→Otypearepredictedtooccuratthethermodynamic\nstates [0 GPa, 1300(100) K] and [3.6(1) GPa, 0 K].\nAdditionally, a representative of the so-called nano-\ntwinned structures recently predicted to occur in BFO24\nhas been analyzed in this work. This phase is found to\ndisplay elastic and vibrational features that are reminis-\ncent of the results obtained for both the RandOstruc-\ntures, and to become energetically more stable than the\nRphase upon raising pandT. The entropyand enthalpy\nof theOphase, however, turn out to be more favorable\nthan those of the studied Cstructure over practically all\nthe investigated p−Tintervals, and as a result we donot observe any direct R → CorC → Otransformation.\nNevertheless, our results cannot be conclusive in this\npointduetothelimitationsofthestudy(onlyonespecific\nnano-twinned structure is investigated) and DFT-related\naccuracy problems that appear when tackling very small\nfree-energy differences (i.e., of order 1 −10 meV/f.u.).\nIn fact, our results seem to support the possibility that\nsome nano-twinned structures may become stable at the\nboundariesbetween RandOphasesinthe p−Tphasedi-\nagram of BFO, or at least exist as long-lived meta-stable\nphases that are likely to be accessed depending on the\nkinetics of the R → Otransformation.\nFinally, we find that a representative of the so-called\nsuper-tetragonal phases of BFO gets energetically desta-\nbilized over the rest of crystal structures by effect of in-\ncreasing temperature, due to the fact that its spectrum\nofphononfrequenciesis globallythe stiffest one. This ex-\nplains why super-tetragonal structures have never been\nobservedinbulkBFO,inspiteofthefactthattheirDFT-\npredicted energies are very close to those of the RandO\nphases. Interestingly, the investigated super-tetragonal\nstructure is also destabilized upon hydrostatic compres-\nsion.\nAs far as we know, our work is the first application\nof the quasi-harmonic free energy method to the study\nof the phase diagram of a multiferroic perovskite sys-\ntem. The main advantages of this approach are that is\ncomputationally affordable, can be straightforwardly ap-\nplied to the study of crystals, and naturally incorporates\nzero-point energy corrections. Among its shortcomings,\nwe note that it can be exclusively applied to the anal-\nysis of vibrationally stable crystal structures; further,\nit only incorporates anharmonic effects viathe volume-\ndependence of the phonon frequencies and correspond-\ning treatment of the thermal expansion, which may be a\nquestionable approximation at high temperatures. Nev-\nertheless, we may think of several physically interesting\n(andcomputationallyverychallenging)situationsinvolv-\ning BFO-related multiferroics in which the present ap-\nproach can prove to be especially useful. A particularly\ninteresting possibility pertains to the study of solid solu-\ntions, i.e., bulk mixtures of two or more compounds, at\nfinite temperatures. By assuming simple (or not so sim-\nple) relationsamongthe freeenergyofthe compositesys-\ntem, the relative proportion between the species, and the\nvibrational features of the integrating bulk compounds,\none may be able to estimate the phase boundaries in the\ncomplicated x-p-Tphase diagrams at reasonably modest\ncomputational effort. In this regard, the BiFeO 3-BiCoO 3\nand BiFeO 3-LaFeO 3solid solutions emerge as particu-\nlarly attractive cases, since the application electric fields\nin suitably prepared materials can potentially trigger the\nswitching between different ferroelectric-ferroelectricand\nferroelectric-paraelectric phases.76,77\nBeyond possible applications, studying the BiFeO 3-\nBiCoO 3solid solution is by itself very interesting. On\nthe one hand, this is a case involving transitions be-\ntween phases that are verydissimilar structurally (super-17\ntetragonal and quasi-rhombohedral), and which have dif-\nferent magnetic orders (C-AFM and G-AFM). Hence, in\nthis case we can expect spin-phonon effects to have a\nlarger impact in the free energy, which would allow us\nto better test the spin-phonon quasi-harmonic approach\nthat we have introduced in the present work. Addi-\ntionally, the treatment of the C-AFM order requires a\nmorecomplicatedmodel ofexchangeinteractions, involv-\ning at least two (preferably three72) coupling constants.\nHence,treatingC-AFMphasesrequiresaextensionofthe\nscheme here presented, so that it can easily tackle more\ngeneral situations. Work in this direction is already in\nprogress within our group.Acknowledgments\nThis work was supported by MINECO-Spain [Grants\nNo. MAT2010-18113 and No. CSD2007-00041] and the\nCSIC JAE-doc program (C.C.). We used the supercom-\nputing facilities provided by RES and CESGA, and the\nVESTA software78for the preparation of some figures.\nThe authors acknowledge very stimulating discussions\nwith Massimiliano Stengel.\n1M. Fiebig, J. Phys. D 38, R123 (2005).\n2M. 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Crystallogr. 41, 653\n(2008)."    },    {        "title": "2007.13994v1.Magnetic_bubbles_in_an_M_type_hexagonal_ferrite_observed_by_hollow_cone_Foucault_imaging_and_small_angle_electron_diffraction.pdf",        "content": "1 \n Magnetic bubble s in an M -type hexagonal ferrite  observed by hollow -\ncone Foucault imaging and small -angle electron diffraction  \nAtsuhiro  Kotani1, Hiroshi  Nakajima1, Atsushi  Kawaguchi1, Yukihiro  Fujibayashi1, Kento Uchihashi1, \nKeiko Shimada2, Ken Harada1,2, and Shigeo  Mori1,* \n \n1Department of Materials Science, Osaka Prefecture University, Sakai, Osaka 599 -8531, Japan  \n2Center for Emergent Matter Science (CEMS), The Institute of Physical and Chemical Research (RIKEN), \nHatoyama, Saitama 350 -0395, Japan  \n \n*E-mail; mori@mtr.osakafu_u.ac.jp  \n \n \nABSTRACT  \nWe report  hollow -cone imaging and small -angle electron diffraction of nanoscale magnetic textures such as  \nmagnetic -striped domains  and magnetic  bubbles of M-type hexagonal ferrite BaFe 10.35Sc1.6Mg 0.05O19. The \nadvantage of the hollow -cone Foucault  method is that magnetic domains with various directions of \nmagnetization  can be visualized under an infocus condition.  Moreover,  the contrast of  magnetic domain walls \nin magnetic bubbles depend s on the inclination angle of the illumination  beam . The combination of small -\nangle electron diffraction and hollow -cone Foucault imaging prove s that magnetization at domain walls \nexhibits in-plane direction s in the magnetic -striped domains and  magnetic bubble s. \n  2 \n 1. Introduction  \nLorentz transmission electron microscopy ( Lorentz microscopy ) is an effective observation method for \nvisualizing nanoscale magnetic textures in real space.1) Since its development2), Lorentz microscopy  has played \na significant role in revealing interesting magnetic domains.  Thus far , several methods have been employed \nfor Lorentz microscopy  including  Fresnel imaging (visualization of domain walls by defocusing), Foucault \nimaging (visualization of domains by selecting magnetic -deflection spots)1),3), differential phase contrast4),5), \nfocal -series reconstruction methods6),7), phase microscopy (especially using hole -free phase plate)8), and electron \nholography9),10). Further , Lorentz microscopy  itself has been improved  to measure magnetic domains \nquantitatively. For example, Foucault imaging combined with small -angle electron diffraction  (SmAED)  can \nbe used to visualize  magnetic  domains under external magnetic  fields and analyze  the magnitude of \nmagnetization.11)–13) Moreover, Foucault imaging using an electron biprism makes it possible to present two \ntypes of 180 ° magnetic domains and remove diffraction effects.14)  \nRecently, we developed  hollo w-cone Foucault (HCF) imaging with SmAED  using a conventional \ntransmission electron microscope .15) HCF imaging can be used to visualize several magnetic domains with \nvarious azimuthal direction s of magnetization by rotating the incident beam. The advantage of HCF imaging \nis that both ma gnetic domains and domain walls are visualized simultaneously with sufficient contrast under \nthe infocus condition. This p revious stud y used HCF imaging revealed  that 90° and 180 ° domains  have in -\nplane magnetization in FeGa . However, HCF imaging has not been applied to other types of magnetic domains; \nfurther, the contrast of HCF imaging has not been clarified  for other domain structures .  \nTherefore , in this study , we appl y HCF imaging to the observation of nanoscale magnetic textures such as \nmagnetic striped domains  and magnetic bubbles  in hexagonal ferrites of BaFe 10.35Sc1.6Mg 0.05O16 (BFSMO)16)–\n20). We chose this compound  for this study  because it has out -of-plane  magnetization in magnetic domains and \nin-plane magnetization in domain wall s, and this type of magnetic domain  has not been observed using HCF \nimaging. Furthermore, BFSMO exhibits magnetic bubbles , which are circularly rotating magnetic structures \nunder external magnetic fields. Although whirling magnetic structures such as magnetic bubbles and \nskyrmions have attracted considerable attention because of intriguing phenomena,21)–29) magnetic bubbles have \nbeen observed by Fresnel imaging (out -of-focus method) . Further, it is difficult to observe them using \nconventional Foucault imaging because they have an all in -plane direction of magnetization due to the rotating  \nstructures.  In this paper , we report the observation results of HCF imaging of BFSMO and reveal the contrast \nreversal of magnetic bubbles that depends on the inclination angle of the illumination beam . Our results  show 3 \n that magnetization is oriented in the  out-of-plane direction inside the magnetic bubbles , and it is oriented in \nthe in -plane direction at the domain wall s. \n \n2. Experimental methods  \nFigure 1 shows a schematic of the configuration used for HCF  imaging  in this study . For magnetic domain \nobservation, the objective lens  was turned off , and then , the current in the objective mini-lens was adjusted so \nthat a crossover (diffraction spots) was formed at the selected -area (SA) aperture plane. The SA aperture \noperates as an inclination angle sele cting aperture in the optics  of the HCF configuration . These conditions are \nsimilar to those of conventional SmAED  optics .11),30) The electron beam irradiate s the specimen with an area \nof approximately 85 m in diameter with tilt angles in the X and Y directions controlled using the beam \ndeflector system placed above the specimen. The circulating electron beam irradiate s the specimen  in all \nazimuthal directions around the optical axis. During HCF imaging, when the incident beam was rotated  around \nthe optical axis , the diffraction pattern rotated synchronously with it . For HCF imaging,  some rotating spots \ncan be selected using the SA aperture  whose diameter of  20 m corresponds  to 2.60  × 10−4 rad. The \nintermediate and projection lens es can be used to adjust the image magnification and camera length of \ndiffraction patterns. The camera length used in this study was 240 m. The HCF images were obtained by \nintegrating the images during the incident beam rotation, which enables us to visualize magnetic domains and \nmagnetic domain walls that cause diffraction spots  in various directions . The details of the optical system are \ndescribed in our previous work.15)  \nThis study employed  two conditions of  the inclination angle of the illumination beam :  \n<  (1) \n–  < <\nIn Eqs. (1) and (2), is the angle of the magnetic  diffraction  spots, is the angle limited by the SA aperture, and  \nis the inclination angle of the i ncident beam . Figure 1(b) illustrates the relationship between th e angles of magnetic \ndiffraction  spots  and the SA aperture , which corresponds to Eq. (1) . The transmitted beam  without  magnetic \ndiffraction (black spot)  can pass through the SA aperture ; some magnetic diffraction  spots are blocked by the \naperture. Hereafter, we refer to this mode as bright -field (BF) HCF  imaging  because the transmitted beam is \nused for visualization.  Conversely,  Fig. 1(c) shows  the situation corresponding to Eq. 2. In this condition, the SA 4 \n aperture blocks the transmitted beam, and thus, it is referred to as  dark-field (DF) HCF  imaging . Note that the \nbeam -convergence semi -angle  should be smaller than  to observe magnetic  diffraction  spots.  \nHCF imaging w as performed using a transmission electron microscope JEM -2100F  (200 kV, JEOL Co. \nLtd., Japan). An external magnetic field was applied using the objective lens.  A single crystal of BFSMO was \ngrown via the floating zone method.18) The c plane , which was oriented  perpendicular to the magnetic easy \naxis, was polished and then thinned by Ar-ion milling.  \n \n3. Results and discussion  \nFirst, we clarify  the relationship between the Fresnel image and SmAED pattern of BFSMO (Fig. 2) for \nHCF imaging . The contrast of the Fresnel image indicates that magnetization is oriented in the out-of-plane \ndirection in the magnetic domain s and in the in -plane direction in the magnetic domain wall s, as reported in \nRef 18). The SmAED pattern recorded with a 240 m camera length shows a transmitted  spot 000 and magnetic  \ndiffraction  spots.  Magnetic  diffraction  spots are formed in a ring shape because the magnetic domains show a \nmaze pattern with all azimuthal direction s. The period calculated from these magnetic spots  (5.84 rad) is 430 \nnm, which corresponds to twice the magnetic -domain width. As indicated by the re d arrows in the inset of Fig. \n2(a), the domain walls have alternating  magnetization s in the  upwards  and down wards  directions  and a \nmagnetic period of approximately 440 nm . This agrees  with the periodic arrangement of the SmAED pattern . \nThis indicat es that the magnetic  diffraction  spots  originate  from the domain walls , as demonstrated in  the \nprevious study31).  \nTo perform the HCF imaging  of magnetic bubbles , we applied an external magnetic field of 2 T and then \nreduced it to zero.  This process change d the striped domains into magnetic bubbles.  Figure 3(a) shows a Fresnel \nimage in which magnetic stripe s and magnetic bubble s coexist  in a remanent magnetization state. The red \narrow s of Fig. 3(b)  indicate the direction s of magnetization  inferred  from the contrast of the Fresnel image. \nThe Fresnel image demonstrates that the in -plane direction component of the magnetization rotates clockwise \nor counterclockwise in the domain wall s of the magnetic bubbles.  Similar to Fig. 2,  the SmAED  pattern depic ts \nring-shaped scattering because of the various azimuthal directions at  the domain walls ; again , diffraction spot s \ncorresponding to the periodic arrangement of domain walls can be seen  clearly . \nThe HCF  image s of magnetic striped domains and magnetic bubbles  are shown in Figs. 3(c) and 3(d). \nFigure s 3(c) and 3(d) were  obtained  under conditions corresponding to  Eqs. (1) and (2), respectively : \nrad, 2.60 × 10-4 radand   15.3 rad for Eq. (1) and    18.8 rad for Eq. (2).  In Fig. 3(c), the 5 \n transmitted spot  000 was selected in addition to some magnetic diffraction  spots  with the SA aperture  (Eq. (1)). \nIn this condition , the domains are depicted as bright areas , while the  domain walls are dark lines , as shown in \nFig. 3(c) . The contrast can  be explained as follows : As demonstrated in Fig. 2, magnetic  diffraction  spots in \nthe SmAED pattern originate from the domain walls , which have in -plane magnetization . The transmitted  spot \n000 w as caused by the vacuum or domains in the specimen with the magnetization  parallel to the incident \nbeam . Thus , the domain walls show as dark lines , compared with the domains themselves , when  some of the \nperiodic spots are blocked  by the aperture  during the hollow cone irradiation process.  \nConversely , Fig. 3(d) shows the HCF ima ge captured using some of the magnetic  diffraction  spots  and by \nexcluding the transmitted spot 000 [Eq. (2)]. Unlike  Fig. 3(c), the ma gnetic domain  walls appear as bright lines \nunder these  experimental condition s. The contrast change is attributed  to the domain walls causing magnetic \nspots that w ere selected with the aperture.   \nThe contrast reversal is  clearly  confirmed in Fig. 4. These  images  are magnified images of BF and DF \nHCF images around magnetic bubbles. The circular domain walls are visualized as dark and bright lines in Fig. \n4(a) and 4(b), respectively.  The intensity profiles along the line X–Y indicate contrast reversal at the domain \nwall.  As explained above, the region s with the in -plane magnetization component can be seen as bright in the \nmagnetic bubbles  because the DF HCF image was formed only using magnetic  spots .  \nMagnetic bubbles comprise domain walls aligned with all azimuthal directions. Thus, conventional \nFoucau lt imaging cannot visualize magnetic bubbles  as it can only depict  domains with one particular direction \nof magnetization . HCF imaging, which has the ability to visualize domains with directional scattering  in \nmultiple directions , thus  support s magnetic domain observation. In addition , Fresnel image s (e.g., Fig. 3b) \nindicate  domain walls as a pair of bright and dark lines . HCF imaging can visualize  magnetic domain walls as \na single bright or dark line  (BF image of Fig. 4a and DF image of Fig. 4b) . Consequently, by combination with \nFresnel imaging, HCF imaging  makes  it possible  to understand  the presence of  domain walls  even when bend \nconto urs and Fresnel  fringes from twin boundaries and edges  exist . \nThese  experimental results can be understood in a similar manner as those in a previous study  of 180° \nmagnetic domains in FeGa15). Noticeably, the relationship between domain  and domain wall contrast in this \nstudy is opposite to that of FeGa. In the 180° domain structure of FeGa, the magnetization is oriented in the \nin-plane  direction inside the domains , and it is oriented in the out -of-plane direction in the dom ain wall. As a \nresult, the domain wall s were  observed as  bright line s in a BF HCF image  and as dark lines in a DF HCF image.  \nConsidering that magnetic  spots were caused by  in-plane magnetic components , the contrast of the previous \nstudy  can be explained  in a similar manner  to that of this study15). In addition,  phase microscopy studies support 6 \n a magnetic domain structure  similar to that obtained by HCF imaging.7),8) Therefore, the present HCF imaging \nexperimental results  demonstrate  that magnetization at the domain walls points to the in -plane direction  in \nBFSMO .   \n \nConclusions  \nMagnetic textures  and the magnetization distribution s of hexagonal ferrite  BFSMO were investigated  \nusing SmAED  and HCF imaging . We demonstrated that  HCF imaging  can visualize  magnetic domain walls \nand magnetic bubbles ; the contrast depends on the inclination angle of the illumination  beam.  In BFSMO , BF \nHCF imaging depict ed domain walls as dark lines , wher eas in DF HCF imaging , they were depicted as bright \nlines.  HCF imaging combined with SmAED demonstrated that  this material exhibits in-plane magnetization  \nin the magnetic domain wall s. \n \n \nAcknowledgments  \nThis work was supported by KAKENHI, Grant -in-Aid for Scientific Research (B) JP18H03475 and (S) \nJP19H05625 . This work was also in part supported by JSPS KAKENHI Grant Number JP19H05814 (Grant -\nin-Aid for Scientific Research on Innovative Areas “Interface Ionics ”).  7 \n References  \n1) J. N. Chapman, J. Phys. D. Appl. Phys. 17, 623 (1984).  \n2) H. W. Fuller and M. E. Hale, J. Appl. Phys. 31, 238 (1960).  \n3) M. De Graef, Introduction to conventional transmission electron microscopy  (Cambridge University \nPress, 2003).  \n4) S. McVitie, D. McGrou ther, S. McFadzean, D. A. MacLaren, K. J. O’Shea and M. J. Benitez, \nUltramicroscopy 152, 57 (2015).  \n5) D. McGrouther, R. J. Lamb, M. 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Phys. 58, 55006 (2019).  \n  9 \n  \nFig. 1. (a) Schematic  of optical  configuration  of hollow -cone Foucault imaging.  Yellow  and red (blue)  lines \nrepresent  the direct and diffracted beams, respectively. Diffraction spots with all azimuthal directions  such as \nthe red and blue lines  can pass through the selected -area aperture t o appear in the hollow -cone Foucault  image.  \n(b) Relation ship between diffraction spots and SA aperture  in a bright -field mode.  (c) Relation ship between \ndiffractio n spots and SA aperture in a dark-field mode. The labels  indicate  the inclination angle of the \nillumination beam , diffraction angle , and angle limited by the selected -area aperture . The angle  \nrepresent s the divergence semi -angle.  (b) and (c) correspond to the condition s of Eqs. (1) and (2), respectively.   \n10 \n  \nFig. 2. (a) Fresnel image and (b) small -angle electron diffraction  (SmAED)  pattern of magnetic striped domains \nin BFSMO. In the inset of (a), the region indicated by the white dotted square is magnified. Red arrows \nrepresent t he directions of the magnetization  estimated from the Fresnel image .  \n  \n11 \n  \n \n \nFig. 3. (a) Fresnel image of a coexisting state of striped domains and magnetic bubbles at room temperature \nunder no magnetic field after applying an external field of 2 T. The inset is a small -angle electron diffraction \npattern (scale bar  20 rad). (b) Magnified image of the area surrounded by the white dotted rectangle  in (a). \nRed arrow s indicate the direction s of magnetization. (c) Bright -field and (d) dark -field HCF images. The insets \nof (c) and (d) show the ranges selected with the selected -area aperture for the HCF imaging  (scale bar 20 rad).  \n \n  \n12 \n  \nFig. 4. (a) Bright -field (BF) HCF image of BFSMO, (b) dark -field (DF) HCF image, and (c) intensity profiles \nalong the red and blue lines (X –Y) in panels (a) and (b). The field of view in panels (a) and (b) is  the same as \nthat of Fig. 3(b). The contrast of the magnetic bubbles  in the BF and DF HCF images are opposite to each \nother.  \n"    },    {        "title": "1707.08451v4.Bismuth_Ferrite_Dielectric_Nanoparticles_Excited_at_Telecom_Wavelengths_as_Multicolor_Sources_by_Second__Third__and_Fourth_Harmonic_Generation.pdf",        "content": "Bismuth Ferrite Dielectric Nanoparticles Excited at Telecom Wavelengths as\nMulticolor Sources by Second, Third, and Fourth Harmonic Generation\nJeremy Riporto,1, 2Alexis Demierre,2C\u0013 edric Schmidt,2Gabriel Campargue,2Vasyl Kilin,2Tadas Balciunas,3\nMathias Urbain,1Andrius Baltuska,3Ronan Le Dantec,1Jean-Pierre Wolf,2Yannick Mugnier,1and Luigi Bonacina2\n1Universit\u0013 e Savoie Mont Blanc, SYMME, F-74000 Annecy, France\n2GAP-Biophotonics, Universit\u0013 e de Gen\u0012 eve, 22 chemin de Pinchat, 1211 Gen\u0012 eve 4, Switzerland\n3Photonics Institute, TU Wien, Gusshausstrasse 27/E387, 1040, Vienna, Austria\nWe demonstrate the simultaneous generation of second, third, and fourth harmonic from a sin-\ngle dielectric Bismuth Ferrite nanoparticle excited by a telecom \fber laser at 1560 nm. We \frst\ncharacterize the signals associated with di\u000berent nonlinear orders in terms of spectrum, excitation\nintensity dependence, and relative signal strengths. Successively, on the basis of the polarization-\nresolved emission curves of the three harmonics, we discuss the interplay of susceptibility tensor\ncomponents at the di\u000berent orders and we show how polarization can be used as an optical handle\nto control the relative frequency conversion properties.\nKeywords: harmonic generation; harmonic nanoparticles; perovskites; bismuth ferrite; frequency\nconversion.\nIntroduction\nThe generation and control of nonlinear parametric sig-\nnals at the nanoscale is paving the way to novel appli-\ncations in imaging, sensing, optoelectronics. To date,\nmost of the research e\u000borts have been concentrated on\nnoble metal nanoparticles and nanostructures1with a fo-\ncus on their second ( \u001f(2))2,3and third order ( \u001f(3))4,5re-\nsponse. Some notable exceptions include the nonlinear\nharmonic generation by semiconductor nanoparticles6,7,\ntwo dimensional materials8{11, and noncentrosymmet-\nric metal oxide nanoparticles (Harmonic NanoParticles,\nHNPs). Dielectric HNPs are attracting growing interest\nbecause of their extremely high nonlinear coe\u000ecients,12\nand robustness of their nonlinear response which - con-\ntrary to noble metal particles - is primarily associated\nwith their bulk properties and negligibly a\u000bected by sur-\nface phenomena.13,14Moreover, the sub-wavelength di-\nmensions of HNPs lift the spectral limitations imposed by\nphase-matching conditions in bulk nonlinear crystals, en-\nabling wide tunability of excitation light and emission of\nmultiple signals at once. Some research groups are work-\ning on the e\u000eciency enhancement of the optical prop-\nerties by engineering hybrid systems based on a HNP-\ncore and a plasmonic-shell tailored for speci\fc spectral\nresonances.15,16\nRecently, we have demonstrated the simultaneous ac-\nquisition of Second and Third Harmonic Generation\n(SHG, THG) by bare individual perovskite Bismuth Fer-\nrite (BiFeO 3, BFO) HNPs.17We showed that the coinci-\ndent acquisition of both harmonics can strongly bene\ft to\nimaging selectivity in optically congested environments18\nfor applications including cell-tracking over long time\nin tissues.19Besides harmonic generation, one can ex-\npect that high \u001f(n)values by HNPs can be exploited for\ndisposing of localized sources of long wavelength radia-\ntion by optical recti\fcation or for generating nonclassi-\ncal states of light, in analogy to what has been demon-\nstrated using other kinds of nanostructures.20{23In thisrespect, the possibility of working e\u000eciently at telecom\nwavelengths (1.5 \u0016m) undeniably constitutes an asset for\na future integration of HNPs as frequency conversion el-\nements and all-optical logic operators24in photonics cir-\ncuits.\nIn this work, we demonstrate that second, third, and\nfourth harmonic (FHG) emitted by an individual BFO\nHNP upon excitation at 1560 nm by an Erbium-doped\n\fber oscillator can be e\u000eciently detected. Moreover,\nwe show how the polarization control of excitation light\nallows tuning the relative intensities of the three har-\nmonics. The simultaneous acquisition of three har-\nmonics from the same individual nanoparticle is - to\nour best knowledge - a unicum to date and, besides\nall the applications we mentioned, HNPs might assume\nthe role of model system for the study of the interplay\namong multiple-harmonics and high harmonic generation\nin solids.25{28\nResults and Discussion\nThe starting evidence motivating this work is the ob-\nservation that single BFO HNPs deposited on a sub-\nstrate in the focus of the laser emit simultaneously at\nthe three harmonics as from the images in Fig. 1A. The\nheat-maps colors are red for SHG (780 nm), green for\nTHG (520 nm), and blue for FHG (390 nm). In the\nfollowing, we \frst present a thorough assessment demon-\nstrating by independent experimental observables [i) im-\nage spot size, ii) spectrum, and iii) excitation intensity\ndependence] that the three emission are genuinely asso-\nciated with di\u000berent nonlinear orders. Successively, we\ndiscuss the polarization-resolved emissions at the di\u000ber-\nent orders which shed light on the tensorial properties\nof the nonlinear susceptibilities and could prospectively\nbe exploited for selective frequency up-conversion from\nshort-wave infrared to the visible.\ni) The Gaussian \fts to the diameters of the particlearXiv:1707.08451v4  [physics.optics]  10 Mar 20182\nlog(Normalized harmonic \nemission intensity)\n100 200300400\nIntensity at focus [GW/cm2]420390360 550520490\nWavelength [nm]810780750Normalized harmonic \nemission intensity\n100200300400\nIntensity at focus  [GW/cm2]SHG THG FHGA\nB\nC\nC'\nLog-Log I2  (1.93)\n I3  (3.01)\n I4  (3.68)\nFIG. 1: A.Images taken at the di\u000berent harmonics of a single\nisolated BFO HNP. The intensity pro\fle obtained at each har-\nmonic (colored dots) is \ftted by a Gaussian function (dashed\nlines). Scale bar 500 nm. B.Normalized harmonic spectra.\nThe interval between two dashed vertical lines corresponds to\n30 nm. C.Normalized power dependence of the intensities of\nthe three harmonic emissions. The continuous lines represent\nthe nominal Intraces with n= 2;3;4. The optimal nvalues\nobtained by \ftting the experimental traces are reported in\nparentheses (\fts not shown). C'.Log-Log representation of\nthe data in panel C.\nimages in Fig. 1A indicate that the FWHM decreases\nwith increasing nonlinearity, as one expects for a di\u000brac-\ntion limited object smaller than the point spread func-\ntion (PSF) at the highest order. The observed widths\nof the PSF range from 673 nm for SHG, to 486 nm for\nTHG and 420 nm for FHG. The average dimensions of\nthe HNPs (\u0019100 nm, Fig. S1) remain therefore out of\nreach at all orders. In the Supplementary Material, wefurther comment these results in the context of the imag-\ning properties of the set-up. ii) In Fig. 1B, we provide\nthe normalized harmonic spectra detected in the forward\ndirection. In the wavelength domain, one expects the\nwidth of the emission to scale as /1\nnpn, withnbeing\nthe nonlinear order. This formula is derived for Gaussian\npulses in the time domain.29Therefore, to apply this es-\ntimation to our traces stemming from a structured spec-\ntral pro\fle at the excitation wavelength, we proceeded by\nvisually determining the broadest Gaussian curves sup-\nported by the excitation and by each harmonic spectrum\n(Fig. S2). This way, we obtained widths for the di\u000ber-\nent harmonics within 10% deviation from the theoretical\nestimation. This procedure, although involving approxi-\nmations, points to a rather complete upconversion of the\nfrequencies in the fundamental spectrum and it is consis-\ntent with the fact that BFO HNPs are smaller than the\ncoherence lengths for each nonlinear order, l(n)\nc. By using\nthe optical constants of BFO derived by Kumar et al.30\nand applying a calculation including the e\u000bect of Gouy\nphase (see Eq. S1) we obtain l(n)\ncvalues in the forward\ndirection spanning from 1 \u0016m forn= 2 to 0.325 and 0.23\n\u0016m forn= 3 and 4, respectively.4In our calculations,\nl(n)\ncdeem larger than HNPs typical dimensions. This im-\nplies that no destructive interference takes place within\nthe particle volume. iii) To complete this preliminary as-\nsessment of multiorder response, in Fig. 1C, we present\nthe harmonic signal strength as a function of the laser\nintensity at the sample, I. Note that for this comparison\nthe signals are normalized at the maximal laser intensity\nof the series, which corresponds to 440 GW/cm2. As dis-\ncussed in the next section, in absolute terms the THG is\nby far the most intense under these excitation/detection\nconditions: roughly 2 orders of magnitude stronger than\nSHG and 4 orders stronger than FHG. In the image, the\nnominal \ftting curves ( i.e.,In,n= 2;3;4) are plotted\nas continuous lines. One can appreciate their fairly good\nagreement with the experimental data. In the legend, we\nreport the optimal values for the exponent nobtained let-\nting this parameter free to vary in the \ftting procedure.\nThe retrieved values are all within 10% deviation from\nthe theoretical values. In Fig. 1C', the data and \ftting\ncurves from panel C are provided in log-log representa-\ntion. Altogether these results obtained by independent\nmeasurements (nonlinear PSF, harmonic spectra, inten-\nsity dependence) support the association of the signals\nfrom single HNPs with three di\u000berent harmonics: SHG,\nTHG, and FHG.\nRelative intensities of harmonic orders.\nA natural question arises concerning the relative in-\ntensities of the three emissions, as one would normally\nexpect a major decrease in signal strength with increas-\ning nonlinear rank, provided that the symmetry require-\nments ( i.e., noncentrosymmetricity) are ful\flled for the\ngeneration of even orders (SHG, FHG,...). Clearly, one3\nγHNP1 HNP2A\nB\nFIG. 2: Polarization sensitive response. A.The shaded re-\ngions represent the emission intensity at the di\u000berent har-\nmonic orders from two isolated BFO HNPs as a function of\nthe polarization angle \r. The dashed purple line are \fts to\nthe traces obtained using the \u001f(2)and\u001f(3)tensors reported in\nSchmidt et al.17for 1064 nm excitation. The corresponding\nEuler angles sets ( \u001e; \u0012;  ) we retrieved are (78\u000e, 38\u000e, 314\u000e)\nfor HNP1 and (68\u000e, 13\u000e, 77\u000e) for HNP2, respectively. B.Al-\nternative representation to highlight the selective frequency\ngeneration obtained by varying the polarization angle \r. The\nimages are created by adding as RGB components the nor-\nmalized SHG, THG, and FHG polarization-resolved traces in\nA.\nshould also take into proper account the di\u000berent in-\ntensity dependence exhibited by signals associated with\neach\u001f(n): higher excitation intensity is expected to\nfavour higher orders as the ratio of two successive har-\nmonics scales as /1\nI.32In a previous Hyper Rayleigh\nScattering experiment at 1064 nm, we determined that\nSHG/THG'40 for 11 GW/cm2excitation.17Therefore,\nwe could expect this ratio to be here 61 when work-\ning at 53-fold larger intensity, viz. 590 GW/cm2(280\npJ/pulse). However, we observe a surprisingly smallerSHG/THG value, of the order of 10\u00002-10\u00003. Sample res-\nonances play an important role in determining the value\nof harmonic ratios:33a resonance was reported at 504 nm\nfor BFO 25 nm thin \flms30supporting the e\u000ecient gen-\neration of THG observed here at 1560 nm excitation (a\ngreen spot is visible by naked eyed on small aggregates).\nThis close-to-resonance condition can also help explain-\ning the very high second order susceptibility reported for\nBFO HNPs excited at 1064 nm, which was estimated to\n160 pm/V.1The FHG/THG ratio is, on the other hand,\nof the order of 10\u00004. Being aware that, among all tech-\nniques, the values extracted by microscopy present the\nlargest uncertainty because they imply averaging the re-\nsponse of individual particles (10 in the present case, Fig.\nS3) with di\u000berent spatial orientations modulating their\nharmonic ratios, we complemented these measurements\nwith additional ones performed on pellets of compressed\nBFO HNPs (Fig. S4A). These measurements were car-\nried out at 1 TW/cm2using a \u0016J laser system, averaging\nthe response of a large ensemble of randomly oriented\nparticles over an elliptic area of 60 \u0002120\u0016m2. By this ap-\nproach, we obtained SHG/THG \u001920 (Fig. S4B) while the\nFHG/THG is\u001410\u00004. Such a large discrepancy among\nthe outcomes of the two methods, in particular for the\nSHG/THG ratio, is not fully clear. On one hand, the\npresence of aggregates in the pellets with dimensions ex-\nceeding the coherent length of BFO can a\u000bect the signal\nin an uneven way throughout the spectral domain. More-\nover, the comparison can be also undermined by the dif-\n\fculty to \fnd a meaningful de\fnition of peak intensity\nencompassing both large particles ensembles and isolated\nobjects substantially smaller than the focal spot size. Fi-\nnally, the di\u000berence observed can be ascribed to the crit-\nical dependence of coherent signals generated by individ-\nual nanostructures on experimental settings ( e.g., N.A.\nand collection angle). This last aspect has been subject of\nmultiple theoretical studies in the plasmonic community\nbased on di\u000berent approaches (method of moments,35\n\fnite elements,36hydrodynamic model37,38). Recently,\nthe hydrodynamic approach has been applied to calcu-\nlate the SHG and THG angular radiation patterns si-\nmultaneously emitted by individual plasmonic nanopar-\nticles, which speci\fcally highlights this sensitivity to de-\ntection parameters showing rather di\u000berent angular emis-\nsion patterns at the two harmonics.39We believe that\nonly a rigorous extension to higher harmonics of Hyper\nRayleigh Scattering on colloidal suspensions can provide\nreliable values for the material.40\nThe comparatively high conversion e\u000eciencies at the\nthird order we observe by both approaches for a non-\ncentrosymmetric material displaying very high quadratic\nnonlinearity such as BFO, can also be potentially as-\ncribed to the presence of multi-step (cascading) processes\ninvolving a succession of purely \u001f(2)phenomena: SHG\nand sum frequency mixing.28In this case, THG would\nresult from !+!= 2!and!+ 2!= 3!,41whereas\nFHG from !+!= 2!and 2!+ 2!= 4!or, alterna-\ntively, from !+!= 2!followed by 2 !+!= 3!and4\n3!+!= 4!.42It is tempting to attribute the compara-\ntively low emission at 2 !to a depletion of this frequency\nused as intermediate \feld for generating 3 !, however dis-\ncerning multi-step from direct higher order nonlinear pro-\ncesses is a complex task, in particular for nanoparticles\nas the absence of macroscopic propagation excludes dis-\ncrimination methods based on phase-matching criteria.41\nThe use of HNPs with controlled size and narrow size\ndistribution or epitaxial thin \flms of variable thickness\ncould help elucidating this aspect in a future series of\nexperiments.\nPolarization properties.\nIn Fig. 2A, we introduce the results on polarization\ndependence for two sub-di\u000braction limited and isolated\nparticles: HNP1 and HNP2. The shaded regions dis-\nplay the intensity of the harmonic emission detected as a\nfunction of the polarization angle of the excitation laser,\n\r. Note that di\u000berently from other works,17,43,44in this\ncase no polarization analyser was set in the detection\narm. The di\u000berences between the response of the two\nHNPs are associated with the di\u000berent spatial orienta-\ntions of their crystal axis with respect to the laboratory\nframe (Euler angles \u001e;\u0012; in Fig. 3). The simple in-\nspection of the polarization resolved traces can provide\nprecious information and it deems useful to discard from\nthe analysis eventual polycrystalline aggregates.43,44In\ngeneral, the SHG traces possess a structure character-\nized by two dominant lobes in agreement with our pre-\nvious observations.17For THG and FHG the side lobes\nbecome more prominent. Interestingly, the orientation of\nthe main lobes is mostly maintained among the even or-\nders (SHG, FHG) while for THG it seems that other ten-\nsor elements become predominant with major changes in\norientation and symmetry. In our previous study,17start-\ning from a known \u001f(2)tensor,30we \ftted the orientation\nof several BFO particles and then used the retrieved Eu-\nler angles to determine the unknown \u001f(3)tensor elements\nby simultaneously \ftting the THG response of several\nHNPs. Here, we use these tensor values for \u001f(2)and\u001f(3)\nto \ft the SHG and THG traces and obtain the Euler\nangles of each particle. The \fts are reported as purple\ndashed lines on the data and the angle sets for HNP1 and\nHNP2 provided in the \fgure caption. Although the \fts\ncorrectly capture the main features of the polarization\ncurves (main lobes angles, presence of orthogonal lobes),\none should be aware that this procedure implies several\napproximations and the result should be considered qual-\nitative in nature and primarily intended to support the\nfact that the BFO point group (3 m) is compatible with\nthe observed traces. In particular, the tensors we apply\nare derived at 1064 nm and not at 1560 nm. Note also\nthat we could not readily extend this approach to \u001f(4)be-\ncause the number of independent elements of this tensor\nprevents the retrieval of a reliable outcome. Finally, we\nhighlight that the possible presence of competing multi-step\u001f(2)processes would undermine the general valid-\nity of this description, which would remain however an\ne\u000bective tool for predicting the polarization dependent\nresponse of BFO HNPs even in presence of concurrent\ndirect and cascaded generation.\nPolarization-based control of relative harmonic\nintensities.\nThe response of the two randomly oriented HNPs sug-\ngests that the choice of the excitation polarization, even\nin absence of any detection analyser, can be used to mod-\nulate the relative intensities of the three emissions for a\ngiven laser polarization angle \r. In Fig. 2B, we graph-\nically emphasize this procedure showing the total emis-\nsion obtained by adding the normalized polarization de-\npendent harmonic components displayed by HNP1 and\nHNP2. This alternative representation shows how, for a\ngiven HNP orientation, speci\fc values of \rare associated\nwith strong simultaneous SH, TH, FH emission (white re-\ngions), with individual harmonics (red, green, blue in our\nrepresentation) and combination thereof (purple, pink...)\nor low emission (dark regions). We speculate that this ap-\nproach could be adapted to precisely oriented BFO HNPs\nand thin \flms with thickness smaller than the shortest\ncoherence length to provide polarization-controlled fre-\nquency converters from the telecom region over the vis-\nible spectrum. Engineered hybrid structures composed\nby HNPs with a plasmonic shell of tailored thickness\nor, alternatively, the choice of materials with tailored\nresonances,33could also be a way to mitigate the large\nconversion e\u000eciency di\u000berences at the three harmonic or-\nders for de\fned applications.15,16Alternatively, one could\nshape the excitation geometry to control to some extent\nthe angular emission pattern at the di\u000berent orders.45\nConclusions\nIn conclusion, we have reported what, to our best\nknowledge, is the \frst demonstration of simultaneous ac-\nquisition of three harmonic frequencies generated by an\nisolated nanoparticle. Notably, our experiment is per-\nformed using a pJ \fber laser at telecom wavelength,\nwhich holds great promise for implementing dielectric\nnonlinear nanophotonics46in optoelectronic circuitry.\nConsidered the novelty of our observation, we \frst thor-\noughly assessed the spectral and imaging properties and\nthe intensity dependence of the emissions to ensure that\nthey are genuinely associated with frequency conversion\nby\u001f(n)(n= 2;3;4) or cascaded \u001f(2)processes. The\nrelative intensities of the three harmonics have been criti-\ncally discussed highlighting the sensitivity of this param-\neter to the measurement method. All estimations point\nto high generation e\u000eciency for THG, likely because of\nthe presence of electronic resonances in the spectrum. Fi-\nnally, we have discussed the excitation-polarization de-5\nForward \nDetect.\nλ/2\nz\ny\nHighN.A.\nObj.xθ\nφψ\nHNP\nFemtosecond\nFiber\nLASER\nEγ\n1500 nm\n1600 nm\nEpi-\nDetect.Interf.Filter\nInterf.Filter3D piezoscannerA\nC\nγB\nFIG. 3: A.Schematics of the experimental set-up. Not\nshown: a spectrometer can be inserted in the forward de-\ntection arm and intereference \flters removed for acquiring\nspectrally resolved traces. B.Measured laser spectrum. C.\nEuler angles de\fning the HNP crystal axis orientation with\nrespect to the laboratory frame.\npendence of the particle emission, demonstrating that\nthis approach opens the way to directly investigating the\ninterplay among nonlinear susceptibility tensors elements\nat di\u000berent orders and modulating the relative strengths\nof three color components (red, green, violet) for photon-\nics applications.\nMethods\nBFO nanoparticles synthesized by the company FEE\nGmbH (Idar-Oberstein, Germany) were obtained as a\nwater stabilized colloidal suspension from the company\nTIBIO (Comano, Switzerland) under a research agree-\nment. The average size is estimated to \u0019100 nm by\ndynamic light scattering (DLS) and transmission elec-\ntron microscopy (Fig. S1). For imaging, a drop of BFO\nsuspension is cast onto a microscope substrate and the\nsolvent let evaporating.\nAs reported in Fig. 3, the light source of the set-up\nis a Telecom femtosecond \fber laser at 1560 nm with a\nrepetition rate of 100 MHz and 100 mW average power(T-Light FC , Menlo Systems). Pulses are compressed\ndown to 90 fs by an optical \fber connected to the laser\noutput. At the \fber output, the beam is collimated in\nthe free space and expanded to a diameter of 6 mm. For\npolarization resolved studies, the linear polarization of\nthe laser is rotated by a \u0015=2 plate mounted on a mo-\ntorized rotation stage. In the case of power dependence\nmeasurements, the laser energy is continuously modu-\nlated through the succession of a \u0015=2 plate and a polar-\nisation analyser. Afterwards, the beam is re\rected by\na 45 degrees short-pass \flter (Chroma) and focused on\na single isolated HNP by a 100 \u0002microscope oil immer-\nsion objective (NA 1.3). The signal generated by the\nparticles can be detected in the backward or forward di-\nrection. In the latter case, the collection objective is a\n40\u0002N.A. 0.6 air objective. HNPs are selected by scan-\nning a (x;y) planar ROI of approximately 20 \u000220\u0016m2\nwith a piezo-stage and carefully adjusting the zposition\nby maximizing their nonlinear signal. Both for epi- and\nforward-detection, narrow bandwidth interference \flters\nare used to select the harmonic spectral region (Thorlabs\nFBH780-10 for SHG, FBH520-40 for THG, FBH400-\n40for FHG and Semrock BrightLine Fluorescence Filter\n387/11 for FHG). Additionally, a scanning spectrome-\nter (Acton SP2300 , Princeton Instruments, 300 g/mm) is\nplaced in the forward detection arm to acquire spectrally\nresolved traces. The measurements are obtained using\ntwo di\u000berent Hamamatsu detectors, selected according\nto their spectral response: H7732-01 low noise side-on\nphotomultiplier tube (185 nm to 680 nm), and H7421-\n50photon counting head with a GaAs photocatode (380\nnm to 890 nm). Alternatively, we use a ultra-low-noise\nsingle photon counting module ( SPD-A-VISNIR , Aurea\nTechnology, Besan\u0018 con, France).\nI. ACKNOWLEDGEMENTS\nWe acknowledge the \fnancial support by Swiss SE-\nFRI (project C15.0041, Multi Harmonic Nanoparticles),\nby the French-Switzerland Interreg programme (project\nNANOFIMT), and by the NCCR Molecular Ultrafast\nScience and Technology of the Swiss National Science\nFoundation. This study was performed in the context of\nthe European COST Action MP1302 Nanospectroscopy.\nWe are grateful to Dr. Davide Staedler at TIBIO SA\n(Comano, Switzerland) and Dr. Daniel Rytz at FEE\nGmbH (Idar-Oberstein, Germany) for synthesizing and\nproviding us colloidally stable BFO HNPs, to Dr. Johann\nCussey from Aurea Technology (Besan\u0018 con, France) for\nproviding us the the single photon counting module and\ntechnological support, and Virginie Monnier (Institut des\nNanotechnologies, Lyon) for the TEM images of BFO\nHNPs.6\n1M. Kauranen and A. V. Zayats, Nature Photonics , 2012, 6,\n737{748.\n2J. I. Dadap, J. Shan, K. B. Eisenthal and T. F. Heinz,\nPhysical Review Letters , 1999, 83, 4045.\n3J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine,\nE. Benichou, C. Jonin and P.-F. Brevet, Nano Letters , 2010,\n10, 1717{1721.\n4M. Lippitz, M. A. van Dijk and M. Orrit, Nano Letters ,\n2005, 5, 799{802.\n5M. Danckwerts and L. 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Galez and J. P.\nWolf, Applied Physics B-Lasers and Optics , 2007, 87, 399{\n403.\n45L. Carletti, A. Locatelli, D. Neshev and C. De Angelis,\nACS Photonics , 2016, 3, 1500{1507.\n46D. Smirnova and Y. S. Kivshar, Optica , 2016, 3, 1241{\n1255.7\nI. SUPPLEMENTARY MATERIAL\nII. BFO HNPS CHARACTERIZATION\n200 nm25\n20\n15\n10\n5\n0 %\n102 34567\n1002 34567\n1000\nSize [nm]\nFIG. 1: TEM image of BFO HNPs and plot of the Dynamic Light Scattering distribution by number.\nA detailed description of the synthesis and properties of the nanoparticles used in this work can be found in Schwung\net al. ,1TEM and DLS representative data of a sample obtained by this protocol are reported in Fig. 1.\nIII. WIDTH OF THE POINT SPREAD FUNCTION (PSF) AT THE DIFFERENT HARMONIC\nORDERS\nTaking into account excitation wavelength and objective numerical aperture, the nominal lateral FWHM of a perfect\nimaging system under linear excitation should be FWHMtheo\nlinear =0.51\u0015/N.A.=612 nm.2For the nonlinear case, Zipfel et\nal.provide the following expression for a two-photon excited \ruorescence emitter: FWHMtheo\n2nd order =2p\nln 20:325\u0015p\n2NA0:91=\n391 nm.3These values cannot be applied here because the resolution is expected to be severely reduced by the fact\nthat we are using an high N.A. oil immersion objective intended for the visible region and not for an excitation at\n1.5\u0016m. Therefore all aberration corrections and optical elements (comprising the matching medium) are far from\noptimal. Indeed, we observe an energy reduction of 75% upon laser transmission through this objective, indicating a\npoor compatibility at this wavelength. By considering that the resolution should be proportional to 1 =pnwheren\nis the nonlinear order, we can readily compute an actual value of \u0019840 nm for the width of the linear PSF, both by\nmultiplying the FWHM FHG (420 nm) byp\n4 and FWHM THG (486 nm) byp\n3. Note that this result supports the\nfact that we are observing a sub-di\u000braction limited emitter at two harmonic orders. The same calculation applied to\nthe FWHM SHG (673 nm) provides a result \u001915% higher. In this series, SHG was epi-detected using the H7421-50\nphoton counting and THG and FHG forward detected by the H7732-01 low noise side-on photomultiplier tube. The\n15% discrepancy can very likely be attributed to the deviation from linear response of the former detector in the\nintensity regime of the measurement.\nNote that the FWHM value of 840 nm was used for the microscopy-based intensity ratio calculation.\nIV. ESTIMATION OF THE WIDTHS OF THE HARMONIC SPECTRA\nIn Fig. 2, we report the normalized spectra of the laser and of the three harmonics generated by a single BFO\nHNP along with Gaussian curves supported by these spectra and determined by visual inspection. On the \fgure we\nprovide the Gaussian FWHM and, in parentheses, the product FWHM \u0001npnwhich should be directly compared with\nthe laser spectrum as discussed in the main text.\nV. CALCULATION OF COHERENT LENGTHS AT DIFFERENT ORDERS\nThe coherence length is estimated using8\nSpectral intensity16001500 800 750\n540520500\nWavelength [nm]420400380360Laser SHG\nTHG FHG81 nm 27 nm\n15.5 nm 10 nm(76.4 nm)\n(80.5 nm) (80 nm)\nFIG. 2: Normalized spectra of laser and di\u000berent harmonics (continuous lines) along with Gaussian curves supported by the\nspectra (dashed lines).\nl(n)\nc=\u0019\nk(n!)\u0000nk(!)\u0000n\u0001kG(1)\nwheren= 2;3;4 for SHG, THG, and FHG, respectively. \u0001 kGis the wave vector corresponding to the Gouy-phase\nshift. The numerical value of \u0001 kGwas estimated at -0.5 \u0019=\u0015by Cheng and Xie for a 1.4 N.A. objective.4\nVI. INTENSITY RATIOS\nA. Measurements on individual particles\nIntensity ratio measurements by individual BFO HNPs were performed using two di\u000berent detectors to minimize\nthe need of e\u000eciency corrections among di\u000berent data sets. As reported in Fig. 3, SHG and THG were measured\nby detector 1 (SPD-A-VISNIR ultra-low-noise single photon counting module, Aurea Technology) and THG and\nFHG by detector 2 (H7732-01 low noise sideon photomultiplier tube, Hamamtsu). The traces highlight the particle-\nto-particle signal intensity variations, which come from di\u000berences in sizes (all signals are expected to scale as the\nparticle volume squared), orientations, and possibly varying radiation patterns. We further con\frmed these results\non magnitude estimation among the di\u000berent nonlinear orders employing a modi\fed set-up with a NA 0.4 re\rective\nAl-coated objective in the forward arm (Newport) and detecting all harmonics by an EM-CCD (Andor, Ixon3) placed\nat the imaging output of the spectrometer.\nB. Ensemble measurements on BFO particle pellets\nIn Fig. 4A, we provide a SHG image of the BFO HNPs pellet surface obtained by a commercial multiphoton\nmicroscope (Nikon A1R-MP) coupled with a Ti:sapphire oscillator (Mai Tai Spectra Physics). The epi-collected\nsignal was processed by a Nikon A1 descanned spectrometer. The image scale bar is 10 \u0016m. One can see how the\nSHG intensity of HNPs is modulated by their diverse orientation and that most of the particles appear as bright\ndi\u000braction limited spots. The emission spectrum averaged over the whole image is reported in Fig. 4B.\nFor comparing relative intensities of the harmonics on BFO HNPs on dry pellets we relied on the laser set up\nreported in Fig. 5. This system delivers \u001980 fs pulses at 1.5 \u0016m generated in an OPA pumped by a 1 kHz 14 mJ\n200-fs Yb:CaF2 CPA laser. The OPA is based on KTA crystals and seeded by a supercontinuum generated in a9\n10 8 6 4 2 0\nParticle numberSignal intensitySHG Det. 1\nTHG Det. 1\nTHG Det. 2\nFHG Det. 2\nFIG. 3: Forward detected signals at the di\u000berent harmonics generated by 10 distinct HNPs on a microscopy substrate. SHG is\nmeasured by detector 1, FHG by detector 2 and THG by both independently.\nbulk YAG plate and delivers 1.5 mJ signal pulses. The signal beam is \fltered out at the OPA output using a\nset of dichroic mirrors, the energy is attenuated using a half-wave plate and a polarizer and then focused onto the\nsample using f=200 mm CaF 2lens at 60\u000eincidence. The harmonic signals are collected in re\rection geometry using a\nSchwarzschild objective ( Re\rX , Edmund Optics), imaged onto the slit of a imaging spectrometer, and detected using\nan EM-CCD (Andor, Ixon3). In Fig. 3C, we present the spectra of the di\u000berent harmonic generated by the pellet.\nThe relative intensities are corrected for CCD exposure time and spectral sensitivity and for grating e\u000eciency and\ncan be quantitatively compared.10\nSHG @ 1000 nmA\n10 µm\n510 500 490Laser spot\nSHG BSignal Intensity\n775 750 725 525 500 475\nWavelength / nm400 375 350SHG THG FHG\nX 5•104X 20C\nFIG. 4: A. SHG image of the surface of BFO HNPs pellet obtained at 1 \u0016m excitation. B. SHG spectrum associated to image in\nA. C. Harmonic spectra generated by the BFO pellet using the KHz laser system tuned at 1.5 \u0016m. The relative intensities are\ncorrected for exposition time and spectral properties of the optical components. The dashed line in A indicates the dimension\nof the focal spot on the sample (at1\ne2) taking into account the 60\u000ebeam incidence.\nYb:CaF2\n1 kH z 14 mJ 220 f sfs KTA OP A (1.5 µm + 3 µm)\nDM\nDM\nM\nM\nLM\nMDM\nDMM\nKTA 1\nλ/2 PolKTA 2\nROImaging\nspec tromet er\nBFO HNPspellets\nFIG. 5: Experimental setup for harmonic generation by BFO HNPs pellets using \u0016J energy pulses from a femtosecond parametric\nampli\fer. DM: dichroic mirror, TFP : thin \flm polarizer, RO: re\rective objective.11\n1Schwung, S.; Rogov, A.; Clarke, G.; Joulaud, C.; Magouroux, T.; Staedler, D.; Passemard, S.; Justel, T.; Badie, L.; Galez, C.\nJournal of Applied Physics 2014 ,116, 114306.\n2Wilson, T. Journal of microscopy 2011 ,244, 113{121.\n3Zipfel, W. R.; Williams, R. M.; Webb, W. W. Nature biotechnology 2003 ,21, 1369{1377.\n4Cheng, J.-X.; Xie, X. S. JOSA B 2002 ,19, 1604{1610."    },    {        "title": "1402.1336v1.Controlled_growth_of_bismuth_ferrite_multiferroic_flowers.pdf",        "content": " 1 Controlled Growth of Bismuth Ferrite Multiferroic F lowers \nB. Andrzejewski a* , K. Chybczy ńska a, B. Hilczer a, M. Błaszyk a, T. Luci ński a, M. Matczak a,b , \nL. K ępi ński c \na Institute of Molecular Physics \nPolish Academy of Sciences \nSmoluchowskiego 17, PL-60179 Pozna ń, Poland \n* corresponding author: bartlomiej.andrzejewski@ifm pan.poznan.pl \nb NanoBioMedical Centre, Adam Mickiewicz University  \nUmultowska 85, PL-61614 Pozna ń, Poland  \nc Institute of Low Temperature and Structure Researc h \nPolish Academy of Sciences \nOkólna 2, PL-50422 Wrocław, Poland \n \nAbstract  — This study reports on the synthesis of \nball-like bismuth ferrite BiFeO 3 nanoflowers by \nmeans of microwave assisted hydrothermal process \nand also on their composition and mechanism of \ngrowth. It turns out that the petals of the nanoflowers \nare composed of the nanocrystals with the size about \n35-39 nm whereas their thickness and size depends \non the concentration of surfactants. The petals \ncontain BiFeO 3 phase and traces of Bi 2O3 oxide and \nmetallic Bi and Fe deposited mainly at their surface. \nAmounts of impurity phases are more pronounced in \nnanoflowers synthesized during short time, and \nbecome almost negligible for longer microwave \nprocessing. The nanoflowers contain also mixed Fe \nvalence, with the Fe 2+ /F e3+ ratio depending on the \ntime of synthesis. The growth and shape of the \nnanoflowers result from the process of diffusion in \nthe initial stages of hydrothermal reaction.  \n \nKeywords-component; bismuth ferrite, multiferroics,  nanoflowers, \nmicrowave assisted synthesis  \nI.  INTRODUCTION \nNanoparticles exhibit a tendency to aggregate durin g the \nsynthesis process in a variety of form, the most in tersting of \nwhich are nanostructures very similar in form to li ving plants. \nAmong them, one can mention the most simple shapes like \nnanoalgae [1], more complex nanodendrites [2], nano grass [3, \n4] nanotrees [4] and of course the most sophisticat ed forms \nlike nanoflowers [5]. Nice SEM images of nanoplants  and \nnanoforms have been awarded by the Materials Resear ch \nSociety on the Science as Art competition organized  twice  \na year [6]. However, these nanostructures are not o nly \nbeautiful but also important for understanding thei r physical nature as well as from the point of view of their  future \napplications. For example, nanoplants like nanoseew eeds and \nnanotrees when sensitized by organic dyes become ef fective \nsolar cells [4, 7]. Nanograss lithium batteries are  heavy-duty \nsource of power for mobile phones [3]. Nanoflowers can be \nused as excellent field emitters [8, 9], they also exhibit high \ncatalytic [10] and photocatalytic activity [11] and  enhanced \ndielectric response [12]. Due to excellent biocompa bility of \nsome nanoflowers they are also important for applic ations in \nmedicine and biology as amperometric [13] and color imetric \n[14] biosensors, cell tags for in-vivo applications  [15], for \ncancer cell recognition, bioimaging [16, 17] and dr ug delivery \n[18]. \nThe dimension of nanoflowers varies from a few doze n of \nnm (about 40 nm for Au nanoflowers) [15] to a few d ozen of \nmicrometers (about 50 µm for SnO 2 flowers) [19]. Thus the \ncommon term “nanoflowers” does not necessarily refe r to the \nexternal size of these objects. It rather comprises  the flower-\nlike materials with characteristic lenghts below 10 0 nm in at \nleast one dimension (for example thickness of the p etals). \nNanoflowers can consist of plate- or sheet-like pet als [20], \nperforated or brush-like ones [21], nanocrystalline  petals [22], \npetals branched into tips [10], nanobelt-like petal s [23], \nnanofibers [12], and even of bundles of nanorods [2 4]. They \nexhibit sometimes unusual morphological details, fo r example \nhollow cores [25], vase like [26] and hexangular sh apes [27] \nor snowflake-like [13, 28] and downy-velvet-flower- like \nnanostructures [29].  \nNanoflowers can be made of various elements like me tals \n[15, 30, 31], carbon, [25] and of compounds of the elements. \nExamples of the latter are metal oxides [9, 12, 20- 23] and \nvarious salts: suphides, tellurides, nitrides and p hosphides [24, \n27, 29, 32-35]. There are also known organic-inorga nic \nnanoflowers [36] and DNA nanoflowers [17].   2 Recently, first nanoflowers of functional materials  like \nmultiferroics have been synthesized [37, 38]. Multi ferroics \nexhibit simultaneously more than one order paramete r in  \na single phase and therefore have large technologic al potential. \nThe most interesting are magnetoelectric (ME) multi ferroics \nwith magnetic and charge ordering and some mutual c oupling \nbetween magnetization and spontaneous polarization.  Bismuth \nferrite BiFeO 3 (BFO) is the best known material which exhibit \nME multiferroic properties at room temperature [39] .  \nIt belongs to rhombohedrally distorted perovskites with R3c  \nspace group and has high ferroelectric Curie temper ature \nTC=1100 K and high Néel temperature TN=643 K.  \nThe ferroelectric (FE) properties result from the o rdering of \nlone electron pairs of Bi 3+ , whereas the antiferromagnetic \n(AFM) G-type ordering of Fe 3+  spins exhibits cycloidal \nmodulation with the period λ=62 nm [40, 41]. It is assumed \nthat a weak FM moment in this compound originates f rom \nDzyaloshinskii-Moriya type interaction which forces  small \ncanting of the spins out of the rotation plane of t he cycloid. \nThe weak FM moment increases when the spin cycloid is \nsuppressed in BFO particles with sizes smaller or c omparable \nto the modulation period λ. The spin ordering in the cycloid \ncan be also modified by strong magnetic field [40, 42].  \nThe paper reports on the process of growth of bismu th \nferrite multiferroic flowers obtained very recently  by \nmicrowave assisted hydrothermal synthesis [43]. The  method \nto obtain high-purity BFO phase and the parameters important \nfor controlling the growth and morphology of BFO \nnanoflowers are discussed. One can expect that thes e materials \ncan be applied in spintronic devices, as THz radiat ion emitters \nor catalysts [44].  \nII.  EXPERIMENTAL \nA.  Sample Synthesis \nPowder-like samples composed of BFO nanolflowers we re \nsynthesized by means of microwave assisted hydrothe rmal \nmethod [43]. The nitrate of bismuth Bi(NO 3)3·5H 2O and iron \nFe(NO 3)3·9H 2O in molar ratio 1:1 were used as the precursors \nfor the synthesis. The precursors were added togeth er with \nNa 2CO 3 into a KOH water solution of a molar concentration of \n6 M. To control the process of growth of BFO nanofl owers \nvarious amounts of polyethylene glycol PEG 2000 wer e added \nto the mixtures. The mixtures were transferred into  a Teflon \nreactor (XP 1500, CEM Corp.), loaded into a microwa ve oven \n(MARS 5, CEM Corp.) and heated at the 200 °C during ts=30 \nmin or 60 min. After the synthesis, the suspensions  of BFO \nnanoflowers were first cooled to room temperature, next \ncollected by filtration kit, rinsed with HNO 3, distilled water \nand placed in a dryer for 2 h. The final products w ere brown \npowders of BFO nanoflowers.  \n \nB.  Sample Characterization \nThe crystallographic structure of the BiFeO 3 nanoflowers \nwere studied by means of X-ray diffraction method ( XRD) \nusing a diffractometer fitted with a Co lamp ( λ=0.17928 nm) and with a HZG4 goniometer in the Bragg-Brentano \ngeometry.  \nScanning electron microscope (SEM) FEI NovaNanoSEM \n650 and transmission electron microscope (TEM) Phil ips \nCM20 SuperTwin were used to study the morphology an d the \nstructure of BiFeO 3 nanoflowers.  \nX-ray photoelectron spectroscopy (XPS) was applied to \nstudy the composition of the BFO nanoflowers. The s pectra \nwere collected at room temperature with UHV (standa rd \npressure of 5 ⋅10 -10 mbar) VG Scienta R3000 spectrometer and \nAlK α radiation (1,486.6 eV). BiFeO 3 in the form of fine \npowder was spilt onto conducting adhesive carbon ta pe fixed \nto a molybdenum support. In every case the binding energy \nwas determined by reference to the C1 s component at the \nenergy of 285 eV and Gaussian-Lorentzian functions were \nused to deconvolute the line shapes.  \nMagnetic measurements were performed using a Quantu m \nDesign Physical Property Measurement System (PPMS) fitted \nwith a Vibrating Sample Magnetometer (VSM) probe.  \nIII.  RESULTS AND DISCUSSION \nAn example of the SEM micrograph of an assembly of \nBFO nanoflowers resulting from the microwave synthe sis \nduring ts=60 min is presented in Fig. 1. The almost regular \nball-like nanoflowers with diameter in the range of  6 – 21 µm \nare composed of fine petals.  \n \nFigure 1.  BFO nanoflowers obtained during ts= 60 min synthesis. \nThe morphological details of a selected BFO nanoflo wer \nobtained in the same synthesis conditions are shown  in Fig. 2. \nOne can observe that the ball-shaped nanoflower con sist of  \na great number of thin crystalline petals arranged \nperpendicularly to the nanoflower surface. \n  3 \n \nFigure 2.  Morphological details of a selected BFO nanoflower;  the diameter \nof the ball-like nanoflower amounts to ∼17 µm .  \nWe have found that the shape of BFO nanoflowers \ndepends on the progress of microwave assisted react ion. If the \nreaction is in an initial stage, and the time ts of the synthesis \ndoes not exceed about the 20 min. the nanoflowers a re \nirregular and composed of a few dozen of the petals . In the \ninitial stage of the synthesis small nanoflower “bu ds” appear, \nas shown in Fig. 3a, which are growing and blooming  (Fig. 3b \nand 3c) as the reaction proceeds. The nanoflowers a re fully \ndeveloped when the reaction is completed, which tak es place \nafter about ts=60 min of processing. The developed \nnanoflowers form regular balls consisting of few hu ndreds of \nplatelet-like petals as shown in Fig. 2 and 3d. Thu s we can \nassume that the growth of the nanoflowers begins fr om only  \na few petals forming central part of flowers, which  is very \npronounced in Figs. 3a and 3b. The succeeding petal s start to \ncrystallize around these flower “buds”.  \n \nFigure 3.   Growth process of BiFeO 3 flowers (the pictures are not in the \nsame scale) \nOne can observe that the individual petals are irre gular \ncrystallites with thickness ranging from a few doze n of \nnanometers to about 500 nm and longitudinal dimensi on of \nfew hundreds nm. The growth mechanism of such thin \ncrystallites is determined by the processes taking place during \nthe hydrothermal synthesis:  Bi(NO 3)3·5H 2O + Fe(NO 3)3·9H 2O + 6KOH ⇒ BiFeO 3 + \n6KNO 3 + 17H 2O.  \nCrucial here are the intermediate stages of this re action during \nwhich bismuth and iron oxides are formed [37]:  \nX(NO 3)3 → X 3+  + 3NO 3- ↑ \nX3+  + 3OH - → X(OH) 3 \n2X(OH) 3 → X 2O3 + 3H 2O \nwhere: X=Bi, Fe.  \nBismuth ferrite compound originates due to the diff usion \nbetween Bi 2O3 and Fe 2O3 oxides occurring in the last stage of \nthe reaction: \nBi 2O3 + Fe 2O3 → 2BiFeO 3 \nThe process of diffusion strongly influences the di rection of \ncrystallographic growth resulting in formation of d endrites and \nfinally in petal-like morphology [45]. The growth o f petals can \nnot be related to Wulff facets theorem [46, 47] bec ause the \nrhombohedral compounds like BFO crystallize without  any \npreferred growth orientation. However, it explains the ball-like \nagglomeration of the petals forming the flowers: th e \nminimization of the surface to volume ratio minimiz es also the \nhigh surface energy of the BFO agglomerates.  \nThe thickness of the petals was found to be highly \ndependent upon the amount of PEG addition. The dist ribution \nof the petals thicknesses can be well fitted using log-normal \nfunction [48]: \n \n(1) \n \nwhere Dm denotes the median thickness of the petal and σ is \nthe distribution width. The histograms of the petal s \nthicknesses together with the fits obtained by mean s of eq. (1) \nare shown in Fig. 4. For the low amount of PEG 2000  used \nduring synthesis i.e. 0% and 0.01% the histograms c an be \nfitted with single log-normal function. However, if  the \nconcentration of PEG is higher, the histograms exhi bits two \nwell resolved maxima and a superposition of two log -normal \nfunctions is necessary to obtain a reasonable fit.  \nThe parameters of the best fits are given in Table I.  \nTable 1 Mean size <D>, median thickness D m and distribution width σ of the \npetals as dependent on the amount of PEG 2000 used in the synthesis \nwt% PEG < D> [nm] Dm1  [nm] σ1 Dm2  [nm] σ2 \n10% 524 348 0.2 642 0.1 \n1% 260 138 0.6 371 0.1 \n0.2% 252 130 0.5 424 0.1 \n0.01% 159 120 0.3 - - \n0% 78 61 0.4 - - \n \n\n\n\n\n\n\n− =\nmDDln exp D)D( f2\n22 21 1\n21\nσ πσ  4 0 50 100 150 200 250 300 010 20 30 40 50 \n  Count Count 0% PEG \n0 100 200 300 400 500 010 20 30 \n0.2% PEG 0.01% PEG \n  Count \n0 200 400 600 800 1000 0510 15 20 \n   \n0 200 400 600 800 1000 010 20 30 40 \n10% PEG 1% PEG \n   \n0 200 400 600 800 1000 1200 0246810 12 14 16 18 Petal thicknesses [nm] \nPetal thicknesses [nm] \n   \n \nFigure 4.  Histograms of the petal thicknesses of selected BFO  nanoflowers.  \nIt is evident, that the nanoflowers synthesized wit hout \nPEG are composed of the finest petals. In this case , the \nthickness of petals varies from about 30 nm to abou t 200 nm, \nas presented in the histogram in Fig. 4a. Most of t he petals \nexhibit thickness between 50 nm and 100 nm with the  mean \nsize 〈D〉≈ 78 nm. For higher amount of PEG 2000 addition the \nmean thickness of petals increases and above 0.2% o f PEG \nused for the synthesis the histograms exhibit two m axima.  \nThe distribution of the petals thicknesses obtained  for the \nhighest amount of PEG (10%) is very wide with two m axima: \nat about 350 nm and 650 nm. We assume that the addi tion of \nPEG promotes two mechanisms of growth of BFO crysta ls in \nthe nanoflowers. The first one is the increase in t he thickness \nof flat crystals/petals. The second mechanism leads  to the \nformation of very thick petals similar to microcube s. Indeed, \nthe nanoflowers obtained for a very high concentrat ion of PEG \nare very dense and composed both of the fine petals  and of the \ncube-like BFO crystals. This is another manifestati on of the \nfact that BFO rhombohedral compound crystallizes wi thout \nany preferred orientation. We should admit that BFO  \nmicrocubes can be also easily obtained by means of \nmicrowave assisted synthesis [38].  \nThe crystalline structure of the petals was also ex amined \nby means of TEM. Results are presented in Figs. 5 a nd 6. The \npetals of nanoflowers obtained during long-time syn thesis (at \nleast ts=60 min) are composed of small nanocrystals (see Fi g. \n5). Selected area diffraction pattern (SAD) present ed in the \ninset to Fig. 5 indicates that these nanocrystals e xhibit well \ncrystallized bismuth ferrite phase. On the other ha nd, the \npetals of the nanoflowers obtained during a short t ime microwave heating ( ts=20 min) contain a great amount of \nsmall BFO nanocrystallites (of a few nm in size) im mersed in \nan amorphous phase. Wide diffraction rings in the S AD \npattern of this petal (inset to Fig. 6) confirm thi s interpretation.  \n \nFigure 5.  TEM micrograph of the biggest crystallites for the BFO \nnanoflowers synthesised during long time ts=60 min. The inset presents SAD \npattern.  \n \nFigure 6.  TEM micrograph for the BFO nanoflowes synthesised d uring short \ntime ts=20 min. The insert to figure shows SAD pattern. \nThe XRD patterns measured at room temperature for t he \npowders containing BFO nanoflowers are presented in  Fig. 7. \nPanel 7a shows the pattern for the powder synthesiz ed during \nthe long time ( ts=60 min). The rest of the panels present the \ndata for the powders obtained after ts=30 min of processing \nand with various concentration of PEG 2000 surfacta nt (from \n0.01 to 10 wt%). The solid lines correspond to the best fits \nobtained by means of Rietveld method and calculated  using \nFULLPROF software to the experimental data represen ted by \nopen points. The lines below the XRD data indicate the \ndifference between the data and the fit. The vertic al sections \ngive the positions of individual Bragg peaks. Analy sis of these \nXRD patterns indicates the presence of the BFO rhom bohedral \nphase with R3c  space group in all samples. The parameters of \n 5 the hexagonal crystallographic cell are given in Ta ble II.  \nIt should be however, noted that XRD studies of nan opowders \nhave a considerable draw-back due to a decrease in the \ndiffraction coherence length. As a result, the decr ease in the \ngrain/crystallite size causes an increase in the ha lfwidth of the \ndiffraction profile. \n40 80 \n40 80 0.01% PEG   2Θ [deg] 0.2% PEG   1% PEG \n Intensity [a.u.] 10% PEG     0% PEG a/ b/ c/ d/ e/ \n \nFigure 7.  The XRD pattern for the powders containing BFO nano flowers.  \n \nTable II The parameters of the hexagonal R3c  crystallographic cell for BFO \nnanoflowers synthesized during ts=30 min with various PEG 2000 content \n \nThe mean size of the crystallites in the powders wa s \ncalculated using the Scherrer’s equation [49]: d=K λ/βcos Θ, \nwhere d denotes the crystallite size, β is the half-width of the \ndiffraction peak (012), Θ stands for the angle corresponding to \nthe position of the Bragg peak and λ is the used wavelength. \nThe value of the constant K in the Scherrer’s equat ion was \nassumed K=0.9. Fig. 8 shows the mean size of the \nnanocrystallites forming the petals versus  the PEG concentration in the solution. One can observe that  the size of \nthe nanocrystallites weakly depends on the PEG conc entration. \nThis is in clear contrast to the strong dependence of the mean \nthickness of the petals Dm on the PEG 2000 content in the \ninitial solution. Thus we conclude that the petals,  regardless of \ntheir thickness, are composed of the same building blocks i.e . \nsimilar in size BFO nanocrystals.  \n \nFigure 8.  Dependence of the crystalline size d on the PEG 2000 \nconcentration  \nSince the XRD pattern indicate almost no impurities , \nsecondary or ternary phases, we performed also elem ental \nanalysis of the BFO nanoflowers by means of EDX map ping \nof elements distribution. The maps are presented in  Fig. 9.  \nFigure 9.  SEM micrograph and the distribution of the elements  (Fe- red \ncolour, Bi- green color) in BFO nanoflowers.  \nThe distribution of Bi and Fe elements is homogeneo us, \neven inside of the BFO nanoflower, where we expecte d to find \nimpurity phases, which may serve as a “glue” for cr ystalline \npetals. More information on the composition of the \nnanoflowers was obtained from XPS analysis. The res ults are \nshown in Figs. 10, 11 and 12. The signal from Bi 4f  core level \nin BFO with maxima corresponding to Bi 4f 5/2 →164.9 eV and wt % PEG a, b [Å]  c [Å]  a,b, g  [deg] d [nm] v [Å] 3 \n10% 5.579 13.865  a=b= 90 o, g= 120 o 39 373.685 \n1% 5.575 13.856  a=b= 90 o, g= 120 o 35 372.966 \n0.2% 5.576 13.857  a=b= 90 o, g= 120 o 35 373.086 \n0.01% 5.576 13.856  a=b= 90 o, g= 120 o 35 373.028 \n0% 5.572 13.847  a=b= 90 o, g= 120 o 36 372.326 \n \nFe  \n \nBi   \nFe, Bi   6 Bi 4f  7/2 → 159.6 eV is shown in Fig. 10. The spectrum was \ndeconvoluted to extract the contributions originati ng from Bi 3+  \nions in BFO compound, Bi 0 in metallic bismuth and Bi 3+  ions \ncomposing Bi 2O3 oxide. In the case of the BFO powder \nsynthesized for the short time ts=20 min, contributions from \nmetallic bismuth and Bi 2O3 oxide are substantial and equal to \nabout 30% and to 20%, respectively. The amount of t he \nBiFeO 3 phase is about 50%, only. These values, however, \ndoesn’t correspond to the bulk composition of BFO \nnanoflowers, because the XPS signal comes from the thin \nlayer (about 5 nm) at the surface of the petals. In deed, the \nXRD data for this sample indicate negligible amount s of \nimpurities (see Fig. 7a). Therefore, it is reasonab le to conclude \nthat the impurities are deposited mainly at the sur face of the \nflowers. The XPS study performed for the flowers ob tained \nwith a long-time synthesis ( ts=60 min) indicates traces of \nmetallic bismuth Bi 0 and Bi 2O3 oxide at the surface. However, \nthe overlapping of the Bi 3+ 4f core level doublet of BFO with \ntraces of the doublet of metallic Bi 04f  core level and also with \nBi 3+ 4f  core level in Bi 2O3 hinders more detailed analysis.  \n168 164 160 156 152 \n 7/2 \n  \nlong synthesis 60 min 155.6 158.5 \n165.2 163.8 Bi 4f \n5/2 \nb/ \n \nshort synthesis 20 min 156.3 159.2 \n160.2 161.8 \n164.5 \n165.5 \nBinding Energy [eV] a/ \n \nFigure 10.  XPS signal from Bi 4f  core level electrons for the sample \nsynthesised during short time ts=20 min a) and long time ts=60 min b)  \nBi 2O3 oxide and metallic Bi and Fe deposited at the surf ace of \nthe BFO flowers are formed during intermediate stag es of \nBFO compound synthesis, as discussed above.  \nThe XPS signal from Fe 2p  core level in BFO is presented in \nFig 11. The total XPS signal for the sample synthes ized during \nshort time ts=20 min shown in Fig. 11a can be deconvoluted \ninto the following set of synthetic peaks in BFO ph ase: \nFe 2p 1/2 →724.9 eV and Fe 2p 3/2 →711.5 eV due to Fe 2+  ions, Fe 2p 1/2 →727.0 eV and Fe 2p 3/2 →713.7 eV due to Fe 3+  ions. \nThe contribution from Fe 02p 3/2  →708.8 eV transition in \nmetallic iron Fe 0 is shifted towards lower energies.  \nThe content of Fe 2+ and Fe 3+  ions in the sample synthesized \nduring short time ts=20 min (Fig. 11a) is almost identical and \nequal to about 51% of Fe 2+  ions and to about 49% of Fe 3+  ions. \n740 736 732 728 724 720 716 712 708 704 long synthesis 60 min Fe 2+ ~58% \nFe 3+ ~42% 1/2 \n  \n706.7 710.2 712.7 \n718.9 723.6 726.3 731.2 \n734.0 Fe 2p \n3/2 b/ \n   \nBinding Energy (eV) 708.8 711.5 713.7 719.5 721.9 733.0 727.0 724.9 \nFe 2+ ~51% \nFe 3+ ~49% a/ \nshort synthesis 20 min \n \n \nFigure 11.  XPS signal from Fe 2p  core level electrons for the sample \nsynthesized during short time ts=20 min a) and long time ts=60 min b)  \nThe XPS signal recorded for the nanoflower processe d during \nlong time ts=60 min is presented in Fig. 11b. The main \ncontributions to the total signal originate from Bi FeO 3 \ncompound expressed as two synthetic peaks due to Fe2+  ions: \nFe 2p 1/2  at 723.6 eV and Fe 2p 3/2  at 710.2 eV; and the other two \nsynthetic peaks corresponding to Fe 3+  ions: Fe 2p 1/2 →726.3 \neV, Fe 2p 3/2 →712.7 eV. The signals from Fe 2+  and Fe 3+  ions  \nare overlapped due to strong multiplet splitting an d shake up \nphenomena (satellites marked in Fig. 11b by „dash-d ot”).  \nThe weak signal caused by metallic iron presence (F e 0) at \n706.7 eV is shifted towards lower energies like in the sample \nprocessed for the short time ts=20 min. The content of Fe 2+ \nand Fe 3+  ions is this time unequal and is about 58% and 42% , \nrespectively. The differences in the ratio between Fe 2+  and \nFe 3+  ions can be caused by the reaction between metalli c Fe \nand Bi 2O3 deposited on the surface of petals. The product of  \nthis reaction BiFeO x should exhibit oxygen deficiency and \nthus higher ratio of Fe 2+  ions. Simultaneously, the content of \nmetallic Fe and Bi 2O3 impurities should decrease as it is really \nobserved in XPS study for the nanoflowers synthesiz ed during \nlong time.   7 The O 1s binding energy of the BiFeO 3 phase determined from \nXPS data presented in Fig. 12 is the same in short and long \nsynthesized samples and equal to ~529.6 eV. Accordi ng to \nZhang et al. [50] this peak corresponds to O 2- ions in the BFO \nlattice.  \n5 3 5 5 3 0 5 2 5 \nb in d in g  e n e rg y  [e V ] O1 s 5 2 9.6 \ns h o rt s y n th e s is a / \n \nFigure 12.  XPS signal from O 1s  core level signals for the sample synthesised \nduring short time ts=20 min a) and long time ts=60 min b)  \nIV.  CONCLUSIONS \nThe BFO flowers synthesized by means of microwave a ssisted \nprocess consists mainly of BiFeO 3 and traces of Bi 2O3 oxide \nand metallic Bi and Fe deposited mainly at the surf ace of \ntheirs petals. The presence  of impurity phases is more \npronounced in the flowers synthesized during the sh ort time \nts=20 min, and almost disappear, if the time of proce ssing \napproaches ts=60 min. For the short synthesis the content of Fe \nions of various valence is comparable i.e . 51% of Fe 2+  and \n49% of Fe 3+  ions. The nanoflowers synthesized during the \nlong time exhibit unequal ratio of Fe ions with var ious \nvalence; 58% of Fe 2+  and 42% of Fe 3+  ions. The samples \nobtained with the synthesis contain also large amou nts of \namorphous or nanocrystalline phase. The nanoflowers  \nsynthesized during long time are composed mainly of  \nnanocrystallites. The petals of the nanoflowers, re gardless \ntheir shape and thickness, are always composed of t he same \n“building blocks” i.e . nanocrystals with the size about 35-39 \nnm. The thickness of the petals can be controlled b y the \namount of the surfactant added, and varies from 78 nm to \nabout 420 nm. The growth and shape of the nanoflowe rs is \ndetermined by the process of diffusion in the initi al stages of \nreaction resulting in petal-like morphology. On the  other hand, \nthe ball-like shape of the nanoflowers allows decre asing total \nsurface energy of the BFO crystallites agglomeratio n.  \nThe BFO nanoflowers were found to exhibit enhanced \nmagnetization [43]. Our XPS studies show that the e ffect may \nbe not only related to the size effect but also to the Fe 2+  ions \napparent at the petal surfaces. The Fe 2+  ions have the magnetic \nmoment of ~ 6.7 µB , which is higher in comparison with that \nof Fe 3+  ions equal to ~5.9 µB.   \n V.  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K. \nMajumder1,3  \n1Laser Biomedical Applications Division,  Raja Ramanna  Centre for Advanced Technology, \nIndore,452013, Madhya Pradesh, India.  \n2BARC Beamlines Section, Raja Ramanna Centre for Advanced Technology, Indore 452013, \nMadhya Pradesh , India . \n3Homi Bhabha National Institute, Training School Complex,  Anushakti Nagar, Mumbai \n400094, Maharashtra, India.  \n \n \n \n \n \n \n \n \n \n \n*Address for Correspondence:  \nE-mail Address:  sonalipra8@gmail.com ; srinusatapathy@gmail.com ; \n \n 2 \n Abstract  \nPoly(vinylidenefluoride -co-trifluoroethylene) (P(VDF -TrFE))  based pyroelectric as well as \nmagnetoelectric materials offer  great promise s for energy harvesting for flexible and wearable \napplications . Hence, t his work focus  on solar energy harvesting as well  as magnetoelectric \nphenomenon in two phase nanocomposite film where the constituting phases are manganese \nferrite (MnFe 2O4) nanoparticles and P(VDF -TrFE)  polymer. Composite  films  have been \nprepared using solution casting technique. X-ray diffraction result shows higher crystallinity \nof these films. The ferroelectric, magnetic and magnetoelectric properties in variation with \napplied field and volume percentage of ferrite nanoparticles have been investigated. The \npreparation condition was optimi zed in such a way that it results improved ferroelectric \npolarization of nanocomposite film after incorporation of small amount of ferrite nanoparticles. \nThe maximum  magnetoelectric -coupling coefficient of about 156 mV/Oe -Cm was obtained for \noptimum  nanocomposite film when DC bias field was applied perpendicular to electric \npolarization direction. From a  pyroelectric device perspective , solar energy harvesting is also \nreported. An open circuit voltage of 5V and short circuit current of order of ~1  nA is \ndemonstrated without any pre amplification. Hence, the combination of magnetoelectric and \npyroelectric properties of nanocomposite film presented here indicate as a perfect candidate for \nsmart materi als, spintronics devices and specified magnetoelec tric-based  applications.  \n \n \n \n \n \n \n \n \n \n \n 3 \n 1. Introduction  \nEnergy harvesting  is referred as energy extraction which allows the capturing of unused \nambient energy such as vibration, light, temperature variations , strain  and converting into \nusable electrical energy. Energy harvesting is a perfect match for low -power portable \nmicroelectronics which depend on a battery power. It can provide cost -effective and \nenvironmental friendl y solutions for low  power applications. The conv ersion of thermal energy \nhas gained more attention in the recent years.  \nThe field of multiferroics has greatly expanded in the last few years, particularly with the \ndiscovery of  magnetoelectric (ME) effect  [1,2] . The requisite for the observance of this effect \nis the coexistence of coupling between  magnetic and electric order parameters . Such ME effect \nhas mutual control on their c oupled electrical polarization and magnetization . ME response in \nsingle -phase ME materials  is very weak and happening only at very low temperatures . \nTherefore , they cannot be implemented  in technological device applications  [3]. Due to \nlimitations in single -phase material, however, composite materials which mainly consist of two \nphases (piezoelectric and magnetostrictive  phase ) have focused here.  \nThe fascinating properties of P (VDF -TrFE) such as  light weight , large pyro -and piezoelectric  \nefficiency, high elasticity,  transparency and flexibility attract  for potential  applications  [4,5] . \nEven if  PVDF  show s low dielectric constant, P(VDF -TrFE) have  high dielectric constant which \nleads to good magnetoelectric  coupling property  [6–8]. Therefore, designing a suitable polymer \nmagnetoelectric nanocomposite material is a subject of intensive research for promising \ntechnological applications.  Experiment al research have sho wn that piezoelectric polymers in \ncomparison to ceramic -based composite ME materials have  solved some problems such as \nfragility, low electrical resistance and high dielectric losses  [9,10] . From a  pyroelectric device \nperspective, ferroelectri c P(VDF -TrFE ) is a fascinating materia l for harvesting thermal energy. \nThis extends the range of potential applications to the biomedical, energy, power, and signal \nprocessing fields.  In this regard, the conversion of mechanical and thermal energies into \nelectrical energy by exploiting piezoel ectric and pyroelectric materials has attracted \nconsiderable attention  [11,12] . \nMany studies on ME properties in ceramic/polymer composites, such a s PZT/P( VDF -TrFE ), \nNi/P(VDF -TrFE ) and alloy /polymer  based composites have limitations to some extent when \nused in nanocomposite due to the large leakage current and  easy oxidization  [13,14] . As a \nreplacement, oxide  based magnetostrictive materials have been proposed for application  in ME 4 \n nanocomposites. Among different magnetic  oxide materials, MnFe2O4 is an economic  \nalternative to the existing alloy -based magnetostrictive materials.  MnFe2O4 has \nmagnetostrictive  coefficients  almost -55 ppm with high Curie temperature  [15,16] . Ferrite  \nnanoparticles are fascinat ing materials  due to their chemical and thermal stability and unique \nstructural, optical, magnetic, electrical, and dielectric properties  [17]. They have  wide potential \ntechnological appli cations in high density magnetic recording  and switching devices [18–20], \nsensor tech nology , photoluminescence, biosensors, magnetic drug delivery, permanent \nmagnets, magnetic refrigeration, magnetic liquids, microwav e absorbers, biomedicine \n(hyperthermia)  [21,22] . Bulk Manganese ferrite (MnFe 2O4) has a partially inverse spinel \nstructure  with 20% of Mn2+ ions located at octahedral (B) sites and 80% located at tetrahedral \nsites [23]. MnFe 2O4 nanoparticles  has attract ed due to its good biocompatibility, high \nmagnetization value , high anisotropy, size -dependent saturation magnetization , heat-resistant, \nenvironmenta lly friendly, non -toxic, high shock resistan t [24]. \nIn this context , we report fabrication of MnFe 2O4/P(VDF -TrFE ) 0–3 nanocomposite  films with \ngood  dielectric,  ferroelectric, magnetic,  magnetoelectric  as a product response of MnFe 2O4 \nnanoparticles and P(VDF -TrFE)  and finally pyroelectric  properties at room temperature . The \nconcentration of MnFe 2O4 nanoparticles significantly influences the ferroelectric properties \nand hence ME and response  of the copolymer matrix. The optimized  nanocomposite films \nshow high coupling coefficient, which is first reported value in P(VDF -TrFE)  matrix . In this \nwork we implemented lock in technique  which was  reported by G. V. Duong et al  [25]. \nTransverse and longitudinal magnetoelectric coupling  of electrically poled films are reported \nwhich were measur ed by keeping magnetization  (M) and polarization (P) in perpendicular and \nparallel manner, respectively. The variation of magnitudes of α with volume % of MnFe 2O4 \nand DC bias field is also emphasized.  Additionally, solar energy harvesting property of this \nnanocomposite film was also tested.   \n2. Experimental  \n  2.1. Materials and methods  \nManganese ferrite (MnFe 2O4) nanoparticles have been synthesized via sol -gel auto combustion \nroute. The starting materials were Manganese nitrate tetrahydrate (Mn  (NO 3)2. 4H 2O) (purity \n99.9985%, Alfa Aesar) and iron nitrate nanohydrate (Fe  (NO 3)3.9H 2O) (purity 99.99%, Alfa \nAesar). These metal nitrates were dissolved in distilled water with continuous stirring to get a \nclear solution. Then citric acid (Alfa Aesar) was added und er continuous stirring and the 5 \n solution was maintained at 70°C for 1 hour. Continuous heating at 200°C leads to brown color \ngel and finally results to formation of manganese ferrite powders. Then the powders were \ngrounded into fine powders.  \nThe nanocomposi te films were prepared using doctor’s blade method [26]. The required \namount of P(VDF -TrFE) (70:30) powder (Poly K ) was first dissolved in N, N-dimethyl \nformamide (DMF) (Fluka) at 70°C using the hot plate assembled with mechanical Teflon  \nstirrer.  Continuous stirring of the solution for nearly two hours leads to complete dissolution \nof the P(VDF -TrFE) powder. Different volume % (0.5, 1 and 2) of m anganese ferrite powders \nwere added to the solution and again continuously stirred for two hours at 100°C. The solution \nwas ultra -sonicated for few hours to ensure the uniform dispersion  of magnetic nanoparticles. \nAfter that, the solution was casted on a c lean glass substrate into a film of uniform thickness \nusing doctor’s blade  by keeping constant velocity and gap between glass substrate and blade . \nThe solvent was evaporated in an oven at 100℃, resulting in a film with a thickness of 40 𝜇𝑚. \n  2.2 Characterization techniques  \nThe phase of the MnFe 2O4 nano powder and MnFe 2O4/P(VDF -TrFE) nanocomposite  films was \nconfirmed by X-ray diffractometer (Rigaku Geigerflex) with Cu  Kα1 (wavelength λ = 1.54 Å) \nas the radiation source  and FTIR spectr ometer  (JASCO -660 plus ). The size of  the nanoparticles \nand the microstructure of composite films  were examined by Carl Zeiss (Sigma -02) field \nemission scanning  electron microscope (FESEM).  The X -ray absorption near -edge structure \n(XANES) were collected in transmission mode  at the synchrotron  beamline -09 RRCAT,  Indore \n(India ) (energy range  4–25 keV , energy resolution 10 keV) . Spectra at the Mn (6539 eV) and \nFe (7112 eV)  K-edges were acquired at room temperature using a Si  (111) double crystal \nmonochromator. Novo -control  Alpha impedance analyzer  was used for t he dielectric studies  \nof the P(VDF -TrFE) and MnFe 2O4/P(VDF -TrFE) nanocomposite films.  Ferroelectric  \nproperties  of these films were  examined  using P –E loop  tracer (Marine India) and Magnetic \nproperties of the  MnFe 2O4 particles and MnFe 2O4/P(VDF -TrFE) nanocomposite  films were \nstudied using S700X SQUID  magnetometer (Cryogenics Ltd, UK). M agnetoelectric  \nmeasurement wa s carried out using lock in amplifier method (Marine  India).  Solar energy \nharvesting study was demonstrated with the help of solar simulator  (Enli Tech.,Taiwan) as \nradiation source  and Keithley 2450 source meter . \n 6 \n 3. Results and Discussions  \n3.1. Structural analysis  \nThe XRD spectra of the MnFe 2O4 nanoparticles are shown in fig.1 (a) . All of the main peaks \nare indexed as the spinel structure of MnFe 2O4 according to JCPDS file #731964  (fig.1 (b)) . \nFig.1 (d) shows  well-fitted Rietveld refinement of (χ2=1.87) MnFe 2O4 nanoparticles. The peak \nprofiles were fitted using Pseudo -Voigt function in Fig. 1(d). The result confirms a cubic spinel  \nstructure with space group symmetry Fd-3m. It is seen that all the nanoparticles show a single -\nphase as no other secondary phase is found in the refinement.  The u nit cell parameters, atomic \nposition, bond length, fitting factor and tolerance factor obtained from refinement are \nsummarized in Table 1. Crystal structure of MnFe 2O4 is plotted by Visualization of Electronic \nStructural Analysis (VESTA) software. Fig ure 1(e) shows the 3D lattice structure of MnFe 2O4, \nwhich is cubic spinel type. The blue, red and green color ball s in figure denotes Mn, Fe and \nOxygen ions, respectively. Tolerance factor is a useful tool to evaluate the phase stability of a \ncertain crystal structure. The tolerance factor (τ) was measured to be 0.82 by using the formula  \n𝜏= √3\n2 * (𝑅𝐵+ 𝑅𝑋\n𝑅𝐴+ 𝑅𝑋)                    (1) \nwhere, RA,RB and RX are ionic radius of A -site, B -site and oxygen ion, respectively. As the \ncalculated tolerance factor value is less than one, this predicts  the phase stability  of synthesized \nspinel structure [27]. The discrepancy (less than 1) in t olerance factor is because the octahedral \nare compressed to constitute the main framework of spinel structure.  \nThe average crystallite size of MnFe 2O4 nanoparticles was calculated to be ~16 nm from X -\nRay line broadening by using Scherer’s equation  \nt = kλ\nβCOSθ                                        (2) \nwhere, t  is average crystallite size, λ  is the wavelength of the Cu -Kα radiation (λ= 1.5406Å), θ \nis the Bragg’s diffraction angle, and β  is the full width at half maximum (FWHM) of the peak \nin the X -ray diffraction pattern [28]. Figure 1 (c) represents the plot s of XRD pattern s of the \nMnFe 2O4/P(VDF -TrFE) nanocomposite films  having different volume fraction of MnFe 2O4 in \nP(VDF -TrFE) . Highly intense diffraction peak at ~20° confirms the presence of 𝛽-phase  in \nP(VDF -TrFE) film . It is clear from the XRD pattern s of MnFe 2O4/P(VDF -TrFE) \nnanocomposite  films  that β -phase P(VDF -TrFE) ex ists in all  nanocomposite films. However, \nthe intensity of P(VDF -TrFE) peak dominates over the manganese ferrite peak in the composite   7 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \nTable 1.  Crystallographic information of MnFe 2O4 nanoparticles obtained  \nfrom Rietveld Refinement at room temperature  \nfilms . This is due to the low content of manganese ferrite nanoparticles in the composite, in \ncomparison to P(VDF -TrFE) content.  The nano size of the manganese ferrite powders  as well \nas the distribution of nanoparticles in the polymer is further confirmed using FESEM \nmeasurement . The size of the MnFe 2O4 nanoparticles was determined with the help of ImageJ \nsoftware. The data obtained from the software was fitted with a lognormal function. Finally, \nwe found a mean particle diameter approximate to 17 nm from the fitted data (inset of Fig.2 \n(a)). This mean d iameter is well matched with the obtained particle size from XRD.  The evenly \ndispersion of nano particles is also observ ed from the FESEM image of nanocomposite. The \nspherulitic type microstructure with self -connected continuous network  of P(VDF -TrFE) was \nnoticed in nanocomposite . Structural parameters of \nMnFe 2O4 nanoparticles  Values obtained from \nRietveld Refinement  \nSpace group  F d -3 m (227)  \nLattice type  F \nStructure  cubic  \na = b = c (Å)  8.3921  \nV(Å3) 591.0502  \nα = β = γ (degrees)  90.00  \nMn (x,y,z)  (0.0010, 0.0010, 0.0010)  \nFe (x,y,z)  (1.6355, 1.6355, 1.6355)  \nO (x,y,z)  (0.7600, 0.7600, 0.7600)  \nd<Mn -O> (Å) 2.0300  \nd<Fe-O>(Å) 1.8100  \nRp (%) 16.9 \nRwp (%) 15.9 \nRexp (%) 11.65  \nχ2 1.87 \n 0.82 8 \n  \nFig.1. (a) X -ray diffraction pattern and (b) JCPDS file of MnFe 2O4 nanoparticles; (c) X -ray \ndiffraction pattern of 0.5, 1 and 2 volumes % MnFe 2O4/P(VDF -TrFE) nanocomposite films; \n(d) Rietveld refinement profile of MnFe 2O4 nanopa rticles; (d)  Crystal structure of MnFe 2O4 \ngenerated from the Rietveld refinement of XRD data using VESTA software.  \n  3.2. FTIR studies  \nFormation of the spinel MnFe 2O4 nano particles  was further  supported by  the wavelength \ndependent transmittance data obtained  using a FTIR  spectrometer . Figure 2(c) shows a FTIR \nspectrum below 1000 cm−1 with common feature s of ferrites.  The peak observed  at 646 cm−1 \nis assigned  to the intrinsic stretching vibration  of the  metal -oxygen at the octahedral and \ntetrahedral sites  [15]. Peak at 1384 cm-1 can be assigned to the carboxylate group on the \nMnFe 2O4 nanoparticles due to citric acid [29]. The presence of band at 1640 cm-1 confirms the \nstretching vibration of -CH 2 while the broad band present at 3356 cm-1 is due to O -H bond of \nwater vapor present in air.  In summary, t he spectroscopic stu dies confirm the formation of \nMnFe 2O4 nanoparticles . In order to further confirm the presence of both Manganese ferrite and  \n9 \n  \nFig.2.  (a) and (b) FESEM image of MnFe 2O4 nanoparticles and 0.5 vol % MnFe 2O4/P(VDF -\nTrFE) nanocomposite  film, respectively; FTIR spectrum of (c) MnFe 2O4 nanoparticles and (d) \npure P(VDF -TrFE) & MnFe 2O4/P(VDF -TrFE) nanocomposites  \nP(VDF -TrFE) in nanocomposite film, FTIR transmittance data  for composite films were \ncollected and compared with the data of pure P(VDF -TrFE) film.  The peak s at 848 and 1289 \ncm-1 confirm  the β-phase of P(VDF -TrFE) in nanocomposite film. It is important to notice tha t \nthose two peaks  which is due to β-phase of P(VDF -TrFE) is  less inten se in pure P(VDF -TrFE)  \nbut enhanced in nanocomposite film.  Hence, in order to compare the percentage fraction of β-\nphase , the relative fraction of  β-phase was quantified as 79% and 97% for pure P(VDF -TrFE) \nand 0.5 vol % nanocomposite , respectively using the formula  \nF(β) = A(β)\nK(β)\nK(α) ∗A(α)+ A(β)                       (3) \nwhere, A(β) and A( α) are the absorb ance at 84 8 and 76 4 cm-1, respectively, K(β) and K (α) are \nthe absorption coefficients at the  respective wav e numbers, whose values are 7.7 × 104 and 6.1  \n10 \n × 104 cm2 mol-1, respectively . This confirms that inclusion of ferrite nanoparticles into polymer \nmatrix promotes the growth of  β crystalline phase in nanocomposite . In addition to this, t he \ncarboxylate group on the MnFe 2O4 nanoparticles  which is denoted by the peak at 1342 cm-1 is \nalso present in nanocomposite and not in pure P(VDF -TrFE). Hence IR spectra confirms the \npresence of b oth the phases in nanocompos ite films.  \n  3.3. XANES  spectroscopy  \nFig.3.  Normalized XANES spectra recorded at (a) Mn K -edge and (b) Fe K -edge of MnFe 2O4 \nnanoparticles  \nThe cation distribution between tetrahedral  and octahedral sites is an important  point to study \nas it influences the magnetic properties of MnFe 2O4 nanoparticles . The bonding environment \nof Mn and Fe was studied by means of X -ray absorption  fine structure  spectroscopy. Generally,  \nferrite nanoparticles  prepared at lower temperatures  (here combustion method) present a non -\nequilibrium cation distribution.  Figure 3 show s the room temperature XANES spectra for the \nas synthesized MnFe 2O4 nanoparticles recorded at the Mn and Fe K-edges , respectively  and in \nthe same conditions for the standard  oxides . The XANES spectra were processed in the usual \nway to obtain normalized absorbance.  Information on the oxidation state of Mn and Fe  in \nMnFe 2O4 nanoparticles has been obtained by comparing the main absorption edge position \nwith those of standard reference oxides . The Mn K -edge XANES spectra (Fig. 3a) shows that \nthe absorption edge of MnFe 2O4 nanoparticles  is in between the position of MnO and MnO 2. \nThis indicates  that the average oxidation state of Mn in as synthesized MnFe 2O4 nanoparticles \nis higher than 2+, presenting as mixed Mn2+ and Mn3+ oxidation state . It is also noticed a \nshoulder peak (black vertical arrow ) at around 6552 eV  for MnFe 2O4 nanoparti cles. It is related \n11 \n to the Jahn−Teller effect, which results in an elastic distortion of the octahedral sites in order \nto energetically compensate rearrangement of the electronic distribution of octahedrally \ncoordinated Mn3+ cations  [30,31]. In Fig . 3b, the XANES spectra of the MnFe 2O4 nanoparticles \nat the Fe K -edge  are compared with those of the  standard FeO, Fe 3O4 and α-Fe2O3. The \nabsorption edge position of the MnFe 2O4 nanoparticles is almost match with standard Fe3O4 \nwhile slightly  less than the edge position of α-Fe2O3, indicating the presence of both Fe2+ and \nFe3+ in synthesized MnFe 2O4 nanoparticles . It was also noticed that t he intensity of the pre -\nedge peak is very similar in  MnFe 2O4 nanoparticles and α-Fe2O3. The presence of pre-edge \npeak suggest s a fraction  of Fe3+ in non-symmetrical  sites (tetrahedral site)  i.e., the nanoparticles  \nhave degree of inversion.  This is because tetrahedral symmetry is non -centrosymmetric and it \nenables t he characteristic pre -edge peak due to 1s→3d transitions which become dipole \nallowed after mixing of the metal d with 4p states  [32,33]. \nTo determine the valence state of Mn and Fe in MnFe 2O4 nanoparticles , the XANES spectra \nof the standard oxides were analyzed by applying Linear Combination Fitting (LCF) method \nto the spectra of MnFe 2O4 nanoparticles . In this calculation, the XANES spectrum of MnFe 2O4 \nnanoparticles  at Mn K -edge was analyzed considering a l inear combination of XANES spectra \nobtained from MnO, Mn 2O3 and MnO 2. Similarly, in Fe K -edge, a linear combination of \nXANES spectra of Fe 3O4 and FeO were considered for LCF. The quality of the fit corresponds \nto low R factors (< 1%). The LCF fit result in  Mn K -edge (Fig.4a) shows nearly 4% MnO, 78% \nMn 2O3 and 18% MnO 2 in synthesized nanoparticles. The oxidation state of Mn in these oxides \nis like +2, +3 and +4 respectively. Hence, this result indicates that Mn has mixed valence states \nin MnFe 2O4 nanoparticl es where Mn3+ state is more than other Mn2+ and Mn4+ states. Further, \nin Fe K -edge, the LCF fit (Fig.4b) points out the proportions like 96% Fe 3O4 (mixture of +2 \nand +3 oxidation states) and 4% FeO (+2 oxidation state). Such proportions lead that there is \n52% Fe2+ and 48% Fe3+ state in MnFe 2O4 nanoparticles . As the nanoparticles are synthesized \nat low  temperature  method , the thermal energy is too low to overcome the energy barrier to an \nordered cation distribution. This change of cation distribution is consistent with the change of \nunit cell size. It was observed that the lattice constant obtained from Rietveld refinemen t which \nis about 8.39  Å is less than the bulk lattice constant 8.51  Å. It was previously studied that t he \ninverse spinel has a smaller unit cell than its counterpart with normal cation occupancy  [34,35]. \nThis difference in unit cell size has been attributed to a shorter cation -anion bond for Fe2+ in \nthe tetrahedral sites compared to Fe3+ in the octahedral sites, and  to the possible change of  Mn 12 \n Fig.4. Experimental XANES spectra at (a) Mn K -edge and (b) Fe K -edge of MnFe 2O4 \nnanoparticles (blue open spheres) fitted (red full line) using LCF  method . Residual is also \nshown (wine full line)  \noxidation state.  Hence according to the result, the synthesized MnFe 2O4 nanoparticles is in \nmixed state with partially inverted.   \n3.4. Dielectric  studies  \nFigure 5(a) represents the real part of dielectric constant (ε/) of pure P(VDF -TrFE) and  \nMnFe 2O4/P(VDF -TrFE) nanocomposites in the frequency range of 100 Hz to 1 MHz at room  \ntemperature. The permittivity of nanocomposite films enhances from pure P(VDF -TrFE) after \nincorporation of manganese ferrite nanoparticles. This is attributed to increase in  interfacial \npolar ization between polymer matrix and manganese ferrite nanoparticles. However, the \npermittivity was found to be independent of frequency in higher frequency region . The \nfrequency variation of permittivity at different temperatures (in the range from 30°C to 100°C \nwith 10°C interval) for 0.5%, 1%  and 2% MnFe 2O4/P(VDF -TrFE) nanocomposite  films are \nshown in fig. 5(b), (c) and (d),  respectively. It is realized that permittivity of nanocomposite \nfilms increases with increase in temperature due to easy orientation of dipoles as the polymer \nchains breaks with temperature. This trend is also observed at a particular frequency as shown  \nin figures 6(a), (b) and (c) . In low frequency region (figures 5 (b), (c) and (d) ), the increase in \npermittivity with increase in temperature becomes notable. This is mainly attributed to \nMaxwell -Wagner -Siller (MWS) interfacial polarization. Likewise, a s the temperature \nincreases, it expands the interface of polymer and nanoparticles, wh ich ultimately increases the \ninterfacial  polarization.  \n13 \n Fig.5.  (a) Room temperature frequency dependent permittivity of pure P(VDF -TrFE) and \nMnFe 2O4/P(VDF -TrFE) nanocomposite films; (b), (c) and (d) frequency dependent \npermittivity of 0.5, 1 and 2 volume %  nanocomposite film from RT to 100°C, respectively.  \n3.5. Ferroelectric  Properties  \nThe room temperature spontaneous polarization of pure P(VDF -TrFE) and MnFe 2O4 /P(VDF -\nTrFE) nanocomposites were  recorded at frequency 50 Hz (fig. 6(d)). A well -defined saturated \nloop was observed for pure P(VDF -TrFE) at application of 483 kV cm−1 electric field resulting \nto saturation polarization nearly 5.6 μC cm−2 at coercivity 310 kV cm−1. The observed \nferroelectric hysteresis loop for 0.5 vol % nanocomposite indicate s enhanced  ferroelectric  \nproperties.  The ferroelectric properties retain up to 0.5 vol % of manganese ferrite nanoparticles \nin P(VDF -TrFE) polymer. The saturation polarization was found to be increased from pure \nP(VDF -TrFE) to the value 8.2 μC cm−2 with E C = 400 kV cm−1 at application of electric \nfield555 kV cm−1. It is concluded that uniform distribution of nanoparticles in polymer matrix \nassists well -saturated  ferroelectric loops of 0.5 vol %  composite, which is also  consistent  with  \n14 \n  \nFig. 6. (a), (b) an d (c) Temperature dependent permittivity of 0.5, 1 and 2 volume % \nnanocomposite film in the frequency range 100 Hz to 1MHz, respectively; (d) P -E hysteresis \nloop measurement of pure P(VDF -TrFE) and MnFe 2O4/P(VDF -TrFE) nanocomposite films  \nnanoparticles . Hence, enhanced  saturation  polarization and coercive field in compare to pure \nP(VDF -TrFE) were observed  in 0.5 vol % nanocomposite. As the concentration of \nnanoparticles increases in polymer, we observed decrease in saturation polarization. This may \nbe due  to hindrance of nanoparticles in polymer  chain structure , resulting to increase in \nconductivity in composite.  \n3.6. Magnetic  characteri zations  \nTo explore the impact of particle morphologies and phase compositions on the  magnetic \nproperties, MnFe 2O4 nanopar ticles were investigated with S700X SQUID magnetometer. The \nDC magnetization plot measured from 3 to 370 K is shown in Fig. 7(a) . The data were collected \n(at 300 Oe) in zero field cooled (ZFC) and field cooled (FC) protocol.  Fig. 7(b)  exhibits the \nroom temperature M -H hysteresis of MnFe 2O4 nanoparticles ove r a magnetic field range of  \n15 \n   \nFig.7. (a) Temperature and (b) magnetic field dependent magnetization curve of  MnFe 2O4 \nnanoparticles; (c) M -H hysteresis loop of MnFe 2O4/P(VDF -TrFE) nanocomposite films; (d) \ncomparison between saturation polarization and saturation magnetization of nanocomposite \nfilms with concentration of nanoparticles in P(VDF -TrFE)  \n±70 kOe. It is evident from figure that MnFe 2O4 nanoparticles show ferrimagnetic behavior \nwith saturation magnetization nearly 58 emu/g ( ~ 2.2 µB per formula unit) i.e., nearly about \n73% inverse spinel and low coercive field  i.e., nearly 55 Oe. This high percentage of inverse \nspinel is due to high perc entage of Fe3+ ions in tetrahedral site and partial oxidation of Mn ions  \nfrom +2 to + 4 states confirmed from XANES  analysis. Typically, the inverse parameter in \nferrite nanoparticles varies between 0.2 to 0.9 which is highly depend on different synthesis \nmethod, particle size, preparation condition etc. which ultimately causes different cation \ndistribution in tetrahedral and octahedral sites. Fig. 7(c) shows the room temperature M -H of \nMnFe 2O4 /P(VD F-TrFE) nanocomposites. From figure, it is clear that the composites reserve \nthe magnetic  properties of MnFe 2O4 nanoparticles, but the saturation magnetization of the \ncomposites increases with the increase in nanoparticles concentration. This is due to the  \ncontribution of the magnetic moment of individual magnetic particle in polymer. However, the \n16 \n decrease of saturation magnetization of nanocomposites in comparison to MnFe 2O4 \nnanoparticles might be because of magnetic moment of very small amount of nano par ticles \ninvolved per cubic centimeter  and due to quenching of the surface moment of magnetic \nnanoparticles due to presence of non -magnetic polymer phase [36].  \n3.7. Magnetoelectric  properties  \nTo reveal the interaction among electric and magnetic dipoles , magneto -electric coupling \ncoefficient ( αME) of th e nano composite  films  was examined. In this MnFe 2O4 /P(VDF -TrFE) \nnanocomposite  films , the ME effect was generated as p roduct property between \nmagneto strictive and piezoelectric  phases . DC magnetic field dependence of α ME was measured \nby superimposing an AC magneti c field (25 Oe) at frequency 10 kHz  for all nano composites.  \nFor the  applied static magnetic field, strain is generated in magn etic phase due to \nmagnetostrictive property of manganese ferrite. Hence, the corresponding stress  is experienced \nby P(VDF -TrFE) through the interface. Due to the piezoelectric property of P(VDF -TrFE) \npolymer a voltage is induced in the composite. This is k nown as linear magnetoelectric effect, \nwhich is quantified by magnetoelectric voltage coefficient  according to the formula  \nαVME = 𝛿𝑉\n𝑡∗ 𝛿𝐻                 (4) \nwhere, 𝛿𝑉 is the output voltage , t is the thickness of nanocomposite film and 𝛿𝐻 is the AC \nmagnetic field applied.  The magnetoelectric coupling mainly depends on the ferroelectric \nproperties of the piezoelectric phase in composite. In ferrite/polymer composite, ferroelectric \npolarizat ion enhances up to a certain ferrite content depending on the electrostatic interaction \nbetween nanoparticles and polymer (also evidenced from ferroelectric loop in fig  6d and 7d). \nFrom fig 9(d) it was evidenced that the ME voltage coefficient is found to be increases  up to \n0.5 vol % of ferrite nanoparticles. For further high concentration of nanoparticles, the \npiezoelectric  properties decrease because of polymer chain disruption due to nanoparticles \nwhich ultimately leads to decrease in ME voltage  output. In the present study, the ME voltage \ncoefficient was measured in two modes i.e., transverse αV31 (direction between applied DC \nmagnetic field and output voltage is perpendicular) and longitudinal mode αV33 (direction \nbetween applied DC magnetic fi eld and output voltage is parallel ). The ME coefficients of \n0.5%, 1%, and 2% MnFe 2O4/P(VDF -TrFE) electrically poled (poled across the thickness) films \nare shown in the Fig. 9(a), (b) and (c) respectively. The maximum magnitude of α V31 is found \nto be 156.14  mVOe−1cm−1 at 1.21  kOe DC magnetic field  for 0.5% MnFe 2O4/P(VDF -TrFE)   17 \n  \nFig. 8. Schematic illustration of mechanism of ME coupling  \npoled composite film. To understand the phenomenon l et’s consider  the plane of the film be in \nxy plane. Before the measurement the sample was ferroelectrically poled across the thickness \nor along z direction. Hence, the polarization (P) of domains is align ed along z -direction. When \nmagnetic field is applied along x -direction, it creates maximum stress along z -direction.  It is \nobserved that in both transverse and longitudinal case, path followed by α is not identical for \nincreasing and decreasing applied  DC magnetic cycle.  This is due to the fact that in the \ndecreasing cycle of H dc, only poled ferromagnetic domains contributes to remnant \nmagnetization and magnetostriction, which in turn decrease the α value.  \n4. \nME output \nvoltage  \nH \nH \n3. \n1.Non- Poled \nsample  \n+ \nAu \nV \nAu \n- \n2. Electric Poling \nprocess  \nP(VDF -TrFE)  \nMnFe 2O4 \nnanoparticles  \nH \nContracted due to \nnegative \nmagnetostriction  \nCompressed stress \nacting on composite \nalong perpendicular \ndirection to keep \nvolume constant  \n5. \nMaximum stress \nrealised by sample  \nAu \n6. \n H \nAu \n+ \n- 18 \n  \nFig.9.  (a), (b) and (c) magnetoelectric voltage coefficient of 0.5, 1 and 2 volume % \nnanocomposite film in transverse mode (respective inset shows in longitudinal mode), \nrespectively; (d) volume concentration dependent transverse and longitudinal magnetoelectric \nvoltage coefficient  \nIt is important to notice that initially α increases  with DC magnetic field and reaches a \nmaximum and then falls with further increase of DC bias field. This is due to the fact that the \noutput voltage is proportional to piezomagnetic coefficient (q) accord ing to the equation  \nα = q *d                                     (5)  \nwhere, q is the piezomagnetic coefficient of magnetic phase and d is the piezoelectric \ncoefficient of electric phase. In general, piezomagnetic coefficient increases with increase in \nthe bias field H dc. At optimum DC bias field, it attains maximum and then saturates with further \nincrease in H dc. Hence, α follows the trend accordingly. The behaviour of α V31 is same to that \nof α V33, except in magnitude (α V33 ~ 16 mVOe−1cm−1). Since, manganes e ferrite has negative \nmagnetostriction coefficient (~ 55 ppm), nanoparticles contract along the field direction when \nexternal static magnetic field was applied.  \n19 \n  \nTable 2. Comparison of present work with the ferroelectric polarization and ME coefficient \nvalues of the reported 0 -3 ferrite/P(VDF -TrFE) magnetoelectric nanocomposites  \nTo keep the volume conserved, particles have to elongate along the perpendicular direction of \napplied field. Therefore, maximum strain was developed in perpendicular direction of applied \nmagnetic field and hence the ME voltage coefficient  was obtained max imum for transverse  \nmode in compare to longitudinal mode. Comparison of the ferroelectric and magnetoelectric \nresponse of the present nanocomposite with the reported  0-3 ferrite/P(VDF -TrFE) \nmagnetoelectric nanocomposites is shown in Table  2. It is clear fr om the results that the present \nMnFe 2O4/P(VDF -TrFE)  nanocomposite films  possess excellent magnetoelectric and \nferroelectric  response at room temperature, when compared with the reported other \nferrite/P(VDF -TrFE) nanocomposites.  \n3.8. Solar  energy harvesting  \nP(VDF -TrFE) is a pyroelectric material , a subclass of piezoelectric materials. This material \nrespon ds to a time -varying temperature gradient and produce a change of polarization . P(VDF -\nTrFE) possess es spontaneous polarization due to non-centrosy mme tric crystal structure . This \ngives rise to a permanent dipole moment along the crystal axis  [44]. The work ing principle of \npyroelectric material is ; when a time varying thermal flux applied  let say increasing the \ntemperarure i.e. dT/dt > 0 , the individual dipole will  randomly align due to thermal agitation  \nas the temperature rises up to thermal equilibrium . As a Consequen ce, the net polar ization  of \nthe material decreases.  This decrease in polarization results in the decrease in surface bound  \nSl. No   \n0-3 nanocomposite  Remnant \npolarization  \n(μCcm−2) ME coefficient  \n(mV cm−1Oe−1)  \nReferences  \n1 MnFe 2O4_P(VDF -TrFE)  6.5 156.14  In this work  \n2 BaTiO 3_P(VDF –TrFE)  6.03 22.2 [37] \n3 CoFe2O4_P(VDF -TrFE)  17 42 [38] \n4 NiFe2O4_P(VDF -TrFE)  5.8 90 [39] \n5 BaFe 12O19_P(VDF -TrFE)  - 15 [40] \n6 SmFeO 3_P(VDF -TrFE)  - 55 [41] \n7 NiFe 2O4_P(VDF -TrFE)  1.6 136.4  [42] \n8 CoFe 2O4_P(VDF -TrFE)  5 47.1 [43] 20 \n free charge s which generates a current due to an electrical imbalance across the polar axis if \nthe surfaces are connecte d to electric circuit.  This is known as pyroelectric  current. Similarly, \nupon reducing the temperature  (dT/dt < 0) , the dipoles are rest ored to their initial positions. \nThis results an increase in the net polarization which leads to reverse the flow of cur rent [45]. \nHowever, if dT/d t = 0 , then no current will be generated.  This build s the basis of pyroelectric \nenergy harve sting . \n \nFig.10. (a) Schematic of pyroelectric solar energy harvesting ; (b) Lateral schematic view of \nsample  used for solar energy harvesting ; (c) Picture of synthesized MnFe 2O4/P(VDF -TrFE) \nnanocomposite film  \nTo analyse the solar energy harvesting performance of MnFe 2O4/P(VDF -TrFE) nanocomposite \nfilm, the optimum nanocomposite film (0.5 vol % MnFe 2O4/P(VDF -TrFE) ) was selected which \nshows excellent ferroelectric  and ME properties.  Sample having surface area of 2 ×2 cm2 with \nan active area of 1 ×1 cm2 is placed normal to the incident solar radiation of ph otovoltaic device. \nSolar radiation is blocked using a chopper in order to attain cooling. During this process, \ncooling process happens through natural convection created by the temperature gradient. First \nof all, the bottom and top surface of the film were  silver coated by sputtering to make a \nconductive medium. T henceforth , the silver coated film electrically poled along perpendicular \ndirection of the film.  After that the film was kept at 20 cm distance normal to the solar radiation \nof power about 40 W/m2 (using solar simulator) . Connections were taken from the top and \nbottom surface of the sample and connected to digital storage oscilloscope (DSO) using BNC \nconnector  to measure the output voltage difference between the top and bottom electrodes.  \n21 \n  \n             Fig.11. Pyroelectric output voltage in (a) DC and (b) AC coupled with DSO  \nFig 10(a) shows the schematic of the above arrangement.  The output peak to peak was recorded \nacross load resistance of 1 M Ω by applying  different frequencies  using the chopper . Figure 11 \nshows the generated output voltage in response to temperature change caused by chopper at \nfrequencies 10, 50, 100 and 200 Hz . The output voltage s are recorded at same time scale with \nDC and AC coupling mode in DSO . \n \n22 \n  \nFig.12. Solar energy harvesting (a)  open cir cuit voltage  and (b) short circuit current at 10 Hz  \nIn DC coupling mode  (fig. 11a), the total charge develops between two electrodes are collected \nas output. Hence in DC coupling, the sample behaves like a capacitor and the magnitude of \nvoltage increases with decreasing the frequency due to increase in temperature gradient on the \nsample surface. Whereas, in AC coupling mode  (fig. 11b), the sample behaves like current \nsource and the output voltage follow the shape of pyroelectric current. Therefore,  the output \nwas collected across the load resistance as I ×R. Further, the open circuit voltage and the short \ncircuit current was measured using Keithley 2450 source meter without any preamplification  \nof current  (Figure 12) . Hence , the maximum output power was measured to be 2 .5 nW \n(equivalent to 2.5 mW for 200  TΩ output resistor) . Reports are available on PVDF based \npyroelectric energy harvesting with different load resistances [46–48]. Zabek (2017) et al  \ndemonstrated PVDF/graphene ink structure for maximum 20 V open circuit voltage at high \nload resistance 200 T Ω (Using Keithley high impedance analyzer) . In our case, the maximum \nopen circuit voltage of 5 V was found at load resistance 10 G Ω. This amplitude of output \nvoltage at low load resistance  (in comparison to 200 T Ω) in this magnetoelectric \nMnFe 2O4/P(VDF -TrFE) nanocomposite film will be better than previous PVDF based reported \nresults.  \n \n \nConclusions  \nA novel  MnFe 2O4/P(VDF -TrFE) nanocomposite film  was prepared to demonstrate \nmagnetoelectric and  pyroelectric properties . MnFe 2O4/P(VDF -TrFE)  nanocomposite film with \n23 \n optimum 0.5 volume %  exhibited improved ferroelectric properties  when compared to pure \nP(VDF -TrFE) film . It is concluded that multiferroic properties of nanocomposite film depends \non various factors like processing condition, microstructure , size of ferrite nanopar ticles, \nconcentration of ferrit e in polymer , interface and ferroelectric properties in order to get perfect \nmagnetoelectric coupling in ferrite/polymer nanocomposite. By optimizing all  these \ncharacteristics , a transverse ME voltage  coefficient of 156.4 mVcm-1Oe-1 at an optimum DC \nbias of 900 Oe is achieved which is very much encouraging for ME device applications.  \nFurther, due to the excellent ferroelectric properties, it provides rapid temperature gradient \nduring heating, hence transforming solar energy to electri cal energy through pyroelectric effect \nof these films.  \n \nAcknowledgement  \nAuthors thank fully acknowledge  RRCAT (Indore) and HBNI, Mumbai (Sanction No. \nDAE/LBAD/5401 -00-206-83-00-52, LT830006) for financial support. The authors are \nthankful to Shri Prem K umar  for XRD measurement and  Mrs. Rashmi Singh  for FESEM. Dr. \nAasiya  Shaikh  is acknowledged for her help in FTIR measurement . One of the authors, is \ngreatly thankful to Dr. Shilpa Tripathi, Md. 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The emergence of superlattice periodicities at me tal to insulator transitions in hole \ndoped perovskite oxides responds to a rearrangement  of the local atomic structure, and electron \nand spin density distribution. Originally, the ioni c model based on a checkerboard- type atomic \ndistribution served to describe the low temperature  charge and orbital ordered (COO) phases \narising in half- doped manganites. In the last year s, the exploitation of resonant x- ray \nscattering (RXS) capabilities has shown the need to  revisit these concepts and improve the \npicture. Yet, we have realised that COO is a more c ommon phenomenon than expected that can \nbe observed in a wide range of doping levels. Here we compare the experimental data recently \ncollected by RXS on La 0.4 Sr 1.6 MnO 4 (x=0.60) and La(Pr) 1/3 Sr 2/3 FeO 3 (x=0.67). The first shows \na COO phase similar to that found in the x=0.5 samp le but angular peak positions vs. T denotes \nthe incommensurability of superlattice reflections.  Meanwhile, the analysis of the \ncommensurate CO phase in the studied ferrite underl ines the role of the structural changes also \ninvolving La(Sr) and O atoms.  \n1.  Introduction \nThe metal- insulator (MI) phase transition in trans ition metal (TM) oxides continues to draw the \nattention of the scientific community in condensed matter physics. Within the assumed linkage of this \nprocess to lattice structure, charge distribution a nd spin and orbital magnetic moments, the precise r ole \nof the any of the corresponding order parameters ha s not been fully detailed yet. Among these \nmaterials, Mn- based compounds are one of the most studied families due to the interest on the \ncolossal magnetoresistive properties they can exhib it. An ionic model was proposed by Goodenough \n[1] to explain the different magnetic structures pr esent in La 1-xAxMnO 3 (A: alkali- earth) as a function \nof hole doping (x), i.e. the relation of Mn 3+ : Mn 4+  ions. For the particular case of x=0.5, this ratio  is 1:1 \nand would facilitate their periodic array in the cr ystal lattice in the low temperature charge localis ed \nphase thus leading to the so- called charge orderin g (CO) and orbital ordering (OO) phenomena. Since \nthis figure reproduced reasonably well many of the macroscopic properties of these materials it did no t \nsignificantly changed so far. Nevertheless, the dev elopment of new experimental techniques such as  \nthe resonant x- ray scattering (RXS) currently allo ws us to characterise the local geometry and \nelectronic structure of TM atoms in charge localise d phases, combining the atomic selectivity that \nprovides x- ray absorption and crystal sites contra st characteristic of diffraction [2]. The pioneer w orks \non half- doped perovskites such as La 0.5 Sr 1.5 MnO 4 and Nd 0.5 Sr 0.5 MnO 3 [3,4] confirmed the existence of \ntwo well differentiated sites occupied by two Mn io ns (approximated as Mn 3+  and Mn 4+ ) in the CO  \n \n \n \n \n \nphase (below ~200 to 250 K). In agreement with prev ious results by x- ray diffraction [5], they \nshowed that these sites order in the crystal lattic e following zig-zag chains (as a checkerboard) \npropagating in the ab plane and leading to the appe arance of a superlattice modulation originated in \nthe charge segregation. Associated to it and with a  doubled periodicity, single occupied e g orbitals of \nMn 3+  ions would order following a CE- type scheme. This  model has been generalised for half- doped \nR1-xAxMnO 3 ( R: lanthanide), layered (or ~2D structure) R1-xA1+x MnO 4 compounds, intermediate R2-\n2x A1+2x MnO 7 series and similarly to other Co or Ni based oxide s [6,7]. Nevertheless, in the last years \nan increasing number of both theoretical and experi mental works have cast doubts on the validity of \nthe ionic approximation. There have been detected t wo main problems. First, the charge segregation \nbetween dissimilar TM sites has been found to be mu ch smaller than one electron at the time that the \ncovalence with nearest O atoms has been remarked [8 -10]. And secondly, CO phases have been found \nin doped manganites out of half- doping condition w here the number of crystallographic sites do not \nmatch with the calculated expected ratio of +3/+4 i onic states [11]. It is convenient to remark here t hat \nthe term CO is commonly used in the broadest sense,  i.e. it comprehends the cases of a charge \nsegregation between sites, no matter its magnitude.  In spite of these results, CO systems based on \ndifferent TMs continue to be generally described in  terms of integer ionic states (TM n+ /TM m+ ), those \nwhere m-n=1 being the most widely studied. In this sense, focusing on systems showing CO phases \nwith a ratio of sites ≠1 and/or m-n≠1 appears to be a direct way towards a better knowl edge of the \nphysics of electron localisation. \nElectron and x- ray diffraction results on La 1-xCa xMnO 3 and La 1-xSr 1+x MnO 4 have shown that the \nCO and OO modulations become incommensurate with th e crystal lattice for x>0.5 [12,13] except for \nx~0.67 or 0.75, probably due to the fact that it ca n be expressed as a simple rational number (2/3, 3/ 4) \nand facilitates the array of two Mn sites with diss imilar oxidation state in a 1:2 or 1:3 relation. Fo r \n0.5<x<0.67, the modulation vector depends approxima tely linearly on x, which has been interpreted in \nterms of a charge density wave (CDW) of Mn e g electrons. Indeed, electron diffraction results ha ve \ndiscarded the existence of charge stacking faults ( i.e. a bimodal distribution of Mn sites) and indica ted \nthe presence of a superlattice with a uniform perio dicity [14]. \nCharge ordering has also been subject of deep inves tigations in Fe oxides, where we can find a \nsimilitude with manganites. First studies by Mössba uer and neutron diffraction [15,16] showed that \nthe T driven charge localised phases in La 1-xAxFeO 3 (A: Ca, Sr) occur in a wide doping range (x<0.7). \nFurther results obtained by transmission electron m icroscopy demonstrated that the superlattice \npropagation vector q direction is found to be locked to (1/2 0 0) for x <0.5 and parallel to the [111] \ncubic perovskite direction for x>0.5 [17]. For x=2/ 3, it becomes commensurate with a (1/3 1/3 1/3) \nperiodicity and it was postulated to be originated in a …AABAAB… sequence of two differentiated \nFe ions, identified as A=Fe 3+  and B=Fe 5+ . Yet, the same propagation vector would be obtaine d if we \nconsider B’=Fe 4+ , leading to a …AB’B’AB’B’… sequence. As we show fu rther through this paper, \nthe charge localisation is here strongly coupled to  the stabilisation of lattice distortions affecting  not \nonly Fe atoms but also R and O. \nIn this work we summarise resonant x- ray scatterin g data collected on the over- doped layered \nmanganite La 0.4 Sr 1.6 MnO 4 (x=0.6) at the Mn K edge and La(Pr) 1/3 Sr 2/3 FeO 3 (x=2/3) at the Fe K edge. \nWe analyse the CO and OO phases with respect to the  half- doped and under- doped systems. In the \nfirst case, superlattice reflections are found for the wave vectors qv,h+~(h+ε, h+ε, 0) and qw,h+~(h+2ε, \nh+2ε, 0), where 2 ε=1-x, h is integer and considering the tetragonal I4/mmm  space group. In particular, \nwe present data for h=1, 2. For ferrites, using the  primitive cubic perovskite structure notation (tho ugh \nreal structure is discussed later) q’CO,h’ =(h’/3 h’/3 h’/3) p, equivalent to q’CO,h’ =(0 0 2h’) hex  in hexagonal \ncoordinates ( R-3c  space group). Though expected to be equivalent, di fferent spectra are recorded for \nH=2, 4, 5. A structural model here presented explai ns these divergences. \n2.  Experimental details \nSingle crystals of La 0.4 Sr 1.6 MnO 4 (LSMO-214) and R1/3 Sr 2/3 FeO 3 (R: La, Pr) (L(P)SFO) were grown by \nthe floating zone method as described elsewhere [18 ,19]. They were characterised by x- ray powder  \n \n \n \n \n \ndiffraction and electrical and magnetic measurement s to control their quality. For RXS experiments, \nthe Mn crystal was carefully cut along a surface no rmal to the [110] direction. Meanwhile, the ferrite s \nwere cut perpendicular to the [111] p. Data were recorded at the ID20 beamline in the ES RF in \nGrenoble, France. Incident x- rays were monochromat ised to energies close to the Mn and Fe K \nabsorption edges by means of a Si (1 1 1) crystal. Samples were mounted on a four (+1) circles \ndiffractometer working in vertical scattering confi guration (i.e. incident σ polarization) and equipped \nwith a He close- cycle refrigerator for low T measu rements. The additional circle in the diffractomete r \nserves for φ azimuthal angle rotation around the scattering vec tor Q. The origin corresponds to \ncrystallographic [001] and [-110] p parallel to the incident beam direction for LSMO-2 14 and LSFO, \nrespectively. Polarisation of the scattered radiati on was analysed ( σ’ or π’) by means of a Cu (2 2 0) \n(Mn K) and MgO (2 2 0) (Fe K) single crystals at ~4 5 deg., radiation collection being performed by an \navalanche photodiode. All spectra have been correct ed for absorption. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1. Left panel: (1.6 1.6 0) σσ’ at φ=-5° (solid line), -50 ° (dashed) and -80 ° (dotted). The inset \nshows the intensity at the maximum of the resonance  as a function of φ. Right panel: qv,1(2)+ reflections \nin σ-π’ channel. Fluorescence (circles) is also shown. In set: azimuthal evolution at the pre- peak \n(crosses) and main resonance (circles). Minima are zero. All spectra have been recorded at 80 K. \n \n3.  Results and discussion \nEnergy dependent spectra for the qw,2-~(1.6 1.6 0) reflection in the σ-σ’ channel and the four possible \nqvi+ (i=1,2) in the rotated σ-π’ channel  in LSMO-214 ( ε=0.2) at the Mn K edge are shown in Figure 1. \nIσ-σ’ was absent in the latter reflections while a very weak σ-π’ contribution (near the detection limit) \ncannot be discarded in the (1.6 1.6 0) once having subtracted the analyser crystal leakage (not shown \nhere). The left panel shows that there exist both a  resonant and off- resonant signal for qw,2- in the \nwhole studied range of φ. The intensity at the resonance is ~ 10 -6 that of the Bragg (220) reflection at \nthe same energy. By analogy to checkerboard model a s established for the half- doped compound, this \nreflection would be the counterpart of the (h/2 h/2  0), with h odd, associated to CO. The results \nindicate the existence of a small charge disproport ionation between Mn sites and a Thomson \ncontribution due to little shifts of atoms from the ir ideal crystallographic positions in the I4/mmm \nsystem. The intensity at the maximum of the main re sonance (after off- resonant contribution \nsubtraction) shows little variations with the azimu thal angle as plotted in the inset. This is in cont rast \nto the observations reported in other half- doped c ompounds [10]. These changes could be ascribed to \neither the different dimensionality or the incommen surate character of this reflection. In the right \npanel, qv,1(2)+~(0.8 0.8 0), (1.2 1.2 0), (1.8 1.8 0) and (2.2 2.2  0) are shown after normalisation to the 6.53 6.54 6.55 6.56 6.57 6.58 0.0 0.4 0.8 1.2 \n-120 -60 0 60 \n  \nφ (degrees)  (0.8 0.8 0) \n (1.2 1.2 0) \n (1.8 1.8 0) \n (2.2 2.2 0) \n  Norm. Intensity (arb. units) \nE (keV) 6.53 6.54 6.55 6.56 6.57 6.58 0.0 5.0x10 -5 1.0x10 -4 1.5x10 -4 2.0x10 -4 \n-120 -90 -60 -30 0 30 0.0 4.0x10 -5 8.0x10 -5    \nφ (degrees) \n  Intensity (arb. units) \nE (keV) 6.53 6.54 6.55 6.56 6.57 6.58 0.0 0.4 0.8 1.2 \n-120 -60 0 60 \n  \nφ (degrees)  (0.8 0.8 0) \n (1.2 1.2 0) \n (1.8 1.8 0) \n (2.2 2.2 0) \n  Norm. Intensity (arb. units) \nE (keV) 6.53 6.54 6.55 6.56 6.57 6.58 0.0 5.0x10 -5 1.0x10 -4 1.5x10 -4 2.0x10 -4 \n-120 -90 -60 -30 0 30 0.0 4.0x10 -5 8.0x10 -5    \nφ (degrees) \n  Intensity (arb. units) \nE (keV)  \n \n \n \n \n \nsame scale. Following the analogy to the x=0.5 case , these would be ATS reflections [20] whose \norigin might be in the x 2-z2/y 2-z2 OO. The identical main resonances confirm their sa me dipolar \ncharacter while at the pre-peak a Q dependence is observed. Thus, it is interesting to  note that its shape \ncan be grouped into couples, qv,1+ ~ qv,2+  and qv,1-~ qv,2-. On the other hand, the inset shows that the \nazimuthal evolution at both the pre- peak and main peak follows a ~sin 2 φ dependence with coincident \nzero minima as reported for the half- doped compoun d [3]. \nFigure 2 plots temperature dependences of the diffr action angle 2 θ of the superlattice (1.8 1.8 0) \nand the Bragg (2 2 0) atomic reflections. First, OO -kind reflection (1.8 1.8 0) peak is broadened, \nwhich is indicative of a shorter correlation length  than for nuclear reflections. Second, we notice on  \ntheir commensurability. The contraction of a tetragonal lattice parameter when lowering T is re flected \nin the drift of the angular position of (220) refle ction to larger angles, about 0.2 ° from 240 to 100 K. In \ncontrast, any variation is hardly detectable in (1. 8 1.8 0). This would mean that Miller indexes are \nincommensurate to the crystal lattice and T depende nt. Indeed, the actual (h k l) found in the \nexperiment for qv and  qw are typically deviated ~10 -2 Å-1 from ideal rational values. A lock- in to the \nlatter has not been observed in the range from 80 t o 220 K. It turns out evident that these results ma ke \ndifficult the direct applicability of the checkerbo ard model as understood for manganites with x=0.5 \nbut rather agree with a picture where the electroni c charge density is strongly coupled to the lattice , \npropagating as a CDW. \nWe do not discuss here on the magnitude of the form al charge segregation between inequivalent \nMn sites. It remains as the subject for a future de eper analysis. \n \nFigure 2. Temperature dependence of 2 θ angular position in (220) Bragg and (1.8 1.8 1.8) superlattice \nreflections. Color (arbitrary units) varies linearl y with intensity from blue to red scale. \n \nIn the case of La(Pr) 1/3 Sr 2/3 FeO 3, the variation of 2 θ in (4/3 4/3 4/3) p superstructure confirms the \ncommensurate character of the CO phase (not shown h ere). In Figure 3 (a) we see the q’CO,H =(h’/3 h’/3 \nh’/3) p energy scans in the σ-σ’ channel for h’=2, 4, 5 in PSFO. Resonances are ob served at the Fe K \nedge energy for h’=4, 5, while (2/3 2/3 2/3) p shows a “valley”. Off- resonant intensities are al so highly \ndependent on Q. It is maximal for h’= 4 and much weaker for h’= 2 , 5. The comparison to LSFO [21] \nreveals a similar shape in all spectra though reson ances (h’=4, 5) are less marked in Pr sample and 2θθ θθ angle (degrees) \n220 \n180 \n140 \n100 T (K) (2 2 0) σσ σσσσ σσ’ (1.6 1.6 0) σσ σσσσ σσ’ \n (1.8 1.8 0) σσ σσπ’ \n87.8  88  88.2  88.4  76  77  78  79   \n \n \n \n \n \nrelative intensity for h’=2 is quite different. A s maller resonant signal correlates to previous resul ts \narguing that the charge segregation between differe nt Fe sites decreases proportionally to R cation size \n[22]. We recently demonstrated that a lattice modul ation involving not only Fe but also La(Sr) and O \natoms is necessary in order to reproduce RXS spectr a in LSFO. Furthermore, these results combined \nto x- ray absorption data permitted to discard the integer CO model based on a …+3+3+5+3+3+5… \nsequence. Simulations converged to a real average c harge disproportionation of 0.6 e -, i.e. formal \nFe +3.3  and Fe +3.9  ordered along the [111] cubic direction in a 1:2 r atio in combination with a \ndisplacement of La(Sr)O atomic planes in antiphase to Fe [21,23]. Spectra in figure 3 suggest that a \nsimilar but not identical structural model would be  necessary to fit PSFO curves. All studied \nsuperlattice reflections disappear at T~180 K. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3. (a) Energy dependence of superlattice ref lections in PSFO at 100 K. (b) Energy profile of \nLSFO (4/3 4/3 4/3) p for φ=-15 (solid), 30 (dashed) and 75 (dotted) degrees. \n \nCompared to Mn oxides, no OO- kind superstructure h as been found in R1/3 Sr 2/3 FeO 3 ferrites. \nMoreover, energy scans of the (h’/3 h’/3 h’/3) p reflections are identical as a function of the azi muth \nangle in R1/3 Sr 2/3 FeO 3 ferrites, as demonstrated in figure 3(b). In manga nites, the appearance of the \nOO- kind superstructure  is usually explained in te rms of the partial occupation of lowest energy e g \norbitals, in direct correlation to the development of a Jahn- Teller distortion in Mn atoms that incre ases \na Mn-O pair bond length. This type of distortion se ems to be absent in ferrites. The description of Fe4+  \n(3d 4) as a low spin system without e g electrons is in agreement. The comparison of CO/OO  phases in \nlayered and non- layered samples can help to decoup le the role of geometrical and electronic order \nparameters. Actually, due to the different dimensio nality of the lattice, the e g orbitals involved in the \nOO formation in R0.5 A0.5 MnO 3 and La 0.5 Sr 1.5 MnO 4 are thought to be different [24].   \n4.  Conclusions \n \nWe have investigated the charge and orbital ordered  phases in La 0.4 Sr 1.6 MnO 4 and La(Pr) 1/3 Sr 2/3 FeO 3 \nby RXS at the Mn and Fe K absorption edges. The tra nsition metal is in a formal intermediate \noxidation state in both samples, i.e Mn +3.60  and Fe +3.67 , respectively. In the case of the Mn oxide, \nrecorded CO and OO spectra show reminiscences from the well known half- doped (x=0.5) parent \ncompound. However, in comparison to the latter, pea ks intensities are strongly reduced (a factor ~10 2) \nand considerably broadened (unpublished data) due t o a shorter correlation length. A Mn bimodal \ndistribution model considering charge- stacking fau lts or COO phase correlated “melting” islands \nwould be then plausible. Though, the incommensurate  character these superlattice reflections appear to  7.10 7.11 7.12 7.13 7.14 0510 \n0.0 0.4 0.8 1.2  \n(2/3 2/3 2/3)  x5 (5/3 5/3 5/3) Intensity (arb. units) \nE (keV) (4/3 4/3 4/3) \n \n7.120 7.125 7.130 7.135 7.140 6912 \n  \nE (keV) (a) (b) \n7.10 7.11 7.12 7.13 7.14 0510 \n0.0 0.4 0.8 1.2  \n(2/3 2/3 2/3)  x5 (5/3 5/3 5/3) Intensity (arb. units) \nE (keV) (4/3 4/3 4/3) \n \n7.120 7.125 7.130 7.135 7.140 6912 \n  \nE (keV) (a) (b)  \n \n \n \n \n \nhave (fig.2) seems hardly reconcilable with a model  where CO is only supported by Mn atoms. Other \nexperimental results point also to discard this pos sibility and support a continuous charge modulation , \ni.e. a CDW model [14]. This could commensurate to t he lattice for particular rational values of hole \ndoping, as x=1/2 and 2/3. \nCO is also observed in R1-xSxFO in a wide range of hole- doping, where similar c ommensurate- \nincommensurate coupled electron- lattice modulation s have been proposed to explain the observed \nphenomenology. For x=2/3, we have shown that Pr bas ed sample modulated crystal structure in the \nCO phase must be very similar to that we proposed f or the La sample [21]. Here we argued that small \ncooperative displacements of Fe, La(Sr) and O atoms  are necessary to account for the measured RXS \nspectra and that Fe charge segregation was not very  significant. \nWe realise that the rich phenomenology of COO phase s in mixed valence transition metal oxides \nstill leaves open questions, and further experiment al and theoretical efforts must be done in order to  \nimprove the actual picture. \nAcknowledgements \nThe authors wish to thank ESRF for beamtime grantin g and financial support from Spanish MICINN \nFIS2008-03951 and DGA Camrads projects. \nReferences \n[1]  Goodenough J B 1955 Phys. Rev. 100  564 \n[2]  Materlik G et al 1994 Resonant Anomalous X-Ray scattering, Theory and App lications , \nAmsterdam: Elsevier \n[3]  Murakami Y et al 1998 Phys. Rev. Lett. 80  1932  \n[4]  Nakamura K et al 1999 Phys. Rev. B 60  2425 \n[5]  Radaelli P G et al 1997 Phys.Rev. B 55  3015 \n[6]  Zaliznyak I A et al 2000 Phys. Rev. Lett. 85  4353 \n[7]  Kajimoto R et al 2003 Phys.Rev. B 67  014511 \n[8]  García J et al 2001 J. Phys.: Condens. Matter 13  3243  \n[9]  Ferrari V et al 2003 Phys. Rev. Lett. 91  227202 \n[10]  Herrero-Martín et al 2004  Phys. Rev. B 70  024408  \n[11]  Subías G et al 2008 J. Phys.: Condens. Matter 20  235211 \n[12]  Larochelle S et al 2001 Phys. Rev. Lett. 87  095502  \n[13]  Chen C H et al 1997 J. Appl. Phys. 81  4326 \n[14]  Loudon J C et al 2005 Phys. Rev. Lett. 94  097202 \n[15]  Takano M et al 1983 Bull. Inst. Chem. Res., Kyoto U niv. 61  406 \n[16]  Battle P D et al 1990 J. Solid State Chem. 84  271 \n[17]  Li J Q et al 1997 Phys. Rev. Lett. 79  297 \n[18]  Herrero-Martín J et al 2005 Phys. Rev. B 72  085106  \n[19]  Blasco J et al 2008 J. Crystal Growth 310  3247 \n[20]  Dmitrienko V E et al 2005 Acta Cryst. A 61  481 \n[21]  Herrero-Martín J et al 2009 Phys. Rev. B 79  045121  \n[22]  Park S K et al 1999 Phys. Rev. B 60  10788 \n[23]  Blasco J et al 2008 Phys. Rev. B 77  054107 \n[24]  Wilkins S B et al 2005 Phys. Rev. B 71  245102 "    },    {        "title": "1905.10321v2.Photovoltage_from_ferroelectric_domain_walls_in_BiFeO__3_.pdf",        "content": "Photovoltage from ferroelectric domain walls in BiFeO 3\nSabine K ¨orbel and Stefano Sanvito\nSchool of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland∗\nWe calculate the component of the photovoltage in bismuth ferrite that is generated by ferroelectric domain\nwalls, using first-principles methods, in order to compare its magnitude to the experimentally measured pho-\ntovoltage. We find that excitons at the ferroelectric domain walls form an electric dipole layer resulting in a\ndomain-wall driven photovoltage. This is of the same order of magnitude as the experimentally measured one,\nbut only if the carrier lifetimes and di ffusion lengths are larger than previously assumed.\nKeywords: Domain walls, Excitons, Ferroelectric domains, Photovoltaic e ffect, Ferroelectrics, Oxides, Perovskite, DFT +U,\nDensity functional calculations, First-principles calculations\nIntroduction It has long been debated if and to what ex-\ntent ferroelectric domain walls contribute to the photovoltaic\neffect (PVE) in ferroelectric oxides like BaTiO 3and BiFeO 3.\nOne possible origin of the PVE in BiFeO 3is the so-called\nbulk photovoltaic e ffect (BPVE) [1–6]. The BPVE is a phe-\nnomenological term describing any photovoltaic e ffect that\ntakes place in the homogeneous interior of the material, as op-\nposed to interface e ffects. The BPVE in ferroelectrics such as\nBiFeO 3and BaTiO 3has been ascribed to noncentrosymmetric\nscattering or relaxation of electrons and holes after photoexci-\ntation in noncentrosymmetric crystals, resulting in a net shift\nof charge carriers (“shift current”) [2, 3, 5, 6]. The BPVE de-\npends on the polarization direction of the incoming light [4].\nThis dependence was indeed observed in the case of BiFeO 3,\nand it was hence concluded that the BPVE is at the origin of\nthe photovoltaic e ffect in BiFeO 3[7, 8].\nAlternatively a domain-wall driven PVE (DW-PVE) has\nbeen proposed [9–11]. The argument in favor of the DW-PVE\nis a variation in the ferroelectric polarization of the atomic\nlattice at the domain walls. It has been suggested that this po-\nlarization variation gives rise to internal electrostatic fields at\nthe domain walls. If true, this would mean that ferroelectric\ndomain walls could separate photogenerated charge carriers in\nthe same way as p-njunctions, without the need for p- and n-\ntype doping, and it would be possible to align arbitrarily many\nsuch junctions in series and add up the individual voltages cre-\nated by each single junction. However, the DW-PVE theory\nneeds to postulate a local ferroelectric polarization not only\nof the spatially discrete atomic lattice, but also for the con-\ntinuum of the valence electrons. But is it possible to define\nand determine the electric polarization of an arbitrary section\nof a crystal? Only then can this local polarization induce an\nelectric field. Such approach was adopted in previous first-\nprinciples studies of voltage steps /drops at ferroelectric do-\nmain walls in PbTiO 3[12] and BiFeO 3[9] and yielded electro-\nstatic potential drops (electronic potential steps) ranging from\n0.02 to 0.2 V per domain wall for the dark state (without illu-\nmination). We will demonstrate below that this polarization-\nbased approach does not yield the correct sign of the poten-\ntial step /drop and the photoinduced charge density at domain\nwalls in BiFeO 3. In contrast to the BPVE, the DW-PVE is in-\ndependent of the polarization direction of the incoming light.\nSome studies [6, 13] take the middle ground by assumingthat the BPVE and the DW-PVE may exist simultaneously and\ncan be cooperative or antagonistic depending on the system\ngeometry. The argumentation here is that, besides the char-\nacteristic angular dependence of the BPVE with respect to\nthe polarization of the incoming light, there is a polarization-\nindependent o ffset in the photocurrent, which might be as-\ncribed to a DW-PVE originating in an electrostatic field at the\nferroelectric domain walls [6], and /or in a locally modified\nBPVE, caused by the local crystal-structure modifications at\nthe domain walls [13].\nWhereas the atomistic structure of ferroelectric domain\nwalls in BiFeO 3is accurately known thanks to electron mi-\ncroscopy [14], experimental spectroscopy of photoelectrons\nat ferroelectric domain walls with atomic resolution is to our\nknowledge not possible today, but is perfectly within reach of\nfirst-principles calculations based on density-functional the-\nory (DFT). In fact, optical excitations and the spatial distri-\nbutions of the photoexcited charge carriers in molecules and\nsolids are nowadays routinely investigated by means of many-\nbody perturbation theory [15–18], such as the Bethe-Salpeter\nequation, or time-dependent density-functional theory. In the\ncase of ferroelectric domain walls such studies would be hard\nto undertake because of the large system size needed to model\na domain wall. However, there are approximate methods with\na favorable balance between accuracy and computational cost,\nsuch as the excitonic ∆-self-consistent-field ( ∆SCF) method,\nwhich yields more than qualitative agreement with highly ac-\ncurate many-body methods [19], but can be performed at the\nsame computational cost as a DFT ground-state calculation.\nThe excitonic ∆SCF approach has been applied to study exci-\ntons in various systems, including organic dyes [20], polymers\n[19, 21], and surfaces [22]. Here we use it to directly deter-\nmine the magnitude of the domain-wall driven photovoltaic\neffect in BiFeO 3, as given by the electronic potential induced\nby excitons localized at the domain walls.\nMethods We focus on the 71° and the 109° domain wall,\nfor which the direction of the ferroelectric polarization in\nadjacent domains di ffers by about 71° and by 109°, respec-\ntively. In rhombohedral perovskites like BiFeO 3there exists\nalso a 180° domain wall; however, due to its symmetry it\nshould be photovoltaically inactive, and hence we do not con-\nsider it here. The DFT calculations were performed with the\nvasp code [24], using the projector-augmented wave (PAW)arXiv:1905.10321v2  [cond-mat.mtrl-sci]  31 Aug 20202\nFIG. 1. (Color online) 71° domain wall: (a) 280-atom supercell\nwith two domain walls, (b) all three components of the ferroelectric\npolarization P, (c) polarization component Psperpendicular to the\ndomain wall, and (d) polarization-based electronic potential energy\nVpb\neand electronic potential energy Vedirectly obtained from DFT\n(thick black lines), shifted to zero and smoothened [23]. The dashed\nlines in (d) denote open-circuit conditions.\nmethod and pseudopotentials with 5 (Bi), 16 (Fe), and 6 (O)\nvalence electrons, respectively. We employed the local-spin-\ndensity approximation, and corrected the 3 dstates of Fe with\na Hubbard- Uof 5.3 eV following Dudarev’s scheme [25].\nThe Uvalue for Fe was taken from the “materials project”\ndatabase [26] and it is optimized for oxide formation ener-\ngies, but it also yields an optical band gap of 2.54 eV close to\nthe experimental one of ≈2.7–2.8 eV [27–32]. This computa-\ntional setup yields structural properties of BiFeO 3in reason-\nable agreement with experiment [23, 33–37]. The reciprocal\nspace was sampled with 2 ×5×3k-points for the 71° wall\nand with 2×5×5k-points for the 109° wall. Plane-wave\nbasis functions with energies up to 520 eV were used. We\nemployed a supercell approach with periodic boundary condi-\ntions, such that each supercell contained 120–280 atoms and\ntwo domain walls. Both the atomic positions and cell parame-\nters were allowed to relax until the energy di fference between\nsubsequent ionic relaxation steps fell below 0.1 meV . Our cal-\nculated structural properties of the domain walls are similar to\nFIG. 2. (Color online) The same as Fig. 1 for the 109° domain wall.\nthose in previous theoretical [14, 38–40] as well as experimen-\ntal [14] works [23]. Excitons were modeled with the excitonic\n∆SCF method [20], namely by occupying the valence states\nwith altogether ( N−NX) electrons and the conduction states\nwith NXelectrons, where NandNXare the total number of\nelectrons and excitons in the supercell, respectively. This was\ndone in every iteration step of the electronic self-consistency\ncycle, using the same density functional for minimizing the\nenergy as in a ground-state calculation. The geometries of the\nsystems in the presence of an exciton were optimized when\nconsidering excitonic polaron states. Supercells containing\n120 atoms were used in this case. The excitonic ∆SCF method\nis suitable for exploring low-lying excited states only, whereas\naccess to higher-lying excited states such as Rydberg states\ncan be obtained using constrained DFT [41 ?] or by maxi-\nmizing the similarity of the excited-state orbitals to reference\norbitals [42, 43]. In order to obtain the photovoltage per do-\nmain wall, VDW\nphoto, we first determine the photovoltage profile,\nVphoto, which is given by the di fference in electronic potential\nbetween the excited state, VX\ne, and the ground state, VGS\ne:\nVphoto(s)=/bracketleftBig\nVX\ne(s)−VGS\ne(s)/bracketrightBig\n/(−|e|), (1)\nwhere sis the coordinate perpendicular to the domain wall.\nThe domain-wall contribution VDW\nphotois then equal to the am-\nplitude of the spatial variation of the photovoltage profile at3\n-0.006-0.004-0.0020.0000.0020.0040.006̺excess (e/˚A)\nP(a)\n-0.10.00.10.20.3\n0 5 10 15 20 25 30\ns(˚A)(b)instantaneous relaxed\ne−\nh+\nDW\ninstantaneous relaxed\ne−\nh+\nFIG. 3. (Color online) Smoothened densities of excess electrons\nand holes for excitons ( X) at the 71° domain wall before (“in-\nstantaneous”) and after (“relaxed”) polaron formation: (a) for an\nXdensity of 0.1 Xper 120-atom supercell (a planar Xdensity of\n≈2.3×1013X/cm2) a large polaron forms, (b) for an Xdensity of 1 X\nper 120-atom supercell ( ≈2.3×1014X/cm2) a small polaron forms.\nGray bars and arrows indicate ferroelectric domains.\nthe domain wall (compare Fig. 6),\nVDW\nphoto=Vphoto/parenleftBig\ns−\nDW/parenrightBig\n−Vphoto/parenleftBig\ns+\nDW/parenrightBig\n, (2)\nwhere s+/−\nDWare positions to the right and to the left of the\ndomain wall. We calculate both VX\neandVGS\nein the ground-\nstate structure. By doing so, the electronic screening is al-\nready included. We still need to consider the screening by the\nlattice, which we calculate ab initio [23]. Note that, since\nwe are working with DFT, Veis the Kohn-Sham potential.\nFurthermore, the use of periodic boundary conditions implies\nthat we are calculating the short-circuit potential. The open-\ncircuit potential is then obtained from the short-circuit poten-\ntial by adding a constant gradient that compensates the poten-\ntial slope in the domain interior [12]. The photovoltages are\nextrapolated [23, 44] to avoid finite-size e ffects.\nResults and discussion We begin the discussion with\nthe electronic potential at the ferroelectric domain walls in\nthe ground state (without excitons). Figure 1 depicts the\nferroelectric polarization, the electronic potential energy per\nelectron, Ve, and the polarization-based potential energy per\nelectron, Vpb\ne, as calculated from the polarization variation\nat the wall including screening [23]. If the polarization-\nbased approach was valid, Vpb\neshould yield the same potential\nstep/drop at the wall as Ve.Vehas a potential minimum on the\nright-hand side of the wall. Accordingly, we expect excess\nelectrons to accumulate on the right-hand side of the wall.\nThe magnitude of the electronic potential drop (the electro-\nstatic potential step) at the wall is 0.13 eV (extrapolated [23]).\n-0.050.000.050.100.15̺excess (e/˚A)P(a)\n-0.0010.0000.0010.0020.003\n0 5 10 15 20 25 30\ns(˚A)(b)1.2·1014X/cm2\nDWe−\nh+(0.5X/SC)\n2.3·1012X/cm2\n(0.01X/SC)\n1234567\n0.0 0.2 0.4 0.6 0.8 1.00 5 10 15 20EX−EGS(eV/X)\nnX(1/supercell)nDW\nX(1013/cm2)\nlargeX smallXncritX\ncoexistence\nnmetX(c)FIG. 4. (Color online) Smoothened densities of excess electrons and\nholes for exciton ( X) densities in the coexistence region of the large\n(dashed line) and the small (solid line) Xpolaron in a 120-atom su-\npercell (SC) with 71° domain walls. (a) For a planar Xdensity nDW\nXof\n≈1.2×1014X/cm2the small Xpolaron is stable and the large Xpo-\nlaron is metastable. (b) For a planar Xdensity of≈2.3×1012X/cm2\nthe large Xpolaron is stable and the small Xpolaron is metastable.\n(c) Formation energies of the small (small filled circles) and the large\n(large empty circles) Xpolaron. The large Xpolaron is stable for X\ndensities below ncrit\nXand between ncrit\nXandnmet\nXit is metastable.\nVpb\neexhibits an electronic potential step at the wall instead of\na potential drop, and the magnitude of the potential variation,\n45 meV , is too small. For the 109° domain wall (Fig. 2) the\npotential has a pronounced minimum inside the domain wall,\nand a very small slope in the domain interior corresponding to\nan electronic potential drop of about 17 meV [23]. Vpb\nelacks\nthe minimum inside the wall, and exhibits an electronic po-\ntential step instead of a potential drop, and the magnitude of\nthe potential variation, 56 meV , is too large. We conclude that\nthe polarization-based approach fails to provide the correct\nsign and magnitude of potential steps /drops at domain walls\nin BiFeO 3.4\n101102103104105\n10−310−210−1100101ldiff(nm)\nIlight(W/cm2)\nsmall\nXlarge\nX1 sun\n1000 suns\ndtypical\nDW\ncoexistence\nFIG. 5. (Color online) Excitonic phase diagram. The solid blue line\nindicates the critical light intensity as a function of the carrier dif-\nfusion length ldiffaccording to Eq. (6) in the Supplemental Material;\nthe dashed blue line marks the upper boundary of the coexistence re-\ngion of the small and the large exciton polarons. Orange vertical bars\nmark the intensity of sunlight and thousandfold concentrated sunlight\nabove the band gap of BiFeO 3, and the gray horizontal bar marks a\nrange of typical domain-wall spacings dDWfound in the experimental\nliterature.\n-0.10-0.050.000.050.10\n0 10 20 30 40 50 60 70 80Vphoto (V)\ns(˚A)-12.4-12.3-12.2-12.1-12.0Ve(eV)\nVDW\nphoto ,SCVDW\nphoto ,OC(b)VGS\ne VX\neDW PVDW\nphoto ,SCVDW\nphoto ,OC\nVGS\n(a)\nFIG. 6. (Color online) (a) Smoothened potential energy of electrons\nat the 71° domain wall in the ground state ( VGS\ne) and in the excited\nstate ( VX\ne) in a 280-atom supercell with 0.1 excitons. (b) Instanta-\nneous photovoltage profile Vphoto[cf. Eq. (1)] for short-circuit (solid\nline) and open-circuit conditions (dashed line). Green arrows show\nthe magnitude of the domain-wall photovoltage VDW\nphoto.\nNext, we will add an electron-hole pair (an exciton) to our\nsupercell. Depending on the exciton density, two di fferent\ntypes of exciton polarons form, as depicted in Fig. 3. For low\nexciton densities, a large exciton forms with photoelectrons\nand -holes localized on opposite sides of the domain wall re-\nsulting in an excitonic dipole moment. Note that the sign ofthe photoinduced charge density (electron on the right-hand\nside of the wall, hole on the left) is the same as one would\nexpect based on the electronic potential drop (the electrostatic\npotential step) in the ground state, but it is opposite to what\none would expect based on the polarization-based potential\n(electron on the left-hand side of the wall, hole on the right).\nThe same is true for individual electrons and holes [45]. At\nhigh exciton densities a small, almost concentric exciton po-\nlaron forms (a self-trapped exciton) with a negligible dipole\nmoment. Figure 4 shows the densities of excess electrons and\nholes for di fferent densities of exciton polarons in the 120-\natom supercell, and the excitation energy per exciton of the\nlarge and the small exciton polaron. There is a critical exciton\ndensity above which the large exciton polaron transforms into\na small exciton polaron (above which the formation energy of\nthe small exciton polaron becomes lower than that of the large\nexciton polaron), and a coexistence region in which either the\nlarge or the small exciton polaron is stable and the other one\nis metastable. Since only the large exciton configuration will\nlead to a sizable photovoltage, we determine next under which\nexperimental conditions the large exciton forms.\nIn Ref. 46 it was found that the photocurrent in BiFeO 3fol-\nlows a rate equation, ˙ n=g−n/τ,where nis the photocarrier\ndensity, gis the photocarrier generation rate, and τ≈75µs is\nthe photocarrier lifetime. Here we adopt the same rate equa-\ntion for the exciton density nX, which in the steady state is\ngiven by\nnX=gτ=Ilight(1−R)τ\nEphoton dfilm, (3)\nwhere Ilightis the intensity of the light in W /cm2,dfilm=100 nm\nis the thickness of the experimentally studied films in Refs. 7\nand 11, Ephoton=3.06 eV is the photon energy of the laser used\nin the experiment in Ref. 7, and R≈0.27 is the reflectiv-\nity calculated at this photon energy from first principles. In\nthe following we assume that 100% of the penetrating light is\nabsorbed [23], in line with experiment [47]. We assume fur-\nther that all photocarriers within a distance ldiff(the di ffusion\nlength) from the domain wall reach the domain wall. Fig-\nure 4(b) shows that the critical exciton density from the super-\ncell calculation is ncrit\nX≈0.22Xper supercell, corresponding\nto an planar exciton density of ≈5.2×1013/cm2. This is equal\nto a photoelectron density of 3.7 ×1018/cm3if one assumes a\ndiffusion length of 140 nm. From Eq. (3) we obtain the crit-\nical light intensity for the transition between large and small\nexciton polarons [23], which is depicted in Fig. 5. In the case\nof natural sunlight and a di ffusion length near typical domain-\nwall spacings of a few hundred nanometers, the large exci-\nton forms, which should give rise to a photovoltage, whereas\nin the case of a thousandfold concentrated sunlight, such as\nin a concentrator solar-cell setup, the small exciton polaron\nforms, which should contribute less to the photovoltage, if at\nall. The small exciton polaron might be detectable with pho-\ntoluminescence spectroscopy as a state inside the band gap,\nor as a drop in the photovoltage at illumination intensities of5\n10−310−2\n10−310−210−1100Vphoto (V)\nIlight(W/cm2)10−310−210−1100101\nIlight(W/cm2)10−310−210−1100101102103104105\nIlight(W/cm2)\nsmall X\nlargeX\ncoexistence(a) ( b)\nVtotal ,expt .\nphoto ,71◦\nVDW ,theo\nphoto ,71◦\nVDW ,theo\nphoto ,109◦(c)\nFIG. 7. (Color online) Measured photovoltage per domain of BiFeO 3films with 71° domain walls from Ref. 11 and calculated domain-wall\nphotovoltage of the 71° and the 109° domain walls as a function of the illumination intensity for di fferent photocarrier life times τand di ffusion\nlengths ldiff: (a)τ=75µs,ldiff=140 nm; (b) τ=75µs,ldiff=10 nm; (c)τ=1 ns, ldiff=140 nm. Solid lines are an extrapolation; dotted lines\nare a guide to the eye.\nthe order of 0.1–1 W /cm2or higher, depending on the carrier\ndiffusion length.\nIn the following we consider moderate light intensities, at\nwhich the large exciton forms. The domain-wall photovolt-\nage is the spatial potential variation induced by the large ex-\nciton at the domain wall, depicted in Fig. 6. The potential\ngenerated by the excitons (partially) compensates the elec-\ntronic potential drop (the electrostatic potential step) at the\ndomain wall. The relation between exciton density and light\nintensity is the same as before, now we use the parameters\nof the experimental photovoltage measurement from Ref. 11\nin Eq. (3) ( Ephoton =3.31 eV , R=0.25, domain-wall spac-\ningdDW=140 nm). These parameters are similar to those used\nabove to determine the critical exciton density of the phase\ntransition. As an upper limit we consider a di ffusion length\nequal to the domain-wall spacing, ldiff=dDW=140 nm [23].\nWe also consider an estimated lower limit of ldiff=10 nm simi-\nlar to that assumed in Ref. 11, and a shorter carrier lifetime\nof 1 ns, similar to that reported in Ref. 48. The resulting\nopen-circuit photovoltage contribution per domain wall, to-\ngether with the experimental results from Ref. 11, are depicted\nin Fig. 7 and extrapolated to lower light intensities using a\npower law [23]. In the case of the 109° domain wall we can\nonly give a possible range (shaded area) [23]. The experimen-\ntal photovoltage is the total photovoltage of a film with 71°\ndomain walls, consisting of all photovoltaic e ffects (bulk and\ndomain-wall e ffects), divided by the number of domains. The\nexperimental conditions are well inside the range in which the\nlarge exciton forms (marked by the vertical solid line), for\nwhich our approach should be valid. The calculated domain-\nwall photovoltage matches the experimentally measured one\nonly in the most optimistic scenario [Fig. 7(a)], in which we\nassume a photocarrier density of 1016/cm3to 1018/cm3, a car-\nrier lifetime of 75 µs, and a carrier di ffusion length of 140 nm.For comparison, in Ref. 11 a carrier density of about 1012/cm3\nto 1013/cm3, a carrier di ffusion length of 8 nm, and a lifetime\nof 35 ps were assumed, and in Ref. 48 a lifetime of 1 ns. If we\nassume such conditions [Figs. 7(b) and 7(c)], the DW-PVE is\norders of magnitude too small to account for the major part of\nthe measured photovoltage in BiFeO 3. The domain-wall pho-\ntovoltage may be further reduced through screening by free\ncharge carriers and /or point defects that accumulate at the do-\nmain walls.\nConclusion We have analyzed the contribution of ferro-\nelectric domain walls to the photovoltage in BiFeO 3using\nfirst principles methods. In general we find that the ferro-\nelectric polarization profile does not allow one to determine\nthe correct sign and magnitude of the electrostatic potential at\nthe domain walls. Instead the electronic potential should be\ndirectly determined from ab initio calculations. The domain-\nwall driven photovoltages can be as large as the experimen-\ntally measured total photovoltages (up to ≈10 mV per domain\nwall), and may therefore be responsible for a large portion\nof the photovoltaic e ffect in BiFeO 3. This, however, is true\nonly if the carrier lifetimes and carrier di ffusion lengths are of\nthe order of≈100µs and≈100 nm, respectively, which is or-\nders of magnitude larger than previously assumed. Otherwise,\nthe domain-wall photovoltage is orders of magnitude smaller,\nand then the major fraction of the photovoltage should origi-\nnate from other e ffects, for example bulk e ffects. Furthermore,\nthere is a transition from a large to a small exciton polaron at\nhigh illumination intensities of the order of 0.1–1 W /cm2or\nhigher, which might be experimentally detected as a drop in\nthe photovoltage at high light intensities, or as a state inside\nthe band gap in photoluminescence spectroscopy.6\nACKNOWLEDGEMENT\nThis project has received funding from the European\nUnion’s Horizon 2020 research and innovation programme\nunder the Marie Skłodowska-Curie Grant Agreement No.\n746964. Computation time and support provided by the Trin-\nity Centre for High Performance Computing funded by Sci-\nence Foundation Ireland is gratefully acknowledged. We also\nthank the Irish Centre for High-End Computing (ICHEC) for\nthe provision of computational facilities and support. We are\ngrateful to Subhayan Roychoudhury (Trinity College Dublin)\nfor sharing his knowledge of excited-state density-functional\ntheory. Graphics were made using gnuplot andvesta [49].\n∗skoerbel@uni-muenster.de\n[1] V . M. Fridkin and B. N. Popov, “Anomalous photovoltaic e ffect\nin ferroelectrics,” Sov. Phys. Usp. 21, 981 (1978).\n[2] Wolfgang Kraut and Ralph von Baltz, “Anomalous bulk pho-\ntovoltaic e ffect in ferroelectrics: a quadratic response theory,”\nPhys. Rev. B 19, 1548 (1979).\n[3] Ralph von Baltz and Wolfgang Kraut, “Theory of the bulk pho-\ntovoltaic e ffect in pure crystals,” Phys. Rev. B 23, 5590 (1981).\n[4] V .M. 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Crystallogr. 44, 1272–1276 (2011).Photovoltage from ferroelectric domain walls in BiFeO 3:\nSupplemental Material\nSabine K ¨orbel and Stefano Sanvito\nSchool of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland∗\nCalculated ground-state properties of BiFeO 3\nideal\nExpt perovskite this work\na(Å) 5.57330(5)d; 5.57882(5)b; 5.5006\n5.5876(3)a; 5.58132(5)c; (-1.3%)\nc(Å) 13.84238(16)d; 13.867(1)a; 13.507\n13.86932(16)b; 13.87698(15)c; (-2.4%)\nc/a 2.4817a; 2.48370d;√\n6 2.4556\n2.48607b; 2.48633c≈2.4495 (-81%)\nV(Å3)372.36d; 373.06f; 373.83b; 342.75\n374.37c; 374.94a; 375.05e(-8%)\nzFe 0.22021(09)b; 0.22046(8)d; 0.25 0.227\n0.22067(8)c; 0.2209(5)e; (-20%)\n0.2209(6)f; 0.2212(15)a;\nxO 0.443(2)a; 0.44506(22)d; 0.5 0.436\n0.4453(8)f; 0.44582(22)c; (+12%)\n0.4460(8)e; 0.44694(28)b\nyO 0.012(4)a; 0.01700(28)c; 0.0 0.018\n0.0173(12)e; 0.01789(28)d; (±0%)\n0.01814(35)b; 0.0185(13)f\nzO 0.9511(5)f; 0.9513(5)e; 0.0 0.961\n0.95152(11)d0.95183(14)b; (-15%)\n0.95183(11)c; 0.9543(20)a\nα(°) 59.34c; 59.35b; 59.35e; 60 59.89\n59.39d; 59.39f; 59.42a(-81%)\nω(°) 12.2e; 12.3b; 12.4c; 0 14.4\n12.5a; 12.5f; 12.6d(+14%)\nP≈100g; 94\naMoreau et al. [1]. Single crystal X-Ray and neutron powder di ffraction at\nroom temperature.\nbPalewicz et al. [2]. Neutron powder di ffraction at 298 K.\ncPalewicz et al. [3]. Neutron powder di ffraction at 298 K.\ndPalewicz et al. [3]. Neutron powder di ffraction at 5 K.\neFischer et al. [4]. Neutron powder di ffraction at 293 K.\nfFischer et al. [4]. Neutron powder di ffraction at 4.2 K.\ngLebeugle et al. [5]. Single crystals, R3cphase, room temperature.\nTABLE I. Measured (ordered by size) and calculated structure pa-\nrameters of BiFeO 3in the R3cphase: Hexagonal lattice constants a\nandc,c/aratio, cell volume V, fractional atomic coordinates, rhom-\nbohedral cell angle α, octahedral rotation angle ω, and ferroelectric\npolarization P. Errors (in brackets) are with respect to the closest ex-\nperimental value. Errors of atomic coordinates, α, and c/aare those\nof ferroelectric distortions from the ideal perovskite structure.\nTable I contains our calculated structural parameters of\nBiFeO 3in the R3cphase in comparison with experimental\ndata. Lattice parameters deviate from experiment by about2%, typical for LDA calculations. Ferroelectric displacements\nof atoms from the ideal perovskite structure deviate by up\nto 20%. Ferroelectric distortions of the unit cell ( c/aratio\nand rhombohedral cell angle) are severely underestimated by\n≈80%, nevertheless the ferroelectric polarization is in good\nagreement with experiment, indicating that ferroelectric prop-\nerties are well captured by our computational setup.\nCalculated domain-wall properties\nIn Table II we compare our calculated DW formation ener-\ngies and widths for di fferent supercell sizes with those avail-\nable in the literature. For historical reasons, we include the\n180° wall, although we do not consider it in the main article.\nNote that Ref. 6 found close agreement between domain-wall\nstructures seen in electron miscroscopy and those calculated\nfrom first principles. This is true in particular for the domain-\nwall widths, which are as narrow as about one atomic layer\nin the case of the 109° domain wall and two atomic layers in\nthe case of the 180° domain wall, both according to experi-\nment and to first-principles calculations (see Fig. 1 and 2 in\nRef. 6). The 71° domain wall is slightly broader and extends\nover about three atomic planes. We fitted the polarization and\ntilt profiles PandAwith a tangens hyperbolicus to extract the\nDW widthξP,\nPr(s)=P∞\nrtanh[( s−s0)/ξP], (1)\nwhere Pris the rotating component of the polarization, P∞\nris\nits asymptotic value far away from the domain wall, sis the\ncoordinate perpendicular to the domain wall, and s0is the po-\nsition of the domain wall. In this section, di fferent from below\nand the main article, polarization profiles are calculated from\nionic positions and formal ionic charges (Bi3+, Fe3+, and O2−).\nTests indicated that applying the more sophisticated Born ef-\nfective charges changed the resulting polarization essentially\nonly by a prefactor. The DW energies are close to those of\nDi´eguez [7] and do not change very strongly with supercell\nsize. Polarization and tilt profiles (Fig. 2) are also converged\nwith respect to supercell size.\nFigure 1 shows the layer-resolved ionic sublattice displace-\nments compared to a paraelectric reference structure with-\nout octahedral tilts. The displacement profiles are atomically\nsharp.arXiv:1905.10321v2  [cond-mat.mtrl-sci]  31 Aug 20202\nE(mJ/m2)\ntype natoms this work Lit.ξP(Å)ξA(Å)\n71° 80 167 152a, 156d2.94±0.01 3.4±0.7\n71° 100 170 2.94±0.05 2.4±0.4\n71° 120 172 167a, 143.2c2.94±0.04 2.6±0.3\n71° 160 128b\n109° 80 60 62a, 53d1.8±0.2 0.4\n109° 100 60 2.0±0.1 0.4\n109° 120 63 62a, 52.9c1.9±0.1 0.1\n109° 160 33b\n180° 80 94 74a, 71d1.63±0.06 –\n180° 100 86 1.67±0.05 –\n180° 120 84 82a, 81.9c1.67±0.04 –\n180° 140 86 1.67±0.04 –\n180° 160 98b\naDi´eguez et al. [7], LDA +U,U=4 eV\nbWang et al. [6], GGA +U,U=7 eV , J=1 eV\ncChen et al. [8], GGA +U,U=3 eV\ndRenet al. [9], LDA +U,U=3.87 eV\nTABLE II. DW energies Eand DW widths ( ξP: polarization, ξA: tilt\nwall width) as function of the number of atoms natoms contained in\nthe supercell. Errors are standard errors of the fit.\nElectronic potential step at the domain wall\nFigure 3 depicts the magnitude of the electrostatic open-\ncircuit potential step at the domain wall in the ground state\n(without excitons). These data are also listed in Table III. The\n−∆VDW\ne(mV) dDW(Å)\nnatoms 71° 109° 71° 109°\n160 187 -18.3 22.1 31.2\n200 159 -5.8 27.6 39.0\n240 150 2.9 33.2 46.8\n280 157 5.6 38.7 54.6\n∞ 133 17.3∞∞\nTABLE III. Electrostatic open-circuit potential step −∆VDW\neof the\n71° and 109° domain walls without excitons as a function of the\ndomain-wall distance dDW(the number of atoms in the supercell,\nnatoms), as depicted in Fig. 3.\npotential converges to a positive number for both walls, so that\nit has the opposite sign compared to the polarization-based\npotential. We obtain the limit ∆VDW\ne(∞)of the potential step\nfor large domain-wall distances dDWby fitting with a power\nlaw:\n∆VDW\ne(dDW)= ∆VDW\ne(∞)+c·dp\nDW, (2)\nwhere ∆VDW\ne(∞),c, and pare fit parameters. The resulting\nexponents are p71° DW =−2.00 and p109° DW =−2.07.\n-0.3-0.2-0.10.00.10.20.30.4\n-8 -6 -4 -2 0 2 4 6 8∆u(Å)\ns−s0(Å)(a) 71◦\n-0.3-0.2-0.10.00.10.20.30.4\n-10 -5 0 5 10∆u(Å)\ns−s0(Å)(b) 109◦\n-0.3-0.2-0.10.00.10.20.30.4\n-8 -6 -4 -2 0 2 4 6 8∆u(Å)\ns−s0(Å)(c) 180◦\nO Bi FeFIG. 1. The three cartesian components of the atomic displacements\nfrom the paraelectric, tilt-free phase in each atomic layer (a) for\nthe 71° domain wall, (b) for the 109° domain wall, and (c) for the\n180° domain wall in the coordinate system spanned by the supercell\nvectors: pseudocubic [110], [ ¯110], and [001] directions (71° wall);\n[100], [011], and [0 ¯11] (109° wall); [1 ¯10], [110], and [001] (180°\nwall). Components that nominally remain constant across the do-\nmain wall are transparent. s0is the position of the domain wall. The\nsupercells contained 120 atoms.\nCalculation of the polarization-based potential\nThe ferroelectric polarization was calculated from ionic dis-\nplacements from ideal perovskite positions using direction-\naveraged Born e ffective charges Z∗of the R3cphase ( Z∗\nBi=\n4.865, Z∗\nFe=3.886, Z∗\nO=−2.917). First a smooth curve was\nfitted to the polarization profile perpendicular to the domain3\n-60-40-200204060\n-10 -5 0 5 10P(µC/cm2)\ns−s0(Å)(b) 109◦-40-200204060\n-8 -6 -4 -2 0 2 4 6 8P(µC/cm2)\ns−s0(Å)(a) 71◦\n-80-60-40-20020406080\n-10 -5 0 5 10P(µC/cm2)\ns−s0(Å)(c) 180◦\n80 atoms\n100 atoms\n120 atoms\n140 atoms\nfit\nFIG. 2. Polarization Pprofiles for di fferent supercell sizes (a) for\nthe 71°, (b) for the 109°domain wall, and (c) for the 180° wall. Com-\nponents that do not change sign at the domain wall are drawn trans-\nparent. Dashed transparent lines are the bulk reference. The profiles\nwere fit according to Eq. (1). s0is the position of the domain wall.\nwall [ Psin Fig. 1 (c) and Fig. 2 (c) in the main article] using\nthe program gnuplot. We employed a symmetric fit function\nof the form\nPs(s)=P(0)\ns−P(1)\ns\ncosh2/parenleftBigs−s0\nξ/parenrightBig, (3)\nwhere P0\ns,P1\ns,s0, andξare fit parameters. The polarization-\nbased electronic potential was then calculated as (compare\n-0.050.000.050.100.150.20\n120 160 200 240 280−∆VDW\ne(eV)\nnatoms\n71◦\n109◦\nfitFIG. 3. Electrostatic potential step (electronic potential drop) at the\n71° and the 109° domain wall without excitons as a function of the\nnumber of atoms natoms in the supercell (which is proportional to the\ndomain-wall distance).\nEq. (8) in Ref. [10])\nVpb\ne(s)=−1\nεss/integraldisplays\ns−(Ps(s/prime)−Ps(s−)) ds/prime, (4)\nwhere s−is a position in the domain interior on the left hand\nside of the domain wall, and εssis the calculated electronic\ndielectric constant for an electric field perpendicular to the\ndomain-wall plane (see below). For Ps(s) we used the fitted\ncurve from Eq. (3).\nElectronic and lattice screening\nThe screening (the real part of the static dielectric constant\nε) is calculated ab initio using the primitive rhombohedral cell\nand a k-point mesh of 10×10×10 points.εis diagonal in the\ncoordinate system spanned by the hexagonal lattice vectors\n(the pseudocubic [111], [ ¯110], and [ ¯1¯12] directions, parallel\nand perpendicular to the ferroelectric polarization). The elec-\ntronic contribution εehas the eigenvalues 7.4, 8.1, and 8.1,\nthe lattice (ionic) contribution has the eigenvalues 21, 37, and\n37. In order to obtain the screening in the direction sperpen-\ndicular to a domain wall, one needs to rotate εand obtains\nεe\nss=7.6,εion\nss=26, andεtotal\nss=34 for the 71° domain wall\nandεe\nss=7.9,εion\nss=32, andεtotal\nss=40 for the 109° domain\nwall. Both in the case of the polarization-based potential in\nthe ground state and in the case of the photovoltage, screening\nis included a posteriori . For the polarization-based potential\nwe useεss\ntotal, for the photovoltages we use an e ffective dielec-\ntric constant εeff\nssthat replaces the electronic screening εe\nssthat\nis automatically included by the total (electronic and lattice)4\n-0.006-0.004-0.0020.0000.0020.0040.006̺excess (e/˚A)P(a)\n0.00.1\n0 5 10 15 20 25 30 35 40 45\ns(˚A)(b)instantaneous relaxed\ne−\nh+DW\ninstantaneous relaxed\ne−\nh+\nFIG. 4. Smoothened densities of excess electron and hole for exci-\ntons ( X) at the 109° domain wall before (“instantaneous”) and after\n(“relaxed”) polaron formation: (a) for an Xdensity of 0.1 Xper 120-\natom supercell (a planar Xdensity of≈3.3·1013X/cm2) a large\npolaron forms, (b) for an Xdensity of 1 Xper 120-atom supercell\n(≈3.3·1014X/cm2) a small polaron forms.\nscreening:\nεeff\nss=εion\nss+εe\nss\nεess. (5)\nThe e ffective dielectric constant is εeff\nss=4.4 for the 71° DW\nandεeff\nss=5.0 for the 109° DW.\nExcitonic densities\nFigure 4 shows the densities of excess electron and hole for\nan exciton at the 109° domain wall. Similar to the case of\nthe 71° wall (see Fig. 3 in the main article), a large or small\nexciton polaron forms depending on the exciton density.\nPhotovoltage\nFigures 5 and 6 show all calculated photovoltage profiles\nfor the 71° and the 109° domain wall in the 280-atom super-\ncell.\nThe open-circuit (OC) voltage profiles were obtained from\nthe short-circuit (SC) voltage profiles by adding a constant\ngradient such that in the domain interior the resulting OC volt-\nage slope vanishes, as depicted in Fig. 7.\nTable IV contains the calculated short-circuit and open-\ncircuit domain-wall photovoltages obtained with di fferent su-\npercell sizes (di fferent domain-wall distances), and those ex-\n-0.06-0.04-0.020.000.020.040.06\n0 10 20 30 40 50 60 70 80Vphoto (V)\ns(˚A)-12.4-12.3-12.2-12.1-12.0Ve(eV)\n0.001 X0.005 X0.01X0.02X0.05X0.1X1X\n(b)GSDW PVGS\n(a)FIG. 5. (a) Ground-state and excited-state potential profile for ex-\nciton densities from 0.001 to 1 excitons ( X) per 280-atom supercell\nwith 71° domain walls, (b) photovoltage profiles for these densities.\n-0.010.000.01\n0 20 40 60 80 100Vphoto (V)\ns(˚A)-12.4-12.3-12.2-12.2-12.1Ve(eV)\n0.005 X0.01X0.02X0.05X0.1X\n(b)GSDW PVGS\n(a)\nFIG. 6. The same as Fig. 5 for the 109° domain wall. Exciton densi-\nties range from 0.005 Xto 0.1 Xper supercell.\ntrapolated to the limit of infinitely large domain-wall dis-\ntances. In the case of the 71° domain wall the extrapolated\nvoltages (numbers in boldface in Tab. IV) are depicted in\nFig. 7 in the main article. In the case of the 109° domain wall it\nis not possible to perform such an extrapolation because there\nis no strongly confining potential slope. Instead we depict the\nrange of photovoltages between that of the largest supercell\n(280 atoms), and the largest photovoltage as a function of su-\npercell size (numbers in boldface in Tab. IV).\nParameters used in the rate equation\nLight penetration depth in a BiFeO 3film The penetration\ndepth calculated from our first-principles absorption coe ffi-\ncient (which is similar to the one measured in Ref. 11) is about\n33 nm at the photon energy of 3.06 eV that was used in experi-\nment, such that about 95% of the penetrating light is absorbed\nin a 100 nm thick film. In Ref. 12 the penetration depth was\nestimated to be 50 nm, which would result in 86% of the pen-\netrating light being absorbed.5\n-0.8-0.6-0.4-0.20.0\n0 20 40 60 80 100 120 140Vphoto (V)\ns(˚A)SC\nOC\nDWIav\nFIG. 7. Short-circuit (SC) photovoltage profile (thin solid line),\nopen-circuit (OC) photovoltage profile (dashed line) for a 280-atom\nsupercell with 71° domain walls, and the interval Iav(thick solid line)\nin which the average voltage slope was compensated by a constant\ngradient.\nCarrier di ffusion length It is di fficult to accurately esti-\nmate the photocarrier di ffusion length ldiff, therefore two dif-\nferent numbers are considered in the main article. An upper\nboundary should be the length of a ferroelectric domain.\nExciton density at the domain wall The planar exciton\ndensity nDW\nXis given by the number of excitons NXand the\ndomain-wall area ADWcontained in the supercell, nDW\nX=\nNX/(2ADW). The domain-wall area is ADW≈42.7 Å2for the\n71° domain wall and ADW≈30.2 Å2for the 109° domain wall.\nAssuming that all photocarriers within ldiffreach the domain\nwalls, the exciton density can be expressed as nX=nDW\nX/ldiff,\nwhere nDW\nXis the planar exciton density at the domain wall.\nThen Eq. (3) in the main article becomes\nIlight=nDW\nXEphoton dfilm\nldiff(1−R)τ. (6)\nExtrapolation of domain-wall photovoltages to large\ndomain-wall distances\nFigure 8 shows the open-circuit photovoltage as a function\nof the distance between two 71° domain walls dDW. The pho-\ntovoltage does not yet converge for our employed dDW(super-\ncells with up to 280 atoms), therefore we extrapolate it to the\nlimit of large dDWby means of a fit function:\nVDW\nphoto(dDW)=VDW\nphoto(∞)−∆VDW\nphotoe−b d3/2\nDW, (7)\nwhere VDW\nphoto(∞),∆VDW\nphoto, and bare fit parameters. VDW\nphoto(∞)\nis the extrapolated domain-wall photovoltage that is used in\nthe main article. The fit function was chosen based on the ex-\nponential decay of the wave function of a particle in a linear\npotential (Airy function [13]). The data for the largest consid-\nered Xdensity of 1 Xper supercell could not be extrapolated\nto large dDWand were therefore left out of the analysis.\nIn the case of the 109° domain wall, see Fig. 9, the domain-\nwall photovoltage increases, then decreases with increasing71° DW 109° DW\nVDW\nphoto(mV) dDW VDW\nphoto(mV) dDW\nnX natoms SCusOC usOC (Å) SCusOC usOC (Å)\n0.005 120 1.47 3.68 0.829 16.6 2.47 -0.281 -0.0558 23.4\n160 22.1 31.2\n200 3.86 11.0 2.48 27.6 5.77 12.4 2.46 39.0\n240 5.76 15.1 3.40 33.2 6.47 15.7 3.12 46.8\n280 7.85 18.1 4.08 38.7 11.7 5.36 1.06 54.6\n∞ 23.9 5.38∞ ∞\n0.01 120 2.90 7.28 1.64 16.6 4.85 0.03 0.00596 23.4\n160 4.03 10.3 2.32 22.1 31.2\n200 7.16 20.4 4.60 27.6 10.9 21.0 4.17 39.0\n240 10.8 29.1 6.56 33.2 14.2 26.0 5.17 46.8\n280 14.4 36.0 8.11 38.7 8.49 6.13 1.22 54.6\n∞ 52.9 11.9∞ ∞\n0.02 120 5.66 14.2 3.20 16.6 8.92 -0.39 -0.0775 23.4\n160 22.1 31.2\n200 13.3 37.5 8.45 27.6 17.6 35.4 7.03 39.0\n240 20.1 54.7 12.3 33.2 20.2 33.3 6.62 46.8\n280 26.5 68.9 15.5 38.7 11.4 8.69 1.73 54.6\n∞ 111 25.0∞ ∞\n0.05 120 13.2 33.2 7.48 16.6 17.1 1.58 0.314 23.4\n160 22.1 31.2\n200 28.2 79.1 17.8 27.6 28.7 50.9 10.1 39.0\n240 41.1 113 25.5 33.2 33.6 53.8 10.7 46.8\n280 52.0 138 31.1 38.7 22.7 3.95 0.785 54.6\n∞ 186 41.9∞ ∞\n0.1 120 23.8 59.7 13.5 16.6 26.3 -3.04 0.604 23.4\n160 28.5 77.0 17.3 22.1 21.8 20.7 4.11 31.2\n200 44.2 123 27.7 27.6 38.0 54.9 10.9 39.0\n240 59.4 162 36.5 33.2 38.2 35.4 7.03 46.8\n280 69.3 180 40.6 38.7 24.2 23.4 4.65 54.6\n∞ 193 43.5∞ ∞\n1 120 83.8 165 37.2 16.6 23.4\n160 83.8 173 39.0 22.1 31.2\n200 90.9 175 39.4 27.6 39.0\n240 98.1 189 42.6 33.2 46.8\n280 102 172 38.8 38.7 54.6\nTABLE IV . Calculated short-circuit (SC) and open-circuit (OC)\ndomain-wall photovoltages VDW\nphotofor di fferent domain-wall distances\ndDW(different numbers of atoms natoms in the supercell) and extrapo-\nlations to infinitely large domain-wall distances as a function of the\nnumber of excitons nXper supercell for the 71° and the 109° domain\nwalls. Unscreened voltages are marked by the subscipt “us”, such as\nSCus. The photovoltages printed in boldface are the ones drawn in\nFig. 7 in the main article.\ndomain-wall distance. Here we cannot easily extrapolate to\nlarge domain-wall distances, instead we consider the maxi-\nmum photovoltage as a function of supercell size as an upper\nlimit. As a lower limit we take the photovoltage of the largest\nconsidered domain-wall distance.6\n0.000.050.100.150.20\n120 160 200 240 28016.6 22.1 22.6 33.2 38.7VDW\nphoto (eV)\nnatomsdDW(˚A)\n1X\n0.1X\n0.05X\n0.02X\n0.01X\n0.005 X\nFIG. 8. Open-circuit photovoltage of the 71° domain wall as a\nfunction of the number of atoms natoms in the supercell and of the\ndistance between domain walls, dDW(data points from Table IV).\nSolid lines are fits with Eq. (7), dotted lines are a guide to the eye.\nExtrapolation of domain-wall photovoltages to low light\nintensities\nThe photovoltages shown in Fig. 7 in the main article were\nextrapolated to low exciton densities (low light intensities I)\nwith a power law of the form\nVDW\nphoto(I)=VDW\nphoto(I0)/parenleftBiggI\nI0/parenrightBiggp\n(8)\nthat was optimized using the three data points corresponding\nto the lowest light intensities. In the case of the 71° domain\nwall p≈1.14. The photovoltage data of the 109° DW ex-\nhibit too much noise to safely extrapolate them to low light\nintensities.\nSmoothening of charge densities and potential\nThe rapid and strong oscillations of charge densities and\nelectronic potential at atomic nuclei, which obscure varia-\ntions on a larger scale, were smoothened in the following way:\nFirst the charge density or potential was averaged in a plane\nparallel to the domain-wall plane [in the r-tplane, compare\nFigs. 1(a) and 2(a) in the main article], then a sliding-window\naverage over an sintervall of one atomic plane spacing, like in\nRef. [10], was applied to the excess charge carrier densities.\n0.000.020.040.060.080.10\n120 160 200 240 28023.4 31.2 39.0 46.8 54.6VDW\nphoto (eV)\nnatomsdDW(˚A)\n0.1X\n0.05X\n0.02X\n0.01X\n0.005 XFIG. 9. The same as Fig. 8 for the 109° domain wall. Note the\nchange of scale.\nIn order to smoothen the potential a low-pass filter was ap-\nplied that removed wavelengths up to one atomic plane spac-\ning. The two smoothening methods should be roughly equiv-\nalent, and it is only for historical reasons that one was used for\nthe densities and the other for the potential.\n∗skoerbel@uni-muenster.de\n[1] J. M. Moreau, C. Michel, R. Gerson, and W. J. James, “Fer-\nroelectric BiFeO 3X-ray and neutron di ffraction study,” Journal\nof Physics and Chemistry of Solids 32, 1315 – 1320 (1971).\n[2] Andrzej Palewicz, R Przeniosło, Izabela Sosnowska, and\nAW Hewat, “Atomic displacements in BiFeO 3as a function of\ntemperature: neutron di ffraction study,” Acta Crystallographica\nSection B: Structural Science 63, 537–544 (2007).\n[3] A Palewicz, I Sosnowska, R Przenioslo, and AW Hewat,\n“BiFeO 3crystal structure at low temperatures,” Acta Physica\nPolonica-Series A General Physics 117, 296 (2010).\n[4] P Fischer, M Polomska, I Sosnowska, and M Szymanski,\n“Temperature dependence of the crystal and magnetic struc-\ntures of BiFeO 3,” Journal of Physics C: Solid State Physics 13,\n1931 (1980).\n[5] Delphine Lebeugle, Doroth ´ee Colson, A Forget, and Michel\nViret, “Very large spontaneous electric polarization in BiFeO 3\nsingle crystals at room temperature and its evolution under cy-\ncling fields,” Appl. Phys. Lett. 91, 022907 (2007).\n[6] Yi Wang, Chris Nelson, Alexander Melville, Benjamin Winch-\nester, Shunli Shang, Zi-Kui Liu, Darrell G Schlom, Xiaoqing\nPan, and Long-Qing Chen, “BiFeO 3domain wall energies and7\nstructures: a combined experimental and density functional the-\nory+U study,” Phys. Rev. Lett. 110, 267601 (2013).\n[7] Oswaldo Di ´eguez, Pablo Aguado-Puente, Javier Junquera, and\nJorge ´I˜niguez, “Domain walls in a perovskite oxide with two\nprimary structural order parameters: First-principles study of\nBiFeO 3,” Phys. Rev. B 87, 024102 (2013).\n[8] Yun-Wen Chen, Jer-Lai Kuo, and Khian-Hooi Chew, “Polar or-\ndering and structural distortion in electronic domain-wall prop-\nerties of BiFeO 3,” J. Appl. Phys. 122, 075103 (2017).\n[9] Wei Ren, Yurong Yang, Oswaldo Di ´eguez, Jorge ´I˜niguez,\nNarayani Choudhury, and L. Bellaiche, “Ferroelectric Domains\nin Multiferroic BiFeO 3Films under Epitaxial Strains,” Phys.Rev. Lett. 110, 187601 (2013).\n[10] B Meyer and David Vanderbilt, “Ab initio study of ferroelectric\ndomain walls in PbTiO 3,” Phys. Rev. B 65, 104111 (2002).\n[11] V ˇZelezn ´y, D Chvostov ´a, L Pajasov ´a, I Vrejoiu, and M Alexe,\n“Optical properties of epitaxial BiFeO 3thin films,” Applied\nPhysics A 100, 1217–1220 (2010).\n[12] Marin Alexe and Dietrich Hesse, “Tip-enhanced photovoltaic\neffects in bismuth ferrite,” Nature Communications 2, 256\n(2011).\n[13] Wolfgang P. Schleich, Quantum Optics in Phase Space, Ap-\npendix E (WILEY-VCH Verlag Berlin GmbH, 2001)."    },    {        "title": "1504.03405v1.Magnetic_charges_and_magnetoelectricity_in_hexagonal_rare_earth_manganites_and_ferrites.pdf",        "content": "Magnetic charges and magnetoelectricity in hexagonal rare-earth manganites and\nferrites\nMeng Ye\u0003and David Vanderbilt\nDepartment of Physics & Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA\n(Dated: June 16, 2021)\nMagnetoelectric (ME) materials are of fundamental interest and show broad potential for tech-\nnological applications. Commonly the dominant contribution to the ME response is the lattice-\nmediated one, which is proportional to both the Born electric charge Zeand its analogue, the\ndynamical magnetic charge Zm. Our previous study has shown that exchange striction acting on\nnoncollinear spins induces much larger magnetic charges than those that depend on spin-orbit cou-\npling. The hexagonal manganites RMnO 3and ferrites RFeO 3(R= Sc, Y, In, Ho-Lu) exhibit strong\ncouplings between electric, magnetic and structural degrees of freedom, with the transition-metal\nions in the basal plane antiferromagnetically coupled through super-exchange so as to form a 120\u000e\nnoncollinear spin arrangement. Here we present a theoretical study of the magnetic charges, and of\nthe spin-lattice and spin-electronic ME constants, in these hexagonal manganites and ferrites, clari-\nfying the conditions under which exchange striction leads to an enhanced Zmvalues and anomalously\nlarge in-plane spin-lattice ME e\u000bects.\nPACS numbers: 75.85.+t,75.30.Et, 75.70.Tj, 75.47.Lx\nI. INTRODUCTION\nThe cross-coupling between magnetic, electric, and\nelastic properties can lead to a plethora of novel and\nprofound physical phenomena, with potentially broad\nand innovative applications. Magnetoelectric (ME) ef-\nfects are those in which the electric polarization Pre-\nsponds to an applied magnetic \feld H, or magnetization\nMresponds to an applied electric \feld E. The ME cou-\npling (MEC) between magnetic and electric properties\nhas motivated intense experimental and theoretical in-\nvestigations in bulk single crystals, thin \flms, composite\nlayers, and organic-inorganic hybrid materials in recent\nyears.1{11\nAt the linear-response level, the linear MEC tensor \u000b\nis de\fned as\n\u000b\f\u0017=@P\f\n@H\u0017\f\f\f\nE=\u00160@M\u0017\n@E\f\f\f\f\nH; (1)\nwhere indices \fand\u0017denote the Cartesian directions\nand\u00160is the vacuum permeability. From a theoretical\npoint of view, the linear ME e\u000bect can be decomposed\ninto electronic (frozen-ion), ionic (lattice-mediated), and\nstrain-mediated responses.11Each term can be further\nsubdivided into spin and orbital contributions based on\nthe origin of the induced magnetization. As the orbital\nmoment is usually strongly quenched on the transition-\nmetal sites, most phenomenological and \frst-principles\nstudies have focused on the spin-electronic12and the\nspin-lattice13{15contributions. The lattice response can\nbe written, following Ref. 13, as\n\u000blatt\n\f\u0017= \n\u00001\n0\u00160Ze\nm\f(K\u00001)mnZm\nn\u0017; (2)\n(sum over repeated indices implied), i.e., as a matrix\nproduct of the dynamical Born electric charge Ze, the\ninverse force-constant matrix K\u00001, and the dynamicalmagnetic charge Zm, wheremandnare composite la-\nbels for an atom and its displacement direction. \n 0is the\nunit cell volume. Note that Zmis the magnetic analog\nof the dynamical Born charge, and is de\fned as\nZm\nm\u0017= \n 0@M\u0017\n@um\f\f\f\nE;H;\u0011=\u0016\u00001\n0@Fm\n@H\u0017\f\f\f\nE;u;\u0011; (3)\nwhereumis an internal displacement, Fmis an atomic\nforce, and\u0011is a homogeneous strain. In principle, Zm\nhas both spin and orbital parts, corresponding respec-\ntively to spin and orbital contributions to M\u0017, or Zee-\nman and p\u0001Aterms induced by H\u0017, but we shall focus\non the spin part in the following. Our previous \frst-\nprinciples study has shown that exchange striction acting\non noncollinear spin structures induces much larger mag-\nnetic charges than when Zmis driven only by spin-orbit\ncoupling (SOC). Therefore, exchange striction provides a\npromising mechanism for obtaining large MECs.16\nThe hexagonal manganites RMnO 3and ferrites RFeO 3\n(R= Sc, Y, In, and Ho-Lu) form an interesting class\nof materials exhibiting strong couplings between electric,\nmagnetic, and structural degrees of freedom.17A series\nof \frst-principles15,18{20and phenomenological21studies\nhave greatly enhanced our understanding of the coupled\nproperties. The ferroelectricity is induced by the struc-\ntural trimerization, and the direction of the spontaneous\npolarization is related to the trimerization pattern.19An\ninteresting \\cloverleaf\" pattern formed from interlock-\ning domain walls between structural and ferroelectric do-\nmains has been found in hexagonal RMnO 3and is now\nunderstood in terms of Landau theory.21{23Hexagonal\nRMnO 3andRFeO 3have rich magnetic phase diagrams\nand show considerable potential for manipulation and\npractical applications.24{26The magnetic order has two\ndi\u000berent origins, with the transition-metal Mn3+or Fe3+\nsublattices ordering \frst, often followed by ordering of\nthe rare-earth ions R3+at lower temperature. The mag-arXiv:1504.03405v1  [cond-mat.mtrl-sci]  14 Apr 20152\nnetic anisotropy is easy-plane and easy-axis for 3 dand 4f\nspins respectively; the 3 dmoments are antiferromagneti-\ncally coupled through superexchange so as to form a 120\u000e\nnoncollinear arrangement in the x-yplane, while the 4 f\nrare-earth moments are collinear along the hexagonal z\naxis.\nThe low-temperature magnetic phases of RMnO 3and\nRFeO 3allow a linear ME e\u000bect to be present. The\nrecently developed ME force microscopy technique has\nbeen used successfully to observe the ME domains in\nErMnO 3.27In that work, a large ME component \u000bzz\u001813\nps/m has been measured at 4 K, which is below the Mn3+\nordering temperature of 81 K but above the Er3+or-\ndering temperature of 2 K. However, \frst-principles cal-\nculations predict that the SOC-induced spin-lattice \u000bzz\narising from the Mn3+order is 0.7-1.0 ps/m.15This dis-\ncrepancy suggests that the dominant ME e\u000bect in the\nhexagonal ^zdirection is mediated by the Er3+4felec-\ntrons in ErMnO 3. The in-plane ME e\u000bect, which has\nnot been measured or calculated, has a di\u000berent ori-\ngin. It is dominated by an exchange-striction mecha-\nnism on the Mn3+sublattice, because the noncollinear\nspin pattern is sensitive to the lattice distortion. Thus,\nhexagonalRMnO 3andRFeO 3are good candidates to\nshow exchange-striction enhanced magnetic charges and\nanomalously large spin-lattice MECs.\nIn this work, we use \frst-principles density-functional\nmethods to study the magnetic charges and the spin-\ninduced MECs arising from the 3 delectrons in hexagonal\nHoMnO 3, ErMnO 3, YbMnO 3, LuMnO 3, and LuFeO 3.\nFor the transverse magnetic charge components and\nMECs, we also provide a comparison between results in-\nduced solely by exchange striction and ones including\nSOC. Our results con\frm that the exchange striction\ngreatly enhances the in-plane magnetic charges, while the\nSOC contribution is minor for most components except\non Mn atoms. However, the e\u000bect of SOC on the MECs\nis surprisingly large in many cases. This occurs because\nthe exchange-striction contribution tends to be reduced\nby cancellations between modes, while the SOC contribu-\ntion is mainly associated with a few low-frequency modes.\nThe in-plane ME responses are discussed case by case,\nand the conditions under which exchange striction leads\nto anomalously large in-plane spin-lattice MECs are clar-\ni\fed.\nThe paper is organized as follows. In Sec. II A and\nII B we introduce the geometric structure and magnetic\norder of hexagonal RMnO 3andRFeO 3. In Sec. II C\nwe analyze the tensor symmetries of the Born charges,\nmagnetic charges and MECs in two di\u000berent magnetic\nphases ofRMnO 3andRFeO 3. The computational de-\ntails are described in Sec. II D. The results and discussion\nof Born charges, magnetic charges and MECs in RMnO 3\nand LuFeO 3are presented in Sec. III. We summarize and\ngive our conclusions in Sec.IV.\nR\nMn/Fe\nO\nR\n1\nR\n2\nO\nP1\nO\nT2\nO\nT1\nO\nP2\na\nb\nc\n(a)\na\nb\nR\n1\nR\n2\n(\nb\n)FIG. 1. Structure of ferroelectric hexagonal RMnO 3(6 f.u.\nper primitive cell). (a) Side view from [110]. (b) Plan view\nfrom [001]; dashed (solid) triangle indicates three Mn3+con-\nnected via Op 1to form a triangular sublattice at z= 0\n(z= 1=2).\nII. PRELIMINARIES\nA. Hexagonal RMnO 3\nAbove the structural transition temperature Tc\u0018900 -\n1500 K, the hexagonal manganites RMnO 3(R= Sc, Y,\nIn, and Ho-Lu) are paraelectric insulators. The space\ngroup is P6 3/mmc with two formula units (f.u.) per\nprimitive cell. Below Tc, the size mismatch between\nthe small-radius R3+ion and the large MnO 5bipyra-\nmid leads to an inward tilting of the three corner-shared\nMnO 5polyhedra and an associated \\one-up/two-down\"\nbuckling of the R3+ion layer, as shown in Fig. 1. The\ntransition triples (\\trimerizes\") the unit cell, lowers the\nstructural symmetry to P6 3cm, and induces ferroelec-\ntricity. As the induced polarization is nonlinearly cou-\npled to the trimerization, these systems are improper\nferroelectrics.18,19,21\nThe Mn3+magnetic order develops below the N\u0013 eel\ntemperature TNof\u001870 - 130 K. The in-plane Mn-O-Mn3\n(a)\na\nb\nMn\n3+\n at z=1/2\nMn\n3+\n at z=0\n(\nb\n)\nFIG. 2. Magnetic phases of hexagonal RMnO 3andRFeO 3.\nMn3+ions form triangular sublattices at z= 0 (dash line) and\nz= 1=2 (solid line). (a) A 2phase with magnetic symmetry\nP63c0m0; spins on a given Mn3+layer point all in or all out.\n(b) A 1phase with the magnetic symmetry P6 3cm, with Mn3+\nspins pointing tangentially to form a vortex pattern. The A 1\nand A 2phases di\u000ber by a 90\u000eglobal rotation of the spins.\nThe B 1and B 2phases can be obtained from A 2and A 1by\nreversing the spins on the dashed triangles.\nsuperexchange determines the noncollinear 120\u000eantifer-\nromagnetic (AFM) order on the Mn3+triangular lattice.\nOn the other hand, the inter-plane Mn-O- R-O-Mn ex-\nchange, which is two orders of magnitude weaker than\nthe in-plane exchange, modulates the relative spin direc-\ntions between two consecutive Mn planes.15,24At tem-\nperatures lower than \u00185.5 K, the rare-earth ions with\npartially \flled 4 fshells develop collinear spin order along\nthe hexagonal zdirection. For the Mn3+order, there are\nfour distinct magnetic phases, namely A 1(P63cm), A 2\n(P63c0m0), B 1(P60\n3cm0), and B 2(P60\n3c0m). The linear\nME e\u000bect exists only in A 1and A 2phases. The A 1and\nA2phases are shown in Fig. 2; the B 1and B 2phases\ncan be obtained from A 2and A 1by reversing the spins\non the dashed triangles. From previous experiments,\nit is known that at zero temperature without a mag-\nnetic \feld, HoMnO 3is in the A 1phase, while ErMnO 3,\nYbMnO 3, and LuMnO 3are not in either A phase. Under\na weak magnetic \feld along the ^ zdirection, ErMnO 3and\nYbMnO 3undergo a transition into the A 2phase.24{26\nB. Hexagonal RFeO 3\nEpitaxially grown thin-\flm hexagonal RFeO 3has a\nsimilar structure as hexagonal RMnO 3, with improper\nferroelectricity below \u00181000 K. Replacing Mn3+with\nFe3+introduces larger spin moments and stronger super-\nexchange interactions in the basal plane. In a recent\nexperiment, AFM order has been found to develop at\nTN= 440 K followed by a spin-reorientation transition\nbelowTR= 130 K in LuFeO 3.28It has also been con-\n\frmed that below 5 K, the magnetic structure of LuFeO 3\nis that of the A 2phase.29\nC. Symmetry\nOur purpose is to understand the mechanisms that\ngenerate large magnetic charges that may in turn in-\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nequal component\nequal magnitude with opposite sign \n(a)\n(\nb\n)\n(c)\n(d)\n(\ne\n)FIG. 3. Symmetry patterns of Born charges, magnetic charges\nand ME tensors in RMnO 3andRFeO 3. (a) Tensor form of\nthe ME coupling in the A 2phase, Born charges on R1and\nOP1sites in either A phase, and magnetic charges on the\nsame sites in the A 2phase. (b) Tensor form of the Born and\nmagnetic charges on R2and O P2sites in either A phase. (c)\nTensor form of the Born charges on Mn, Fe, O T1, and O T2\nsites lying on an Mymirror plane in either A phase, and of\nthe magnetic charges on the same sites in the A 2phase. (d)\nTensor form of the ME coupling in the A 1phase, and of the\nmagnetic charges on R 1and O P1sites in the A 1phase. (e)\nTensor form of the magnetic charges on Mn, Fe, O T1, and\nOT2sites lying on an Mymirror plane in the A 1phase.\nduce anomalously large spin-lattice MECs. Therefore,\nwe focus on the A 1and A 2magnetic phases, shown in\nFig. 2, which allow a linear MEC to exist. ErMnO 3,\nYbMnO 3, and LuMnO 3actually adopt other phases as\ntheir ground-state magnetic order at low temperature.\nNevertheless, we include them for purposes of comparison\nwhen calculating the properties of the hexagonal RMnO 3\nmaterials in the A 2phase. We also study LuFeO 3in the\nA2phase, and for HoMnO 3we study both the A 1and\nA2phases.\nThe A 1and A 2phases have the same P6 3cm structural\nsymmetry, so the forms of the atomic Born charge tensors\nin the two phases are the same. The Born charges for\nR1and O P1take the tensor form shown in Fig. 3(a),\nwhile those of R2and O P2have the symmetry pattern\nshown in Fig. 3(b). For the Mn, Fe, O T1, and O T2sites\nlying on a vertical Mymirror plane, the Born charges\nare as given in Fig. 3(c); for the partner sites related by\nrotational symmetry, the tensors also need to be rotated\naccordingly.\nThe symmetry forms of the atomic magnetic charge\ntensors can be derived from the on-site magnetic point\nsymmetries. For the A 1phase, the magnetic space group\nis P6 3cm and the magnetic charges of R1and O P1take\nthe forms given in Fig. 3(d); those for R2and O P2have\nthe tensor symmetry shown in Fig. 3(b); and for Mn,\nFe, O T1, and O T2they can be written in the form of\nFig. 3(e). For the A 2phase, the magnetic group is\nP63c0m0; all the improper operators are associated with\nthe time-reversal operation, so the magnetic charges have\nthe same tensor forms as the Born charges.\nA symmetry analysis of the structure and the mag-4\nTABLE I. Atomic Born charge tensors Ze(in units ofjej) for\nLuMnO 3and LuFeO 3in the A 2phase. TM = Mn or Fe.\nLuMnO 3LuFeO 3 LuMnO 3LuFeO 3\nZe\nxx(Lu1) 3:61 3:79Ze\nxz(OT1) 0:19 0:11\nZe\nzz(Lu1) 4:12 3:94Ze\nzz(OT1)\u00003:19\u00003:21\nZe\nxx(Lu2) 3:66 3:84Ze\nxx(OT2)\u00001:90\u00002:15\nZe\nyx(Lu2) 0:13 0:15Ze\nzx(OT2)\u00000:20\u00000:19\nZe\nzz(Lu2) 3:96 3:88Ze\nyy(OT2)\u00001:85\u00002:13\nZe\nxx(TM) 3 :17 2:96Ze\nxz(OT2)\u00000:18\u00000:11\nZe\nzx(TM) 0 :44 0:21Ze\nzz(OT2)\u00003:33\u00003:30\nZe\nyy(TM) 3 :26 3:01Ze\nxx(OP1)\u00003:00\u00002:40\nZe\nxz(TM) 0 :07\u00000:02Ze\nzz(OP1)\u00001:54\u00001:61\nZe\nzz(TM) 3 :95 4:16Ze\nxx(OP2)\u00003:05\u00002:45\nZe\nxx(OT1)\u00001:92\u00002:19Ze\nyx(OP2)\u00000:03\u00000:02\nZe\nzx(OT1) 0:25 0:25Ze\nzz(OP2)\u00001:43\u00001:52\nZe\nyy(OT1)\u00002:00\u00002:28\nnetic space group identi\fes the phonon modes that cou-\nple to the electromagnetic \feld. The infrared (IR)-active\nphonon modes that couple to the electric \feld are the\nlongitudinal A1modes and the transverse E1modes,\n\u0000IR= 10A1+ 15E1; (4)\nincluding the three acoustic modes. The magnetization\nis generated by phonon modes that couple to the mag-\nnetic \feld. In the A 1phase, the magneto-active phonon\nmodes are the longitudinal A2modes and the transverse\nE1modes,\n\u0000A1\nmag= 5A2+ 15E1; (5)\nwhere one pair of acoustic E1modes are included. In the\nA2phase, on the other hand, the IR- and magneto-active\nphonon modes are identical, since the magnetic and Born\ncharge tensors have the same form in this case.\nFor the MECs in the A 1phase, as the longitudinal\nIR-active and magneto-active modes are mutually exclu-\nsive, the ME tensor takes the form of Fig. 3(d), which\ndoes not have a longitudinal ME component. For the A 2\nmagnetic phase, the A1andE1modes are both IR-active\nand magneto-active, so that the ME tensor has both lon-\ngitudinal and transverse components and adopts the form\nshown in Fig. 3(a).\nD. First-principles methods\nOur calculations are performed with plane-wave den-\nsity functional theory (DFT) implemented in VASP30us-\ning the generalized-gradient approximation parametrized\nby the Perdew-Burke-Ernzerhof functional.31The ionic\ncore environment is simulated by projector augmented\nwave (PAW) pseudopotentials,32and the 4felectrons\nare placed in the PAW core. We use a Hubbard U=\n4:5 eV andJ= 0:95 eV on the dorbitals of the Mn/Fe\natoms, and the moment on the rare-earth ions are notTABLE II. Eigenvalues of the force-constants matrix (eV =\u0017A2)\nfor IR-active modes in LuMnO 3and LuFeO 3in the A 2phase,\nexcluding translational modes.\nA1modes E1modes\nLuMnO 3 LuFeO 3 LuMnO 3 LuFeO 3\n4:24 3 :48 3 :32 3 :56\n7:44 6 :70 4 :68 4 :62\n8:74 8 :41 6 :73 6 :97\n11:51 11 :47 7 :35 8 :09\n14:01 12 :03 8 :63 8 :83\n15:60 15 :59 9 :56 9 :24\n22:66 20 :53 11 :36 11 :37\n25:87 22 :83 12 :46 12 :46\n35:82 28 :46 13 :02 13 :85\n14:09 14 :92\n16:49 16 :87\n17:37 17 :35\n23:36 21 :19\n37:75 28 :75\nconsidered.15The structures are fully relaxed in the\nDFT+U33calculations with their non-collinear spin ar-\nrangements in two cases, when SOC is present and when\nit is absent. In our noncollinear magnetization calcula-\ntion, a high cuto\u000b energy 700 eV and a tight energy error\nthreshold 1 :0\u000210\u00009eV are necessary to get fully con-\nverged magnetic properties. The Born e\u000bective charge\ntensors and the \u0000-point force-constant matrices are ob-\ntained using linear-response methods in the absence of\nSOC. The dynamical magnetic charges are computed by\napplying a uniform Zeeman \feld12to the crystal and\ncomputing the resulting forces. Polarization is calculated\nusing the Berry phase formalism.34A 4\u00024\u00022 \u0000-centered\nk-point mesh is used in the calculations.\nIII. RESULTS AND DISCUSSION\nA. Born charge and force-constant matrix\nThefelectrons are not included in our calculations for\nthe hexagonal RMnO 3class of materials, so the major\ndi\u000berences between compounds result from the variation\nof the rare-earth radius; the trimerization tends to in-\ncrease as the radius of the rare-earth element decreases.\nBecause of the similarity in the geometric structures, the\ndielectric and phonon properties are almost identical in\ntheRMnO 3compounds, regardless of the magnetic or-\ndering. In Tables I and II we list the Born charge tensors\nand the eigenvalues of the force-constant matrix for the\nIR-active modes of LuMnO 3and LuFeO 3. Only small\ndi\u000berences are observed between LuMnO 3and LuFeO 3,\nre\recting the di\u000berent transition-metal atom. The re-\nsults for the other RMnO 3compounds are quite similar\nto those of LuMnO 3and are given for completeness in\nthe Supplement.5\nTABLE III. Longitudinal magnetic charge components Zm\n(10\u00003\u0016B=\u0017A) ofRMnO 3and LuFeO 3in the A 2phase. All\ncomponents vanish in the absence of SOC.\nHoMnO 3ErMnO 3YbMnO 3LuMnO 3LuFeO 3\nZm\nzz(R1)\u000050\u000053\u000053\u000067 7\nZm\nzz(R2) 14 35 24 16 7\nZm\nxz(TM)\u000092\u000086\u000061\u000067 9\nZm\nzz(TM) 24 1 6 25 2\nZm\nxz(OT1)\u000049\u000044\u000041\u000019 23\nZm\nzz(OT1) 99 81 53 33 22\nZm\nxz(OT2)\u00007\u000012\u000012\u000012 0\nZm\nzz(OT2)\u0000119\u000094\u000064\u000049\u000025\nZm\nzz(OP1)\u0000276\u0000257\u0000230\u0000190 54\nZm\nzz(OP2) 141 140 125 100 \u000035\nB. Magnetization and magnetic charge\nIn the A 2phase, the trimerization induces not only\nan electric polarization, but also a weak ferromagnetism\nin the ^zdirection arising from out-of-plane tilting of the\nMn3+spin moments induced by SOC. The net magneti-\nzations in the 30-atom unit cell for A 2-phase HoMnO 3,\nErMnO 3, YbMnO 3, and LuMnO 3are 0.309, 0.303, 0.292,\nand 0.268\u0016B, respectively. These magnetic moments are\nfound to depend almost linearly on the tilting angle of\nthe MnO 5bipyramids, which takes values of 5.03\u000e, 5.07\u000e,\n5.16\u000e, and 5.21\u000erespectively in these four compounds,\nbut in any case the variation is not very large. In con-\ntrast, the result for LuFeO 3is -0.077\u0016B, which is much\nsmaller and of opposite sign compared with the RMnO 3\nmaterials.\nThe magnetic charges de\fned in Eq. (3) are more sen-\nsitive to the local environment, and now the di\u000berences\nbetweenRMnO 3compounds are more signi\fcant. We\ndivide the magnetic charge components into two groups\nthat we label as \\longitudinal\" and \\transverse\" depend-\ning on whether the coupling is to magnetic \felds along\nthe ^zdirection or in the x-yplane respectively.?\nThe longitudinal magnetic charge components are cal-\nculated with a magnetic \feld directed along ^ z, which\nis roughly perpendicular to the spin directions. These\ncomponents are only non-zero when SOC is considered.\nThe scenario here is similar to the case of a transverse\nmagnetic \feld ( HxorHy) applied to Cr 2O3, since the\nmagnetization is along the zaxis for Cr 2O3. It is there-\nfore not surprising to \fnd that the longitudinal magnetic\ncharges ofRMnO 3and LuFeO 3in Table III are compa-\nrable to the SOC-induced transverse magnetic charges in\nCr2O3.16The longitudinal magnetic charges for O P1and\nOP2in LuFeO 3are opposite to, and about three times\nsmaller than, the ones in RMnO 3. These results explain\nthe di\u000berences between RMnO 3and LuFeO 3regarding\nthe magnitude and the direction of the weak ferromag-\nnetism, which is generated by trimerization distortions\ninvolving vertical displacements of O P1and O P2.\nFor the response to transverse magnetic \felds, bothTABLE IV. Transverse magnetic charge components Zm\n(10\u00002\u0016B=\u0017A) of HoMnO 3in the A 1phase, as computed in-\nculding or excluding SOC.\nTotal No SOC Total No SOC\nZm\nyx(Ho1)\u000025\u000028Zm\nzy(OT1)\u0000188\u0000230\nZm\nxx(Ho2)\u000015\u000018Zm\nyx(OT2)\u000057\u000067\nZm\nyx(Ho2)\u00001 3 Zm\nxy(OT2)\u000020\u000026\nZm\nyx(Mn) 92 54 Zm\nzy(OT2)\u0000192\u0000231\nZm\nxy(Mn)\u000010 2 Zm\nyx(OP1)\u0000483\u0000551\nZm\nzy(Mn) 41 48 Zm\nxx(OP2) 395 461\nZm\nyx(OT1) 23 28 Zm\nyx(OP2) 184 253\nZm\nxy(OT1)\u00007\u00007\nthe \feld and the spins lie in the basal plane, so the dy-\nnamical magnetic charges are driven by both SOC and\nexchange striction. As the exchange-striction strength\ncan exceed that of the SOC by orders of magnitude in\nsome materials, it is worthwhile to understand the rel-\native size of these two e\u000bects in RMnO 3and LuFeO 3.\nIn Tables IV and V we present the transverse magnetic\ncharges induced with and without SOC in the A 1and\nA2phases. It is obvious that the SOC contributions\nare an order of magnitude smaller for many transverse\ncomponents. Similarly, the magnetic charges induced by\nexchange striction are about ten times larger than the\nSOC-driven longitudinal ones in Table III. However, the\nSOC is crucial for the Mn atoms and it even reverses the\nsigns of their transverse magnetic charges.\nC. Magnetoelectric e\u000bect\nWe calculate the spin-lattice MEC from Eq. (2) us-\ning our computed Born charges, force-constant matrices,\nand magnetic charges. The spin-electronic contributions\nare calculated based on the @P=@H version of Eq. (1)\nwith the lattice degrees of freedom frozen. We further\nsubdivide the ME tensor components into longitudinal\nand transverse ones based on the direction of Hrelative\nto the hexagonal axis as before, so that the longitudinal\n(transverse) spin-lattice MEC is calculated using the lon-\ngitudinal (transverse) magnetic charge components. The\nMEC tensor elements allowed by symmetry are the lon-\ngitudinal\u000bzzand transverse \u000bxx=\u000byyones in the A 2\nphase, and only the transverse \u000byx=\u0000\u000bxycomponents\nin the A 1phase.\nIn the \frst part of Table VI, the spin-contributed\nlongitudinal MECs are shown for RMnO 3and LuFeO 3\nin the A 2phase. The MEC from the spin channel is\ndominated by the spin-lattice contribution. Although\nthe longitudinal magnetic charges of LuFeO 3are smaller\nthan forRMnO 3, the spin-lattice MECs j\u000bzzjinRMnO 3\nand LuFeO 3are similar,\u00180.25 ps/m. The results are\ncomparable to those reported for the transverse MEC\nin Cr 2O335and for\u000bzzin ErMnO 315in previous \frst-\nprinciples calculations. In the second part of Table VI, we6\nTABLE V. Transverse magnetic charge components Zm(10\u00002\u0016B=\u0017A) ofRMnO 3and LuFeO 3in the A 2phase, as computed\nincluding or excluding SOC.\nHoMnO 3 ErMnO 3 YbMnO 3 LuMnO 3 LuFeO 3\nTotal No SOC Total No SOC Total No SOC Total No SOC Total No SOC\nZm\nxx(R1)\u000023\u000024\u000021\u000022\u000037\u000040\u000042\u000035\u000036\u000052\nZm\nxx(R2) 6 \u00001 6 3 12 9 14 6 15 24\nZm\nyx(R2) 16 18 11 12 10 10 8 7 \u00009\u000011\nZm\nxx(TM)\u00002 10 \u00007\u000010\u000016\u000021\u000011 1 \u000052\u000043\nZm\nzx(TM)\u000042\u000024\u000038\u000022\u000025\u000034\u000031\u000017\u0000102\u000095\nZm\nyy(TM)\u00005 46 \u00007 32 \u000022 27 \u000032 15 \u000016\u000011\nZm\nxx(OT1) 5 5 6 6 12 16 14 11 0 0\nZm\nzx(OT1) 191 221 150 154 162 178 150 122 128 105\nZm\nyy(OT1) 24 23 22 22 31 33 34 25 15 11\nZm\nxx(OT2) 20 23 16 19 19 22 17 12 25 20\nZm\nzx(OT2) 195 217 140 161 173 189 166 134 130 110\nZm\nyy(OT2)\u000059\u000061\u000048\u000046\u000057\u000060\u000057\u000045\u000041\u000042\nZm\nxx(OP1)\u0000445\u0000510\u0000392\u0000422\u0000532\u0000602\u0000564\u0000499\u0000665\u0000609\nZm\nxx(OP2) 241 234 215 202 298 299 316 247 388 356\nZm\nyx(OP2)\u0000378\u0000422\u0000335\u0000355\u0000466\u0000506\u0000498\u0000427\u0000673\u0000621\nshow the spin-related transverse MECs \u000bxxforRMnO 3\nand LuFeO 3in the A 2phase. The same information is\npresented in graphical form in Fig. 4.\nIt is clear from the comparison between the \frst and\nsecond parts of Table VI that the transverse spin-lattice\nMECs are one order of magnitude larger than the longi-\ntudinal ones, as a result of the exchange-striction mech-\nanism. Surprisingly, Fig. 4(a) shows that the e\u000bect of\nSOC on the exchange striction is profound, even revers-\ning the sign of the spin-lattice MECs in RMnO 3. This\nunusual behavior can be traced mainly to two observa-\ntions about the spin-lattice contributions from di\u000berent\nIR-active modes in the RMnO 3materials. Firstly, the\nTABLE VI. Computed MECs \u000bzz(longitudinal) and \u000bxx\nand\u000byx(transverse) for RMnO 3and LuFeO 3(ps/m). Spin-\nlattice, spin-electronic, and total spin couplings are given as\ncomputed with and without SOC.\nSpin-latt. Spin-elec. Total spin\nTotal No SOC Total No SOC Total No SOC\n\u000bzzin A 2phase\nHoMnO 3\u00000:27 0 0 :06 0\u00000:21 0\nErMnO 3\u00000:26 0 0 :05 0\u00000:21 0\nYbMnO 3\u00000:25 0 0 :06 0\u00000:19 0\nLuMnO 3\u00000:19 0 0 :00 0\u00000:19 0\nLuFeO 3 0:26 0 0 :00 0 0 :26 0\n\u000bxxin A 2phase\nHoMnO 3\u00000:99 5:12 4:10 4:83 3:11 9:95\nErMnO 3\u00001:30 2:40 2:56 3:72 1:26 6:12\nYbMnO 3\u00002:52 1:20 3:72 4:66 1:20 5:86\nLuMnO 3\u00002:60 1:31 3:82 3:50 1:22 4:81\nLuFeO 3\u00002:20\u00001:57\u00000:79\u00000:32\u00002:99\u00001:89\n\u000byxin A 1phase\nHoMnO 3 9:55 4:88 5:24 5:35 14:79 10:23exchange-striction MEC is smaller than expected as a\nresult of a large degree of cancellation between the con-\ntributions from di\u000berent transverse IR-active modes. To\nillustrate this, the mode-by-mode contributions are pre-\nsented for a few selected cases in Table VII. Secondly, the\nsoftest modes are dominated by Mn displacements, pre-\ncisely those for which SOC has the largest e\u000bect on the\nZmvalues, even \ripping the sign of some components. In\n-20246810\n(c)(b)\nspin-latt.  Total\n No SOC(a)\n0246spin-elec. \n-4-20246810121416\nLuFeO3HoMnO3 LuMnO3 YbMnO3ErMnO3total spin\nHoMnO3\n(A1phase)\nFIG. 4. Transverse MECs for RMnO 3and LuFeO 3.\u000bxx\n(ps/m) in the A 2phase and\u000byxin the A 1phase. (a) Spin-\nlattice; (b) spin-electronic; and (c) total spin couplings.7\nTABLE VII. Transverse MEC contributions (ps/m) from IR-\nactive modes for A 2and A 1phases of HoMnO 3and A 2phase\nof LuMnO 3. Results are given in ascending order of force-\nconstant eigenvalues, which are reported in Table II of the\nSupplement.\nA2phase HoMnO 3A1phase HoMnO 3A2phase LuFeO 3\nTotal No SOC Total No SOC Total No SOC\n0:01 0:12 0:25 0:18 0:28 0:39\n\u00001:16 2:62 4:98 2:36\u00000:54\u00000:50\n0:66 2:32 3:59 2:37\u00001:31\u00001:22\n\u00000:51\u00000:35\u00000:32\u00000:48 1:30 1:23\n2:79 3:13 2:87 3:33 3:31 3:12\n0:35 0:21 0:30 0:30 1:84 1:73\n\u00001:88\u00001:85\u00001:35\u00001:90\u00004:43\u00004:11\n1:13 1:25 1:19 1:38\u00002:59\u00002:25\n\u00002:96\u00003:07\u00002:70\u00003:40 1:24 1:13\n0:01 0:13 0:19 0:06\u00001:48\u00001:27\n0:21 0:24 0:21 0:26\u00000:15\u00000:14\n0:36 0:40 0:34 0:42 0:89 0:83\n\u00000:03\u00000:03\u00000:03\u00000:04\u00000:62\u00000:55\n0:02 0:01 0:03 0:03 0:07 0:03\nthis way, it turns out that SOC can result in large rela-\ntive changes in the MEC results. In the case of LuFeO 3,\nthe SOC e\u000bect on the Zmvalues is weak, even for Fe\natoms. Thus, the MEC of LuFeO 3does not change as\ndramatically as that of RMnO 3when SOC is included.\nFrom Fig. 4(b) it can be seen that the spin-electronic\ncontribution is not negligible in the transverse direction,\nand it counteracts the MEC from the spin-lattice channel\nin A 2phaseRMnO 3. The total transverse ME e\u000bect is\nsummarized in Fig. 4(c). Because of the large SOC e\u000bect\nand the cancellation between the lattice and electronic\ncontributions, the total spin MEC \u000bxxis\u00181.2 ps/m in\nmost A 2-phaseRMnO 3compounds, except for HoMnO 3.\nIn HoMnO 3, the cancellation between the spin-lattice\nand the spin-electronic MECs is the weakest of all the\nRMnO 3compounds, resulting in the largest total spin\nMEC of\u00183:1 ps/m in the A 2phase. In LuFeO 3, the\nspin-lattice and spin-electronic terms are all smaller than\ninRMnO 3. However, the cancellation induced by the\nSOC perturbation and the spin-electronic contribution is\navoided, so that LuFeO 3has a large total spin MEC of\n\u0018\u00003 ps/m.\nWe present the MECs for HoMnO 3in the A 1phase inthe last line of Table VI and in Fig. 4. In principle the\nMECs of HoMnO 3in the A 1and A 2phases should be\nthe same without SOC, as the two phases only di\u000ber by a\nglobal spin rotation. This is approximately con\frmed by\na comparison of the corresponding entries for HoMnO 3in\nTable VI. The ME contribution from exchange striction\n(i.e., without SOC) is \u00185 ps/m for both the A 2and A 1\nphases. However, when the e\u000bect of SOC is included, the\nspin-lattice contribution is strongly enhanced by another\n\u00185 ps/m. Furthermore, the spin-electronic MEC has the\nsame sign as the spin-lattice one, which adds \u00185 ps/m\nto the MEC. Therefore, the total spin MEC \u000byxreaches\n\u001815 ps/m, and is the largest in all of the RMnO 3and\nLuMnO 3materials we studied.\nIV. SUMMARY\nIn summary, we have studied the spin-related magnetic\ncharges and MECs for HoMnO 3, ErMnO 3, YbMnO 3,\nLuMnO 3, and LuFeO 3using \frst-principles calculations.\nWe con\frm that the exchange striction acting on non-\ncollinear spins induces much larger magnetic charges\nthan does SOC acting alone. Nevertheless, the e\u000bect of\nSOC on the MECs is surprisingly large, rivaling that of\nexchange striction in many cases. This occurs because\nthe exchange-striction contribution tends to be reduced\nby cancellations between di\u000berent IR-active modes, while\nthe SOC contribution is mainly associated with just a few\nlow-frequency modes with large Mn displacements. We\nalso \fnd that the RMnO 3materials have spin-electronic\nMECs comparable to the spin-lattice ones. Among the\nRMnO 3and LuFeO 3materials we studied, we \fnd that\nthe A 1phase of HoMnO 3is the most promising ME ma-\nterial, with the largest MEC of \u001815 ps/m. Extrapolating\nour conclusions to other hexagonal RMnO 3andRFeO 3\ncompounds that are not included in our calculations, we\npredict that the A 2phase is more promising for the fer-\nrites, while the A 1phase has a stronger MEC for the\nmanganites.\nACKNOWLEDGMENTS\nWe thank Weida Wu for useful discussions. The work\nwas supported by ONR grant N00014-12-1-1035.\n\u0003mengye@physics.rutgers.edu\n1M. Fiebig, J. Phys. D 38, R123 (2005).\n2N. A. Spaldin and M. Fiebig, Science 309, 391 (2005).\n3W. Eerenstein, N. D. Mathur and J. F. Scott, Nature 442,\n759 (2006).\n4Y. Tokura, J. Magn. Magn. Mater. 3101145 (2007).\n5S.-W. Cheong and M. Mostovoy, Nat. Mater. 6, 13 (2007).\n6R. Ramesh and N. A. Spaldin, Nat. Mater. 6, 21 (2007).7K. F. Wang, J.-M. Liu and Z. F. Ren, Adv. Phys. 58, 321\n(2009).\n8M. Fiebig and N. A. Spaldin, Eur. 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Hillebrands\nFachbereich Physik, Nano+Bio Center, and Forschungszentr um OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\nM.P. Kostylev\nSchool of Physics, University of Western Australia, Crawle y, Western Australia 6009, Australia\n(Dated: November 16, 2018)\nOne-dimensional magnonic crystals have been implemented a s gratings of shallow grooves chemi-\ncally etched into the surface of yttrium-iron garnet films. S cattering of backward volume magneto-\nstatic spin waves from such structures is investigated expe rimentally and theoretically. Well-defined\nrejection frequency bands are observed in transmission cha racteristics of the magnonic crystals. The\nloss inserted bythe gratings and therejections bands bandw idthsare studied as afunction of the film\nthickness, the groove depth, the number of grooves, and the g roove width. The experimental data\nare well described by a theoretical model based on the analog y of a spin-wave film-waveguide with\na microwave transmission line. Our study shows that magnoni c crystals with required operational\ncharacteristics can be engineered by adjusting these geome trical parameters.\nPACS numbers: 75.50.Gg, 75.30.Ds, 75.40.Gb\nI. INTRODUCTION\nPeriodically structured magnetic materials such as\nmagnonic crystals (MC) attract special attention in view\nof their applicability for both fundamental research on\nlinear and nonlinear wave dynamics in artificial media,\nand for signal processing in microwave frequency range\n[1]. Similar to sound and light in sonic and photonic\ncrystals, the dispersion characteristics of spin waves in\nmagnonic crystals are strongly modified with respect to\nuniform media. This results in the appearance of fre-\nquency band gaps [2, 3] wherein spin-wave propagation\nis forbidden. A large variety of nonlinear spin-wave phe-\nnomena, aswellasthe dependencyofthespin-waveprop-\nerties both on the magnitude and orientation of a bias\nmagnetic field determine the wide tunability of opera-\ntional characteristics of the magnonic crystals and their\npotential for design of microwave filters, switchers, cur-\nrent controlled delay lines, power limiters, etc [1].\nDepending on the required insertion loss, operating\nfrequency range, dimensions, temperature stability, and\nother performance specifications, periodic structures can\nbe fabricated from either ferrite or ferromagnetic sub-\nstances by means of geometric structuring [1, 4, 5, 6, 7,\n8, 9, 10], metal deposition [11], ion implantation [12], lo-\ncal variations of the bias magnetic field [13], to name but\na few.\nAt the present time, the smallest out-of-band inser-\ntion loss in conjunction with the deepest rejection bands\nhave been observed in the experiments with geometri-\ncally structured yttrium-iron-garnet (YIG) single crystal\nferrite films grown on a gallium-gadolinium substrate by\n∗Electronic address: chumak@physik.uni-kl.de;\nAlso at National Taras Shevchenko University of Kiev, Ukrai ne.means of liquid-phase epitaxy. In this unique material\n[14] the lifetime of spin-wave excitations can exceed a\ncoupleofhundreds nanoseconds, and the spin-waveprop-\nagation path reaches a few centimeters.\nThe surface magnetostatic waves [15] (wave propaga-\ntion direction is perpendicular to the magnetic field ap-\nplied in the film plane) and the forward magnetostatic\nwaves [16] (external field is oriented perpendicular to the\nfilm plane) were previously studied theoretically and ex-\nperimentally in such periodical structures [1, 4, 5, 6].\nIn our recent paper [17] we presented the first experi-\nmental and theoretical results on scattering of backward\nvolume magnetostatic waves (wave propagation direc-\ntion is parallel to the magnetic field applied in the film\nplane) [15] from an one-dimensional structure with peri-\nodic changes of the YIG film thickness.\nIn the present paper we present a detailed study of this\nkind of magnonic crystal. Its main characteristics, such\nas insertion loss in the rejection bands, parasitic loss in\nthe transmission bands, and the frequency bandwidth of\nthe rejection bands, were investigated for crystals hav-\ning different groove depths, widths, and groove numbers.\nOur results show these parameters can be used to opti-\nmize the design of magnonic crystals.\nII. TECHNOLOGY AND EXPERIMENTAL\nMEASUREMENTS\nTo fabricate magnonic crystals, 5.5 µm and 14 µm-\nthick YIG films, which were epitaxially grown in the\n(111) crystallographic plane, were used. Photolitho-\ngraphic patterning followed by hot orthophosphoric acid\netching was used to form the grooves. The lithography\nwas based on a standard photoresist AZ 5214E hard-\nened by UV irradiation and baking which makes it stable\nagainst hot 160oC orthophosphoric acid. Using this pro-2\nFIG. 1: (Color online) Scheme ofamagnonic crystal structur e\nand of the measurement setup used in the experiments.\ncedure we patterned arrays of Nparallel lines ( N= 10\nandN= 20) of widths w= 30µm and interline spac-\nings of 270 µm. Alternatively, arrays where w= 10µm\nspaced by 290 µm were also prepared. In all cases, the\nlattice constant was a= 300µm. The grooves were\ntransversely oriented with respect to the spin-wave prop-\nagation direction. In order to study the dependence of\ncrystal characteristics on the groove depth δthe grooves\nwere etched in 100nm steps from 100nm to 2 µm. The\ngroove depth was controlled by the etching time (etch\nrate was ∼1µm/min) and measured using a profilome-\nter. Anisotropic chemical etching of the YIG crystal\nstructure by orthophosphoric acid [18] was observed: the\netch rate parallel to the film plane was approximately\nten times larger than in the perpendicular direction, so\nthe final groove depth profile along the direction of wave\npropagation had a trapezoidal shape.\nThe inset in Fig. 1 shows the microscope image of\nthe grooves structure of the magnonic crystal with the\ngroove depth δ= 500 nm and groove width w= 30µm.\nAnisotropic etching can be clearly seen.\nIn order to excite and receive the dipolar spin waves,\ntwo microstrip antennas were placed 8mm apart, one in\nfrontofthe grating,andonebehind it (seeFig.1). Abias\nmagnetic field of 1845Oe was applied in the plane of the\nYIG film stripe, along its length and parallelto the direc-\ntion of spin-wave propagation. Under these conditions,\nbackward volume magnetostatic wave (BVMSW) propa-\ngation occurs. A continuous-wave microwave signal was\napplied from a network analyzer (see Fig. 1) to the input\nantenna, and BVMSW transmission characteristics were\nmeasured. The microwave signal power was maintained\nat 1mW in order to avoid any non-linear processes.\nIII. THEORETICAL MODEL\nIn a general case, a theoretical description of the scat-\ntering of dipole spin waves from inhomogeneities is givenby a singular integral equation\nm(r) = 4πˆκ(f,Hi(r),M(r))·/parenleftbigg/integraldisplay\nVˆG(r−r′)m(r′)dr′+h(r)/parenrightbigg\n, (1)\nwheremis the dynamic magnetization, his the mi-\ncrowavefieldoftheinputantenna, ˆG(r−r′)istheGreen’s\nfunctionofdipolemagneticfield,and ˆ κ(f,Hi(r),M(r))is\nthetensorofthemicrowavemagneticsusceptibilitywhich\ndepends on the spin-wave carrier frequency fand on the\nmagnetic parametersofthe film. The latter are functions\nof the position r, since the grooves induce inhomogene-\nity in the internal static field Hiand in the equilibrium\nmagnetization M. The integration is carried over the\nvolume of the magnetic structure, thus the inhomogene-\nity of the film thickness is taken into account. A simple\napproximate solution to Eq.(1) for the reflected wave can\nbe obtained in the frame of the first Born approximation\n(see [19] for details of the approach). Neglecting mag-\nnetic damping in the medium far in front of the groove\nlatticex << x′for the complex reflection coefficient R,\none obtains\nR(k) =iw\nd0F(k,d0,δ)sin(kw)\nkwN/summationdisplay\nn=0exp(−2ikna),(2)\nwhereF(k,d0,δ) =[exp(−kd0)(exp(kδ)−1)−(exp(−kδ)−1)]\nkd0≈\n2δ/d0−kδforkd0<<1, where kis the wavenumber of\nthe incident spin wave, d0is the unpatterned film thick-\nness,δis the groove depth, wis the groove width, ais\nthe lattice period, and Nis the total number of grooves\nin the groove lattice. This formula is in agrement with\nBragg’s diffraction law. The maxima of reflection occur\natk=nπ/aor for the wavelengths 2 a/n. With increas-\ningNthe depth ofthe reflection bands increaseand their\nwavenumber widths decrease. Furthermore, the reflec-\ntion grows with an increase in wandδ/d0, which is con-\nsistent. However,accordingtothisformula,thereflection\nshould decrease with increasing k, which is in contradic-\ntion with the experimental results presented here (see\nSection IV). Thus a more detailed model is necessary.\nTobuild amoreappropriatemodel, we firstnotice that\nthequasi-1Ddipolefield[20]ofthelowestBVMSWthick-\nness mode decays within a distance of a few film thick-\nnesses from its source. Because the width of the grooves\nwis much smaller than a, the spin wave travels as an\neigenmode of a continuous film of thickness d0through\nmost of the lattice ( i.e.between grooves). For the sec-\ntions where the thickness is d0the integral equation re-\nduces to a simple formula (see Eq. (50) in [19]) which\nshows that between the grooves, the transmitted and re-\nflected waves only accumulate phase and decay (due to\nintrinsic magnetic damping). Thus, in order to describe\nthe formationof stop bands, one has to considerthe scat-\ntering of a BVMSW from just one groove. The effect\nof multiple consecutive grooves is obtained by cascad-\ning the structure period using matrices of scattering T-\nparameters and taking interference effects into account.3\nThe most direct way to proceed is to solve this two-\ndimenstional singular integral equation numerically to\nobtain the scattering characteristics of a BVMSW scat-\ntering from a single groove. However this work is beyond\nthe scope of this paper. Another way to treat scattering\nfrom a single groove was previously suggested in Refs.\n[21, 22]. In our short paper [17], we adapted this method\nto the case of a BVMSW. For completeness, we repro-\nduce our theory here, but in more detail. We first con-\nsider the grating as a periodical sequence of sections of\nregulartransmissionlineswith differentpropagationcon-\nstants (different spin-wave wavenumbers) for the same\ncarrier frequency. We neglect the fact that the groove\nedges are oblique, and consider the groove cross-section\nas a rectangle with the same depth and having the same\narea. Inordertodescribespinwavetransmissionthrough\nthe magnonic crystal we use T-matrices which describe\nthe relation between the amplitudes of the wave incident\nonto an inhomogeneity and reflected from it [23]. The\nT-matrix T(1)for a section of unstructured film (section\nof film between neighboring grooves) of length a−whas\ndiagonal components only:\nT(1)=/parenleftbigg\ne(−ik+k′′\n0)(a−w)0\n0 e(ik−k′′\n0)(a−w)/parenrightbigg\n,(3)\nwherekisthe spin-wavewavenumberinthe unstructured\nfilm,k′′\n0=γ∆H/(2vgr)istherateofthespin-wavespatial\ndamping, γis the gyromagnetic ratio, ∆ His the ferro-\nmagnetic resonance linewidth, and vgris the spin-wave\ngroup velocity.\nSimilarly, the T-matrix T(3)for a regular spin-wave\nfilm waveguide with a thickness d=d0−δis\nT(3)=/parenleftbigg\ne(−ik+k′′\ng)wd0/d0\n0 e(+ik−k′′\ng)wd0/d/parenrightbigg\n,(4)\nwherek′′\ngis the spin-wave damping rate for the groove.\nHere we use the fact that the BVMSW dispersion law for\nsmall wavenumbers kd≪1 is practically linear. There-\nfore, the spin-wave wavenumber in the grooves is kd0/d.\nTo describe the loss increase in the pass bands with\nincreasing groove depth, we introduce an empirical pa-\nrameterζwhich accounts for larger contribution of two-\nmagnon scattering processes in the areas which under-\nwent anisotropic etching [18]. Then the damping rate in\nthe grooves can be expressed as k′′\ng=k′′\n0(1+ζδ/d0).\nAt the edges of the grooves the incident wave is par-\ntially reflected back. This is accounted for by the T-\nmatrices for the groove edges. The matrix for the front\nedge is T(2), and that for the rear edge is T(4). Following\n[23], the transmission coefficient through the junction is\n1−Γ. Then one obtains:\nT(2)=/parenleftbigg\n(1−Γ)−1Γ(1−Γ)−1\nΓ(1−Γ)−1(1−Γ)−1/parenrightbigg\n,(5)\nT(4)=/parenleftbigg\n(1+Γ)−1−Γ(1+Γ)−1\n−Γ(1+Γ)−1(1+Γ)−1/parenrightbigg\n.(6)Using the property of T-matrix multiplication one finds\na T-matrix for one period of the structure:\nT = [T(1)·T(2)·T(3)·T(4)]. (7)\nTo obtain the T-matrix for a magnonic crystal with N\ngrooves Tmc, one has to raise T to the N-th power:\nTmc= [T(1)·T(2)·T(3)·T(4)]N. (8)\nThe most important operational parameter of magnonic\ncrystals is the power transmission coefficient. It can be\ndetermined as Ptr= 1/|Tmc\n11|2= 1/|Tmc\n22|2, whereTmc\n11\nandTmc\n22are the matrix elements.\nIn order to make use of this theory, one has to spec-\nify the form of the reflection coefficient Γ. The model\nwe suggest is based on the analogy of the change in the\nfilm waveguiding properties to a change in the charac-\nteristic impedance Zof a microwave transmission line\n[21]. The expression for the complex reflection coefficient\nfor a junction of two microwave lines with characteristic\nimpedances Z0andZcan be written as [23]:\nΓtr.line=Z−Z0\nZ+Z0. (9)\nWe assume that the change of the characteristic\nimpedance of a spin-wave waveguide arising from the\nchange of YIG-film thickness is due to a change of\nthe film’s effective inductance. Then the characteristic\nimpedance is linearly proportional to the propagation\nconstant (to the spin-wave wavenumber in our case), and\nwe obtain a formula for the reflection coefficient for a\nwave incident onto the edge of a groove from the un-\nstructured section of the film:\nΓ =ηd0−(d0−δ)\nd0+(d0−δ)=ηδ\n2d0−δ, (10)\nwhereη >1 is a phenomenological parameter introduced\nin this formulato accountfor eventualfactorswhich were\nnot taken into account in this simplistic model.\nFor the wave incident onto the same junction in the re-\nverse direction, Γ −=−Γ, which has already been taken\ninto account in the expressions for the T-matrices above.\nThe expression Γ −=−Γ means that the wave phase\nchange due to reflection from one and another edge of\ngroove is π.\nIV. RESULTS AND DISCUSSION\nThe experimental BVMSW transmission characteris-\ntics for the unstructured film and for the magnonic crys-\ntals with δ= 300, 600 and 900nm measured with a\nnetwork analyzer are shown in Fig. 2(a). The initial\nYIG film thickness d0is 5.5µm. The results are shown\nfor the groove number N= 20 and the groove width\nw= 30µm. The BVMSW transmission characteristic\nfor the unstructured film has a maximum just below the4\npoint of ferromagnetic resonance. One sees that the in-\nsertion loss is ≈20 dB and is determined by the energy\ntransformation efficiency by the input and the output\nantennas and by the spatial decay of spin waves during\ntheir propagation in the space between the antennas. As\nBVMSW frequency band is bounded above by the ferro-\nmagnetic resonance frequency no spin-wave propagation\noccurs for higher frequencies. With decreasing frequency\nthe BVMSW excitation and reception efficiencies drop.\nThis drop is due to the finite width of the antennas in\nthe direction of BVMSW propagation. Thus for very low\nand very high frequencies, no spin waves can be excited\nand the insertion loss ( ≈50 dB) is determined by the di-\nrectelectromagneticleakagefrom theinput tothe output\nantenna. A small separate peak at the right-hand edge\nof the transmission characteristic is due to excitation of\nthe first width standing mode of film [24].\nIn order to remove the influence of the antennas and\nof the spin-wave spatial decay we calculate the difference\nbetween the logarithms of the transmission characteris-\ntics for the unstructured and the structured films. Fig-\nure 2(b) shows the obtained dependence. The additional\ndotted straight line in this figure indicates the limit of\nthe dynamic range of our experimental setup. The latter\nis found as the difference in the transmission character-\nistics of the microstrip antenna structure that is covered\nor uncovered by a continuous YIG film.\nFrom Fig. 2(b) one sees that a lattice of 20 grooves\nas deep as 300nm leads to the appearance of a set of\nrejection bands (or transmission gaps), where spin-wave\ntransmission is highly reduced. According to the con-\ndition for Bragg reflection, higher-order rejection bands\ncorrespond to largerspin-wavewavenumbers. In the case\nof BVMSWs, the latter corresponds to lower frequencies.\nFrom the depths and the frequency widths ∆ fof the\ngaps, oneseesthatthe efficiencyoftherejectionincreases\nwith increasing order of Bragg reflection. This suggests\nthat BVMSW with smaller wavelengths are more sensi-\ntive to the introduced inhomogeneities.\nBoth Fig. 2(a) and 2(b) demonstrate that an increase\ninδleads to an increase in the rejection efficiency and\nin the frequency width of rejection bands ∆ f. Addition-\nally, a small frequency shift of the minima of transmis-\nsion towards higher frequencies is observed, as well as\nan increase in insertion losses in the transmission (i.e.\nallowed) bands. For δ= 900nm the insertion loss in\nthe whole spin-wave band is so important that almost no\nspin-wave propagation is observed (see Fig. 2(a)) for the\nfilm of 5.5 µm thickness.\nThe results of our numerical computation of Tmcare\nshown in Fig. 2(c). One sees that this model provides\nqualitative agreement with all the experimentally ob-\nserved trends. In particular, it correctly predict the ob-\nserved increase in the rejection efficiency with increas-\ningk. The calculated efficiency of the rejection also in-\ncreases with increasing rejection band order. In order\nto prove this we calculate the transmission character-\nistic for a structure which consists of only one 30 µm\nFIG. 2: (Color online) (a) - BVMSW microwave transmis-\nsion characteristics for an unstructured film (bold line) an d\nfor magnonic crystals with different groove depths δ; (b) -\nmeasured transmission loss inserted by the magnonic crysta l\nstructure; (c) - calculated loss; (d) - calculated transmis sion\ncharacteristics. Parameters of calculation: groove numbe r\nN= 20, width of grooves at their bottom w= 30µm, lat-\ntice constant a= 300µm, film thickness d0= 5.5µm, sat-\nuration magnetization 4 πM0= 1750G, bias magnetic field\nH0= 1845Oe, efficiency coefficient η= 6, resonance line\nwidth ∆ H= 0.5Oe, surface damage coefficient ζ= 30. In\n(c), the dash-dotted line shows the calculated loss inserte d by\na single groove of 300nm in depth.\nwide and 300nm deep groove (see the dash-dotted line\nin Fig. 2(c)). One sees that the efficiency of spin-wave\nreflection from one groove increases with increase in spin\nwave wavelength. It is worth noting that the transmis-\nsion loss inserted by one groove is about 0.1dB. Thus,\nalmost all the energy of the spin waves is transferred\nthrough the groove. Only a very small part (about 3 %)\nis reflected back.\nFig. 2(c) shows that our computation gives the correct\nshape of the transmission characteristics. In particular,\nthere is a good agreement with the frequency widths ∆ f\nof rejection bands and in their frequency shifts upwards\nwith increasing δ. Furthermore, the model properly de-5\nscribes the increase in the parasitic insertion loss in the\ntransmission bands with increasing groove depth.\nThe calculated influence of the groove array on the\ntransmission characteristic of the magnonic crystal is\nshown in Fig. 2(d). The presented curves were found\nas a difference of the experimental transmission charac-\nteristic for a plane film (bold line in Fig. 2(a)) and the\ncalculated groove induced loss (Fig. 2(c)). As a result,\nFig. 2(d) are a theoretical analogue to Fig. 2(a), where\nexperimental results are presented. From comparison of\nthe figures one can concludes that proposed theoretical\nmodel describes experimental results especially good for\nthe small values of the groove depth. With increasing of\nthe groove depth the disagreement increases because of\nthe theoretical model limitations.\nA. Influence of the groove depth\nThe groove depth has a profound influence on the\ntransmission characteristics of the fabricated magnonic\ncrystals (see Fig. 2). In order to investigate this effect\nin Fig. 3 we plot the insertion loss for the first-order\nrejection band, its frequency width, and the parasitic\nloss in the first transmission band for magnonic crys-\ntals of different thicknesses d0and with different groove\ndepths. The central frequency of the first-order gap is\n7160MHz. From Fig. 3 (a) one sees that the rejection\nefficiency strongly increases with increasing the relative\ngroove depth δ/d0. An increase in the parasitic loss in\nthe transmission bands is also observed. By comparing\nthe lossesin the rejectionand transmissionbands, we can\nestimate the optimal value of δ/d0. We define this op-\ntimum value as a situation in which rejection is efficient\nbut parasitic loss is still small (around 3 dB). For both\nfilms we find an optimal value of δ/d0≈0.1.\nBoth our calculation and measurements for different\nfilm thickness show that parasitic losses in transmission\nbands are determined by the relative groove depth only.\nThe situation with the rejection efficiency is more com-\nplicated. From Fig. 3 (a) it can be seen that the exper-\nimental dependencies are slightly different for different\nfilm thicknesses. For the same relative groove depth, re-\njection is larger for thicker films. However, with increas-\ningδthis difference between films with different thick-\nnesses diminishes, and for the largest values of δ/d0the\nexperimental dependencies for different thicknesses col-\nlapse. This suggests that the relative groove depth is the\nleading parameter for the magnonic crystal optimization.\nIt is worth noting that the behavior of the calcu-\nlated curves is slightly different. This suggests that the\naccuracy of our simple theory decreases with increas-\ning groove depth. Processes not taken into account in\nour model probably become more prominent with larger\nδ/d0.\nFig. 3 (b) shows the experimental and calculated fre-\nquency width of the first rejection band. The fre-\nquency width was measured at the distance betweend0=□5.5 m /c109\nd0=□5.5 m /c109\nRelative□groove□depth / /c100d0d0=□14 m /c109\nd0=□14 m /c109\n0.05 0.10 0.151020304050\n01020304050\n0Rejection□band□loss□(dB)\nRejection□band□width□(MHz)1020304050\n0Pass□band□loss□(dB)\n,\n,,,\n,,\n(a)\n(b)\nFIG. 3: (color online) (a) - insertion loss in the first-order re-\njection band (circles) and the pass band (squares) as a func-\ntion of the relative groove depth δ/d0; lines show correspond-\ning calculated dependencies. (b) - experimental (triangle s)\nand theoretical (lines) frequency width of the first-order r e-\njection band as a function of the relative grooves depth. In\nboth panels, filled red symbols and solid red lines show de-\npendencies for the film d0= 5.5µm thick; opened blue sym-\nbols and dashed blue lines are for d0= 14µm. Number of\ngroovesN= 20; lattice constant a= 300µm; groove width\nw= 30µm.\nthe points for which the transmitted signal intensity is\nhalved. Obviously,thesameBraggconditionforthespin-\nwave number in the maximum of rejection is fulfilled for\nthe magnonic crystals based on the films with thickness\n5.5µm and 14 µm, because the arrayhas the same lattice\nconstant 300 µm in both cases. However, the spin wave\ndispersion strongly depends on film thickness [15]. For\nsmall wavewavenumbersits slope increases with increase\nind0. This results in the wider rejection bands observed\nfor the thicker magnonic crystal.\nFrom Fig. 3 (b) one sees that the width of the rejection\nband for the optimal relative groove depth 0.1 is 15 MHz\nand 45 MHz for the 5.5 µm and 14 µm thick films, respec-\ntively. From the point of view of optimizing magnonic\ncrystals, this gives an additional degree of freedom for\ndesigning a crystal which satisfied operational parame-\nters. A required rejection-band bandwidth can be ob-\ntained by adjusting the magnon crystal thickness. The\noptimal rejection efficiency can be obtained by adjusting\nthe relative groove depth.\nFrom Fig. 3 (b) it follows that our theory is not very\naccurate. The agreement is qualitative only. However,\nconsidering the simplicity of our model, the agreement is\nsatisfactory, and provides key trends.6\nRelative□groove□depth / /c100d0N=□10\n0.05 0.10 0.1510203040\n0Rejection□band□loss□(dB)\nRejection□band□width□(MHz)Pass□band□loss□(dB)\n10203040\n010203040\n0N=□20\nN=□10N=□20 ,\n,,,\n,,\n(a)\n(b)\nFIG. 4: (color online) (a) - insertion loss in the first-order re-\njection (circles) and pass (squares) bands as a function of t he\nrelative groove depth δ/d0; lines show corresponding calcu-\nlated dependencies. (b) - experimental (triangles) and the o-\nretical (lines) bandwidth of the first-order rejection band as a\nfunction of relative groove depth. Filled red symbols andso lid\nred lines are for the magnonic crystal with N= 20 grooves;\nopened green symbols and dashed green lines are for the crys-\ntal with N= 10 grooves. Film thickness d0= 5.5µm; lattice\nconstant a= 300µm; groove width w= 30µm.\nB. Influence of the number of grooves\nThe groove number is also important for the optimiza-\ntion of the magnonic crystal. Fundamentally, increasing\nthe groovenumber should increase the efficiency of rejec-\ntion.\nAccording to the developed theoretical model (see\nEq. (8)), doubling the number of grooves should dou-\nble the rejection losses on the log scale. In order to test\nthis theoretical prediction, two crystals with the same\ngeometry but having different grooves numbers N= 10\nandN= 20 were investigated.\nFig. 4 presents the insertion loss and the frequency\nwidth ∆f1of the rejection bands for the magnonic crys-\ntals with the groove numbers N= 10 and N= 20, as\na function of the relative groove depth δ/d0. One sees\nthat reducing the number of grooves by one half results\nslightly decreases the rejection efficiency. One of the pos-\nsibleexplanationforsuchanexperimentalbehaviorcould\nbe a deviation from perfect periodicity in the lattice,\nwhich may saturate such dependence for groove numbers\nlarger than some characteristic value. However, a careful\nexamination of optical images of the quasi-crystal does\nnot reveal any noticeable defects. As such, we suggest\nthis unexpectedly weak dependence may be connected\nwith some peculiarity of the intrinsic spin wave damping,\n,,,\n,,\nRelative groove depth / /c100d0w= 10 m /c109\n0.05 0.10 0.1510203040\n0Rejection band loss (dB)\nRejection band width (MHz)Pass band loss (dB)\n10203040\n010203040\n0w= 30 m /c109\nw= 10 m /c109w= 30 m /c109(a)\n(b)\nFIG. 5: (a) - insertion loss in the rejection band (squares)\nand the pass band (circles) as a function of the groove depth;\nsolid lines show the respective calculated dependencies. ( b)\n- experimental (triangles) and theoretical (line) width of the\nfirst rejection band as a function of the groove depth. In\nboth panels, filled symbols and solid lines are for a magnonic\ncrystal with groove width w= 30µm; opened symbols and\ndashed lines are for a magnonic crystal with groove width\nw= 10µm. Film thickness d0= 5.5µm; lattice constant\na= 300µm; number of grooves N= 20.\non the structure which is not taken into account in our\nmodel.\nFig.4(b) showsexperimentalandcalculatedfrequency\nwidthofthe firstrejectionbandforthemagnoniccrystals\nwith the groove numbers N= 10 and N= 20, as a\nfunction of the relative groove depth. No pronounced\ndependence on the number of groove Nis observed.\nC. Influence of the groove width\nThe groove width is also an important parameter in\nthe design and optimization of magnonic crystals. In an\nideal case, a Bragg reflector should consist of infinitely\nnarrow reflectors, which are periodically placed in space.\nHowever, in our case infinitely narrow grooves will pro-\nduce no reflection. Therefore one has to keep wfinite.\nAs follows from Eq.(2) and from the T-matrix theory,\nan increase in the groove width increases the reflection\ncoefficient of individual grooves. However, this obviously\nintroducesout-of-phasereflections fromthe front and the\nrear edges of the grooves. Dephasing becomes more pro-\nnounced with increasing w. (Indeed, sucha structure can\nbe modeled as a superposition of Braggs reflectors with\nthree different lattice constants ( a,a−w,anda+w) and\nlattice origins which do not coincide in space.)7\nTo investigate the impact of the groove width, a\nmagnonic crystal with a groove width equal to the half\nof the lattice constant ( w=a/2) was fabricated. A suit-\nable model for such a structure should concept in a com-\nbination of four Bragg reflectors: two with a lattice con-\nstanta, and the remaining two with lattice constant a/2.\nSuchastructureischaracterizedbythecoincidenceinthe\nBraggwavenumbersbetweenthetransmissionresonances\nfor thea/2-period Bragg reflector and the reflection res-\nonances for the a-period reflection. Our additional ex-\nperiment investigations proved this model: every even\nrejection band seen for a lattice of narrow (30 µm wide)\ngrooves are absent for the structure with w=a/2.\nFig. 5 demonstrates the insertion loss (upper panel)\nand the frequency width ∆ f1(bottom panel) of the\nfirst rejection band (spin-wave wavelength here is around\n600µm) for the magnonic crystals with groove widths\nw= 10µm andw= 30µm as a function of relative\ngroove depth δ/d0. One sees that tripling the groove\nwidth increases the rejection efficiency by approximately\n4 times in the log scale. Almost the same effect is ob-\nserved for the parasitic loss in the transmission band (see\nsquares in Fig. 5(a)). One can see that the results of the\ncalculation are in good qualitative agreement with the\nexperiment.\nIn Fig. 5(b) experimental and calculated values for the\nwidth ofthe firstrejectionband areshown. Oneseesthat\nfor different groove widths, these plots are very close to\none another. Even thought the difference is small, it\nmay be sufficient to fine-tune the characteristics of the\nmagnonic crystals and enable the obtention of required\nrejection efficiency and bandwidth of rejection bands.\nV. CONCLUSIONS\n1. In this work we experimentally demonstrate that in\nthe BVMSW configuration a one-dimensional magnoniccrystal is characterized by the excellent spin-wave signal\nrejection of more than 30dB. The efficiency of the rejec-\ntion can be controlled by the groove depth and width as\nwell as the number of grooves in the crystal. A simple\ntheoretical model was proposed which is in good qualita-\ntive agreement with the experimental results.\n2. It is found that the optimal groove depth which en-\nsures strong rejection in the rejection bands while main-\ntaining insertion loss in the transmission bands around\n3 dB is approximately 1/10 of the total film thickness.\nDecreasing in the groove depth from this optimal value\nleads to a drop in the rejection efficiency. With its in-\ncreasing parasitic loss in the transmission bands rapidly\ngrows.\n3. The efficiency of the rejection increases with an\nincreasing number of grooves. However, this increase is\nsmaller than our model’s prediction.\n4. When the width of grooves is much smaller than\nthe spin-waves wavelength, increasing the groove width\nleads to a fast increase in the rejection efficiency.\n5. The width of the rejection bands for BMVSWs ex-\nceeds the values for the other spin-wave configurations\n[1, 4, 5, 6]. It can be controlled by the film thickness\nand the groove depth. Varying the film thickness, groove\ndepth, groove width and number of grooves allows the\nengineering of magnonic crystals with optimal character-\nistics.\nFinancialsupportbytheDFGSE1771/1-1,Australian\nResearch Council, and the University of Western Aus-\ntralia is acknowledged.\n[1] K.W. Reed, J.M. Owens, R.L. Carter, Circ. Syst. Signal\nProcess. 4, 157 (1985).\n[2] Yu.V. Gulyayev and S.A. Nikitov, Doklady Physics 46,\n687 (2001).\n[3] P.A. Kolodin and B. Hillebrands, J. Magn. Magn. Mater.\n161, 199 (1996).\n[4] C.G. Skyes, J.D. Adam, and J.H. Collins, Appl. Phys.\nLett.29, 388 (1976).\n[5] J.P. Parekh and H.S. Tuan, Appl. Phys. Lett. 30, 667\n(1977).\n[6] J.P. Parekh and H.S. Tuan, IEEE Trans. Microwave The-\nory Tech. MTT-26 , 1039 (1978).\n[7] M. 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[8] G. Gubbiotti, S. Tacchi, G. Carlotti , N. Singh ,\nS. Goolaup, A. O. Adeyeye , and M. Kostylev, Appl.\nPhys. Lett. 90, 092503 (2007).[9] M. Kostylev, R. Magaraggia, F. Y. Ogrin, E. Sirotkin,\nV. F. Mescheryakov, N. Ross, and R. L. Stamps, IEEE\nTrans. On Mag. 44, No. 10 (2008).\n[10] A. Maeda, M. Susaki, IEEE Trans. On Mag. 42, 3096\n(2006).\n[11] J.M. Owens, J.H. Collins, C.V. Smith, Jr., and I.I. Chi-\nang, Appl. Phys. Lett. 31, 781 (1977).\n[12] R.L. Carter, J.M. Owens, C.V. Smith, Jr. and\nK.W. Reed, J. Appl. Phys. 53, 2655 (1982).\n[13] Y.KFetisov, N.V.OstrovskayaandA.F.Popkov, J.Appl.\nPhys.79, 5730 (1996).\n[14] V. Cherepanov, I. Kolokolov, V. L’vov, Physic Reports\n229, 81 (1993).\n[15] R.W. Damon and J.R. Eshbach, Phys. Chem. of Solids\n19308 (1961).\n[16] R.W. Damon and H. Van De Vaart, J. Appl. Phys. 36,\n3453 (1965).\n[17] A.V. Chumak, A.A.Serga, B. Hillebrands, M.P. Kostylev8\nAppl. Phys. Lett. 93, 022508 (2008)\n[18] J. Basterfield, J. Phys. D: Appl. Phys. 2, 1159 (1969).\n[19] M.P. Kostylev, A.A. Serga, T. Schneider, T. Neumann,\nB. Leven, B. Hillebrands, and R.L. Stamps, Phys. Rev.\nB76, 184419 (2007)\n[20] K.Yu. Guslienko, S.O. Demokritov, B. Hillebrands, and\nA.N. Slavin, Phys. Rev. B 66, 132402 (2002).\n[21] B.A. Kalinikos, private communication.[22] A. Maeda and M. Suzaki, IEEE Trans. on Mag., 42, 3096\n(2006).\n[23] W. Berry, IEEE Trans. Microwave Theory Tech. MTT-\n34, 80 (1986).\n[24] M. P. Kostylev, G. Gubbiotti, J.-G. Hu, G. Carlotti, T.\nOno, and R. L. Stamps, Phys. Rev. B, 76, 054422 (2007)."    },    {        "title": "1204.5410v1.Magnetostatic_Spin_Waves_and_Magnetic_Wave_Chaos_in_Ferromagnetic_Films__III__Numeric_Simulations_of_Microwave_Band_Magnetic_Chaos__Its_Synchronization_and_Application_to_Secure_Communication.pdf",        "content": "arXiv:1204.5410v1  [cond-mat.other]  24 Apr 2012Magnetostatic Spin Waves and Magnetic-Wave Chaos in\nFerromagnetic Films.\nIII. Numeric Simulations of Microwave-Band Magnetic Chaos , Its\nSynchronization and Application to Secure Communication\nA.A. Glushchenko, Yu.E. Kuzovlev, Yu.V. Medvedev, and N.I. Mezin∗\nA.A.Galkin Physics and Technology Institute of NASU,\nul. R.Luxemburg 72, 83114 Donetsk, Ukraine\nAbstract\nSelected results of original numeric simulations of non-li near magnetostatic spin waves and\nmicrowave-frequency magnetic chaos in ferrite films are exp ounded, as third part of the work\nwhose first two parts are recent arXive preprints 1204.0200 a nd 1204.2423 . Especially we consider\ncrucial role of parametric processes in creating the chaos a nd simultaneously obstacles to its syn-\nchronization, and examine some possibilities of good enoug h synchronization (to an extent allowing\nits use for direct secure communication in microwave band).\nPACS numbers: 75.30.Ds, 75.40.Mg, 76.50.+g\n∗Electronic address: kuzovlev@fti.dn.ua\n1Introduction\nThis preprint continues preprints 1204.0200 and 1204.2423 and rep resents some results of\nnumerical simulations of auto-generation magnetic-wave chaos in f errite films, its synchro-\nnization and application to secure communication. A references like ( 6.x) means formula\n(x) from Section 6 placed in 1204.2423. Sections 1-5 are placed in 120 4.0200.\nOur main interest below is\n(i) visual investigation of regular andchaotic non-linear magnetost atic spin wave patterns\nauto-generated through feedback consisting of wire inductors ( antennae), amplifier and may\nbe filters;\n(ii) investigation of those conditions of the auto-generation, and p roperties of generated\npatterns, what aremostly responsible forcharacteristics of res ulting chaotic microwave-band\nelectricsignals(voltagesandcurrents), inparticular, theirpossib ilitiestosynchronize chaotic\npatterns in other similar systems (ferrite film) and thus to serve fo r secure transmission of\ninformation.\n7. NUMERIC SIMULATIONS OF AUTO-GENERATION AND SYNCHRO-\nNIZATION OF MAGNETIC-WAVE CHAOS\n7.1. CHAOTIC AUTO-GENERATION IN FILM WITH LINEAR FEEDBACK.\nIf the voltage (EMF) signal, U(t) , induced in a conductor (antenna) by magnetization\nprecession is amplified and transformed into driving current, J(t) , in another conductor,\nthen auto-generation of MW may take place. The simplest variant of this feedback is drawn\nat plot (A) in Fig.13. It consists of two identical loop inductors and pu rely linear amplifier\nwhose gain and phase shift are with frequency independent. Typica l features of chaotic\ngeneration in such the scheme, at H0= 2 , are presented by Figs.13 and 14.\nThe plots (B) and (E) in Fig.13 show that rather wide-band magnetiza tion chaos is\nproduced, in frequency range about 600 MHz. Position of most inte nsive peak at the voltage\nspectrum well corresponds to frequency of the Damon-Eshbach (DE) surface wave with\nlength λ= 2l= 32Ddictated by the loop width, i.e. to ωDE=ωDE(2πD/λ)\n(≈2.54 GHz at H0= 2 ). The lower edge of the spectrum lies below uniform precession\nfrequency ωu(≈2.12 GHz ). This means that the main-branch bulk waves also are\n2excited. The upper edge of the voltage spectrum is far below the up per frequency of MSW,\n(H0+2π)f0(≈3.25 GHz). Hence, long surface MSW are dominating in magnetization\npatterns.\nThis pattern is seen at plot (A) in Fig.14. The contour plot of its spatia l spectrum ((F)\nin Fig.14) demonstrates tracks of (i) a set of long-wave modes with f requencies about ωu\n, (ii) three times shorter than λDE mode with frequency ωDE(3·2πD/λ) (≈2.95\nGHz), (iii) comparatively short-wave surface modes with non-zero transversal wavenumber,\nky∝ne}ationslash= 0 , and frequencies about ωDE(2πD/λ) , and, besides, (iv) approximately two times\nlonger than λDE mode with frequency ≈ωDE(πD/λ) (≈2.36 GHz).\nInterestingly, the latter mode seems be responsible for lower high p eak in the voltage\nspectrum. Plots (B) and (C) show that, in rough terms, auto-gen eration switches between\ntwo states whose frequencies are about ωDE(2πD/λ) andωDE(πD/λ) , and that at the\nfirst of them the energy is usually lower than at the second (in accor dance with negative\nnon-isochronity, see Sec.5).\nDue to linearity of amplifier, the film itself are forced to take all cares about nonlinear\nsaturation of precession angle. Naturally, this results in complicate d large-amplitude chaos\ncharacterized by wide variety of time scales and intermittency of diff erent regimes, which is\nillustrated by plots (B)-(D)in Fig.13 and (B),(C) and (E) in Fig.14. Low- frequency contents\nof this chaos is better visible in the spectrum of energy, E(t) , which is enriching at least\ndown to 5 MHz. Voltage and magnetization time series yield fractal dim ensiondcor≈3.45\n. Hence, we encounter hyper-chaos governed by not less than fo ur relevant variables.\n7.2. AUTO-GENERATOR WITH NON-LINEAR FEEDBACK.\nAs practice says, in real generators an inner non-linearity of amplifi ers is essential. It\nmanifestsitselfintwoways: (i)asnon-linearityofinputresistancew hicheffectively saturates\ninput voltage, and (ii) as saturation of output power or, equivalent ly, of output driving\ncurrent. Besides, one should take into account frequency depen dence of gain, K(ω) , and\nown impedances (capacities and inductances) of feedback conduc tors.\nCorresponding scheme looks as in Fig.15. Here R0,L0andC0are linear (small-\namplitude) input resistance, ∼50 Ohm, inductance of passive (feedback) antenna, ∼5\nnH, and capacity of active (driving) antenna, ∼0.5 pF, respectively. Since in reality such\nthe circuit is quasi-stationary one (its size is much smaller than length s of EM-waves under\noperation), a simple analysis shows that there is no necessity in addit ional reactive elements.\n30.570.5720.5740.5760.5780.58−0.1−0.0500.050.1S⊥x\n0.3 0.4 0.5 0.600.20.40.6S⊥ envelope\n0.3 0.4 0.5 0.605101520\ntime, µs103⋅P(t)22.5 30123456789\nFrequency, GHzU(t)′s spectrum, a.u.\n12340.511.522.533.544.5\nlog of Cell SizeFractal Dimension\n  Fig.13. Chaotic auto−generation in tangentially magnetized film closed on itself by two loop   \ninductors and linear amplifier with frequency−independent gain and phase shift, ∆Φ, at H0=2Ms. \n(A) Passive loop is the source of voltage, U(t), proportional to dS⊥x/dt. The current, J(t), of active \nloop excites spin precession. (B) S⊥x in vicinity of passive loop. (C) S⊥x′s envelope. (D) Power \nabsorption. (E) Spectrum of S⊥x. (F) Correlation dimension of U(t) and S⊥x occurs dcor≈3.45 .differential\nintegralJ(t) gain ∆Φ U(t) l=16D \nx=64D \n(A) \n(B) \n(C) \n(D) (E) \n(F) ωu ωDE \nH0 \n4S⊥z(x,y)\n2.32.42.52.62.7U′s frequency, GHz\n0.5 1 1.5 200.050.10.150.2\ntime, µsEnergy\n−10 010 20 30 40 50 60−101\ntime, in periodsCorrelation020406001020304050607080\nFrequency, MHzE(t)′s spectrum, a.u.\n−0.6−0.4−0.2 00.20.40.6−0.200.2\nkxDkyD(A) \n(B) \n(C) \n(D) (E) \n(F) \n  Fig.14. Chaotic MSW auto−generation (continued from Fig.13).  (A) Typical S⊥ pattern. (B) Instant                             \nvoltage frequency, ωin(t), obtained from its zero crossings. Strict line shows the surface MSW frequency,                      \nωDE(kD), which corresponds to kx=2π/λ (at ky=0), with λ=2l=32D being wavelength selected by the                \nloop geometry (see Fig.13(A)). (C) Excess energy, E(t). Negative correlation between E(t) and ωin(t)                           \nis clearly seen. (D) Mutual correlation of magnetizations, S⊥, about active and passive loops. (E) Energy                       \nspectrum. (F) Contour plot of S⊥z. Three equi−frequency lines are shown corresponding to surface MSW                            \nin infinite−size film with frequencies ωu (uniform precession), ωDE(2πD/λ) and ωDE(3⋅2πD/λ). \n5For practical high-frequency wide-band amplifiers, typical level o f the input saturation is\nUsat∼1÷3 V, level of output saturation is Jsat∼20÷70 mA, while K0=\nmax|K(ω)| ∼5÷15 , and linear frequency characteristics can be modeled by\nK(ω) =K0ω2\n0(ω2\n0−ω2−igω)−1exp(iωτd), (1)\nwithτddescribing time delay. In numerical simulations, usually values ω0/2π= 3.4 GHz\nandg= 6·109s−1were used, as in INA-32063 wideband silicon RFIC amplifier ( τdis\nsmall as compared with time delays in film and therefore insignificant).\nFig.15 shows how numerical copy of this scheme generates at relativ ely small gain, K0≈\n5 , and moderate output power saturation. We see that likely period ic (although more\nor less complicated) oscillations take place in all the amplitudes and pha ses. If decreasing\ngain and /or output saturation level one may observe more simple os cillations, down to\ntrivial mono-chromatic regime and failure of generation, while increa se results in further\ncomplications and then chaos.\n7.3. ROLE OF ANTENNAE SEPARATION.\nThe Fig.16 illustrates chaotic auto-generation in numerical model co rresponding to\n4.4 mm×0.9 mm×10µm-size film and l= 0.5 mm width of loop inductors, or to a\nproportionally re-scaled system (but not too small), at field H0= 500 Oe , moderate gain\nand relatively high output power saturation. At top of Fig.16 static m agnetization of such\nshaped film, at Ms= 140 Oe and H0= 500 Oe , is presented. Clearly, a noticeable\ndemagnetization takes place at film corners only.\nIn this numerical experiment, two variants of separation, x, of feedback and active\ninductorswereconsidered, with x=x1= 2λandx=x2= 1.8λ, where λ= 2l=\n1 mm is maximum of wavelengths preferably selected by the loop anten nae. Comparison of\nthese cases demonstrates, at plot (D), that auto-generation s pectrum is sensitive not only\nto theλbut also to x. It could be expected, because any phase shift in feedback should\nbe compensated by equal magnetic wave phase difference, kxx, between antennae, with\nkbeing a dominating wave vector. At smooth feedback frequency ch aracteristics, one can\nsuppose that k1xx1≈k2xx2, therefore\nk2x−k1x≈ −k1x(x2−x1)/x1, ω2−ω1≈vg(k2x−k1x), (2)\n600.2S⊥\n510U(t), V\n0.010.015E(t), r.u.\n0.8 0.85 0.9 0.95 1 1.0533.1\ntime, µsωin, GHZ\n  Fig.15. (Top) Principal scheme of auto−generator under numerical simulation.               \n(Below) Regular auto−oscillations at the parameters as follows: H0=500 Oe, γ=0.0007, R0=50 Ohm,                   \nC0=0.5 pF, L0=5 nH, Usat=1 V, Isat=40 mA, d=10 µ, λ=0.1 cm, x=2 λ. (A) Envelope of x−component       \nof spin precession, S⊥x, about passive loop. (B) Envelope of voltage, U(t). (C) Excess energy, E(t). \n(D) Instant voltage frequency from zero crossings. Strict line shows frequency ωDE(2πD/λ). C0 J(t) \nJ′(t) K(ω) \nR0 L0 \nI(t) V(t) \nI′(t) U(t) \nIsat Usat \n(A) \n(B) \n(C) \n(D) \n7whereω2−ω1is corresponding change in frequency and vgis group velocity. According\nto this estimate, transition from x1= 2λtox2= 1.8λmust result in increase of auto-\ngenerationfrequencies, whichagreeswithnumericalsimulation. Ta kingωbethefrequency\nof DE wave, we obtain vg≈4π2D/ω1and then ω2−ω1≈0.032 (in dimensionless\nunits), that is ≈13 MHz , in satisfactory quantitative agreement with plot (D).\nLet us more discuss the voltage spectrum. As in case of linear feedb ack (see above), its\nupper-frequency peak can be surely related to DE mode with kx≈2π/λ. But what is\noriginof thelower-frequency peak? Fromwatching fordynamics of magnetizationpattern, it\nis possible to relate this peak to a long DE mode with wavelength nxwhere 1 /lessorsimilarn/lessorsimilar2 .\nSuch mode also can get suitable phase conditions under feedback de termined by Eq.1.\n7.4. HYPER-CHAOS UNDER NON-LINEAR FEEDBACK.\nNon-linear concurrence between the two dominating wave modes ma nifests itself in\nchaotic jumps of instant (time-local) frequency of auto-generat ion,ωin(t) , as shown by\nplots (A) and (C) in Fig.16. It should be underlined that from the point of view of energy\npattern (quasi-local energy, see Sec.6.14) the same interplay of d ifferent modes looks as the\nbirth and drift of envelope solitons. In these terms, the frequenc y separation of two main\npeaks at plot (D) is nothing but mean number of soliton births per sec ond (see below).\nDue to high power saturation level, the whole picture occurs very ch aotic. Its correla-\ntion dimension obtained from voltage and magnetization time series (s olid and dot lines,\nrespectively) is dcor≈4 (if not greater).\n7.5. WEAK CHAOS IN GENERATOR WITH WIRE INDUCTORS.\nItis interesting howthepicture will bechanged if two-element loopind uctorsarereplaced\nby wires, at the same feedback circuit. Figs.17a and 17b can give the answer for case\nx= 150D(=1.5 mm) with xbeing distance between the wires.\nIn this case, two dominating modes chosen by the system are long DE waves with lengths\n≈2xand≈x. In spatial spectrum of magnetization (in momentum space) the mo de\nwith length ≈x/2 can be detected, but practically it does not contribute to voltage\nspectrum. Besides, as usually, a set of comparatively weak excited equal-frequency modes\nwithky∝ne}ationslash= 0 is present. In real space non-linear magnetization pattern cha racteristic\nrhombic structuring is well visible invoked by dispersing properties of underlying linear MW\neigenmodes. But frequencies of the latter become lowered by non- linearity.\nAt nearly the same output saturation and even greater linear gain, the whole picture is\n8−0.200.2S0x(x,y)\n11.5 22.5 333.13.2ωin, GHz\n11.5 22.5 300.050.1\ntime, µsE(t), r.u.\n0.20.40.60.811.233.13.2\ntime, µsωin, GHz2.9 33.13.2024681012141618\nFrequency, GHzU(t)′s spectra, r.u.\n2 3 41.522.533.544.555.5\nlog of Cell SizeFractal Dimension\n  Fig.16. (Top left) x−component of static magnetization in relatively large−area 440D ×90D film.                 \n(Other) Hyperchaos in two auto−generators, at H0=500 Oe, γ=0.0012, R0=50 Ohm, C0=0.5 pF, L0=5 nH,            \nUsat=1 V, Isat=70 mA, Gain=7, D=10 µ, λ=0.1 cm. The only difference between the two is in inter−loop     \ndistance, x=2 λ and x=1.8 λ. The smaller distance results in higher frequency. Direct lines in (A), (C) and \n(D) show uniform precession frequency, ωu, and surface MSW frequency ωDE(2πD/λ).                x=1.8λx=2λ \nx=2λ x=2λ x=1.8λ ωu ωDE \nDcor≈4 (A) \n(B) \n(C) (D) \ndifferential \nintegral \n9significantly more smooth and less chaotic than in case of loop inducto rs, with dcor≈2.55\nonly! In part, the reason is that both the EMF accepted by wire indu ctor and power\npumped by it are approximately two times smaller than by loop inductor . In this concrete\nexperiment, power absorption P≈ ∝an}bracketle{tJV∝an}bracketri}htT∼25 mW , where JandVare driving\ncurrent and EMF in active inductor, respectively, and ∝an}bracketle{t..∝an}bracketri}htTmeans time average (over a\nfew precession periods).\n7.6. GENERATION OF ENVELOPE SOLITONS.\nIn the middle plot in Fig.17b, typical spatial distribution of the quasi-lo cal energy density\n(Sec.6.14) is shown (to be precise, of eloc/H0whereH0isMsunits). It highlights\npresence of three envelope solitons of whose the first (most left) is in birth, the second is\njust passing through feedback inductor andthe third is dying rema inder of previously passed\nsoliton. Obvious rhombic shaping of the solitons reflects four most p referred directions\nof local energy propagation (corresponding to maximum group velo city),|vgy|/|vgx| ≈\n/radicalbig\nH0/4π(see Sec.5.4).\nNotice that solitons are very sharp (as measured in x-direction): their width is even less\nthan wavelengths of carry MSW enveloped by them! But several wa velengths have time to\npass through such the envelope while it itself displaces on its width.\n7.7. INSTANTON STRUCTURING OF SPIN PRECESSION.\nLet us go back to strong chaos considered above in subsection 4 an d have a look how\nenvelope solitons manifest themselves in local magnetization time ser ies. The Fig.18 shows\nbehavior of spin and of eloc/H0, where elocis quasi-local energy density (Sec.6.14), at a\npoint close to the feedback inductor. Notice that\neloc/H0≈(S2\nx+p2S2\nz)/2≈p|Ψ|2/2, (3)\nwhere Ψ is the wave function (see in Sec.5.3 and Eq.5.9 there), and pis the eccentricity\n(see Eqs.4.6 and 5.10), i.e. can be interpreted as merely squared ang le of precession. Com-\nparison of plots (C) and (D) in Fig.18 visually prove that at fast time sc ale (measured in\nperiods of precession) elocis almost integral of motion.\nWhat most of all impresses in plots (B), (C) and (F) is that eloc(and thus local angle\nof precession) never overlooks to regularly turn into either exact zero or to extremely small\nvalue. If S⊥(r,t) was mere random field (like thermally activated magnetization noise in\nthermodynamical equilibrium) such behavior would be absolutely impro bable. It gives best\n10345U, V\n203040P, mW\n2.962.9833.02ωin, GHz\n1.41.451.51.551.61.651.70.010.0150.020.0250.03\ntime, µsE, r.u.2.95 33.053.1024681012\nFrequency, GHzU′s spectrum, r.u.\n23450.511.522.53\nlog of Cell SizeFractal Dimension\n  Fig.17a. Relatively slightly chaotic oscillations in auto−generator with two wire inductors instead of \nloop ones, at the parameters as follows: H0=500 Oe, γ=0.001, R0=50 Ohm, C0=0.5 pF, L0=5 nH, \nUsat=1 V, Isat=60 mA, Gain=10, D=10 µ. Separation of inductors equals to x=150D=0.15 cm. The primarily \nexcited DE MSW mode has wave length λ=2x. (A) Envelope of voltage on passive inductor, U(t). \n(B) Envelope of power absorption, P(t). (C) Instant frequency from U(t) ′s zero crossings. (D) Excess \nenergy per unit volume. (E) Voltage spectrum. (F) Correlation dimension obtained from the voltage signal \nis dcor≈2.55. Horizontal lines in (C) and dotted lines in (E) correspond to ωu, \nωDE(2πD/λ) and ωDE(4πD/λ), with λ=2x=300D. (A) \n(B) \n(C) \n(D) (E) \n(F) ωu \nωDE(2π/300) \nωDE(2π/150) \n11−0.2−0.15−0.1−0.05 00.050.10.150.2−0.2−0.100.10.2\nkxDkyD00.050.1eloc/H0\n(energy density)\n  Fig.17b.  Continuation to Fig.17a. (Top) Instant S⊥ pattern. Light lines show positions           \nof the wire inductors. (Middle) Typical spatial distribution of eloc/H0=1−S+2πS⊥z2/H0 with \neloc being quasi−local energy density (see main text). The presence of three narrow \nsolitons is seen which possess characteristic rhombic shaping and the length on order of \nmean wavelength. (Bottom)  Contour plot of spatial spectrum of S⊥. The dotted curves are                 \nequi−frequency lines corresponding to ωu, ωDE(2πD/λ), ωDE(4πD/λ) and \nωDE(8πD/λ), with λ=2x.                                                           S⊥(x,y) \n12evidence that we observe results of essentially nonlinear self-orga nization of magnetization\nfield in “clots”. As in nonlinear field theory, spatially local time tracks o f self-organization\nmay be termed instantons.\nAs it was mentioned in Sec.5.4, to be more precise, the eccentricity, p, should be\nrelated to dominating MSW mode (instead of uniform one). In other w ords, the quasi-local\nenergy introduced in Eq.6.6 must be redefined with taking into accoun t non-local (gradient\ndependent) part of energy estimated by Eq.6.7. Corresponding re finement reads\neloc=H0(1−Sy)+πDk0S2\nx+π(2−Dk0)S2\nz∝1−Sy+HmS2\nz/2H0≈(S2\nx+p2S2\nz)/2,(4)\nHm\nH0≡4π(1−Dk0)\nH0+2πDk0, p=/bracketleftbigg\n1+Hm\nH0/bracketrightbigg1/2\nwherek0is module of dominating wave-vector (for loop inductors, |k0| ≈π/l) and the\ncondition Dk0≪1 is taken in mind. For comparatively short-wave chaos, this is bette r\ninvariant of fast motion (precession) than what defined by (6.6).\n7.8. PHASE SLIPS AND NOISE PRODUCTION.\nPlots (D) and (E) in Fig.18 show that when local amplitude of precessio n is passing\nthrough zero its phase can slip by some angle. In contrast to ideal b lack solitons (see\nSec.5.6), this angle can take any value different from ±π. Of course, usually two close\nspins undergo the same slip, but sometimes one or another spin can b e beaten from common\nbehavior. Hardly this fine detail of magnetization dynamics is unambig uously determined\nby a few relevant chaotic variables. Therefore, local phase slips ma y become the source of\nsome amount of noise, in addition to chaos.\nAlthough, by our observations, this noise is rather weak, probably it may create obstacles\nto ideal copying of chaos under synchronization.\n7.9. QUASI-PERIODIC CHAOS.\nFig.19 illustrates example of quasi-periodic chaos obtained in auto-ge nerator with rela-\ntively narrow loop antennae ( l= 0.2 mm) at moderate gain (=9) and output saturation\n(Isat= 55 mA), and antennae separation x= 4l. Dominating wave-length of magneti-\nzation pattern is close to λ= 2lwhile carrier frequency of its oscillations is rather close\nto frequency of DE wave with kx= 2π/λandky= 0 .\n131200 1250 1300 1350 1400 1450 150000.050.1\ntime, in τ0 unitseloc/H000.20.4S⊥ Envelope\n00.050.1eloc/H0\n−0.4−0.200.20.4Local S⊥\n2145 2150 2155 2160−2000200\ntime, in τ0 units0 0.5 100.20.40.60.81\nFrequency, in f0 unitseloc′s spectrum, r.u.390039203940396000.050.1\ntime, in τ0 unitseloc/H0\n  Fig.18.  Instantonic temporal structuring of chaotic magnetization pattern and noise production.                           \n(A) Envelope of local S⊥x at particular point in vicinity of passive inductor. (B) eloc/H0∝|Ψ|2                               \nwhere eloc is quasi−local energy density in the same particular point. It is almost free from high                                     \nfrequencies thus behaving as integral of fast motion. But in slower time scale it regularly turns into                       \nexact zero, demonstrating solitonic self−organization of magnetic energy. (C) Zero of S⊥x(r0,t).                                           \n(D) Simultaneous S⊥z (dot line) and S⊥x (solid line). (E) The zero results in random phase                                           \nslip of spin precession, thus generating phase noise. (F) More zeros. (G) eloc′s spectrum.                                                 (A) \n(B) \n(C) \n(D) \n(E) (F) \n(G) S⊥′s Phase\n14Inthis case, time series of different variables (magnetic energy, ma gnetization and voltage\nenvelopes and phases, instant frequency, etc.) possess charac teristic features of the kind of\nchaotic attractor termed noisy limit circle. The voltage spectrum at plot (F) resembles ruled\nspectra of complex periodic signals (with period being inverse distanc e between neighboring\nspectral lines). Nevertheless, in fact voltage signal is chaotic and its fractal dimension,\ndcor≈3.25 (see plot(G)) even indicates hyperchaos.\nPlot (D) shows that in spite of large magnitude of instant frequency oscillations ( ∼500\nMHz) and therefore very wide total frequency band of generate d signal ( ≈0.9 GHz)\nits phase modulation lies between well certain boundaries not exceed ing≈100o. From\ncomparison of plots (D) and (E) it is seen that positive (negative) slo pe of time-smoothed\nphase trajectory corresponds to smaller (greater) energy, in a ccordance with negative non-\nisochronity of MW.\nWhat here happens? The frequency separation of the three main g roups of lines in plot\n(F),≈270 MHz, is by its sense the frequency of soliton births near active in ductor. Hence,\none envelope soliton is born approximately each 3 .7 ns. The group velocity of DE waves\nat operational frequency (see Eq.5.14) is vg≈107cm/s, therefore time of soliton flight\nfrom active to passive loop is about 8 ns. Mean duration of one cycle o f the quasi-periodic\nmodulation can be estimated from plot (E) as ≈40 ns. Comparing these numbers we\nconclude that two solitons simultaneously are in action and that each cycle contains birth\nand transfer of 10 ÷12 solitons. What is for the cycle duration itself, seemingly it can be\nconnected to the relaxation time, Γ−1.\n7.10. ROLE OF FILM WIDTH TO WAVE LENGTH RATIO.\nIntuitively, one may predict that formation and propagation of env elope solitons is sensi-\ntive to the ratio w/λwherewis width of film and λ(λ≈2lfor two-element loop\nantennae) is dominating wave length in magnetization pattern. The F ig.20a shows that,\nindeed, this is confirmed by numerical simulation of auto-generation in narrow (in the sense\nthatw/lessorsimilarλ) and wide ( w > λ) films.\nIn narrow film, soliton takes all width of a film. In wide film, at very begin ning of next\nsoliton manufacturing one energy clot may be inflated occupying all t he width but soon it\nbreaks into two or three (or even more) solitons each of which take s a part of the width only.\nThen solitons run not in parallel to film boundaries but along one of two easy propagation\ndirections which correspond to maximum group velocity (see Sec.5). Fig.20a demonstrates\n1500.20.4S⊥x(t)\n468U(t), V\n2.533.5 ωin, GHz\n−1000Phase,o\n0.9 0.95 10.020.030.04\ntime, µsE(t)2.5 3 3.50246810\nFrequency, GHzU(t)′s spectrum\n234511.522.533.54\nlog of Cell SizeFractal Dimension\n  Fig.19. Noisy limit circle in auto−generator with the parameters:  H0=400 Oe, γ=0.001, R0=50 Ohm,                            \nC0=0.5 pF, L0=5 nH, Usat=1 V, Isat=55 mA, Gain=9, D=10 µ, loop width l=0.02 cm, loop separation                              \nx=0.08 cm. The primarily excited DE MSW mode has wave length λ=2l. (Top) Instant S⊥x pattern.                            \nWhite lines show positions of two loops. (A) Envelope of local S⊥x in vicinity of left (active) loop.                          \n(B) Envelope of voltage, U(t). (C) U(t) ′s instant frequency and (D) corresponding phase oscillations.                            \n(E) Excess energy. (F) Voltage spectrum is typical one for quasi−periodic signals. (F) dcor≈3.25.                            \n      Horizontal lines in (C) and dotted lines in (F) mark ωu, ωDE(2πD/λ) and ωDE(3⋅2πD/λ). \nThe MSW group velocity vgr≈107 cm/s. Time of soliton flight between loops is x/vgr≈7.5 ns. The circle              \nperiod, ≈0.04 µs (see (E)), is close to relaxation time, Γ−1≈100τ0, and contains ≈11 solitons.         (A) \n(B) \n(C) \n(D) \n(E) (F) \n(G) \ndcor≈3.25 ωu \nωDE(2πD/λ) \nωDE(6πD/λ) \n16this for the case when two soliton chains are created (of course, if three or five soliton chains\nrealize then central of them moves in parallel to boundaries).\nImportantly, in general in wide film (at w/λ/greaterorsimilar3 ) wave fronts (equal-phase lines) of\nmagnetization pattern stronger are curved then in narrow film (es pecially in region where\ndifferent soliton chains meet one another). Therefore, the wider is film the smaller is voltage\n(EMF) per unit length induced in direct antenna, up to that wider film c an produce lower\nvoltage (at the same precession angle) which worsens the feedbac k.\nAs example of unpretentious situation, Fig.21 presents movie of one cycle of chaotic\nsoliton generation in moderately pumped narrow film.\n7.11. ABOUT COHERENCE OF AUTO-GENERATION.\nIt is known that spectrum of any oscillation with arbitrary amplitude m odulation and\narbitrary (let random) but limited phase modulation consists of infinit ely narrow line (at\ncarrier frequency) surrounded by some pedestal. An unlimited pha se modulation that is\naccumulation of phase slips only produces a broadening of the carrie r line.\nIn our numerical simulations of the auto-generation (even concer ning strongly chaotic\nregimes) accumulation of phase slips (phase diffusion) on the averag e did not exceed one-\ntwo periods of carrier frequency per microsecond. Hence, coher ent generation was observed,\nto the extent of calculation duration (apparent broadening of car rier line on spectrum plots\nis artifact of short-time Fourier transform).\nTheoretically, the degree of coherence of chaotic generation is int eresting open question.\nFrom practical point of view, it may be better answered in real expe riments which reflect\nrealities (like temperature instability) not accounted for by numeric al model.\n7.12. SPATIAL DIFFERENTIATION OF AUTO-GENERATION SPECTRUM .\nUnder auto-generation, magnetization in the whole film oscillates coh erently. Due to\nchaotic soliton production, local phases of common oscillation under go chaotic deviations\nwhich however remain limited (usually not greater then ±60o). Local phase slips do not\ndestroy common coherent picture.\nBut, of course, details of pedestal of oscillation spectrum may str ongly depend on what\nvariable is under measuring. This statement is illustrated by Fig.20a.\nTheFig.20acorrespondstowidefilm, with w/λ= 5 andcomparativelylargeseparation\nof loop inductors, x= 3λ. It presents example of rather wideband chaos with complicated\nspectrum which in 6 peaks can be resolved. Again, the distance betw een central (most\n170510152025U(t)\n00.511.52Local S⊥\n2.62.833.23.400.010.020.03S⊥′s wave\nFrequency, GHz2.62.833.23.400.10.20.30.4<S⊥>w\nFrequency, GHz  Fig.20a. Difference of soliton patterns in wide film (with width, w, greater than S⊥′s wavelength, λ)  \nand narrow film (w< λ). Vertical lines show inductors. The inclined lines show directions of most                     \neasy (fast) soliton propagation (tan φ≈(H0/4πMs)1/2) .  (Top) Energy density, eloc, under chaotic       \nauto−generation in wide film. Two soliton chains flow from active loop along easy directions.                              \n(Bottom) In narrow film, one soliton per its width can be placed only.                                                     \n  Fig.20b. Different variables have different spectra under the same chaos. (A) Voltage spectrum                             \nin wide film at w=0.2 cm, λ≡2l=0.04 cm, x=3 λ, H0=400 Oe. (B) Spectrum of local spin precession,                   \n S⊥, about center of film. (C) Spectrum of the expectedly induced plane−wave,  ∫exp(−2πix/λ)S⊥dr/V.        \n(D) Spectrum of width−averaged magnetization, ∫S⊥(x,y)dy/w, at x taken about right−hand edge.                             \nDotted lines mark ωu, ωDE(2πD/λ) and ωDE(3⋅2πD/λ). ’The distance between highest peak and \ntwo peaks marked by → and ← in (A) and (D) equals to inverse time separation of solitons,                        \n≈1/(7 ns). Naturally, spectrum (B) is most broad (up to 1 GHz) and most representing ωu.                                φ \n(A) (B) \n(C) (D) \n18powerful) carrier peak and its closest neighbors equals to inverse m ean time separation of\nsoliton births. Approximately twice larger distance between the car rier and next two neigh-\nbors can be interpreted as second harmonics of the births freque ncy. Naturally, the carrier\npeak is most brightly represented in spectrum of spatial Fourier tr ansform of magnetization\natk={2π/λ,0}(plot (C)) and least brightly in spectrum of local spin precession (plo t\n(B)).\n7.13. CIRCUMFERENTIAL SYNCHRONIZATION OF AUTO-GENERATION .\nNaturally, the question arises about possibility of synchronization o f chaotic auto-\noscillations in one generator by voltage signal taken from another ( at least identical) gen-\nerator. Firstly, we tried to use voltage (EMF), uM(t) , of additional wire inductor placed\nat right-hand end of the master film (i.e. behind the feedback induct or). This voltage was\ncompared with voltage, uS(t) , produced in identical control inductor in the slave film. The\ndifference, uS−uM, was transformed into current, Jsyn(t) , passed through one more wire\ninductor close to the control one, as shown in Fig.22. Notice that in c ase of exact coinci-\ndence,uS−uM= 0 , the synchronizing current turns into zero, Jsyn= 0 , and thus slave\nfilm feels no difference from master film.\nParadoxically, because of almost ideal coherence of chaotic auto- generation, at available\ndurationofnumerical experiments, high-frequency (carrier) sy nchronization isautomatically\nexecuted and thus calls no problem. The problem consists of synchr onization of phase and\namplitude chaotic modulation.\nFilms with w/λ= 3 were investigated. Various linear and non-linear (saturated)\nconnections between uS−uMandJsynwere tested, but none of variants has shown an\nevident superiority. Typical results are illustrated by plots (3)-(8 ) in Fig.22 which show\nthree pieces of synchronization with respect to envelopes of feed back voltages in the master\nand slave, UM(t) andUS(t) (see scheme in Fig.15), and to their instant frequencies.\nUndoubted signs of partial synchronization can be noticed already in 100 ns after be-\nginning of the process. They become better visible if take into accou nt obvious delay of\nthe slave response. Plots (9) and (10) show how the picture looks if correction by delay is\nmade. Most of details of signal UM(t) circulating in feedback device of master generator\nare reproduced by similar signal US(t) in the slave. According to plot (2), high-frequency\n(carrier) phase difference between these signals does not leave th e frame ±80o. Neverthe-\nless, in respect to the modulation there are essential quantitative and sometimes qualitative\n19Fig.21. One cycle of quasi−periodic MSW envelope solitons auto−generation in narrow film \n(w<λ), at H0=500 Oe, λ=2l=0.1 cm, x=2 λ, γ=0.0007, Isat=35 mA, Usat=1 V, Gain=6.7. \n   The circle duration ≈10 ns. The contour plots show instant spatial distributions of quasi−\nlocal energy density, eloc(x,y,t)=1−S+HmS⊥z2/2H0. It is seen that in the greatest part \nof time two solitons are present only.\n20010U\n3.053.1\n010U\n3.053.15\n010U\n33.05\n010\n←   140 ns   →U\n33.05\n←   140 ns   →\n0.5 1 1.5−4−20246\ntime, µs100*∆E, ∆U\n0.5 1 1.5−20020\ntime, µsDelay, ns0.5 1 1.5050100150\ntime, µs∆φ, o\n  Fig.22. High−frequency chaos synchronization in Slave film by voltage signal taken from edge \nof another identical Master film. (1) Voltage difference between edge inductors, uS−uM, produces \nsynchronizing current, Jsyn, influencing the slave. (2) Phase difference between feedback voltages \nUS(t) and UM(t) (see Fig.15). (3) Amplitudes of feedback voltages (in V) and (4) their instant \nfrequencies (in GHz) at start of synchronization. Fat (thin) curves relate to master (slave). \n(5)−(8) Same at two later time intervals. (9)−(10) Comparison of amplitudes and frequencies shown \nin (7) and (8) if accounting for delay of synchronization and energy background. (11) Correlation \nbetween difference of US and UM amplitudes (thin curve) and difference of the slave and master \nenergies (fat curve). (12) Delay via time.uM \nuS \nJsyn (1) (2) \n(3) (4) \n(5) (6) \n(7) (8) \n(10) \n(12) (9) \n(11) \n21distortions. The results are far from copying of chaos. To estimat e them we need in a more\nsoft criterion for partial synchronization.\n7.14. DEFECTS OF SYNCHRONIZATION.\nCareful analysis of data reflected in Fig.22 yields that most part of m isalignment between\nUM(t) andUS(t) can be related to two reasons: (i) not constant but slowly time-va rying\ndelay, ∆ T(t) , and (ii) slowly time-varying amplitude shift which is in close correlation\nwith difference between energies of master and slave film, ∆ E(t) =ES(t)−EM(t) .\nThe latter statement is confirmed by plot (11). Notice that the dela y, as shown at plot (12),\nmay be both positive and negative (which corresponds to anticipatin g synchronization).\nThis findings mean that, approximately, the relation\nUS(t)≈ξ[ES(t)−EM(t)]+UM(t−∆T(t)) (5)\ntakes place with some coefficient ξ. Characteristic time scale of variations in ∆ E(t) ,\nin amplitude shift ξ∆E(t) and in the delay ∆ T(t) too, are about ten times slower then\nvariations in phase and amplitude of UM,S(t) during a cycle of soliton production.\nHence, the conclusion arises that some low-frequency component of the chaotic dynamics\nis not sensitive to synchronizing signal and therefore hinders to sy nchronize fast components\nwhich themselves are rather sensitive. Additional surprising eviden ce for this is that sharp\nswitching from one master signal, uM(t) , to another (may be the same but shifted by\na time) temporarily improves synchronization which then becomes sp oiled at longer time\nscale.\n7.15. SYNCHRONIZATION BY FEEDBACK SIGNAL.\nSeemingly, better synchronization will be achieved if synchronizing s ignal is taken from\nthe“heart” ofgenerator, i.e. fromthe feedback device. Inthes implest variant shown byplot\n(A) in Fig.23, the master feedback voltage mix up with the slave one, w ith some coefficient\nα.\nIn this experiment besides the control voltages, uS(t) anduM(t) , were under compar-\nison taken from wire inductors placed between active and passive loo ps (see plot (B)). At\nα= 0.4 , best results of synchronization with respect to the power abso rption, PM,S(t) ,\namplitude of control voltage, uM,S(t) , and amplitude and instant frequency of feedback\nvoltage, UM,S(t) , are presented by four center plots in Fig.23. Again, here slow time -\nvarying delay between Slave and Master signals and slowly difference in their magnitudes\n22were found, in correspondence with Eq.5. Similar correlation betwee n difference in power\nabsorption and difference in film energies is detected in plot (C).\nPlot (F) allows to see what the events develop in a concrete point mar ked by star at plot\n(B) (which itself represents a snapshot of quasi-local energy pat tern). The new interesting\nobservation follows from the histogram (probability distribution) of delay, at plot (E). Sharp\npeaks in this plots are separated by ≈0.35 ns which is close to period of precession. This\nmay be hint on fine features of synchronization still to be understo od.\nClearly, a kind of rough synchronization takes place, but in details it lo oks even worse\nthan in the above circumferential case.\nTherefore stronger mixing with α= 0.8 was tested. Its results are shown in Figs.24a-\nb (see captures to these figures). According to top four plots in F ig.24a, results occurs\ndefinitely better than at α= 0.4 . This improvement is directly connected to essentially\nsmaller difference between carrier phases of US(t) andUM(t) (plot (4) in Fig.24b) than\nin previous case (plot (D) in Fig.23).\n7.16. MORE ABOUT DEFECTS OF SYNCHRONIZATION.\nFrom the power and feedback voltage time series it is seen that most of maxima and\nminima corresponding to soliton births are well reproduced by slave s ystem. However,\nsynchronization of events in the interspace between driving and fe edback inductors is rather\nbad as registered by the powers, PM,S(t) , and control voltages, uM,S(t) , in Fig.24a .\nThis means that magnetization pattern in slave film imitates that in mas ter film but never\ncopies it.\nThis misalignment can be explained as follows. There is no rigid connectio n between film\nenergy and a number of solitons. Some background (smoothly distr ibuted and slowly vary-\ning) part of energy behaves more or less autonomously and remains out of synchronization.\nPlots (1)-(5) in Fig.24b demonstrate that all the characteristics o f instant (time-local) qual-\nity of synchronization are closely correlated with difference betwee n the energy backgrounds\nin slave and muster films. Plot (C) shows that (in concrete case unde r consideration) char-\nacteristic frequency of chaotic variations of energy background is about 30 MHz, which is of\norder of inverse relaxation time, Γ−1.\n23100150200P(t), mW\n2468U(t), V\n22024026028002468\ntime, nsu(t), V\n2.752.82.852.9ωin, GHz\n01020\n←  400 ns  →700⋅∆E, ∆P\n220240260280020406080\ntime, ns∆φ,o\n−0.500.5\n←  4 ns  →center S⊥x\n−101200.51\ndelay, nsprobability\n  Fig.23. Synchronization of chaos in Slave generator by mixing its feedback (passive loop) \nvoltage, US(t),  with feedback voltage from Master film, UM(t). (A) Mixing with the coefficient α. \n(B) Wide film under investigation. Two left and two right lines show active and feedback loops \nand central line additional control inductor. The four plots below demonstrate best results if α=0.4. \nThe powers, P, feedback voltage amplitudes, U, its frequencies, ωin, and envelopes of voltage on\nthe control inductor, u(t), in Master (fat curves) and Slave (thin curves) films are compared with \ncorrections to time−dependent delay and energies. (C) Correlation of power ′s difference and \nenergies difference. (D)  High−frequency phase difference between US and UM. (E) Watching \nfor phase coherence of spin precession at points marked by the star in (B). (F) The time delay \nhistogram shows that delay is nearly quantized by period of precession, ≈ 0.35 ns. αUM+(1−α)US US \nUM \nα \n(A) (B) \n(C) (D) \n(E) to Slave\namplifier \n(F) α=0.4 \n24Fig.24c presents one more example of the same kind of synchronizat ion, with α= 0.9\n, at wide film ( w/λ= 5 ) in strongly chaotic regime. Under corrections to influence by\nnon-synchronized energy background, the picture of synchron ization with respect to power\nabsorption looksenough good(plot (D)). But thewhole picture has no significant differences\nfrom previous case.\n7.17. MAGNETIC TURBULENCE.\nThe Figs.25a-billustrate what happens at too powerful auto-gene ration which is achieved\nunderweakoutputsaturation, Isat= 100 mA,althoughatrelativelysmalllineargain(=4).\nIn spite of that film is narrow ( w/λ= 2.5 ), extremely chaotic shapeless energy pattern is\nformed by irregularly scattered solitons (see top picture in Fig.25b) .\nIn this case (which may be named turbulent magnetic chaos), notice able chaotic drift\n(diffusion) of the carrier phase takes place due to phase slips, as sh own at plot (A) in\nFig.25a. Plots (F)-(H) and (C)-(E) present well expressed slips in lo cal magnetization and\ninsuch the integral characteristics as feedback voltage. Accord ing to plot (B), corresponding\nslow chaotic variations of carrier frequency are excellently anti-co rrelated with the energy\n(as the non-isochronity implies).\nTwo interest things should be appointed.\n(i) In contrast to commonly strong chaos, with dcor≈3.9 if estimated from feedback\nvoltagetimeseries (seeplot(C)inFig.25b), theenergybehaves muc hmoreregularlyandcan\nbe characterized by own fractal dimension, dcor≈2.7 . Possibly, this is just manifestation\nof above mentioned autonomy of the energy background.\n(ii) Very slow chaotic variations in energy (and thus in the carrier fre quency and other\nrelated values) are observed, with characteristic frequencies do wn to 3 MHz (plots (D) and\n(E) in Fig.25b).\n7.18. CHAOTIC PULSE AUTO-GENERATION.\nSlight decrease in linear gain (from 4 to 3) at the same film at same outp ut saturation\nresults in new interesting regime of auto-generation which is charac terized by sharp pulses\nof the power absorption, P(t) , and voltages in feedback and driving loops, U(t) and\nV(t) .\nThis example is illustrated by Figs.26a-b. Theoretically, P=∝an}bracketle{tJV−IU∝an}bracketri}htT, where\nI(t) andJ(t) are currents in feedback and driving loops (see scheme in Fig.15), a nd\n∝an}bracketle{t..∝an}bracketri}htTmeans averaging over period of precession. The part ∝an}bracketle{tJV∝an}bracketri}htTis power absorption\n25150 P(t), mW\n34567U(t), V\n2.82.852.92.95ωin, GHz\n360 380 400 420 440 46001020\ntime, nsu(t), V\n−0.500.5center S⊥x\n−1 0 100.20.40.60.811.2\ndelay, nsprobability\n   Fig.24a.  Synchronization of strong chaos in wide film by mixing feedback voltages                         \n(see Fig.23,A) at  α=0.8. Other parameters: H0=400 Oe, λ=2l=0.04 cm, x=2.5 λ, γ=0.0007. \nTypical time series of power absorption, P(t), feedback voltage envelope, U(t), its instant frequency,        \nωin(t), envelope of voltage on the control wire inductor (see Fig.23,B), u(t), and magnetization      \nat film center, S⊥x, are shown. Fat curves relate to Master. Dotted thin curves are actual Slave       \nseries. Solid thin curves are the latters but corrected by subtracting (i) time−variable delay of             \nsynchronization and (ii) difference of slowly varying energy backgrounds. The right−hand bottom               \nplot is histogram (probability distribution) of the delay.                                                    −0.500.5\n←  4 ns  →center S⊥x\n26−0.0100.01∆E(t)\n−40−2002040∆P(t), mW\n203040∆φu(t),o\n0102030∆φU(t),o\n0.1 0.2 0.3 0.4−202\ntime, µs−Delay, ns\n2.62.833.201234\nFrequency, GHzUM&US′s spectra2.62.833.201\nFrequency, GHz(US−UM)′s spectrum0 50 100\nFrequency, MHz∆E′s spectrum\n    Fig.24b.  Continuation from Fig.24a. Synchronization of strong chaos (see Fig.23,A) at  α=0.8. \n(1) Energy difference between Slave and Master films, ∆E(t). (2) Slow (time−smoothed) component, ∆P(t), \nof their difference in power absorption. (3) Microwave−band phase difference, ∆φu(t), between voltages uS(t) \nand uM(t) on the control inductors (see Fig.23). (4) The same, ∆φU(t), for feedback voltages US(t) and UM(t). \n(5) Time delay of synchronization with respect to fast variations in power absorption caused by soliton ′s \nbirths. Rather pronounced correlation between all these variables are evident. (A) Spectra of  US(t) and \nUM(t). (B) Spectrum of  US(t)−UM(t−δt) , where δt, δt≈0.01 ns, is time delay equivalent of ∆φU(t)′s mean \nvalue in (4). (C) Spectrum of ∆E(t). Its characteristic frequency, ≈30 MHz, coinsides approximately \nwith central peak bifurcation in (B) and with energy relaxation rate, 2 Γ≈2γ(H0/Ms+2π)/τ0. (1) \n(2) \n(3) \n(4) \n(5) (A) (B) (C) \n27−2 −1 0 1 2−2−1012\nkxλ/2πkyλ/2π\n−0.4−0.200.20.4\n←  4 ns  →\n50150P, mW\n←  250 ns  →\n−40−200204060\n←  250 ns  →2.82.933.1\n←  40 ns  →\n   Fig.24c.  Synchronizing hyperchaos (with dcor≈3.5) in wide film (w/ λ=5), at  α=0.9, H0=400 Oe, \nλ=2l=0.04 cm, x=3 λ. (A) Local energy pattern, eloc(x,y,t). The lines mark inductors (fat) and \nx−coordinates distanced by λ (thin). (B) Spatial Fourier transform of eloc(x,y). Clearly, separation \nof energy clots (solitons) equals to λ, while their width (in x−direction) is of order of λ/2. (C) Example \nof local phase slip in Master ′s precession (star at (A)) not imitated by Slave. (D) Power absorption by \nMaster film (fat), by Slave film (dotted thin) and that shifted accounting for non−constant delay and \nslow energy fluctuations (solid thin). (E) ∆φU (1), ∆φu (2), delay in units of τ0/7 (3), and 2000 ⋅∆E (4). \nOne can see mutual correlation of these time series. (F) Example of bad synchronization in \nrespect to instant frequency of u(t). (A) \n(B) (C) \n(D) \n(E) (F) \nωin of u(t) S⊥x(star) \n1 \n2 \n3 \n4 active\nloop \nfeedback loop \ncontrol inductor \n28from driving loop which can be written as ∝an}bracketle{tJV∝an}bracketri}htT=|J||V|cos(φV−φJ) , with |J|and\n|V|being envelopes and φVandφJphases. Fat curve and thin curve close to it at plot\n(5) in Fig.26b show pulsations of Pand∝an}bracketle{tJV∝an}bracketri}htT, respectively, while the low-amplitude\ncurve here represents ∝an}bracketle{tIU∝an}bracketri}htTand shows that the feedback loop is consuming film’s energy.\nIn this case, again (i) essential difference between common chaos ( withdcor≈3.3\n) and particular energy chaos (with dcor≈2.2 ), and (ii) slow energy variations (with\ncharacteristic frequencies ∼2 MHz if not less) take place.\nAt last, due to relatively simple and expressively shaped chaos and re latively small fractal\ndimension in this case, we may try do draw three-dimensional projec tion of corresponding\nchaotic attractor. Two variants of the projection are shown in Fig .28.\n7.19. ON POWER THRESHOLD OF CHAOS.\nIn numerical simulations, we can set any gain and output power as we want. In practice,\nsmall-sized high-frequency amplifiers have rather limited output pow er. Therefore, consider\nrough analytical estimate of the smallest (threshold) power absor ption by film which is\nsufficient for chaotic auto-generation.\nTime averaging of the energy conservation law (6.3), if combined with the relations (6.4)\n(atE0= 0 as it was seen) and (6.6) (i.e. for not too short-wave chaos), yie lds\n∝an}bracketle{tP∝an}bracketri}htT≈H0Γ∝bardblSx∝bardbl2≈H0pΓ|Ψ|2=ωuΓ|Ψ|2, (6)\nwhere Ψ is the wave function considered in Sec.5, pis eccentricity of uniform precession,\nand∝an}bracketle{tP∝an}bracketri}htTis mean power absorption per unit volume (in dimensionless units). Hen ce, in\naccording with the estimates of the minimum |Ψ|’s value, Amin, which initiates wave\ninstability (see Sec.5.5), we can estimate power level (per unit volume ) which is sufficient\nfor beginning of chaos as\nmin∝an}bracketle{tP∝an}bracketri}htT≈ωuΓ2/|κ| (7)\nIn real units, for the total threshold power absorption, Pthr, in a film with volume V\nthis formula implies\nPthr≈2πM2\nsVfuΓ2/|κ|, (8)\nwherefuis uniform precession frequency (in s−1) and Γ remains dimensionless.\n29−500−400−300−200−1000100Phase drift\n0.60.811.21.41.61.822.22.4−0.0200.02Energy, ∆ωin\nµs\n0200400600P, mW\n00.050.1eloc(r0)\n0102030Voltages, V\n−0.500.5S⊥x(r0)\n−1000100200Phases\n←   60 ns   →−0.200.2S⊥z(r0)\nFig.25a. Turbulence in strongly pumped film at H0=400 Oe, λ=2l=0.08 cm, x=2 λ, w/λ=2, \nγ=7⋅10−4,  Isat=100 mA, Usat=1 V, Gain=4. (A) Slow drift of phases of the feedback,   \ncontrol and active loop voltages. Slope of the drift is in close connection with slow energy            \nvariations. Both are characterized by frequency ≈20 and besides lower frequencies.                \n(B) E(t)−<E> (fat light curve) as compared with time−smoothed instant frequency of V(t) (thin           \ndark curve). (C−E) Example of phase slip of feedback voltage. (F−H) Fragment of spin                    \nprecession picture in the point r0=(x0,y0) marked by star in Fig.25b. The arrow shows                \nthe phase slip moment.                                                                                  ←   8 ns   →U(t)′s phase\nu(t)′s phase ∆ωin≈−∆E(A) \n(B) \n(C) \n(D) \n(E) (F) \n(G) \n(H) U(t) \nU(t)′s phase V(t)′s phase \n303 4 5 6246\nLog of Cell SizeDimension\n00.5 11.5 200.51\nτ, µs\n050100150200E(t)′s spectrum\nFrequency, MHz\n2.52.62.72.82.9 33.13.2U(t)′s spectrum\nFrequency, GHz050100150200250P′s&U2′s spectra\nFrequency, MHz\n   Fig.25b.  Turbulence in strongly pumped film at H0=400 Oe, λ=2l=0.08 cm, x=2 λ, w/λ=2.5,                                                                                                                        \nγ=7⋅10−4, Isat=100 mA, Usat=1 V, Gain=4. (A) Distribution of energy clots is non−regular. (B) But energy                                                \ncorrelation function, KE(τ), discovers rather regular slow fluctuations in total film ′s excitation level.                                                                                                          \n(C) Correspondingly, fractal dimension of energy signal (fat curves), dcor≈2.7, is smaller than of U(t) ′s                                                                                                                            \nenvelope (thin curves), dcor≈3.9. (D) The energy spectrum has characteristic frequencies ≈3 MHz                                                                                                                        \n(see inset) and ≈20 MHz. (E) The spectra of power absorption (dark curve) and of U2(t) (light curve)                                                                                                            \nhave also peaks at ≈100 MHz and ≈170 MHz related to mean period of soliton ′s births, ≈8 ns.                                                                                                                        0102030KE(τ) dcor≈3.9 \ndcor≈2.7 (A) \n(B) (C) \n(D) (E) \n(F) ωu ωDE(2πD/λ) U(t) \nωDE(2⋅2πD/λ)  V(t) \n31In YIG film, at H0≈550 Oe (i.e. ≈4 in dimensionless form) |κ| ≈1.4 and\nfu≈3.2·109s=1. If take Γ = 0 .007 , which corresponds to approximately 1 Oe half-\nwidth of linear FMR, and the volume be 7 mm ×2 mm×10 micron then Eq.8 gives\nPthr≈180 mW. However, the estimate is rather sensitive to energy losses , and at 0.5 Oe\nhalf-width of FMR it turns into ≈40 mW. Probably, this is still overestimated value since\nnot the whole volume must be critically excited at threshold pump.\nConclusion\nWe have reported selected results of work which was made in 2002-2 003years. Additional\ninformation about this work, - in the form of HTML presentation (als o made at that time)\nwithfigures, movies andcommentstothem, -canbeobtainedbydow nloading andunzipping\nfilehttp://yuk-137.narod.ru/experience/mfs_2002.zip . This presentation includes\nalso some information about our (together with our colleagues) firs t attempts of practical\ngeneration and synchronization of magnetic-wave chaos and its pr operties in comparison\nwith that observed in numerical simulations.\nOn the latter subjects and, besides, physical properties of our f errite (YIG) films and first\npractical applications of electro-magnetic chaos, generated with the help of these films, to\ndirect microwave-frequency secure communication, see also our r ecent journal articles [1,2].\nPrincipal conclusions what can be extracted from the aforesaid ma terial are as follow.\n(i) There are two main machanisms of the magnetic-wave chaos auto -generation. One\nis creation of non-regular chains of (“gray”) envelope solitons com posed (in in-plane mag-\nnetized films under our above consideration) primarily with ‘surface” (“Damon-Eshbach”)\nmagnetostatic spin waves (MSW). Another mechanism is parametric energy transfer from\nthese waves to relatively short “backward” MSW (“bulk” MSW) and r eversed process.\n(ii)Thefirstmechanismdominatesatnottoolarge(in-plane)magnet izingfield(therefore,\ngenerates lower carrier frequencies) but it needs in much smaller lev el of energy consumption\n(dissipation) in film and, hence, much smaller amplification in a feedback circuit. Besides,\nimportantly, it produces much more “rich” chaos, which allows, in prin ciple, to “hide” a\nlarge amount of information.\n(iii) However, these advantages of the “parametric” chaos (as co mpared with the “soli-\n32tonic” one) are accompanied with probable instability of the paramet ric interaction between\nsurface and backward MSW, which may take form of explosive and irr eversible transfer of\nenergy from surface to short backward waves. A degree of the ir reversibility is as large as\ngreat is number of excited backward wave modes per one exciting su rface mode. At some\nregimes of auto-generation one can observe from time to time fast and practically full energy\nswaps into short backward waves and, consequently, sharp and lo ng-standing suppressions\nof auto-generation (sometimes, under definite conditions, it may b reak for periods up to\nnearlyµs).\nThis phenomenon (in the presentation, we called it “parametric collap se”) was observed\nboth in numerical and real experiments (with good correspondenc e between them, though\nnumeric procedures inevitably loss most short waves). Naturally, it can create strong obsta-\ncles to synchronization of chaos, because the breaks of auto-ge neration imply uncontrollable\nand hardly reproducible in “slave” system) phase slips.\n(iv) Therefore, to avoid the “parametric collapse” but, at the sam e time, exploit para-\nmetric processes as much as possible, it is necessary to introduce s ome auto-regulation of\nfeedback amplification. It can help in generation of such output elec tromagnetic signals\nwhat possess not too strong amplitude modulation, at strong enou gh phase and frequency\nchaotic modulation, and cause satisfactory “synchronous chaot ic response” in slave system.\n(v)Fractal(correlation)dimensionsofbothhigh-frequencycha oticoscillationsofdifferent\nvariables (local magnetization values, e.m.f.’s in antennae, and voltag es and currents in\nfeedback circuit) and low-frequency variables (amplitudes, phase s and “instant frequencies”)\nalmost always lie in interval from 2 to 4 and usually between 3 and 4, thu s demonstrating\n“hyper-chaos”. This numeric result well agrees with fractal dimen sions of low-frequency\n(amplitude and phase) signals obtained from real chaotic generato rs. At the same time,\nsome of low-frequency variables, both real and numerical (e.g ‘qua si-local energy density”\nand total “excess” magnetic energy) can show dimensions nearly b y unit lesser, from interval\n2÷3.\n(vi) These facts, seemingly, help to understand why, in spite of not high fractal dimen-\nsion of the our chaos, we never observe “ideal synchronization”, instead seeing only what\ncan be termed “generalized synchronization”. The matter is that, in the schemes under\ninvestigation, all the control and feedback signal are directly cou pled with the “surface”\n(“Damon-Eshbach”) MSW modes only. This is obvious from compariso n of voltage-current\n33spectra, - whose frequency band always lies near or higher the unif orm precession frequency,\n- and numerically found spectra of local magnetization values (local precession) which can\nbe concentrated mainly at twice lower frequencies or even four time s lower ones (because of\ntwo successive parametric divisions of frequency), down to absolu te lower bound of MSW\nspectrum (moreover, sometimes, - e.g. after the parametric colla pse, - become concentrated\njust near this bound).\nTherefore, itisnotsurprising ifoneof4effective relevant degrees offreedomofoursystem\n(the number 4 follows from fractal dimension between 3 and 4), - wh ich is responsible for\nmagnitude of short backward MSW and their energy exchange with s urface MSW, - stays\nfar out of control we used.\nIf this is true explanation of the generalized character of synchro nization, then we have\nchances to improve the described results.\n(vii) It appeared undoubt that even rather rough numeric simulatio ns are useful not only\nfor adequate theoretical of real experiments, but also for their planning.\nClearly, the described work leaved many unanswered questions and hypotheses waiting\nfor numeric and experimental examination. This will subject of sepa rate papers.\nREFERENCES\n[1] N.I.Mezin, Pis’ma v ZhTF 37, No.23, 61 (2011) (in Russian; translated to English\nby AIP).\n[2] N.I.Mezin, A.A.Grishchenko, and Yu.E.Kuzovlev, Pis’ma v ZhTF (in p ress).\n34345671234\nLog of Cell SizeDimension\n0100200P(t), mW\n0.90.95 11.05051015\ntime, µsVoltages, V\n   Fig.26a. Auto−generation of power absorption pulses in moderately pumped film at H0=400 Oe,                              \nλ=2l=0.08 cm, x=2 λ, w/λ=2.5, γ=7⋅10−4, Isat=100 mA, Usat=1 V, Gain=3. (Top) Quasi−local  \nenergy pattern. (Left) Power pulsations, envelopes of feedback (thin dark) and active loop (fat light)                       \nvoltages, U(t) and V(t), and their time−smoothed instant frequencies. (Right) Fractal dimension of                           \nenergy, E(t), occurs ≈2.2 versus that of U(t) being ≈3.3. Power and voltage spectra show                         \nthat typical period of soliton births is ≈ 10 ns.                                                                      2.6 2.8 300.20.40.6 U(t)′s spectrum\nFrequency, GHz0100200300400500P(t)′s spectrum\nFrequency, MHz\n0.40.60.81−50050\ntime, µs∆ωin, MHz0102030E(t)′s spectrumU(t) V(t) \nωu ωDE(2πD/λ) dcor≈3.3 \ndcor≈2.2 u1(t) u2(t) \n350510u1,2 (V)\n2.62.83ωin 1,2 (GHz)\n2.62.83ωin V,U (GHz)\n010V,U (V)\n0100200Power (mW)\n90092094096098010001020104010601080−4504590135ΦVJ,ΦVU\n−50050<∆ωin>, MHz\n0.30.40.50.60.70.80.911.1−20020\ntime, µs\n<∆P>,<∆|J|>ns\n   Fig.26b (continuation to Fig.26a). Dynamics of the power pulses formation. (1) Amplitudes of \nvoltages on two control wire inductors shown at Fig.26a(A), u1(t) (fat curve) and u2(t) (thin), and \n(2) their instant frequencies. (3) Instant frequencies of V(t) (fat) and U(t) (thin), and (4) their \namplitudes. (5) Power absorption (fat), |V(t)| ⋅|J(t)|⋅cos(φV−φJ) (thin), and power consumption by \nfeedback loop (marked by arrows). (6) Phase differences between V(t) and J(t) (light) and V(t) \nand U(t) (dark). (7) Slow deviations of V ′s (fat) and U ′s (thin) frequencies. (8) Slow \nvariations of exciting current magnitude (fat) and power (thin). 1 \n2 \n3 \n4 \n5 \n6 \n7 \n8 \n36−80−60−40−20020406080 −500506.26.46.66.877.27.47.67.8\n|V|⋅|J|⋅cos Φslow|V|⋅|J|⋅sin Φslowωin\n−10010203040−100−50050100 66.577.58\nΦV−ΦJ\nPωin\nFig.28. Two different three−dimensional representations (and their 2−D projections) of chaotic                            \nattractor of the system illustrated by Fig.26a−b. (Top) Absolute value of radius−vector in horizontal                        \nplane is chosen be product of the active loop voltave and exciting current amplitudes, |V(t)|                             \nand |J(t)|, while its angle is scanned ∝Φslow≡∫<ωin>dt with < ωin> being time−smoothed \nslow part of ωin (shown at the left bottom of Fig.26a). (Bottom) Coordinates in horizontal plane                   \nare chosen be power absorption, P(t), and phase difference, ΦV−ΦJ, between V(t) and J(t).                         \n37"    },    {        "title": "1009.3371v1.Investigation_of_multiple_echo_signals_formation_mechanism_in_magnet_at_excitation_by_two_arbitrary_radio_frequency_pulses.pdf",        "content": " \n \nInvestigation of multiple echo signals formation mechanism in magnet \nat excitation by two ar bitrary radio -frequency pulses.  \n \nМ.D. Zviadadze *, G.I. Mamniashvili , R.L.  Lepsveridze , \nAndronikashvili Institute of Physics, 6 Tamarashvili  Str., Tbilisi, 0177, Georgia  \nA.M.  Аkhalkatsi , \nJavakhishvili Tbilisi State University, 3 Chavchavadze Ave. Tbilisi, 0128, Georgia  \n \n*Corresponding author  \nE-mail address:  m.zviadadze@mail.ru ,    m.zviadadze@aiphys ics.ge  \n \n \nАbstract \n \nThe quantum -mechanical calculations of intensities and time moments of appearance of multiple \nspin echo signals of excitation of nuclear spin system  of magnet by two arbitrary width radio -\nfrequency pulses were carried out. This method was used by us earlier at consideration of multiple-\npulse analogs of single -pulse echo in multidomain magnets upon sudden jumps of the effecting \nmagnetic field in the rotating coordinate system during the action of radio -frequency pulse.  \nThe formation mechan isms of echo signals are discussed. The appearance of four  primary  \nstimulated echo signals is predicted. The total number of echo signals at fixed parameters of radio -\nfrequency pulses does not exceed thirteen ones.  \nTheoretical conclusions are in  compliance with experiments carried out on lithium ferrite. As it was \nestablished by us earlier in this magnetic dielectric, in difference from ferrometals, it is observed \nvery short relaxation times of single -pulse and two -pulse stimula ted echoes, and the contribut ion of \nradio -frequency pulse fronts distribution mechanism is insignificant. For this reason lithium ferrite \nis a good material for the experimental verification of theoretical conclusions in experimental \nconditions most close to the theoretical model.  \nKey words : multiple echo, wide pulse, lithium ferri te, cobalt  \n \n \n \n \n  \n2 \n  \n1. Introduction  \n \nIt is known that in the magnetically ordered systems the formation of NMR res ponses after the \nexcitation of  nuclear spin  systems (NSS) of a sample by RF pulse sequence is defined  by two main \nreaso ns: inhomogeneous broadening (IB) of the spectroscopic transitions (so -called Zeeman \ninhomogeneous broadening) and inhomogeneous distribution of the radio -frequency ( RF) field \nenhancement factor on nuclei (so -called Rabi IB), caused by th e hyperfine interaction between \nelectron and nucle ar spins [1]. \nFor investigation s of the echo -responses of Hahn spin -systems (further we consider conditions \nwhere the dynamic frequency shift does not play a role [ 1]) it was  devoted very many works [2 -8]. \nTheoretical and experimental definition of time moments of the echo -signal appeara nce, their \nnumber and intensity makes it possible to identif y the echo formation mechanisms in different \nmaterials what is important for the practical applications.  \n2. Statement  of problem  \nAt starting time moment 0=t  the equilibrium NSS of magnet is exc ited by the first RF pulse of  \nlength 1τ. Then during time 1ττ> (delay time ) a free evolution of NSS takes place, after w hich in \ntime moment ττ+=1t  the s econd RF pulse of  length τττ+>1 2  is applied . After termination of \nthe second pulse in the time moment 2 1τττ++=t  = t1\n1tt> the process of free induction starts again \nand on the time s cale  the multiple echo sig nals are observed.  \nIt is supposed  that pulses have generally different frequencies ()2,1ω , amplitude s ()2,1\n1ω  (in the \nfrequency units ) and lengths 2,1τ. It is understood also that in the time moment of a pulse \napplica tion it takes place the jumplike  change of hyperfine field on nuclei: )2,1(\nnj nj H H→  (as \nexample, due to the displacement of domain wal l as result of magnetic pulse influence [ 9], or due to \nother reason ), what results in the jumplike  change of resonance frequency of the −jisochromat  \n)2,1( )2,1(\nnj Ijnj I j H H γωγω =→=  (it is believed that usually Hnj >> Ho\nω). \nIn the coordinate system rotating with RF field frequency  around z axis (RCS) NSS is described \nby the density matrix ()t*ρ  and Liouville equation [ 10] \n()()[]t Httit*,* *ρρ=∂∂ ,      (1) \nwhere  \n()() ∑ ++∆= +=\njSNx\njz\njj SL t H I I Ht H H H1* **, * ω  ,  ()()()tIi HtIi t Hz\nSL z SL ω ω − = exp exp*;   (2)  \nSLH - spin-lattice interaction (we do not need the full form of it ), SNH  - is the Suhl -Nakamura \ninteraction:  \n∑ = =\n≠−+\n\nkk k kk SN U UIIU H , .  \n3 \n  \nAt further consideration we neglect the interaction of nuclear spins from different isochromats,  \ntherefore we could present SNH  in the form:  \n()() ∑ ∑\n≠−+= ≈\njN\nkkk j SN j SN SNj\nIIU H H H\n , ,                                           (3)  \nwhere the summing over ,k is spread on spins of the −jth isochromat (their num ber is jN). It is \nobvious that  \n()[]0 ,=j SNz\njHI .                 (4) \nValue ωω−=∆j j  is the resonance detuning ; j n j Hγω=  is the Zeeman frequency of the −jth \nisochromat ; 1 1 HIηγω=  is the Rabi frequency (the Rabi IB is not taken into account ); −η the RF \nfield enhancement factor ;y\njx\nj jzyx\nj iI I I I ±=±,,, is nuclear spin operators belong to the −jth \nisochromat.  \nAs it is seen from H amiltonian *H in form (2) the nucle ar spins in RCS experience influence of the \neffective field:  \n()I j j i k H γω /1\n+∆=  .                                                              (5)  \nThe action of two different RF pulses on NSS cou ld be schematically presented in the following \nway (Fig.1) : \n \nFig.1. Schematic presentation of the influence of two wide RF pulses on NSS  τ+τ>ττ>τ1 2 1, , \n1,2,3,4 - pulse fronts. \n \nThe problem is in the solution of the Liouville equation (1) in regi ons IVI− at the initial condition  \n() 1/ 1 *0 0Tr I t\njz\njj L t \n\n\n−≅=∑ =ωβ ρ ρ  ,     (6)  \n4 \n  \nallowing for the condition of concatenation in time moments 2 1 1 1 , , ττττττ +++ : \n() ()()()()ττρττρτρτρρρ +=+ = =1*\n10 1*\n2 1*\n1 1*\n10 0*\n1 , , 0 ,        (7) \n()()2 1 2 2 1 20 τττρτττρ ++=++* *, \nproviding the continuity of solution . \nDuring the action of RF pulses the spin -lattice interaction ()SNx\nSL H t H<<  is neglected . \nThe NSS Hamiltonians in regions I (α=1) and III(α=2) are presented by expressions . \n()()( ) ∑+∆= +=\njx\njz\nj j SL t I I Ht H H Hα\nα α α αα ω1* *\n,* *, ,  2,1=α .                               (8) \n*\nαtH is obtained from *\ntH in (2) with help of substitutions  \n() () ()() () ()α α α α\nαα αγωωωω ωωωωnj I j j j j j H=→ −=∆→∆→ → , , ,1 1                   (9)  \nIn regions II and IV Hamiltonian )(*\n0tH is obtained from *\ntHat ω1\n() () ∑∗ ∗++∆=\njSL SNz\njj t H H I tH0=0 and takes form : \n                                                  (10)  \nFrom  the form of *\nαH it follows that during the action of −αth pulse the nuclear spins are \ninfluenced by t he effective magnetic field  \n()()I j j i k H γωα\nαα/1)( \n+∆= , 2,1=α ,                                      (11) \nwhere  ik\n, are ort hs of xz, exes in the RCS . \n \n3. Solution of Liouville equation  \n \nRegion I \nThe solution of equation  \n()()[]t Htti*\n1*\n1*\n1,ρρ=∂∂       (12) \nat the initial condition ( 6) takes form : \n() \n\n\n\n−= tHiexp tHiexpt* * *\n1 0 1 1 ρ ρ                  (13)  \n5 \n  \nLet us transform expression (13) to a more convenient form for calculations introducing w ith thi s \naim a unitary operator  \n()()(). , / cos, exp121\n12\n1 1 1 1 1 1 1 j I j j j j j\njy\njj y H I i U γω θ θ ≡+∆=ΩΩ∆=\n\n\n=∑   (14) \nThe operator 1yU realizes a rotation of coordinate system around y axis on 1jθ angles different for \ndifferent isochromats.  At this rotation  for each isochromat the z axis of “new” system becomes \ndirected along the effective magnetic field  1jH\n from (11). \nIt is easy to show that  \n{ }1 1 1 1 1 1 1 1 1 y y SN y z y SN y z y*UUHU H~U H UH~U H+ + ++ =+ = ,  ∑Ω=\njz\njj z I H~\n1 1 .                       ( 15) \nNeglecting in the “modified” interaction +\n1 1 y SN y UHU  by the nonsecular, i.e. noncommut ing with \n1~\njH, terms (it is supposed that 1\n2\n22/1\n2 22\n1 ,1~−\n\n=>> TT TrITrH\nzSN\nSNωω  is “homogeneous” NMR \nspectrum width) we obtain  \n1 1 1 1 1 1 1 ,SN z y y*H~H~H~,UH~U H += =+, () ∑∑\n≠−+⋅ + =\njN\nkkj k j k SNj\nII II U H\n  \n} sin {12\n1 1, θ θλ .                   (16)  \nAllowing for (13) and (16), we obtain finally  \n()1 1 1 01 1 1 1 y y y j*UtH~iexpU UtH~iexpUt \n\n\n− =+ +\n ρ ρ .             (1 7) \n \nRegion II \nIt is necessary to solve the equation  \n()()()[]t ,tHtti*\n10 010ρρ∗+\n=∂∂                                                                (18) \nwith Hamiltonian (10) at initial condition  \n()()1*\n1*\n01\n1τρ ρ\nτ=\n=tt .                                           (19) \nLet us write Hamiltonian ()tH*\n0 in form  \n() ()() () ∑ += +∆=\nj*\nSL SNz\njj*t H HtV,tVI tH0 , \nand pass to a n ew presentation   \n6 \n  \n() ()() ()\n\n\n−∆−\n\n\n−∆ = ∑ ∑\njz\njj*\njz\njj tI i expt tI iexpt'1 10 1 τ ρτ ρ .                        (20)  \nIt is evident, that  \n()()()1*\n1 1*\n101' τρτρ ρτ===tt .     (21) \nFor the operator ()t'ρ  one obtains equation  \n()()()[]t',t'Vtt'i ρρ=∂∂ ,  () ()() ()\n\n\n−∆−\n\n\n−∆ += ∑ ∑\njz\njj*\nSL\njz\njj SN tI i expt H tI iexp Ht'V1 1 τ τ  . (22)  \nSolving equatio n (22) at initial condition ( 21) and using (20), we find out  \n() ()()()() ()\n\n\n−∆\n\n\n−∆−= ∑ ∑+\njz\njj*\njz\njj*tI iexp,tU ,tU tI i expt1 1 1 1 1 1 10 τ ττρττ ρ  \n() ()\n\n\n\n−=∫t\n'dt't'ViexpT ,tU\n11\nττ.                                                (23)  \nEvidently, the expression ()()1 1 , exp ττ tU tI i\njz\njj\n\n\n−∆−∑  is the evolution operator for NSS in the \ntime interval τττ +≤≤1 1t  between pulses  \n \nRegion III \nThe solution of equation  \n()()[]t,Htti* **\n2 22ρρ=∂∂             (24)  \nat initial condition  \n()()ττρττρ +=+1 10 1 2* *,              (25)  \nin analogy with (1 7) takes the form  \n() ()() ()\n−− + \n\n−−− =+ +ττ ττρττ ρ1 2 2 1 102 1 2 2 2 tH~iexpU U tH~iexpUty*\ny y*\n .                      (26) \n \n \n  \n7 \n  \nRegion IV \nSolution o f equation  \n()()()[]t ,tHtti* **\n20 020ρρ=∂∂                        (27) \nat initial condition  \n()()2 1 2 2 1 20 τττρτττρ ++=++* *          (28)  \nin analogy with (23), takes the form  \n() ( )( )()\n( ) ( ). exp ,, exp\n2 1 2 11*\n2 2 1 2 1*\n20\n\n\n\n−−−∆ ++⋅⋅++ ++\n\n\n−−−∆−=\n∑∑\n+\njz\njjjz\njjj\ntI i tUtU tI i t\nτττ ττττττρττττττ ρ\n            (29) \n3.  \n4. General expression for free induction decay (FID) signal  \n \nThe FID signal is proportional to the value of transverse nuclear magnetization  \n()(){} ( ) [ ]⋅−−−∆= = ∑+\njjti It SptI2 1*\n20 exp τττ ρ  \n()( )( ) { }2 1 2 1 2 1*\n2 , , τττττττττρ ++ ++ ++ ⋅+ +tUI tU Spj .                                  (30) \nFor obtaining (30) we use the relation  \n( ) ( ) ( ) { } ∑ ∑ ∑+ +��−−∆=\n\n−−−∆−\n\n−−−∆ jj j\njz\njj\njz\njj I ti tI i I tI i2 1 2 1 2 1 exp exp exp τττ τττ τττ  \nAs far as SN SL H H<< , one could take  in (30)  \n( ) ( )\n\n−−−−≅++2 1 2 1 exp , τττ τττ t HitUSN.                            (31)  \nMaking in (30) the substitution 2 1 1 1 τττ++=+→ t,ttt  and allowing for (3 1), we obtain   \n()()()(){} ∑+∆=+\njj*\nj tIt Sptiexp ttI12 1 ρ , () \n\n−\n\n=+ +tHiexpItHiexptISN j SN j .                      (32)  \nIn formula (32) time t is counted from a time moment of the second pulse termination 1t. \nAccording ly to (26)  \n8 \n  \n() ()2 22 2 1 102 22 2 2 y y*\ny y*U H~iexpU U H~iexpUt \n\n+ \n\n− =+ +τ ττρτ ρ .                       (33) \nThe substitution of (3 3) to (32) gives : \n()()() () ∑\n\n\n− \n\n+ ∆=+++ +\njy j y y*\nj H~iexpUtIU H~iexpU Sptiexp ttI22 2 2 22 2 1 10 1 τ τ ττρ .                  (34)  \nA further simplification of expression (3 4) is made by the approximation \n()()+ +−=j j ITt tI2/ exp ,                         (35)  \nwhich is  equivalent to the Lorentz approximation of the correlation , \n(){}{} ()2/ exp1/ 1/ Tt SpIISp SptIISp −⋅ =+− +−, \nfrequently used in practice [11 ]. In this approximation the transverse magnetization in region III \ncoincides with the solution of Bloch equations for this re gion. As a result, one obtains:  \n() () ∑\n\n\n− ⋅\n\n+ \n\n\n\n\n\n\n−∆ =+++ +\njy y j y y j U HiUIU HiU SptTi ttI2 22 2 2 22 2 1*\n10\n21~exp~exp1exp τ τ ττρ .  (36)  \nIt was shown in work [10] that \n() () ().exp exp~exp~exp\n2 2 22*\n2 2*\n2 2 22 2 2 22 2\nz\nj j j j j jj y y j y y\nI I IHiI HiU HiUIU HiU\nγβατ τ τ τ\n++==\n\n−\n\n=\n\n− \n\n\n− ++ ++ +\n                (37)  \nAllowing for (37) the expression (3 6) is transformed to the form : \n() ()() () ()[ ] { } ∑ ++ + \n\n\n\n\n\n\n−∆ =+− +\njz\nj j j j j j*\nj I I I SptTi exp ttI2 2 2\n1 10\n211γβαττρ .  (38) \nAccordingly to (23),  \n() ()()()\n\n\n∆ + +\n\n\n∆−=+ ∑ ∑+\njz\njj*\njz\njj*I iexp, U , UI i exp τ ττττρττττ ττρ1 1 1 1 1 1 1 10 . (39) \nSubstituting (39) into (38) and allowing for the relation  \n()±\n′′′±\n′′′ ∆±=\n\n\n∆−\n\n\n∆ ∑ ∑ j j\njz\njj j\njz\njj I i I i I I i τ τ τ exp exp exp , \nwe obtain s \n()()()()()()()()() [ { ∑ +∆− +∆ + −∆ =+− + + −\njj j j j j j j i I i I U SptT i ttI τ βτ αττττρ exp exp , exp2 2\n1 1 1*\n11\n2 1   \n9 \n  \n()]()}1 12,τττγ + + UIz\nj j .                                                    (40) \n \nLet us use approximations  \n()()()2 1 1 1 1 / exp , , T I UI Uj j τ ττττττ −≅+ +± ± +, \n[]()()()1\n1 1 1 1 1 / exp , ,− +−−+≅+ + T II I UI Uz\njz\njz\njz\nj τ ττττττ .                       (41 ) \nThe first one of them is equivalent to (35), the second one provides coincidence of the longitudinal \nmagnetization with the soluti on of Bloch equations in the region II. The value z\njI0 is an \nequilibrium value of the longitudinal component of nuclear spin in the high-temperature \napproximation:  \n() ()131\n0 + −= = SS I Sp Ij Lz\njz\nj ωβ ρ  .     (42) \nAs a result, for t he FID signal (40) we obtain the value  \n()()()()()()()()()( ) [{ ∑ + −∆− +−∆ −∆ =+−− − − + −\njj j j j j j j j I T i I T i I SptT i ttI τ βτ ατρ1\n22 1\n22\n1*\n11\n2 1 exp exp exp  \n() ()()() [ ]}12 2/ exp T I I Iz\njz\nj jz\nj j τ γγ −−++ .                                                 (43)  \nSubstituting in (43) the value ()1 1τρ* from (17) and carrying out calculatio ns, we obtain finally the \nFID signal in the general form:  \n()()[]()() ( )()()()[] { ∑ +−∆ +−− −∆ =+− ∗ −\njj j j j jz\nj T i T tT i I ttI τ γατ γ1\n21 2\n12 1\n2 1 exp / exp1 exp  \n()()()[ ]()()()}11 2 1\n2*1 2/ exp exp T T ij j j j j τγγτ γβ − +−∆− +−.                                     (44)  \nAt 0=τ  it is obtained the value  \n()()[]()()()()()(){ } ∑ ++ −∆ =++−\njj j j j j j jz\nj tT i I tI1 2 *1 2 1 2 1\n2 2 1 exp γγγβγα ττ , \nwhich ex actly coincides with formula (21) from work [10], as one should expect . Similar to that as \nthe situation with one stepwise change of effective field jH\n [10] in time moment 1τ=t  after the \nswitching on a RF pulse with duration 2 1τττ+=  at moment 0=t , turned out to be analogous  to \nthe excitation of NSS by three short -time pulses [ 12] (three signals of two -pulse echo (TPE) (12), \n(13), (23) and one signal of stimulated echo (123)  – design ations are taken from work [12]), the \nconsidered in this work case of two arbitrary pulses appeared to be analogous to the excitation of \nNSS by four short -time pulses [ 13]. Substituting values of factors ()()()()()1 1 2 2 2\nj j j j j , , , , γγγβα  from \nwork [10] in (44), one could find out that expression for ()1ttI+  contains 20 signals: two \ninduction signals, corresponding time moments 1τ=t , 2 1 1 τττ++==tt  and 18 echo signals, the \ntiming of which is presented in the table:    \n10 \n  \nTable.  The timing  of echo signals arrangement  \net - the appearance times of echo in case of presence of sudden jumps; ) (j ja etΩ=Ω - echo appearance \ntimes in absence of jumps; )0 ; (1=Ω=Ω ωj ja et - echo appearance times in case of four sh ort RF pulses \n[13] (in our designations).  \n \n  et ) (j ja etΩ=Ω  )0 ; (1=Ω=Ω ωj ja et  \n1 (12) τττ −∆Ω−Ωj j j /) (22 11 τττ −∆−Ωj j /) (2 1 τττ−−2 1 \n2 ((12)3)  j j j ∆Ω+Ω− /) (22 11τττ  j j ∆+Ω− /) (2 1τττ  2 1τττ−−  \n3 (123)  j j j ∆Ω−Ω /) (22 11ττ  j j ∆−Ω /) (2 1ττ  2 1ττ−  \n4 (23) j j∆Ω− /22ττ  j j∆Ω− /2ττ  2ττ−  \n5 (13) j j j ∆Ω−Ω+ /) (22 11τττ  j j ∆−Ω+ /) (2 1τττ  2 1τττ−+  \n6 ((12)34)  j j∆Ω− /11ττ  j j∆Ω− /1ττ  1ττ−  \n7 (124)  j j∆Ω /11τ  j j∆Ω /1τ  1τ \n8 ((13)4)  τττ −∆Ω−Ωj j j /) (11 22 τττ −∆−Ωj j /) (1 2 τττ−−1 2 \n9 (234)  τ τ τ \n10 ((23)4)  ττ−∆Ωj j /212 ττ−∆Ωj j/2 ττ−2 \n11 (134)  ττ+∆Ωj j/11 ττ+∆Ωj j/1 ττ+1 \n12 ((123)4)  j j j ∆Ω−Ω /) (11 22ττ  j j ∆−Ω /) (1 2ττ  1 2ττ−  \n13 (((12)3)4)  τττ −∆Ω+Ωj j j /) (11 22 τττ −∆+Ωj j /) (2 1 τττ−+2 1 \n14 (34) j j∆Ω /22τ  j j∆Ω /2τ  2τ \n15 ((12)4)  τττ +∆Ω−Ωj j j /) (11 22 τττ +∆−Ωj j /) (1 2 τττ+−1 2 \n16 (1234)  j j j ∆Ω+Ω /) (11 22ττ  j j ∆+Ω /) (2 1ττ  2 1ττ+  \n17 (24) ττ+∆Ωj j/22 ττ+∆Ωj j/2 ττ+2 \n18 (14) τττ +∆Ω+Ωj j j /) (11 22 τττ +∆+Ωj j /) (2 1 τττ++2 1  \n11 \n  \n \nThe echo signals could be classified in the following way [1 3]: \n1) six primary signals TPE (12), (13), (14), (23), (24), (34), formed in pairs by fronts 1, 2, 3, 4 \n(see Fig.1). ((12) and (34) are single -pulse echo (SPE) signals from the first and the second \nRF pulses, correspondingly);  \n2) four secondary signals of TPE ((12)3), ((12)4), ((13)4), ((23)4), formed by the primary echo \nsignal with following fronts ; \n3) four signals of primary stimulated  echo (123), (124), (134), (234), formed by three fronts ; \n4) two signals of the secondary TPE (((12)3)4), ((123)4), formed by echo signal ((12)3) and \nstimulated echo signal (123) with front 4, respectively;  \n5) one signal of “complicated” stimulated echo ((12)34) , formed by three influences : (12) and \n3 and 4 fronts ; \n6) one signal (1234) which is specific for four -pulse sequence and does not formed by one of \nthe above noted mechanisms . \nFrom the presented table it follows that at fulfilling conditions  \nτττττ +>>1 2 1,       (45) \nthe m aximum number of echo  signals which could be usually observed after a time moment of the \nsecond pulse termination , as in  works [6,13], is equal to thirteen (it is not observed the first echo \nsignals (12), ((12)3), (123), (23), (13), the appearance time mome nts of which are arranged from \nleft side in respect to the moment 2 1 1 τττ++=t ). But the real number of the observed signals and \ntime moments of their appearances depend s on relation between 2 1,,τττ  and val ues of jumps . \nIf instead of (45) one takes condition  \n2 2 1 ,τττττ <+> , \nthen the maximum  number of the observed echo signals is twelve (it is not observed signals (23), \n((12)34), ((12)3), ((12)4), ((123)4),  (((12)3)4). But the real number of the obse rved echo signals and \ntime moments of their appearances depend on relation between 2 1,,τττ  and values of jumps.  So, \nas example, at conditionally chosen by us values 7 , ,22 1 === ττττ for four short -time similar \npulses [13] signals (124)  and ((13)4) and (134)  and ((123)4) appears in pairs in the same place, \ntherefore, it is observed 11 , but not 13 echo signals.  \nSo, one could control the number of ech o signals and their arrangement  changing 1 2 1 ,,,ωτττ . It \nshould be particular ly noted that these changes do not influence the position of stimulated echo \n(234) ( τ=et ). At 0=τ   one observes four echo signals from work [10].  \nFor shortness, we do not present here all expressions for thirteen echo  signals but restrict ourselves \nwith four most interesting echo amplitudes (numbering of echo amplitudes corresponds to their \nposition in the table).   \n12 \n  \n()\n\n\n\n−+−⋅ =→∑ 1\n21\n222\n22\n1 ,6 exp2/ sin sin sin21((12)34)\njj j j\njz\nj eT TtI Aττθθθ . \nIn conditions of (45) the signal of “complicated” stimulated echo ((12)34) is int eresting due to the \nfact that from it there starts the spectrum of echo signals and, besides this, at sufficiently large \nvalues ()1\n1ω, for which 0 /11≤∆Ω−−j j et ττ , it is not observed. This signal could be used for \nevaluation of 1ω values.  \n\n\n\n+−⋅ =→∑\n212\n1 ,9 exp sin2sin41(234)TtI Aj j\njz\nj eτθθ . \nThe primary  stimulated echo (234) is notable by fact that the time moment of its appearance t e=τ  \ndepends only on the delay time, but its amplitude does not depend on T\n()()()\n\n\n\n−−−−⋅ =→∑\n12\n22\n1\n21\n222\n2 12\n,16 exp2/ sin sin sin211234T T T TtI A\nj jj j j\njz\nj eτττθθθ1 \n. \nThe echo  signal (1234)  could be used for measurements of spin -relaxation time T\n() () [ ]()( )2\n2 2 2 12\n22\n2 ,18 / / exp sin / exp1 sin sin 14j j j j\njz\nj e T Tt I A τ θπτ θθ −−⋅⋅−− =→∑1 \n. \nThe primary (TPE) (14) is the last observed echo signal at arbitrary times 2 1,,τττ  and jumps.  \nIt should be noted that from thirteen observed e cho signals only four ((124), ((123)4), (34) and \n(1234), depend on T 1\nSN SL H H<< which is the consequence of the duration of RF pulses and condition \n. \n5. Experimental results  \n \nLet us present now the experimental results of investigation of mult iple echo in lithium ferrite and \ncobalt formed at excitation by two wide arbitrary width RF pulses. Experiments were carried out \nusing the nuclea r spin echo spec trometer and sample of lithium ferrite, enriched by 57\nWe use approach developed in work [10] where it  was investigated multiple -pulse  analogs of SPE \nobtained at jumplike changes o frequency and amplitude of R F pulses, in the limits of the  RF pulse \nlength. It was turned out that RF pulse fronts and locations of jumplike changes of НFe isotop e, \nand cobalt described in [ 10]. Nuclear spin echo signals were averaged by a “Tektronix 2430 А”. \nThe choice of sample is caused by the different rol e of pulse fronts in these materials stipulating the \nexistence of two different mechanisms of SPE formation – multiple – pulse and distor tion ones \n[14].  \neff in RCS in \nlimits of RF pulse action have their qualitative analogs in the  exciting RF pulses of the Hahn echo \nmethod. In this approach the value of Н eff direction change in RCS is an analog of deflection angle \nof the nuclear magnetization vector under the influence of two RF pulses in Hanh method.  In \nframes of this approach, the excitation by two wide RF pulses could be considered to be equiv alent \nto the excitation by a  complicated  single -pulse when besides fronts of RF pulse in l imits of its  \n13 \n  \nlength there are two jumplike chan ges of Н eff\n in RCS, with the amplitude of RF pulse between them \nbeing zero.  This complicated single -pulse excitation has its analog in the four -pulse excitation in \nthe Hahn method [13,15].  \nOscillograms of multiple echo signals at excitation by two wide RF pulses  in lithi um ferrite are \npresented in Fig.2.  \n \nFig.2 . a) at fNM = 71 М Hz. RF pulse durations τ 1=8 µs, τ2=5 µs and time interval between them τ 12=9 µs. \nb) at excitation by four short τ р\n=1 µs RF pulses coinciding with the edges of \nwide pulses. c), d) correspond to the case of two equal-length RF pulses.   \nThe repetition period of RF pulse pairs is optimal for observation of multiple echo. The upper beam shows \necho signal amplitudes from NMR receiver in dependence of time and lower beam – signals from video -\ndetector showing the shape, amplitude and duration of RF pulses . \nIn Fig. 2а it is presented the osc illogram s of multiple echo signal obtained at excitation by two RF \npulses of different durations and in Fig.2b –  the oscill ograms of its four -pulse analog  in case when \nfour short RF pulses coincide with fronts of two wide RF pulses .  \nIn Fig.2c and 2 d it is presented corresponding  oscillograms of multiple echo and its four -pulse \nanalogs in lithium ferrite for two equal length RF pulses. In this case the picture is essen tially \nsimplified and similar the one considered in [ 10]. \nSimilar  oscillograms for cobalt are presented in Fig.3 a,b.  \n \nFig.3. Multiple echo signals in cobalt at excitation by two wide RF pulses with frequency f NMR = 217 М Hz \n(a) τ1=10 µs, τ2=7 µs, τ12=16 µs;  (b) τ1 = τ2= τp =10 µs, τ12=16 µs.  \n14 \n  \nThe multiple echo signals oscillograms in lithium ferrite [6], for multiple echo signals in \n59СоСо FeNi  and 51\nVFe obtained on a wide -band coherent NMR spectrometer in  the mode of phas e-\nsensitive detection at 4.2 К, Fig.4. The multiple echoes in both cases hav e thirteen components.  \n \nFig.4. Signals of multiple echo (phase -sensitive detection at 4.2 K) in (a) 59CoCoFeNi on 220 MHz frequency, \nrepetition rate 10 msec, τ1=12 µs,  τ2=8 µs,  τ=51 µs.(b) 51VFe, frequency 98 MHz, repetition period 5 msec, τ 1=20 µs, \nτ2=8 µs, τ=74 µs [6].  \nAt the same time the multiple echo components in cobalt differ significantly from ones in lithium \nferrite by the ir significantly longer width s, and its main compo nent coinciding with TPE signal in \nthe limit o f short RF pulses has the shape  of two -hump Mims echo signal [2]. In difference with this \nin lithium ferrite it is observed considerably narrow echo signals with their shape and intensity \ndepending on the repet ition freque ncy of pairs of RF pulses si milarly to those observed in [14 ] for \nsignals of SPE formed by the multiple pulse mech anism.  \nEven more large difference is revealed at analysis of relaxation characteristics of multiple echo \ncomponents in lithiu m fer rite and cobalt. It is  simple r to compare them in case of two equal length \nwide RF pulses. This situation was for the first time considered in [3] on example of FeV. In this \nwork it was observed unusually fast relaxation rate for one of main components of multiple echo \n(C2 in designations of [ 3]). The room temperature measurements gave rates of 1, 1⋅10-2 and 0.3⋅10-3 \nµs-1, correspondingly, for the three pro cesses: C 2 decay, spi n-spin and spin- lattic e relaxation . \nSimilar study carried  out in work [ 8] for lit hium ferrite gave results c lose to [3] and made it possible \nto clear out the nature of strongly relaxing along with RF pulse duration increase component which, \nas it was turned out, was changing synchronously with single -pulse and two -pulse stimulated echo  \nsignals. As compared with work  [3], they ha ve larger intensit ies providing th e possibility to carry \nout relaxation measurements. It was established that this component possessed relaxation rate close \nto ones of SPE and stimulated TPE which are formed in l ithium ferrite by the multiple -pulse \nmechanism [8]. In the same time, the relaxation rate of corresponding component  in Co was close \nto the one of SPE in cobalt which  in correspondence with the distort ion mechanism effect ive in this \nmaterial  has value close to Т 2 (0.5 ÷ 0.8 Т2\nOn the basis of presented results it is possible therefore to carry out the conclusion on sim ilarity of \nmultiple echo formation mechanisms in lithium ferrite and in materials studied in [6].  [14]).  \n15 \n  \nThe presence of so short relaxation times of one of the main component of multiple echo at increase \nof RF pulse duration could be understood taking into accoun t the fact, firstly noted in [16 ], that the \nSPE sig nal in frames of the nonresonant  formation mechanism of SPE could have more short \nrelaxation time s then Т2 because the dephasing conditions  of isochromat s in the ef fective RF field \nat the action of RF pulse differ from ones during the process of isochromat reph asing  after the \ntermination of RF pulse.  \n6. Conclusion  \nThe method of quantum -mechanical calculatio ns developed earlier by us to study the multiple -pulse \nanalog  of single -pulse echo in magnets upon sudden jumps of the effective magnetic fields in \nrotating coordinate system duri ng the action of RF pulse was used  for calculations of intensities and \ntime m oments of appearance of multiple echo signals at excitation the nuclear spin system of \nmagnets by two arbitrary width RF pulses.  \nThe formation mechanisms of multiple echo signals and influence of relaxation processes are \ndiscussed. The total number of e cho signals predicted t heoretically and  confirmed experimentally is \nthirteen.  \nSpin echo signals formed at excit ation by four short RF pulses coinciding with fronts of two wide \nRF pulses were generated as analogs of multiple echo signals. Experiments were carried out in lithium ferrit e where the RF pulse front distor tions are insignificant and compared with ones in \ncobalt where the contribution of distortion mechanism in our experimental conditions is significant . \nFor this reason lithium ferrite is appropria te material for the experimental verification of theoretical \nresults in experimental conditions most close to the theoretical model.  \nAcknowledgments  \nThe work is supported by the Georgian National Science Foundation N GNSF/ST07/7- 248 Grant.  \nReferences  \n1. M.I. Kurkin and E.A.  Turov. NMR in magnetically ordered substances  and its application . \nMoscow .: Nauka , 1990  (244 p.) . \n2. W.B.  Mims. Spin echoes from broad resonance lines with high tuning angles. Phys. Rev. 1966, v.  \n141, N  2, p.499 –  502. \n3. R.W.N. Kinnear , S.J. Campbell, D.H.  Chaplin. Structure in nuclear spin echoes from \nferromagnets.  Phys . Lett. 1980, v . А76, N 3-4, p.311-314. \n4. A.E. Reingardt , V.I. Tsifrinovich , O.V.  Novoselov and V.K.  Mal'tsev . Multiple echo in \nferromagnetic cobalt -bearing alloys.  Fiz. Tverd. Tela, 1983, v. 25, pp. 3163- 3164. [ Phys. Solid \nState 1983, v. 25, pp. 1823- 1824] . \n5. I.G. Kiliptari. Multiple echo in magnets at unequal durations of exciting pulses. Fiz. Tverd. Tela, \n1992, v. 34, N5, pp. 1418- 1424 [Sov. Phys. Solid State 1992, v.  34, N8, 1346- 1352)].  \n6. L.N. Shakhmuratova, D.K.  Fowler, D.H.  Chaplin.  Multistructural character of nuclear spin echo \nsignals formed from multidomain ferromagnets: theory and experiment.  Journal of Magnetism \nand Magnetic Materials (MMM), 1998, v. 177- 181, pp. 1476- 1477.   \n16 \n  \n7. V.S. Kuz'min,  V.M.  Kolesenko , and E.P.  Borbotko .\n8. A.M. Akhalkatsi , G.I.  Two-pulsed nuclear echo signal in \nmagnetically ordered media. Fiz. Tverd. Tela, 2008, v. 50, N11, pp. 2043- 2049 [ Sov. Phys. \nSolid State, 2008, v.50, N11, pp. 2133 -2140] . \nMamniashvili , Z.G. Shermadini, T.G.  Gavasheli, T.O.  Gegechkori, \nW.G.  Clark . Multip le NMR spin echoes in magnets: t he echo structure and potential \napplications. / Journal of Appli ed Physics, 2009, v.105, N7, p. 07D303- 1 - 07D303- 3. \n9. A.M. Akhalkatsi , G.I. Mamniashvili. The si gnals of spin nuclear  echo upon a  combined  action of \nmagnetic and radiofrequency pulses in magnets .Phys. Met. Metallogr ,1998,v .86, №  5, 461 -463. \n10. A.M. Akhalkatsi , M.D. Zviadadze , G.I. Mamniashvili , N.M. Sozashvili , A.N. Pogorely, \nO.M. Kuzmak . Multipulse analogs of single -pulse echo signals in multidomain magnetics. Phys. \nMet. Metallogr , 2004, v. 98, № 3, 252 -260. \n11.L.L. Buishvili, E.B.  Volzhan, and K.O.  Khutsishvili . Quantum statistical theory of a one -pulse \necho. Fiz. Tverd. Tela, 1982, v. 24, N10, pp. 3184 -3186 [ Sov. Phys. Solid State , 1982, v.24, pp. \n1808- 1810].  \n12. A. Loesche. Nuclear Induction . Moscow, 1963  [A. Lösche , Kerninduktion , Deutscher Verlag \nder Wissenschaften, Berlin, 1957].  \n13. Т.Р. Das, D.K. Roy. Spin echoes with four pulses –  an extension to n pulses. Phys. Rev. 1955,  \nv. 98. p. 525 – 531. \n14. A.M.  Akhalkatsi, G.I. Mamniashvili, T.O.  Gegechkori, S.  Ben-Ezra. On formation mechanism \nof 57Fe single -pulse echo in lithium ferrite . Phys. Met. Metallogr,  2002, v.94, N1,  pp.33 -40. \n15. G.J.B . Crowford. Spin echoes with four pulses. Phys. Rev. 1955, v.99, pp. 600. \n16. V. P. Chekmarev , M. I. Kurkin, and S. I. Goloshchapov , Mechanism of formation of single -\npulse echo in Hahn spin systems. Zh. Eksp. Teor. Fiz., 76 (1979), 1675; [Sov. Phys. JETP 1979, \nv.49, N 5, p.851 -855]. "    },    {        "title": "1703.07533v1.Fabrication_and_magnetic_control_of_Y3Fe5O12_cantilevers.pdf",        "content": "arXiv:1703.07533v1  [cond-mat.mes-hall]  22 Mar 2017Fabrication and magnetic control of Y 3Fe5O12cantilevers\nYong-Jun Seo1,2,∗Kazuya Harii1,3, Ryo Takahashi1,3, Hiroyuki Chudo1,3, Koichi\nOyanagi4, Zhiyong Qiu2, Takahito Ono5, Yuki Shiomi1,4, and Eiji Saitoh1,2,3,4\n1Spin Quantum Rectification Project, ERATO, Japan Science an d Technology Agency, Aoba-ku, Sendai 980-8577, Japan\n2WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan\n3Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan\n4Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan and\n5Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan\nWe have fabricated ferrite cantilevers in which their vibra tional properties can be controlled by\nexternal magnetic fields. Submicron-scale cantilever stru ctures were made from Y 3Fe5O12(YIG)\nfilms by physical etching combined with use of a focused ion be am milling technique. We found that\nthe cantilevers exhibit two resonance modes which correspo nd to horizontal and vertical vibrations.\nUnder external magnetic fields, the resonance frequency of t he horizontal mode increases, while that\nof the vertical mode decreases, quantitatively consistent with our numerical simulation for magnetic\nforces. The changes in resonance frequencies with magnetic fields reach a few percent, showing that\nefficient magnetic control of resonance frequencies was achi eved.\nSpin mechanics [1], which explores interplay between\nmagnetism and mechanical motion, is a young research\nfield emerging along with the advance in spintronics [2].\nClassical examples of such phenomena are the Einstein-\nde Haas effect [3] and its inverse effect, the Barnett effect\n[4]. In the Einstein-de Haas effect, mechanical rotation is\ninduced by transfer of angular momentum from magne-\ntization to mechanical ones. To detect mechanical effects\ninduced by spins, a cantilever structure provides one of\nthe most suitable tools [5–10]. A cantileveris a long rigid\nplate of which one end is supported tightly but the other\nend can mount a load. Because of their high sensitivity\n[11, 12], cheapness, and ease of fabrication in large areas,\ncantilever structures have been essential in spin mechan-\nics [1, 5–10].\nIn commercial devices e.g.micro-electro-mechanical\nsystems (MEMS), cantilevers are mostly fabricated on\nsilicon wafers. Silicon is the most common semiconduct-\ning material on the earth, and widely used as a base\nmaterial in the semiconductor industry. Silicon can-\ntilevers thereby have great advantage since nanofabrica-\ntion techniques developed in the semiconductor industry\ncan be harnessed effectively. However, recent develop-\nment in state-of-the-art nanofabrication techniques such\nas a focused ion beam (FIB) method enables wide ma-\nterial choice as ingredients of cantilevers, such as mag-\nnetic, piezoelectric, and ferroelectric materials. Can-\ntilevers made of such functional materials are promising\nfor exploration of new features in minute cantilever de-\nvices.\nIn this study, we have fabricated ferrimagnetic can-\ntilevers using garnet ferrite Y 3Fe5O12(YIG). YIG is\na typical magnetic insulator [13–18] with excellent mi-\ncrowave properties, and thus has widely been used in\nmagnonicsand spintronics fields [2]. However,direct fab-\nrication of YIG cantilevers has not been reported yet,\n∗Electronic address: seo@imr.tohoku.ac.jpalthough magnetic control of cantilever properties is ex-\npected owing to the strong spontaneous magnetization of\nYIG. In addition to functionality asmagnetic cantilevers,\na marriage between MEMS technology and spintronics\nwill acceralate the study of spin mechanics. As shown in\nthe following, we successfully fabricated a YIG cantilever\nwith a Pt mirror in situusing an FIB milling technique,\nand demonstrated efficient control of resonant frequen-\ncies by using small external magnetic fields.\nFigure 1 shows the fabrication process of our YIG can-\ntilever. A YIG cantilever with a Pt mirror was fab-\nricated using a dual beam FIB/SEM system (Versa3D\nDualBeam; FEI Company). The starting material is a\nYIG film with 3 µm thickness grown on a gadolinium\ngallium garnet (GGG) substrate. A cantilever structure\nwas patterned by the FIB milling, as shown in Fig. 1(b).\nThe depth of the milling was about 6 µm, which is much\ngreater than the thickness of the YIG layer. In order\nto improve the reflectivity of the laser light used in the\nDoppler vibrometry, a Pt film was deposited on the head\nof the cantilever in situusing the FIB deposition, as\nshown in Fig. 1(c). Then, the base of the cantilever was\nmilled away by the FIB milling at the angle of 38 degrees\nfrom the film plane, as shown in Fig. 1(d). This process\nwas repeated for the other side (Fig. 1(d)), and then\nthe YIG cantilever structure was obtained. A fabricated\nYIG cantilever with a Pt mirror is shown in Fig. 1(e);\nthe size is 0 .8µm in width, 0 .9µm in thickness, and 80\nµm in length. The cantilever is not completely symmet-\nric, as shown in a cross-sectional image in Fig. 1(e). For\ncomparison, non-magnetic Gd 3Ga5O12cantilevers were\nfabricated using the same method.\nVibration spectra of the fabricated cantilevers in the\ndirection normal to the cantilever ( zaxis in Fig. 2(a))\nwere measured with a laser Doppler vibrometer (MSA-\n100-3D;PolytecInc.) atroomtemperature, asillustrated\nin Fig. 2(a). Here, the cantilever vibration is mainly\ndriven by thermal energy of the cantilever, but other\nminor mechanisms such as residual vibrational/acoustic\nexcitation and electrical noise may exist. The measure-2\nFIG. 1: (a)-(d) Schematic illustrations of the fabrication pro-\ncess of YIG cantilever. (e) SEM images of the fabricated\nYIG cantilever. Cross sectional image at the head part is als o\nshown in (e).\nment was performed in a high vacuum of 10−4Pa to im-\nprove sensitivity. External magnetic fields were applied\nto the cantilever samples using electromagnets along the\nperpendicular direction to the cantilever within the film\nplane (xaxis in Fig. 2(a)).\nFigure 2(b) shows the frequency dependence of dis-\nplacement, D, of the YIG cantilever. In the frequency\nrange from 60 to 80 kHz, two sharp peaks are observed;\nthe frequencies of the peaks are 64 .656 kHz and 72 .516\nkHz. From a numerical simulation using COMSOL Mul-\ntiphysics software [19], we assigned these peaks as res-\nonance modes of the cantilever. The lower resonance\nfrequency (64 .656 kHz) corresponds to the horizontal vi-\nbrational mode, while the higher one (72 .516 kHz) the\nvertical vibrational mode. Though the laser beam was\nset to be perpendicular to the film plane [Fig. 2(a)],\nsmall distortion in cantilever shape enables the detection\nof the horizontal vibration mode. In commercial can-\ntilevers, since the cantilever thickness is much less than\nthe cantileverwidth, the resonancefrequencies ofthe two\nmodes are significantly different; it is noted that the res-\nonance frequency in cantilevers is known to be propor-\ntional to the cantilever thickness. In contrast, since the\nwidth and thickness of our YIG cantilever are similar,\nboth the horizontal and vertical modes were observed in\nFIG. 2: (a) A schematic illustration of measurement setup.\n(b)Frequencydependenceof displacement measured along th e\ndirection normal to the film plane (denoted by D) for the YIG\ncantilever. In this frequency range, two resonance modes,\nhorizontal and vertical modes, are observed as sharp peaks.\nthe similar frequency range. This argument is supported\nbythe factthat the ratioofthetworesonancefrequencies\n(= 72.516kHz/64.656kHz) almost coincides with that of\nthecantileverthicknesstothewidth(= 900nm /800nm).\nFrom the displacement measurement without applying\nmagnetic fields shown in Fig. 2, the quality factor ( Q)\nof the YIG cantilever was estimated to be 1000. Using a\nrelation of the minimum detectable force [22, 23]\nδFmin=/radicalBigg\n4kkBT\n2πf0Q, (1)\nthe minimum detectable force from the cantilever size is\nestimated to be 5 ×10−16N for the horizontal and the\nvertical resonance mode. Here, k,kB,T, andf0are the\nspring constant, the Boltzmann constant, the cantilever\ntemperature, and the resonance frequency, respectively.\nThespringconstant koftheYIGcantileverisdetermined\nusing the relation of k=EI/L3, whereEis The Young’s\nmodulus of YIG [20, 21], Iis momentum of inertia simu-\nlated from the cantilever dimension using the COMSOL\nsoftware[19], and Lis length of the cantilever. The value\nofkis calculated to be 6 mN/m. The obtained min-\nimum detectable force shows that highly sensitive YIG\ncantilevers which can detect forces as small as 100 aN\n(= 10−16N) were fabricated. This minimum detectable\nforce is much less than that used commercially in the\natomic force microscopy (AFM).\nMagnetic field dependence of the resonance frequen-\ncies is shown in Fig. 3. With increasing magnetic-field\nstrength, theresonancefrequenciesofboththe horizontal\nand vertical modes change clearly in Figs. 3(a) and 3(b),\nalthough the peak shape (i.e. the Qfactor) is almost\nconstant with magnetic fields. As shown in Fig. 3(a),\nthe resonance frequency of the horizontal mode steeply\nincreases, as the magnetic field is raised from 0 G to 390\nG. Above 390 G, the increase in the resonance frequency\nwith magnetic fields tends to be almost saturated. At\n1060 G, the resonance frequency is 66 .753 kHz, higher3\nFIG. 3: Frequency dependence of the displacement (denoted\nbyD) measured in the frequency range around the resonance\nfrequencies of (a) horizontal and (b) vertical modes in the\nYIG cantilever, and (c) horizontal and (d) vertical modes in a\nnon-magnetic GGG cantilever. (e) The frequency shifts with\nmagnetic fields are plotted as a function of external mag-\nnetic fields for the YIG and GGG cantilevers. Fits to the\nexperimental results for the YIG cantilever (solid and dash ed\ncurves) are also shown.\nthan that in zero magnetic field by about 2 kHz. Also\nfor the vertical mode, the similar strong magnetic field\ndependence is observed especially at low magnetic fields,\nas shown in Fig. 3(b). However, on the contrary to the\nhorizontal mode, the resonance frequency of the vertical\nmode decreases with increasing magnetic fields. Hence,\nthe clear difference in the response to magnetic fields is\nobserved between the two modes.\nThe magnetic field dependence of the frequency shifts\nis summarized in Fig. 3(e). The shifts are observed also\ninnegativemagneticfieldsfortheYIGcantilever,andthe\nmagnetic field dependence is clearly even with respect\nto magnetic fields for both the horizontal and vertical\nmodes. The maximal magnitudes of the frequency shifts\nreachafew percent, indicatingthatthe efficientmagnetic\ncontrol of the resonance frequencies is achieved by small\nmagnetic fields.\nTo examine the origin of the frequency shifts,\nwe performed similar experiments for a non-magnetic\nGd3Ga5O12cantilever. The frequency dependence of D\nfor the GGG cantilever is shown in Figs. 3(c) and 3(d).\nAlso for the GGG cantilever, the resonance modes corre-\nsponding to horizontal and vertical modes are observed;\nthe resonance frequencies are different from those for the\nYIG cantilever because the cantilever sizes are a little bit\ndifferent. As shown in Figs. 3(c) and 3(d), under exter-\nnal magnetic fields up to 1000 G, no frequency shifts are\nobserved either for the horizontal or vertical mode in the\nGGG cantilever. This results show that the frequency\nshifts observed in the YIG cantilever are related to the\nspontaneous magnetization in YIG.\nAs an origin of the frequency shifts in the YIG can-\ntilever, let us first consider magnetostriction effects un-\nder external magnetic fields. In the YIG cantilever, the\nFIG. 4: (a) Contour plot of stray-magnetic-field profile sim-\nulated around the YIG cantilever. The x-component of the\nstray field at the point [ x,z], (Hx[x,z]) is mapped. The\nsimulation was performed in the cross section around the\nhead part of the YIG cantilever under B= 1000 G, where\nBis a unidirectional external magnetic field applied along\nthex-axis. (b),(c) The spatial change in the stray field\n∆Hx≡Hx[x,z]−Hx[0,0] (b) in the xdirection at z= 0\n(∆Hx[x,0] =Hx[x,0]−Hx[0,0]) and (c) in the zdirection at\nx= 0 (∆Hx[0,z] =Hx[0,z]−Hx[0,0]). The point at [0, 0] is\nset at the center of the YIG cantilever (see (a)). Hx[0, 0] are\n7.70 Oe, 101 Oe, and 1030 Oe under B= 10 G, 100 G, and\n1000 G, respectively.\nmagnetostrictioneffect might affect the vibrationalprop-\nerties. However, the megnetostriction coefficient for YIG\nis as small as 10−4% [24, 25], and thus the possible fre-\nquencyshiftsduetomagnetostrictioneffectsareexpected\nto be much smaller than the observed shifts (100%).\nTherefore, the magnetostriction in the YIG cantilever is\nnot likely to explain the large frequency shifts observed\nin Figs. 3(a) and (b).\nSince the fabricated YIG cantilever is surrounded by\nthe YIG film, spacial gradients of the stray fields around\nthe cantilever should affect the vibrational properties\nthrough the magnetic force gradients. When the magne-\ntization of the YIG cantilever is uniformly aligned in the\nmagnetic-field direction (defined as the xdirection), the\nmagneticforcegradientforthe idirection( i=x,y,z)[26]\nis given by\n∂FMag\n∂i=∂2(/vectorM·/vectorH)\n∂2i=Mx∂2Hx\n∂2i≡keff,(2)\nwhere/vectorMis the magnetization and /vectorHis the magnetic\nfield. Owing to this change in the effective spring con-\nstantkeffofthe cantileverbythe magneticforcegradient,4\nthe frequency shift [27] is expected to be observed, i.e.\n∆f=1\n2π/parenleftBig/radicalbigg\nk+keff\nmeff−/radicalbigg\nk\nmeff/parenrightBig\n. (3)\nHere,kis the spring constant in zero magnetic field and\nmeffis the effective mass of the cantilever. As shown\nin eq. (2) and eq. (3), the frequency shift due to the\nmagnetic force gradient is caused by the changes in Mx\nand∂2Hx\n∂i2with magnetic fields.\nWe performed a numerical simulation for the mag-\nnetic force gradient using COMSOL Multiphysics soft-\nware. Figure 4(a) shows a contour plot of the simulated\nmagnetic-field profile in the cross section around the tip\nof the YIG cantilever. Because of the magnetization in\nthe YIG film surrounding the cantilever, the stray field\naround the cantilever has spatial gradients even under\nthe unidirectional magnetic field. Besides, with increas-\ning magnetic field strength, the stray field along the x\ndirection ( Hx) increases owing to the strong magnetiza-\ntion of the surrounding YIG film. Here, the zaxis is\ndefined as the direction normal to the cantilever, and the\nmagnetic field is applied along the xaxis. The horizontal\nand vertical vibrations depend on Mx∂2Hx\n∂x2andMx∂2Hx\n∂z2,\nrespectively. In the horizontal ( x) direction, the sign of\nthe stray-field curvature∂2Hx\n∂x2is positive, and its mag-\nnitude increases with increasing external magnetic fields,\nas shown in Fig. 4(b). Thus, accordingto eq. (2) and eq.\n(3), the positive keffresults in the valley-like magnetic-\nfield dependence of the frequency shift, as shown in Fig.3(e). In contrast, in the vertical ( z) direction, the mag-\nnetic field Hxis strongest at [ x,z] = [0,0]. In this case,\nas shown in Fig. 4(c),∂2Hx\n∂z2is negative and decrease with\nincreasing magnetic fields, which turns out to give rise to\nthe peak-like magnetic-field dependence of the frequency\nshift shown in Fig. 3(e). The full calculation of the mag-\nnetic field dependences of Mxand∂2Hx\n∂i2(i=x,z) quan-\ntitatively explains the magnitudes and the signs of the\nfrequency shifts for the horizontal and vertical modes, as\nindicated by the solid and dashed curves, respectively, in\nFig. 3(e). Hence, the large frequency shifts observed in\nthe YIG cantilever induced by the magnetic fields can be\nexplained by magnetic force gradients produced by the\nsurrounding YIG film.\nIn summary, we have reported on the fabrication and\nthe magnetic control of YIG cantilevers. Under the ex-\nternal magnetic fields, the frequencies of the two res-\nonance modes of the cantilever are shifted clearly; the\nshifts at 1000 G reach a few percent. The efficient mag-\nnetic control of the resonance frequency is well explained\nby magnetic force gradients from the surrounding YIG\nfilm. Since YIG has been typically used for study of var-\nious spin current phenomena, the YIG cantilever would\nbe useful for the mechanical detection of spin currents.\nWe thank S. Maekawa, M. Ono, M. Matsuo, Y.\nOikawa, and T. Hioki for fruitful discussions. This work\nwas supported by ERATO, Spin Quantum Rectification\nProject.\n[1] J.E. LosbyandM. R.Freeman, arXiv:1601.00674 (2016).\n[2] S.Maekawa, S.O.Valenzuela, E.Saitoh, T.Kimura, Spin\nCurrent, Oxford University Press, (2012).\n[3] A. Einstein and W. J. de Haas, Royal Neth. Acad. Arts\nSciences (KNAW) 18, 696 (1915).\n[4] S. J. Barnett, Phys. Rev. 6, 239 (1915).\n[5] T. M. Wallis, J. Moreland, and P. Kabos, Appl. Phys.\nLett.89, 122502 (2006).\n[6] P. Mohanty, G. Zolfagharkhani, S. Kettemann, and P.\nFulde, Phys. Rev. B 70, 195301 (2004).\n[7] G. Zolfagharkhani, A. Gaidarzhy, P. Degiovanni, S. Ket-\ntemann, P. Fulde, and P. Mohanty, Nature Nanotech. 3,\n720-723 (2008).\n[8] J. A. Boales, C. T. Boone, and P. Mohanty, Phys. Rev.\nB93, 161414(R) (2016).\n[9] J. E. Losby, F. Fani Sani, D. T. Grandmont, Z. Diao, M.\nBelov, J. A. J. Burgess, S. R. Compton, W. K. Hiebert,\nD. Vick, K. Mohammad, E. Salimi, G. E. Bridges, D. J.\nThomson, M. R. Freeman, Science 350, 798 (2015).\n[10] M. Wu, N. L.-Y. Wu, T. Firdous, F. F. Sani, J. E. Losby,\nM. R. Freeman, and P. E. Barclay, Nature Nanotech.\n(2016).\n[11] T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin,\nK. Wago, and D. Rugar, Appl. Phys. Lett. 71, 288-290\n(1997).\n[12] J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali,and A. Bachtold, Nature Nanotech. 7, 301-304 (2012).\n[13] Z. Zhang, P. C. Hammel, and P. E. Wigen, Appl. Phys.\nLett.68, 2005 (1996).\n[14] K. Wago, D. Botkin, C. S. Yannoni, and D. Rugar, Appl.\nPhys. Lett. 72, 2757 (1998).\n[15] M. M. Midzor, P. E. Wigen, D. Pelekhov, W. Chen, P.\nC. Hammel, and M. L. Roukes, J. Appl. Phys. 87, 6493\n(2000).\n[16] V. Charbois, V. V. Naletov, J. B. Youssef, and O. Klein,\nAppl. Phys. Lett. 80, 4795 (2002).\n[17] V. V. Naletov, V. Charbois, O. Klein, and C. Fermon,\nAppl. Phys. Lett. 83, 3132 (2003).\n[18] O. Klein, V. Charbois, V. V. Naletov, and C. Fermon,\nPhys. Rev. B 67, 220407 (2003).\n[19] https://www.comsol.com/\n[20] D. F. Gibbons and V. G. Chirba, Phys. Rev. 110, 770-\n771 (1958).\n[21] H. M. Chou and E. D. Case, Materials Science and En-\ngineering, 100, 7-14 (1988).\n[22] J. A. Sidles, J. L. Garbini, K. J. Bruland, D. Rugar, O.\nZuger, S. Hoen, and C. S. Yannoni, Rev. Mod. Phys. 67,\n249-268 (1995).\n[23] U. Durig, O. Zuger, and A. Stalder, J. Appl. Phys. 72,\n1778 (1992).\n[24] A. E. Clark, B. DeSavage, W. Coleman, E. R. Callen and\nH. B. Callen, J. Appl. Phys. 34, 1296-1297 (1963).5\n[25] A. B. Smith and R. V. Jones, J. Appl. Phys. 34, 1283-\n1284 (1963).\n[26] K. Babcock and V. Elings, IEEE Trans. Magn., 30, 4503-\n4505 (1994).[27] A. M´ endez-Vilas and J. D´ ıaz, Modern Research and Ed-\nucational Topics in Microscopy , FORMATEX 805-811\n(2007)."    },    {        "title": "1208.2195v2.Resonant_features_of_planar_Faraday_metamaterial_with_high_structural_symmetry.pdf",        "content": "arXiv:1208.2195v2  [physics.optics]  14 Mar 2013EPJ manuscript No.\n(will be inserted by the editor)\nResonant features of planar Faraday metamaterial with high\nstructural symmetry\nStudy of properties of a 4-fold array of planar chiral rosett es placed on a ferrite substrate\nSergey Y. Polevoy1, Sergey L. Prosvirnin2,3, Sergey I. Tarapov1, and Vladimir R. Tuz2,3\n1Usikov Institute of Radiophysics and Electronics, Nationa l Academy of Sciences of Ukraine, 12, Proskura St., Kharkiv 6 1085,\nUkraine\n2Institute of Radioastronomy, National Academy of Sciences of Ukraine, 4, Krasnoznamennaya St., Kharkiv 61002, Ukrain e\n3School of Radio Physics, Karazin Kharkiv National Universi ty, 4, Svobody Square, Kharkiv 61077, Ukraine\nReceived: date / Revised version: date\nAbstract. The transmission of electromagnetic wave through a planar c hiral structure, loaded with the\ngyrotropic medium being under an action of the longitudinal magnetic field, is studied. The frequency\ndependence of the metamaterial resonance and the angle of ro tation of the polarization plane are obtained.\nWe demonstrate both theoretically and experimentally a res onant enhancement of the Faraday rotation.\nThe ranges of frequency and magnetic field strength are define d, where the angle of polarization plane\nrotation for the metamaterial is substantially higher than that one for a single ferrite slab.\n1 Introduction\nIt is known, that bulk chiral artificial structures [1], [2]\nmanifest a reciprocal optical activity. The typical con-\nstructiveobject of3D chiralmedia is aspirallyconducting\ncylinder. The concept of chiralityalsoexists in two dimen-\nsions.Aplanarobjectissaidtobe 2Dchiralifit cannotbe\nsuperimposed on its mirror image unless it is lifted from\nSend offprint requests to :the plane. For instance, an array of metallic rosettes is an\nexampleofsuchanobject. Hetchand Barron[3,4], Arnaut\nand Davis [5,6] were the first who introduced planar chiral\nstructures into the electromagnetic research. However, 2D\nchirality does not lead to the same electromagnetic effects\nwhich are conventional for 3D chirality and, so, it became\na subject of special intense investigations [7,8,9].\nPlanar chiral materials are quite simple structures in\nmanufacturing. However, in contrast to traditional fre-2 Sergey Y. Polevoy et al.: Resonant features of planar Farad ay metamaterial with high structural symmetry\nquency selective surfaces, they provide an additional twist\nparameter to control electromagnetic properties. Besides,\nin some particular cases, quasi-2D planar chiral metallic\nstructures can be asymmetrically combined with isotropic\nsubstrates to distinct a reciprocal optical response inher-\nent to true 3D chiral structures. In such metamaterials,\nat normal incidence of the exciting wave, an optical ac-\ntivity appears only in the case, when their constituent\nmetallic elements have finite thickness, which provides an\nasymmetric coupling of the fields at the air and substrate\ninterfaces [10].\nFrom the viewpoint of possible applications in micro-\nwave and THz frequency bands, it is known that the thin-\nner metallic elements of planar structures, so they are\neasier in fabrication. Thus, knowledge about optical prop-\nerties of metamaterials based on the thin planar chiral\nstructures are especially important.\nThe results of a detailed study of polarization trans-\nformations caused by an array of the perfectly conducting\ninfinitely thin planar chiral elements are presented in [11].\nIn thiswork,the opticalresponseofplanarchiralmetama-\nterials with four-fold symmetry was studied in the case,\nwhen the arrays are placed on an isotropic dielectric sub-\nstrate. One of the results obtained in this study is an ar-\ngue that the 2D chiral planar structures do not change\nthe polarization state of the normally incident wave in\nthe main diffraction order.This theoreticalconclusionwas\nconfirmedwithnumericaldataobtainedbyasimulationin\nthe case of arrays made of infinitely thin metallic rosettes\nplaced on a dielectric substrate.Frombothfundamentalandapplicationpointsofview,\nthe planar metamaterials placed on a ferrite substrate [12]\nand layered ferrite-dielectric structures [13,14] are quite\ninteresting objects because they can be used successfully\ntodesignnon-reciprocalmagneticallycontrollablemicrowave\ndevices based on the Faraday effect. On the other hand,\nmagneto-opticallyactive substrate can be serve as a sensi-\ntive element for THz magnetic near-field imaging in meta-\nmaterials[15].Thepolarizationrotationofanear-IRprobe\nbeam revealed in the substrate measures the magnetic\nnear-field.\nA general theoretical approach is used in [2] to predict\nelectromagneticpropertiesofuniaxialcompositeswithfour-\nfold inclusions in the form of planar chiral gammadions\ncombinedwithferriteellipsoids.Itneedstwopseudo-vectors\nto describe the system. The first vector is a bias magnetic\nfield and the second one is a vector defining the hand-\nedness of the gammadion shape. They are pseudo-vectors\n(axial vectors) because being time-odd. As a result of the\ntheory, these composite systems are bi-anisotropic non-\nreciprocal media described by specific constitutive equa-\ntions of the same kind as that ones used in the moving\nchiral media.\nHowever, it is necessary to clarify the effect of the par-\nticles handedness (and the corresponding pseudo-vector\nin the theory) on the system properties and the degree\nof reciprocal rotation. As it has been mentioned above, it\nis important at least in the case of metallic planar chiral\nparticles which have small thickness in comparison with\nthe wavelength. The theoretical and experimental studiesSergey Y. Polevoy et al.: Resonant features of planar Farada y metamaterial with high structural symmetry 3\nof the particle handedness effect are extremely important\nin this point and are the subject of the present research.\nThus, the purpose of this paper is to study both the-\noretically and experimentally the resonant properties of\nplanargyrotropicmetamaterials(arraysofmetallicrosettes\nplaced on a ferrite substrate) depending on the value of\nstatic magnetic field strength. The field is applied nor-\nmally to the structure plane, i.e., the systems are con-\nsidered in the Faraday geometry. The periodic cell size\nof the studied metamaterials is chosen in such a way that\nthehigh-qualityfactorresonancesappearin thestructures\nspectra in the millimeter waveband.We consider metama-\nterials based on a 4-fold symmetry array which consisted\nof thin metallic rosettes. As a main result of our study the\nessential resonant enhancement of the Faraday rotation is\ndemonstrated both theoretically and experimentally for\nthe metamaterial. This effect is substantially higher than\nthat one for a single ferrite slab.\n2 Structures under study and theoretical\napproach\nThe metamaterialbeing under investigationis designed as\na layered structure, which consists of a planar chiral peri-\nodic structure placed on a ferrite plane-parallel slab with\nthickness 0.5 mm. The chiral structure is made of fiber-\nglass (ε′= 3.67, tanδ= 0.06) with a thickness 1.5 mm,\none side of which is covered with copper foil. The foil\nside of this layered structure is patterned with a periodic\narray which square unit cell consists of a planar chiralrosette (see Fig. 1). The ferrite slab is leaned to this ar-\nray of metallic elements. Two samples of each kind (i.e.\nright-handed and left-handed elements) of gyrotropic pla-\nnar metamaterial 60 ×60 mm2which are differed by the\nperiod of the rosette arrayhave been performed. Sample 1\nof both right-handed and left-handed kinds has the period\nd= 5 mm and the radius of rosette arcs a= 1.66 mm,\nwhereassample 2 has d= 4 mm and a= 1.33mm, respec-\ntively. The angular size φand the width wof the copper\nstrips which form the rosettes for all samples are identical.\nWe applied the ’resonant model’ of ’saturated’ ferrite\n[16,17] to calculate the ferrite constitutive parameters in\nthe case when the static magnetic field H0is more strong\nthan the field of the saturation magnetization 4 πMS, and\nthe ’non-resonant model’ of ’non-saturated’ ferrite [18,19]\nif the fieldH0is less than 4 πMS.\nWhen the field strength is larger then 4 πMSwe use\ncommon expressions for permittivity and permeability for\nz-axis biased ferrite [16,17], assuming the ferrite material\nis magnetically saturated and taking into account the di-\nelectric and magnetic losses\nεf=ε,ˆµf=\nµ iβ0\n−iβ µ0\n0 0µz\n, (1)\nwhere\nµ= 1+4π(χ′−iχ′′), β= 4π(K′−iK′′), µz= 1,(2)\nχ′=ω0ωm[ω2\n0−ω2(1−α2)]D−1,\nχ′′=ωωmα[ω2\n0+ω2(1+α2)]D−1,(3)4 Sergey Y. Polevoy et al.: Resonant features of planar Farad ay metamaterial with high structural symmetry\nFig. 1.(Color online) The periodic array of planar chiral\nelements placed on a dielectric substrate: (a) the photo;\n(b) the square unit cell of the periodic array ( dis the\nperiod of the structure) with a metallic element shaped as\nthe planar chiral right-handed rosette ( ais the radius of\narc,φ= 120 deg is its angular size, w= 0.267 mm is the\nwidth of copper strips which form the rosette).\nK′=ωωm[ω2\n0−ω2(1+α2)]D−1,\nK′′= 2ω2ω0ωmαD−1,(4)\nD= [ω2\n0−ω2(1+α2)]2+4ω2\n0ω2α2,\nωm=γ4πMS,(5)\nω0is the frequency of ferromagnetic resonance (FMR),\nαis the dimensionless damping constant, γis the gyro-\nmagnetic ratio, MSis the saturation magnetization. Weuse the Gaussian system of units. The ferrite material of\nbrandL14Hischaracterizedbythe followingsetofparam-\neters:ε= 13.2−i0.0697,α= 0.0285,ωm/2π= 14.2 GHz.\nThevalueωmcorrespondstothesaturationmagnetization\nfield of 4πMS= 4800 Oe.\nWhen the field strength H0is smaller than 4 πMS, the\nexperiment can be well described using the non-resonant\n’non-saturated’ ferrite model [18,19]. Let us note that in\nthe non-saturated model, the current magnetization M\nis a function of the static magnetic field strength M=\nM(H0). The elements of the tensor ˆ µf(1) are represented\nby empirical expressions [19]:\nµ=µdem+(1−µdem)(M/MS)3/2,\nµz= (µdem)P, P= (1−M/MS)5/2,\nβ=−γ4πM/ω, µ′′=µ′′\nz=β′′= 0,(6)\nwhereµdemis the permeability of completely demagne-\ntized ferrite, which properties can be calculated using the\ntwo-domainmodel [18] forfrequencies ω>γ(Hr+4πMS):\nµdem=1\n3+2\n3/radicaligg\n(ω/γ)2−(Hr+4πMS)2\n(ω/γ)2−H2r,(7)\nwhereHris the strength of field matched to the rema-\nnent magnetization. For the used ferrite brand, it is Hr=\n3500 Oe. The dependence of the components of the per-\nmeability tensor of ferrite versus the static magnetic field\nstrength are presented in Fig. 2 for the frequency f=\nω/2π= 30 GHz.\nForathinferriteslabmagnetizednormallytoitsplane,\nthe FMR frequency ω0is defined by the well-known for-\nmula [17]:\nω0=γ|H0−4πMS|. (8)Sergey Y. Polevoy et al.: Resonant features of planar Farada y metamaterial with high structural symmetry 5\nFig. 2.(Color online) (a) Theoretical dependences of the\ncomponents of permeability tensor for the thin ferrite slab\nversus the normally applied static magnetic field at f=\n30GHz;(b)thesamedependencesdetailedforsmallstatic\nfields by ’non-resonant’ ferrite model.\nThe dependence of FMR frequency versus the static mag-\nnetic field strength is shown in Fig. 4. Note that the for-\nmula (8) is rigorous when the field strength H0is larger\nthan 4πMS. When the field strength is less than 4 πMS,\nthe frequency of FMR may be somewhat lower due to the\nfact that the ferrite changes in the multidomain state and\na violation of its magnetic order grows as the static field\nstrength decreases (see the dashed line in Fig. 4). On the\nsame reason, the FMR linewidth should grow as the field\nstrength decreases.\nAs the field strength decreases below 4 πMS, the do-\nmain structure appears in the ferrite and its magnetic\nstate demonstrates a certain disorder. Note that in this\ncase, the values of the diagonal components of the ˆ µf, i.e.\nthe valueµ, tends to permeability of completely demag-\nnetized ferrite µdem(7). This value is not equal to zero\n(Fig. 2b). The latter is reasonable, because when domains\ndisorder, then their contribution to the integral magneti-zation decreases. However, the magnetization of each do-\nmain is a positive value, in spite of the external field is\ndirected along the domain magnetic moment or against\nit. Contributions to the diagonal components µfrom all\ndomains are added and it tends to some constant when\nthe field strength decreases. A quite different behavior is\nobserved for the off-diagonal component β. As the field\nstrength decreases, the domains, which magnetic moment\nis directed along the external field, and domains, which\nmagnetic moment is directed opposite to the field, give\na different sign for the contribution to the β(the non-\nreciprocal Faraday effect). Thus, contributions of all do-\nmains to the off-diagonal components βare subtracted\nandβtends to zero as the field strength decreases. Note\nthat when the field strength is less than 4 πMS, the cor-\nrect count of the magnetic disorder of domain structure\nin the ferrite should lead to the gradual change of the\ncomponents µandβ.\nThe fields, intensities, and polarization characteristics\nof the electromagnetic waves diffracted by the array of\nrosette-shapedelementswerecalculatedusingthefullwave\nmethoddescribedearlierin[12].Thisapproachisbasedon\nthe method of moments for solution of the vector integral\nequation for surface currents induced by the electromag-\nnetic field on the array elements [20]. The last ones are\nassumed to be perfectly conducting and infinitely thin.\nThe equation was derived with boundary conditions that\ndemand a zero value for the tangential component of the\nelectric field on metal strips. In our calculations, we used6 Sergey Y. Polevoy et al.: Resonant features of planar Farad ay metamaterial with high structural symmetry\nthe Fourier transformations of fields and surface current\ndistributions.\n3 Experiment and data analysis\nThe experimental setup [14] consists of the structure un-\nder study, which is placed between two matching rect-\nangular horns (transmitting and receiving ones) fitted to\nthe Vector Network Analyzer Agilent N5230A. Horns are\nsituated on the axis passed normally to the plane of the\nstructure (Fig. 3a). Using the Network Analyzer the S-\nparameters, namely S21- the transmission coefficient for\nthe structure in the frequency range 22-40 GHz, can be\ndetected and analyzed by the special computer software.\nFor measurements in a longitudinal static magnetic\nfield, the structure and horns are positioned between the\npoles of the electromagnet to provide the orientation of\nthe components of electromagnetic field ( E,H) and static\nfield (H0) as it is shown in Fig. 3b. The electromagnet\npoles have axial holes, that allow one to place horns inside\nthe magnetic system. The poles diameter is 120 mm and\nthe distance between them is less than 30-90 mm. Note\nthat due to such sufficiently large poles diameter, the in-\nhomogeneity of the static magnetic field in the structure\narea does not exceed 3-5 %, which is quite enough to pro-\nvide experiments with high quality. The static magnetic\nfield strength is controlledbya computer. Amoredetailed\ntechnique of such a kind fully automated experiment one\ncan find in [14].\nFirst of all, let us mention that the experimental study\nof transmission of normally incident wave through twoFig. 3. (Color online) Experimental setup: (a) the\noverview; (b) the scheme of experiment.\nkinds of planar chiral arrays differed by sign of chiral-\nity was carried out in both cases of free standing arrays\nand arrays placed on ferrite substrate. It was shown that\nthere is not any difference in the intensity of transmitted\nfield and polarization transformations obtained for these\ntwo samples. Thus the experimental evidence of indistin-\nguishabilityofthese propertieshasbeen demonstrated be-\ntweentwoenantiomorphouskinds ofplanarchiralsamples\nconsisted of right-handed and left-handed thin metallic\nrosettes in the case of normally incident wave. This prop-\nerty was argued theoretically before in [11,12].\nThus, at the normal incidence of the exciting wave,\nthe complex layered structure being a thin planar chi-Sergey Y. Polevoy et al.: Resonant features of planar Farada y metamaterial with high structural symmetry 7\nral metallic array placed on the normally magnetized fer-\nrite substrate (or the isotropic dielectric substrate) does\nnot manifest any appearance of the property related to\n3D chiral objects. It is an impressive observation because\nthe symmetry is broken in the direction orthogonal to the\nstructure plane and we deal with the object which has a\nvolume chiral geometry. The reason is in a very small dif-\nference between the fields existed on the array interfaces\nwith free space and the substrate in the case, when the\nconsidered array has a small thickness in comparison with\na wavelength.Afinite thickness ofmetallic elements ofthe\narray is a prerequisite to make asymmetrically coupling\nfields at the air and substrate interfaces and to observe an\neffect of volume chirality of such structure [10].\nOnthebasisofthetheoreticalapproachdescribedabove,\nwe have defined numerically the transmission spectrum of\nthe structure under study. The characteristic frequency\nranges where the transmission demonstrates a minimum\nand the resonant behavior exists (the metamaterial reso-\nnancedip frequency fr) wasdetermined. Theseresonances\nare caused by metallic elements of the structure. In the\ncase of linearly y-polarized normally incident plane elec-\ntromagneticwave,the dependence of fronthe staticmag-\nnetic field strength has been calculated for two values of\nthe planar chiral structure period d(see Fig. 4). Besides\nthat, the dependence of the FMR frequency on the static\nmagnetic field strength for the thin ferrite slab used in ex-\nperiments ( f0(H0) =ω0/2π) is plotted in the same figure.\nOne can see that: (i) the variation of the metamate-\nrial resonant dip frequency ( dfr/dH0) is as stronger asFig. 4. (Color online) Theoretical dependence of the\nmetamaterial resonance dip frequency on the static mag-\nnetic field strength for two values of period of the planar\nchiral structure. The solid line denotes the dependence of\nFMR frequency of the ferrite on the static field according\nto the expression (8). The same dependence but corrected\nin the region of small field is presented by the dashed line.\nthe frequency of this resonance is closer to the FMR fre-\nquencyf0. This fact is caused, obviously, that near the\nFMR the value of the real part of the diagonal compo-\nnentsofthepermeability µconsiderablyincreases.Inturn,\nµis uniquely connected with the value of the resonant fre-\nquency related to array; (ii) in the range of magnetic field\nstrength from 12500Oe up to 15000Oe, tworesonant dips\n(i.e. two values of resonant frequency for the same value of\nthe magnetic field strength) are observed. Such scenario is\ncausedbytheeffect ofresonancenotonlydiagonalcompo-\nnents of the permeability but off-diagonal ones as well. In\nparticular, it is known [16,17], that in the vicinity of FMR\nfrequency, the eigenwave propagation constant of the lon-\ngitudinally magnetized ferrite can acquire more than one8 Sergey Y. Polevoy et al.: Resonant features of planar Farad ay metamaterial with high structural symmetry\nvalue (in the given case, it is two). To be specific, let us\ncall the area, where the resonant frequency of array and\nFMR frequency are close enough to each other as an ’in-\nteraction area’; (iii) as the structure period increases, the\nresonant frequency of response dips decreases.\nComparison of experimental data and theoretical con-\nclusions has been made in the field range 0-6500 Oe. In\nparticular, the qualitative agreement between experimen-\ntal and calculated data for the dependence of metama-\nterial resonance dip frequency fron the magnetic field\nstrength (for the d= 5 mm) is revealed(Fig. 5). When the\nmagnetic field strength exceeds the value corresponding\nto the saturation magnetization field (4 πMS= 4800 Oe),\nthe derivation dfr/dH0changes sign. It is related to the\nmentioned above effect, namely the presence of low-field\nmode(with df0/dH0<0)intheFMRspectrum[16],when\nthe field strength is less than 4 πMS. However, as it was\nexpected, the slope of the experimental frequency depen-\ndence of the metamaterial resonance dip on the magnetic\nfieldstrengthisabitsmallerthanthatonepredictedinthe\ntheory. This difference can be explained by the fact that\nthe magnetically disordered domains appear in the struc-\nture. The maximal value offrequencyshift ofthe metama-\nterial resonance on the magnetic field strength (triangle\nmarkers in Fig. 5) is about 900 MHz. The origin of the\ndivergence between theoretical and experimental data is\nnon-equality of actual and theoretical values of the ferrite\nconstitutive parameters and their frequency dispersion.\nIn order to verify the nonreciprocal properties of the\nmetamaterials under study, the experimental analysis ofFig. 5.(Color online) The dependence of the metamate-\nrial resonance dip frequency on the static magnetic field\nstrength for planar chiral structure d= 5 mm.\nthe electromagnetic wave transmission for the case where\nthe angle between the plane of polarization of transmit-\nting and receiving horn is ψ= 45 deg. It can be seen\n(Fig. 6) that both character and magnitude of the shift\nof metamaterial resonance dip frequency depend strongly\non the static magnetic field direction. Thus, the nonrecip-\nrocal properties of the investigated planar metamaterial\nare demonstrated. Let note, that for ψ= 90 deg this de-\npendence has the symmetric form as was expected. The\nlast observationis yet another proofofan independence of\nthe metamaterial response on the handedness of metallic\nrosettes.\nFor a more detailed study of the polarization proper-\nties of the metamaterial under study we have performed\nthe experimental and numerical analysis of the polariza-\ntion rotation (more exactly, of the rotation of main axis of\nthe polarization ellipse) of the wave transmitted through\nthe structure with respect to the linearly polarized inci-\ndent wave. Theoretical dependences of the angle of po-Sergey Y. Polevoy et al.: Resonant features of planar Farada y metamaterial with high structural symmetry 9\nFig. 6.(Color online) Measured metamaterial resonance\ndip frequency of gyrotropic planar chiral metamaterial\nversus the static magnetic field strength for d= 5 mm\nandψ= 45 deg.\nFig. 7.(Color online) Theoretical dependences of the po-\nlarizationrotationangle oftwo different metamaterialres-\nonant modes versus the static magnetic field strength for\ntwo values of the structure period d.\nlarization rotation θr(H0) on the magnetic field strength\nfor two resonant modes of the metamaterial and for two\nvalues of its period dare shown in Fig. 7.\nThe points marked by squares correspond to the high-\nfrequency modes (hf-modes, located to the left of depen-dencyf0(H0) in Fig. 4), and the points marked by circles\ncorrespond to the low-frequency modes (lf-modes, located\nto the right of dependency f0(H0)). It is easily seen that\nthe structure with a smaller period rotates the plane of\npolarization on the greater angle than the structure with\nthe large period. This may be caused by higher quality\nfactor of resonant modes in the structure with the smaller\nperiod that occurs due to increase of the summary surface\nof metallic elements when the period decreases.\nOne can see while the field strength tends to zero, the\nrotation angle decreases to zero as well for both modes.\nThis fully coincides with used theoretical models of ferrite\npermeability (Fig. 2), where it was shown that the off-\ndiagonal component βwhich is responsible for polariza-\ntion rotation tends to zero as the field strength decreases.\nThis occurs, as mentioned above, due to the compen-\nsation of the effect of multidirectional domains orienta-\ntion on the rotation angle. However, let us note, that\nin the ’interaction area’ (where H0is from 12500 Oe up\nto 15000 Oe) polarization rotation angles increase drasti-\ncally.It can be seen that for high-frequencymodes (square\nmarkers) the maximum of θreachesθr≈ −50 deg. For\nlow-frequencymodes(circlemarkers),thisdependencelooks\nmonotone(underthegivenfieldstrength),andreachesthe\nmaximum values at θr≈50 deg.\nSuch resonant-like behavior of θroccurs obviously in\nthe ’interaction area’ due to increasing the values of off-\ndiagonal components of the ferrite permeability (Fig. 2a)\nin the vicinity of FMR.10 Sergey Y. Polevoy et al.: Resonant features of planar Fara day metamaterial with high structural symmetry\nThe resultsofexperimentalverificationofdependences\nθr(H0) (Fig. 7) and fr(H0) (Fig. 4) are summarized in\nFig. 8. To provide clear demonstration of the effect of ge-\nometrical parameters of the metamaterial under study on\nits polarization properties, experimental data are shown\nfor: (i) the polarization rotation angle θof linearly polar-\nized wavetransmitting through a ferrite slab (Fig. 8a); (ii)\nthe polarizationrotationangle θoflinearlypolarizedwave\ntransmitting through planar chiral structure loaded with\na ferrite slab when the period is chosen to be d= 5 mm\n(Fig. 8b).\nOne can see that the surface plotted for the ferrite slab\n(Fig. 8a) is much smoother than that one for the array\nstructure loaded with ferrite slab (Fig. 8b). The mono-\ntonic growth of θfrom 0 deg to 15 deg with increasing\nfield strength from 0 Oe to 6500 Oe for all frequencies is\noccurredfortheferriteslab.Apresenceofmoderatedipsis\ncaused by the impossibility to provide the perfect match-\ning of elements of the experimental setup. Also, for the\nplanar chiral array loaded with ferrite slab, a monotonic\ngrowth ofθon the field strength takes a place. However,\nnear to the frequency of the metamaterial resonance dip\n(fr= 25.5−26.5 GHz (Fig. 5)), this dependence acquires\na pronounced resonant character, and for θ→θrachieves\nsignificantlyhighervaluesthanthatonefortheferriteslab\n(up toθr≥45 deg).\nIt can be seen that the value θr(Fig. 8b) also depends\non the magnetic field strength, and the maximum of θr\nis observed at H0≈4800 Oe (i.e. in the transition area\nfrom saturated ferrite model to unsaturated one). In thisFig. 8.(Color online) Experimental dependences of the\npolarization rotation angle θas a function of frequency\nand static magnetic field strength for: (a) ferrite slab; (b)\nferrite loaded by planar chiral structure with period d=\n5 mm.\nregion the real part of permeability has extreme (Fig. 2a),\nwhich explains the extreme in the dependency of θr(H0).\nTheoreticalandexperimentalcurvesforthechiralstruc-\nture loaded with the ferrite slab are similar in shape and\nexhibit a character extreme in the vicinity of the field\nstrength close to the saturation magnetization, as it is ex-\npected from the general representations.Sergey Y. Polevoy et al.: Resonant features of planar Farada y metamaterial with high structural symmetry 11\nThe distinct feature ofthe planarchiral Faradaymeta-\nmaterial (i.e. the resonant planar arrayloaded with ferrite\nslab) is larger sensitivity of its polarization properties to\nthe static magnetic field strength with that one of the\nsame ordinary ferrite slab. This phenomenon can be ex-\nplainedbythe fact thatthe resonantcharacterofthe mag-\nnetic permeability component of ferrite (or their strong\nfrequency dispersion) is applied on the resonant charac-\nter of oscillations in the planar chiral structure (strong\nfrequency dispersion of the effective material parameters\nof the chiral structure), which takes a place in the ’inter-\naction area’. Note that a similar situation, known as the\namplification of the Faraday effect have been detected by\nthe authors in the millimeter wave range before, but in\nmore simple resonant structures (the open resonator [21],\nthe photonic crystal [14]). However, in the case consid-\nered here, we are dealing with the structure being planar\nresonant metamaterial that promises the similar effect in\nthe very thin structure. The needed resonant properties of\nthin metamaterial slab are imparted by complex shaped\nmetallic rosettes. The complex shape ofarrayparticlesen-\nables us to achieve resonant response of the structure in\nthe wavelength less than pitch of the array. The 4-fold\nsymmetry planar chiral rosettes are chosen to clear the\nway to design the polarization insensitive array structure\nat least at normal incidence of the exciting wave. Thus we\ncan produce sub-wavelength resonant structures suitable\nfor such promising applications as planar metamaterial\nwhich is controllable by static magnetic field.4 Conclusion\nThe transmission of electromagnetic waves of millimeter\nrangethroughthelayeredmetamaterialformedbytheres-\nonant planar chiral structure loaded with the gyrotropic\nmedium has been studied both experimentally and theo-\nretically. Namely: (i) the dependence of frequency of the\nmetamaterial resonant response and the angle of polar-\nization rotation on the longitudinal static magnetic field\nare detected, and a satisfactory agreement between the\ntheory and experiment is demonstrated; (ii) the range of\nfrequencies and magnetic field strength where the angle of\npolarization rotation by the metamaterial appears essen-\ntially higherthan that one related to a single ferrite slab is\ndefined; (iii) at the normal incidence of the exciting wave,\nthe independence of this metamaterial response on hand-\nedness of its planar chiral thin metallic elements has been\nverified; (iv) the usage of arrayswith high structural sym-\nmetry based on planar chiral particles enables additional\nmeanstoproducesub-wavelengthresonantmetamaterials,\nwhich have small size of the periodic cell and controllable\nproperties by static magnetic field.\nReferences\n1. B.Z. Katsenelenbaum, E.N. Korshunova, A.N. Sivov, A.D.\nShatrov, Physics-Uspekhi 40(11), 1149 (1997)\n2. A. Serdyukov, I. Semchenko, S. Tretyakov, A. Sihvola,\nElectromagnetics of Bi-anisotropic Materials: Theory and\nApplications (Gordon and Breach Science Publishers, Am-\nsterdam, 2001)\n3. L. Hecht, L.D. Barron, Chem. Phys. Lett. 225, 525 (1994)12 Sergey Y. Polevoy et al.: Resonant features of planar Fara day metamaterial with high structural symmetry\n4. L. Hecht, L.D. Barron, Journal of Molecular Structure\n348, 217 (1995)\n5. L.R. Arnaut, L.E. Davis, On planar chiral structures , in\nProgress in Electromagnetic Research Symposium (PIERS\n1995)(Seattle, WA, 1995), p. 165\n6. L.R. Arnaut, J. Electromagnetic Waves and Applications\n11, 1459 (1997)\n7. S.L. Prosvirnin, Analysis of electromagnetic wave scat-\ntering by plane periodical array of chiral strip elements ,\ninProceedings of 7-th Intern. Conf. on Complex Me-\ndia ”Bianisotropics-98” , edited by A.F. Jacob, J. Reinert\n(Braunschweig, Germany, 1998), pp. 185–188\n8. S. Zouhdi, G.E. Couenon, A. Fourrier-Lamer, IEEE Trans.\nAntennas Propagat. 47(6), 1061 (1999)\n9. S.L. Prosvirnin, N.I. Zheludev, Phys. Rev. E 71, 037603\n(2005)\n10. T. Vallius, K.Jefimovs, J. Turunen,P.Vahimaa, Y.Svirko ,\nAppl. Phys. Lett. 83(2), 234 (2003)\n11. S.L. Prosvirnin, N.I. Zheludev, J. Opt. A: Pure Appl. Opt .\n11, 074002(10) (2009)\n12. S.L. Prosvirnin, V.A. Dmitriev, The European Physical\nJournal - Applied Physics 49(3), 33005(5) (2010)\n13. M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P.B.\nLim, H. Uchida, O.A. Aktsipetrov, T.V. Murzina, A.A.\nFedyanin, A. B.Granovsky, J. Phys. D: Appl. Phys. 39(8),\n151 (2006)\n14. A.A. Girich, S.Y. Polevoy, S.I. Tarapov, A.M. Merzlikin ,\nA.B. Granovsky, D.P. Belozorov, Solid State Phenomena\n190, 365 (2012)\n15. N. Kumar, A.C. Strikwerda, K. Fan, X. Zhang, R.D.\nAveritt, P.C.M. Planken, A.J.L. Adam, Optics Express\n20(10), 11277 (2012)16. R.E. Collin, Foundation for Microwave Engineering ,\n2nd edn. (Wiley-Interscience, New York, 2001)\n17. A.G. Gurevich, Ferrites at Microwave Frequencies (Hey-\nwood, London, 1963)\n18. E. Schlomann, J. Appl. Phys. 41(1), 204 (1970)\n19. J. Green, F. Sandy, IEEE Trans. on Microwave Theory\nand Techniques MTT-22 (6), 641 (1974)\n20. S.L. Prosvirnin, Journal of Communications Technology\nand Electronics 44(6), 635 (1999)\n21. S.I. Tarapov, Y.P. Machekhin, A.S. Zamkovoy, Magnetic\nResonance for Optoelectronic Materials Investigating (Col-\nlegium, Kharkov, 2008)"    },    {        "title": "0908.4383v1.Magnetic_dipolar_mode_vortices_and_microwave_subwavelength_metamaterials.pdf",        "content": "Magnetic-dipolar-mode vortices and microwave subwavelength \nmetamaterials \n \nE.O. Kamenetskii *, M. Sigalov, and R. Shavit   \n  \nDepartment of Electrical and Computer Engineering, \nBen Gurion University of the Negev, Beer Sheva, Israel \n \nAugust 28, 2009  \nAbstract \n \nThere has been a surge of interest in the s ubwavelength confinement of the electromagnetic \nfields. It is well known that in optics the subw avelength confinement can be obtained due to \nsurface plasmon (quasielec trostatic) oscillations. In this paper we propose to realize the \nsubwavelength confinement in microwaves due to dipolar-mode (qua simagnetostatic) magnon \noscillations in ferrite particle s. Our studies of interactions between microwave electromagnetic \nfields and small ferrite particles with magnetic -dipolar-mode (MDM) osci llations show strong \nlocalization of electromagnetic energy. MDM oscill ations in a ferrite disk are origins of \ntopological singularities resulti ng in Poynting-vector vortices and symmetry breakings of the \nmicrowave near fields. We show that new su bwavelength microwave metamaterials can be \nrealized based on a system of interacting MD M ferrite disks. The volume- and surface-wave \npropagation of electromagnetic signals  in the proposed dense metama terials will be characterized \nby topological phase variations. The MDM-particle-met amaterial concept ope ns a significantly \nnew area of research. In part icular, there is a perspectiv e for creation of engineered \nelectromagnetic fields with uni que symmetry properties.   \n \nIntroduction \n \nThe possibility of compressing electromagnetic fields in space to a degree much better than \npredictable by classical diffraction theory has gain widespread attention. For example, the \nlocalization of electromagnetic energy is consider ed as a very important phenomenon in context \nof applications in optical sensing and optical data communication. Use of the optical near-field \ncharacterization technique should reduce the size gap between optical and electronic devices [1, \n2]. It makes possible to downscale established ante nna design into the optical frequency regime \n[3, 4, 5]. In such optical structur es, subwavelength confinement of the light takes place due to the \nresonant interaction between the quasielectrostatic oscillations of  electrons in metal nanoparticles \nand planar films and the electromagnetic field.     In this paper we propose use of small ferrite particles for subwavelength confinement in microwaves. The localization of electromagnetic  energy takes place due to the resonant \ninteraction between quasi-2D ferrite disks wi th magnetic-dipolar [or magnetostatic (MS)] \noscillations and the external el ectromagnetic fields. It is well  known that for a given microwave \nfrequency, the wavelength of magne tic-dipolar waves in confined ma gnetic structures is two-four \norders of magnitude less than the wavelength of free-space elec tromagnetic (EM) waves [6]. As \na result, one has characteristic sizes of MDM resonators much less than the free-space EM \nwavelength. In most cases of the present- study experiments, an excitation of MS-wave \noscillations in thin-film ferrite samples is realiz ed by microwave electric currents in microstrip \ntransducers [7]. An interaction of the external EM fields with  the MS-wave resonators is not \nusually a subject of these investig ations. At the same time, histor ically, the first evidence for MS-\nwave oscillations was obtained by White and Solt in a microwave experiment of interaction of a  2small ferrite sphere with the cavity electromagnetic fields [8]. Shortly after this experiment, \nDillon showed unique multiresonance MDM spectra fo r a ferrite disk in a microwave cavity [9]. \nYukawa and Abe gave further consideration of  this phenomenon in a ferrite disk [10]. For \nexplanation of effective multiresonance interactions between the external EM fields and MS-wave oscillations, different mechanisms had been  discussed [11, 12]. It was found later that a \nconvincing model of such inter actions arises from unique spectral properties of MDMs in a \nferrite disk. A spectral theory of MDMs in a quas i-2D ferrite disk developed in Refs. 13 – 15, \ngives proper explanations of the known experime ntal results. One of im portant conclusions of \nthe spectral theory – the fact th at the MDMs in a ferrite disk can be excited not only by external \nmicrowave magnetic fields but also by external microwave electric fields – was confirmed in \nnew microwave experiments [16 – 18].     In this paper we show that strong 3D lo calization of electromagnetic energy by small ferrite \ndisks appears due to the vortex behavior of  MDM oscillations. The particles with MDM \noscillations are origins of the P oynting-vector vortices of the mi crowave near fields abutting to \nthe disk surfaces. The Poynting-vector vortices are the regions with topological singularities. \nSuch topological singularities are well studied in optics. For exam ple, optical beams with phase \nsingularities are robust structures  with respect to perturbations . In such beams, one has a \ncirculating flow of energy resulting in 2D conf inement of electromagnetic energy in transversal \ndirections [19]. In the near-field optics, due to phase sin gularities one obtains subwavelength \ntransmission through narrow slits [20], novel supe rlenses [21], and superre solution process in \nmetamaterials [22].      The subwavelength confinement and near-field  manipulation of the electromagnetic fields is \none of the main aspects attracted the concept of metamaterials [23 – 27]. We show that new \nsubwavelength microwave metamaterials can be r ealized based on a system  of interacting MDM \nferrite disks. An array of these MDM particles with evanescent-tail chiral interactions will \nrepresent a new type of a microwave magnetic  metamaterial. There are singular-microwaves \nmetamaterials: The MDM particles are singular  points with topological  charges and symmetry \nbreaking properties. As a whole, concerning the problem of microwave magnetic metamaterials, \nit is worth noting also that in some recent publ ications, interesting re sults on tunable photonic-\nband-gap structures with ferrite rods have been  shown [28, 29]. These st ructures, however, do \nnot have any effects of subwavel ength confinement. On the other hand, in the periodic spin-wave \nstructures called magnonic crysta ls, there are no effects of inte ractions with the external \nmicrowave EM fields [30].   \nMagnetic-dipolar modes in a quasi-2D ferrite disk \n \nA quasi-2D MDM ferrite disk is an open resonator with a high Q-factor. The spectral problem \nfor MDM oscillations is formulated for the magn etostatic-potential wave functions [13 – 15]. It \nwas shown that the MDM disk particle can be modeled as a combined structure of a normal \nelectric (anapole) and an in-plane rotating ma gnetic moments [31, 32]. As a very important \nproperty in spectral characterizat ions, there are vortex states of  MDMs in a ferrite disk. The \nspectral theory gives two types of vortices. Fo r MDMs with the anapole- moment properties, one \nhas chiral-edge-state vortices [15], while for MDMs with the ro tating-magnetic-dipole properties \nthere are the power-flow-density vor tices [32]. One of important f eatures of the modes with the \nrotating-magnetic-dipole properties is a very  good correspondence between analytical and \nnumerical (based on the HFSS EM simulati on program – software produced by ANSOFT \nCompany) results of the mode characterization [31, 32]. This allows analyzing numerically the \nstructures with such modes. For different types of the vortex states one has different mechanisms \nof the evanescent-tail interact ions between the MDM particle s. An analytical model of \ninteracting particles was develope d in Ref. [33]. In the present paper, the field structures of  3interacting ferrite disks are studied numerically based on th e HFSS program. We analyze the \nEM-field confinement effects of the near-fie ld properties of single and interacting MDM \nparticles.  \nMDM vortices in a quasi-2D ferrite disk and microwave subwavelength confinement  \n \nTo show the microwave subwavelength confin ement effect, we analyze numerically the \nPoynting-vector distributions and the EM field structures in a rectangular waveguide with an \nenclosed MDM ferrite disk. Then we extend our an alysis of the fields for a composition of MDM \nferrite disks.       For our studies we use a ferrite disk with  the following material parameters: the saturation \nmagnetization is  1880  4=\nsMπ G and the linewidth is Oe 8.0=∆H . The disk diameter \nis 3=D mm and the disk thickness is 05.0=t mm. Generally, these da ta correspond to the \nsample parameters used in microwave experiments [10, 16 – 18]. The disk is normally \nmagnetized by a bias magnetic field  49000=H Oe and is placed inside a 10TE -mode \nrectangular waveguide symm etrically to its walls. The waveguide walls are made of copper. For \na waveguide with a ferrite disk, a numerical analysis gives a multiresonance frequency characteristic of the reflection coefficient. This characteristic is represented in Fig. 1 (a). The \nresonance peaks are designated in succession by numbers n. An insertion in an upper right-hand \ncorner of Fig. 1 (a) shows geomet ry of a structure with  notations of power flows of incident (\niPr\n), \nreflected (rPr\n), and transmitted (tPr\n) waves. The analytically derived spectral peak positions for \nrotating-magnetic-dipole modes [31, 32] repr esented on Fig. 1 (b) are in quite good \ncorrespondence with the numerically obtained spectra l peak positions in Fig. 1 (a). An insertion \nin the right-hand part of Fig. 1 (b) illustrates a model of a MDM ferrite disk with a rotating \nmagnetic dipole mpr [31].  \n    The EM-field confinement effects due to the MDM oscillations in a ferrite disk become \nevident from the pictures of the Poynting vector distributions. Figs. 2 (a),  (b), and (c) show the \nPoynting vector distributions for th e fields inside a waveguide on the xz vacuum plane situated at \nthe distance of 150 mkm above an upper plane of a ferrite disk (in further consideration, we will \nconventionally call this vacuum plane as plane A) . The pictures in Figs. 2 (a) and (c) correspond \nto the first ( f = 8. 5225 GHz) and the second ( f = 8. 6511 GHz) resonances, while the picture in \nFig. 2 (b) is at the frequenc y between the resonances ( f = 8. 5871 GHz). It is obvious that at the \nMDM resonant frequencies there are vortices of the Poynting vect or distributions with strong \nsubwavelength confinement of th e electromagnetic energy. No such confinement is observed at \nnon-resonance frequencies. The fact that the MDM vortices are the origins of the EM field \nconfinement can be illustrated, additionally, by the pictures of the Poynti ng vector distributions \ninside a ferrite disk. These pictures, shown in Fi gs. 2 (d), (e), and (f) (which are placed in \ncorrespondence with Figs. 2 a, b, c), give evidence for the MDM power flow vortices at \nresonances. For non-resonance frequencies,  no such MDM vortices are observed.   \n    In Figs. 3 (a), (b), and (c) we show, respec tively, the magnetic field di stributions on plane A at \nthe first-resonance frequency f = 8. 5225 GHz, at non-resonance frequency f = 8. 5871 GHz, and \nat the second-resonance frequency f = 8. 6511 GHz. To watch the dynamics, the fields are \nrepresented for two phases: o0 =tω  and o90 =tω . It is evident that at MDM resonances there \nare strong field concentrations and symmetry vi olations. No field enha ncement and no symmetry \nviolation occur at non-resonance frequencies. As  a very important property of the observed \npictures of the magnetic field di stributions, there is an evident rotating-magnetic-dipole behavior \nat resonance frequencies [see Figs. 3 (a) and (c)]. Because of such rotating-magnetic-dipole  4behavior, predicted in Refs. [22, 23], the fields ar e symmetry violated and strongly confined near \nthe disk surfaces.  \nMDM vortices in a chain of quasi-2D ferri te disks and microwave subwavelength \nconfinement  \n \nThe observed strong localization of EM energy is due to the effect when the precessing magnetic \nmoments in a ferrite disk interact collectively, by oscillating in the MDM resonance, with the \nmicrowave fields. At the MDM resonance states, the vortex rings act as traps, providing purely \nsubwavelength confinement of the EM fields. Th e interaction between the MDM ferrite disk and \nthe electromagnetic field has two consequences. First, the presen ce of a circul ating flow of \nconcentrated energy presumes exis tence of angular EM momentum directed perpendicular to the \ndisk plane. The second consequence is that the fi elds outside a ferrite di sk are evanescent fields \nin nature: they decay exponentially with distance  from the ferrite surface. All these properties \noffer the potential for developing new types of s ubwavelength microwave metamaterials. As an \ninitial stage of an anal ysis of the proposed subwavelength mi crowave metamaterials, we consider \nhere a chain of ferrite magnetic-dipolar-vortex pa rticles. In such a chain, the evanescent-tail \ncoupling between adjacent MDM re sonators induces transverse dynamics and should constitute \nan effective waveguide structure.      In our numerical study, the chain of three quasi-2D normally  magnetized ferrite disks is placed \ninside a \n10TE -mode rectangular waveguide symmetrically to its walls. The chain is oriented \nalong a waveguide axis. The disk diameters are 3 mm and distances between the disk axes are \n3.2 mm. By virtue of quasi-magneto static interactions between the disks, there are splittings of \nMDM resonance peaks. Fig. 4 shows such splitting in the reflecti on coefficient characteristic for \nthe first resonance peak [see Fig. 1 (a)]. For our analysis of a disk ch ain, we will choose two \nresonance frequencies designated in Fig. 4 as 1f′ and ″\n1f. In Figs. 5 (a) and (b), we show the \nPoynting vector distributions on pl ane A at resonance frequencies 1f′= 8.5248 GHz and ″\n1f= \n8.5356 GHz, respectively. The pictures of the Poynti ng vector distributions inside every ferrite \ndisk in Figs. 5 (c) and (d), show that the obser ved effect of strong subwavelength confinement of \nthe electromagnetic fields by a disk chain is due  to the MDM-resonance vortex behaviors of the \nparticles.       For a ferrite-disk chain, the magnetic field di stributions on plane A at  resonance frequencies  \n1f′= 8.5248 GHz and ″\n1f= 8.5356 GHz  and for two phases: o0 =tω  and o90 =tω  are shown in \nFig. 6. There is evident field enhancement in a region near a disk chain. One can clearly observe \nthe rotating-magnetic-dipole behavior of every fe rrite disk in the chain. At two resonance \nfrequencies, there are different relations between phases of magnetic  fields in  ferrite disks. At \nresonant frequency 1f′, all disks oscillate in phase [see Fi g. 6 (a)]. At re sonant frequency ″\n1f, \nthere are in-phase oscillations for two extreme di sks, while an interior disk oscillates in an \nopposite phase [see Fig. 6 (b)].         Here we considered a terminated chain of quasi-2D ferrite disks. Th ere are standing magnetic-\ndipolar waves along such a chain.  It is evident that for an infinite structure, an interaction \nbetween a chain of magnetic-di pole particles and the electromagnetic field will give the \npropagation behavior of magnetic -dipolar waves along the chain. At the same time, in the \ndirections perpendicular to the chain, th e fields will exponentially decay.      \n \n \n  5Future directions and challenges \n \n     1. In optics, it was recently shown that  an array of evanescent-tail-coupling nanoparticle \nplasmonic resonators is an effective waveguide structure with low losses [2]. This is the demonstration of non-diffraction-limited guidi ng of electromagnetic energy over micron- and \nsubmicron distances. Due to the heightened local fiel ds surrounding plasmoni c-resonator guiding \nstructures, such optical device s have potential ap plications not only in photonics and \ntelecommunications but also in lo calized biological sensing of mo lecules. Similarly to a system \nof coupled plasmonic resonators in optics, we have here effective non-diffraction-limited \nmicrowave waveguides. An arra y of MDM-disk resonators c oupled by quasi-magnetostatic \nevanescent tails will be an eff ective microwave waveguide struct ure with low losses. Symmetry \nbreakings of near fields in such structures wi ll allow localized sensing of chiral biological and \nchemical objects in microwaves.         2. Interaction of microw ave fields with a MDM ferrite-d isk results in Poynting-vector \nsingularities. The power-flow-density vortices in a chain of ferrite disks [s ee pictures in Figs. 5 \n(c) and (d)] are well localized topological exci tations that do not pertur b the fields at large \ndistances from the particles. Topological pointlik e solutions with corele ss 3D textures – the \nSkyrmions – are well known in nuclear and elemen tary particle physics [34]. Stable pointlike \nSkyrmions can be observed in a trapped Bose-Ein stein condensate [36]. Based on an array of \ninteracting MDM ferrite disks,  one can realize s ubwavelength microwave photonics crystal \nstructures with channeling of Skyrmion-like t opological excitations (channeling of EM power-\nflow vortices).     3. Interactions between microwave electr omagnetic fields and quasi-2D ferrite disks with \nMDM oscillations can be limited not only to the electromagnetic wave propagation in structures \nwith subwavelength dimensions. Dipolar-mode magnoni cs of ferrite particle s can also help to \ngenerate and manipulate microwave electromagnetic  radiation. The observed in Figs. 1 (a) and 4 \nstrong reflections of electromagnetic power (in comparison with characteristics of an empty \nwaveguide) at resonance frequenc ies of a MDM ferrite disk or a MDM ferrite-disk chain are due \nto certain phase relations between the phases of rotating magnetic dipoles and the phases of \nmicrowave magnetic fields in a rectangular waveguide. As a cogent  argument, one can \npresuppose that special microwave structures with rotating magnetic  fields can be realized so \nthat the phase of a rotating magne tic dipole of a MDM ferrite disk, enclosed in such a microwave \nstructure, will be in the opposite phase with respect to the phase of a microwave magnetic field. \nIn this case, the rotational energy of a magnetic di pole of the ferrite-disk particle can be used for \ncreation of an effective microw ave subwavelength antenna. The mi crowave radiation in such a \nsubwavelength antenna will appear as a result of collective interaction of precessing electron \nspins in a high- Q-factor MDM ferrite resonator with a magn etic field of a microwave structure. \n ------------------ ------------------ ----------------- ------------------ --------------- ------------- \n* Email: kmntsk@ee.bgu.ac.il \n[1] W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003). \n[2] S. A. Maier and H. A. Atwater, J. Appl. Phys. 98, 011101 (2005). \n[3] L. Novotny, Phys. Rev. Lett. 98, 266802 (2007). \n[4] A. Alú and N. Engheta, Phys. Rev. Lett., 101, 043901 (2008). \n[5] P. Ghenuche et al, Phys. Rev. Lett. 101, 116805 (2008). \n[6] A. Gurevich and G. Melkov, Magnetic Oscillations and Waves  (CRC Press, New York, \n1996). \n[7] W. S. Ishak and K.-W. Chang, IEEE Trans. Microwave Theory Tech. 34, 1383 (1986). \n[8] R. L. White and I. H. Solt, Jr., Phys. Rev. 104, 56 (1956).  \n[9] J.F. Dillon Jr., J. Appl. 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D' Aguanno et al, Phys. Rev. A 77, 043825 (2008).  \n[23] J.B. Pendry, Phys. Rev. Lett. 85, 3966 (2000). \n[24] N. I. Zheludev, Nature Mat. 7, 420 (2008). \n[25] M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006). \n[26] K. Aydin, I. Bulu, and E. Ozbay, Appl. Phys. Lett. 90, 254102 (2007). \n[27] V. M. Shalaev, Nature Photon. 1, 41 (2007). \n[28] L. Kang et al, Opt. Express 16, 17269 (2008);  \n[29] S. Liu et al, Phys. Rev. B 78, 155101 (2008). \n[30] V. V. Kruglyak et al, J. Appl. Phys. 98, 014304 (2005). \n[31] E.O. Kamenetskii, M. Sigalov,  and R. Shavit, J. Appl. Phys. 105, 013537 (2009).  \n[32] M. Sigalov, E.O. Kamenetskii, and R. Shavit, J. Phys.: Condens. Matter 21, 016003 (2009). \n[33] E. O. Kamenetskii, J. Appl. Phys. 105, 093913 (2009). \n[34] T. H. R. Skyrme, Proc. R. Soc. London A 260, 127 (1961). \n[35] C. M. Savage and J. Ruostekoski, Phys. Rev. Lett. 91, 010403 (2003). \n  Figure captions\n \n \nFig. 1 . MDM resonances of a quasi-2D ferrite disk. ( a) The numerically obtained multiresonance \nfrequency characteristic of th e reflection coefficient for a wa veguide with an enclosed MDM \nferrite disk. The inset shows geom etry of a structure with notati ons of power flows of incident \n(iPr\n), reflected (rPr\n), and transmitted (tPr\n) waves. ( b) The analytically derived spectral peak \npositions for MDMs. The inset illustrates a mode l of a MDM ferrite disk with a rotating \nmagnetic dipole mpr.  \n \nFig. 2 . The field confinement originated from  the MDM vortices in a ferrite disk. ( a) The \nPoynting vector distributi ons for the fields on the xz vacuum plane situated at the distance of 150 \nmkm above an upper plane of a ferri te disk (this vacuum plane is conventionally called as plane \nA) at the frequency ( f = 8. 5225 GHz) of the first resonance. ( b) The same at the frequency ( f = \n8. 5871 GHz) between the resonances. ( c) The same at the frequency ( f = 8. 6511 GHz) of the \nsecond resonance. ( d) The Poynting vector distributions inside  a ferrite disk at the frequency of \nthe first resonance. ( e) The same at the frequency between resonances. ( f) The same at the \nfrequency of the second resonance.    7Fig. 3 . Evidence for correlation between strong field concentrations and symm etry violations at \nMDM resonances. ( a) The magnetic field distributions on plane A at the frequency ( f = 8. 5225 \nGHz) of the first resonance for two phases: o0 =tω  and o90 =tω . (b) The same at the frequency \n(f = 8. 5871 GHz) between the resonances. ( c) The same at the frequency ( f = 8. 6511 GHz) of \nthe second resonance. It is evid ent that at MDM resonances ther e are strong field concentrations \nand symmetry violations.  \nFig. 4.  Splittings of the MDM resonance peaks in a chain of quasi-magnetost atically interacting \nferrite disks. In an analysis of a disk chain, two resonance frequencies 1f′= 8.5248 GHz and \n″\n1f= 8.5356 GHz are chosen. \n \nFig. 5 . The field confinement originated from the MD M vortices in a chain of interacting ferrite \ndisks. ( a) The Poynting vector distribution on plane A at resonance frequency 1f′= 8.5248 GHz. \n(b) The same at resonance frequency ″\n1f= 8.5356 GHz. ( c) The pictures of the Poynting vector \ndistributions inside ever y ferrite disk in a chain at resonance frequency 1f′= 8.5248 GHz. ( d) \nThe same at resonance frequency ″\n1f= 8.5356 GHz.    \n \nFig. 6 . The magnetic field distributions vortices in  a chain of interacting ferrite disks. (a) The \nmagnetic field distributions on plane A at the resonance frequency 1f′= 8.5248 GHz for two \nphases: o0 =tω  and o90 =tω . (b) The same at the resonance frequency ″\n1f= 8.5356 GHz. \nThere is evident field enhancement in a region near a disk chain. One can clearly observe the \nrotating-magnetic-dipole behavior of every ferrite  disk in the chain. At resonant frequency 1f′, \nall disks oscillate in phase . At resonant frequency ″\n1f, there are in-phase oscillations for two \nextreme disks, while an interior disk oscillates in an opposite phase.     \n               8\n \n \n \n \nFig. 1 . MDM resonances of a quasi-2D ferrite disk. ( a) The numerically obtained multiresonance \nfrequency characteristic of th e reflection coefficient for a wa veguide with an enclosed MDM \nferrite disk. The inset shows geom etry of a structure with notati ons of power flows of incident \n(iPr\n), reflected (rPr\n), and transmitted (tPr\n) waves. ( b) The analytically derived spectral peak \npositions for MDMs. The inset illustrates a mode l of a MDM ferrite disk with a rotating \nmagnetic dipole mpr.  \n \n  \n \n   9\n \n \n \nFig. 2 . The field confinement originated from  the MDM vortices in a ferrite disk. ( a) The \nPoynting vector distributi ons for the fields on the xz vacuum plane situated at the distance of 150 \nmkm above an upper plane of a ferri te disk (this vacuum plane is conventionally called as plane \nA) at the frequency ( f = 8. 5225 GHz) of th e first resonance. ( b) The same at the frequency ( f = \n8. 5871 GHz) between the resonances. ( c) The same at the frequency ( f = 8. 6511 GHz) of the \nsecond resonance. ( d) The Poynting vector distributions inside  a ferrite disk at the frequency of \nthe first resonance. ( e) The same at the frequency between resonances. ( f) The same at the \nfrequency of the second resonance.      \n \n   \n  10\n \n                                                     ( a) \n                        \n \n                                                      ( b) \n                        \n   \n                                                      ( c) \n \n   \nFig. 3\n. Evidence for correlation between strong field concentrations and symm etry violations at \nMDM resonances. ( a) The magnetic field distributions on plane A at the frequency ( f = 8. 5225 \nGHz) of the first resonance for two phases: o0 =tω  and o90 =tω . (b) The same at the \nfrequency ( f = 8. 5871 GHz) between the resonances. ( c) The same at the frequency ( f = 8. 6511 \nGHz) of the second resonance. It  is evident that at MDM res onances there are strong field \nconcentrations and symmetry violations. \n  11\n \n \n \nFig. 4.  Splittings of the MDM resonance peaks in a chain of quasi-magnetost atically interacting \nferrite disks. In an analysis of a disk chain, two resonance frequencies 1f′= 8.5248 GHz and \n″\n1f= 8.5356 GHz are chosen.  \n \n                 12\n \n                                                       ( a)                                                        (c) \n    \n \n                                                     ( b)                                                       ( d) \n \n \nFig. 5 . The field confinement originated from the MD M vortices in a chain of interacting ferrite \ndisks. ( a) The Poynting vector distribution on plane A at resonance frequency 1f′= 8.5248 GHz. \n(b) The same at resonance frequency ″\n1f= 8.5356 GHz. ( c) The pictures of the Poynting vector \ndistributions inside ever y ferrite disk in a chain at resonance frequency 1f′= 8.5248 GHz. ( d) \nThe same at resonance frequency ″\n1f= 8.5356 GHz.     13\n \n                                                           ( a) \n \n \n                                                              ( b) \n \n \nFig. 6 . The magnetic field distributions vortices in  a chain of interacting ferrite disks. ( a) The \nmagnetic field distributions on plane A at the resonance frequency 1f′= 8.5248 GHz for two \nphases: o0 =tω  and o90 =tω . (b) The same at the resonance frequency ″\n1f= 8.5356 GHz. \nThere is evident field enhancement in a region near a disk chain. One can clearly observe the \nrotating-magnetic-dipole behavior of every ferrite  disk in the chain. At resonant frequency 1f′, \nall disks oscillate in phase . At resonant frequency ″\n1f, there are in-phase oscillations for two \nextreme disks, while an interior disk oscillates in an opposite phase.     \n "    },    {        "title": "0904.0332v1.Current_controlled_dynamic_magnonic_crystal.pdf",        "content": "arXiv:0904.0332v1  [cond-mat.other]  2 Apr 2009Current-controlled dynamic magnonic crystal\nA. V. Chumak,∗T. 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, University of Western Australia, Crawle y, Western Australia 6009, Australia\n(Dated: October 21, 2018)\nWe demonstrate a current-controlled, dynamic magnonic cry stal. It consists of a ferrite film whose\ninternal magnetic field exhibits a periodic, cosine-like va riation. The field modulation is created by\na direct current flowing through an array of parallel wires pl aced on top of a spin-wave waveguide. A\nsingle, pronouncedrejection bandinthespin-wavetransmi ssion characteristics is formed duetospin-\nwave scattering from the inhomogeneous magnetic field. With increasing current the rejection band\ndepth and its width increase strongly. The magnonic crystal allows a fast control of its operational\ncharacteristics via the applied direct current. Simulatio ns confirm the experimental results.\nPACS numbers: 75.50.Gg, 75.30.Ds, 75.40.Gb\nSpin wavesin magnetic materialsattract special atten-\ntion because of their potential application as information\nunits in signal processing devices. Digital spin wave logic\ndevices [1, 2] as well as devices for analogous signal pro-\ncessing [3, 4, 5] can be fabricated based on spin waves. It\nhas been shown that the spin-wave relaxation, one of the\nmain obstacles for spin-wave application, can be over-\ncome by means of parametric amplification [5, 6].\nThe study of spin waves in magnetic materials is also\ninteresting from a fundamental point of view. The inter-\naction of the numerous existing spin-wave modes in fer-\nromagnetic samples [7] as well as nonlinear effects such\nas soliton formation [8, 9] are just some examples.\nMagnoniccrystals,whicharedefinedasartificialmedia\nwith a spatially periodic variation of some of their mag-\nnetic parameters, constitute a research field which con-\nnects fundamental physics with application [10, 11, 12,\n13, 14]. They arethe analogueof photonic crystalswhich\noperate with light. The spectra of spin-wave excitations\nin such structures are considerably modified compared\nto uniform media and exhibit features such as full band\ngaps where spin waves are not allowed to propagate.\nPromising functionalities arise by taking advantage of\nthe dynamic controllability and by potentially changing\nthecharacteristicsofthemagnoniccrystalfasterthanthe\nspin-wave relaxation time: even the simple possibility to\n”switch” a periodical inhomogeneity on and off imme-\ndiately offers a method to trap and release a spin wave\npacket. This can be exploited for instance in information\nstorage.\nHere, we present a first realization of such a dynamic\nmagnonic crystal. It is based on spin-wave propagation\nin an yttrium iron garnet (YIG) film placed in a period-\nically varying, dynamically controllable magnetic field.\nThe magnetic field is created by the superposition of a\n∗Electronic address: chumak@physik.uni-kl.de\nYIGInput\nantennaCurrent\nOutput\nantenna\nMagnetic fieldSpin wave\nFIG. 1: (Color online) Sketch of the magnonic crystal struc-\nture used in the experiments.\nspatially homogeneous bias magnetic field with the lo-\ncalized Oersted fields of current carrying wires which are\nplaced in an array layout close to the YIG film surface\n[15]. By controlling the direct current in the wires the\nfield modulation is adjusted and the spin-wave transmis-\nsion is changed from full transmission for no applied cur-\nrent to a transmission showing a distinct, 30 MHz-wide\nstop band for an applied current of 1 .25 A. The dynamic\ncontrollability constitutes a major difference to previous\nrealizationsofmagnoniccrystalswithaperiodicallyvary-\ning magnetic field [16].\nPrevious studies focused on the interaction of propa-\ngating spin-wave packets with the Oersted field of a sin-\ngle current carrying wire or a set of two wires at most\n[17, 18, 19, 20, 21]. It was shown that the spin-wave\ntransmission can be effectively changed by varying the\nvalue of the direct current. However, the appearance of\na pronounced frequency stop-band, for which spin-wave\ntransmission is prohibited (while it remains almost unaf-\nfected outside the band), is only observed for larger wire\nnumbers.\nA sketch ofthe experimental section is shown in Fig. 1.\nIt consisted of a 5 µm-thick YIG film which was epitax-\nially grown on a gallium gadolinium substrate. A bias\nmagnetic field of 4 π·1.6 Am−1was applied along the2\nYIGwaveguidesothattheconditionsforthepropagation\nofbackwardvolumemagnetostaticwaves(BVMSWs) are\ngiven.\nTo achieve a periodic modulation of the magnetic field\nan array of connected, parallel wires was designed. The\nwire structure was patterned by means of photolithog-\nraphy on an aluminium nitride substrate with high ther-\nmalconductivityinordertoavoidheating. Thestructure\nconsistsof 40wires of75 µm width with a 75 µm spacing.\nThewirearraywasplacedabovetheYIG film insucha\nway that the wires ran perpendicularly to the spin-wave\nwaveguide. Thus, the magnetic Oersted field produced\nby each of the current carrying wire segments is oriented\nin first approximation parallel to the bias magnetic field.\nIn the experiment the individual wires were connected\nto form a meander structure [15] where the current in\nneighboringwiresflowsinoppositedirections(seeFig.1).\nThus, a magnonic crystal with a lattice constant a=\n300µm and 20 repetitions was fabricated.\nAnother possible configuration would have all currents\nflowinginthesamedirections(”multi-stripstructure”)so\nthat forallwiresthe Oerstedfieldshaveidenticalorienta-\ntion. Thisisanadvantagesinceourpreviousstudieshave\nshowntheexistenceoftwophysicallydifferentregimesfor\nthe different field orientations: When the Oersted field\ndecreases the internal field one implements the spin-wave\ntunneling regime [18]. When the internal field is locally\nincreasedtheconditionsforresonantspin-wavescattering\n[19] can be fulfilled for which the spin-wave transmission\ndepends non-monotonically on the applied current and\nexhibits a strong frequency dependence [20].\nHowever, the meander structure has important advan-\ntages: (i) It produces a much stronger field modulation\nbecause the in-plane components of the Oersted fields for\nneighbouring wires are oriented in opposite directions.\n(ii) It ensures that the magnetic field averaged over the\nstructure remains constant for any current magnitude.\nTwo microstrip antennas were placed, one in front and\nanother one behind the wire structure (see Fig. 1) in or-\nder to excite and detect BVMSWs. A network analyzer\nconnected to the input and output antennas was used to\nmeasure the spin-wave transmission characteristics.\nIn order to minimize the electromagnetic coupling be-\ntween the current carrying wire segments and the spin\nwaves, a 100 µm thick SiO 2spacer was placed between\nthe YIG film and the wire structure. Note, that the spin-\nwave dipole field decays exponentially with the distance\nform the film surface while the Oersted field around the\nwires scales with the inverse distance between the wire\nand the film surface. The distance of 100 microns be-\ntween the wire array and the film surface proved to be\nlarge enough to avoid any disturbance of the spin-wave\npropagation by the meander conductor, but still small\nenough to ensure an efficient modulation of the magnetic\nfield in the film by the current field.\nExperimental results are shown in Fig. 2. The dotted\ncurves in the panels show the transmission characteris-\ntics without direct current applied to the wires. TheyFIG. 2: (Color online) Spin-wave transmission characteris tics\nin a uniform magnetic field (dotted curves) and in a magnetic\nfield which is periodically modulated by the current Iapplied\nto the wires (solid curves). The dashed curves are calculate d\nas the product of the experimentally observed transmission\nin a uniform field with the transmission coefficient obtained\nfrom the numerical simulations (where in addition the lim-\nited dynamic range of the experimental setup is taken into\naccount).\nare typical for BVMSWs, limited by the ferromagnetic\nresonance frequency towards high frequencies and by the\nantenna excitation efficiency from the opposite side. The\nminimal transmission loss of about 35 dB is determined\nby the spin-wave excitation/reception e��ciency of the\nmicrowave antennas and by the spin-wave relaxation pa-\nrameter of the ferrite film.\nFigure 2(a) shows that the application of a current\nI= 0.25 A to the structure results in the appearanceof a\npronouncedrejectionbandatafrequency f1≈6510MHz\nwhere the transmission of spin waves is prohibited. The\nrejection band already appears for a current as small as\n80 mA. With an increase in the current the rejection\nbanddepth increasesrapidlyandfor0 .25Ait reachesthe\ndynamic rangeof the experimental setup which is limited\nmainly by the direct electromagneticleakagebetween the\nmicrostrip antennas. A further increase in the current\napplied to the wires results in a pronounced broadening\nof the rejection band (see Fig. 2(b) and Fig. 2(c)).\nWe emphasize one particularly interesting feature of3\nFIG. 3: (Color online) (a) Experimentally obtained width of\nfirst rejection band as a function of the current applied to\nthe wires. The lines and filled circles represent the results\nof numerical simulations. (b) Calculated critical current for\nwhich a band gap with a suppression ≥90% is obtained for\ndifferentnumbersofparallel wire segments. The line indica tes\na fit with a decaying exponential function.\nthemagnoniccrystalpresentedhere. Practicallyonlyone\nrejection band is formed. This is not true for magnonic\ncrystals consisting of an array of grooves on the YIG film\nsurface [13] where multiple rejection bands are formed.\nAs shown by our calculations, for the magnonic crys-\ntal studied here the spatially periodic modulation of the\nmagnetic field is close to cosinusoidal. For an ideal har-\nmonicvariationonlyonerejectionbandshouldexistsince\nthe reflection amplitude is proportional to the Fourier\ncomponent of the inhomogeneity profile corresponding\nto twice the spin-wave wave vector as seen from Eq. (6)\nin [19].\nThe presence of only one rejection band is an advan-\ntage for applications in a microwave filter device. An-\nother positive aspect of the presented dynamic crystal is\nthatpracticallynolossesoccurforfrequenciesoutsidethe\ninduced stop bandwith increasingcurrent(seeFig. 2(c)).\nIn the groove-structure-basedmagnonic crystal [13] such\nundesired parasitic increase of losses in the transmission\nbands was observed for larger groove depths.\nIn order to investigate the dynamic properties of the\nmagnonic crystal additional experiments with a pulsed\ndirect current supplied to the meander structure were\nperformed. The microwave frequency which was applied\nto the input antenna to excite the spin-wave signal was\nchosen inside the rejection band ( f= 6.51 GHz). The di-\nrectcurrentsuppliedtothemeanderstructurewaspulsed\nwith a duration of 50 ns and a strength of 0 .5 A. The ob-\ntained results show that the spin-wave transmission can\nbe dynamically turned on and off with a transition time\nfor the magnonic crystal of approximately 50 ns.\nThemeasuredrejectionbandwidth asafunctionofthe\napplied current is shown in Fig. 3(a). It was measured\nfor the first rejection band at the power level where the\nspin-wave intensity decreases to one tenth, i.e. −10 dB,\nof its value. One can see that the band width can betuned from 5 MHz for 125 mA current to 31 MHz for\n1.25 A and exhibits a linear behavior with respect to the\napplied current. The possibility of a dynamical control\nof the rejection band width seems to be promising for\nthe design of a dynamic stop-band microwave filter. The\ncenter frequency of the rejection band can be controlled\nby means of an applied magnetic field.\nThe experimental results were confirmed by numerical\nsimulations. We useda 1-dimensionalapproach,in which\nthe dipole field was expressed via a Green’s function and\nthe magnetic field was averaged over the film thickness.\nDetails on the model can be found in [19]. The obtained\nfrequency-dependent transmission curves show a single,\nwell pronounced stop-band for low currents which coin-\ncides well with the experiment (see Fig. 2). The calcu-\nlated rejection efficiency, given by the depth of the stop\nband, reached up to −150dB which exceeds the dynamic\nrange in the experiment greatly.\nTwo effects are observed if the number of parallel wire\nsegments in the simulation is decreased: Firstly, the\nachieved stop-band width for a given current decreases\nslightly. Secondly, the rejection efficiency decreases dra-\nmatically. As a consequence, the desired signal suppres-\nsion (e.g. one tenth of the transmission for no applied\ncurrent) is only reached for higher currents which results\nin the behavior of the rejection band gap width shown in\nFig. 3(a). Figure 3(b) summarizes the calculated depen-\ndence of the critical current necessary to obtain a 10 dB\nsignal suppression on the number of parallel wire seg-\nments. As can be seen a larger number of wires exponen-\ntially reducesthe criticalcurrentwhich hasa particularly\npronounced effect on the current if the number of wires\nis smaller than 20.\nIn conclusion, we presented a current-controlled\nmagnonic crystal whose operational characteristics can\nbe tuned dynamically within a transition time of 50 ns.\nThe spatiallyperiodic Oerstedfield ofameander conduc-\ntor located in the vicinity of YIG film surface results in a\npronounced modification of spin wave dispersion which\nleads to the appearance of spin-wave rejection bands.\nThe width of the main rejection band varies linearly with\nthe magnitude of applied direct current and can be tuned\nin the range from 5 MHz to 30 MHz. Numerical simula-\ntions are in good qualitative agreement with the experi-\nment. Overall, the presented dynamic magnonic crystal\nis promising for the investigation of linear and nonlinear\nspin-wave dynamics and can be used as a dynamically\ncontrolled microwave stop-band filter.\nFinancial support by the DFG project SE 1771/1-\n1, the Matcor Graduate School of Excellence, the Aus-\ntralian Research Council, and the University of Western\nAustralia is acknowledged. Special acknowledgments go\nto the Nano+Bio Center, TU Kaiserslautern. T. Neu-\nmann would like to thank especially Robert L. Stamps\nand the University of Western Australia for their assis-\ntance during his research stay.4\n[1] T. Schneider, A. A. Serga, B. Leven, B. Hillebrands,\nR. L. Stamps, and M. P. Kostylev, Appl. Phys. Lett.\n92, 022505 (2008).\n[2] K. Lee and S. Kim, J. Appl. Phys. 104, 053909 (2008).\n[3] J. D. Adam, Proc. IEEE 76, 159 (1988).\n[4] Yu. V. Kobljanskyj, G. A. Melkov, A. A. Serga, V. S.\nTiberkevich, andA.N.Slavin, Appl.Phys.Lett. 81, 1645\n(2002).\n[5] A. A. Serga, A. V. Chumak, A. Andre, G. A. Melkov,\nA. N. Slavin, S. O. Demokritov, and B. Hillebrands,\nPhys. Rev. Lett. 99, 227202 (2007).\n[6] E. Schl¨ omann, J. J. Green, and U. Milano, J. Appl. Phys.\n31, 386S (1960).\n[7] B. A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013\n(1986).\n[8] B. A. Kalinikos, N. G. Kovshikov, and A. N. Slavin,\nJETP Lett. 38, 413 (1983).\n[9] S. O. Demokritov, A. A. Serga, V. E. Demidov, B. Hille-\nbrands, M. P.Kostylev, and B. A.Kalinikos, Nature 426,\n159 (2003).\n[10] K. W. Reed, J. M. Owens, and R. L. Carter, Circ. Syst.\nSignal Process. 4, 157 (1985).\n[11] Yu. V. Gulyaev, S. A. Nikitov, L. V. Zhivotovskii, A. A.\nKlimov, Ph. Tailhades, L. Presmanes, C. Bonningue,\nC. S. Tsai, S. L. Vysotskii, and Yu. A. Filimonov, JETP\nLetters77, 567 (2003).[12] 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[13] A. V. Chumak, A. A. Serga, B. Hillebrands, and M. P.\nKostylev, Appl. Phys. Lett. 93, 022508 (2008).\n[14] Z. K. Wang, V. L. Zhang, H. S. Lim, S. C. Ng, M. H.\nKuok, S. Jain, and A. O. Adeyeye, Appl. Phys. Lett. 94,\n083112 (2009).\n[15] A. N. Myasoedov and Y. K. Fetisov, Sov. Phys. Tech.\nPhys.34, 666 (1989).\n[16] A. V. Voronenko, S. V. Gerus, and V. D. Haritonov, Sov.\nPhys. J. 31, 76 (1988).\n[17] A. A. Serga, T. Neumann, A. V. Chumak, and B. Hille-\nbrands, Appl. Phys. Lett. 94, 112501 (2009).\n[18] 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[19] M. P. Kostylev, A. A. Serga, T. Schneider, T. Neumann,\nB. Leven, B. Hillebrands, and R. L. Stamps, Phys. Rev.\nB76, 184419 (2007).\n[20] T. Neumann, A. A. Serga, B. Hillebrands, and M. P.\nKostylev, Appl. Phys. Lett. 94, 042503 (2009).\n[21] U.-H. Hansen, M. Gatzen, V. E. Demidov, and S. O.\nDemokritov, Phys. Rev. Lett. 99, 127204 (2007)."    },    {        "title": "1511.07747v1.In_situ_resonant_photoemission_and_X_ray_absorption_study_of_the_BiFeO3_thin_film.pdf",        "content": "In situ  resonant photoemission and X-ray absorption study of the BiFeO 3 \nthin film \nAbduleziz Ablata, Mamatrishat Mamat a∗,Yasin Ghupurb, Rong Wub, Emin \nMuhemmedb, Jiaou Wangb, Haijie Qianb, Rui Wub, Kurash Ibrahimb∗ \n \naSchool of physical Science and Technology, Xinjiang University, Urumqi 830046 \nChina \nbBeijing Synchrotron Radiation Facility, Institute of High Energy Physics, CAS, \nBeijing 100049, China  \nAbstract   \nMultiferroic bismuth ferrite (BiFeO 3) thin films were prepared by \npulsed laser deposition (PLD) technique. Electronic structures of the film \nhave been studied by  in situ  photoemission spectroscopy (PES) and x-ray \nabsorption spectroscopy (XAS). Both the Fe 2 p PES and XAS spectra \nshow that Fe ion is formally in +3 valence state. The Fe 2 p and O K edge \nXAS spectra indicate that the oxygen octahedral crystal ligand field splits \nthe unoccupied Fe 3 d state to t 2g↓ and e g↓ states. Valence band Fe 2 p-3d \nresonant photoemission results indicat e that hybridization between Fe 3 d \nand O  2p plays important role in the multiferroic BiFeO 3 thin films.  \n1. Introduction \nMultiferroics are a group of ma terials that a same system \nsimultaneously having different fo rms of ferroic properties such as \nferroelectricity, ferromagnetism, and fe rroelasticity [1]. In multiferroics, \n                                                        \n∗ corresponding author \n E-mail Address: kurash@ihep.ac.cn  (K. Ibrahim) \n mmtrxt@xju.edu.cn  (M.Mamat) magnetization can be tuned by app lied electric field, and electron \npolarization by applied magnetic field. These effects called \nmagnetoelectric effect[2]. Few multiferroic materials, YMnO 3, PbVO 3, \nBiCrO 3, BiMnO 3 and BiFeO 3 (BFO), exhibit as of natural occurring \nphases [3-5]. Among them, the BFO is the only material that shows \nmagnetoelectric effects at room temper ature[6]. The BFO is considered to \nbe a promising candidate for electr onic device application benefitting \nfrom its high Curie ( TC～825 °C) and Néel ( TN～370 °C) phase transition \ntemperatures[4]. Compared to its bulk form ，BFO thin film shows \napparently improved ferroelectric and magnetization characters[7].    \nUnderstanding the multiferroic pr operties of BFO in terms of its \nelectronic structure is important, because both the electric and magnetic \nproperties closely relate to it. An ope n problem with BFO is that it loses \neasily the ferroic characters with high current leakage. The current \nleakage was attributed to small amount of Fe2+ ion or oxygen vacancies. \nTo solve this problem, there has been investigation with different growth \ntechniques and various doping methods [8-10]. Decrease or elimination \nof the Fe2+ ion impurity is usually judged through the trends of decreased \ncurrent leakage and enhanced ferroel ectric hysteresis loops. Experimental \ninvestigations in term s of valence band electr onic structure are rare \n[11,12]. On the other hand, results in  the electronic band structure of BFO \nby theoretical calculation are controve rsial [13,14]. Density of state analysis [13] shows that the Bi 6 p state locates above 4eV of the gap, and \nthe origin of the ferroelect ricity is ascribed as O 2 p–Bi 6 p dynamic \nhybridization. Other studies  [14] show the Bi 6 p state has a contribution \non both valence and cond uction bands through hybrid ization with both Fe \n3d and O 2 p states. In the current work we aim at the investigation of the \nvalence band states of the BFO by means of photoemission method to \nuncover possible contributing routes in regard to the controversial \nconclusions.   \nWe first prepared BFO thin films by PLD method for in situ  PES \nand XAS measurement. The advantage of thin film over bulk state is at \nleast threefold. Firstly , thin film can free from charging effect by \ncontrolling film thickness that is a main unwanted factor in the PES \nmeasurements. Secondly, thin films can be grown by thickness \ncontrolling, as well as with slightly varied stoichiometry by regulating \nambient oxygen partial pressure. In the last, the in situ  prepared thin films \ncan free from surface contamination th at is crucial for PES measurements, \ndue to the measurement it self is highly surface sen sitive. The results show \nthe films prepared under highe r oxygen partial pressure is near to \nstoichiometry, and the Fe ion is form ally in valence-three state. Oxygen K \nedge XAS and vale nce band resonant photoemi ssion spectroscopy (RPES) \nthrough Fe 2 p core level excitation indi cate the conduction and valence \nbands are mainly of Fe 3 d and O 2 p hybridization state. 2. Experiment \nCeramic BFO  target of 20 mm diameter with 2 mm thickness  has \nbeen prepared by sintering 1.01:1 mixtures of Bi 2O3 (99.99 at.%) and \nFe2O3 (99.99 at.%) at 820 °C, following the routine way of solid state \nreactions. Slight excess of Bi 2O3 has the role to compensate the \npreferential loss of bismuth during sintering. X-ray diffraction (XRD) \nresult of the target material indicates that the sample is in single phase of \nBFO. The BFO films are prepared by PLD method on the Pt \n(111)/TiO 2/SiO 2/Si (100) and quartz substrat es in a chamber connected \nto the photoemission system at 4B9B  beam line of Beijing Synchrotron \nRadiation Facility [15]. \nPrior to deposition, the Pt (111)/TiO 2/SiO 2/Si (100) substrates were \npreheated at 700 °C to eliminate surface contaminations. To grow  near \nstoichiometric BFO films, the subs trates were kept at 550°C and the \nfilms were grown at different oxyge n partial ambient pressures. The \nlaser fluence was 2.3J/cm2 with pulse repetition rate 1.5Hz. A \nstoichiometric film was obtaine d at oxygen partial pressure Po 2=5.6 Pa, \nconsistent with reported results [ 16,17]. After deposition, the sample \nwas transferred to the PES chambe r under a background pressure of \n~10-8 Pa. The overall energy resoluti on was 0.2–0.7 eV, depending on \nthe selected monochromatic photon energies. Photoemission spectra \nwere calibrated to the in-substrate Pt 4 f7/2 peak at 71.2 eV. The XAS measurement carried out with tota l electron yield detection mode.  \n3. Results and Discussion \n3.1 PES characteriza tion of the films \nUsually, PES measurement can be used  to check whether a sample is \nstoichiometric or not and to estimate relative atomic ratios in the system. \nDetermination of ionic valence state is through indirect way by measuring \nenergy positions of concerned elements ’ main peaks and related satellite \npeaks, by inspecting the ratios of p eaks intensity of constitute atoms. \nFig.1 illustrates a wide range scan PES spectrum of the BFO film at 900 \neV that gives overall status informa tion about the constituent elements of \nthe system. The dominating signals ar e of core-level photoemission peaks \nand Auger lines of Bi, Fe, and O atoms. Absence of the C 1 s signal at \nabout 285 eV binding energy indicates  that the surface is free from \ncontamination, thanks to the in situ operations. Characteristic core-level \nphotoemission peaks of Fe 2 p3/2 711.3 eV, Fe  2p1/2 725.4 eV,  Bi 4p3/2 \n680.7 eV, Bi 4 d5/2 441.4 eV, Bi 4 d3/2 465.7 eV, Bi 4 f7/2 159.3 eV, Bi 4 f5/2 \n164.6 eV, O 1 s 530.3 eV  and Pt 4 f at 71.2 eV from s ubstrate, and Auger \ndecay channels of Fe L 3M23M23, Fe L 3M23M45, Fe L 3M45M45, O KVV are \nlabeled on the spectrum. \nThe O 1 s narrow range scan spectrum ( Fig.2 a) with a higher energy \nresolution and better statistics show asymmetric peaks near 530 eV. The \nspectrum can be fitted by two peaks locating at lower and higher binding energies (LBE and HBE), respectively.  The LBE peak is attributed to the \nO 1s of perfect BFO phase, while th e HBE one relates with oxygen \ndefects in samples [18]. Th e peak area of the LBE is  larger than that of \nHBE, indicating the sample was in  low oxygen vacancy, the BFO is \nmainly with perfect stoi chiometric constituent. \nThe Fe 2 p core level photoemission spectrum  (Fig.2 b) subjects to be \napplied to determine the valence state of Fe ions, where it splits into Fe \n2p3/2 at 711.3 eV and Fe 2 p1/2 at 725.4 eV as result of spin–orbit coupling \neffects. The Fe 2 p3/2 peak shape resulted in fitting is normally inspected \nto determine if the cation is in uni-v alence or in multi-valence state, \nwhere the Fe 2 p3/2 is decomposed into symmetric superposition \ncomponents of Fe2+ and Fe3+[19]. But recent litera ture report suggests \nthat decomposing the Fe 2p into symmetric components is not feasible \n[20]. The way to fix the formal vale nce state is by sat ellite peak of Fe3+ \nand Fe2+ in PES, because of different d orbital electron configurations, \nrelaxation of the Fe2+ and Fe3+ show satellite peaks at 6 eV and 8 eV, \nrespectively, above their 2 p3/2 main peaks [16]. The measured spectrum in \nFig.2(b), a satellite pe ak 8eV above the Fe 2 p3/2 711.3 eV, confirms that \nthe iron atom is formally in Fe3+ state.  In the limits of the PES \nmeasurement sensitivity to molar porti on in system, there is no apparent \nevidence showing up fo r the existence of Fe2+ ion in the underlying BFO \nthin film.  The Bi 4 f7/2 and 4 f5/2 spin-orbit coupling components locating at \n159.3 eV and 164.6 eV respectively, in  Fig.2 (c), are consistent with \nliterature values of Bi3+[21]. There are no shoulders  appearing either at \nhigher or lower binding energy sides as being clues for the existence of \nhigher than Bi3+ oxidized or lower than Bi3+ reduced metallic bismuth \nstates[22]. The estimated mo lar ratio of Bi to Fe fr om the peak area of the \nBi 4f and Fe 2 p, which are normalized to r espective atomic orbital cross \nsections at the specific excitation energy[ 23], is near to 1:1 indicates that \ncomposition of the thin  film under proper Po 2 pressure maintained almost \nthe same with its target precursor. \n3.2. O K edge XAS and Valance band photoemission  \nThe XAS provides information on th e excitation of a core electron \ninto unoccupied states as function of  photon energy, that is about a cross \nsectional variation of measured phot oelectron DOS versus photon energy. \nIt implies unoccupied density of state and crystal field sp litting effect, and \nmost importantly provide signatur e on hybridization between atoms. \nThe left and right pane ls of Fig.3 show the valence band PES and O K \nXAS spectra of the BFO thin film . Valence band PES represents the \noccupied DOS distribution below Fe rmi level. In Fig 3(a) the VB \nspectrum can be divided into tw o main block structures labeled as  α, β, γ, \nδ from Fermi edge to below ~9 eV  and a broad peak  at ~12 eV. The \nfeature around 12 eV originates from Bi 6 s states according to the DOS result by first principle calculation [2 4]. In the coming section, we will \ndiscuss an enhanced feature of the four α, β, γ, δ structures by resonant \nphotoemission at Fe L edge.  \nThe O K-edge XAS spectra in Fig 3(b) provides information on the \nhybridization of O 2 p with Fe 3 d states.  Under  ionic picture, the O 2 p is \nconsidered fulfilled with six electr ons. Then following the definition of \nXAS, we should observe any absorption effect at the O 1s region, but \nactually we do observe a copious stru cture in this region. The observed \nresults imply a certain portion of empty O 2 p orbitals with nontrivial \nprobability due to hybridization between the Fe 3 d [25], the hybridization \nresults in dynamic char ge transfer from O 2 p to Fe 3 d. The oxygen K \nedge further manifests an extended st ructure up to at least 17 eV above \nthe threshold, implying an electronic structure resulted in various effects \nsuch as ligand field splitting,  hybridization of the oxygen 2 p and \ntransition metal 3 d states. The region near to the absorption edge, labeled \nas a/, a and b, the most sensitive region to th ose effects, is expanded and \nshown as inset in Fig.3. Am ong them the features a/ and a have the key \nimportance for understanding inte r-atomic interactions. The a/ at ~530.8 \neV is attribut ed to the O 2 p+Fe 3d t2g↓ characteristic states,  the peak a to \nthe O 2 p + Fe 3 d eg↓ states, where ↓ denotes the minority spin states.  \nReason for how that O 2 p + Fe 3 d hybridization leads to the status \nquo a/, a structure is twofold. Firstly, the XAS is a local process in which an electron is promoted to an unocc upied electronic state, which couples \nto the original core level restricted by the electric dipole selection rule by \nparity consideration, which states  that the change in the angular \nmomentum quantum nu mber can only take ∆L=±1 values, while the spin \nkeeps unchanged [26]. For the excitation of an O 1 s electron of l =0, the \n∆L=±1 requirement means that only O 2 p character l =1 can be reached, \nand an inter-atomic O 1 s—Fe 3 d transition is not a llowed. In terms of \nionic picture for oxygen in oxides, this  prohibits, theoretically, occurrence \nof O 1 s—O 2 p absorption. These apparent idealized situations do not \nhold regarding the experimental reality, by showing up a wide peak envelope in 529~535 eV region with fine inte rnal structures as of \ncomplicated interaction effects. The se experimental facts imply that the \nrigid ionic picture is not approp riate for understanding the observed \nresults, and the results re veal that there exist str ong interactions of O ions \nwith the central transition metal Fe  ions. The interactions open up \nchannels available for the transition between O 1 s and Fe 3 d-like orbitals. \nSuch interaction (hybridization) bri ngs about transfer of certain amount \nof electron density, i.e., dynami c charge transfer from O 2 p to Fe 3 d \norbital that creates ho le-state in the O 2 p. The hole-state creation on the \nligand side is instrument al for understanding the experimental features \nobserved at the O K edge. Secondly,  the observed ~1.4 eV distance \nbetween the a\n/ and a peaks in the O 1 s XAS, a similar value to the crystal-field splitting 10D q induced between the two empty states t 2g↓ and \neg↓, is almost the same as observed in the Fe 2 p XAS spectra in Fig.4. \nThese imply that they are of Fe 3 d character and provide a solid base for \nthe concept of dynamic charge transfer through hybridization.   \nA common point for the peak b is to ascribe it as of Bi 6 sp \ncharacter[14], that is due to inte raction between the full filled O 2 p and \npartial occupied Bi 6 sp states allow to form a certain degree of covalent \nbonding between the O 2 p and Bi 6 sp. There exists also controversial \npoints of view regarding the b’s origin [27], wher e refers the peak b to as \nresult of the interactions  between the unoccupied O 2 p and Bi 5 d states. \nThe peak c at ~542 eV derives from hybridization of O 2 p and Fe \n4sp[28].  \n3.3. Fe 2p XAS and Fe 2p- 3d resonant photoemission  \nFig.4 shows the Fe 2 p XAS spectrum of BFO thin film together with \nthose of reference oxides of formal trivalent Fe 2O3(Fe3+), divalent FeO \n(Fe2+) [29], and plus the atomic multip let calculation result. The XAS \ndata of Fe 2O3 and FeO were shifted by ～3 eV to align with the \nmeasured one. Calculation performed with atomic multiplet model in \noctahedral symmetry for di fferent 10Dq values varyi ng from 1.0 to 1.8eV, \na multiplet splitting energy range 10Dq suited for the 3d transition metal \n2p edges [30]. The Slater integrals wer e scaled to 80% of  the associated \natomic values. For simulate lifetime  effects and instrument resolution, discrete multiplets were broadened w ith a Lorentzian of 0.3 eV and a \nGaussian of 0. 3 eV at both L3 and L2 edges. A good agreement achieved \nbetween the calculation and experime nt for a value at 10Dq~1.6 eV. \nThe measured energy separation between t 2g↓ and e g↓ states, both for \nL3 and L2 levels, about 1.4 eV corresponds  to the crystal field splitting \nenergy 10Dq. It leads the Fe 3 d5 electrons to high-spin configuration \nt2g3eg2 (6A1g) [31]. Resemblance of the BFO line shapes to that of Fe 2O3, \nand a significant deviation from FeO in dicate the Fe ions are formally in \ntrivalent (Fe3+:3d5) state, confirming the ev idence drawn from the PES. \nResonant photoemission (RPES) spec tra shown in Fig.5(a) recorded \nbetween photon energy 702 eV and 718 eV spanning the Fe 2p 3/2 \nthreshold region of XAS in Fig.4 to look at the va lence electronic \nstructure relation of Fe with its surrounding ligand O atoms. The most \nresonantly enhanced spectrum at 710.5 eV and two off-resonance spectra \nbelow and above the threshold are shown in Fig.5(b) to contrast resonant \nand off-resonant effects. The spectr a are normalized to the storage ring \ncurrent and the spectrum sweeping numbers. The vale nce band spectra \nconsist of four main  structure labeled as α, β, γ and δ, the same as those \nobserved in Fig.3(a).  \nAn overall fuzzy enhancement in sp ectral intensity of the peaks and \nthen decreasing when the excitati on energy pass through the Fe 2p 3/2 \nthreshold region is clear in the spectra  shown in Fig.5(b), that reflect the sensitiveness of the Fe 3 d characteristic partial DOS to the threshold \nexcitation. A fuzzy broad block of p eaks, which simultaneously varies all \ntogether in lack of independently enhanced peak shapes below 0-14 eV \nFermi level, reaches maximum at least with a factor of ten at excitation \nenergy 710.5 eV relative to those exc ited below and above at this energy. \nThe valence band peaks photoemission intensity increase with such fuzziness as whole, inst ead of observing a specif ic peak enhancement, \nreveals at least two kinds of info rmation. The first is that the \nphotoemission in the on-resonance re gion has apparently additional \ncontribution source from that of photoemission in the off-resonance \nregion. The second remarkable point re veals that the resonantly enhanced \nintensity shape of valence band s strongly relates to surviving \nenvironment of Fe 3 d electrons, instead of the Fe 3 d partial density of \nstate per se . \nThe resonant photoemission enhancemen t in an order of magnitude at \nthreshold excitation is a generic situa tion for most elemen ts in condensed \nmatter systems, and its origin can differ for different systems. One can \nsay at least one common cause about the significant difference between \nthe off-resonance and on-resonance is that the former mainly results in \nthe single-photon single-channel process, and the latter through \nsingle-photon multiple-channel proces ses. Here the single-photon is \nreferred to as monochromatized si ngle energetic photons, single-channel process means the measured DOS corresponds to single-photon \nsingle-channel (or direct) photoemissi on process. While the single-photon \nmultiple-channel (or indirect) processe s make an additional aggregative \ncontribution from multiple-channel de-excitations against single photon \nexcitation, which are in addition to the DOS results  in the single-photon \nsingle-channel process.  \nThe above discussed processes are schematically shown in Fig.6. \nAmong them, the single-photon single-ch annel process refers to as the \ndirect photoemission ①. In this process, whenev er the photon energy is \nbelow or above the threshold region,  the measured intensity of the \nvalence band comes only contribution fr om direct photoionization of the \nvalence state electrons, no other processes are count able for it. When the \nexcitation photon energy match with the potential energy of certain inner \nshell, the resonantly excited electron from that shell leads to the processes \n② or ③. These last two processes are indirect photoemissions, their \ncontributions add up to th at of the normal process ①, at the end they \ngive the measured results where the in tensity has an enormous increase. \nThe single-photon multiple-channel pro cesses at the threshold region are \nactually the summation of ① + ② + ③ + . . .. Here in the both \nprocesses ② and ③, the neutral initial state [ndvbp3 26], where the vb \nrefers to all the rest valence ba nd electrons except that of Fe 3 dn, the Fe \n2p state electron absorbs a photon and excites into intermediate [1 53 2+ndvbp ]* or [1 2 53 32+ndvbsp ]* excited neutral states by creating a \nprimary vacancy in the Fe 52p core shell and prom oting one electron to \n13+ndvb  state. These states then de-excit e through corresponding channels \ninto respective final states. The *1 5] 3 2[+ndvbp  state de-excites into \n−−] 3 2[1 6 ndvbp  final state through super-Cost er-Kronig channel [32], and \nthe *1 2 5] 3 32[+ndvbsp  state into −]3 32[1 6 ndvbsp  through Coster-Kronig \nprocess [33]. These are the main origins for the observed intensity \nincrease at threshold excitati ons in photoemission measurement. \nUnravel electronic structural or igin of the density of state \nenhancement as a whole with the thre shold excitation in the valence band \nregion, where the valence band states marked as α, β, γ and δ, requires to \nunderstand the relationship between Fend3 and the rest vb electrons in \nthe region. In other words, need  to have an appreciation of \nFend3surviving environment. The resona nt valence band  photoemission \nmeasured by excitation of inner-she ll electron which leaves a core-hole \nbehind play the role as of pump-probe, by wh ich one is able to inspect the \nelectronic structural interaction of constituent elements in the valence \nregion. As shown in Fig.5(a), the fo ur feature through enhanced in order \nscale, they are with as whole but w ithout a single peak with apparent \nsharpening, seemingly if they were a single body. It implies there have \nexist a strong hybridization between Fe 3 d with its surrounding \nenvironment, such as O 2 p and Bi 6 p states.  One can approximately assign the four feature after carefu lly compare the on and off resonance \nspectrum in Fig.5(b). According to the energy diagram of Fe 3 d5 \nstates[31], the resonan tly enhanced feature  α is assigned to e g↑(↑\ndenotes the majority sp in) states of Fe 3 d, feature β to the t 2g↑ states \nwith a weakly hybridized O 2 p state (mostly of Fe 3 d like character). The \nenergy separation between these two stat es consistent with the energy \nseparation between the unoccupied e g↓ and t 2g↓ states as shown in \nFig.3(b). The feature γ to the hybridization of Fe 3 d-O 2p bonding states \nsince the intensity of this fe ature increased through Fe 2 p-3d threshold \nand reach a maximum at on resonance region.  \nThe above discussed resonant phot oemission investigation results \nshow that the whole ranges of ndvb3 bands have mixed with Fe 3 d, O 2 p \nand Bi 6 p states. In our prev ious work about BiFe 1-xMn xO3, also support \nthere have a strong hybridi zation between the Fe 3 d states with O 2 p \nstates in pure BFO and the hybridiza tion decreased with increasing Mn \ncontent[34]. The resonate photoemissi on spectra in Fig.5 strongly support \nour assumption th at whole the marked region ( α,β,γ and δ) have a Fe 3 d \ncharacters through hybridization with other orbitals.  \n4. Conclusion  \nIn conclusion, the electronic struct ure of multiferroic BFO has been \ninvestigated by using the PES and XAS. The measured PES survey \nspectrum, Fe 2 p PES and XAS spectra show that  the Fe ions are formally trivalent (3d5: t2g3↑eg2↑). The measured valen ce band RPES and O K-edge \nXAS spectra show that the valen ce band and conduction band mainly \nconsists of Fe 3 d and O 2 p states through hybridization, and d-d transition \nbetween the valance ba nd and conduction band play a key role in \nelectronic properties of BFO. \n \nAcknowledgements \nThis study was financially supported by Nat ional Natural Science \nFoundation of China (Grant No.11164026,  61464010, 11375228, \n61366001), University Research Pr oject of Xinjiang Uyghur Autonomous \nRegion (XJEDU2014I002) and Doctoral  Startup Foundation of Xinjiang \nUniversity (BS130110). The author s also thank beamlines 4B9B and \n4B7B of BSRF for providing the beam time. R. Wu acknowledge \nsupports under the proj ects 11204303 by NSFC. \n Reference \n[1] W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature 442, 759 (2006). \n[2] M. Fiebig, T. Lottermoser, D. Frohlich, A. V . Goltsev, and R. V. Pisarev, Nature 419, 818 (2002). \n[3] N. A. Hill, J Phys Chem B 104, 6694 (2000). \n[4] R. Ramesh and N. A. Spaldin, Nature Materials 6, 21 (2007). \n[5] N. Kumar Swamy, N. Pavan Kumar, P. V . Reddy, M. Gupta, S. S. Samatham, D. Venkateshwarulu, \nV . Ganesan, V . Malik, and B. K. Das, J Therm Anal Calorim 119, 1191 (2015). \n[6] R. S. Fishman, J. H. Lee, S. Bordács, I. Kézs márki, U. Nagel, and T. Rõõm, Physical Review B 92, \n094422 (2015). \n[7] J. Wang  et al. , Science 299, 1719 (2003). \n[8] Z. C. Quan, W. Liu, H. Hu, S. Xu, B. Sebo, G.  J. Fang, M. Y . Li, and X. Z. Zhao, Journal of \nApplied Physics 104 (2008). \n[9] S. K. Singh, K. Maruyama, and H. Ishiwara, Applied Physics Letters 91 (2007). \n[10] S. K. Singh, H. Ishiwara, and K. Maruyama, Applied Physics Letters 88 (2006). \n[11] S. Das, S. Basu, S. Mitra, D. Chakravorty, and B. N. Mondal, Thin Solid Films 518, 4071 (2010). \n[12] L. You, N. T. Chua, K. Yao, L. Chen, and J. L. Wang, Physical Review B 80, 024105 (2009). \n[13] S. J. Clark and J. Robertson, Applied Physics Letters 90 (2007). \n[14] H. Wang, Y. Zheng, M. Q. Cai, H. T. Huang, and H. L. W. Chan, Solid State Communications 149, \n641 (2009). \n[15] K. Ibrahim  et al. , Physical Review B 70, 224433 (2004). \n[16] L. Bi, A. R. Taussig, H. S. Kim, L. Wang, G. F. Dionne, D. Bono, K. Persson, G . Ceder, and C. A. \nRoss, Physical Review B 78, 104106 (2008). \n[17] H. Bea, M. Bibes, S. Fusil, K. Bouzehouane, E. Jacquet, K. Rode, P. Bencok, and A. Barthelemy, \nPhysical Review B 74, 020101 (2006). \n[18] C. Rath, P. Mohanty, A. C. Pandey, and N. C. Mishra, Journal of Physics D-Applied Physics 42 \n(2009). [19] Z. Quan, H. Hu, S. Xu, W. Liu, G. Fang, M. Li, and X. Zhao, Journal of Sol-Gel Science and \nTechnology 48, 261 (2008). \n[20] A. T. Kozakov, A. G. Kochur, K. A. Googlev, A. V . Nikolsky, I. P. Raevski, V . G . Smotrakov, and V . \nV . Yeremkin, Journal of Electron Spectroscopy and Related Phenomena 184, 16 (2011). \n[21] H. Akazawa and H. Ando, Journal of Applied Physics 108 (2010). \n[22] C. Jovalekic, M. Pavlovic, P. Osmokrovic, and L. Atanasoska, Applied Physics Letters 72, 1051 \n(1998). \n[23] J. J. Yeh and I. Lindau, Atom Data Nucl Data 32, 1 (1985). \n[24] A. T. Kozakov, K. A. Guglev, V . V . Ilyasov, I. V . Ershov, A. V . Nikol'skii, V . G. Smotrakov, and V . \nV . Eremkin, Physics of the Solid State 53, 41 (2011). \n[25] J. van Elp and A. Tanaka, Physical Review B 60, 5331 (1999). \n[26] R. D. Cowan, The theory of atomic structure and spectra (Univ of California Pr, 1981), V ol. 3. \n[27] T. J. Park, S. Sambasivan, D. A. Fischer, W.  S. Yoon, J. A. Misewich, and S. S. Wong, J Phys \nChem C 112, 10359 (2008). \n[28] R. Saeterli, S. M. Selbach, P. Ravindran, T. Grande, and R. Holmestad, Physical Review B 82 , \n064102 (2010). \n[29] J. P. Crocombette, M. Pollak, F. Jollet, N.  Thromat, and M. Gautiersoyer, Physical Review B 52, 3143 (1995). \n[30] E. Stavitski and F. M. F. de Groot, Micron 41, 687 (2010). \n[31] M. Abbate  et al. , Physical Review B 46, 4511 (1992). \n[32] L. C. Davis and L. A. Feldkamp, Physical Review B 23, 6239 (1981). \n[33] W. Bambynek, C. D. Swift, Cras eman.B, H. U. Frend, P. V . Rao, R. E. Price, H. Mark, and R. W. \nFink, Rev Mod Phys 44, 716 (1972). \n[34] Abduleziz Ablat, Emin Muhemmed, Cheng Si, Jiaou Wang, Haijie Qian, Rui Wu, Nian Zhang, \nRong Wu, and Kurash Ibrahim Journal of nanomaterials 2012 , 3238 (2012). \n \n \n  \nFigure Caption \n \nFig.1. PES survey spectrum of BFO\n films onto a silicon substrate. \n Fig.2. PES survey spectrum of (a)O 1s, (b)Fe 2p and (c) Bi 4f  lines of the BFO\n films. The molar \nratio of Bi and Fe was calculated by normalized peak area of Fe Fe 2p 3/2 and Bi 4f 7/2 (see text for \ndetails). \n \nFig.3 Valence band photoelectron and O 1s X-ray absorption spectrum of the BFO  films. The \ninsets at right panel shows an e xpended image of the XAS spectrum. \n \nFig.4 Comparison of the Fe 2p XAS of BFO films to those of FeO[29], Fe 2O3[29] and atomic \nmultiplet calculation. \n \nFig.5 (a) Valence band spectrum of BFO  recorded with photon energy across the Fe 2p-3d \nresonance. (b) On resonance (hv=710.5eV) and off resonance (hv=702.3eV and 717.5eV) PES \nspectra of BFO.   \nFig.6 Schematic expression of (1) direct photoemission and (2), (3) indirect photoemission \nprocesses that make major contributions to the measured valence band DOS distribution at around \nFermi level upon excitation of a core level electron in threshold region.  \n  \nFig.1  \n \n \n  \n \n  \nFig.2\n \n \n \n  \n \n \n \n  \n \n \n \n \nFig.3 \n \n  \n \n \n \n \n \n  \n \n  \n \n Fig.3 \n \n  \n \n \n \n  \n \n  \n \n \n \n  \n \n \n \n  \n \n  \n \nFig.4 \n \n \n Fig.5 (a) and (b) \n        \n \n \n Fig.6  \n \n \n \n "    },    {        "title": "1409.1811v1.Theoretical_assessment_of_boron_incorporation_in_nickel_ferrite_under_conditions_of_operating_nuclear_pressurized_water_reactors__PWRs_.pdf",        "content": "arXiv:1409.1811v1  [cond-mat.mtrl-sci]  5 Sep 2014Theoretical assessment of boron incorporation in nickel fe rrite\nunder conditions of operating nuclear pressurized water re actors\n(PWRs)\nZs. R´ ak,∗E. W. Bucholz, and D. W. Brenner\nDepartment of Materials Science and Engineering,\nNorth Carolina State University, Raleigh, 27695-7907\n(Dated: March 26, 2021)\nAbstract\nA serious concern in the safety and economy of a pressurized w ater nuclear reactor is related to\nthe accumulation of boron inside the metal oxide (mostly NiF e2O4spinel) deposits on the upper\nregions of the fuel rods. Boron, being a potent neutron absor ber, can alter the neutron flux causing\nanomalousshiftsandfluctuationsinthepoweroutputofther eactorcore. Thisphenomenonreduces\nthe operational flexibility of the plant and may force the dow n-rating of the reactor. In this work\nan innovative approach is used to combine first-principles c alculations with thermodynamic data to\nevaluate the possibility of B incorporation into the crysta l structure of NiFe 2O4, under conditions\ntypical to operating nuclear pressurized water nuclear rea ctors. Analyses of temperature and pH\ndependence of the defect formation energies indicate that B can accumulate in NiFe 2O4as an\ninterstitial impurity and may therefore be a major contribu tor to the anomalous axial power shift\nobserved in nuclear reactors. This computational approach is quite general and applicable to a\nlarge variety of solids in equilibrium with aqueous solutio ns.\n∗zrak@ncsu.edu\n1I. INTRODUCTION\nA major impediment that prevents pressurized water reactors (P WRs) from operating at\nhigher duty and longer cycles is the accumulation of boron within meta l oxide scales that\ndeposit on the upper spans of fuel assemblies. Boron, in the form o f boric acid (H 3BO3), is\nadded to PWR coolant to control the neutron flux while lithium hydrox ide (LiOH) is also\ndosed to control the acidity of the coolant and to reduce corrosio n. Because of the large\nneutron absorption cross section of10B, a small amount of accumulated B is sufficient to\ncause an abnormal decrease in the neutron flux, which shifts the p ower output toward the\nbottom half of the reactor core. This phenomenon, known as axial offset anomaly (AOA),\nhasbeen observed inhigh-duty reactorsthat runlong fuel-cycles . In extreme form, AOA can\ndecrease the reactor shutdown margin sufficiently to force major power reduction leading to\nsubstantial economic losses [1, 2].\nAOA modeling efforts traditionally assume that boron deposition within the metal oxide\nscales (which are commonly referred to as CRUD, an acronym for Ch alk River Unidentified\nDeposits) occurs predominantly through precipitation of lithium bor ate compounds such as\nLiBO2, Li2BO7, Li2B4O7[2–7]. Although the retrograde solubility of these borates could\nexplain the “lithium return” experienced during plant shutdown, the y have never been\nobserved experimentally in PWR CRUD. Using M¨ ossbauer spectrosc opy together with XRD\non CRUD scrapes recovered from high duty PWRs, Sawicki identified the precipitation\nof Ni2FeBO5as a possible mechanism for B deposition [8, 9]. Mesoscale CRUD models\ndeveloped by Short et al., assume that supersaturation of boric acid leads to precipitation\nof boron trioxide (B 2O3) within the CRUD [10].\nInrecentwork, wecombined ab initio calculationswithexperimental formationenthalpies\nto investigate the incorporation of B into the structure of nickel f errite (NiFe 2O4, NFO) as\na potential new mechanism for B deposition within CRUD [11]. Assuming s olid-solid (and\nsolid-gas) equilibrium between nickel ferrite and elemental reservo irs of Fe, Ni, B (and O 2\ngas) we found that it is thermodynamically favorable for B to form se condary phases with\nFe, Ni, and O ( e. g.B2O3, Fe3BO5, and Ni 3B2O6) instead of entering the NFO structure as\na point defect. Building on previous works [12–14], the present stu dy attempts to deal with\nthe same question, however, here the defect formation energies are evaluated assuming solid-\nliquidequilibriumbetweenNFOandthesurroundingaqueoussolutionof Ni, Feanddissolved\n2boric acid (H 3BO3). To set up solid-liquid equilibrium, the chemical potentials of individual\naqueous species are defined as a function of temperature, press ure, and concentration and\nare linked to the chemical potentials of the ionic species in solid. This ne w scheme allows for\nthe evaluation of defect formation energies under conditions that are specific to operating\nnuclear PWRs. The approach is quite general and applicable to a large variety of solids in\nequilibrium with aqueous solutions.\nII. DEFECT FORMATION ENERGIES UNDER SOLID-LIQUID EQUILIBRI UM\nThe first-principles calculations required for the present study ha ve been carried out\nwithin the Density Functional Theory (DFT) using the same computa tional parameters and\ncrystal models that are specified in Ref. 11.\nThe formationof a defect in a crystalline solid can be regarded as an e xchange of particles\n(atoms and electrons) between the host material and chemical re servoirs. The formation\nenergy of a defect Din charge state qcan be written as [15, 16]\n∆Hf(Dq) =E(Dq)−E0+/summationdisplay\niniµi+q/parenleftBig\nEF+Edef\nVBM/parenrightBig\n. (1)\nIn Eq. (1), E(Dq) andE0are the total energies of the defect-containing and defect free\nsolids, calculated within the DFT. The third term on the right side of Eq . (1) represents the\nchange in energy due to the exchange of atoms between the host c ompound and the chemical\nreservoir, where µiis the atomic chemical potential of the constituent i(i= Ni, Fe, or B).\nThe quantity nirepresents the number of atoms added to ( ni<0) or removed from ( ni>0)\nthe supercell. The quantity EFis the Fermi energy referenced to the energy of the valence\nband maximum (VBM) of the defective supercell, Edef\nVBM. This value is calculated as the\nVBM energy of the pure NFO, corrected by aligning the core potent ial of atoms far away\nfrom the defect in the defect-containing supercell with that in the defect free supercell [16].\nThe quantity qrepresents the charge state of the defect, i. e.the number of electrons\nexchanged with the electron reservoir with chemical potential EF.\nUnder solid-liquid equilibrium conditions, the chemical potentials of the ionic species in\nthe solid, µizi, are equal to the chemical potential of the aqueous species in the saturated\nsolution, µizi,aq. To derive an expression for defect formation energy, Eq. (1) ha s to be\nwritten in terms of ionic chemical potentials, µizi, instead of atomic chemical potentials, µi.\n3This can be accomplished by adding and subtracting the term/summationtext\ninizi(EF+E0\nVBM) from\nEq. (1). This term can be interpreted as the energy necessary to exchange electrons between\ntheelectronreservoir andtheatomicspecies inthepureNFO.Ifwe combine thisenergywith\nthe atomic chemical potential (third term in Eq.(1)) we obtain the ion ic chemical potential\nof species in NFO:\n/summationdisplay\niniµi+/summationdisplay\ninizi/parenleftbig\nEF+E0\nVBM/parenrightbig\n=/summationdisplay\niniµizi. (2)\nIn Eq. (2) zirepresents the ionic charge, i. e. the number of electrons exchanged with\nthe electron reservoir to create the ionic species in NFO, and E0\nVBMis the energy of the\nVBM in the pure NFO to which the energy of the electron reservoir ( EF) is referenced.\nUsing Eq. (2) together with the solid-liquid equilibrium condition, ( µizi=µizi,aq), the defect\nformation energy becomes:\n∆Hf(Dq) = ∆E(Dq)+/summationdisplay\niniµizi,aq−/summationdisplay\ninizi/parenleftbig\nEF+E0\nVBM/parenrightbig\n+q/parenleftBig\nEF+Edef\nVBM/parenrightBig\n(3)\nwhere ∆E(Dq) is the energy difference between the defect containing and defec t free su-\npercells. Therefore, to calculate the defect formation energy, t he chemical potentials of the\naqueous species have to be evaluated. The scheme described abov e has the advantage that it\ndecouples the ionic charge from the charge state of the defect; c harge neutrality is achieved\nthrough exchange of electrons with the electron reservoir with en ergy equal to the Fermi\nlevel.\nIII. AQUEOUS CHEMICAL POTENTIALS\nThe chemical potential of the aqueous ions, µizi,aq, can be written as the sum of the\nstandard chemical potential, µ0\nizi,aq, and a temperature dependent term\nµizi,aq=µ0\nizi,aq+RTlnaizi. (4)\nIn Eq. (4), R= 8.314 J/K·mol is the universal gas constant and aiziis the activity of\nthe ionic species iziin the aqueous solution. In the present case, because NFO is weakly\nsoluble in water, the activity of the ionic species can be approximated by the concentration\nof ions in the solution.\n4The standard chemical potentials of aqueous cations can be evalua ted using thermochem-\nical data combined with theoretical total energies calculated within the DFT framework. If\nwe consider the reaction\nM(solid)+zH+(aq)→Mz+(aq)+z\n2H2(gas), (5)\nthe equilibrium condition can be written as\nµ0\nM,solid+zµ0\nH+,aq=µ0\nMz+,aq+z\n2µ0\nH2,gas (6)\nTherefore, µ0\nMz+,aqcan be expressed using the standard Gibbs energy of formation of ions\nin aqueous solution as\nµ0\nMz+,aq= ∆G0\nf/parenleftbig\nMz+,aq/parenrightbig\n+µ0\nM,solid+z/parenleftbigg\nµ0\nH+,aq−1\n2µ0\nH2,gas/parenrightbigg\n. (7)\nThe Gibbs energies of formation required for Eq. (7) are obtained f rom that SUPCRT\ndatabase [17, 18] which uses the revised Helgeson-Kirkham-Flower s equation of state to pre-\ndict the thermodynamic behavior of aqueous species at high temper ature and pressure [19].\nThe chemical potentials of the solid phases µ0\nM,solidare usually approximated by the total\nenergy per atom of the elemental solid calculated within the DFT fram ework. However, as\npointed out in earlier work, this approach suffers from incomplete er ror cancellation when\ntotal DFT energies of physically and chemically dissimilar systems are c ompared [20–23].\nTherefore, to compute the elemental-phase chemical potentials o f the Fe, Ni, B, and O, we\nextend the database of 50 elemental energies published by Stevan ovicet al.[21] to include\nB. To do this we add 26 B-containing binaries to the large fitting set of 252 compounds that\nhavebeenusedbyStevanovic et al.andsolvetheoverdetermined system of278equationsfor\n51 elements using a least-square approach, as described in Refs. 2 1 and 22. The calculated\nDFT energies and experimental formation enthalpies of the 26 B-co ntaining binaries are\nlisted in Table I, while the 51 elemental-phase chemical potentials are g iven in Table II in\nthe Appendix.\nThe last term in parenthesis in Eq. (7) can be evaluated using a Born- Haber-type cycle\nof hydrogen. The formation of an aqueous proton in water, H+(aq), is described by the\nreaction 1 /2H2(gas)→H+(aq)+e−(gas). The path of this reaction can be envisioned as\ndissociation of the H 2molecule followed by ionization of the H atom and dissolution of the\n5TABLE I. DFT energies and experimental enthalpies of format ion of 26 B-containing binaries that\nhave been added to the fitting set in Ref. 21, to calculate the e lemental-phase chemical potentials.\nTheoretical enthalpies of formation are also listed.\nCompound DFT energy (eV/atom) ∆ Hexp(eV/atom) ∆ Htheor(eV/atom)\nB2S3 -5.58 -0.52 -0.56\nB2O3 -8.02 -2.64 -2.57\nB6O -7.14 -0.78 -0.84\nBN -8.79 -1.31 -1.32\nBP -6.45 -0.60 -0.44\nCr3B4 -7.10 -0.42 -0.30\nCrB -7.12 -0.41 -0.27\nCrB2 -8.00 -0.43 -1.25\nFe2B -6.37 -0.35 -0.12\nFeB -6.50 -0.38 -0.17\nHfB2 -8.07 -1.14 -1.18\nMgB2 -5.11 -0.32 -0.36\nMgB4 -5.80 -0.22 -0.33\nMn2B -6.85 -0.32 -0.10\nMn3B4 -6.96 -0.35 -0.28\nMnB -7.01 -0.37 -0.30\nMnB2 -6.95 -0.33 -0.29\nNbB2 -7.45 -0.61 -0.78\nNi2B -4.96 -0.22 -0.32\nNi3B -4.67 -0.23 -0.27\nNi4B3 -5.25 -0.27 -0.34\nNiB -5.44 -0.24 -0.32\nTaB2 -8.12 -0.72 -0.76\nTiB -6.91 -0.83 -0.84\nTiB2 -7.30 -1.09 -1.06\nZrB2 -7.58 -1.12 -1.20\n6proton in water. Therefore, the quantity required in Eq. (7) can b e calculated as the sum\nof free energies of atomization, ionization, and solvation of H. All of these free energies are\navailable in the literature and the free energy of formation of aqueo us proton has recently\nbeen evaluated at 298K and 1 bar as 4.83 eV [24]. This value is used in th e present study.\nTo evaluate the chemical potential of aqueous species (Eq.(4)) an d the defect formation\nenergies (Eq. (3)), the remaining quantity that has to be determin ed is the ionic concentra-\ntion in aqueous solution. This could be obtained assuming a saturated solution with respect\nto NFO and using experimental solubility data combined with charge ne utrality require-\nments. However, because the experimental data related to NFO s olubility is limited, in the\npresent work we use [Fe3+] = 4.17×10−13and [Ni2+] = 1.66×10−14mol/dm3, concentrations\nthat are characteristic to an operating PWR [3, 4].\nTo assess the validity of the scheme described above regarding the standard chemical\npotentials as well as the predictive power of the elemental-phase ch emical potentials listed\nin Table II, we evaluate the Gibbs free energies of formation of NiO an d NiFe 2O4based on\nthe following reactions:\nNi2++H2O→NiO+2H+(8)\nNi2++2Fe3++4H2O→NiFe2O4+8H+(9)\nUnder equilibrium conditions, employing Eq. (7) to express the chemic al potentials of\nNi2+and Fe3+, and considering that µ0\nH2+1/2µ0\nO2=µ0\nH2O(with 1/2µ0\nO2=µ0\nO), the change\nin Gibbs energies of reactions (8) and (9) are:\n∆G0(NiO) =µ0\nNiO,s−∆G0\nf/parenleftbig\nNi2+,aq/parenrightbig\n−∆G0\nf(H2O)−µ0\nNi,s−µ0\nO (10)\n∆G0(NiFe2O4) =\n=µ0\nNiFe2O4,s−∆G0\nf(Ni2+,aq)−2∆G0\nf(Fe3+,aq)−4∆G0\nf(H2O)−µ0\nNi,s−2µ0\nFe,s−4µ0\nO\n(11)\nIn Eqs. (10) and (11) µ0\nNiO,sandµ0\nNiFe2O4,srepresent the chemical potentials of solid\nNiO and NiFe 2O4and they are approximated by the total DFT energies per formula u nit\n(f. u.). The Gibbs free energies of formation of aqueous Ni2+and Fe3+as well as the free\n7/s50/s56/s48 /s51/s48/s48 /s51/s50/s48 /s51/s52/s48 /s51/s54/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48\n/s78/s105/s70/s101\n/s50/s79\n/s52\n/s32/s32/s71/s105/s98/s98/s115/s32/s102/s114/s101/s101/s32/s101/s110/s101/s114/s103/s121/s32/s40/s107/s74/s47/s109/s111/s108/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s78/s105/s79\nFIG. 1. (Color online) Gibbs free energies of formation of Ni O and NFO from aqueous ions. The\nexperimental values, indicated by squares (for NiO) and tri angles (for NFO) at 298 and 333 K, are\nin good agreement with the calculated values.\nenergy of water are taken from the SUPCRT database [17, 18], while the elemental-phase\nchemical potentials of solid Ni, Fe and gaseous O are listed in Table II. T he reaction energies\nat 0.1 MPa, calculated using Eqs. (10) and (11), are plotted as a fun ction of temperature\nin Fig. 1. The calculated values are in good agreement with the available experimental free\nenergies at 298 and 333 K [25], as illustrated in Fig. 1. In general the experimental val-\nues (∆G0\n298(NiO) = 69.04, ∆G0\n333(NiO) = 66.19, and ∆ G0\n298(NiFe2O4) = 55.66 kJ/mol)\nare slightly higher than the calculated ones (∆ G0\n298(NiO) = 65.95, ∆G0\n333(NiO) = 63.28,\nand ∆G0\n298(NiFe2O4) = 52.29 kJ/mol), with the exception of NFO at 333K where the\nagreement between experimental (∆ G0\n333(NiFe2O4) = 36.04 kJ/mol) and theoretical val-\nues (∆G0\n333(NiFe2O4) = 36.96 kJ/mol) is remarkably good. This indicates that the above\nmethod is adequate for evaluation of aqueous chemical potentials a nd free energies of reac-\ntion and formation.\nTo evaluate the possibility of B incorporation into the crystalline stru cture of CRUD,\nwe compute the formation energies of B-related point defects in NF O, assuming solid-liquid\nequilibrium based on the method described above. The point defects investigated here\nare substitutional and interstitial B impurities. The crystal and de fect models used in the\npresent work areidentical to those described in Ref. 11. Three ty pes of substitutional defects\n8are possible: B can substitute for a tetrahedral or octahedral F e atom ( BFeTorBFeO) or\nit can occupy an octahedral Ni site ( BNiO). Similarly, there are three different interstitial\nsites in the spinel structure that can be occupied by B: one octahe dral site ( BO) and two\ntetrahedral sites( BT1andBT2). While both T 1and T 2are tetrahedrally coordinated by O\natoms, the T 1site has four nearest neighbor (NN) cations and the T 2site has two cations\nas NNs. The interstitial defects are illustrated in Fig. 1 in Ref. 11.\nIV. SUBSTITUTIONAL B DEFECTS IN NFO\nIn the primary coolant of a PWR, B is present in the form of boric acid ( H3BO3).\nTherefore, the process of B incorporation into NFO as a substitut ional impurity can be\nenvisioned as the addition of one H 3BO3molecule to the NFO followed by the removal of a\ncation (Fe3+or Ni2+) and three hydroxyl (OH−) ions from the NFO supercell. This process\ncan be described by the reaction:\nNFO+H3BO3→NFO(B)+Fe3+/Ni2++3(OH−) (12)\nTo calculate the formation energies of a substitutional defect, Eq . (3) can be applied\nwithnH3BO3=−1,nFe3+/Ni2+= 1 and nOH−= 3.\nBecause the method for the formation energy calculation is similar fo r all B-related de-\nfects, we will describe the details for the susbtitutional B at Fe site (BFeT/O), and for the\nother defects we only present the results. In the case of BFe, taking into consideration that\nthe ionic charges are zFe3+= 3,zOH−=−1, andzH3BO3= 0, the fourth term on the right\nside of Eq. (3) cancels out, indicating that the charge neutrality of the solution is maintained\nafter the substitution takes place. This makes sense because in th is process one Fe3+ion in\nNFO is replaced by one B3+ion. Thus the expression for defect formation energy can be\nwritten as:\n∆Hf/parenleftbig\nBq\nFeT/O/parenrightbig\n= ∆E/parenleftbig\nBq\nFeT/O/parenrightbig\n+µFe3++3µOH−−µH3BO3+q/parenleftBig\nEF+Edef\nVBM/parenrightBig\n(13)\nAs described in Section III, the chemical potential of aqueous spe cies can be expressed as\nthesumofthestandardchemicalpotentialandatemperaturede pendentterm. Thestandard\nchemical potential of aqueous Fe3+can be readily evaluated using Eq. (7). In a similar way,\n9considering the reactions H2+ 1/2O2→OH−+H+andB+ 3/2H2+ 3/2O2→H3BO3,\nthe standard chemical potentials of OH−and H 3BO3can be represented as:\nµ0\nOH−= ∆G0\nf/parenleftbig\nOH−/parenrightbig\n+µ0\nO+µ0\nH2−µ0\nH+ (14)\nµ0\nH3BO3= ∆G0\nf(H3BO3)+µ0\nB,s+3\n2µ0\nH2+3\n2µ0\nO (15)\nInserting the expressions of the standard chemical potentials (E qs. (14) and (15)) into Eq.\n(13), the formation energy of substitutional B at an Fe site can be written as:\n∆Hf/parenleftbig\nBq\nFeT/O/parenrightbig\n= ∆E/parenleftbig\nBq\nFeT/O/parenrightbig\n+∆G0\nf(Fe3+)+3∆G0\nf(OH−)−∆G0\nf(H3BO3)+µ0\nFe,s−\n−µ0\nB,s+RT(ln[Fe3+]+3ln[OH−]−ln[H3BO3])+q/parenleftBig\nEF+Edef\nVBM/parenrightBig\n(16)\nTocalculatetheformationenergy, asexpressed byEq. (16), the concentrationsofH 3BO3and\nOH−are needed. The former is approximated by the experimental value of 1400 ppm\nB (2.26×10−2mol/dm3) at the beginning of the cycle [3, 4] and the latter is estimated\nfrom the ionization constant of water, pKw=−log([H+][OH−]). Using a semi-empirical\nequation [26], pKwis calculated for a temperature range of 25 to 350◦C and is combined\nwith the pH=−log[H+] of the primary coolant to obtain the hydroxyl concentration as\n[OH−] = 10pH−pKw. This approach allows us to include the pH dependence into the cal-\nculations. Equation (16) indicates that the formation energy depe nds on the charge state\nof the defect (q) and the energy of the electron reservoir (Ferm i level, E F). Both of these\nquantities are kept as parameters, with q ranging from -2 to 2 and w ith EFtaking values\nwithin the band gap of NFO. In the present work we employ the electr onic structure of NFO\nas calculated in Ref. [11]; thus the allowed values of E Frange from zero, corresponding to\nenergy of the valence band maximum (VBM), to 1.3 eV, correspondin g to the conduction\nband minimum (CBM).\nTo calculate the energy required to incorporate B at a Ni site, Eq. ( 3) can be applied\nwithnH3BO3=−1,nNi2+= 1,nOH−= 3 and zH3BO3= 0,nNi2+= 2,nOH−=−1:\n∆Hf(Bq\nNi) = ∆E(Bq\nNi)+µNi2++3µOH−−µH3BO3+/parenleftbig\nEF+E0\nVBM/parenrightbig\n+q/parenleftBig\nEF+Edef\nVBM/parenrightBig\n(17)\n10The main difference between formation energies of BFeandBNi, as described by Eqs. (13)\nand (17), is related to the presence of the extra term ( EF+E0\nVBM)in the latter. This term\nis necessary to restore the charge neutrality of the process whe rein Ni2+in NFO is replaced\nby B3+.\nApplying the method described above, the formation energy of a su bstitutional B at a\nNi site in NFO can be written as:\n∆Hf(Bq\nNi) = ∆E(Bq\nNi)+∆G0\nf(Ni2+)+3∆G0\nf(OH−)−∆G0\nf(H3BO3)+\n+µ0\nNi,s−µ0\nB,s−/parenleftbig\nµ0\nH+−1/2µ0\nH2/parenrightbig\n+(EF+E0\nVBM)+q/parenleftBig\nEF+Edef\nVBM/parenrightBig\n+\n+RT(ln[Ni2+]+3ln[OH−]−ln[H3BO3])\n(18)\nThe formation energies of neutral B defects (q=0) as a function o f temperature, calculated\nfor pH = 7 and E F= 0, 0.5, and 1.0 eV areillustrated in Fig. 2 (a), (b), and(c), respect ively.\nA general trend, characteristic to all defects, is that the forma tion energies increase with\ntemperature, indicating that B incorporation in NFO becomes more e nergetically favorable\nas the coolant temperature decreases during plant shutdown. As regards to the substitu-\ntional defects, in the entire temperature range, the formation e nergies are relatively high,\nsuggesting that the incorporation of B in NFO as a substitutional de fect is unlikely. As illus-\ntrated in Fig. 2 (b) and (c), the high formation energy values are ma intained for all values of\nEFwithin the bandgap. In the case of neutral defects (q=0), any ch ange in the position of\nEFonly affects the formation of energy of BNiwhile the energies of BFeTandBFeOremain\nunchanged. This is because, unlike BFeTandBFeO, the formation of BNirequires electron\nexchange with the electron reservoir to maintain the neutrality of t he substitution process.\nAs evident from Eq. (17), as theE Fvalue increases (Fermi level closer to CBM) moreenergy\nis needed to add electrons to the electron reservoir, therefore t he formation energy of BNi\nincreases.\nV. INTERSTITIAL B DEFECTS IN NFO\nThe process of B incorporation into NFO as an interstitial defect ca n be described as the\naddition of one H 3BO3molecule to NFO followed by the removal of three OH−groups:\nNFO+H3BO3→NFO(B)+3/parenleftbig\nOH−/parenrightbig\n(19)\n11/s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48/s45/s50/s45/s49/s48/s49/s50/s51/s52/s53/s54\n/s66\n/s70/s101/s95/s79\n/s66\n/s70/s101/s95/s84\n/s66\n/s78/s105\n/s66\n/s84/s49\n/s66\n/s79\n/s66\n/s84/s50\n/s32/s32/s72\n/s102/s40/s68/s113\n/s41/s32/s40/s101/s86/s41/s40/s97/s41\n/s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48/s45/s50/s45/s49/s48/s49/s50/s51/s52/s53/s54 /s40/s98/s41\n/s66\n/s70/s101/s95/s79\n/s66\n/s78/s105\n/s66\n/s84/s49\n/s66\n/s79/s66\n/s84/s50\n/s66\n/s84/s50\n/s32/s32\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48/s45/s50/s45/s49/s48/s49/s50/s51/s52/s53/s54\n/s66\n/s70/s101/s95/s84/s69\n/s70/s61/s48/s46/s53/s32/s101/s86\n/s40/s99/s41\n/s66\n/s70/s101/s95/s79\n/s66\n/s84/s49\n/s66\n/s79\n/s32/s32/s69\n/s70/s61/s48 /s69\n/s70/s61/s49/s46/s48/s32/s101/s86\nFIG. 2. (Color online) Temperature dependence of the B-rela ted defect formation energies in NFO,\nas calculated for (a) E F= 0, (b) E F= 0.5 eV, and (c) E F= 1.0 eV. The most stable defects are\ninterstitial B at tetrahedral and octahedral sites.\nThe formation energies can be calculated by employing Eq. (3) with nH3BO3=−1 and\nnOH−= 3:\n∆Hf/parenleftBig\nBq\nT1/T2/O/parenrightBig\n= ∆E/parenleftBig\nBq\nT1/T2/O/parenrightBig\n+3∆G0\nf(OH−)−∆G0\nf(H3BO3)−µ0\nB,s−\n−3/parenleftbig\nµ0\nH+−1\n2µ0\nH2/parenrightbig\n+3(EF+E0\nVBM)+q/parenleftBig\nEF+Edef\nVBM/parenrightBig\n+\n+RT(3ln[OH−]−ln[H3BO3])(20)\nThe calculated values for the neutral (q=0) interstitial defects a s a function of temper-\nature, for pH = 7 and E F= 0, 0.5, and 1 eV, are illustrated in Fig. 2 (a), (b), and (c).\nThe formation energies exhibit a strong dependence on the E Fvalue. This is also evident\nfrom Eq. (20), where the term ( EF+E0\nVBM) is multiplied by a factor of three. This is\nbecause during the process of B incorporation as an interstitial imp urity in NFO, three\nOH−groups are released (see Eq. (19)), and therefore to maintain ch arge neutrality three\nelectrons must be added to the electron reservoir. Thus, the clos er the E Fto the CBM,\nthe higher the formation energy of the interstitial defects. Neve rtheless, for all values of\nEFwithin the NFO bandgap, the incorporation of B as an interstitial def ect (BT2andBO)\nis more energetically favorable than the creation of any substitutio nal B defect. The most\n12/s45/s51/s48/s48/s48/s51/s48/s48\n/s32/s32\n/s86\n/s70/s101/s95/s84\n/s45/s51/s48/s48/s48/s51/s48/s48/s32\n/s32/s86\n/s70/s101/s95/s79\n/s32/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s47/s115/s112/s105/s110/s41\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52/s45/s51/s48/s48/s48/s51/s48/s48/s32\n/s86\n/s78/s105\n/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\nFIG. 3. (Color online) Total DOS calculated for vacancy cont aining NFO. All three defects (tetra-\nhedral Fe vacancy – upper panel, octahedral Fe vacancy – midd le panel, and Ni vacancy – lower\npanel) introduce defect states at the top of the VBM.\nremarkable feature, however, is that for values of E Fbelow the midgap ( EF≤0.5 eV) the\nformation energies of BT2andBObecome negative, implying that B can be incorporated\nin substantial quantities in p-type NFO. Furthermore, at lower tem peratures, the formation\nenergies decrease considerably, suggesting that during theshut down of aPWR, theabsorbed\nB becomes energetically more stable within the CRUD.\nExperimentally it has been observed that NFO accommodates nonst oichiometry with\nFe/Ni ratios above and below 2.0 [27, 28]. More recently, investigatio ns of thermophysical\nproperties of NFO in the temperature range of 300-1300K also rev ealed that minor non-\nstoichiometry is responsible for variations in the Curie temperature relative to previously\n13/s53 /s54 /s55 /s56 /s57/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s40/s98/s41\n/s54/s48/s48/s75\n/s53/s48/s48/s75\n/s52/s48/s48/s75\n/s32/s32\n/s112/s72/s51/s48/s48/s75/s69\n/s70/s61/s48/s69\n/s70/s61/s48/s46/s53/s32/s101/s86/s69\n/s70/s61/s49/s46/s48/s32/s101/s86\n/s53 /s54 /s55 /s56 /s57/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53 /s40/s97/s41\n/s32/s32\n/s112/s72/s32 /s72\n/s102/s40/s66\n/s84/s50/s41/s32/s101/s86/s54/s48/s48/s75\n/s53/s48/s48/s75\n/s52/s48/s48/s75\n/s51/s48/s48/s75\n/s53 /s54 /s55 /s56 /s57/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s40/s99/s41\n/s32/s32\n/s112/s72/s54/s48/s48/s75\n/s53/s48/s48/s75\n/s52/s48/s48/s75\n/s51/s48/s48/s75\nFIG. 4. (Color online) pH dependence of the formation energy ofBT2, calculated at various tem-\nperatures and E F= 0.5 eV. At higher pH values the incorporation of B into NFO be comes less\nenergetically favorable.\nreported literature values [29]. To assess the electrical propertie s (n-type or p-type) associ-\nated with the nonstoichiomentry, we examine the electronic struct ure of vacancy-containing\nNFO. Specifically, we look at the nature of defect states introduce d by tetrahedral and oc-\ntahedral Fe vacancies ( VFeTandVFeO) and Ni vacancy ( VNi) in NFO. From the calculated\ndensity of states (DOS), illustrated in Fig. 3, it is apparent that all t hree vacancies introduce\nacceptor states right above the VBM, contributing to p-type con ductivity in NFO. Because\nincorporation of B is favored in p-type NFO, this finding reinforces t he idea that the B\nadded to the primary coolant might be absorbed by the CRUD as inter stitial impurities.\nOne method to control corrosion in a PWR is to tune the pH of the prim ary coolant.\nBecause the most stable defect in our study is the interstitial B ( BT2), we analyze the pH\ndependence of its formation energy. This is illustrated in Fig. 4 for va rious temperatures\nand three different values of EF. It is noticeable that the formation energy of BT2increases\nstrongly with pH, suggesting that a slightly basic coolant might preve nt B uptake by CRUD\nas an interstitial defect. However, in the case of strongly p-type NFO, as illustrated in\nFig. 4 (a), even at elevated pH the formation energy of BT2is negative, indicating that B is\nenergetically stable inside the crystal structure of the CRUD.\n14VI. SUMMARY\nAn innovative approach that combines first-principles calculations w ith thermodynamic\ndata, has been used to evaluate formation energies of B-related d efects in NFO assuming\nchemical equilibrium with aqueous solutions. The approach has the ad vantage that the\nionic charge is decoupled from the charge state of the defect. Cha rge neutrality is achieved\nthrough exchange of electrons with the electron reservoir that h as energy equal to the Fermi\nlevel. This allows for the investigation of defects whose charge stat es are different from the\nionic charge added to or removed from the system. Furthermore, the scheme extends the\n0K DFT results to higher temperatures and pressures and includes pH and concentration\ndependence.\nThe method has been employed to investigate the energetics of B st ability in NFO as a\nsubstitutional or interstitial impurity. Calculations have been carr ied out assuming solid-\nliquid equilibrium between NFO and an aqueous solution at conditions of t emperature,\npressure, and Ni2+, Fe3+and H 3BO3concentrations that are characteristic to the primary\ncoolant of a PWR. The results indicate that in the temperature rang e of 300 to 600 K, the\ninterstitial B impurities are thermodynamically more stable than the s ubstitutional defects.\nThe formation energies exhibit a strong dependence on the position of the Fermi level (E E)\nwithin the bandgap of the host NFO, p-type NFO being more favorab le for B incorporation.\nAnalysis of the electronic DOS associated with vacancies in NFO indicat e that both Fe and\nNi vacancies generate p-type conductivity, suggesting that non stoichiometric NFO, that\nis predominantly present in PWR CRUD, is favorable for accommodatin g B interstitials.\nThe examination of the pH dependence of the defect stability indicat es that a basic PWR\ncoolant might be necessary to mitigate the B uptake by the CRUD. Th e results of the\npresent investigation reveal that under operating PWR conditions , B is stable in NFO as\nan interstitial impurity, therefore it can accumulate in the atomic st ructure of CRUD and\ncan be a possible large contributor to AOA.\nACKNOWLEDGMENTS\nThis research was supported by the Consortium for Advanced Simu lation of Light Water Re-\nactors(CASL,http://www.casl.gov),anEnergyInnovationHub(h ttp://www.energy.gov/hubs)\n15for Modeling and Simulation of Nuclear Reactors under U.S. Departme nt of Energy Con-\ntract No. DE-AC05-00OR22725. The computational work has bee n performed at NERSC,\nsupported by the Office of Science of the US Department of Energy under Contract No.\nDE-AC02-05CH11231.\nAppendix: Elemental-phase chemical potentials\nTABLE II. Calculated chemical potentials of elemental subs tances in their conventional reference\nphase.\nElement µ0\nM,solid(eV) Element µ0\nM,solid(eV) Element µ0\nM,solid(eV)\nAg -0.86 Ge -4.21 Pt -3.97\nAl -3.12 Hf -7.55 Rb -0.70\nAs -4.92 Hg -0.18 Rh -4.79\nAu -2.27 In -2.39 S -3.99\nB -6.56 Ir -5.99 Sb -4.24\nBa -1.40 K -0.83 Sc -4.71\nBe -3.45 La -3.72 Se -3.44\nBi -4.27 Li -1.68 Si -5.12\nCa -1.70 Mg -1.11 Sn -3.85\nCd -0.66 Mn -6.85 Sr -1.20\nCl -1.60 N -8.37 Ta -8.96\nCo -4.79 Na -1.09 Te -3.18\nCr -7.14 Nb -6.91 Ti -5.59\nCu -2.00 Ni -3.69 V -6.49\nF -1.68 O -4.71 Y -4.90\nFe -6.10 P -5.47 Zn -0.93\nGa -2.51 Pd -3.14 Zr -6.02\n16[1]PWR Axial Offset Anomaly (AOA) Guidelines, Revision 1 , EPRI, Palo Alto, CA, 1008102\n(2004)\n[2]Rootcause Investigation of Axial Power Offset Anomaly , EPRI, Palo Alto, CA, TR-108320\n(1997).\n[3]Modeling PWR Fuel Corrosion Product Deposition and Growth Pr ocesses, EPRI, Palo Alto,\nCA, 1009734 (2004).\n[4]Modeling PWR Fuel Corrosion Product Deposition and Growth Pr ocess: Final Report , EPRI,\nPalo Alto, CA, 1011743 (2005).\n[5]Axial Offset Anomaly (AOA) Mechanism Verification in Simulat ed PWR Environments ,\nEPRI, Palo Alto, CA, 1013423 (2006).\n[6]Pressurized Water Reactor (PWR) Axial Offset Anomaly Mechan ism Verification in Simulated\nPWR Environments , EPRI, Palo Alto, CA, 1021038 (2010).\n[7] S. 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Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989).\n[21] V. Stevanovic, S. Lany, X. W. Zhang, and A. Zunger, Phys. Rev. B85, 115104 (2012).\n[22] S. Lany, Phys. Rev. B 78, 245207 (2008).\n17[23] A. Jain, G. Hautier, S. P. Ong, C. J. Moore, C. C. Fischer, K. A. Persson, and G. Ceder,\nPhys. Rev. B 84, 045115 (2011).\n[24] M. Todorova and J. Neugebauer, Phys. Rev. Applied 1, 014001 (2014).\n[25] K. Fujiwara and M. Domae, in Proceedings of 14th Interna tional Conference on the Properties\nof Water and Steam, edited by M. Nakahara et al. (Maruzen co., Ltd, Kyoto, 2004), p. 581.\n[26] A. V. Bandura and S. N. Lvov, J Phys Chem Ref Data 35, 15 (2006).\n[27] A. E. Paladino, J. Am. Ceram. Soc. 42, 168 (1959).\n[28] H. M. Obryan, F. R. Monforte, and R. Blair, J. Am. Ceram. S oc.48, 577 (1965).\n[29] A. T. Nelson, J. T. White, D. A. Andersson, J. A. Aguiar, K . J. McClellan, D. D. Byler, M.\nP. Short, and C. R. Stanek, J. Am. Ceram. Soc. 97, 1559 (2014).\n18"    },    {        "title": "2302.06532v1.Improper_ferroelectricity_in_ultrathin_hexagonal_ferrites_film.pdf",        "content": "Improper ferroelectricity in ultrathin hexagonal ferrites film   Xin Li,1 Yu Yun,1 Xiaoshan Xu1,2*  1Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588, USA 2Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, USA *Corresponding author: Xiaoshan Xu (X.X.)   Abstract:             The suppression of ferroelectricity in ultrathin films of improper ferroelectric hexagonal ferrites or manganites has been attributed to the effect of interfacial clamping, however, the quantitative understanding and related phenomenological model are still lacking. In this work, we report the paraelectric-to-ferroelectric phase transition of epitaxial h-ScFeO3 films with different thickness through in-situ reflection high-energy electron diffraction (RHEED). Based on the interfacial clamping model and the Landau theory, we show that the thickness-dependence of the ferroelectric Curie temperature can be understood in terms of the characteristic length of interfacial clamping layer and the bulk Curie temperature. Furthermore, we found that the critical thickness of improper ferroelectricity is proportional to the characteristic length of interfacial clamping layer. These results reveal the essential role of mechanical clamping from interface on the improper ferroelectricity of hexagonal ferrites or manganites, and could serve as the guidance to achieve robust improper ferroelectricity in ultrathin films.     Introduction             The two-dimensional (2D) ferroelectricity has attracted intensive research interests recently due to novel mechanisms for stabilizing polar order as well as their great potentials to device scalability.[1-3]. In principle, epitaxial films of bulk ferroelectrics can also be scaled down to several unit cells. However, for proper ferroelectrics, such as BaTiO3 (BTO), PbTiO3 (PTO) [4-8]  of perovskite structures, if the depolarization field is not fully screened, there may exist critical thickness below which ferroelectricity is quenched. The critical-thickness problem can be absent in improper ferroelectrics. For example, in improper ferroelectric hexagonal manganites (h-RMnO3, R=Sc,Y, Ho-Lu ) or ferrites (h-RFeO3), ferroelectric order originates from the linear coupling of polarization with non-polar structural distortion. This mechanism enables ferroelectricity in the ultrathin limit[9-11] ,which is comparable to the range of  thickness for 2D ferroelectricity.            On the other hand, recent studies revealed an interfacial clamping layer in hexagonal manganite and ferrite thin films [9,10] , which may affect the ferroelectricity significantly in the ultrathin limit. The transition temperature from paraelectric to ferroelectric phase was observed to decrease with reducing thickness and the interfacial clamping effect suppresses polarization within the first 2 unit cells (uc) in h-RMnO3 film[9]. Despite these observations, how the interfacial layer determines the temperature and thickness dependence of ferroelectricity has not been studied systematically, the ultrathin films of h-RFeO3 could provide the chance to further clarify the general role of interfacial clamping layer.            Improper ferroelectric h-RFeO3 is formed by triangle lattice of FeO5 bipyramids sandwiched by rare earth layers[11,12] . The triangle lattice of FeO5 rotates 60 deg for alternative half unit cell, as shown in Fig.1 (a). The collective displacement of FeO5 bipyramids corresponds to the K3 mode distortion. The K3 mode causes the imbalanced displacements of ions along the c axis (Γ!\" mode), leading to the ferroelectricity in h-RFeO3. The ferroelectricity is called improper since the primary order parameters are (𝑄,𝜙#), in which Q is the magnitude of in-plane displacement and 𝜙# is the rotation angle of apical oxygen relative to the coordinates of FeO5 bipyramids. The induced spontaneous polarization is proportional to 𝑄$cos\t(3𝜙#) in h-RFeO3.            As shown in Fig.1(b), below Tc, the energy landscape of ferroelectric h-RFeO3 is Mexican-hat shape with six-fold symmetry based on the Landau theory. For the ground states, the order parameter 𝜙#\tcan only take discrete values %&$ where n is integer. When the temperature increases, the magnitude of Q for the ground states decrease gradually, which also leads to the suppression of polarization[13]. Above Tc, the ferroelectric phase transforms to the paraelectric phase (Q = 0), as shown by in Fig.1(c) based on Landau theory[14]. In additional to temperature-dependent phase transition, decreasing thickness is also expected to trigger the ferroelectric-to-paraelectric transition considering the interfacial clamping, as depicted in Fig.1(d). While the thermally driven transition is intrinsic to the temperature dependence of free energy, the thickness-driven transition is expected to be an extrinsic effect due to the elastic disruption of polar order at the film/substrate interface.           Here we focus on h-ScFeO3 films, in which the smaller radius of Sc ion leads to stronger K3 distortion with Q ≈ 1 angstrom[15,16]. We show that Tc of the h-ScFeO3 films, inferred from characteristic diffraction streaks of the in-situ RHEED patterns, decreases when the film thickness reduces. The thickness dependence of Tc and a critical thickness of ferroelectricity (𝜁'() can be well modelled with the Landau theory using the characteristic length of the interfacial clamping layer (𝜁)) and the bulk Curie temperature (𝑇*) as two key parameters. These results elucidate that, in hexagonal ferrites and manganites, the interfacial-clamping-originated critical thickness is expected to affect scalability when it is comparable to the thickness of 2D ferroelectric layers.   Results and Discussion:                 To reveal the origin of thickness-dependent scaling of Tc, the epitaxial h-ScFeO3 films with different thickness were grown using pulsed laser deposition (PLD), followed by annealing at high temperature. The detailed growth conditions can be found in the previous work [15]. As shown in Fig. 2(a), the 2θ scans of x-ray diffractions indicate that h-ScFeO3 films along the (0001) direction were formed on both Al2O3 (0001) and SrTiO3 (STO)(111) substrates without impurity phase. Moreover, the in-plane epitaxy relationship can be inferred by comparing the RHEED patterns of film with substrate in Fig. 2(b). Specifically, the in-plane [100] direction of h-ScFeO3 film is parallel to [11200] of Al2O3 and [2011] of STO, respectively.             In h-RFeO3, paraelectric-to-ferroelectric phase transition is accompanied by the tripling of the in-plane unit cell for the ferroelectric phase (P63cm) compared with the paraelectric phase (P63/mmc), as indicated in Fig. 1(d). Since the separation of the RHEED streaks is inversely proportional to the in-plane lattice constant[17-19], RHEED patterns with the electron beam along the h-ScFeO3 [100] direction provides a measurement for tripled unit cell of ferroelectric phase with P63cm symmetry. Specifically, the RHEED pattern of the paraelectric phase consists of the (0,0) and (±𝑛,0) diffraction streaks. The tripled in-plane unit cell of the ferroelectric phase leads to the formation of weaker diffraction streaks at the ±(3n+1)/3 and ±(3n+2)/3 positions along h-ScFeO3 [001] direction, as shown in Fig. 2(b).            Fig. 3(a) to (c) display two-dimensional RHEED patterns of 2, 5 and 17 unit cell (uc) h-ScFeO3/Al2O3 films respectively, which were captured at room temperature, below which are the normalized line profiles of RHEED intensity after integrated along vertical direction, with background subtracted. For the 17 uc film, the intensity of the weak streaks gradually increases when the temperature decreases, indicating the existence of thermal driven paraelectric-to-ferroelectric phase transition. When the thickness is 2 uc, as shown in Fig. 3(a), no weak streaks could be identified down to the room temperature. The missing weak streaks suggests that the mechanical boundary conditions at the interface, or interfacial clamping, modifies the energy-minimum state and fully suppress the K3 distortion in the interfacial layers.            To trace the change of Tc with respective to the film thickness, the temperature dependence of RHEED intensity of the weak streaks was analyzed quantitatively. Fig. 4 shows the intensity of the weak streaks, normalized using the (1,0) and (-1,0) streaks, as a function of temperature for various film thickness for h-ScFeO3 films grown on both Al2O3 and STO substrates. As shown in Fig. 4(a), there is no obvious transition for the intensity of the weak streak for the 2 uc film, so the ferroelectric Tc may be lower than the room temperature. All the other films, with thickness  ranging from 4 to 41 uc, exhibit a transition of the weak streak intensity above room temperature. More importantly, Tc increases when the thickness increases.           To interpret the key factors determining the thickness-dependent scaling of Tc in ultrathin h-RFeO3 film and the potential critical thickness, we treat the h-RFeO3 near the interface as interfacial clamping layer, where the magnitude of structural distortion (Q) is suppressed completely at the beginning, and increases gradually when rare earth layer is away from the interface[10,11]. Considering the free energy of the structural distortion and the elastic energy at the same time within the framework of Landau theory[14,20] (see details in supplementary), the structural distortion with thickness and temperature can be written analytically as:   \t\t\t\t\t\t\t\t\t\t\t\t\t\t𝑄(𝑧,𝑇)=𝑄∞(𝑇)+\",-./\"!\"($)0+1\t,-./\"!\"($)0\t\t,𝑤𝑖𝑡ℎ⎩⎨⎧𝑄3(𝑇)==\"4&5\t=1−66'=𝑄)=1−66'𝜁\t(𝑇)==7\"4&\t=6'6'\"6=𝜁)=6'6'\"6              (1)  in which 𝑎)<0 and b>0 are coefficients of Landau theory, k corresponds to stiffness coefficient, 𝑇8 is the Curie temperature of bulk state, and 𝜁)\tis the characteristic length of interfacial clamping layer at T = 0 K. By integrating the energy density with the thickness of ferroelectric layer (𝑡'), the total energy can be expressed as:  \t\t𝐹(𝑡'\t,𝑇)=−4&!6(𝐶1(𝑡',𝑇)BC\t𝑇−D\t𝑇*+!76(4&\t9!(;),6)9+(;),6)\tFG𝑄3(𝑇)!+6(!∗#&*9$(;),6)9+(;),6)\t𝑄3(𝑇)?H\t\t\t\t(2)\t with \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t⎩⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎧𝐶1(𝑡!,𝑇)=./1−exp4−𝑧𝜁(𝑇)71+\texp4−𝑧𝜁(𝑇)79\"\t𝑑𝑧#!$𝐶2(𝑡!,𝑇)=.(2𝜁∗𝑒𝑥𝑝4−𝑧𝜁(𝑇)7)\"41+\texp4−𝑧𝜁(𝑇)77%\t𝑑𝑧#!$\t𝐶3(𝑡!,𝑇)=./1−exp4−𝑧𝜁(𝑇)71+\texp4−𝑧𝜁(𝑇)79%\t𝑑𝑧\t#!$   \t\t\t\t\t\t\t\t\t\t.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(3)\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  When the coefficient before 𝑄∞(𝑇)!\tterm is less than zero, minimizing 𝐹(𝑡'\t,𝑇) results in finite 𝑄3(𝑇) corresponding to the ferroelectric order. At T=Tc, this coefficient is zero, i.e.,  \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t𝛼(𝑡',𝑇)=𝑇−𝑇*C1−2𝜁)!\t9!(;),6)9+(;),6)G=0                                                   (4) (see details in supplementary).              The phase diagram of  𝛼(𝑡',𝑇)\t for h-ScFeO3 films is\tplotted in Fig. 5(a), with 𝜁) and 𝑇* inferred in Fig.5(b). Since 𝛼(𝑡',𝑇)<0\t\t(>0) corresponds to the ferroelectric (paraelectric) state of the h-RFeO3 films, the curve of 𝛼(𝑡',𝑇)=0indicates that the critical thickness of ferroelectricity increases with temperature, which is consistent with the observation of temperature-dependent corrugation of rare earth layers in h-YMnO3/YSZ film [10]. Moreover, as implied by Eq. (4), the dependence of Tc on 𝑡'\tis influenced by both 𝑇*\tand 𝜁), which are two independent parameters that can be extracted from experimental observations, such as in-situ RHEED patterns in this work. As shown in Fig. 5(b), the experimental Tc of both h-ScFeO3 and h-YMnO3 films decreases with smaller thickness, and the trend is slower in the h-ScFeO3 films. By fitting the experimental data with Eq. (4), we find for h-YMnO3/YSZ film, that Ts = 814 ±\t 183 K and 𝜁) = 0.93 ± 0.44 uc. For the h-ScFeO3 films, we find Ts = 681 ±\t 28 K and 𝜁) = 0.68 ± 0.18 u.c. Therefore, the slower suppression of Tc with thickness in h-ScFeO3 films can be attributed to smaller 𝜁).            It should be noted that the phenomenological model here does not consider the transition of single domain to multi-domain states with thickness, since the elastic energy is much larger than depolarization energy [13], as well as possible variation of stiffness constant at the interface, which may contribute to the minor discrepancy. Moreover, based on Eq. (4), the ferroelectric critical thickness  (𝜁'() can be defined as the interception of 𝛼(𝑡',𝑇)=0  with T = 0 K, as shown in Fig.5 (c), and the ratio of 𝜁'(/𝜁) = 2.25 is a fixed value, which is  independent of materials (see details in supplementary Section 3). This ratio indicates that the absence of ferroelectric critical thickness (𝜁'(=0) or unsuppressed corrugation of initial rare-earth layer in h-RFeO3 or h-RMnO3 films can only exist when there is no interfacial clamping layer (𝜁) =0) or when the films become freestanding. Moreover, the improper ferroelectricity under ultrathin limit could be artificially controlled by choosing the characteristic length of  interfacial clamping layer through different interface.  Conclusion               In summary, through quantitative study of in-situ RHEED patterns during paraelectric-to-ferroelectric phase transition, the thickness-dependent scaling of ferroelectric Tc in epitaxial h-ScFeO3 films was revealed for the first time. Based on interfacial clamping layer, a phenomenological model from Landau theory was introduced to interpret this scaling behavior and reveal the correlation between ferroelectric critical thickness and characteristic length of interfacial clamping layer in ultrathin h-RFeO3 or h-RMnO3 film. These results serve as the guidance to achieve robust improper ferroelectricity in h-RFeO3 through interfacial engineering for future study.   Experimental section  The h-ScFeO3/Al2O3 (0001) and h-ScFeO3/SrTiO3 (111) epitaxial thin films were grown using pulsed laser polarization (PLD) with base pressure lower than 3×10-7 mTorr, a repetition rate of 2 Hz. Before the deposition, the substrates were pre-annealed at 700 ºC for 1 hour. During the growth, the substrate temperature was kept at 700 ºC - 920 ºC, specifically, the films were grown at low temperature first (~ 700 ºC) then annealed at high temperature (~ 920 ºC). The growth oxygen pressure is 10 mTorr. The in-situ RHEED was used to monitor ferroelectric to paraelectric phase transition of h-ScFeO3 film after the growth, the incident angle of electron beam is kept fixed during the process of reducing temperature, and the raw images of RHEED are analyzed based on python program.  The crystal structure and the thickness of the epitaxial h-ScFeO3 films were measured by XRD (Rigaku SmartLab Diffractometer).    Acknowledgements   Funding: This work was primarily supported by the National Science Foundation (NSF), Division of Materials Research (DMR) under Grant No. DMR-1454618 and by the Nebraska Center for Energy Sciences Research. The research was performed in part in the Nebraska Nanoscale Facility: National Nanotechnology Coordinated Infrastructure and the Nebraska Center for Materials and Nanoscience, which are supported by the NSF under Grant No. ECCS- 2025298, and the Nebraska Research Initiative.    Reference:  1.M. Osada1, T. Sasaki. APL Mater. 7, 120902 (2019)  2. C. Cui, F. Xue, W.-J. Hu, L.-J. Li. Npj 2D Mater. Appl. 2, 18 (2018).  3. M. Wu, P. Jena. Wiley Interdiscip. Rev. Comput. Mol. Sci. 8, e1365 (2018).  4. D. D. Fong, G. B. Stephenson, S. K. Streiffer, J. A. Eastman, O. Auciello, P. H. Fuoss, C. Thompson.         Science 304, 1650–1653 (2004).  5. D. D. Fong, A. M. Kolpak, J. A. Eastman, S. K. Streiffer, P. H. Fuoss, G. B. Stephenson, Carol Thompson, D. M. Kim, K. J. Choi, C. B. Eom, I. Grinberg, and A. M. Rappe. Stabilization of Monodomain Polarization in Ultrathin PbTiO3 Films. Phys. Rev. Lett. 96, 127601(2006).  6. Jia, CL., Nagarajan, V., He, J. Q. et al. Nature Mater 6, 64–69 (2007).  7. Junquera, J., Ghosez, P. Nature 422, 506–509 (2003).  8.  Y. S. Kim, D. H. Kim, J. D. Kim, Y. J. Chang, and T. W. Noh. Appl. Phys. Lett. 86, 102907 (2005)  9.  N. Sai, C. J. Fennie, A. A. Demkov. Phys. Rev. Lett. 102, 107601(2009).  10.  J. Nordlander,  M.Campanini, M.D.Rossell, et al. Nat. Commun 10, 5591 (2019).  11. Y. Yun, et al. 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Room-Temperature Multiferroic Hexagonal LuFeO3 Films. Phys. Rev. Lett.110, 237601 (2013)  19. X. Zhang et al. Effect of interface on epitaxy and magnetism in h- RFeO3/Fe3O4/Al2O3 films (R = Lu, Yb). J. Phys.: Condens. Matter 29,164001 (2017).  20.  C. X. Zhang, K. L. Yang, P. Jia, H. L. Lin, C. F. Li, L. Lin, Z. B. Yan, J.-M. Liu. J. Appl. Phys. 123, 094102 (2018).                        Fig.1 (a) Atomic structure of h-RFeO3 and the rotation angle of apical oxygen atoms (𝜙#). (b) Representative energy landscapes of h-RFeO3 at different temperature, assuming Tc as 750 K. (c) The schematic diagram for the temperature dependence of the order parameter Q and the polarization P=Q3cos\t(3𝜙#) for bulk-state h-RFeO3, based on the Landau theory. (d) Schematic diagrams for the temperature and thickness driven phase transition in h-RFeO3 films and related atomic structures in a-b plane.    \n  \n Fig.2 (a) 2θ scan for h-ScFeO3/Al2O3 and (b) h-ScFeO3/STO films. (b) RHEED patterns of ferroelectric phase (P63cm symmetry) and paraelectric phase (P63/mmc symmetry) and in-plane epitaxy relationships for h-ScFeO3/Al2O3 and h-ScFeO3/STO films.    \n  \n  Fig.3 (a) to (c) RHEED images of h-ScFeO3/Al2O3 films with thickness of  2, 5, 17 uc at room temperature. (d) Normalized profiles of RHEED intensity with different thickness and temperature for h-ScFeO3/Al2O3 films.   \n  \n Fig.4 Normalized RHEED intensity of (1/3, 0) streak with temperature for (a) h-ScFeO3/Al2O3 and (b) h-ScFeO3/STO films with different thickness.   \n  \n Fig.5 (a) Calculated phase diagram of coefficient α in Eq. (4) with temperature and film thickness as independent variables, the blue solid blue line corresponds to α = 0. (b) Measured thickness-dependent Tc of h-ScFeO3/STO, h-ScFeO3/Al2O3 and h-YMnO3/YSZ films and related fitting based on Eq. (4), the data of h-ScFeO3 comes from in-situ RHEED in this work and the data of h-YMnO3 comes from Ref. [9]. (c) Phase diagram of α near T= 0 K and the comparison of 𝜁) and 𝜁'(.    \n"    },    {        "title": "1803.09677v1.Enhancement_of_the_magnetoelectric_effect_in_multiferroic_CoFe__2_O__4__PZT_bilayer_by_induced_uniaxial_magnetic_anisotropy.pdf",        "content": "1\nEnhancement of the magnetoelectric effect in multiferroic\nCoFe 2O4/PZT bilayer by induced uniaxial magnetic anisotropy\nAlex Aubert1, Vincent Loyau1, Fr´ed´eric Mazaleyrat1, and Martino LoBue1\n1SATIE UMR 8029 CNRS, ENS Paris-Saclay, Cachan, 94235 FRANCE\nDOI : 10.1109/TMAG.2017.2696162\nIn this study we have compared magnetic, magnetostrictive and piezomagnetic properties of isotropic and anisotropic cobalt\nferrite pellets. The isotropic sample was prepared by the ceramic method while the sample exhibiting uniaxial anisotropy was\nmade by reactive sintering using Spark Plasma Sintering (SPS). This technique permits to induce a magnetic anisotropy in cobalt\nferrite in the direction of the applied pressure during SPS process. Sample with uniaxial anisotropy revealed a higher longitudinal\nmagnetostriction and piezomagnetism compared to the isotropic sample, but the transversal magnetostriction and piezomagnetism\nwere dramatically reduced. In the case of magnetoelectric layered composite, the magnetoelectric coefficient is directly related to the\nsum of the longitudinal and transversal piezomagnetic coefficients. These two coefficients being opposite in sign, the use of material\nexhibiting high longitudinal and low transversal piezomagnetic coefficient (or vice versa) in ME devices is expected to improve\nthe ME effect. Hence, ME bilayer devices were made using isotropic and anisotropic cobalt ferrite stuck with a PZT layer. ME\nmeasurements at low frequencies revealed that bilayer with anisotropic cobalt ferrite exhibits a ME coefficient three times higher\nthan a bilayer with isotropic cobalt ferrite. We also investigated the behavior of such composites when excited at resonant frequency.\nIndex Terms —Magnetoelectric, Magnetostriction, Magnetic anisotropy, Spark Plasma Sintering, Resonance\nI. I NTRODUCTION\nTHE magnetoelectric (ME) effect has raised great in-\nterest in the recent years because of its potential use\nin smart electronic application [1]–[4]. Beside the research\nfor intrinsinc magnetoelectric alloys, relevant advances have\nbeen reached in the study of magnetostrictive-piezoelectric\nheterostructure composite. In this case, the magnetoelectric\ncoupling is due to the magnetic-mechanical-electric transform\nthrough the interface between layers. The electromagnetic\ncoupling results from the dynamic mechanical deformation\nof the ferromagnet which induces a variation of polarization\nin the piezoelectric layer. Hence, the magnetoelectric effect\nmainly arises from the dynamic magnetostriction, i.e. the\npiezomagnetic coefficient qmof the magnetic material.\nThe piezomagnetic coefficient is defined as the slope of\nthe magnetostrictive coefficient qm=d\u0015=dH , and is the\nmeaningful parameter to investigate for sensors and actuators.\nFor magnetoelectric purposes, the magnetoelectric coefficient\nin the transverse direction \u000b31depends on the sum of the lon-\ngitudinalqm\n11=d\u001511=dH and the transverse qm\n21=d\u001521=dH\npiezomagnetic coefficients of the magnetic layer [5]–[8]. This\nexplains why researches on magnetoelectric layered composite\nare usually focused on good magnetostrictive materials such\nas Terfenol-D, nickel ferrite or cobalt ferrite associated with\nlead zirconate titanate (PZT) [5], [6], [9], [10].\nHowever, magnetic materials used in magnetoelectric de-\nvices are usually isotropic. In magnetostrictive properties,\nthis results in a ratio between maximum longitudinal and\ntransverse magnetostriction of 2:1. Moreover, the isotropy\nof the material implies that longitudinal \u001511and transverse\n\u001521magnetostriction are of opposite sign. The same behavior\noccurs for piezomagnetic properties, longitudinal qm\n11and\ntransverseqm\n21piezomagnetic coefficient are opposite in sign\nand the maximum jqm\n11jis two times higher than jqm\n21j. Thus,by summing up these two coefficient qm\n11+qm\n21, it leads to\na piezomagnetic coefficient qmPtwo times lower than qm\n11,\neventually resulting in a low magnetoelectric coefficient \u000b31\nsince it depends directly on qmP. Hence, to increase the\nmagnetoelectric effect, one must enhance qmPwhich is possible\nby improving qm\n11and keeping qm\n21low and vice versa.\nThe most common approach to enhance the longitudinal\npiezomagnetic coefficient ( qm\n11) and decrease the transverse\npiezomagnetic coefficient ( qm\n21) is to induce uniaxial anisotropy\nin the material. This can be done in cobalt ferrite by magnetic-\nannealing [11]–[14], which consists in applying a strong\nmagnetic field during annealing between 300 and 400\u000eC.\nA rearrangement of Co and Fe ions in the crystal structure\nleads to a uniaxial anisotropy parallel to the direction of\nthe magnetic annealing field. Recently, we proposed [15]\nanother technique to induce uniaxial anisotropy in cobalt\nferrite, by means of a reaction under uniaxial pressure using\nSpark Plasma Sintering (SPS). SPS process [16] is used to\nmake the reaction [17] and/or the sintering [18] of oxide-\nbased materials. During this process, high uniaxial pressure\nis applied while pulsed electric current heats up the die and\nthe ceramic. It has been shown that using SPS to activate\nthe reaction and the sintering of cobalt ferrite permitted to\ninduce a uniaxial anisotropy along the direction of the applied\npressure [15].\nIn this study, magnetic, magnetostrictive and piezomagnetic\nproperties are compared between cobalt ferrite with uniaxial\nanisotropy made by SPS, and isotropic cobalt ferrite made\nby the ceramic method. The ME effect is then compared for\nCoFe 2O4/PZT bilayer using isotropic and anisotropic cobalt\nferrite. The advantage of cobalt ferrite with uniaxial anisotropy\nfor magnetoelectric purpose is shown in different frequency\nranges.arXiv:1803.09677v1  [cond-mat.mtrl-sci]  26 Mar 20182\nII. E XPERIMENTAL DETAILS\nA. Samples fabrication\nPolycristalline cobalt ferrite were prepared by two different\nmethods. In both cases, nanosize oxides ( <50 nm) Fe 2O3and\nCo3O4(Sigma-Aldrich) were used as precursors in adequate\nmolar ratio. Oxides were mixed in a planetary ball mill during\n30 min at 400 rpm, and then grinded during 1 hour at 600 rpm.\nIn the first method, cobalt ferrite was made by the classic\nceramic method. The mixture was first calcined at 900\u000eC\nduring 12 hours to form the spinel phase, and then grinded at\n550 rpm during 1 hour. After uniaxial compaction at 50 MPa\nin a cylindrical die of 10 mm diameter, sample was sintered\nat 1250\u000eC during 10 hours. This sample will be referred\nas CF-CM. In the second method, Spark Plasma Sintering\n(SPS) was used to make the reaction and the sintering (reactive\nsintering) of the cobalt ferrite. The reaction was performed at\n500\u000eC for 5 min followed by the sintering stage at 750\u000eC for\n3 min, both under a uniaxial pressure of 100 MPa. This sample\nwill be referred as CF-SPS. Both methods resulted in cobalt\nferrite with a large majority of spinel phase ( >91 % ) [15].\nThe final shape of both samples is identical, a disk of 10 mm\ndiameter and 2 mm thick.\nTo make magnetoelectric samples, cobalt ferrite disks were\nbonded on commercial PZT disks (Ferroperm PZ27) of 1 mm\nthick and 10 mm diameter using silver epoxy (Epotek E4110).\nThe piezoelectric samples are polarized along the thickness di-\nrection. The magnetoelectric bilayer is then a disk of thickness\n3 mm and 10 mm diameter.\nB. Measurement procedure\nThe magnetic measurements were carried out with a vi-\nbrating sample magnetometer (VSM, Lakeshore 7400) up\nto a maximum field of 800 kA/m. The ferrite disks were\ncut into 8 mm3cubes to compare the measurements in\nthe three directions of the Cartesian coordinate system (see\ninset in Figure 1). Magnetostriction measurements were per-\nformed at room temperature using the strain gauge (Micro-\nMeasurements) method with an electromagnet supplying a\nmaximum field of 700 kA/m. The gauges were bonded on the\npellets’ surface along the direction (1) and the magnetic field\nwas applied in the directions (1) and (2) in the plane of the disk\n(see inset in figure 2). Hence, longitudinal \u001511and transverse\n\u001521magnetostriction coefficients were obtained. The magne-\ntoelectric coefficient is measured as function of a continuous\nmagnetic field HDCproduced by an electromagnet applied\nin the transverse direction (1) of the bilayer magnetoelectric\nsample. A small external AC field is superimposed in the same\ndirection (1 mT, 80 Hz) produced by Helmoltz coils (see\ninset in Figure 4). The magnetoelectric voltage is measured\nwith a lock-in amplifier (EG&G Princeton Applied Research\nModel 5210) having an input impedance of 100 M \nfor\nlow frequencies. At resonant frequency, the magnetoelectric\nvoltage is measured with an oscilloscope. Compliances were\nmeasured using the ultrasonic velocity measurements along the\nthickness direction of the disk using the pulse-echo technique\n(longitudinal and shear waves) at 20 MHz.\nMagnetic polarization (T)(a) CF-CM\n(1) (2) (3)\n(1)(2)(3)0.6\n0.4\n0.2\n0\n-0.2\n-0.4\n-0.6\nInternal Field (kA/m)(1) (2) HA\n(3) EA(b) CF-SPS\n0 400 800 -400 -8000.6\n0.4\n0.2\n0\n-0.2\n-0.4\n-0.60.6\n0.4\n0.2\n0\n-0.2\n-0.4\n-0.6\n0.6\n0.4\n0.2\n0\n-0.2\n-0.4\n-0.6Fig. 1. Hysteresis loop M-H of samples (a) CF-CM and (b) CF-SPS cut into\ncube shape. Measurements are done in the three directions of the cube (1),\n(2) and (3) as represented on the drawing.\nIII. R ESULTS AND DISCUSSION\nA. Magnetism\nIn Figure 1, we show the magnetic polarization as a function\nof the internal field, by taking into account the magnetometric\ndemagnetizing factor of a cube ( Nm= 0:2759) ) [19]. For\nCF-CM (in Figure 1 (a)), the three hysteresis loops exhibit\nsimilar behavior in the three directions of the cube, indicating\nthe isotropy of the material. By opposition, for CF-SPS (in\nFigure 1 (b)), the measurements show that the remanent mag-\nnetization in the direction (3) is higher than for the direction\n(1) and (2) of the cube. Indeed, the remanent magnetic moment\nreaches 301mT along the easy axis while it is 205mT along\nthe hard axis. This behavior indicates a uniaxial anisotropy\nin the direction (3). This particular direction corresponds to\nthe direction of the pressure applied during the SPS process,\nconfirming reactive sintering under applied pressure as an\neffective method to induce a uniaxial anisotropy in cobalt\nferrite [15].\nB. Magnetostriction and Piezomagnetism\nIn Figure 2, magnetostrictive measurement of CF-CM and\nCF-SPS in the longitudinal and transverse direction are re-\nported (see inset in Figure 2). As expected, cobalt ferrite\nwith uniaxial anisotropy exhibits a different behavior from the\nisotropic cobalt ferrite. Indeed, CF-CM shows a maximum lon-\ngitudinal magnetostriction \u001511of -204 ppm and a maximum3\nMagnetostriction (ppm)100\n50\n0\n-50\n-100\n-150\n-200\n-250\nApplied Field (kA/m)\n11CF-CM\n21CF-CM\n11CF-SPS\n21CF-SPS\n0 100 200 300 400 500 600 700(1)(2)100\n50\n0\n-50\n-100\n-150\n-200\n-250\nFig. 2. Magnetostriction curves of CF-CM and CF-SPS are represented\nin black and red respectively. The solid line ( \u001511) corresponds to the\nmeasurement when the applied field is along the direction (1) and the dash\nline (\u001521) when the applied field is along the direction (2). The strain gauge\nis bonded along the direction (1) for all measurements as represented on the\ndrawing.\ntransverse magnetostriction \u001521of 76 ppm, which are usual\nvalues for isotropic CoFe 2O4[14]. For CF-SPS, the maximum\nlongitudinal magnetostriction has increased to -229 ppm while\nthe transverse magnetostriction has dramatically reduced to\n12 ppm and then becomes negative at a given applied field.\nThis type of curves is typical for cobalt ferrite after magnetic\nannealing showing an induced uniaxial anisotropy [12], [13].\nThis leads to a ratio between maximum longitudinal and trans-\nverse magnetostriction of 19:1 while it is approximatively of\n2:1 for isotropic materials. Hence, as expected, the longitudinal\nmagnetostriction of the anisotropic cobalt ferrite is enhanced\nand the transverse magnetostriction is reduced compared to\nthe isotropic ceramic.\nIntroducing uniaxial anisotropy was also found to improve\nthe longitudinal strain derivative qm\n11=d\u001511=dH while reduc-\ning the transverse strain derivative dm\n21[12], [13]. In Figure 3,\nthe magnetic field derivative of the magnetostrictive curves\nare represented in the longitudinal and transverse direction for\nCF-CM and CF-SPS. The sum of both qmP=qm\n11+qm\n21is\nalso plotted. The maximum longitudinal strain derivative for\nCF-CM is -0.73 nm/A while it was increased to -1.3 nm/A\nfor CF-SPS. For the transverse direction, the maximum strain\nderivative for CF-CM is 0.3 nm/A while it was reduced to\n0.1 nm/A for CF-SPS. By summing up these two piezomag-\nnetic coefficient, the strain derivative calculated for CF-CM\nis -0.45 nm/A while improving to -1.2 nm/A for CF-SPS. As\nqm\n11andqm\n21are opposite in sign, the improvement of the sum\nqmPfor cobalt ferrite with induced uniaxial anisotropy CF-SPS\nis mainly due to the low transverse strain derivative qm\n21, a\ndirect consequence of the low transverse magnetostriction \u001521\nof the sample. Moreover, the applied field required to reach the\nmaximumqmPis reduced for CF-SPS when compared to CF-\nCM from 300 kA/m to 155 kA/m. Thus, besides increasing\nqmPby about a factor of three, the uniaxial anisotropy also\nreduces to half the required applied field to reach the maximum\nvalue, which is of great importance to make sensors with high\nsensitivity while requiring low applied fields.\nq\nApplied Field (kA/m)0 100 200 300 400 500 600 700d\nFig. 3. Piezomagnetic curves deduced from magnetostrictive measurement\nfor CF-CM and CF-SPS in black and red respectively. The solid line ( q11=\nd\u001511=dH ) corresponds to the strain derivative in the direction (1) and the\ndash line (q21=d\u001521=dH ) to the strain derivative in the direction (2). Line\nwith square symbol represents the sum of q11andq21.\nC. Magnetoelectric Effect\nTo evaluate the potential of these ferrites in magnetoelectric\napplications, CF-CM and CF-SPS were bonded on PZT disks\nto obtain magnetoelectric bilayers. Magnetoelectric voltage\nwas measured as function of a DC magnetic field applied in\nthe transverse direction of the bilayer disk while a small AC\nfield (1 mT, 80 Hz) was superimposed in the same direction.\nHere, low frequency was used to avoid any resonance effect.\nThe transverse magnetoelectric coefficients \u000b31were hence\ndeduced from the piezoelectric voltage measured along the\nthickness direction. The magnetoelectric setup is represented\nin the inset of Figure 4. The magnetoelectric coefficient mea-\nsured for CF-CM/PZT \u000bCF\u0000CM\n31 and CF-SPS/PZT \u000bCF\u0000SPS\n31\nare shown in Figure 4. The magnetoelectric effect observed\nfor the bilayer with CF-SPS is about three times higher than\nthe one observed in the bilayer with CF-CM. A maximum\nmagnetoelectric coefficient of 26 mV/A and 80 mV/A are\nobtained for the CF-CM/PZT and CF-SPS/PZT respectively.\nMoreover, this maximum value is reached at much lower\napplied field, 120 kA/m for \u000bCF\u0000SPS\n31 when compared to\n275 kA/m for \u000bCF\u0000CM\n31 . These results agree well with the\npiezomagnetic coefficient deduced from the magnetostrictive\ncruves. Indeed, the magnetoelectric model derived at low\nfrequency [5] shows the dependance of \u000b31onqmP:\n\u000b31=\u0011(qm\n11+qm\n21)de\n31\n\u000f33\u0002\n(se\n11+se\n21) +\u0011\r(sm\n11+sm\n21)\u0003\n\u00002(de\n31)2\n\u00021\n1 +Nr\u001f(1)\nwhere\u0011is the mechanical coupling factor, de\n31is the transverse\npiezoelectric coefficient, \u000f33is the dielectric permittivity, sij\nare the compliance, \r=\u0017e\n\u0017m=te\ntm, withteandtmas the\nthickness of PZT and ferrite respectively, \u001fthe susceptibility\nandNrthe demagnetizing factor which depends on the ferrite\nshape.\nHere, both bilayers have the same geometry and mechanical\nproperties. Indeed, compliance were measured for CF-CM,\ngiving :s11= 6.74 nm2/N ands21= –1.97 nm2/N; and for\nCF-SPS :s11= 6.44 nm2/N ands21= – 1.96 nm2/N. Hence,4\nMagnetoelectric Coefficient (mV/A)100\n80\n60\n40\n20\n0\n0 100 200 300 400 500 600 700100\n80\n60\n40\n20\n0\nH\nVACPDC\nHAC\nPZTCF\nCF-CM\n31\nCF-SPS\n31\nApplied Field (kA/m)\nFig. 4. Transversal magnetoelectric coefficient \u000b31as function of DC applied\nfield for a bilayer (2/1) of CF-CM/PZT and CF-SPS/PZT in black and red\nrespectively. The AC applied field HACis of 1 mT at 80 Hz.\nthe meaningful parameter at low frequency behavior should\nbe the piezomagnetic coefficient. This explains why a ratio of\nthree is found between CF-CM and CF-SPS for the maximum\nmagnetoelectric coefficient \u000b31, as it was for the piezomag-\nnetic coefficient qmP. This also demonstrates that to optimize\nthe transverse magnetoelectric effect \u000b31at low frequency, a\nlowq21=d\u001521=dH is needed, and a possible way to reach\nit is to use materials exhibiting uniaxial anisotropy.\nMagnetoelectric measurements were also performed as\nfunction of the frequency of the AC magnetic field as plot-\nted in Figure 5. As reported in several papers [20]–[22], a\nbilayer with PZT of 1 cm diameter has an electromechanical\nresonance (EMR) around 300 kHz. The resonance in ME\ncoefficient occurs when the AC field is tuned to EMR. This\nis what we observed in the magnetoelectric response of both\nCF-CM/PZT and CF-SPS/PZT, where the main resonance was\nfound at 317 kHz and 314 kHz respectively (Figure 5). This\nresults in a magnetoelectric coefficient increased to 7.5 V/A\nfor CF-CM/PZT, which is 300 times higher than the coefficient\nmeasured at low frequency. For CF-SPS/PZT it was increased\nto 11 V/A, “only” 138 times higher when compared to low\nfrequency.\nThe model developped by Filippov [8] for a bilayer structure\nwith disks at the resonant frequency highlights the direct de-\npendance of the magnetoelectric coefficient on the sum of qm\n11\nandqm\n21as for low frequencies. However, in our case, the ratio\nbetween the two bilayers CF-CM/PZT and CF-SPS/PZT for\nthe magnetoelectric coefficient at resonant frequency is of 1.5\nand not 3 as it was at low frequency. At the EMR, mechanical\nparamaters should be mainly involved in the magnetoelectric\ncoupling compared to the piezomagnetic coefficient. But, as\nwas said before, mechanical properties of CF-CM and CF-\nSPS are very close, validated by the compliances values. Also,\nresonant frequency for both bilayer are identical, indicating\nsimilar mechanical behavior. So, the meaningful parameter at\nEMR in this case seems to be either the damping factor [8],\nalso named mechanical loss factor [20], or the mechanical\ncoupling coefficient. These parameters might depend on the\nFrequency (kHz)0 100 200 300 400 500024681012\n024681012Magnetoelectric Coefficient (V/A)CF-CM\n31\nCF-SPS\n31α\nαFig. 5. Transversal magnetoelectric coefficient \u000b31as function of frequency\nfor a bilayer (2/1) of CF-CM/PZT and CF-SPS/PZT in black and red\nrespectively.\nmicrostructure of the cobalt ferrite. Here, CF-CM has lower\nrelative density (90 %) than CF-SPS (97 %) because SPS\nsintering allows very dense materials [15]. Moreover, CF-\nCM has much larger grain size ( \u00184\u0016m) than CF-SPS\n(<100 nm ), because SPS permits very short time process,\nhence the grain growth does not occur [15], [17]. These mi-\ncrostructure properties could affect the damping factor, and/or\nthe mechanical coupling coefficient, explaining the difference\nin amplitude found for the magnetoelectric coefficient between\nthe two bilayers CF-CM/PZT and CF-SPS/PZT at the resonant\nfrequency.\nSome minor peaks are also present at other frequencies\nsuch as 172 kHz, 212 kHz and 448 kHz for CF-CM/PZT and\n165 kHz and 425 kHz for CF-SPS/PZT. These peaks might be\na consequence of the structure used here, which is a bilayer. In\nfact, if the mechanical coupling at the interface is not perfect,\nit can results in a minor improvement of the magnetoelectric\neffect at other frequencies than EMR for bilayers [23].\nIV. C ONCLUSION\nIn summary, magnetic, magnetostrictive and piezomagnetic\nproperties are compared for isotropic and anisotropic cobalt\nferrite disks. Isotropic behavior was observed for cobalt ferrite\nmade by the ceramic method while anisotropic properties\nwere found for cobalt ferrite made by reactive sintering at\nSpark Plasma Sintering. This has a direct effect on the mag-\nnetostrictive behavior and particularly in the piezomagnetic\ncoefficient, were the maximum qmP=d\u0015P=dH obtained\nwas three times higher for CF-SPS than for CF-CM and\nfor a lower magnetic applied field. As the magnetoelectric\neffect is expected to depend mainly on the sum qmP, the\nmaximum magnetoelectric coefficient obtained at low fre-\nquency for the bilayer CF-SPS/PZT is three times higher\nthan for CF-CM/PZT. This result points out the importance\nof investigating at both piezomagnetic coefficient qm\n11andqm\n21\nto determine if a magnetic material has good magnetoelectric\npotential. Measurement at the resonant frequency show that5\nmagnetoelectric effect for anisotropic coblat ferrite was 1.8\ntimes higher than for isotropic cobalt ferrite. 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Phys. , vol. 101,\nno. 8, p. 083902, 2007.\n[23] D. A. Filippov, U. Laletsin, and G. Srinivasan, “Resonance magneto-\nelectric effects in magnetostrictive-piezoelectric three-layer structures,”\nJ. Appl. Phys. , vol. 102, no. 9, p. 093901, 2007."    },    {        "title": "2010.00856v1.Effect_of_tensile_stresses_on_bainitic_isothermal_transformation.pdf",        "content": "Effect of tensile stresses on bainitic isothermal transformation  \n \nT. J. Su 12, M. Veaux 1, 3, E. Aeby -Gautier  1, S. Denis  1, V. Brien  1 and P. Archambault  1 \n \n1 LSG2M, UMR 7584 du CNRS, INPL, École des Mines, Parc  de Saurupt, 54042 Nancy cedex, France  \n2 Beijing lnstitute of Technology, Beijing 100081, China  \n3 PSA, Centre Technique de Belchamp, 25420 Voujeaucourt, France  \n \nAbstract : The effects of tensile stresses on isothermal bainitic transformation were studied in \nthe case of a 35MV7 steel. T he modification of transformation k inetics and  the p resence  of \ntransformation  plasticity is shown i n a first step.  Furthermore the effect of stress  on the \nmorphological m odifica tions o f the ferrite laths is illustrated. The role of the stress o n these \nchanges  is analysed.  \n \n1. INTRODUCTION  \nIt is well established that when transformation occurs in the presence of stresses, the transformation  kinetics, the mechanical \nbehaviour as well as the morphology of the product phases are modifie d [1,2]. If  those interactions between stresses and diffusion \ndependent or martensitic transformations are known and  clarified, less results exist in the case of the bainitic transformation. The \nmechanical behaviour, in term of  the transformation plasticity in bainite, as w ell as the microstructural modifications associated \nwith the  transformation under stress are discussed in very few studies [3, 4]. For the prediction of the  microstructure formation as \nwell as the calculations of the internal stresses during quenching of s teels [5],  a good knowledge of the modifications of the bainitic \ntransformation mechanisms is required to  understand and further model the interactions of stresses on the phase transformations.  \n \n2. EXPERIMENTAL PROCEDURES  \nThe influence of an applied tensil e stress on the isothermal transformations of a 35MV7 steel was studied.  AU tests were \nperformed with our in -house thermomechanical simulator DITHEM which allows to apply  simultaneously controlled thermal and \nmechanical cycles. Radiation furnace or inducti on heating and  controlled cooling by gas (helium) sprayed on the specimen are \nused. The mechanical load is imposed by  an hydraulic jack. The diameter and gage length of the specimen are respectively 3 mm \nand 20 mm. After  austenitization at 1100C during 10 min, the specimen was cooled to the isothermal holding temperature  at a \nrate of 20'ces to 80°C/s, depending on the heating process. The stress was then applied when  reaching the transformation \ntemperature (450 or 350C) and maintained until transformation w as completed. Applied stresses were ranging from 0 to 235 MPa. \nTEM observations were performed on  some specimens of which the transformation conditions are given in Table 1. At the end of \nthe transformation, the specimen was quenched to room temperature. T he time at which transformation was  completed is largely \ndependent on the transformation temperature and the applied stress [6] as shown in  Table 1. Thin foils were cut perpendicular to \nthe direction of the tensile stress and examined by  transmission electron microscopy (Philips CM200 at 200 kV).  \n \n \n \n3. RESULTS  3. 1. Effect of stress on the transformation kinetics and the mechanical behaviour  \n \nFigure 1 shows the gage length variations measured during the isothermal transformation under various  stresse s at 350C. When \nincreasing the stress, an acceleration of the transformation is observed : the  incubation period is shortened and the transformation \nrate increases  [1]. Similar results were obtained for the  transformation at 450C. Moreover, the transformat ion plasticity deformation \nassociated with the transformation u nder stress is clearly evidenced. The evolution of the strain obtained when the  transformation \nis completed, is plotted versus the normalized stress (ratio applied stress / yield stress of  the parent austenite) in Fig. 2, for the \ntwo transformation temperatures. For a normalized stress less than  unity, the results are actually close, whatever the \ntransformation temperature. Moreover, the increase of  the transformation plasticity with stress is n on linear : the slope d/d \nincreases for stresses above the  yield stress of austenite, especially for the lower transformation temperature. These phenomena \nare detailed and discussed in [6].  \n \n \n3. 2. Analysis of the microstructures  \n \nFirst SEM observations  of the microstructure formed at 350C under stress have shown that each  austenite grain appears to \ntransform into fewer variants of bainite, giving the microstructure a less random  appearance [6]. For the samples transformed \nunder very high stress (normali zed stress of 2. 7), each  austenite grain has been transformed into a single orientation. These \nresults are consistent with those of  Bhadeshia et al. and Matsuzaki et al. [3, 4]. The measurements of the width of the plates \nformed at  different temperatures in the case of the 35MV7 steel showed that there is no large change of the plate  width with the \nincrease of the applied stress [6].  \n In Fig. 3 are reported TEM micrographs of specimens transformed at 350 C with stress (350 -128) and  without stress (350 -0). The \nmicrostructure of specimen 350 -0 is typical of lower bainite (Fig. 3 (a)). This  structure consists of aggregates of platelets and \ncarbides precipitated within these sub -units. To  accommodate the interna l stress caused by the dis placive transformation, several \nsheaves with different  orientations form within a previous austenite grain. The area of a sheave and the size of a platelet can be  \nvaried by the section effect. Approximately, the length of the sub -units is about 1 micro metre to 3 micro metres , and the  width \nabout 0. 2 micro metres . When transformed under stress, plates grow much l onger, with a length of over 10 micro metres  and a \nwidth between 0. 3 micro metres  and 0. 9 micro metres  as shown in Fig. 3 (b). Ferrite units exhibit a preferential  orientation as \ncompared with those of Fig. 3 (a). This is consistent with previous optical and SEM  observations [4, 6].  \n \n \n \nFig. 4 and Fig . 5 show the TEM micrographs of specimens transformed at  450'C respectively  without stress (450 -0) and with \nstress (450 -144).  Microstructure of 450 -0 is typica1 of upper  bainite (Fig. 4), consisting of ferrite plates and  carbides primarily \nprecipitated between these  plates. The length  of the sub -units is about 5 gm to 10 gm, and the width of about 0. 3 gm to 0. 9  \ngm. The tensile stress bas a large effect on the microstructure evolution, as exhibited in Fig. 5.  Ferrite units in 450 -144 are much \ncoarser and  have diverse morphologies. The  length of the  units increases and can reach 10 micro metres to 20 micro metres , as \nwell a s the width between 1 to 2 micro metres . Morphologies of the units can be plate -like (Fig. 5(a)) as without stress can be \nplate -like (Fig5  (b)) or needle -like. \nAt last c arbides are more difficult to detect. In some areas carbides can be observed as in the upper bainite  transformed without \nstress, while in other places the carbides are no more evidenced. A careful analysis  reveals that fine carbides are observed inside \nthe ferrite plates [7].   \n \n4. DISCUSSION AND CONCLUSION  \n \nThe occurrence of transformation plasticity is associated with two basic mechanisms, i. e. the anisotropic  plastic slip due to the \ninternal  and applied stresses [8], and the preferential orientation of the transformation product by the applied stress [9]. For the transformation conditions reported here, the  applied stress is near the yield stress of the parent phase. The observations clearly \nevidence that a  preferential orientation is observed. Thus part of the transformation plasticity is due to ferrite units  orientation. \nMoreover, a careful analysis of the crystallographic orientation along some plates shows that  a continuous disorientation exists in \nthe fe rrite units, which can be associated with large deformation  inside the plates during the transformation [7].   \n \n \nDislocations were also observed in the parent austenite. This strong evidence of plasticity in both the  parent and product phases \nas well as the  preferential morpholog ic orientation of the ferrite units show that  the two transformation plasticity mechanisms have \nto be considered for the bainitic transformation. The  modifications in the morphology, the increase in length and in width of the \nplates,  reveal that the plate's  growth is modified. The acceleration of the transformation kinetics can be explained by a modification \nof the nucleation and growth rates. An increase in the nucleation rate alone, as observed for diffusion  controlled transformatio ns \n[10] or for strain induced martensite [11], would lead to smaller ferrite units.  The present observations show that the ferrite sub -\nunits are longer, their growth is thus favored by the  stress, rather than their nucleation. As carbides are mainly in the  ferrite units \nwhen transformation occurs  under stress, one can think that the growth rate enhancement is mainly due to the interfacial kinetics  \nmodification (easier accommodation of the transformation strain by emission of dislocations) rather than  to the  increase in the \ncarbon diffusion rate.  \n \n \nAcknowledgements  The authors are grateful to the support of PSA Peugeot -Citroën and to the French Chinese Program for  \nScientific Cooperation (PRA 98 -MX 98 -04). \n \nReferences  \n[1] : Denis S., Gautier E., Simon A. and Beck G., Mater. Sci. Technol., Vol. 1 (1985) p 805.  \n[2] : Gautier E., Zhang J. S. and Zhang X. M., J. Phys. IV, Colloque C8, Vol. 5 (1995) p 41.  \n[3]: Bhadeshia H.K.D.H., David S. A., Vitek J. M. and Reed R. W., Mater. Sc i. Technol. Vol. 7 (1991)  \np686.  \n[4]: Matsuzaki A., Bhadeshia H. K. D. H. and Harada H., Acta Met. Mater. Vol. 42 (1994) p 1081.  \n[5] : Denis S., Archambault P., Gautier E., Simon A., Beck G., J. of Materials Engineering and  \nPerformance, Vol. 11 (2002) p. 92. \n[6] : Veaux M., Louin J. C., Houin J. P., Denis S., Archambault P., Gautier E., J. Phys. IV, France 11  \n(2001) Pr4 -181-188. \n[7] : Su T. S., Denis S., Aeby -Gautier E., To be published  \n[8] : Greenwood G. W. and Johnson R. H., Proc. Roy. Soc., Vol. 283A (1965) p 403.  \n[9] : Magee C. L., Ph. D. Thesis, Carnegie Mellon University, Pittsburg, PA (1965).  \n[10] : Gautier E. and Simon A. PTM 87, International Conference Phase Transformations'87 Univer sity \nof Cambridge 6 -10 July 1987, The Institute of Metals, Ed. G. W. LORIMER, p 451.  \n[11]: Oison G.B. and Cohen M. Metall. Trans. A 6A (1975) pp 791 -795. "    },    {        "title": "1808.03432v2.Role_of_disconnections_in_mobility_of_the_austenite_ferrite_inter_phase_boundary_in_Fe.pdf",        "content": "Role of disconnections in mobility of the austenite-ferrite inter-phase boundary in Fe\nPawan Kumar Tripathi,1Sumit Kumar Maurya,1and Somnath Bhowmick1,\u0003\n1Dept. of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India\n(Dated: January 3, 2019)\nAustenite ( \r-Fe, face centered cubic (FCC)) to ferrite ( \u000b-Fe, body centered cubic (BCC)) phase\ntransformation in steel is of great signi\fcance from the point of view of industrial applications.\nIn this work, using classical molecular dynamics simulations, we study the atomistic mechanisms\ninvolved during the growth of the ferrite phase embedded in an austenite phase. We \fnd that the\ndisconnections present at the inter-phase boundary assist in growth of the ferrite phase. Relatively\nsmall interface velocities (1.19 - 4.67 m/s) con\frm a phase change via massive transformation mech-\nanism. Boundary mobilities obtained in a temperature range of 1000 to 1400 K show an Arrhenius\nbehavior, with activation energies ranging from 30 - 40 kJ/mol.\nI. INTRODUCTION\nDuring the process of iron and steel making, as molten\nFe is cooled, \frst it solidi\fes to \u000e(BCC) allotrope of iron\nat a temperature of 1811 K. This is followed by solid-\nsolid phase transformations, initially from \u000e-Fe to\r-Fe\n(FCC) at 1667 K and \fnally from \r-Fe to\u000b-Fe (BCC)\nat 1185 K. The latter is very important, because the mi-\ncrostructure and mechanical properties of Fe-alloys are\ngoverned by the amount of austenite ( \r-Fe) and ferrite\n(\u000b-Fe) present after the transition. Being a very com-\nplex process, governed by several extrinsic (composition,\nrate of cooling etc.) and intrinsic (nucleation, inter-phase\nand grain boundary mobility, relative orientation of the\ntwo phases etc.) factors, the atomistic mechanisms in-\nvolved during the phase transition are not clearly under-\nstood yet. Based on several experimental studies, it has\nbeen established that the nature of the transition is ei-\nther martensitic or massive.1,2The former is a di\u000busion-\nless transformation, which takes place via a coordinated\nmovement of atoms by a distance less than the inter-\natomic spacing. On the other hand, massive transforma-\ntion occurs via nucleation and growth of the ferrite phase\nat the expense of the austenite phase, driven by Gibbs\nfree energy change.\nIn order to describe the kinetics of the \r-\u000btransfor-\nmation, mainly two types of models have been proposed\nin the literature; di\u000busion controlled growth model3and\ninterface controlled growth model.4In reality, transfor-\nmations are mixed in nature, starting as an interface\ncontrolled process and following the initial stages of\nnucleation and growth, a relatively slow di\u000busion con-\ntrolled process takes over.5,6The interface controlled\nphase transformation is characterized in terms of intrinsic\nmobility of the inter-phase boundary and values ranging\nfrom 10\u00006to 10\u00009m-mol/(J-s) have so far been reported\nin the literature.2,6,7Boundary mobilities are also known\nto show an Arrhenius behavior, with activation energy\nreported to be\u0019140 kJ/mol.2,6,7\nSince the nucleation and growth of the ferrite phase\nstarts at the \r-\u000binter-phase boundary, orientations\nof the two phases at the interface play a crucial\nrole in transformation. Several orientation relation-ships (OR) between the FCC and BCC phase have\nso far been proposed in the literature. This includes\nBain,8Nishiyama-Wasserman (NW),9Kurdjumov-Sachs\n(KS),10Greninger-Troiano (GT)11and Pitsch.12Other\nthan the Bain and Pitsch, interface is formed between\nthe two closest packed planes of the two phases, i.e.,\n(111) FCCk(110) BCC. Pitsch OR is exactly opposite to\nthis, with (111) BCCk(110) FCC. In case of Bain OR,\ninterface is formed between the (001) plane of both the\nphases. Among all the ORs, NW and KS are more often\nreported in case of iron and steel.13\nBecause of its length and time scale, interface con-\ntrolled\r-\u000bphase transformation can be investigated by\natomistic calculations14{17and several studies related to\nmassive and martensitic transformations based on clas-\nsical molecular dynamics simulations have been reported\nso far.18{25During martensitic transformations, interface\nvelocities are found to be very high, ranging between\n200-700 m/s in case of Bain and KS ORs at di\u000berent\ntemperatures.20On the other hand, much smaller inter-\nface velocities (0.7-3.4 m/s) are obtained in case of mas-\nsive transformation, as reported for a \r\u0000\u000binterface of\nNW type.21,22A comparison between the NW and KS\nORs reveals planar and needle like growth of the ferrite\nphase taking place at the respective interfaces, the former\nbeing ten times slower than the latter.23Bi-directional\ntransformations are also reported in case of NW orienta-\ntion, with signi\fcant di\u000berence of interface velocity be-\ntween the\r\u0000\u000b(24 m/s) phase change and vice versa\n(240 m/s).24\nInterestingly, in many of the computational studies\nmentioned above, some kind of defect (like a free sur-\nface, stacking faults, twin boundaries, steps present at\nthe\r\u0000\u000binterface etc.) is present in the initial structure,\nwhich assists the phase transformation. Motivated by\nthis, we focus on a particular type of defect, known as dis-\nconnections. This a type of interfacial defect having both\ndislocation and step-like character.26{28Disconnections\nare reported to be observed at the inter-phase bound-\naries of several ferrous and non-ferrous materials.29{33\nThey are also reported to play important role during the\nphase transformation.27,28,34{36In this paper we investi-\ngate role of disconnections during the austenite to ferritearXiv:1808.03432v2  [cond-mat.mtrl-sci]  2 Jan 20192\ntransformation in pure-Fe, using classical molecular dy-\nnamics simulations. Our study clearly shows that the\ndisconnections located at the austenite-ferrite interface\nfacilitate the growth of the \u000b-Fe phase. We also calcu-\nlate the velocity and mobility of the \u000b\u0000\rinterface and\nthe values suggest a massive transformation from \r-Fe to\n\u000b-Fe.\nThe paper is organized as follows: in Sec. II we discuss\nthe simulation details, which include A) a discussion on\ninteratomic potential, B) calculation of driving force for\nthe phase transformation, C) crystallographic description\nof the simulation box and D) calculation of interface ve-\nlocity and mobility. This is followed by a detailed discus-\nsion of the main results obtained in this wrok in Sec. III\nand the paper is concluded in Sec. IV.\nII. SIMULATION DETAILS\nA. Interatomic Potential\nAll the calculations are performed using classical\nmolecular dynamics (MD) simulations, as implemented\nin Large-scale Atomic/Molecular Massively Parallel Sim-\nulator (LAMMPS) package.37Interaction among metal\natoms are well approximated by EAM (embedded atom\nmethod) potentials. In this work we use the empirical\npotential developed by Ackland et al.38Using this po-\ntential, calculated values of lattice parameter, cohesive\nenergy, vacancy formation energy and elastic constants\nare found to be in very good agreement with experi-\nmental data, as well as density functional theory (DFT)\nbased predictions [see Table I]. However, there are two\ndrawbacks of this potential. First, it overestimates the\nmelting point [see Table I], a fact already reported in the\nliterature.39Second, the BCC phase remains more sta-\nble than FCC up to the melting temperature. In reality,\na BCC to FCC phase transition is observed at 1185 K\nin the experiments, which can not be captured by this\nempirical potential. Despite these limitations, Ackland\npotential has been used in numerous MD studies on Fe,\nincluding the FCC to BCC phase transition.21,22,40,41\nB. Driving Force for the Phase Transformation\nIt is well known that the driving force behind mas-\nsive transformation is the reduction of the Gibbs free\nenergy (\u0001G\r\u0000\u000b) as Fe transforms from the austenite to\nthe ferrite phase. In order to calculate \u0001 G\r\u0000\u000b, we \frst\ncalculate \u0001 GL\u0000S, the free energy di\u000berence between the\nliquid (L) and solid (S) phase using the Gibbs-Helmholtz\nequation,\n\u0001GL\u0000S\nT=ZTm\nTHS(T)\u0000HL(T)\nT2dT; (1)\nwhereHis the enthalpy, which is a function of temper-\natureTandTmis the melting point. The calculation isTABLE I. Comparison of physical properties estimated using\nthe empirical potential proposed by Ackland et al.(1997)38\nwith DFT calculations or experimentally measured values.\nProperty Experiment\nor DFTAckland et\nal.(1997)\na (\u0017A), BCC at T=0 K 2 :855a2.866\na (\u0017A), FCC at T=0 K 3 :658a3.680\nTm(K) 1812a2358\nEcoh(ev/atom) -4.316a-4.316\nEv\nf(ev/atom) 1 :84a1.89\nC11(GPa) 242 :00b243.39\nC12(GPa) 146 :50b145.03\nC44(GPa) 112 :00b116.00\naValues taken from Mendelev et al.(2003)42\nbValues taken from Hirth and Lothe (1968)43\nTABLE II. Driving force for \r-\u000btransition in iron, calculated\nusing the Ackland potential.\nTemperature(K) 1000 1200 1400\n\u0001G\r\u0000\u000b(ev/atom) 0.0198 0.0183 0.0168\n\u0001G\r\u0000\u000b(kJ/mole) 1.912 1.767 1.622\ncarried out separately for both \u000band\rsolid phases from\n1800 to 2200 K at an interval of 100 K [see Fig. 1]. In case\nof the solid phase, the simulation box (of size 10 \u000210\u000210)\nis equilibrated at a given temperature and zero pressure\nusing a NPT ensemble and the enthalpy at that particu-\nlar temperature ( HS(T)) is given by the potential energy\nof the system. On the other hand, in case of the liquid\nphase (simulation box size 10 \u000210\u000210), \frst the system\nis melted at 3000 K and then rapidly cooled to a lower\ntemperature using a NPT ensemble. Rapid cooling en-\nsures that the liquid like structure is maintained even be-\nlow the melting point and \fnally the system is subjected\nto a NVT run (same temperature at which the liquid is\n-0.06-0.05-0.04-0.03-0.02-0.0100.01\n1700 1800 1900 2000 2100 2200 2300ΔG (ev/atom)\nTemperature (K)BCC\nBCC-fit\nFCC\nFCC-fit\nFIG. 1. Free energy di\u000berence between the liquid and solid\nphase (\u0001GL\u0000S) for both\u000b-Fe and\r-Fe. Free energy di\u000ber-\nence between the solid phases (\u0001 G\r\u0000\u000b) is calculated from the\nvertical di\u000berence between the two lines.3\ncooled) to get the potential energy, equal to the enthalpy\nof the liquid phase at a given temperature ( HL(T)). In\norder to estimate the melting point, we use the coexis-\ntence method (developed by Morris and Song44) andTm\nfor the BCC and FCC phase are found to be 2358 K and\n2237 K, respectively.\nCalculated values of \u0001 GL\u0000Sare illustrated in Fig. 1,\nalong with a linear \ft of \u0001 GL\u0000Sas a function of tem-\nperature, as shown below-\n\u000fFCC: \u0001GL\u0000S= 6:495\u000210\u00005T\u00000:1469,\n\u000fBCC: \u0001GL\u0000S= 7:245\u000210\u00005T\u00000:1742.\nFinally, the free energy di\u000berence between the solid\nphases (\u0001G\r\u0000\u000b) is obtained from the vertical di\u000berence\nbetween the two lines at any given temperature. The\nlines shown in Fig. 1 are extrapolated to get the value\nof \u0001G\r\u0000\u000bat lower temperature, lying in the range of\n1000 K to 1400 K. As reported in Table II, in the tem-\nperature range of 1000 to 1400 K, \u0001 G\r\u0000\u000bvalues pre-\ndicted by EAM potential lie in the range of 1.912 to 1.622\nkJ/mole. Although the value dips a bit with increasing\ntemperature, the system is still far from the ferrite to\naustenite transformation even at 1400 K. This is a well\nknown drawback of the Ackland potential, which prefers\nferrite over the austenite phase all the way to the melting\ntemperature. Comparing with the experimental results,\nit is found that the numerical values of \u0001 G\r\u0000\u000bobtained\nfrom the Ackland potential in this study [see Table II]\nare equivalent to the actual free energy di\u000berence at a\ntemperature range of roughly 700-750 K.45Thus, driv-\ning force applied in this study is not completely out of\nrange.\nC. Simulation Box Details\nSince our goal is to study the FCC to BCC phase tran-\nsition, we must have both the phases present in the be-\nginning. For this purpose, initially we create a BCC and\na FCC box separately. The crystallographic directions\nparallel to the box edges and sizes of the boxes in terms\nof number of atomic planes present along a particular\ndirection are reported in Table III. Note that, the cross-\nsection (yz plane) of both the boxes are chosen such that\nthe area mismatch of the individual interfaces remains\nless than 0.5% after we join the two phases at a later\nstage. A larger mismatch of cross-section between the\ntwo phases should be avoided, as it leads to high stresses,\nwhich can signi\fcantly a\u000bect the transformation process.\nFirst, both the boxes are equilibrated separately for 2 ns\nusing a NVT ensemble to bring all the atoms in ther-\nmal equilibrium. This is followed by a 6 ns run using a\nNPxT ensemble for the purpose of volume equilibration,\nwithout altering the interface area.\nAfter equilibration, boxes are joined in a sequence\nof BCC-FCC-BCC, making a sandwich like structure\nFIG. 2. (a) Shape of the simulation box after joining the\nseparately equilibrated BCC (blue) and FCC (green) phases,\nforming a sandwich like BCC-FCC-BCC structure. The \fgure\nis prepared using the Ovito software,46which uses common\nneighbor analysis to distinguish between the FCC and BCC\nphase. (b)-(e) Atomic con\fguration of the BCC-FCC inter-\nfaces considered in this work. (b) Atomically \rat interface\nbetween the (110) plane of BCC and (111) plane of FCC, as\nper NW OR. Steps or disconnections appear when FCC phase\nis rotated with respect to the zaxis from the ideal NW OR\nby (c) 3:11\u000e, (d) 4:04\u000eand (e) 5:77\u000e. Large gaps between the\ntwo phases are shown for the sake of visual clarity. In reality,\nspacing between BCC and FCC region are taken to be the\naverage of BCC (110) and FCC (111) inter-planar distance.\n[see Fig. 2]. After joining the two phases, the simu-\nlations box remains fully periodic, having no free sur-\nface and there are two BCC-FCC interfaces within the\nbox. Interface is formed parallel to the yzplane and\nthe growth direction (of the BCC phase) is perpendicu-\nlar to the interface (along the xaxis). Since NW is one\nof the most commonly observed orientation relationship4\n(111)\n(111)l\nh\nθ[hkl]\nL[11-2]\n[111][hkl]\nθ\nθ=3.11, L=17d, HP=(998), [hkl]=[44-9]\nθ=4.04, L=13d, HP=(776), [hkl]=[33-7]\nθ=5.77, L= 9d, HP=(554), [hkl]=[22-5]o\no\noTP=(111)\nFIG. 3. A schematic diagram of the disconnections present\nin the FCC phase, as shown in Fig. 2(c)-(e). In all the cases,\n(111) forms the terrace plane (TP), but the habit plane (HP,\nshown by the dotted line) changes depending on \u0012, the tilt\nangle from the ideal NW OR. [hkl] and [11 \u00162] direction lies in\nthe HP and TP, respectively and the angle between the two\ndirections equals \u0012.~lis the ledge vector and step height h\nis measured from the HP. As shown in Fig. 2, every step is\nmonoatomic. Lis the length of the TP, consisting of 17, 13\nand 9 atomic rows and dis the diameter of Fe atom.\nbetween the BCC and FCC phase, we select this among\nvarious possibilities (as mentioned in Sec. I) to create the\naustenite-ferrite interface in this work. NW is a semi-\ncoherent interface between the closest packed planes of\nBCC and FCC phase, described as (110) BCCk(111) FCC\nand [001] BCCk[1\u001610]FCC. In case of ideal NW orientation\nrelationship, an atomically \rat interface is created be-\ntween the BCC (110) and FCC (111) plane [see Fig. 2(b)].\nKeeping the BCC phase \fxed, we further tilt the FCC\nphase about the zaxis (parallel to the [1 \u001610] direction),\nwhich creates some equally spaced steps or disconnec-\ntions in the FCC phase at the interface [see Fig. 2(c)-\n(e)]. As shown in the diagram, the FCC phase is tilted\nwith respect to the ideal NW orientation relationship by\nan angle of 3 :11\u000e[Fig. 2(c)], 4 :04\u000e[Fig. 2(d)], and 5 :77\u000e\n[Fig. 2(e)]. Evidently, number of steps at the interface\nincreases with increasing tilt angle. Further details re-\ngarding the crystallographic directions parallel and per-\npendicular to the BCC-FCC interface are given in Fig. 2\nand Table III. While joining the two phases, spacing be-\ntween BCC and FCC region are taken to be the average\nof BCC (110) and FCC (111) inter-planar distance.\nA schematic diagram of the disconnections present at\nthe FCC phase [see Fig. 2(c)-(e)] is presented in Fig. 3.\nIn all the three cases, (111) forms the terrace plane (TP),\nbut the habit plane (HP, shown by the dotted line in the\n\fgure) changes, depending on the tilt angle from ideal\nNW OR. For \u0012= 3:11\u000e;4:04\u000e;5:77\u000e, the corresponding\nHPs are found to be (998), (776) and (554), respectively.\nAs shown in Fig. 3, [hkl] and [11 \u00162] direction lies in theTABLE III. Crystallographic orientations parallel to the x,y\nandzdirection of the simulation boxes used in this work.\nPhase Direction OrientationSize (no.of\natomic planes)Tilt angle\nBCCx\ny\nz[110]\n[\u0016110]\n[001]16\n42\n54-\nFCCx\ny\nz[111]\n[11\u00162]\n[1\u001610]180\n37\n600\u000e\nFCCx\ny\nz[998]\n[44\u00169]\n[1\u001610]180\n37\n603.11\u000e\nFCCx\ny\nz[776]\n[33\u00167]\n[1\u001610]180\n37\n604.04\u000e\nFCCx\ny\nz[554]\n[22\u00165]\n[1\u001610]180\n37\n605.77\u000e\nHP and TP, respectively and the angle between these\ntwo directions is equal to the tilt angle ( \u0012) from ideal\nNW OR. The tilt angle ( \u0012) is also equal to the angle\nbetween the [111] direction and a vector perpendicular\nto the HP. Ldenotes the length of the TP, consisting of\n17, 13 and 9 atomic rows, when the HP is (998), (776)\nand (554), respectively. The ledge vector is marked as ~l\nand since the BCC phase is terminated by a \rat inerface,\n~lis also equal to the Burgers vector of the disconnection\n(de\fned as the sum of individual ledge vectors of the\ntwo phases).22,26As illustrated in Fig. 2, every step is\nmonoatomic. The step height ( h) is measured from the\nHP and it is approximately equal to 2.4 \u0017A.\nAfter creating the simulation box with both the phases\npresent in it [see Fig. 2(a)], we \fnally run the dynamics\nusing a NP xT ensemble until the FCC phase completely\ntransforms into the BCC phase [see Fig. 4]. Depend-\ning on the temperature, it takes around 4 to 15 ns for\nthe transformation to complete. The transformation can\nalso be tracked by monitoring the change of potential\nenergy, which continuously decreases as the fraction of\nBCC phase increases with time [see Fig. 5 and Fig. 6].\nD. Interface Velocity\nThe speed ( v) at which the austenite-ferrite interface\nmoves can be estimated from the rate of change of po-\ntential energy\u0000dE\ndt\u0001\nduring the transformation [see Fig. 5]\nusing the following equation:\nv=\n2aLdE\ndt; (2)\nwhereais the area of interface connecting the two phases,\nLis the latent heat of solid-solid phase transformation\n(enthalpy di\u000berence per atom between the \u000band\rFe)\nand \n is the volume per atom in the FCC phase. The\nfactor 2 in the denominator takes into account the two5\nFIG. 4. Evolution of BCC phase (blue) at 1000 K, when the\nFCC phase (green) is rotated by an angel of 4.04\u000efrom the\nideal NW OR. The snapshots are taken at 0, 5, 10 and 15 ns.\nFCC-BCC interfaces present in the simulation box. The\nvelocity (~ v) at which the austenite-ferrite interface moves\nis proportional to the driving force for the phase transi-\ntion (\u0001G\r\u0000\u000b),\n~ v=~M\u0001G\r\u0000\u000b; (3)\nwhere~Mis the interface mobility. Using the calculated\nvalues ofvand \u0001Gm, we further estimate the numerical\nvalue of interface mobility, which is related to the acti-\nvation energy required for one atom present in the FCC\nphase to cross the inter-phase boundary due to thermal\n\ructuations and get attached to the BCC phase. The\nactivation energy ( Q) is calculated from the following\nequation\n~M=~M0exp\u0012\n\u0000Q\nRT\u0013\n; (4)\nwhereRis the universal gas constant.\nIII. RESULTS AND DISCUSSIONS\nAmong the four di\u000berent BCC-FCC interfaces [see\nFig. 2 and Table III], no phase transformation is observed\nin case of ideal NW OR. However, when the FCC region\nis tilted with respect to the ideal NW OR, FCC to BCC\nphase transformation is indeed observed. As shown in\nFig. 4, growth of the BCC phase (blue) starts from both\nend of the simulation box and the FCC phase (green) is\ntransformed in due course of time, ultimately converting\nthe entire box to a BCC phase. Lack of phase transfor-\nmation in case of ideal NW type interface is probably due\nto the absence of any defect sites, which can assist the\ngrowth of the BCC phase. On the other hand, in case of\nother orientations (tilted with respect to the ideal NW\nOR), steps or disconnections are present at the interface\n[see Fig. 1], which is found to facilitate the growth of theBCC phase. This is going to be discussed in detail later\nin this section.\nTracking the \r\u0000\u000bphase transformation can simply\nbe done by monitoring the potential energy of the sys-\ntem as a function of time. Since \rhas higher free energy\nthan that of \u000b[see Fig. 1], the potential energy of the\nsystem is going to decrease as the former is transformed\ninto the latter phase. This is shown in Fig. 5 (a)-(c)\nfor three di\u000berently oriented FCC phases at three dif-\nferent temperatures. For a given orientation, the phase\ntransformation is faster at higher temperature [see Fig. 5\n(a)-(c)]. This is because activation energy required for\nan atom in the \rphase to detach from its parent FCC\nlattice, cross the interface and attach to the BCC lattice\nof the\u000bphase is provided by thermal \ructuations and\nthis process is facilitated at higher temperature.\nIt would also be interesting to compare the rate of\ntransformation among three di\u000berent orientations of the\nFCC phase at a given temperature. Change of potential\nenergy as a function of time during the transformation\nis plotted in Fig. 6 (a), (b) and (c) for 1000, 1200 and\n1400 K, respectively. Clearly, higher the tilt of the FCC\nphase with respect to the ideal NW OR, faster is the\nrate of transformation to the BCC phase. As already\nshown in Fig 2 (c)-(e), higher tilt angle with respect to\nthe ideal NW OR results more steps or disconnections in\nthe FCC side of the interface. Thus, these steps must be\nplaying some important roles during the phase transfor-\nmation process. Considering the fact that no transfor-\nmation is observed in case of atomically \rat ideal NW\nOR, as well as phase transition rate being enhanced with\nincreasing number of steps at the interface, it appears\nthat the steps or disconnections assist the growth of the\nBCC phase. This hypothesis is further con\frmed by tak-\ning snapshots of the simulation box at various time steps\nduring the transformation. Three such con\fgurations,\none each for every orientation considered in this paper,\nare shown in Fig. 7 (a)-(c), where only the BCC phase\nis illustrated for the sake of visual clarity. Comparing\nwith Fig. 2(c)-(e), we conclude that depending on the\nnumber of disconnections present initially, there are as\nmany locations from which new layers of ferrite phase\nstarts to grow during the transformation. Note that, the\ndisconnections present at the inter-phase boundary re-\nmain untill the whole simulation box is converted to the\nferrite phase. Moreover, it is also observed that a new\nset of disconnections develop during the process and the\ninterface movement takes place via the lateral motion of\nthese disconnections. This is very similar to the process\ndescribed by Song and Hoyt.22\nAfter uncovering the atomistic mechanism of austen-\nite to ferrite phase transformation, we now estimate the\nspeed at which the austenite-ferrite interfaces move dur-\ning the transition. Interface velocity is calculated using\nEq. 2, where\u0000dE\ndt\u0001\nis taken to be the slope obtained from\na linear \ft of the potential energy pro\fles during the\ntransformation [see Fig. 5 and Fig. 6]. Calculated values\nof interface velocity (reported in Table IV) lie in the range6\n(a) (b) (c)\n-4.18-4.15-4.12-4.09\n4 8 12Potential Energy (ev/atom)\nTime (ns)1400 K\n1200 K\n1000 K\n-4.15-4.12-4.09\n4 8 12Potential Energy (ev/atom)\nTime (ns)1400 K\n1200 K\n1000 K\n-4.15-4.12-4.09\n4 8 12Potential Energy (ev/atom)\nTime (ns)1400 K\n1200 K\n1000 K\nFIG. 5. Change of potential energy as the FCC transforms to the BCC phase at 1000, 1200 and 1400 K. The FCC phase is\nrotated by (a) 5 :77\u000e, (b) 4:04\u000eand (c) 3:11\u000efrom the ideal NW OR. Evidently, the transformation takes lesser time at higher\ntemperature.\n(a) (b) (c)\n-4.12-4.09\n4 8Potential Energy (ev/atom)\nTime (ns)998\n776\n554\n-4.15-4.12\n4 8Potential Energy (ev/atom)\nTime (ns)998\n776\n554\n-4.18-4.15\n4 8 12Potential Energy (ev/atom)\nTime (ns)998\n776\n554\nFIG. 6. Change of potential energy as the FCC transforms to the BCC phase at (a) 1000, (b) 1200 and (c) 1400 K. A\ncomparison is shown among three orientations of FCC phase with respect to the ideal NW OR; 998 (3 :11\u000e), 776 (4:04\u000e), 554\n(5:77\u000e). Clearly, the transformation takes lesser time at higher angle.\nTABLE IV. Interface velocities calculated (using Eq. 2) at\ndi\u000berent orientations and temperatures. These numbers are\ncalculated by taking average of the values obtained from eight\nindependent simulations starting with di\u000berent initial veloci-\nties for each of the temperature and orientation. The unit of\ninterface velocity is m/s.\nOrientation T=1000 K T=1200 K T=1400 K\n5.77\u000e1.98\u00060.29 3.24\u00060.28 4.67\u00060.21\n4.04\u000e1.46\u00060.09 2.59\u00060.19 4.25\u00060.15\n3.11\u000e1.19\u00060.15 2.28\u00060.21 4.11\u00060.23\nof 1.19 to 4.67 m/s, depending on the temperature and\norientation of the austenite phase. Note that, interface\nvelocities reported in Table IV are calculated by averag-TABLE V. Interface mobilities calculated (using Eq. 3) at\ndi\u000berent orientations and temperatures. Similar to Table IV,\ndata from eight independent calculations are averaged to get\nthe values of mobility for each of the temperature and orien-\ntation. The unit of mobility is 10\u00003m-mol/(J-s).\nOrientation T=1000K T=1200K T=1400K\n5.77\u000e1.0\u00060.15 1.8\u00060.16 2.9\u00060.13\n4.04\u000e0.8\u00060.05 1.5\u00060.11 2.6\u00060.09\n3.11\u000e0.6\u00060.08 1.3\u00060.12 2.5\u00060.14\ning the values obtained from eight independent simula-\ntions starting with di\u000berent initial velocities for each of\nthe temperature and orientation. Comparing with the\nvalues reported in the literature,20,21interface velocities7\n(a) (b) (c)\nFIG. 7. Growth of the ferrite phase initiates at the disconnec-\ntions present at the inter-phase boundary. Atoms belonging\nto the ferrite (BCC) phase are only shown in this \fgure, while\nthe snapshot is taken at some intermediate time step during\nthe transformation. With respect to the ideal NW OR, the\naustenite phase in this particular case is tilted by (a) 5 :77\u000e,\n(b) 4:04\u000eand (c) 3:11\u000e. Depending on the number of steps\nor disconnections present at the interface [see Fig. 2(c)-(e)],\nthere are as many sites from which the growth of the ferrite\nphase starts.\nobtained in the present work are signi\fcantly lesser than\nthat of martensitic transformation, but similar to that of\nmassive transformation. This further con\frms the trans-\nformation in the present work to be massive in nature.\nAs expected, interface velocity for any particular orienta-\ntion increases with temperature because higher thermal\nenergy helps the atoms to cross over from the austenite\nto the ferrite site. Interestingly, at a given temperature,\ninterface velocity increases as the austenite phase is tilted\nfurther away from the ideal NW OR. This is possibly be-\ncause, with increasing number of steps or disconnections,\nthere are more sites from which the growth of the ferrite\nphase can take place; leading to faster movement of the\nboundary at higher tilt angles.\nFinally, we estimate the mobility of the interface\nduring the austenite-ferrite transformation using Eq. 3.\nDriving force and interface velocity data are taken from\nTable II and Table IV, respectively. Calculated values\nof mobility are reported in Table V. Since interface ve-\nlocity increases and \u0001 G\r\u0000\u000bdecreases with temperature,\nmobility for a given interface orientation enhances with\nincreasing value of T. Interestingly, at a particular tem-\nperature, mobility increases with the angle of tilt of the\naustenite phase with respect to the ideal NW OR [see\nTable V]. This is directly related to the enhancement of\nthe interface velocity with the tilt angle at any particular\ntemperature, as reported in Table IV. Activation energy\nQ[see Eq. 4] is estimated from the slope of the ln M\nvs.1\nTline, as illustrated in Fig. 8 for all three di\u000berent\norientations of the austenite phase. Clearly, Qdecreases\nwith increasing tilt angle from the ideal NW OR. The\nnumerical values of the activation energy are found to be\n0.00010.0010.01\n 0.0007  0.0008  0.0009  0.001Interface mobility (mmolJ-1s-1)\nT-1554\n776\n998FIG. 8. Interface mobility plotted as a function of inverse\nof the absolute temperature for di\u000berent orientations of the\naustenite phase with respect to the ideal NW OR. The slopes\nof the \ftted lines give the value of activation energies [see\nEq. 4].\n29.62, 35.60 and 40.63 kJ/mol, when the austenite phase\nis tilted by 5.77\u000e, 4.04\u000eand 3.11\u000e, respectively.\nComparing with experimental data,2,6,7calculated val-\nues of activation energies are found to be 3 to 4 times\nlower in our simulations, which means faster transition\nfrom the austenite to the ferrite phase. There can be sev-\neral reasons behind this anomaly. Firstly, we simulate\npure Fe, while most of the experiments are for Fe-C-X\n(where X can be Mn, Ni etc.) type of alloys and alloying\nelements can slow down the rate of transformation. Sec-\nondly, in our study \u000b\u0000\u000band\r\u0000\rgrain boundaries are\nabsent. In reality, there exist a network of grain bound-\naries, which can hinder the mobility of \u000b\u0000\rinter-phase\nboundary. Thirdly, since we are using the Ackland po-\ntential, the driving force for the phase transition is in the\nhigher side. In reality, driving force is very small close\nto the\u000b\u0000\rtransition temperature (1185 K), which can\nnot be captured by this particular potential. However,\nthe atomistic mechanism of growth of the ferrite at its\ninterface with the austenite phase is unlikely to be de-\npendent on the choice of the empirical potential. We\nbelieve that our results are correct in this regard.\nIV. SUMMARY AND CONCLUSIONS\nIn conclusion, we present a detailed analysis of the roles\nplayed by disconnections, which appear as steps at the\n\u000b\u0000\rinter-phase boundary, during the austenite to fer-\nrite phase transformation. Based on calculated values\nof interface velocities (1.19-4.67 m/s) and mobilities (30-\n40 kJ/mol), we identify the mechanism of \r\u0000\u000btran-\nsition studied in this paper as massive transformation.\nWe clearly show that the disconnections act as centers\nfrom which the ferrite phase starts to grow. 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We obtain a very\nstrong (moderate) tendency for cation inversion in NiFe 2O4(CoFe 2O4), in agreement\nwith experimental bulk studies. This preference for the inverse sp inel structure is\nreduced by tensile epitaxial strain, which can lead to strong sensitiv ity of the cation\ndistribution on specific growth conditions in thin films. Furthermore, we obtain\nsignificant energy differences between different cation arrangeme nts with the same\ndegree of inversion, providing further evidence for recently prop osed short range B\nsite order in NiFe 2O4.\na)Electronic mail: fritschd@tcd.ie\n1The spinel ferrites CoFe 2O4(CFO) and NiFe 2O4(NFO) are insulating ferrimagnets with\nhigh magnetic ordering temperatures and large saturation magnet izations.1,2This combina-\ntion of properties is very attractive for a number of applications, s uch as magneto-electric\nheterostructures and spin-filter devices.3–7These applications require the growth of high\nquality thin films of CFO and NFO on suitable substrates. However, th eelectronic and mag-\nnetic properties of the corresponding films can depend strongly on substrate, film thickness,\nand specific preparation conditions, and eventually differ drastically from the corresponding\nbulk materials. For example, both increased and decreased satura tion magnetizations have\nbeen reported for thin films of CFO and NFO grown on different subst rates at different\ngrowth temperatures.8–10It has been suggested that the large increase in magnetization ob-\nserved in some NFO films is due to the presence of Ni2+on the tetrahedrally coordinated\ncation sites of the spinel crystal structure.8,9\nThe spinel crystal structure (space group Fd¯3m) contains two inequivalent cation sites,\nthe tetrahedrally-coordinated Asites (Td) and the octahedrally coordinated Bsites (Oh).\nIn thenormalspinel structure, AandBsites are both occupied by a unique cation species.\nIn theinversespinel structure, the more abundant cation species (Fe3+in the present case)\noccupies the tetrahedral Asites and 50% of the octahedral Bsites, whereas the remaining\n50% ofBsites are occupied by the other cation species (Co2+or Ni2+in the present\ncase). In practice, site occupancies can vary between these two cases, depending on specific\npreparationconditions, andtheinversion parameter λmeasures thefractionofless abundant\ncations on the Bsite sublattice, i.e. λ= 0 for the normal spinel structure and λ= 1 for\ncomplete inversion. Since in the ferrimagnetic N´ eel state of CFO an d NFO the magnetic\nmoments of the AandBsublattices are oriented antiparallel to each other, small changes\ninλcan lead to significant changes in magnetization.\nHere, we use first principles density functional theory to clarify wh ether epitaxial strain\ncan influence the distribution of cations over the two different catio n sites in CFO and NFO.\nSuch epitaxial strain is generally incorporated in thin films due to the m ismatch of lattice\nconstants between the film material and the substrate, and ofte n leads to drastic changes of\nproperties compared to the corresponding bulk materials.2\nIn order to accommodate different arrangements of cations on th e tetrahedral and oc-\ntahedral sites in our calculations, corresponding to different degr ees of inversion, we use a\nunit cell described by body centered tetragonal lattice vectors c ontaining four formula units\n2(f.u.) ofCFO/NFO.Theunstrained cubiccasecorrespondsto c/a=√\n2. Weareconsidering\nconfigurations corresponding to λ={0,0.5,0.75,1}, and in each case (except for the unique\ncaseλ= 0) we compare at least two different inequivalent cation arrangeme nts. Similar\nto our previous investigation we fix the internal coordinates of the cations to their ideal\nvalues within the cubic spinel structure and fully relax the remaining in ternal anion param-\neters.11,12We then introduce epitaxial strain by constraining the “in-plane” lat tice constant\na, and relax the “out-of-plane” lattice constant cand all internal anion parameters. We\napply strains ranging from −4% to +4% relative to the relaxed alattice constant. All\nour calculations are performed using the projector-augmented w ave method13implemented\nin the Vienna ab initio simulation package (VASP).14We employ the generalized gradient\napproximation (GGA) according to Perdew, Burke and Ernzerhof15together with the Hub-\nbard “+U” correction according to Dudarev et al.,16andUeff= 3eV applied to the dstates\non all transition metal cations.\nThe calculated energy differences with respect to the normal spine l structure for the\nunstrained case are shown in Figs. 1(a) and 1(c) for CFO and NFO, r espectively. Differ-\nent cation arrangements are denoted by their corresponding spa ce group symmetry. For\nλ= 0.75 the resulting symmetries are very low and thus Pm/Pm∗mark two inequivalent\nconfigurations with the same space group. It can be seen that for both CFO and NFO the\ntotal energy decreases with increasing inversion, so that the fully inverse spinel structure\n(λ= 1) is energetically most favorable. The calculated energy differenc e between the normal\nspinel structure andthemost favorable inverse configurationis0 .37eV (1.78eV) per two f.u.\nfor CFO (NFO), in good agreement with the value of 0.339 eV reporte d for CFO by Hou et\nal.17The much larger preference for the inverse spinel structure of N FO compared to CFO is\nconsistent with the experimental observation that NFO samples us ually exhibit complete in-\nversion, whereas the exact degree of inversion in CFO depends on t he heat treatment during\nsample preparation and can vary between 0.76-0.93.1,18The same energetic preference also\nfollows from a simple ligand-field analysis of the Ni2+and Co2+cations within octahedral\nand tetrahedral coordination.19,20However, it can be seen from our first principles results\nthat there are also significant energy differences between differen t cation arrangements cor-\nresponding to the same value of λ. This indicates the importance of other factors such as\nhigher order ligand-field effects and local structural relaxations. We note that configurations\nin which the Co (Ni) cations are clustered together, i.e. configuratio nPm∗forλ= 0.75 and\n30.00 0.25 0.50 0.75 1.00Inversion parameter λ\n-0.4-0.3-0.2-0.10.00.1∆E [eV / 2 f.u.]P4m2\nImmaP4122Pm\nP1PmmaR3m Fd3m(a)\nPm*8.2 8.4 8.6 8.8Lattice constant a [Å]\n-0.4-0.3-0.2-0.10.00.1\n∆E [eV / 2 f.u.](b)\n0.000.25 0.50 0.75 1.00\nInversion parameter λ-2.0-1.5-1.0-0.50.0∆E [eV / 2 f.u.]Fd3m\nPmma\nR3m\nP1Pm\nP4m2\nImmaP4122(c)\nPm*\n8.2 8.4 8.68.8\nLattice constant a [Å]-2.0-1.5-1.0-0.50.0\n∆E [eV / 2 f.u.](d)\nFIG. 1. (Color online) Calculated energy differences ∆ Erelative to the normal spinel structure\nfor different cation arrangements corresponding to different i nversion parameters λfor CFO (a)\nand NFO (c). Variation of ∆ Ewith in-plane lattice constant afor CFO (b) and NFO (d) in the\nvarious configurations. The symbols in the right panels labe l different configurations and mark the\ncorresponding equilibrium lattice constant.\nP¯4m2 forλ= 1, are energetically less favorable than configurations where Co ( Ni) cations\nare distributed more uniformly, in agreement with similar findings of Ho uet al.17\nNext we turn to the question of whether the cation distribution and degree of inversion\n4can be influenced by epitaxial strain. Figs. 1(b) and 1(d) show the e nergy differences of\nthe strained structures relative to the energy of the strained no rmal spinel at the same in-\nplane lattice constant for CFO and NFO, respectively. It can be see n that the equilibrium\nlattice constants decrease slightly with increasing λ. In addition, while full inversion is\nmost favorable for all in-plane lattice parameters, the energy diffe rences between different\nconfigurations decrease for larger in-plane lattice constants. Fo r the case of NFO the energy\ndifference between normal and inverse configuration reaches a ma ximum for a≈8.3˚A and\nthen decreases again for smaller in-plane lattice constants. From t hese results one can\nexpect that CFO and NFO thin films under tensile strain are more likely t o exhibit reduced\ninversion compared to unstrained or compressively strained films (a ssuming that they are\notherwise grown under similar conditions). However, it is unclear whe ther the calculated\nmoderate changes in the relative energies will indeed have a noticeab le effect, or whether\nthe actual cation distribution in thin films is rather dominated by kinet ic effects related to\nspecific growth conditions.\nWe also note that recent Raman investigations of both NFO single cry stals21and NFO\nthin films22have provided evidence for short range cation order on the Bsites compatible\nwithP4122 symmetry (or equivalently P4322). Indeed, we find this to be the lowest energy\nconfiguration for both CFO and NFO over the whole investigated str ain region. The energy\ndifference compared to the Immaconfiguration in the unstrained structures is 28 meV\n(26 meV) per two f.u. for CFO (NFO). The structural preference forP4122 symmetry\nis slightly increased by tensile epitaxial strain in the case of NFO, wher eas for CFO the\ncorresponding energy difference is rather independent of strain.\nWe now investigate the influence of cation inversion on the electronic structure of CFO\nand NFO. We obtain insulating ground states for all considered confi gurations. However,\nthere is a strong tendency of the GGA+ Ucalculations to converge to higher energy states\nwith low-spin and/or conducting character, depending on the initial positions for the struc-\ntural relaxation.23Fig. 2 shows the densities of states (DOS) for CFO and NFO corresp ond-\ning to the lowest-energy configurations for each λ. No significant differences in electronic\nstructure for different cation arrangements with the same λhave been observed. The grad-\nual exchange of Co2+(Ni2+) cations from the Asite with Fe3+cations from the Bsite with\nincreasing inversion, is reflected in the DOS by a decreasing intensity of Co (Ni) T dand\nFe Ohpeaks, and a corresponding increasing intensity of Co (Ni) O hand Fe T dpeaks. In\n5addition, the band gap of CFO (NFO) increases from 0.22 eV (0.35 eV) forλ= 0 to 1.24 eV\n(1.26 eV) for λ= 1.\nThe spin-splitting of the conduction band minimum (CBM), which is impor tant for the\nspin filter efficiency of magnetic tunnel junctions containing CFO or N FO as active barrier\nmaterials, is 0.47 eV for both CFO and NFO in the fully inverse structur e. This is signifi-\ncantly smaller than the value of 1.28 eV (1.21 eV) for CFO (NFO), repo rted by Szotek et\nal.,24and is in goodagreement to recent experimental estimates in the te ns of meV for CFO-\ncontaining junctions.7The CBM spin-splitting in CFO increases to 0.66 eV for λ= 0.75,\nwhich is closer to the value of λ∼0.8 observed recently in thin CFO films.18\nIn summary we have analyzed the effect of epitaxial strain on the ca tion distribution in\nthe spinel ferrites CFO and NFO using first principles total energy c alculations. Using the\nGGA+Uapproach we obtain insulating electronic ground states for all degr ees of inversion\nand cation arrangements, and for all considered values of epitaxia l strain. We find a strong\npreference for the fully inverse structure in NFO, and a somewhat weaker tendency towards\ncation inversion in CFO, consistent with experimental observations . Tensile epitaxial strain\nreduces this preference somewhat, which can lead to a stronger s ensitivity of the cation\ndistribution on growth conditions. Furthermore, for both NFO and CFO we find the (fully\ninverse)Bsiteorderedarrangementwith P4122symmetrytobeenergeticallymostfavorable,\nconsistent with recent experimental results for NFO.21,22Our results provide a reference for\nthe interpretation of experimental data on CFO and NFO thin films, a nd thus contribute\nto a better understanding of these materials as part of magnetic t unnelling junctions and\nspin-filter devices.\nThisworkwassupportedbyScienceFoundationIrelandunderRef. SFI-07/YI2/I1051and\nmade use of computational facilities provided by the Trinity Centre f or High Performance\nComputing (TCHPC) and the Irish Centre for High-End Computing (I CHEC).\nREFERENCES\n1V. A. M. Brabers, in Handbook of Magnetic Materials , Vol. 8, edited by K. H. J. Buschow\n(Elsevier, New York, 1995) Chap. 3, pp. 189–324.\n2Y. Suzuki, Annu. Rev. Mater. Res. 31, 265 (2001).\n3H. Zheng, J. Wang, S. E. Lofland, Z. Ma, L. Mohaddes-Ardabili, T. Z hao, L. Salamanca-\n6Riba, S. R. Shinde, S. B. Ogale, F. Bai, D. Viehland, Y. Jia, D. G. Schlom , M. Wuttig,\nA. Roytburd, and R. Ramesh, Science 303, 661 (2004).\n4N. Dix, R. Muralidharan, J.-M. Rebled, S. Estrad´ e, F. Peir´ o, M. Va rela, J. Fontcuberta,\nand F. S´ anchez, ACS Nano 4, 4955 (2010).\n5M. G. Chapline and S. X. Wang, Phys. Rev. B 74, 014418 (2006).\n6U. L¨ uders, M. Bibes, K. Bouzehouane, E. Jacquet, J.-P. Contou r, S. Fusil, J.-F. Bobo,\nJ. Fontcuberta, A. Barth´ el´ emy, and A. Fert, Appl. Phys. Let t.88, 082505 (2006).\n7A. V. Ramos, M.-J. Guittet, J.-B. Moussy, R. Mattana, C. Deranlot , F. Petroff, and\nC. Gatel, Appl. Phys. Lett. 91, 122107 (2007).\n8U. L¨ uders, M. Bibes, J.-F. Bobo, M. Cantoni, R. Bertacco, and J. Fontcuberta, Phys.\nRev. B71, 134419 (2005).\n9F. Rigato, S. Estrad´ e, J. Arbiol, F. Peir´ o, U. L¨ uders, X. Mart´ ı, F. S´ anchez, and J. Fontcu-\nberta, Mat. Sci. Eng. B 144, 43 (2007).\n10J. X. Ma, D. Mazumdar, G. Kim, H. Sato, N. Z. Bao, and A. Gupta, J. Appl. Phys. 108,\n063917 (2010).\n11D. Fritsch and C. Ederer, Phys. Rev. B 82, 104117 (2010).\n12D. Fritsch and C. Ederer, J. Phys.: Conf. Ser. 292, 012104 (2011).\n13P. E. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994).\n14G. Kresse and J. Furthm¨ uller, Comput. Mat. Sci. 6, 15 (1996).\n15J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).\n16S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, a nd A. P. Sutton, Phys.\nRev. B57, 1505 (1998).\n17Y. H. Hou, Y. J. Zhao, Z. W. Liu, H. Y. Yu, X. C. Zhong, W. Q. Qiu, D. C . Zeng, and\nL. S. Wen, J. Phys. D 43, 445003 (2010).\n18J. A. Moyer, C. A. F. Vaz, E. Negusse, D. A. Arena, and V. E. Henr ich, Phys. Rev. B\n83, 035121 (2011).\n19D. McClure, J. Phys. Chem. Solids 3, 311 (1957).\n20J. D. Dunitz and L. E. Orgel, J. Phys. Chem. Solids 3, 318 (1957).\n21V. G. Ivanov, M. V. Abrashev, M. N. Iliev, M. M. Gospodinov, J. Mee n, and M. I. Aroyo,\nPhys. Rev. B 82, 024104 (2010).\n22M. N. Iliev, D. Mazumdar, J. X. Ma, A. Gupta, F. Rigato, and J. Font cuberta, Phys.\nRev. B83, 014108 (2011).\n723D. Fritsch and C. Ederer, to be published.\n24Z. Szotek, W. M. Temmerman, D. K¨ odderitzsch, A. Svane, L. Pet it, and H. Winter, Phys.\nRev. B74, 174431 (2006).\n8-505DOS [eV-1]λ=0.0 Co (Td)CoFe2O4\n-505λ=0.0 Ni (Td)NiFe2O4\n-505DOS [eV-1]λ=0.75 Fe (Td)-505λ=0.5 Ni (Oh)\n-505DOS [eV-1]λ=0.5 Co (Oh)\n-505λ=0.75 Fe (Td)\n-8 -6 -4 -2 0 2\nE [eV]-505DOS [eV-1]λ=1.0 Fe (Oh)\n-8 -6 -4 -2 0 2\nE [eV]-505λ=1.0 Fe (Oh)\nFIG. 2. (Color online) Total and projected DOS per formula un it for CFO (left panels) and NFO\n(right panels) for the lowest energy configurations corresp onding to different λ. Thedstates of\n(Co, Ni) ( Oh) and (Td) are shown as solid and dashed red (black) lines, whereas dstates of Fe\n(Oh) and (Td) are shown as solid and dashed green (dark gray) lines, respe ctively. The total DOS\nis shown as shaded gray area in all panels. Majority (minorit y-) spin projections correspond to\npositive (negative) values.\n9"    },    {        "title": "2306.17011v1.Registration_between_DCT_and_EBSD_datasets_for_multiphase_microstructures.pdf",        "content": "Registration between DCT and EBSD datasets for multiphase microstructures\nJames A. D. Balla,b, Jette Oddershedec, Claire Davisd, Carl Slaterd, Mohammed Saida, Himanshu Vashishthaa, Stefan Michalikb,\nDavid M. Collinsa\naSchool of Metallurgy and Materials, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom\nbDiamond Light Source Ltd., Harwell Science and Innovation Campus, Didcot, OX11 0DE, United Kingdom\ncXnovo Technology ApS, Galoche Alle 15, Køge, 4600, Denmark\ndWMG, University of Warwick, Coventry, CV4 7AL, United Kingdom\nAbstract\nThe ability to characterise the three-dimensional microstructure of multiphase materials is essential for understanding the interaction\nbetween phases and associated materials properties. Here, laboratory-based di ffraction-contrast tomography (lab-based DCT), a\nrecently-established materials characterization technique that can determine grain phases, morphologies, positions and orientations in\na voxel-based reconstruction method, was used to map part of a dual-phase steel alloy sample. To assess the resulting microstructures\nthat were produced by the lab-based DCT technique, an electron backscatter di ffraction (EBSD) map was collected within the same\nsample volume. To identify the two-dimensional (2D) slice of the three-dimensional (3D) lab-based DCT reconstruction that best\ncorresponded to the 2D EBSD map, a novel registration technique based solely on grain-averaged orientations was developed – this\nregistration technique requires very little a priori knowledge of dataset alignment and can be extended to other techniques that\nonly recover grain-averaged orientation data such as far-field 3D X-ray di ffraction microscopy. Once the corresponding 2D slice\nwas identified in the lab-based DCT dataset, comparisons of phase balance, grain size, shape and texture were performed between\nlab-based DCT and EBSD techniques. More complicated aspects of the microstructural morphology such as grain boundary shape\nand grains less than a critical size were poorly reproduced by the lab-based DCT reconstruction, primarily due to the di fference in\nresolutions of the technique compared with EBSD. However, lab-based DCT is shown to accurately determine the centre-of-mass\nposition, orientation, and size of the large grains for each phase present, austenite and martensitic ferrite. The results reveals a\ncomplex ferrite grain network of similar crystal orientations that are absent from the EBSD dataset. Such detail demonstrates that\nlab-based DCT, as a technique, shows great promise in the field of multi-phase material characterization.\nKeywords: Diffraction-contrast tomography, Crystallographic texture, 3D characterization, Grain morphology, Steel\n1. Introduction\nUnderstanding the deformation behaviour of multiphase\npolycrystalline structural alloys, such as α/β titanium alloys\nfor compressor discs in aeroengines or high strength dual phase\n(ferritic–martensitic) steels for automotive, load bearing chassis\ncomponents, is vital for guiding the design of future materials.\nThe micromechanical material behaviour is intimately linked\nto the material microstructure, not only to the phase fractions,\nbut also to the phase specific grain size distributions (i.e. fine-\ngrained /coarse-grained /bimodal, narrow /wide), grain shapes (i.e.\nequiaxed /needles /plates), and textures. These features will have\nan associated distribution throughout the material in 3D; this\nmay be uniform or heterogeneous, often inherited from the prior\nprocessing.\nInteraction between phases, such as load shedding, is a criti-\ncal attribute that must be well known for predicting failure initi-\nation and deformation evolution. For probing the load sharing\namong phases, experimental techniques such as far-field 3DXRD\n(3-Dimensional X-ray Di ffraction) /HEDM (High Energy X-ray\nDiffraction Microscopy) have proved to be excellent methods\nEmail address: D.M.Collins@bham.ac.uk (David M. Collins)as they are sensitive to the phase, center of mass position, crys-\ntallographic orientation, and lattice distortions (and hence grain\naveraged type-II stress) of every grain, non-destructively, in\n3D [1,2]. Non-destructive experimental techniques to map out\nthe 3D grain structure comprise near-field 3DXRD /HEDM e.g.\n[3,4,5,6] and synchrotron DCT [ 7,8,9,10]. Obtaining such\ndata without the need to access national or international facilities\nis also possible via lab-based DCT [ 11,12,13,14], which is\nhighly attractive if it has the capability to accurately describe\nmicrostructures of engineering alloys.\nFor a multi-phase material, lab-based DCT has been used\nto good e ffect to elucidate hydrogen embrittlement in a duplex\nstainless steel consisting of a dual-phase ferrite-austenite mi-\ncrostructure [ 15]. Here, an old version of the reconstruction\nengine, GrainMapper3D ™, was used to reconstruct each phase\nseparately. The software constrained the analysis to provide only\ngrain sizes, center of mass positions, and crystallographic orien-\ntations, while the movement of di ffraction spots was used, after\nhydrogen infusion, to qualitatively evaluate stress evolution.\nEBSD (electron backscatter di ffraction) can be used to char-\nacterise a 2D region on the sample surface of a polycrystalline\nmaterial (e.g. [ 16,17,18,19,20]), with comparably better\nPreprint submitted to Materials Characterization June 30, 2023arXiv:2306.17011v1  [cond-mat.mtrl-sci]  29 Jun 2023spatial resolution than 3D techniques. The use of 3D-EBSD,\nwhere an individual EBSD scan is acquired with successive se-\nrial sectioning (e.g. [ 21,22,23,24]), is attractive. However,\nbeing a destructive method, in-situ studies are impossible with\n3D-EBSD, which limits its applicability to study deformation.\nEBSD is used routinely for investigating multiphase structural\nmaterials [ 25], while only a very limited number of 3D space\nfilling grain maps of dual phase materials exist, e.g. [26].\nTo directly compare grain center-of-mass data to space fill-\ning 2D /3D grain maps, or grain maps to each other, a regis-\ntration approach is needed. Often, the registration technique\nused during the comparison is not specified [ 8,10]. In some\ncases, both measurement techniques are performed at the same\nfacility and therefore use the same reference frames and length\nscales, so a post-mortem registration is unnecessary [ 27]. For\ncases where algorithmic registration is required, a range of dif-\nferent dataset registration strategies have been employed, such\nas manual alignment [ 11], plane fitting using porosity data [ 28]\nand misorientation minimization [ 29,30]. As the reconstructed\ngrain maps are inherently multi-dimensional, and /or multimodal,\nvisualization packages such as PolyProc offer dataset filtration\nand analysis capability [31].\nFor microstructures comprising multiple constituent phases,\nthere is no registration algorithm developed to date that can han-\ndle a combination of center of mass or space filling data, for each\nphase present, in both 2D and 3D. Hence, this study seeks to\nimplement such a registration algorithm and test its performance\non a di fficult case, namely a two-phase metastable austenitic\nstainless steel with an austenite and martensitic-ferrite structure\nmapped by both EBSD and lab-based DCT. A direct one-to-one\ncomparison between the two methods is considered unreach-\nable due to a morphologically complex, fine-grained dual-phase\nmicrostructure of the sample – some of these microstructural\nfeatures are outside of the detection limits of the DCT method\n(10–40µm[32]). However, comparing statistical properties of\nthe 2D-EBSD map to the nearest 2D slice in the 3D-DCT is\nreachable. This can be determined by the registration between\nthe datasets. In this study, the corresponding properties are also\nderived for a full 3D-DCT volume to ascertain the advantages\nand disadvantages of EBSD versus DCT for the grain mapping\nof multiphase samples.\n2. Experimental Method\n2.1. Material\nThe alloy studied was a two-phase austenitic metastable\nstainless steel alloy with the composition given in Table 1.\nTable 1: Experimental alloy composition\nElement C Ni Cr Mn Si P S Fe\nwt.% 0.04 7 19 2 1 0.04 0.03 Bal.\nThe alloy was cast as a 10 kg billet ( 80×30×210 mm ) and\nhot rolled at 1050 °C in a 3:1 ratio. This was followed by an\nannealing heat treatment of 1250 °C for12 h, then a quench in\n246 mm12 mm\nApertureDetector\nRotation\nstageSourcea\nb\nXZ\nY\nFigure 1: DCT data collection projection geometry ( a) and example detector\nimage showing di ffracted peaks ( b).\nair to room temperature. A small dog-bone shape specimen\nwas machined from the billet with a 0.5×0.5 mm2gauge cross-\nsection and a 2.39 mm gauge length. For the purposes of this\nstudy, the sample was measured in a simple static condition;\nonly the microstructure within the gauge section was of interest.\n2.2. DCT data collection\nThe DCT scans were collected on a ZEISS Xradia 520 Versa\nX-ray microscope equipped with a LabDCT Pro module and\na flat-panel extension. An accelerating voltage of 110 kV was\nused with a power of 10 W . A flat-panel detector ( 75µmpixels)\nwas used to collect the di ffracted X-ray signal in projection\ngeometry with a source-sample distance of 12 mm and a sample-\ndetector distance of 246 mm , as shown in Figure 1a. This gives a\ngeometric magnification factor of 21.5. A150×750µm2beam-\ndefining aperture was placed between the beam and the sample\nto limit the exposed sample region, while the direct beam was\nblocked with a beamstop. A helical phyllotaxis scan strategy\n[33] was employed to scan a ∼1 mm -tall region of the sample\ngauge section. 851projections were captured, see example in\nFigure 1( b), each with a 60 sexposure time, for a total scan time\nof 16 h 45 min.\n1601 absorption contrast X-ray tomography (ACT) projec-\ntions (the finetomography scan) were also taken with a 1 s\nexposure time and a 5µmvoxel size to define the absorption\nmask required for the grain reconstruction process. Finally, a\ncoarse whole-sample tomography scan was performed with a\n13µmvoxel size, 801ACT projections and a 0.5 sexposure time.\nThe reconstructed coarse andfineACT volumes are presented\n22.5 mm\n2.39 mm0.5\nmm\n0.5\nmmFigure 2: Original sample design (left), 13µmcoarse whole-sample tomogra-\nphy reconstruction (centre), 5µmfinegauge-only tomography reconstruction\n(centre).\nin Figure 2. The reconstruction process utilised to generate the\nFigure is outlined in Appendix A.\n2.3. DCT reconstruction\nTo reconstruct the final 3D grain map, a prototype version\nof GrainMapper3D allowing simultaneous indexing of multi-\nple phases was used to process the DCT raw images. This is\nan extended version of the fast geometric indexing outlined by\nBachmann et al. [34], assigning to each voxel in space both the\nphase and the orientation giving the highest completeness score,\nwhere completeness is the ratio between the observed and ex-\npected number of reciprocal vectors associated with the solution.\nA region of 660×655×930µm3was reconstructed with a 5µm\nvoxel size, for a total grid of 132×131×186 voxels . Grains\nwere defined using a 0.25°misorientation threshold between ad-\njacent pixels. This yielded 1888 austenite and 685ferrite grains.\nThe final result comprised 3D maps of orientation, grain ID,\nphase ID and completeness, a selection of which are shown in\nFigure 3.\n2.4. EBSD data collection\nThe sample was mounted in conductive bakelite, polished\nto a0.04µmsurface finish using colloidal silica, then electro-\npolished at 20 °C with an 80:20 mixture of ethanol and per-\nchloric acid at 15 V for20 swith a flow rate of 10 l min−1. The\nsample was examined with a JEOL 7000 field emission gun\nscanning electron microscope (FEG-SEM) equipped with an\nOxford Instruments Nordlys EBSD detector to collect an EBSD\nmap across the full width of the sample gauge. A 1.25µmstep\nsize at a 13 nA probe current and a 20 keV beam energy was\nused. The EBSD scan and indexing was performed using the\nOxford Instruments software AZtec.\n2.5. EBSD post-processing\nThe EBSD map dataset was imported into the MTEX MAT-\nLAB library [ 35]. First, the dataset was cropped to the geometry\nof the sample. Next, the dataset was segmented into individual\n[111]\n[110] [100]\nXZ\nYFerrite\nAustenite\n0.5 mm\n0.5 mm0.9 mma b\nc dFigure 3: Reconstructed DCT maps. ( a,b) whole sample; ( c) austenite phase\nonly; ( d) ferrite phase only. ( a,c,d) are coloured by IPF- Zorientation; ( b) is\ncoloured by phase.\n3200 μm[111]\n[110] [100]\nX\nY\nFerrite\nAustenite\n200 μm 200 μmAustenite FerriteFigure 4: Reconstructed EBSD maps of sample, with IPF- X(axial) orientation colouring (top) and phase colouring (middle). Individual phases are also plotted\n(bottom) with IPF- Xcolouring.\n4grains. A first pass segmented grains by pixel orientation, with a\n5°tolerance. Then, grains with less than 20 contributory pixels\nwere marked as unindexed to exclude grains with potential inac-\ncuracies with grain mean orientation or centroid position. The\nfirst segmentation was then repeated with the updated dataset\nto regenerate the EBSD grain IDs. Then, the EBSD map was\ndenoised to fill unindexed pixels within individual grains using\nan MTEX denoising method with a half-quadratic filter [ 36].\nFinally, the grains were re-segmented to re-associate the updated\npixels to the grains. Grain ID and phase ID maps, along with\ngrain-averaged orientations as Euler angles, were exported from\nMATLAB to an HDF5 file to facilitate further processing with\nPython. The processed EBSD map of the sample, comprising\n750austenite and 648ferrite grains, is shown in its entirety\nin Figure 4, with both IPF- Xorientation colouring and phase\ncolouring.\n3. Registration\n3.1. Importing EBSD and DCT data\nA registration procedure was devised to locate the 2D slice\nwithin the 3D DCT data that best corresponded to the EBSD im-\nage plane. First, both EBSD and DCT datasets were imported us-\ning the pymicro Python library [ 37], to generate Microstructure\nclass instances. The pymicro library stores grain orientations as\na single 3×3orientation matrix per grain ( g) transforming a vec-\ntor in the sample reference frame (− →Vs) into the crystal reference\nframe (− →Vc), as per Equation 1.\n− →Vc=g− →Vs (1)\nDue to di fferences in grain orientation and array axis conventions\nbetween GrainMapper3D, MTEX and pymicro, DCT grain ID,\ngrain orientations and phase ID information were verified using\nthe reconstruction report generated by GrainMapper3D, and\nEBSD grain ID, grain orientations and phase ID information\nwas verified using the MTEX-processed datasets.\n3.2. Initial transformation\nThe longitudinal axis of the sample in the original EBSD\ndataset was parallel to the XEaxis of the EBSD reference frame.\nIn the DCT dataset, the sample longitudinal axis was parallel to\ntheZDaxis of the DCT reference frame. Consequently, a new\nrotated EBSD reference frame was devised such that the EBSD\nsample longitudinal axis was made parallel to the new Zaxis\n(ZR). Given a vector in the original EBSD reference frame (−→VE),\na rotation matrix Rwas defined that transforms the vector into\nthe equivalent vector in the rotated reference frame (− →VR), as per\nEquation 2.− →VR=R−→VE (2)\nTo represent the EBSD grain orientations ( gE) in the new ref-\nerence frame, we must right-multiply by the transform of the\nrotation matrix, as per Equation 3.\ngR=gER⊺(3)3.3. Initial matching grain search\nOnce the EBSD and DCT datasets were approximately aligned\nby applying this initial transformation, an initial search for\nmatching grain pairs was performed. A Python function based on\nthematch_grains method of the pymicro Microstructure\nlibrary was devised to search for matching EBSD grains within\nthe DCT dataset, as per Algorithm 1. The EBSD microstructure\ninstance was filtered to keep only austenite grains, as initial\nobservations of the crystal orientations revealed a highly tex-\ntured martensitic-ferrite phase, which may have generated false\nmatches due to grouping of ferrite grains in orientation space.\n750austenite EBSD grains remained after this filtration. The\nDCT microstructure instance was similarly filtered, leaving 1888\naustenite grains.\nAlgorithm 1: A Python function to find matching\ngrains between EBSD and DCT microstructures.\ndeffind matching grains( EBSD grains ,\nDCT grains ,mistol):\nData:\nEBSD grains : a list of EBSD grains\nDCT grains : a list of DCT grains\nmistol: a tolerance in misorientation (degrees)\nResult:\nbest matches : a list of matched DCT grain IDs\nfor each EBSD grain\n/* Create empty array with the same\nlength as the number of EBSD grains\n*/\nbest matches =np.empty(len( EBSD grains ));\n/* Iterate through input EBSD grains */\nforEBSD index ,EBSD grain in\nenumerate( EBSD grains ):\ngR=EBSD grain rotated orientation matrix;\nbest misorien =mistol;\nbest match =-1;\n/* Iterate through input DCT grains\n*/\nforDCT grain inDCT grains :\ngDCT=DCT grain orientation matrix;\n/* Use pymicro method to check\nmisorientation between grain\norientations */\nmisorien =misorientation( gR, gDCT);\nifmisorien<best misorien :\nbest misorien =misorien ;\nbest match =DCT grain ID;\nbest matches [EBSD index ]=best match ;\nWith a small misorientation tolerance ( 3°), only 222match-\ning austenite DCT grains were found. Additionally, the matching\nDCT grains did not lie on a specific YZDplane, which would be\nanticipated for legitimate matches. It was then theorised that a\nmisorientation remained between the rotated EBSD and DCT\nreference frames, larger than a 3°misorientation tolerance would\n5allow for. This misorientation was attributed to misalignment be-\ntween the EBSD spatial and grain orientation reference frames,\ncompounded with spatial distortions introduced by the large\nfield-of-view map collected at low magnifications. Repeating\nthe search with a wider tolerance of 12°also yielded no specific\nmatching plane signal, likely due to the significantly increased\nnoise floor.\n3.4. Corrective rotation search\nApplying a specific corrective rotation to each EBSD grain\norientation to realign the EBSD reference frames before search-\ning for matches would negate any remaining misorientation be-\ntween the datasets and lead to the discovery of legitimate grain\nmatches. A search through rotation space was therefore required\nto determine the corrective rotation to apply to each EBSD grain\norientation. To perform the search, a global optimization strategy\nwas employed. A modified grain matching function was devised,\nbased on Algorithm 1. Before determining the misorientation\nbetween grain pairs, gRwas right-multiplied by a candidate cor-\nrective rotation matrix Crepresenting a specific point in rotation\nspace. Rotation space was parameterized by three successive\nelemental rotation matrices following the Proper Euler angle\nZXZ convention with angles ( α,β,γ ) yielding a rotation matrix\nCas per Equation 4:\nC=Z(α)X(β)Z(γ)\n=cαcγ−cβsαsγ−cαsγ−cβcγsαsαsβ\ncγsα+cαcβsγcαcβcγ−sαsγ−cαsβ\nsβsγ cγsβ cβ(4)\nwhere sandcrepresent sinandcosrespectively, and XandZ\nrepresent elemental rotation matrices about fixed-frame axes\n[38]. Defining the corrective rotation matrix in this way yields a\n3D search space through ( α,β,γ ). A search for grain matches\nwith a tight misorientation tolerance should achieve a maximum\nnumber of matches at a specific value of ( α,β,γ ), corresponding\nto the real corrective rotation that must be applied to the EBSD\ngrains in order to bring the EBSD and DCT reference frames\ninto coincidence.\nIn order to e fficiently perform this search through rotation\nspace, the previously defined grain matching algorithm (Algo-\nrithm 1) was modified to include the three angles of rotation\n(α,β,γ ) in an array as an input argument, then calculate the\ncorresponding corrective rotation matrix C, right-multiply the\nEBSD grain orientation matrix gR, and search for matches. The\nalgorithm was also modified to return only the fraction of missed\nmatches (using the number of matching DCT grains found, and\nthe number of input EBSD grains as the maximum number of\npotential matches) as a floating-point number between 0and 1.\nThis way, the returned number would be equal to 1if no matches\nwere found, and would decrease towards a minimum of 0if all\ninput EBSD grains had a corresponding DCT match.\nWith an objective function now defined, the PySwarms\nPython library [ 39] was employed to perform a global mini-\nmization over ( α,β,γ ) space. A search space of −20°to20°\nin each dimension was employed. To speed up the evaluation\nof each point in rotation space, only the 10largest austeniteEBSD grains were used. A small misorientation tolerance of 2°\nwas used to minimise the likelihood of the algorithm returning\na false positive match. A global Particle Swarm Optimisation\n(PSO) based minimisation search was performed with the pa-\nrameters as per Table 2. The reader is referred to the original\ndefinitions of the PSO for further detail on the optimisation\nparameters [ 40,41]. The search was parallelised across 20\ncores of an AMD Ryzen 9 3900x CPU. The global Particle\nSwarm Optimisation algorithm returned a rotation sequence of\nCPSO=Z(−3.23°)X(9.04°)Z(3.35°).\nTable 2: Global Particle Swarm Optimisation parameters.\nParameter Value\nc1 0.5\nc2 0.3\nw 0.9\nNparticles 20\nNiters 50\n3.5. Local corrective rotation optimisation\nOnce approximate values for ( α,β,γ ) were found that max-\nimised the number of matches between EBSD and DCT grains,\na local optimisation was performed to further refine these values.\nTo do this, the original matching algorithm, Algorithm 1, was\nused, still with only the 10largest austenite EBSD grains, to\nget the matching EBSD and DCT grain ID pairs after applying\nthe optimised rotation Cto the EBSD grains. Once the list of\nmatching grain pairs was obtained, a new objective function\nwas obtained that returns the mean misorientation between each\ngrain pair in the matched grains list after applying C. This way,\nmore accurate ( α,β,γ ) values would result in a reduced mean\nmisorientation. For the local optimisation search, the minimize\nfunction from the scipy.optimize Python library was used\n[42]. The limited-memory Broyden–Fletcher–Goldfarb–Shanno\n(L-BFGS) nonlinear optimization method was automatically se-\nlected [ 43]. The search space was bounded to −4°to4°in each\ndimension around the result determined by the PSO method. A\nfinal optimised rotation of Cfin=Z(−3.46°)X(10.41°)Z(3.09°)\nwas determined.\n3.6. DCT slice determination\nOnce an optimised corrective rotation matrix was deter-\nmined, a final search for matching grain pairs was performed,\nusing both ferrite and austenite phases of all DCT and EBSD\ngrains, and a misorientation tolerance of 1°. This match was\nperformed with both the uncorrected and corrected EBSD grain\norientations, generating two lists of matched grain pairs. The\nresults of this grain-matching algorithm were then explored by\nplotting the Xcoordinate of the centroids of the DCT grains\nthat were returned by the grain-matching algorithm, both before\nand after applying the optimized corrective rotation to the input\nEBSD grains, as shown in Figure 5.\nThe position in the DCT reference frame of all matched DCT\ngrains in both these lists was then investigated - as the matched\n60.00.51.01.52.0(a)\n−0.3−0.2−0.1 0.0 0.1 0.2 0.3\nMatched DCT grain centroid Xposition (mm)020406080(b)CountFigure 5: Histograms of Xcoordinates of centroids of matched DCT grains\ngenerated from matching algorithm before ( a) and after ( b) correcting EBSD\norientations.\nDCT grains in the corrected grain pair list were all roughly co-\nplanar with a consistent X-axis of their centroid position, the\nnearest corresponding DCT microstructure slice was extracted\nand could therefore be compared directly to the transformed\nEBSD grain map. From the peak of the histogram in Figure 5 (b),\nanXslice position of X=0.22 mm from the centre-of-mass of\nthe DCT scan was determined, corresponding to a 3D array slice\nindex of 109.\n3.7. DCT slice registration\nAs the EBSD map was taken over a larger region of the sam-\nple than the DCT map, it was necessary to accurately determine\nthe crop required to generate a new EBSD map representing\nonly the region explored by the DCT scan. To do this, the 2D\ncentroids of the matching grain pairs were determined in the\nreference frames of the full EBSD map and the matched DCT\nslice respectively. With a list of matching 2D coordinate posi-\ntions, a rigid transformation was performed to determine the\ntranslation required to transform the DCT grain 2D centroids to\nthe EBSD grain centroids. This translation was then rounded to\nthe nearest EBSD integer pixel, and applied to the origin point\nof the 2D EBSD map to generate a cropped EBSD sub-region\nwith new grain centroids that closest matched the corresponding\nDCT centroids. Figure 6 shows the final cropped EBSD map\nwith corrected grain orientations compared to YZslice 109of\nthe DCT dataset.\nAfter limiting matched DCT grains to those that appeared\nin the YZslice, a total of 166matching grain pairs were found\nbetween the EBSD and DCT grain maps. With all registrations\nperformed, the embedding of the DCT and EBSD grain maps\nYZ\nFerrite\nAustenite[111]\n[110] [100]\na\nb\nc\ndFigure 6: DCT map, YZ slice 109of130(a,c) vs cropped EBSD map ( b,d),\ncoloured by IPF- Z(a,b) and phase ( c,d). Select DCT grains are outlined ( a)\nwith their corresponding EBSD matches ( b).\n7XZ\nY0.5 mm[111]\n[110] [100]\nNDRD\nTDFigure 7: EBSD (centre) and DCT (right) IPF- Zorientation maps embedded in sample geometry from coarse tomography (left).\nwithin the sample geometry (as generated from the tomography\ndata) could then be performed, as presented in Figure 7. Details\nof the processing pipeline used to generate the tomography\nmodel are available in Appendix A.\n4. Results and Discussion\nWith both datasets reconstructed and a registration performed,\ndirect statistical descriptions of and comparisons between the\nDCT and EBSD grain maps were undertaken. For all subsequent\ncomparisons, calculations were performed for both the full as-\nreconstructed EBSD and DCT grain maps (labelled as EBSD\n(full) and DCT (full) respectively), the EBSD region cropped\nto the region that overlaps the DCT volume (labeled as EBSD\n(crop)) as well as the extracted DCT slice. It is sometimes valu-\nable to distinguish between the DCT 2D slice (similar to an\nEBSD map, labelled as DCT (2D slice)), and a subset of the 3D\nDCT dataset that contains only the grains that appeared within\nthe 2D slice (labelled as DCT (3D slice)).\n4.1. DCT 3D grain map\nThe reconstructed 3D DCT orientation and phase maps in\nFigure 3 show an equiaxed microstructure with both austenite\nand ferrite phases present. Referring to Figure 3( b), austenite is\nthe majority phase and ferrite is distributed uniformly throughout\nthe volume. The grains from the individual phases presented\nin Figure 3( c) and ( d) show that the volumes of austenite and\nferrite are dominated by grains measuring ∼100µm. Coloured\nby the IPF- Zorientation, ferrite possesses distinct bands of\nsimilarly oriented grains, as evident with those shown as red\n([100] parallel to Z) and green ([110] parallel to Z). Thesemacrozones extend along the gauge direction (Z) with a width\nof three or four grains.\n4.2. EBSD 2D grain map\nThe processed 2D EBSD map of the entire sample, shown\nin Figure 4, displays a dual-phase austenite (approximately\nequiaxed) and martensitic ferrite (laths) microstructure with\nthe former being the majority phase as in the 3D DCT grain map.\nGiven the higher spatial resolution of the EBSD measurements\nover DCT, the grain morphologies are more accurately deter-\nmined. Features such as twins, for example, are clearly evident\nin the austenite phase in the EBSD map. Similar to the DCT,\nthe ferrite has an interconnected network of grains, as is evident\nby several neighbouring grains of the same phase. The macro-\nzone feature, as was very clear in the 3D DCT reconstruction, is\nnot replicated in the EBSD 2D slice. It is further evident from\nthe EBSD that the grain shape for both phases is complex; this\naspect will be quantified in a later section.\n4.3. Phase balance\nFigure 8 shows the phase balance comparisons, calculated by\npixel area or volume fraction, for both EBSD and DCT datasets.\nA significant di fference in per-voxel phase fraction is observed\nbetween the EBSD and DCT grain maps. The EBSD map sug-\ngests a microstructure that is approximately 40 % ferritic by area,\nbut DCT datasets average only around 20−25 % ferritic by area\nor volume.\nThe larger austenite volume fraction in the DCT experi-\nment may be related to the lattice distortions caused by residual\nstressed in the austenite grains, which make the di ffraction spots\nfrom this phase streak radially (see Figure 1( b)) leading to over-\nestimated austenite grain sizes. Furthermore, according to the\n8EBSD (full)EBSD (crop)DCT (full)\nDCT (2D slice) DCT (3D slice)0.00.20.40.60.81.0Phase fractionFerrite\nAusteniteFigure 8: Phase balance comparisons between EBSD (full), EBSD (cropped),\nDCT (full), DCT (2D slice) and DCT (3D slice).\nhigh resolution EBSD map (see Figure 4 and the corresponding\ngrain size distributions in Figure 9) the ferritic grains tend to be\nsmaller than the austenitic grains. In fact most of these are so\nsmall that they fall below the detection limit of the DCT method,\nwhich is of the order 10–40µm[32] for undeformed samples\nwith well defined di ffraction spots displaying no radial streaking,\nhence probably somewhat poorer for the present sample. Thus\nit is possible that a large number of smaller ferrite grains were\njust missed in the DCT experiment, and their corresponding\nvoxels assigned to nearby austenite grains during the reconstruc-\ntion, creating the observed phase imbalance. To additionally\nexclude the contribution of small surface grains being missed\nby the DCT reconstruction (and therefore leading to larger-than-\nexpected grains at the edges of the sample), the same phase\nbalance calculation was performed for a trimmed DCT dataset\nwith 20µmof data trimmed in the XandYdirections – this\nhad little e ffect on the phase balance results, thereby excluding\nsurface e ffects as a primary cause of the di fferences between\nDCT and EBSD techniques.\n4.4. Grain size\nFigure 9 shows grain size distributions of the full EBSD and\nDCT datasets, as well as the extracted DCT slice, and phase-\nspecific grain diameter distributions from EBSD (crop) and DCT\n(2D and 3D slice) datasets. Both 2D DCT grain circle-equivalent\ndiameters (as estimated from only the 2D DCT slice) and 3D\nsphere-equivalent diameters (as estimated from all 3D DCT\ngrains that appear in the 2D slice) are provided.\nIt is evident that there is a significant di fference in grain size\npresentation between DCT and EBSD techniques. EBSD grains,\nboth in the full and cropped grain maps, possess significantly\n0.000.010.020.030.040.050.06\nEBSD (full) circle equiv.\nEBSD (crop) circle equiv.\nDCT (full) sphere equiv.\nDCT (2D slice) circle equiv\n0.000.010.020.030.040.050.06\nEBSD (crop) Austenite\nEBSD (crop) Ferrite\n0.000.010.020.030.040.050.06\nDCT (2D slice) Austenite\nDCT (2D slice) Ferrite\n0 20 40 60 80 100 120\nDiameter ( µm)0.000.010.020.030.040.050.06\nDCT (3D slice) Austenite\nDCT (3D slice) FerriteProbability densityFigure 9: Grain size distributions from EBSD and DCT datasets.\n9smaller diameters overall, with a peak diameter of around 15µm.\nEBSD grain diameters have an approximately log-normal dis-\ntribution. In contrast, DCT grains present substantially larger\non average, with a larger distribution of sizes, and a peak in the\nfull dataset of approximately 40µm. Additionally, there were\nvery few EBSD grains observed with diameters ≥60µm, but a\nsignificant fraction of DCT grains in the full dataset have diam-\neters larger than this. To ensure that this e ffect was not solely\ncaused by the size underestimation common in 2D sections of\n3D volumes [ 44], DCT grain diameters were calculated for the\nfull 3D volume as well as how they appear in the 2D slice.\nRegarding the phase specific grain size distributions, ferrite\ngrains appear smaller than austenite grains in the cropped EBSD\nmap. The reasons for this di fference between phases being much\nless pronounced for the DCT map, both for 2D and 3D slices, are\nundoubtedly the influence of lattice distortion in the austenite\nphase and the grain size detection limit of the DCT technique,\nas described above for the phase fractions. A challenging grain\nmorphology is also a likely reason for DCT grain appearing\nlarger in size as outlined further in the following section.\n4.5. Grain shape\nThe compactness, as calculated by taking the ratio between\nthe grain area (or volume) and the area (or volume) of the com-\nplex hull of the grain, provides a metric for grain morphology,\nwith values close to 1 representing smoother, more circular\ngrains. After determining grain size distributions for Figure 9,\ngrains consisting of fewer than 10pixels ( 12.5µmfor the EBSD\ndataset, 50µmfor the DCT dataset) were removed from each\ndataset prior to compactness calculations to avoid non-physical\nconvex hull results. Figure 10 shows phase-specific grain com-\npactness distributions from EBSD (crop) and DCT (2D and 3D\nslice) grain maps.\nIt is evident from the EBSD map that a large number of\nfairly complex grain boundaries are present, primarily between\ngrains with di fferent phases, which is common for duplex steels\n[45,46]. Quantifying the phase specific grain shapes of the\nEBSD map in Figure 10, the austenite grains appear slightly\nmore compact than the ferrite grains, indicating a simpler austen-\nite grain morphology. This is to be expected due to the thermal\nhistory of the sample - the austenite grain morphology is inher-\nited from the annealing stage, but the ferrite grew rapidly during\nthe quenching stage.\nInterestingly, the DCT reconstruction appears to capture the\nsame compactness distribution for the austenite grains in the 2D\nslice as for the EBSD map. This is likely because the ferrite laths\ntake up only very small area fractions of the austenite convex\nhulls, and with this approximation the 2D shapes of the austenite\ngrains appear similar between EBSD and DCT. Contrary to this,\nthe DCT 3D slice compactness of the austenite grains is signifi-\ncantly di fferent. This marked di fference in compactness between\nDCT 2D and 3D slice highlights the need for 3D characterisation\ntechniques (rather than traditional 2D techniques such as EBSD)\nin order to accurately capture the complicated grain morphology\npresent in this alloy system.\nThe ferrite grains appear more compact and with a much\nnarrower distribution of compactness than the austenite grains in\n0123456\nEBSD (crop)\nDCT (2D slice)\n0123456\nEBSD (crop) Austenite\nEBSD (crop) Ferrite\n0123456\nDCT (2D slice) Austenite\nDCT (2D slice) Ferrite\n0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0\nGrain compactness0123456\nDCT (3D slice) Austenite\nDCT (3D slice) FerriteProbability densityFigure 10: Grain compactness distributions for EBSD and DCT datasets.\n10the DCT 2D and 3D slices, contrary to the observations made for\nthe EBSD map. The aforementioned complex grain boundaries\nbetween phases observed in the EBSD map clearly pose a chal-\nlenge for the DCT reconstruction - many of the intricate features\nat the edge of the grain boundaries are only a few pixels in size,\nand, owing to the di fferences in technique resolution, are poorly\nrepresented in the DCT slice. More accurate grain morphologies\nhave been observed in prior lab-based DCT studies [ 11,12,15],\nand there is no reason to suspect that the challenges encountered\nhere are related to the new dual phase indexing algorithm rather\nthan the complexity of the sample microstructure.\n4.6. Texture\nFigure 11 shows comparisons in austenite and ferrite texture\nbetween EBSD and DCT grain maps. It is clear that the austenite\nphase is broadly untextured in both the DCT and EBSD datasets.\nA substantial{1 1 1}⟨1 1 2⟩texture is observed in the ferrite phase\nin the EBSD (full) scan in Figure 11, typical for rolled body-\ncentered cubic materials [ 47]. Interestingly, a clear di fference\nbetween the ferrite texture is observed between EBSD and DCT\ntechniques. This may be caused by a more complicated 3D\ntexture which the DCT (full) plot is capturing - a close look at\nFigure 3 dsupports this hypothesis as large regions of the DCT\nvolume have broadly di fferent ferrite textures. This explains\nthe more complicated presentation of texture visible in the DCT\n(full) Ferrite subplot of Figure 11. However, when grains being\nplotted are filtered to just those in the registered slice, as shown\nin the DCT (slice) Ferrite subplot, the result matches much closer\nwith the EBSD (crop) Ferrite subplot. This indicates that the\nDCT technique is accurately determining sample texture of both\nphases, but the more complicated 3D texture of the ferrite phase\nis poorly captured by a single 2D slice, demonstrating the need\nfor 3D characterisations to accurately determine sample texture.\n4.7. Registration\nFigure 6 demonstrates the validity of the proposed regis-\ntration technique to determine the location of the 2D slice in\nthe DCT volume that best corresponds to the EBSD map. A\nsubstantial number of larger grains in the cropped EBSD dataset\ncan be identified within the DCT 2D slice. Grain orientations as\nrepresented by their IPF colour appear well matched between\ntechniques, and many matching grain pairs can be visually iden-\ntified solely by their similar centre-of-mass position and IPF\ncolour.\nThe proposed registration algorithm has a number of advan-\ntages when compared to other registration techniques defined\nin the literature. PolyProc , for example [ 31], includes an ad-\nvanced registration technique that utilises genetic algorithms to\niteratively align two 3D datasets by maximising the number of\nshared voxels between datasets using the shape of the sample.\nThis involves a search through 6D space as both translations and\nrotations are explored. Due to the increase in e fficiency from the\nparallelised genetic algorithm search, the PolyProc registration\ntechnique is relatively fast, taking approximately 10 min to align\ntwo 3D datasets. PolyProc is well-suited to alignment of 3D\ndatasets, but can only be applied in the case where the input\ndatasets are both 3D and recorded with the same apparatus.\n6DCT (full) Austenite DCT (full) Ferrite\nEBSD (full) Austenite EBSD (full) Ferrite\nDCT (slice) Austenite DCT (slice) Ferrite\nEBSD (crop) Austenite EBSD (crop) Ferrite5\n4\n3\n2\n1\n0TDRDFigure 11:{1 1 1}orientation distribution functions from all EBSD and DCT\ngrains.\n11The registration technique employed by Renversade et al.\n[30] minimises the mean misorientation between corresponding\nvoxels in two di fferent datasets by modifying the magnification,\ndistortion, shear, rotation and translation, leading to a 9D search,\nwhich they performed using non-linear optimisation. This is a\nrobust technique and was successfully applied to register a 2D\nEBSD slice within a DCT volume. However, the technique is\nlikely significantly slower to apply than the PolyProc technique,\ndue to the high computational cost of misorientation calculation\nbetween a large number of pixels, although the problem is easily\nparallelised by splitting the misorientation calculations across\ndifferent processor cores.\nIn contrast, the technique applied in this paper relies only on\nthe average crystallographic orientation of grains (a 3D search)\nwhich makes it highly applicable to a range of materials charac-\nterisation techniques such as 3DXRD, EBSD, and HEDM. As\na subset of 10grains were selected for both the parallel PSO\nsearch and final L-BFGS search, this registration technique is\nfast, with the entire slice determination procedure taking less\nthan 5 min.\n4.8. Outlook\nFirst established at the European Synchrotron Radiation Fa-\ncility in 2008 [ 7,8], then subsequently implemented at other\nsynchrotrons such as SPring-8 [9] and Soleil [10], synchrotron-\nbased DCT has proved to be a excellent tool for revealing metal-\nlurgical phenomena related to deformation and microstructure.\nWhilst DCT performed at a synchrotron facility has a number of\nbenefits; foremost is the high-intensity of the X-ray sources, of-\nfering rapid acquisition rates necessary for time-resolved studies,\ndifficulties include significant lead times and set-up costs, lim-\nited availability and accessibility restrictions. Such constraints\nare alleviated with lab-based DCT results; initially developed\nin 2013 [ 11] and later made commercially available on select\nZEISS X-ray microscopes under the name LabDCT ™[12,48].\nImprovements in grain shape reconstructions [ 34], acquisition\nstrategies [ 33,49] and the data reconstruction pipeline [ 50] have\nbeen significant. If one considers synchrotron-based DCT as the\nbenchmark to which lab-based DCT will target, future advance-\nments must collectively consider developments to hardware, data\ncollection and post-processing.\nIn this study, several di fferences have been identified be-\ntween the characterisations performed by EBSD and DCT tech-\nniques. The complicated grain morphologies caused by the\ndual-phase microstructure, combined with lattice distortions\nin one of the phases and the presence of a number of small\ngrains in the other, presented significant challenges for the DCT\ngrain reconstruction process. However, important phase specific\nfeatures were observable including a subsurface ferrite grain\nnetwork of similarly oriented grains; an observation that was ab-\nsent in the EBSD map. It is suggested that the technique is well\nsuited for the observation of multi-phase systems, particularly\nfor well chosen microstructures with coarse grains of simple\nmorphologies. There is no reason why in-situ lab-based DCT\nexperiments cannot be performed on such multi-phase materials,\nparticularly for experiments that are unfeasible at synchrotrons\n(e.g. extended duration experiments, high-risk pilot studies, orexperiments that demand a rapid lead-time from conception to\nresults). There is significant promise that new science will be\npossible using the lab-based DCT method on a compendium of\nmulti-phase, polycrystalline materials across several disciplines.\n5. Conclusions\nIn this study a dual-phase steel sample was analysed using\ntwo techniques - DCT and EBSD. A novel registration tech-\nnique was developed and used to identify the 2D slice within\nthe 3D DCT volume that best corresponds to the EBSD grain\nmap. Many larger EBSD and DCT grains with matching IPF\ncolours were visually identified, demonstrating the success of\nthe registration algorithm in determining the correct 2D DCT\nslice. The algorithm is fast, and requires only grain-averaged\norientations, so could therefore be adopted by similar techniques\nsuch as 3DXRD and HEDM.\nFrom the EBSD scan, a complicated grain morphology was\nobserved, especially at the interfaces between the austenitic and\nferritic phases where a lath-like morphology was present with\ncharacteristic lengths on the order of 10–20µm. These fine mi-\ncrostructural details were similar in size to the size detection\nlimit of the DCT technique, which influenced a number of di ffer-\nences in microstructural statistics such as phase fractions, grain\nsize and shape distributions, as outlined:\n1.The sample was measured as 40 % ferritic by area using\nEBSD data. DCT techniques showed a ferrite phase frac-\ntion of 20 % both by area and volume. This discrepancy\nis ascribed to di fferences in austenite and ferrite grain size\n- smaller ferrite grains may have been missed by the DCT\nreconstruction and incorrectly assigned to neighboring\naustenite grains, skewing the phase balance results.\n2.Grain diameters measured with EBSD appeared signifi-\ncantly smaller than diameters measured with DCT. Fur-\nthermore, a di fference in grain diameter distributions was\nobserved between ferrite and austenite phases in the EBSD\nscan, with ferrite grains appearing smaller in diameter.\nThis di fference was also observed in the DCT datasets but\nto a much lesser degree.\n3.In the EBSD dataset, ferrite grains were less spherical\nthan austenite grains, whereas the DCT dataset reports\nferrite grains as more spherical.\nWhile the austenite phase was broadly untextured in both\nEBSD and DCT scans, a strong {1 1 1}⟨1 1 2⟩rolling texture was\nobserved in the ferrite phase with EBSD. In contrast, a more\ncomplicated texture was observed in the ferrite phase with DCT.\nThis di fference was resolved when plotting the texture of just the\ngrains in the 2D DCT slice. This indicates a complex 3D texture\nof columnar networks of similarly orientated ferrite grains in the\nsample; an important microstructural feature absent from the 2D\nmeasurements, demonstrating the need for 3D characterisations\nto accurately determine texture.\n6. Datasets and Code Access\nExample datasets and analysis code used in this study are\navailable upon request.\n127. Acknowledgments\nJames Ball, Stefan Michalik, and David Collins acknowledge\nthe financial support from the Diamond Light Source and the\nUniversity of Birmingham for a PhD studentship. Claire Davis\nand Carl Slater acknowledge the Engineering and Physical Sci-\nences Research Council for support via grants EP /P020755 /01\nand EP /V007548 /1.\n13References\n[1]P. Sedm ´ak, J. Pilch, L. Heller, J. Kope ˇcek, J. Wright, P. 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An STL file describing this\nsurface was exported from Paraview, and imported into Blender\n[53]. Finally, within Blender, the Decimate modifier was used\nwith the Planar setting to simplify the geometry of the flat faces\nof the sample and reduce the number of triangles in the mesh.\n15"    },    {        "title": "2309.10676v1.Remanence_Increase_in_SrFe___12__O___19___Fe_Exchange_Decoupled_Hard_Soft_Composite_Magnets_Owing_to_Dipolar_Interactions.pdf",        "content": "Citation: Guzmán-Mínguez, J.C.;\nGranados-Miralles, C.; Kuntschke, P .;\nde Julián Fernández, C.; Erokhin, S.;\nBerkov, D.; Schliesch, T.; Fernández,\nJ.F.; Quesada, A. Remanence Increase\nin SrFe 12O19/Fe\nExchange-Decoupled Hard-Soft\nComposite Magnets Owing to\nDipolar Interactions. Nanomaterials\n2023 ,13, 2097. https://doi.org/\n10.3390/nano13142097\nAcademic Editor: George C.\nHadjipanayis\nReceived: 8 June 2023\nRevised: 7 July 2023\nAccepted: 15 July 2023\nPublished: 18 July 2023\nCopyright: © 2023 by the authors.\nLicensee MDPI, Basel, Switzerland.\nThis article is an open access article\ndistributed under the terms and\nconditions of the Creative Commons\nAttribution (CC BY) license (https://\ncreativecommons.org/licenses/by/\n4.0/).\nnanomaterials \nArticle\nRemanence Increase in SrFe 12O19/Fe Exchange-Decoupled\nHard-Soft Composite Magnets Owing to Dipolar Interactions\nJesús Carlos Guzm án-Mínguez1, Cecilia Granados-Miralles1\n, Patrick Kuntschke2, César de Juli án Fern ández3\n,\nSergey Erokhin4, Dmitry Berkov4\n, Thomas Schliesch2, Jose Francisco Fern ández1and Adri án Quesada1,*\n1Electroceramic Department, Instituto de Cer ámica y Vidrio, CSIC, 28049 Madrid, Spain;\njesus.guzman@icv.csic.es (J.C.G.-M.); c.granados.miralles@icv.csic.es (C.G.-M.)\n2Max Baermann GmbH, 51429 Bergisch Gladbach, Germany; t.schliesch@max-baermann.de (T.S.)\n3Istituto dei Materiali per l’Elettronica e il Magnetismo, CNR, 43124 Parma, Italy; cesar.dejulian@imem.cnr.it\n4General Numerics Research Lab, 07743 Jena, Germany\n*Correspondence: a.quesada@icv.csic.es\nAbstract: In the search for improved permanent magnets, fueled by the geostrategic and environmen-\ntal issues associated with rare-earth-based magnets, magnetically hard (high anisotropy)-soft (high\nmagnetization) composite magnets hold promise as alternative magnets that could replace modern\npermanent magnets, such as rare-earth-based and ceramic magnets, in certain applications. However,\nso far, the magnetic properties reported for hard-soft composites have been underwhelming. Here,\nan attempt to further understand the correlation between magnetic and microstructural properties\nin strontium ferrite-based composites, hard SrFe 12O19(SFO) ceramics with different contents of Fe\nparticles as soft phase, both in powder and in dense injection molded magnets, is presented. In addi-\ntion, the influence of soft phase particle dimension, in the nano- and micron-sized regimes, on these\nproperties is studied. While Fe and SFO are not exchange-coupled in our magnets, a remanence that\nis higher than expected is measured. In fact, in composite injection molded anisotropic (magnetically\noriented) magnets, remanence is improved by 2.4% with respect to a pure ferrite identical magnet.\nThe analysis of the experimental results in combination with micromagnetic simulations allows us\nto establish that the type of interaction between hard and soft phases is of a dipolar nature, and\nis responsible for the alignment of a fraction of the soft spins with the magnetization of the hard.\nThe mechanism unraveled in this work has implications for the development of novel hard-soft\npermanent magnets.\nKeywords: hard-soft composites; permanent magnets; ferrites; dipolar interactions\n1. Introduction\nThe challenge of constantly improving the performance of permanent magnets has\nbeen ongoing for decades. Lately, the geostrategic and environmental issues associated\nwith rare-earth elements (REE) as raw materials have brought forward the need to develop\nalternative permanent magnets that could, at least partially, substitute them [ 1]. One of the\nstrategies to obtain REE-free substitutes is to fabricate composite magnets, based on a hard\nmagnetic phase with high coercivity and a soft phase with high magnetization, in which\nthe magnetic coupling between hard and soft phases leads to an increase in remanence and\nenergy product, which is one of the figures of merit of permanent magnets [2–6].\nThe most pursued strategy consists of exploiting the exchange-coupling between\nhard and soft particles in order to gain remanence while avoiding a detrimental coercivity\nloss [ 7,8]. According to the model, provided that the soft particles are below a given\nsize threshold [ 9] and that structural coherency at the interface exists, effective exchange\ncoupling should align at remanence the spins of the soft with the magnetization of the hard\nat reasonable coercivity penalties. However, we recently demonstrated that not only is it\nextremely challenging to meet the structural requirements associated with this approach,\nNanomaterials 2023 ,13, 2097. https://doi.org/10.3390/nano13142097 https://www.mdpi.com/journal/nanomaterialsNanomaterials 2023 ,13, 2097 2 of 12\nbut in addition, robust exchange-coupling may lead to a collapse of the coercivity [ 10–12],\nas a consequence of the lowered onset for domain wall propagation [13].\nFocusing on systems based on hexaferrites, such as SrFe 12O19, as the hard magnetic\nphase, results so far have not met the expectations for exchange-coupled hard-soft com-\nposite magnets [ 14]. In fact, more than 30 years after the pioneering work by Kneller and\nHawig [ 15], the effect is not being exploited by the industry. When using Fe as the soft\nphase, works in the field have reported increases in magnetization without a significant\nloss of coercivity [ 16–19] in exchange-coupled powders. Unfortunately, the samples stud-\nied were neither densified nor magnetically oriented, which does not enable extracting\nconclusions on the real remanence of the composites. It is important to explain as well, that\nalthough single-phase behavior in the demagnetization curves is usually associated with\neffective exchange-coupling between hard-soft particles, experiments have shown that the\ncoercive field distribution associated with broad particle size distributions in the composite\nmay lead to single-phase curves in the absence of exchange-coupling [11,12].\nMore recently, the possibility of exploiting dipolar interactions within the composite\nmagnets to improve their remanence, instead of exchange-coupling, has been discussed [ 10].\nIn fact, composites consisting of SFO and FeCo nanowires (in which the dipolar interaction,\nthrough shape anisotropy, plays a crucial role) have been reported to present a significant\nincrease in remanence and energy product [20].\nHere, the microstructural and magnetic properties of magnetically oriented SFO/Fe\ncomposite powders and the corresponding anisotropic dense magnets are studied as a\nfunction of soft phase content and particle size. The magnetic orientation assures the\nmaximum remanence as required for magnets. Scanning Electron Microscopy (SEM) is\nemployed to reveal particle sizes and geometric distribution of hard and soft phases,\nwhile demagnetization curves measured with Vibrating Sample Magnetometry (VSM)\nand Permagraph instruments detail the magnetic properties of the magnetically aligned\npowders and magnets. Analyzed in combination with micromagnetic simulations, the\ncorrelation between microstructure and magnetic properties will be established.\n2. Materials and Methods\nThe raw materials used in this study were commercial strontium ferrite (SrFe 12O19,\n99.99%) powders supplied by Max Baermann GmbH (Germany-China) (Bergisch Gladbach,\nGermany) and various commercial Fe particles (99.99%) with different particle sizes ranging\nfrom 50 nm to 11 \u0016m. The hard-soft composite samples compositions were prepared by\nincorporating 5, 10 and 15 vol% of soft phase to SrFe 12O19(SFO) powders, using different\nsizes of Fe particles. All samples were studied both in the form of oriented powders and\nin the form of injection molded magnets. The preparation of the magnetically oriented\npowder samples for the magnetic characterization consisted of dispersing 30–50 mg of\npowder in a bonding glass inside a capsule under an externally applied magnetic field\nH = 0.3 T generated by a NdFeB N42 magnet. The particles, although in proximity, were\nisolated from one another after this procedure. The injection molded oriented magnets\nwere fabricated at the company Max Baermann GmbH following their usual industrial\nproduction method. The injection-molded pieces were squares of 2 cm in lateral size and\n4 mm in height. In order for the demagnetizing boundary conditions to be comparable, all\nmagnet samples had the same sizes and shapes.\nThe mixing was carried out by means of a low-energy dry mixing method [ 21]. The\ndry dispersion process consisted of shaking SrFe 12O19/Fe particle mixtures in a 60 cm3\nnylon container for 5 min at 50 rpm using a tubular-type mixer (Mixer/Mill, 8000D, SPEX\nSample Prep., Metujen, NJ, USA). The particle size and morphology of the powders were\nevaluated using secondary electron images of field emission scanning electron microscopy,\nFE-SEM (Hitachi S-4700, Hitachi, Tokyo, Japan). Structural analysis was performed by X-\nray diffraction (XRD) with a Bruker D8 diffractometer using Cu K \u000bradiation ( \u0015= 1.5418 Å )\nand a Lynxeye XE-T detector. Subsequent Rietveld analysis of the XRD data was performedNanomaterials 2023 ,13, 2097 3 of 12\nby FullProf Suite [ 22]. Thermogravimetric analysis (TGA) in air was employed to deter-\nmine the oxidation temperature of the Fe powders between RT and 900\u000eC, using a TA\nInstruments Q50 system. The magnetic properties of the samples were measured using\nthe custom-made VSM described in reference [ 23], applying a maximum H-field of 1.4 T,\nand a Permagraph Instrument (Permagraph C, Magnet-Physik, Cologne, Germany). The\nmeasurements were carried out at room temperature, and the sensitivity of the instrument\nwas 0.1 Am2/kg and 0.025 mT, according to specifications.\nFor the micromagnetic simulations, we employed a micromagnetic algorithm initially\ndeveloped for the simulation of the magnetization distribution of magnetic nanocom-\nposites [ 24,25]. The magnetization distribution in an isolated sphere of soft phase with\ndiameter D under the influence of an external magnetic field (produced by the hard phase)\nis simulated by considering the four standard contributions to the total magnetic energy: ex-\nternal field, magnetic anisotropy, exchange and dipolar interaction. The following material\nparameters (typical for Fe) were used for simulating the soft material: saturation magneti-\nzation Ms = 220 Am2/kg, cubic magnetocrystalline anisotropy Kcub = 20 kJ/m3with easy\naxis along the <100> directions and the exchange stiffness constant Abulk= 21 pJ/m [10] .\nThe particles under study had D = 15–50 nm and were discretized in the non-regular mesh\nwith the typical mesh element size of 3 nm. Open boundary conditions were applied in\nall simulations.\n3. Results and Discussion\n3.1. Hard-Soft Composite Powders\nFigure 1 shows the SEM micrographs corresponding to the starting SFO powder and\nthe Fe powders with three different particle sizes. The particle distribution in the SFO\npowders is, as previously reported [ 26], of a bimodal nature, with larger particles in the\n2–5\u0016mrange surrounded by smaller particles in the 200–500 nm range. All SFO particles\nhave platelet shapes, as expected. The SEM characterization of the three Fe powders\nreveals that the smaller-sized powder, Figure 1b, is formed by particles with an average\nsize of 50 nm that are arranged in contact with each other, presumably for electrostatic and\nmagnetostatic reasons. The powder depicted in Figure 1c again reveals a bimodal particle\nsize distribution, where 100–200 nm Fe particles coexist with larger 1–2 \u0016m particles. For\nsimplicity, we refer to this powder in the following as 1 \u0016m Fe powder. The larger-sized Fe\npowder, shown in Figure 1d, presents an average particle size of 11 \u0016m.\nAs the samples were fabricated and manipulated in air conditions, the XRD patterns\nof the Fe powders were measured in order to study their oxidation state. Figure 2a shows\nthe pattern and refinement of the 50 nm Fe powders as a selected example. From the\nrefinement of the patterns for each Fe powder, we concluded that, for 50 nm Fe particles,\nthe powders were composed of 77 wt% Fe, 17 wt% FeO and 6 wt% Fe 3O4. For 1 \u0016m Fe\npowders, the sample consisted of 79 wt% Fe, 15 wt% FeO and 6 wt% Fe 3O4. For 11 \u0016m\nparticles, the XRD refinement detected 100 wt% Fe. Figure 2b presents the TGA of the\n50 nm Fe powders. It could be inferred that the onset for oxidation of the powders was\napproximately 370\u000eC, which hinted at the effectiveness of the original oxide surface layer\nin preventing further oxidation.\nThe saturation magnetization ( Ms) values measured for the Fe powders reveal\nMs = 175 Am2/kg (50 nm), Ms = 186 Am2/kg (1 \u0016m)andMs = 187 Am2/kg (11 \u0016m). These\nvalues are consistent with the Fe wt% extracted from XRD given that the theoretical satura-\ntion magnetization value for pure Fe is Ms = 220 Am2/kg [ 27], except for the value of the\n11\u0016m powder, which suggests that it might be partially oxidized as well. We speculate\nthat the lower reactivity of the larger Fe particles leads to oxide layers that are amorphous,\nand therefore, undetectable for XRD.Nanomaterials 2023 ,13, 2097 4 of 12\nNanomaterials 2023 , 13, x FOR PEER REVIEW  4 of 12 \n \n  \nFigure 1. SEM micrographs showing particle size and morphology of the starting ( a) SFO and Fe \npowders with ( b) 50 nm, ( c) 1 µm and ( d) 11 µm average particle sizes . \n \nFigure 2. (a) XRD pattern and corresponding Rietveld refinement for the 50 nm Fe powder.  † and * \ndenote diffraction maxima from FeO and Fe 3O4 respectively. (b) TGA of the 50 nm Fe powder per-\nformed in air heating between RT and 900 °C. \nThe saturation m agnetization ( Ms) values measured for the Fe powders reveal Ms = \n175 Am2/kg (50 nm), Ms = 186 Am2/kg (1 µm) and Ms = 187 Am2/kg (11 µm). These values \nare consistent with the Fe wt% extracted from XRD given that the theoretical saturation \nmagnetization valu e for pure Fe is Ms = 220 Am2/kg [27], except for the va lue of the 11 µm \npowder, which suggests that it might be partially oxidized as well. We speculate that the \nFigure 1. SEM micrographs showing particle size and morphology of the starting ( a) SFO and Fe\npowders with ( b) 50 nm, ( c) 1\u0016m and ( d) 11\u0016m average particle sizes.\nNanomaterials 2023 , 13, x FOR PEER REVIEW  4 of 12 \n \n  \nFigure 1. SEM micrographs showing particle size and morphology of the starting ( a) SFO and Fe \npowders with ( b) 50 nm, ( c) 1 µm and ( d) 11 µm average particle sizes . \n \nFigure 2. (a) XRD pattern and corresponding Rietveld refinement for the 50 nm Fe powder.  † and * \ndenote diffraction maxima from FeO and Fe 3O4 respectively. (b) TGA of the 50 nm Fe powder per-\nformed in air heating between RT and 900 °C. \nThe saturation m agnetization ( Ms) values measured for the Fe powders reveal Ms = \n175 Am2/kg (50 nm), Ms = 186 Am2/kg (1 µm) and Ms = 187 Am2/kg (11 µm). These values \nare consistent with the Fe wt% extracted from XRD given that the theoretical saturation \nmagnetization valu e for pure Fe is Ms = 220 Am2/kg [27], except for the va lue of the 11 µm \npowder, which suggests that it might be partially oxidized as well. We speculate that the \nFigure 2. (a) XRD pattern and corresponding Rietveld refinement for the 50 nm Fe powder. †and\n* denote diffraction maxima from FeO and Fe 3O4respectively. ( b) TGA of the 50 nm Fe powder\nperformed in air heating between RT and 900\u000eC.\nFigure 3 shows the magnetization curves of Fe/SFO-oriented composite powders\nfabricated with 50 nm sized Fe particles with soft phase contents of 5 vol%, 10 vol% and\n15 vol%, as well as the curves of the individual SFO and Fe (50 nm) phases. From the\nindividual SFO and Fe phases curves, we can see that the hard phase (SFO) presents a\ncoercive field of 324 kA/m, while the saturation magnetization ( Ms) value is 69 Am2/kg.\nThe soft phase (Fe) shows coercivity at ( HC) ~2 kA/m and Ms = 186 Am2/kg. This value\nis lower than expected for Fe (~220 Am2/kg) [ 28], which is due to the fact that these Fe\nnanoparticles are protected by a Fe 3Si layer at their surface, as indicated by the supplier,Nanomaterials 2023 ,13, 2097 5 of 12\nwhich lowers Ms. The presence of this layer excludes the hypothesis of exchange-coupling\nbetween Fe and SFO powders in the mixtures. Regarding the nanocomposites, for the\nsample with 5 vol% Fe (red curve), Ms = 75 Am2/kg and HC= 312 kA/m. In the sample\nwith 10 vol% Fe (green curve), HC= 290 kA/m and Ms = 84 Am2/kg. Finally, the sample\nwith 15 vol% Fe (blue curve) presents HC= 264 kA/m and Ms = 91 Am2/kg. As expected, in\nhard-soft composites, coercivity decreases, while Ms increases with increasing soft content.\nBased on the steep drop in HCfor 15 vol%, we consider 10 vol% Fe the sample that presents\nthe most competitive compromise in magnetic performance (based on HCand Ms).\nNanomaterials 2023 , 13, x FOR PEER REVIEW  5 of 12 \n \n lower reactivity of the larger Fe particles leads to oxide layers that are amorphous , and \ntherefore , undetectable for XRD.  \nFigure 3 shows the magnetization curves of Fe/SFO -oriented composite powders fab-\nricated with 50 nm sized Fe particles with soft phase contents of 5 vol%, 10 vol% and 1 5 \nvol%, as well as the curves of the individual SFO and Fe (50 nm) phases. From the indi-\nvidual SFO and Fe ph ases curves, we can see that the hard phase (SFO) presents a coercive \nfield of 324 kA/m, while the saturation magnetization ( Ms) value is 69 Am2/kg. The soft \nphase (Fe) shows coercivity  at (HC) ~2 kA/m and Ms = 186 Am2/kg. This value is lower \nthan expected  for Fe (~220 Am2/kg) [28], which is due to the fact that these Fe nanoparticles \nare protected by a Fe 3Si layer at their surface, as  indicated by the supplier, which lowers \nMs. The presence of this layer excludes the hypothesis of exchange -coupling between Fe \nand SFO powders in the mixtures. Regarding the nanocomposites, for the sample with 5 \nvol% Fe (red curve) , Ms = 75 Am2/kg and HC = 312 kA/m. In the sample with 1 0 vol% Fe \n(green curve), HC = 290 kA/m and Ms = 84 Am2/kg. Finally, the sample with 1 5 vol% Fe \n(blue curve) presents HC = 264 kA/m and Ms = 91 Am2/kg. As expected , in hard-soft com-\nposites, coercivity decreases , while Ms increases with increasing soft content. Based on \nthe steep drop in HC for 15 vol%, we consider 1 0 vol% Fe the sample that presents the most \ncompetitive compromise in magnetic performance (based on HC and Ms). \n \nFigure 3. Magnetization  vs. applied field c urves of SFO/Fe composite powders using 50 nm Fe par-\nticles and for different soft phase concentrations, including the curves of the individual SFO and Fe \nphases.  \nIt is important to note that the S  shape observed in the second quadrant of the de-\nmagnetizatio n curve of the composites (more evident for the 1 5 vol% sample but dis-\nplayed by the other two as well), as detailed in the inset of Figure 3, strongly hints at an \nabsence of exchange -coupling between hard and soft magnetic phases [11,12,15,20]. Com-\nposites fabricated with 1 µm and 11 µm diameter Fe particles present similar trends (not \nshown). Table 1 summarizes the magnetic properties of the bonded magnets studied.  \n  \nFigure 3. Magnetization vs. applied field curves of SFO/Fe composite powders using 50 nm Fe\nparticles and for different soft phase concentrations, including the curves of the individual SFO and\nFe phases.\nIt is important to note that the S shape observed in the second quadrant of the demag-\nnetization curve of the composites (more evident for the 15 vol% sample but displayed\nby the other two as well), as detailed in the inset of Figure 3, strongly hints at an absence\nof exchange-coupling between hard and soft magnetic phases [ 11,12,15,20]. Composites\nfabricated with 1 \u0016m and 11 \u0016m diameter Fe particles present similar trends (not shown).\nTable 1 summarizes the magnetic properties of the bonded magnets studied.\nTable 1. Magnetic parameters for the oriented powder samples.\nSample Hc (kA/m) MR(Am2/kg) MS(Am2/kg) Type\nSFO 323 65.1 68.7 Oriented Powder\nSFO + 5 vol% Fe 50 nm 301 62.5 76.4 Oriented Powder\nSFO + 10 vol% Fe 50 nm 287 63.5 86 Oriented Powder\nSFO + 15 vol% Fe 50 nm 259 61.3 93.1 Oriented Powder\n100% Fe 50 nm 3 3.7 174.6 Oriented Powder\nRemanent magnetization ( MR) presents an interesting behavior, as shown in Figure 4.\nIn a multiphase magnetic material, in the absence of coupling, remanence is an additive\nproperty [28], the value of which can be calculated using the expression:\n[Mr]exp=Mr,hard\u0003wt%hard+Mr,so f t\u0003wt%so f t (1)Nanomaterials 2023 ,13, 2097 6 of 12\nNanomaterials 2023 , 13, x FOR PEER REVIEW  6 of 12 \n \n Table 1. Magnetic parameters for the  oriented powder samples.  \nSample  Hc (kA/m)  MR (Am2/kg) MS (Am2/kg) Type  \nSFO 323 65.1 68.7 Oriented \nPowder  \nSFO + 5 vol% Fe 50 nm  301 62.5 76.4 Oriented \nPowder  \nSFO + 1 0 vol% Fe 50 nm  287 63.5 86 Oriented \nPowder  \nSFO + 1 5 vol% Fe 50 nm  259 61.3 93.1 Oriented \nPowder  \n100% Fe 50 nm  3 3.7 174.6  Oriented \nPowder  \nRemanent magnetization ( MR) presents an interesting behavior, as shown in Figure \n4. In a multiphase magnetic material , in the absence of coupling, remanence is an additive \nproperty  [28], the value of which can be calculated using the expression:  \n[𝑀𝑟]𝑒𝑥𝑝=𝑀𝑟,ℎ𝑎𝑟𝑑∗𝑤𝑡%ℎ𝑎𝑟𝑑+ 𝑀𝑟,𝑠𝑜𝑓𝑡 ∗𝑤𝑡%𝑠𝑜𝑓𝑡 (1) \n \nFigure 4. Remanence ( MR) as a function of Fe content. The graph shows two groups of three curves, \ncorresponding to the measured values of the composites fabricated with 50 nm and 1 µm Fe pow-\nders and the values calculated by the linear combination of the MR values of Fe and SFO individual \nphases, for both oriented and non-oriented powders.  \nUsing the MR values for Fe and SFO measured in Figure 2, and given the assumption \nthat no exchange -coupling occurs in these composite powders, Figure 3 shows the ex-\npected (calculated) MR values together with the values experimentally measured \nFigure 4. Remanence ( MR) as a function of Fe content. The graph shows two groups of three curves,\ncorresponding to the measured values of the composites fabricated with 50 nm and 1 \u0016m Fe powders\nand the values calculated by the linear combination of the MRvalues of Fe and SFO individual phases,\nfor both oriented and non-oriented powders.\nUsing the MRvalues for Fe and SFO measured in Figure 2, and given the assumption\nthat no exchange-coupling occurs in these composite powders, Figure 3 shows the expected\n(calculated) MRvalues together with the values experimentally measured (extracted from\nFigure 3). By first analyzing the case of non-oriented powders, it can be observed that\nthe calculated and measured values practically coincide for 50 nm Fe particles, and slight\ndeviations are measured for the 1 \u0016m Fe size. We attribute these fluctuations/deviations to\nthe random arrangement of non-oriented particles.\nA very different scenario occurs for magnetically oriented powders. As can be ob-\nserved, for all Fe contents and 50 nm and 1 \u0016m particle sizes, the measured remanence is\nlarger than the expected calculated value. This evidence excludes the hypothesis of total\ndecoupling between hard and soft phases and suggests a magnetizing-type coupling that is\nonly activated in the magnetically oriented state. The plausible reasons for this observation\nwill be discussed later in the manuscript.\n3.2. Injection-Molded Hard-Soft Composite Magnets\nIn order to investigate the potential of these composite powders as dense magnets,\nanisotropic (magnetically aligned) injection-molded permanent magnets were fabricated at\nthe Max Baermann GmbH pilot production line. Fe content of 10 vol% was selected, and\nthree different types of injection-molded permanent magnets were fabricated using the\nthree Fe powders presented above with different particle sizes.\nFigure 5 shows the demagnetization curves, measured in a closed loop in a perma-\ngraph instrument, for the three Fe particle sizes and for a single soft phase content ( 10 vol% ),\nas well as a reference 100% ferrite sample. Figure 5a shows that coercivity decreases and\nthe squareness of the demagnetization curve is significantly affected. While squareness\nis mainly lost, it is important to remark that, in contrast with the VSM curves measuredNanomaterials 2023 ,13, 2097 7 of 12\nin Figure 3 in powders, the smooth shapes of the demagnetization curves in Figure 4 are\nindicative of a system behaving as a single magnetic phase, suggesting an interparticle\ncoupling between two phases.\nNanomaterials 2023 , 13, x FOR PEER REVIEW  7 of 12 \n \n (extracted from Figure 3). By first a nalyzing the case of non -oriented powders, it can be \nobserved that the calculated and meas ured values practically coincide for 50 nm Fe parti-\ncles, and slight deviations are measured for  the 1 µm Fe size. We attribute these fluctua-\ntions/deviations to the random arrangement of non -oriented particles.  \nA very different scenario occurs for magnetica lly oriented powders. As can be ob-\nserved, for all Fe contents and 50 nm and 1 µm particle sizes, the measured remanence is \nlarger than the expected calculated value. This evidence excludes the hypothesis of total \ndecoupling between hard and soft phases and  suggests a magnetizing -type coupling that \nis only activated in the magnetically oriented state. The plausible reasons for this obser-\nvation will be discussed later in the manuscript.  \n3.2. Injection-Molded Hard -Soft Composite Magnets  \nIn order to investigate  the potential of these composite powders as dense magnets, \nanisotropic (magnetically aligned) injection -molded permanent magnets were fabricated \nat the Max Baermann GmbH pilot production line. Fe content of 1 0 vol% was selected, \nand three different types of injection -molded permanent magnets were fabricated using \nthe three Fe powders presented above with different particle sizes.  \nFigure 5 shows the demagnetization curves, measured in a closed loop in a perma-\ngraph instrument, for the three Fe particle sizes  and for a single soft phase content (1 0 \nvol%), as well as a reference 100% ferrite sample. Figure 5a shows that coercivity decreases \nand the squareness of the demagnetization curve is significantly affected. While square-\nness is mainly lost, it is importan t to remark that, in contrast with the VSM curves meas-\nured in Figure 3 in powders, the smooth shapes of the demagnetization curves in Figure \n4 are indicative of a system behaving as a single magnetic phase , suggesting  an interpar-\nticle coupling between two phases.  \nIn Figure 5b, we observe that for pure  SFO (black line), remanent polarization JR = \n0.248 T, for the composite with 50 nm Fe particles, JR = 0.255 T, for 1 µm Fe particles JR = \n0.255 T and lastly for 11 µm , JR = 0.250 T. As for the powders, while a linear combination \nof hard and soft remanences should lead to a ~10% decrease  in remanence in the compo-\nsite with respect to the pure ferrite magnets, we measure a 2.4% increase in remanence, \nwith respect to the pure ferrite magnet, for  50 nm and 1 µm Fe particle sizes and for a 1 0 \nvol% soft content; while a 0.8% increase is measured for 11 µm particles. Hence, we ob-\nserve an anomalous non -monotonous variation in MR and HC with the particle size in-\ncrease.  \n \nFigure 5. (a) Magnetic polarization J as a function of the applied magnetic field Heff of injection -\nmolded composite magnets with 1 0 vol% Fe content for three different Fe particle sizes. (b) Rema-\nnence values for the four samples as a function of particle size.  \nFigure 5. (a) Magnetic polarization Jas a function of the applied magnetic field Heff of injection-\nmolded composite magnets with 10 vol% Fe content for three different Fe particle sizes. ( b) Rema-\nnence values for the four samples as a function of particle size.\nIn Figure 5b, we observe that for pure SFO (black line), remanent polarization JR= 0.248 T ,\nfor the composite with 50 nm Fe particles, JR= 0.255 T, for 1 \u0016m Fe particles JR= 0.255 T\nand lastly for 11 \u0016m,JR= 0.250 T. As for the powders, while a linear combination of hard\nand soft remanences should lead to a ~10% decrease in remanence in the composite with\nrespect to the pure ferrite magnets, we measure a 2.4% increase in remanence, with respect\nto the pure ferrite magnet, for 50 nm and 1 \u0016m Fe particle sizes and for a 10 vol% soft\ncontent; while a 0.8% increase is measured for 11 \u0016m particles. Hence, we observe an\nanomalous non-monotonous variation in MRand HCwith the particle size increase.\nTable 2 summarizes the magnetic properties of the bonded magnets studied.\nTable 2. Magnetic parameters and porosity of the injection moulded magnets.\nSample HC(kA/m) J(T) Type Porosity (%)\nSFO 219.6 0.248\u00060.0025 Bonded Magnet 3.8\nSFO + 10 vol% Fe 50 nm 144.8 0.255\u00060.0025 Bonded Magnet 6.2\nSFO + 10 vol% Fe 1 \u0016m 143.6 0.255\u00060.0025 Bonded Magnet 6.2\nSFO + 10 vol% Fe 11 \u0016m 184.7 0.25 \u00060.0025 Bonded Magnet 6\nThe porosity of the bonded magnets was calculated by measuring the density of the\nsamples and using the theoretical densities. The pure ferrite magnet presents the lowest\nporosity 3.2%, while the composite magnets have porosities between 6–6.2%. Given that\nvolume magnetization Mdepends on porosity paccording to the formula M= (1\u0000p)MS,\nthis entails that the increase in Jobserved cannot be explained by porosity changes.\nIt is also worth noting that the injection molding process is carried out at 250\u000eC, which\nis below the temperature at which the Fe powders oxidize, according to Figure 2b, and\ntherefore, we expect no oxidation of the soft phase.\nFigure 6 shows the SEM characterization of the surface of the injection-molded magnet\nmade with 10 vol% 11 \u0016m Fe particles. The size and morphology of both phases in the\nsystem SFO/Fe can be distinguished, where the smaller SFO particles embedded withinNanomaterials 2023 ,13, 2097 8 of 12\nthe polymer form a percolated matrix that surrounds the larger Fe particles, which seem\nto be isolated. Figure 6b in particular clearly shows the presence of a void around the Fe\nparticle in the center of the micrograph that prevents it from being in direct contact with the\nsurrounding SFO matrix. Although the reasons behind the formation of this microstructure\nhave not been investigated, we speculate that the fluid dynamics during the process of\npolymer wetting may be affected by the presence of the significantly larger Fe particles.\nA crucial consequence of this absence of direct contact is that we can again discard that\nthe increase in remanence is due to effective exchange-coupling at the interface [ 15]. This\nobservation emphasizes the surprising single-phase behavior of the demagnetization curve\nof the composite magnets and the increase in remanence observed.\nNanomaterials 2023 , 13, x FOR PEER REVIEW  8 of 12 \n \n Table 2 summarizes the magnetic properties of the bonded magnets studied.  \nTable 2. Magnetic parameters and porosity of the injection moulded magnets.  \nSample  HC (kA/m)  J (T) Type  Porosity (%)  \nSFO 219.6  0.248 ± 0.0025  Bonded Magnet  3.8 \nSFO + 10 vol% Fe 5 0 nm 144.8  0.255 ± 0.0025  Bonded Magnet  6.2 \nSFO + 10 vol% Fe 1 µm  143.6  0.255 ± 0.0025  Bonded Magnet  6.2 \nSFO + 10 vol% Fe 11 µm  184.7  0.25 ± 0.0025  Bonded Magnet  6 \nThe porosity of the bonded magnets was calculated by measuring the density of the \nsamples and using the theoretical densities. The pure ferrite magnet presents the lowest \nporosity 3.2%, while the composite magnets have porosities between 6–6.2%. Given that \nvolume magnetiz ation M depends on porosity p according to the formula M = (1 − p)MS, \nthis entails that the increase in J observed cannot be explained by porosity changes.  \nIt is also worth noting that the injection molding  process is carried out at 250 °C, \nwhich is below the temperature at which the Fe powders oxidize , according to Figure 2b, \nand therefore, we expect no oxidation of the soft phase.  \nFigure 6 shows the SEM characterization of the surface of the injection -molded mag-\nnet made with 1 0 vol% 11 µm Fe particles. T he size and morphology of both phases in the \nsystem SFO/Fe can be distinguished, where the smaller SFO particles embedded within \nthe polymer form a percolated matrix that surrounds the larger Fe particles, which seem \nto be isolated. Figure 6b in particular clearly shows the presence of a void around the Fe \nparticle in the center of the micrograph that prevents it from being in direct contact with \nthe surrounding SFO matrix. Although the reasons behind the formation of this micro-\nstructure have not been invest igated, we speculate that the fluid dynamics during the pro-\ncess of polymer wetting may be affected by the presence of the significantly larger Fe par-\nticles. A crucial consequence of this absence of direct contact is that we can again discard \nthat the incre ase in remanence is due to effective exchange -coupling at the interface [15]. \nThis observation emphasizes the surprising single -phase behavior of the demagnetization \ncurve of the composite magnets and the increase in remanence observed.  \n \nFigure 6. SEM images taken on the surface of the magnet at (a) ×1000 and ( b) ×6000 magnifications \nshowing  the microstructure of injection -molded magnets with 1 0 vol% Fe particles of 11 µm.  \n3.3. Micromagnetic Simulations  \nA micromagnetic study of magnetization reversal in SFO/Fe samples was performed \nusing an approach specifically developed for modeling the magnetization distribution in \nnanocomposites. The details of this simulation technique can be found in [24,25]. In all \npresented simulations, a cubical modeling volume with sides measuring  200 nm was dis-\ncretized into 400 ,000 mesh elements, each sized about 3 nm. Th is high-performance \nFigure 6. SEM images taken on the surface of the magnet at ( a)\u00021000 and ( b)\u00026000 magnifications\nshowing the microstructure of injection-molded magnets with 10 vol% Fe particles of 11 \u0016m.\n3.3. Micromagnetic Simulations\nA micromagnetic study of magnetization reversal in SFO/Fe samples was performed\nusing an approach specifically developed for modeling the magnetization distribution\nin nanocomposites. The details of this simulation technique can be found in [ 24,25]. In\nall presented simulations, a cubical modeling volume with sides measuring 200 nm was\ndiscretized into 400,000 mesh elements, each sized about 3 nm. This high-performance\ncalculation not only allows us to recover the details of magnetization distribution inside\nthe crystallites, but also enables us to study a significant number of different crystallites,\nwhich is important for investigating magnetic interactions between them. All modeled\nsamples have 20 vol% porosity. Four standard contributions to the total micromagnetic\nenergy are taken into account: external magnetic field, anisotropy, exchange coupling and\nmagnetodipolar interaction energies. Periodic boundary conditions are used.\nWe performed simulations changing the particle diameter between 15 to 60 nm and\nfor three soft phase concentrations: 5%, 10% and 15% (volume fractions of magnetic\nmaterial). The exchange coupling between crystallites was set to zero and anisotropy\naxes were oriented in the initial direction of the magnetic field. For every parameter\nset, the magnetization reversal of the composite was modeled and the corresponding\ndemagnetization curves were calculated. In this manner, remanence was extracted from\nevery curve and is presented in Figure 7 as a function of the soft phase concentration for\neach crystallite size. In all cases, the calculated remanence decreases with both increasing\nsoft content and increasing particle size, as expected in hard soft composites [ 3,4,29,30].\nThe larger the soft particle size, the steeper the decrease in remanence.Nanomaterials 2023 ,13, 2097 9 of 12\nNanomaterials 2023 , 13, x FOR PEER REVIEW  9 of 12 \n \n calculation not only allows us to recover the details of magnetization distribution inside \nthe crystallites, but also enables us to study a significant number of different crystallites, \nwhich is important for investigating magnetic inte ractions between them. All modeled \nsamples have 20 vol% porosity.  Four standard contributions to the total micromagnetic \nenergy are taken into account: external magnetic field, anisotropy, exchange coupling and \nmagnetodipolar interaction energies. Periodic  boundary conditions are used.  \nWe performed simulations changing the particle diameter between 15 to 60 nm and \nfor three soft phase concentrations: 5%, 10% and 15% (volume fractions of magnetic ma-\nterial). The exchange coupling between crystallites was set to zero and anisotropy axes \nwere oriented in the initial direction of the magnetic field. For every parameter set, the \nmagnetization reversal of the composite was modeled and the corresponding demagneti-\nzation curves were calculated. In this manner, remanence was extracted from every curve  \nand is presented in Figure 7 as a function of the soft phase concentration for each crystal-\nlite size. In all cases, the calculated remanence decreases with both increasing soft content \nand increasing particle size, as expected in hard soft composites [3,4,29,30]. The larger the \nsoft particle size, the steeper the decrease in remanence.  \n \nFigure 7. Micromagnetically calculated values of remanence in SFO/Fe composites as a function of \nFe content and for different Fe particle sizes.  \nAs stated above, our micromagneti c approach allows us to present the evolution of \nthe magnetization distribution of individual iron crystallites in the sample. The spin con-\nfiguration of Fe nanoparticles of diameters between 15 –50 nm was simulated and the re-\nsults are shown in Figure 8. It can be observed that Fe particles only behave as magneti-\ncally single -domain for diameter D = 15 nm. For larger diameters, such as the particles \nused in the composites fabricated here, a vortex configuration forms, with the external \nspins (those closer to t he surface of the particle) forming a closed circular loop while the \ncentral spins are aligned, as if an aligned magnetic rod was located at the center of the Fe \nparticles. Figure 8b portrays this by showing an augmented section of the spins inside a \n50 nm Fe particle. These calculations only qualitatively agree with the theoretical thresh-\nold between single and multidomain regimes defined by the coherent size (around 24 nm) \nbut also by the domain wall length (around 65 nm) [25] and they illustrate the magnetic \nvortex/multidomain structure of Fe particles even in  the nanodomains.  \nFigure 7. Micromagnetically calculated values of remanence in SFO/Fe composites as a function of\nFe content and for different Fe particle sizes.\nAs stated above, our micromagnetic approach allows us to present the evolution\nof the magnetization distribution of individual iron crystallites in the sample. The spin\nconfiguration of Fe nanoparticles of diameters between 15–50 nm was simulated and\nthe results are shown in Figure 8. It can be observed that Fe particles only behave as\nmagnetically single-domain for diameter D = 15 nm. For larger diameters, such as the\nparticles used in the composites fabricated here, a vortex configuration forms, with the\nexternal spins (those closer to the surface of the particle) forming a closed circular loop\nwhile the central spins are aligned, as if an aligned magnetic rod was located at the center\nof the Fe particles. Figure 8b portrays this by showing an augmented section of the spins\ninside a 50 nm Fe particle. These calculations only qualitatively agree with the theoretical\nthreshold between single and multidomain regimes defined by the coherent size (around\n24 nm) but also by the domain wall length (around 65 nm) [ 25] and they illustrate the\nmagnetic vortex/multidomain structure of Fe particles even in the nanodomains.\nIt has been demonstrated, by the shape of the demagnetization curves of the powders\nand the micrographs of the magnets showing voids around Fe, that no exchange-coupling\ntakes place between SFO and Fe particles. Under this circumstance, the increase in rema-\nnence experimentally measured for the composite samples (in both powder and injection\nmolded magnet forms) can only be explained by a certain degree of alignment of the spins\nof the soft phase with the magnetization of the hard, which happens even if Fe particles are\nin a multidomain state.\nBased on the difference between measured and calculated (using expression 1) rema-\nnence in the oriented powders (~10% on average) and the MRof Fe and SFO, it can be\ninferred that the fraction of Fe spins that actually aligned with the hard phase at remanence,\nand thus contribute to the overall increase in the remanence of the magnet, is approximately\n4%. Looking at the spin configuration in Figure 7, we suggest that a plausible explanation\nis that the internal spins of the vortex structure are aligned with the internal field created by\nthe hard SFO phase; i.e., due to the dipolar interaction between the hard and soft phase. The\nself-demagnetizing field in the Fe particles, proportional to the Ms of Fe, easily overcomes\nthe internal field created by SFO, especially near the particle surface due to the minimiza-\ntion of the magnetostatic energy, which makes the spins of the Fe particle circularly curl to\nminimize the stray fields. However, the internal spins in the vortex structure are subjected\nto far inferior self-demagnetizing fields and they are, therefore, more likely to align with the\nhard particles. This alignment will be parallel or antiparallel depending on the geometricNanomaterials 2023 ,13, 2097 10 of 12\ndistribution of the field lines inside the magnet, which in turn depends on the distance and\ngeometric arrangement of SFO and Fe particles. It is nevertheless safe to assume that, given\nthe parallel alignment of the magnetization of all SFO particles inside the magnetically\noriented bonded magnet, the internal magnetic fields will lead to a net alignment of the\nsoft spins in the direction parallel to the magnetization of the hard. This mechanism is\nconsistent with the small (~4%) fraction of Fe spins that are estimated to be aligned in the\nmagnet and the fact that the remanence increase, with respect to the theoretically expected,\nis observed irrespective of the Fe particle size.\nNanomaterials 2023 , 13, x FOR PEER REVIEW  10 of 12 \n \n  \nFigure 8. (a) Micromagnetically simulated images of the spin configuration of isolated Fe particles \nof different diameters between 15 –50 nm. (b) Detail of the spin configuration of a 50 nm diameter \nFe particle.  \nIt has been demonstrated,  by the shape of the demagnetization curves of the powders \nand the micrographs of the magnets showing voids around Fe, that no exchange -coupling \ntakes place between SFO and Fe particles. Under this circumstance, the increase in rema-\nnence experimentally mea sured for the composite samples (in both powder and injection \nmolded magnet forms) can only be explained by a certain degree of alignment of the spins \nof the soft phase with the magnetization of the hard, which happens  even if Fe particles \nare in a multido main state.  \nBased on the difference between measured and calculated (using expression 1) rem-\nanence in the oriented powders ( ~10% on average) and the MR of Fe and SFO, it can be \ninferred that the fraction of Fe spins that actually aligned with the hard phase at rema-\nnence, and thus contribute to the overall increase in the remanence of the magnet, is ap-\nproximately 4%. Looking at the spin configuration in F igure 7, we suggest that a plausible \nexplanation is that the internal spins of the vortex structure are aligned with the internal \nfield created by the hard SFO phase ; i.e., due to the dipolar interaction between the hard \nand soft phase. The self -demagnetiz ing field in the Fe particles, proportional to the Ms of \nFe, easily overcomes the internal field created by SFO , especially near the particle surface \ndue to the minimization of the magnetostatic energy, which makes the spins of the Fe \nparticle circularly curl to minimize the stray fields. However, the internal spins in the vor-\ntex structure are subjected to far inferior self -demagnetizing fields and they are , therefore , \nmore likely to align with the hard particles. This alignment will be parallel or antipara llel \ndepending on the geometric distribution of the field lines inside the magnet, which in turn \ndepends on the distance and geometric arrangement of SFO and Fe particles. It is never-\ntheless safe to assume that, given the parallel alignment of the magnetiz ation of all SFO \nparticles inside the magnetically oriented bonded magnet, the internal magnetic fields will \nlead to a net alignment of the soft spins in the direction parallel to the magnetization of \nthe hard. This mechanism is consistent with the small ( ~4%) fraction of Fe spins that are \nestimated to be aligned in the magnet and the fact that the remanence increase , with re-\nspect to the theoretically expected , is observed irrespective of the Fe particle size . \n4. Conclusions  \nThe magnetic properties and the microstructure of SrFe 12O19/Fe hard-soft composites, \nin powders and injection -molded magnet form, have been studied as a function of soft \nphase content -between 5 –15 vol%- and soft particle size -between 50 nm –11 µm. While \ncoercivity decreases with soft ph ase concentration, as expected in hard -soft composites, a \nFigure 8. (a) Micromagnetically simulated images of the spin configuration of isolated Fe particles\nof different diameters between 15–50 nm. ( b) Detail of the spin configuration of a 50 nm diameter\nFe particle.\n4. Conclusions\nThe magnetic properties and the microstructure of SrFe 12O19/Fe hard-soft composites,\nin powders and injection-molded magnet form, have been studied as a function of soft phase\ncontent -between 5–15 vol%- and soft particle size -between 50 nm–11 \u0016m. While coercivity\ndecreases with soft phase concentration, as expected in hard-soft composites, a remanence\nthat is larger than expected is measured in both oriented powders and oriented-bonded\nmagnets. In fact, the hard-soft composite injection-molded magnets present a 2.4% increase\nin remanence with respect to identically prepared pure ferrite magnets, for all particle sizes\nexplored. The lack of exchange-coupling between hard and soft phases, evidenced by the\nabsence of direct contact between SFO and Fe particles seen in the microstructure of the\nmagnets and the shape of the demagnetization curves, points at dipolar interactions as the\ncause for the remanence increase observed. The micromagnetic simulations performed\nreveal that a vortex spin configuration can form in spherical Fe soft particles with diameters\nabove 15 nm. We suggest that the spins at the core of the vortex align with the hard phase,\nexplaining the observation and the fact that it occurs for all particle sizes studied and only\nwhen the particles are magnetically oriented. These results open pathways to improving the\nremanence in hard-soft ferrite-based composites in the absence of exchange-coupling, which\nwould be of great interest as the strict requirements associated with effective exchange-\ncoupled would not have to be met, which in turn enhances the applicability of the method at\nan industrial level. This has ramifications as well in the ultimate development of hard-soft\npermanent magnets with enhanced performance, of any composition.\nAuthor Contributions: Conceptualization, A.Q., C.d.J.F. and J.F.F.; methodology, J.C.G.-M., C.G.-M. ,\nP .K. and T.S.; software, D.B. and S.E.; formal analysis, C.G.-M. and J.C.G.-M.; experiments, C.G.-M. ,\nP .K., T.S. and J.C.G.-M.; simulations, D.B. and S.E.; writing—original draft preparation, A.Q.;\nwriting—review and editing, all authors; visualization, C.G.-M. and A.Q.; supervision, A.Q., C.d.J.F.\nand J.F.F.; funding acquisition, A.Q., C.d.J.F. and J.F.F. All authors have read and agreed to the\npublished version of the manuscript.Nanomaterials 2023 ,13, 2097 11 of 12\nFunding: This work was supported by the European Commission through Project H2020 No. 720853\n(AMPHIBIAN). It is also supported by the Spanish Ministry of Science and Innovation through\nGrants RTI2018-095303-BC51, RTI2018-095303-A-C52, PID2021-124585NB-C33, TED2021-130957B-\nC51 funded by MCIN/AEI/10.13039/501100011033, by “ERDF A way of making Europe” and by the\n“European Union NextGenerationEU/PRTR”. Financed by the European Union—NextGenerationEU\n(National Sustainable Mobility Center CN00000023, Italian Ministry of University and Research\nDecree n. 1033—17/06/2022, Spoke 11—Innovative Materials and Lightweighting). C.G.-M. acknowl-\nedges financial support from grant RYC2021-031181-I funded by MCIN/AEI/10.13039/501100011033\nand by the “European Union NextGenerationEU/PRTR”. A.Q. acknowledges financial support from\nMICINN through the Ram ón y Cajal Program (RYC-2017-23320). The opinions expressed are those\nof the authors only and should not be considered as representative of the European Union or the\nEuropean Commission’s official position. 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MDPI and/or the editor(s) disclaim responsibility for any injury to\npeople or property resulting from any ideas, methods, instructions or products referred to in the content."    },    {        "title": "2303.04541v1.Construction_and_Testing_of_a_Common_Mode_Choke_for_Cryogenic_Detector_Pre_Amplifiers.pdf",        "content": "arXiv:2303.04541v1  [astro-ph.IM]  8 Mar 2023Construction and Testing of a Common Mode Choke for Cryogeni c\nDetector Pre-Amplifiers\nMathias Richerzhagena, Joshua Hopgooda\naEuropean Southern Observatory (ESO), Karl-Schwarzschild -Str. 2, 85748 Garching bei M¨ unchen, Germany\nAbstract. Common-mode choke inductors are useful tools for resolving grounding issues in large detector systems.\nUsing inductive components on cryogenic pre-amplifier boar ds has so far been prevented by the poor low-temperature\nperformance of common ferrite materials such as NiZn and MnZ n. Recently developed nanocrystalline and amorphous\nferrite materials promise improved performance up to the po int where using magnetics at liquid mitrogen temperatures\nbecomes feasible. This research applies the work of Yin et al .1on characterizing ferrite materials by constructing and\ntesting a common mode choke inductor for use on detector pre- amplifiers for the ELT first generation instruments.\nKeywords: common-mode choke, large instruments, grounding issue, de tector system, cryogenic electronics, nanocrys-\ntalline ferrite material .\n1 Introduction\nCommon mode chokes are a well-understood instrument for imp roving immunity to electromag-\nnetic disturbances as described among many in Ref. [ 2]. As passive components consisting of two\nwindings on a common ferromagnetic core, chokes provide hig h impedance for common mode\nsignals while having significantly lower impedance to diffe rential signals. These properties make\nthem efficient at preventing the coupling of common mode dist urbance signals into electronic cir-\ncuits through power or data lines, thus reducing the occurre nce of a class of artifacts summarized\nas “Grounding Issues” during integration and testing. In cr yogenic detector systems, the use of\ncommon mode chokes on power and signal inputs may be especial ly advantageous since the detec-\ntor and associated pre-amplifier board are typically locate d far away from other control electronics\n(i.e. the detector controller). For the ELT first generation instruments METIS,3HARMONI4and\nMICADO5the warm control electronics cabinet and cold pre-amplifier are separated by cables of\nup to 5m length. Ideally, differential power and signal inpu ts of a cryogenic pre-amplifier would\nbe common mode filtered at the pre-amplifier. According to Ref . [6], the use of common mag-\n1netic core materials is not feasible in cryogenic systems du e to the significant loss in magnetic\npermeability of the core at liquid nitrogen temperatures. R ecently-developed nanocrystalline and\namorphous core materials have been shown to perform much bet ter at low temperatures as eval-\nuated in Ref. [ 1]. This research aims at applying the results of Ref. [ 1] to construct a common\nmode choke inductor suitable for cryogenic detector system s and comparing its performance to\nclassic NiZn or MnZn based cores. The inductor is characteri zed in terms of its impedance over\nfrequency curve and not its magnetic properties for easier u se in electronics engineering design\ndocumentation.\n2 Method\n2.1 Construction\nFour common mode choke samples are constructed on commercia lly available toroidal cores.\nFig 1 Segmented Winding Common Mode Choke\nTable 1 Common Mode Choke Sample Parameters\nSample Core Windings\nA ½ W¨ urth 74270113 NiZn 2x 6 Windings AWG22 Magnet Wire\nB V AC L2009-W914 Nanocrystalline 2x 6 Windings AWG22 Magnet Wire\nC ½ W¨ urth 74270113 NiZn 2x 12 Windings AWG26 Magnet Wire\nD V AC L2009-W914 Nanocrystalline 2x 12 Windings AWG26 Magnet Wire\n2Fig 2 Common Mode Choke Samples\nSamples are constructed using nanocrystalline and traditi onal NiZn based core materials and\nplaced on a test coupon made from perforated circuit board. F or the NiZn core, a larger core is split\nin half to achieve a comparable magnetic cross section to the nanocrystalline core. Windings are\nadded manually using a segmented winding pattern as shown in Figure 1. A bifilar winding pattern\ncould be chosen as an alternative, but the segmented techniq ue allows for some spatial separation\nbetween the two windings making the effect of an insulation f ailure less severe since a winding-\nto-winding short circuit is less likely. The resulting indu ctor samples are shown in Figure 2with\nconstruction parameters listed in Table 1.\n2.2 Testing\nFig 3 Common Mode Test Setup\nThe test samples are connected to a vector network analyser ( OMICRON Lab - Bode 100)\nconfigured in one-port impedance measurement mode as shown i n Figure 3and Figure 4to mea-\n3Fig 4 Differential Mode Test Setup\nFig 5 Liquid Nitrogen Immersion Setup\nsure common mode and differential impedance over frequency both at room temperature (approx-\nimately 293K) and liquid nitrogen temperature (77K at stand ard pressure). The chosen vector\nnetwork analyser is capable of directly plotting the impeda nce magnitude trace over frequency.\nThe supported frequency range is up to 40MHz. For the cold tem perature test, the entire sample is\nmanually submersed in liquid nitrogen as shown in Figure 5until steady state is reached.\n3 Results\nTest results are presented as Bode plots of impedance magnit ude over frequency in Figure 6since\nthese plots are often found in the data sheets of commercial c ommon mode chokes which allows\ncomparison with other parts.\nIt is observed that at room temperature, all samples show the typical impedance curve of com-\nmon mode choke inductors. In differential mode the impedanc e is clearly lower than in common\n4Fig 6 Measurement Results\nmode since magnetic flux induced by both windings cancels out . A clear resonance peak caused\nby the parasitic winding capacitance is present. In common m ode operation, the nanocrystalline\nchokes exhibit higher impedance than the traditional core m aterials. This is in line with expecta-\ntions compared to commercially available nanocrystalline inductors. The performance advantage\nis offset by higher cost. At cryogenic temperature it become s apparent that the common mode\nimpedance of the traditional NiZn cores degrades until the p oint where it is almost indistinguish-\nable from the differential mode impedance. This indicates t hat the core material is no longer\neffective, confirming the result stated in Ref. [ 1]. The nanocrystalline cores maintain performance\nexcept for an insignificant decrease in impedance. It is also observed that the DC resistance visible\n5in the low frequency range of the differential impedance cur ves decreases with temperature, as\nexpected.\n4 Discussion\nThe experiments show that common mode choke inductors based on commercially-available nanocrys-\ntalline core materials are suitable for operation at liquid nitrogen temperatures since the core mate-\nrial maintains its magnetic properties at cold temperature . Commercial inductors based on similar\ncore materials may be considered for application in detecto r control systems operating at cryogenic\ntemperatures.\nThe tested common mode chokes are most suitable for decoupli ng the preamp power supply\nlines due to the chosen core size and winding wire gauge. It ma y be advantageous to also common-\nmode filter signal inputs in the future. At the time of writing , no small-size nanocrystalline cores\nwere available for purchase that allow manual construction of sufficiently small data line common\nmode chokes. With the cooperation of a commercial magnetics manufacturer, it may be possible to\nadapt smaller signal line common mode chokes, possibly in su rface mount technology form factors,\nto nanocrystalline core materials. Due to the general perfo rmance advantage of nanocrystalline\ncores even at room temperature it is expected that magnetic c omponents using this technology will\nsoon become more widely available.\nLong-term reliability remains to be demonstrated, but it is promising that none of the cores or\nwindings were damaged by repeated immersion in liquid nitro gen during this work. In general, it\nis advisable to cool down ferrite materials slowly due to the risk of cracking. To prevent contami-\nnation of the cryostat in case a core does fracture it is also a dvisable to use epoxy coated cores to\nkeep fragments from separating.\n6Before inclusion in a scientific detector system, other para meters of the chokes such as winding\nresistance, DC bias current, thermal effects, and self-res onance need to be characterised. Consid-\neration of these parameters should be part of the standard de sign procedure for a common mode\nchoke and are expected to be unlikely to change at cold temper atures based on the test results\npresented here.\nReferences\n1 S. Yin, M. Mehrabankhomartash, A. J. Cruz, et al. , “Characterization of inductor magnetic\ncores for cryogenic applications,” in 2021 IEEE Energy Conversion Congress and Exposition\n(ECCE) , 5327–5333 (2021).\n2 J. J. Goedbloed, Electromagnetic compatibility , Prentice Hall, New York (1992). Translation\nof: Electromagnetische compatibiliteit.\n3 B. R. Brandl, M. Feldt, A. Glasse, et al. , “METIS: the mid-infrared E-ELT imager and spectro-\ngraph,” in Ground-based and Airborne Instrumentation for Astronomy V , S. K. Ramsay, I. S.\nMcLean, and H. Takami, Eds., 9147 , 914721, International Society for Optics and Photonics,\nSPIE (2014).\n4 N. A. Thatte, I. Bryson, F. Clarke, et al. , “HARMONI: first light spectroscopy for the ELT:\ninstrument final design and quantitative performance predi ctions,” in Ground-based and Air-\nborne Instrumentation for Astronomy VIII , C. J. Evans, J. J. Bryant, and K. Motohara, Eds.,\n11447 , 114471W, International Society for Optics and Photonics, SPIE (2020).\n5 R. Davies, J. Schubert, M. Hartl, et al. , “MICADO: first light imager for the E-ELT,” in\nGround-based and Airborne Instrumentation for Astronomy V I, C. J. Evans, L. Simard, and\nH. Takami, Eds., 9908 , 99081Z, International Society for Optics and Photonics, S PIE (2016).\n76 H. Gui, R. Chen, J. Niu, et al. , “Review of power electronics components at cryogenic temp er-\natures,” IEEE Transactions on Power Electronics 35(5), 5144–5156 (2020).\nMathias Richerzhagen is a detector electronics engineer at the European Southern Observatory.\nHe received his Engineering Diploma from RWTH Aachen Univer sity in 2012. His current re-\nsearch includes development of the detector controller for the ELT as well as some work in cryo-\nelectronics.\nBiographies and photographs of the other authors are not ava ilable.\nList of Figures\n1 Segmented Winding Common Mode Choke\n2 Common Mode Choke Samples\n3 Common Mode Test Setup\n4 Differential Mode Test Setup\n5 Liquid Nitrogen Immersion Setup\n6 Measurement Results\nList of Tables\n1 Common Mode Choke Sample Parameters\n8"    },    {        "title": "1709.09965v1.Magnetization_and_Anisotropy_of_Cobalt_Ferrite_Thin_Films.pdf",        "content": " 1 Magnetization and Anisotropy of Cobalt Ferrite Thin Films  \n \nF. Eskandari1,2, S.B. Porter1, M. Venkatesan1, P. Kameli1,2, K. Rode1 and J.M.D. Coey1 \n \n1School of Physics and CRANN, Trinity College, Dublin 2. Ireland.  \n2Department of Physics, Isfahan University  of Technology, Isfahan 84156 –83111, Iran.  \n \n \nAbstract:  \n \nThe magnetization of thin films of cobalt ferrite frequently falls far below the bulk value of \n455 kAm-1, which corresponds to an inverse cation distribution in the spinel structure with a \nsignifican t orbital moment of about 0.6 µ B that is associated with the octahedrally -coordinated \nCo2+ ions. The orbital moment is responsible for the magnetostriction and magnetocrystalline \nanisotropy, and its sensitivity to imposed strain. We have systematically inv estigated the \nstructure and magnetism of films produced by pulsed -laser deposition on different substrates  \n(TiO 2, MgO, MgAl 2O4, SrTiO 3, LSAT, LaAlO 3) and as a function of temperature (500 -700C) \nand oxygen pressure (10-4 – 10 Pa). Magnetization at room -temperature ranges from 60 to 440 \nkAm-1, and uniaxial substrate -induced anisotropy ranges from +220 kJm-3 for films on \ndeposited on MgO  (100)  to -2100 kJm-3 for films deposited on MgAl 2O4 (100) , where the room -\ntemperature anisotropy field reaches 14 T.  No rearrangement of high -spin Fe3+ and Co2+ cations \non tetrahedral and octahedral sites can reduce the magnetization below the bulk value, but a \nswitch from  Fe3+ and Co2+ to Fe2+ and low -spin Co3+ on octahedral sites will reduce the low -\ntemperature magnetization to 120 kAm-1, and a consequent reduction of Curie temperature can \nbring the room -temperature value to near zero. Possible reasons for the appearance of low -spin \ncobalt in the thin films are discussed.  \n \nKeywords; Cobalt ferrite, thin films, pulsed -laser deposition, low -spin Co3+, strain engineering \nof magnetization.  \n  2 1 Introduction  \nThe spinel ferrites are an important family of insulating ferrimagnetic oxides, widely \nused as soft high -frequenc y magnetic materials. Their general formula is MFe 2O4 where the \niron is ferric Fe3+ and M is a divalent transition metal cation, such as Mg2+, Mn2+, Fe2+, Co2+, \nNi2+ or ½(Li++Fe3+) [1]. All the ferrites are ferrimagnetic insulating with a high Curie \ntemperature Tc, and. with the exception of Fe 3O4 all are insulating. None except CoFe 2O4 \nexhibits much anisotropy.  \nThe cubic spinel structure, space group Fd 3̅m illustrated in Fig. 1, is formed of a cubic \nclose -packed array of ox ygen anions slightly displaced from their ideal positions, with the \ncations occupying tetrahedral 8 a sites [A -sites] , which have cubic  4̅3m point symmetry and \noctahedral 16 d sites {B -sites} , which have trigonal 3̅m point symmetry.  There are 56 atoms in \nthe unit cell. The ‘normal’ cation distribution has the divalent cations on A -sites, but the \n‘inverse’ distribution is more common, where one of the two ferric ions occupies the A -sites, \nand the  other ferric ion and the  M2+ cations are on B -sites. The ferri te of interest to us here is \nCoFe 2O4 (CFO), which usually has a near -ideal inverse cation distribution in the bulk  \n[Fe3+]{Co2+Fe3+}O4,        (1)              \nwith Fe3+ cations occupying the 8 a sites. The cation distribution can be modified by heat \ntreatme nt [2], and quenching increa ses the occupancy of A -sites by Co2+. The accepted value \nof the lattice parameter is 839.2 pm, although the value varies slightly with the sample \nstoichiometry and preparation method.  \nFe3+ (3d5; t2g3eg2) has S = 5/2 and a spin moment of 5 µ B. High -spin Co2+ (3d7 t2g5eg2) \nhas S = 3/2 and a spin moment of 3 µ B, but in B -sites the cobalt can also ha ve a significant \nunquenched orbital moment of ~ 0. 6 µB [3-5], which is responsible for the strong cubic \nanisotropy K1c ≈ 290 kJm-3 with <100> easy directions. Although the moment of an isolated  3 Co2+ ion aligns along a local <111> trigonal axis [3], the resultant bulk anisotropy lies along \n<100> [6]. The net moment is about 3.6 µ B per formula at room temperature (RT), and the \nmagn etization of bulk samples is 455 kAm-1 or 86 Am2kg-1 (86 emu/g) , based on the X -ray \ndensity of 5290 kgm-3. The ferrimagnetic Néel temperature of CoFe 2O4 is 790 K, so the ground \nstate, T = 0 values are slightly higher.  \nAnother consequence of the unquenched orbital moment on the Co2+ is an \nexceptionally -large magnetostriction. Originally measured by Bozorth in 1955 [7], the value \nof (3/2)100 for a Co 0.8Fe2.2O4 crystal was found to be -885 ppm, corresponding to  a \ntetragonality ( a|| - a)/a of almost 1 %. The tetragonality has been measured in unsaturated bulk \nmaterial by synchrotron X -ray diffraction [8]. A similar value of magnetostricton ( -845 ppm) \nwas recorded recently for a crystal of nominal composition CoFe 2O4 [9], but, the \nmagnetostriction of a Co -rich crystal, Co 1.1Fe1.9O4, was much less, ( -375 ppm) [7], and the \nvalues can be quite variable.  A result of the magnetostriction is that an i mposed strain ε along \n<100> leads to a uniaxial anisotropy [10]   \nKu ≈ (3/2)100εE        (2) \nwhere E is Young’s modulus (C 11), which is 257 GPa [11]. A 1% biaxial compression therefore \nleads to an easy -plane anisotropy Ku ≈ -2 MJm-3. In an alternative formulation for biaxially \nstrained cubic films is [12]  \nKu = (3/2)100 (C11- C12) (a|| - a)/a     (3) \nwhere C 12 = 106 GPa [11]  \nThe magnetic anisotropy of thin films of CoFe 2O4 is exceptionally -sensitive to substrate -\ninduced stra in [10, 13 -18].  4 Thin films of CoFe 2O4 with good chemical and mechanical stability have attracted \nsome attention as potential perpendicular recording media [19], as tunnel -barrier spin filters \n[20-22] due to the spin -dependent bandgap [23], as magneto -optic media [1] and as \nmagnetostrictive films [24]. \nThere have been many reports of preparation of CoFe 2O4 in thin film form using pulsed -\nlaser deposition (PLD) [10, 13 -15, 19, 24 -31], and the literatur e also includes reports of films \nproduced by rf sputtering [16, 17, 32 -34], MBE [18, 35 -37], CVD [38, 39]   and ALD [40]. \nAuthors use a variety of substrates, deposition temperatures, oxygen pressures, laser fluence, \nsample thickness and thermal treatments, and find a wide range of magnetization, anisotropy \nand hysteresis. A summary of some o f the earlier PLD work is provided in Table 1. What is \nremarkable is that the magnetization found at room -temperature is usually much less than the \nbulk value, and only comes close to it in a few instances. This is puzzling, because we cannot \nlower the mag netization by rearranging the Co2+ and Fe3+ cations on A - and B -sites in a \ncollinear ferrimagnetic structure.  \nIn this work, we have systematically investigated the effects of thin film growth \nconditions on structural and magnetic properties of CFO thin fil ms deposited by PLD on \nvarious substrates, principally with the aim of explaining the anomalously small values of \nmagnetization that are usually found in CFO films, but often ignored by plotting the y -axis of \nthe magnetization curves in reduced units. We d iscuss our results in terms of the presence of \nlow-spin Co3+ ions on B -sites, such as are found in Co 3O4 [41].  \n   5 Table 1. Reports of magnetization of CoFe 2O4 films prepared by pulsed laser deposition.  \n  \nSubstrate  Substrate \nTemperature, \nTs  (oC) Thickness \nt (nm)  Oxygen \nPres sure \n P (Pa) Magnetization  \nM (kAm-1)* Reference  \nMgO (100)  600 \n800 400 4 335 \n426 [25] \nSTO//CoCr 2O4 \n 600/1000p  140-430 0.1 380 [10] \nMAO  500/1000p  65-900 0.1 ? [26] \n \nMgO (100)  \nSTO (100)  700 \n550 80 10-5 180 – 220 \n370 \n [13] \n \nAl2O3 (0001)  \nSiO 2 800 \n550 40-200 \n33 0.2 ? \n260 [19] \n \n \nSTO (100)  500-700 70 0.002 -0.01 ? [27] \n \nMAO(100)  175-690 200-220 1.3  \n(15% \nozone)  450 ( 5K)  [28] \n \n \nMgO(100)  \nSTO(100)  450 200 1.3 300 \n140/4 80 [29] \n \n \nSi(100)/SiO 2 250-600 \n250 135 2.9 \n0.7-7.0 \n 130-270 \n130-220 [24, 30]  \n \nMgO(100)  \nSTO(100)  \nBTO(100)  \nLAO(100)  ? 13-100 7 260 \n390 \n210 \n280 \n [14] \n \n \nMgO(100)  \n 400 50-400 2 100-185 [15] \n \n \nPt(111)  550-750 247-290 9 180-220 [31] \n \n* 1 kAm-1 is equivalent to 1 emu/cc or 1.26 G  p post annealed  \n   6 2. Experimental Methods  \nThe films were deposited by PLD onto various single -crystal substrates  from a sintered \nCoFe 2O4 target prepared by sol -gel synthesis. A KrF excimer laser (248 nm wavelength with \n25 ns pulse width, Lambda Physics) was used to ablate the ceramic target with a laser fluence \nof about 2  J cm-2 and a pulse repetition rate of 10 Hz.  The distance between the substrate and \ntarget was fixed at 6 cm. Before deposition, the chamber was evacuated to 2×10-4 Pa and the \nsubstrate was heated to 900 C for 1 hour, followed by cooling to the required deposition \ntemperature Ts. After deposition, the films were cooled to room temperature at a rate of about \n5C min-1 at constant oxygen pressure.  They were normally deposited at 600  C, but some \nfilms wer e deposited at higher or lower temperature (500 -700C). The oxygen pressure was \nvaried in the range 2 × 10-4 – 10 Pa. In addition, we have deposited films on many other \nsubstrates – MgO (100), MgO (110), MgO (111), SrTiO 3 (STO) (001), MgAl 2O4 (MAO) (001),  \nLaAlO 3 (LAO) (001), (La,Sr)(Al,Ta)O 3 (LSAT) (100) and TiO 2 (001).  The objective was to \ndetermine optimum conditions, which would yield films of the highest quality, magnetization \nand anisotropy.  Reflection  high-energy electron diffraction (RHEED) was obse rved during and \nafter film growth.  2 X-ray diffraction (XRD), -scans and reciprocal space mapping \n(RSM) analysis were carried out to check the crystallinity, orientation and strain of the thin \nfilms using a Bruker D8 X -ray diffractometer (Cu K α1; λ = 154.05 pm). Film thicknesses were  \ndetermined by small -angle X -ray reflectivity (XRR) and confirmed in some cases by \nellipsometry. Morphology of the films was examined using contact -mode atomic force \nmicroscopy (AFM), and composition was checked by energy dispersive X -ray analysis (EDX). \nMost magnetic measurements were made using a 5  T SQUID magnetometer (MPMS 5 XL, \nQuantum Design) on films mounted in clear plastic straws with magnetic field applied parallel \nor perpendicular to the film plane. Measurements on selected films were made using a  7 vibrating -sample magnetometer in fields of up to 14  T (PPMS, Quantum Design) when the \nanisotropy field was very large.  \n \n3. Experimental Results  \n        We have examined 60 films of cobalt ferrite. Since there are so many experimental \nvariables, we make  progress by changing them one at a time — substrate temperature, oxygen \npressure, laser fluence, substrate material. Film thickness was generally in the range 30 – 50 \nnm; we did not investigate ultra -thin films, but we included one 15 nm film on MgO(100) under \npreferred conditions (600oC, vacuum).  Firstly, in order to find the optimum substrate \ntemperature ( Ts) for CFO thin film growth, two sets of samples were prepared  on MgO(100) \nsubstra tes, one under vacuum (2×10-4 Pa) and the other in an oxygen pressure of 2 Pa . In each \ncase the ma gnetization was greatest when Ts ≈ 600oC. The lattice parameter of MgO, which \nhas an ideal cubic close -packed oxygen lattice, is 421 pm or slightly more than half that of \nCFO (839 pm). The X -ray diffraction patterns of CFO thin films deposited under  vacuu m on \nMgO (100) substrates at different Ts of 500 - 700 C are shown in Fig. 2a,b. T he films are all \nhighly -oriented along the [00l] direction, as evidenced by the single CFO  (008) peaks visible \nin Fig 2. Our -scans showed four -fold in -plane symmetry, indi cating that the film growth on \nMgO is quasi -epitaxial. The CFO films on MgO are under in -plane tensile stress, so  the out -\nof-plane lattice parameter ( a = 837 pm) is smaller than the cubic bulk value . The tetragonality \nis ≈ -0.6 %. In Fig. 2b, it can be se en that the CFO  peak width decreases on increasing Ts from \n500 to 650 C.  \n        The oxygen pressure inside the chamber during film deposition is another important factor \ncontrolling the film structure and magnetization. Therefore, we deposited the CFO fi lms at \ndifferent oxygen pressures, ranging from vacuum (2×10-4 Pa) to 10 Pa at a fixed substrate  8 temperature of 600 C. All these films grow epitaxially. Fig. 1c shows that the position of the \n(008) peak shifts to lower angles with decreasing oxygen partial  pressure. Thus, the out -of-\nplane lattice parameter a and the tetragonality decrease with increasing oxygen partial \npressure.  \n      The strain state of the films has been investigated in more detail by reciprocal space \nmapping (RSM) around the MgO (113) and CFO (226) peaks for samples grown at 600 C in \nvacuum . The MgO substrates are often twinned, but different in -plane and out -of-plane lattice \nparameters are seen for CFO, as well as nonuniformity of the out -of-plane parameter across \nthe film thickness fo r all except the deposition in vacuum. The RSMs of Fig. 3 correspond to \nCFO films which are fully strained in the plane of the film with a|| = 842 pm. The out -of-plane \nlattice parameter a is uniform across the film thickness (56 nm) for CFO deposited at 6 00C \nin vacuum  (Fig. 3a) and it is equal to 837 pm.   There is an increasingly broad distribution of a \nwith increasing oxygen pressure. The double MgO spots seen for films deposited in 2 or 10 Pa \nof oxygen are simply due to substrate twins, but the vertica l streaking of the CFO (226) \nreflection indicates a heterogeneous distribution of a across the thickness. In 2 Pa, the \nvariation is from 832 to 838 pm, whereas for the 10 Pa film of similar thickness, the variation \nis from 832 to 842 pm. The vertical stra in is progressively relaxed in oxygen, but not in vacuum. \nFigure 3 also shows RHEED patterns of the same three films, which establish good crystal \nquality . RMS r oughness measured by AFM was < 1 nm. The RHEED indicates that the disorder \nof the surface incre ases with increasing oxygen pressure. This is suggested by the \ndisappearance of the faint Kikuchi lines and broadening of the diffraction pattern into streaks \ncorresponding to a decrease in long -range order at the surface in the plane of the film as the \npressure is increased to 2 Pa, and the collapse of the pattern into spots at 10 Pa, possibly  9 indicating the existence of ordered islands, or the result of transmission through small \ncrystalline structures on the surface.  \n The composition of selected films ha s been checked by EDX analysis. The ceramic \ntarget was stoichiometric, with an Fe:Co ratio of 1.98(2). Analysis at multiple points on four \nfilms deposited on MgO showed that their composition was uniform, but with a slightly \ndifferent Fe:Co ratio of 1.80(2 ) indicating that the films  are Co -rich.  \n            Figure 4 show s room -temperature magnetization (M -H loops) of CFO thin films \ndeposited on MgO (001) in vacuum at substrate temperatures of 500 and 600 C. The saturation \nmagnetization in both cases is sig nificantly less than the value of 86 Am2kg-1 (455 kAm-1) \nmeasured for the bulk ceramic target used for the PLD. Plots of Ms vs Ts for films deposited in \nvacuum and in a 2 Pa pressure of oxygen are shown in Fig. 5.  The moments in vacuum are \nlarger, but in  both cases, the maximum falls at Ts = 600C, which is why we limit further studies \nto films produced at this temperature. We find that there is some effect of laser fluence. \nIncreasing the fluence from 2.1 to 2.7 Jcm-2 leads to an increase of Ms from 397 kAm-1 to 453 \nkAm-1, which is the bulk valu e. The magnetization of these films on MgO lies perpendicular \nto the film plane, with coercivity of up to 400 kAm-1 for the high -moment films. Coercivity in \nthe low -moment films is greater. The magnetization curve of the 453 kAm-1 sample, which we \ncan rega rd as a benchmark, is included in Figure 8. There is no hysteresis when the field is \napplied in -plane, but four -fold anisotropy is associated with the tetragonal symmetry of the \nfilm.  The intrinsic perpendicular anisotropy Ku can be estimated from the eq uation  \n Ku  =  K -Ks        (4) \nwhere the parameter Kis deduced from the area between the perpendicular and parallel \nmagnetization curves, and the shape anisotropy Ks is -½μ0Ms2  The anisotropy energy \nassociated with each term is of the form E = Ksin2θ, where θ is the angle between the direction  10 of uniform magnetization and the film normal.  From the data in Fig. 4, Ms =  397 kAm-1 so Ks \n= - 99 kJm-3. The measured value of Kis 127 kJm-3, hence it follows that Ku = 226 kJm-3. \nSubstantially larger values of Ku have been reported by Niizeki for samples with very similar \nmagnetization curves using torque measurements [42] or simple estimates of the anisotropy \nfield µ 0Ha = 2Ku/Ms [16]. Our estimates of Ku from magnetization curves that exhibit hysteresis \ncan underestimate Ku by up to 35% because of effects of magnetic viscosity in hysteretic  easy \ndirection.  \n Figure 6 shows magnetization curves for samples deposited at 600 C in two diffe rent \noxygen pressures. The oxygen decreases the magnetization, as summarized in Fig. 7. The \nmagnetization of the sample grown in an oxygen pressure of 10 Pa is only 60 kAm-1. This is \npartly because the Curie temperature of the sample is rather low. The oxy gen pressure \ndependence is opposite to that found in NiFe 2O4 and YIG [43, 44] . From the temperature -\ndependence of Ms, Tc is estimated to be about 450 K. The magnetization of these low -moment \nfilms on MgO is perpendicular to the plane, and the coercivity becomes very large at low \ntemperature, exceeding 5 T at 100 K. The 15nm film prepared at 600 C in vacuum is different \nto all the other, thicker films, insofar as the moment lies in -plane. Its magnetization is 389 \nkAm-1. \n Next, we turn to results for the other substrates. Films deposited at 600 C in vacuum \non the other cuts of MgO, (111) or (110), show lower magnetization than on (100), 280 kAm-\n1 and 254 kAm-1, respectively, with less pronounced anisotropy. The results for ot her \nsubstrates, shown in Figure 8, are more interesting. Films on TiO 2 (100)  and LAO  (100)  \nexhibited magnetization of 444 kAm-1 and 412 kAm-1 respectively, close to the bulk value, but \nthese substrates have a large lattice mismatch of ±10% so there is no e pitaxy. The films are  11 polycrystalline. Magnetization lies in -plane and the anisotropy is due to shape, with negligible \nKu contribution from substrate -induced lattice strain.  \n        The films on spinel, MgAl 2O4 substrates, are subject to 3.7% lattice mism atch, leading to \nstrong perpendicular expansion. The RSM illustrated in Fig. 9 shows that the in -plane lattice \nparameter does not follow the template provided by the spinel lattice, but there is nonetheless \na broad, correlated distribution of lattice param eters centred at a|| = 831 pm and a = 850 pm, \ncorresponding to an in -plane compression of  1%, and a similar perpendicular expansion. The \ntetragonality is 2%. Magnetization curves plotted in Fig. 8 show a magnetization of CFO on \nMAO of 315 kAm-1, with a coercivity of 380 kAm-1 and an enorm ous in -plane anisotropy \nestimated from the area between the curves as Ku = -1.12 MJm-3, far exceeding the shape \nanisotropy Ks = - 62 kJm-3.  The anisotropy field μ 0Ha is 14 T, and the anisotropy energy, \nestimated simply as -½μ0HaMs is -2.2 MJm-3. Both Ms and Ku are reduced in magnitude in a 2 \nPa oxygen atmosphere, to 262 kAm-1 and -0.58 MJm-3, respectively. Films deposited on LSAT \nbehave similarly, with strong negative Ku = -0.65 MJm-3 for films made in vacuum.  \n       STO is different. Here the film is ori ented, but the magnetization  is only 40% of the bulk \nvalue. The perpendicular lattice parameter expands a little, but the net anisotropy is \nperpendicular to the plane, as for MgO, but very weak. The effect of substrate strain is 20 – 40 \ntimes less than for  MAO or LSAT, and of opposite sign. The anisotropy and magnetization of \nthe films deposited on different substrates is summarized in Table 2.  \n3 Discussion.  \n3.1 Anisotropy.  \nWe confirm the idea, originally proposed for films on MgO and MAO [10, 36]  and \nconfirmed by electronic structure and crystal field calculations [23, 45 -49]that the magnetic \nanisotropy in cobalt ferrite films is largely governed by substrate -induced strain. All our films  12 show 001 oriented growth, except those deposited on TiO 2 (001) or LaAlO 3 (100) substrates, \nwhich have in -plane lattice parameters that are completely mismatched to that of CoFe 2O4, \nbeing about 10% bigger or smaller. In these two cases, there is effec tively no substrate strain, \nand shape anisotropy dominates (Table 2). The magnetization of these unconstrained films is \nclose to that of bulk CFO. They exhibit easy -plane magnetization with Ks ~ 100 kJm-3 and \nisotropic coecivity, Hc ≈ 120 kAm-1.  \nTable 2. Anisotropy of cobalt ferrite films deposited at 600 C in vacuum on different 100 \noriented substrates.  \nSubstrate  a Parameter \n(pm)  Growth  Ms \n(kAm-1) Ks \n(kJm-3) Ku \n(kJm-3) Anisotropy  \nTiO 2 919 Unconstrained   413  -107    ~ 0 Shape  \nin-plane  \nMgO  842 Quasi -epitaxial   397    -99    226 Strain  \nperpendicular  \nMgAl 2O4 808 Oriented, \nstrained   317    -63 -1120  Strain  \nin-plane  \nSrTiO 3 782 Oriented   183    -21   ~ 30 Strain, \nweakly  \nperpendicular  \nLSAT  775 Oriented, \nstrained   270    -46   -650 Strain  \nin-plane  \nLaAlO 3 758 Unconstrained   444  -123     ~ 0 Shape  \nIn-plane  \nThe blue line separates the parameters that are greater than or less than that of CFO, 839 pm.  \nNext come the MAO and LSAT substrates where there is still a large lattice mismatc h, \nbut there is now clear 001 axis orientation of CFO, with the wide spread of correlated a|| and \na parameters, shown in Fig. 9. The CFO is significantly strained, and according to Eq.3 the \ntetragonality of 2 .2% should lead to uniaxial hard -axis anisotropy of -2.9 MJm-3 on account of \nthe cobalt magnetostriction, However, the magnitude of 100 in CFO is often much less than \nthe frequently -cited 590 ppm [7], and the resulting Ku will be correspondingly reduced. For  13 example, a value s = 225 ppm reported for polycrystalline CFO [50] corresponds to 100 =  \n256 ppm . The tetragonali ty of CFO on LSAT is 1.9 %. \nThe origin of 100 is the high -spin Co2+ on B-sites, so we can expect that there are \nsignificant variations in the amount of this ion in different films, which provides  a link between \nanisotropy and magnetization. High -spin Co2+ on A -sites does not contribute to the anisotropy \nbecause there is no orbital moment in cubic 4̅3m point symmetry. In DFT calculations, cation \ndistributions were fo und to be sensitive to epitaxial strain [51]. Furthermore, a study of the \ninfluence of substrate strain on Ku in CoCr 2O4, a spinel with the normal cation distribution, \nreveals that its sign is opposite to that for CoFe 2O4 [52].   \nMagnetoelastic strain -related anisotropy of CFO on MAO has been discussed \npreviously [34], including  a study of 5 nm MBE films which were found to grown epitaxially \non MAO [36], but the extremely large easy -plane anis otropy we have observed on spinel is \nunprecedented; the anisotropy field is more than double anything reported previously [36, 5 3]. \nThat for CFO on LSAT is also very large. The magnetization of CFO on both substrates is \nabout 2/3 of the bulk value.  \nSTO is a different story. Here the perpendicular 004 and 008 X -ray reflections are \nbroad, but the perpendicular lattice parameter of 841.8 pm indicates a dilation of just 0.3%. \nFurthermore the magnetization is much reduced, to  40% of the bulk value. The shape \nanisotropy is therefore weak, Ks = -21 kJm-3, and it appears from the magnetization curves in \nFig 8 that the uniaxial anisotropy Ku is of similar magnitude but opposite sign, leading to a \nfeeble perpendicular anisotropy Kfor the film. The anisotropy mechanism based on the \nmagnetostriction of B -site Co2+ does not apply in this case. We think the small moment signals \na cation distribution that is quite different to that of bulk CFO.   14 The films grown on MgO at 600oC in vacuum are of good epitaxial character according \nto the RSM of Fig 3a, and the RHEED pattern of Fig. 3d which shows traces of Kikuchi fringes. \nRMS surface roughness for the 56 nm thick film is  0.7 nm. The films have 85 % or more of the \nbulk magnetization, and the tetragonality of -0.6% corresponds to a positive Ku of 800 kJm-3 \naccording to eq. 3, significantly more  than our measured value of Ku of 226 kJm-3 (Table 2). \nThe discrepancy may arise because the value of (3/2)100 in our films is reduced to about 250 \nppm, or because  our method of evaluating Ku from the magnetization curves underestimates \nthe value.  Torque measurements generally  give greater  values  [16, 34 ]. Figure 1 2 summarize s \nthe relation between tetragonality  or perpendicular strain  and anisotropy Ku in the films.  \nWe have not performed a systematic study of anisotropy vs film thickness for CFO \nfilms on MgO, but we found that a 15 nm film was easy -plane, whereas films with 3 0 ≤ t ≤ 70 \nnm are all easy -axis. A low -moment 5 nm MBE film was also found to be easy -plane [36]. \nEasy -axis anisotropy persists up to 300 nm, but the thickest PLD films become easy -plane \nagain as the strain is eventually relaxed, and the structure  become s cubic [54]. We should \ntherefore include another term i/t in the anisotropy expression (4) to take account of the \ninterface an isotropy; K= Ku-Ks + i/t. From the first spin reorientation thickness, we \nestimate that i is easy -plane and approximately - 6 mJm-2.  Figure 10 sketches the evolution \nof the anisotropy of CFO films on MgO, marking the dominant anisotropy term in e ach of three \nthickness regimes.  \n3.2 Magnetization.  \nA remarkable feature of the cobalt ferrite films produced by PLD and other methods is \nthe magnetization, which frequently falls short of the bulk value and is remarkably sensitive to \nsample preparation con ditions. A well -known consequence of thermal treatment of bulk ferrites \nis to induce deviations from the ideal inverse cation distribution [Fe3+]{Fe3+Co2+}O4 by moving  15 some Co2+ ions over to A -sites, and an equal number of Fe3+ ions onto B -sites. The formu la is \nthen   \n[Fe3+1-x Cox]{Fe3+1+x Co2+1-x }O4      (5) \nand the moment is thereby increased from 3.5µ B per formula unit to 3.5(1 + x)µ B, assuming an \nA-site Co moment of 3µ B.  In any case, no permutation of these cations can reduce  the net \nmagnetization of a collinear ferrimagnetic spin structure. The slight excess of cobalt detected \nin the EDX analysis of the films, which was independent of the preparation method, does \nreduce the magnetization a bit. The inverse formula is then [Fe3+]{Fe3+1-y Co2+1+y }O4, where \nan Fe/Co ratio of 1.8 corresponds to y = 0.07. The magnetization falls to 80% of the bulk value, \nbut permuting the cations only increases the value.  \nHow then can we get the low values of room -temperature magnetizati on seen in Table \n2 and Figs 5,7 . The average of Ms for 35 different CFO films was 240 kAm-1, just 53% of the \nbulk value. There are several possibilities to consider, namely low magnetic ordering \ntemperatures, noncollinear magnetic structures including the effects of antiphase boundaries, \nlow-spin cobalt and oxygen stoichiometry  \nMagnetic ordering temperature . The great majority of our magnetization measurements were \nmade at room -temperature. The reason for not measuring systematically at low temperature is \nthe Curie law paramagnetism of iron  impurities in the MgO substrates used for more than half \nthe depositions. The effect is small and linear above 100 K, but increasingly important and \nnonlinear below. Since Tc of CFO is 790 K, thermal effects at RT/ Tc  ≈ 0.37 are expected to \nreduce the magnetization by about 5% [55]. We checked the Curie temperature of the 600oC \nsample with the smallest magnetization, prepared in 10 Pa oxygen pressure (Fig 7), which was \nestimated to be roughly 460 K, from magnetization measured at temperatures up to 380 K.  For \nall other samples with reduced moment, such as those produced in an oxygen pressure of 2 Pa,  16 Tc is higher and finite temperature effect s have little impact on the  magnetization at  room -\ntemperature.  \nNoncollinear magnetic structures. An explanation commonly advanced for the low moments \nof spinel ferrite films deposited on MgO is that antiphase boundaries [17, 36, 37, 5 6] are \nincorporated into CFO during film growth. Originally proposed to explain the low \nmagnetization and slow approach to saturation of magnetite [57, 58], an antiphase boundary \nforms where two crystallites that have nucleated independently on MgO  (100) grow together. \nThere is therefore a plane of antiferromagnetic B – O – B interacti ons at the interface which \nare hard to overcome when the crystallites themselves are of nanoscale dimensions. These \neffects may well be present in our films grown at low temperatures, but the observation of near -\nbulk values of magnetization for samples gro wn at 600 C in vacuum (Fig 8a) and the high \ncrystalline quality of these films (Fig 3a) suggest that antiphase bondaries  may be important \nonly in films grown  at low Ts. These antiphase boundaries are not expected in films on MAO \nwhich has the same spinel s tructure and crystal symmetry as CFO, although  there is another \ntype associated with misfit dislocations that has been identified in Fe 3O4 films on MAO [59]. \n       Another possible reason for a reduced moment in an inverse spinel would be a noncollinear \nor canted spin structure of the B -site cations, which form the majority sublattice. The spins in \nzinc ferrite, for example, freeze in a random arra ngement below about 15 K, but ZnFe 2O4 is a \nnormal spinel, with nonmagnetic Zn2+ cations on the A -sites. The presence of A-site Fe3+ in \ninverse spinels ensures that the antiferromagnetic 135o A – O – B superexchange interactions \nare strong, and determine th e collinear Néel state [60].  \nThe next and most pl ausible explanation for the reduced moment relates to the spin \nstate of the cobalt. In the bulk, B -sites are populated by Fe3+ and high -spin Co2+, which is \nstabilized by the trigonal crystal field at the distorted octahedral sites, which have 3̅m symmetry   17 on account of the deviation of the oxygen 32 e site special position  parameter  u from 3/8. One \npossibility is that substrate -induced strain switches the Co2+ on B -sites to a low -spin state, 3 d7 \nt2g6eg1 with S = ½ and a spin moment of 1µ B. The Curie tempera ture may be similar, but the \nmagnetization will be much lower, ~ 130 kAm-1 or 24 Am2kg-1. Some admixture with the \nnormal spinel configuration [Co2+]{Fe3+2}O4 is possible, but as usual it will increase the \nmoment. Cation size, crystal -field stabilization en ergy and temperature are all factors that \ndetermine the spin state, but Co3+ is often low -spin on octahedral sites, where it adopts a stable \n3d6 t2g6 configuration with no moment, S = 0 [61]. The normal cobalt spinel Co 3O4 has \nnonmagnetic low -spin Co3+ on B-sites [41, 6 2]. Another possibility is therefore the replace ment \nof Fe3+ and high -spin Co2+ on B -sites by Fe2+ and low -spin Co3+. This also has the effect of \nreducing the net moment per formula to 1 µ B but it makes A -site Fe3+ the majority sublattice. \nFigure 11 illustrates the spin states of Co ions on the two site s in a one -electron energy level \npicture. The splitting of the t2g triplet  on octahedral B-sites corresponds to compression  of the \noctahedra  along the local <111>  axis [3] . Table 3 compares the ionic radii and the spin moments \nin tetrahedral and octahedral sites. It can be seen there that the mean B -site radius of 70 pm in \nthe inverse structure is reduced to 65 pm or 66.5 pm in with the first or second of these cobalt \nlow-spin hypotheses, marked in blue or green on the table, respectively. Any substrate which \nexerts imperfectly -relaxed compressive strain, those below the blue line in Table 2, may be \nexpected to favour some low -spin cobalt, which can account for the reduced moments observed \non MAO, STO and LSAT , and on MgO prepared in oxygen . \nThe correlation of the low moment observed in CFO films with l ow-spin cobalt means \nthat strain -induced anisotropy Ku is correspondingly reduced. Low -spin cobalt can provide an \nexplanation for the anomalous behaviour of the films on STO. They have a low moment of 185 \nkAm-1 and a greatly reduced anisotropy with a posit ive sign (see Table 2). Their low moment  18 suggests conversion of much of the cobalt to a low -spin state, which will reduce the strain - \ninduced anisotropy, and may even change its sign [16, 45, 48] . Furthermore, depositing films   \n \nTable 3.  Ionic radii (in pm) and spin states of Fe and Co cations on in tetrahedral or octahedral \noxygen sites  \n Fe2+ Fe3+ Co2+(Hs) Co3+(Hs) Co2+ (Ls) Co3+ (Ls) \n \nTetrahedral \nIonic r adius  \n \nspin  \n58    \n \n \n2  \n49/49/49     \n \n \n5/2  \n56   \n \n \n3/2  \n47    \n \n \n2  \n49           \n \n \n3/2  \n45  \n \n \n1 \n \n \nOctahedral \nIonic radius  \n \nspin  \n78 \n \n \n2  \n65/65  \n \n \n5/2  \n75 \n \n \n3/2  \n61    \n \n \n2  \n65 \n \n \n1/2  \n55 \n \n \n0 \n \n \non substrates like STO or MgO with an undeformed cubic oxygen sublattice may modify \noxygen special position parameter u, hence the trigonal distortion of the oxygen around the B -\nsites, thereby reducing the orbital moment and the magnetostriction 100 that controls Ku. \n Finally, we consider the effect of oxygen pressure, which strongly reduces the moments \nof films on MgO (Fig 7) and MAO. The spinel lattice is known to accommodate excess oxygen \nby means of B -site cation vacancies, oxidizing divalent  cations there to the trivalent state. The \ntextbook example is γFe2O3, which is really the cation -deficient spinel  \n[Fe3+]{Fe3+5/3 ☐ 1/3}O4,        (6) \nwhere ☐ is a vacant B -site. A similar effect in CFO leads to a formula [Fe3+]{Fe3+7/9 Co3+8/9 \n☐ 1/3}O4. If the Co3+ is low -spin, the ferrimagnetic net moment is 1.1 µ B and the A -sites form  19 the majority sublattice. Furthermore, nonmagnetic low -spin Co3+ means that 11/18 of the B -\nsites are nonmagnetic, which will reduce the magnetic ordering temperature to 350  K or less. \nWe can therefore expect oxygen pressure to reduce both Ms and Tc, although admittedly E q 6 \nrepresents an extreme case.  \n The plot in Fig. 12 shows Ku vs tetragonality.  The films p repared in vacuum with Ms \n> 300 kAm-1 show roughly a linear rela tion. These films have mostly high spin Co2+ on B-sites, \nwhich provides the strain -sensitive anisotropy. However, the two films with a low moment Ms \n< 200 kAm-1 (MgO at 2 Pa, STO) are different, we think because of the important fraction of \nlow-spin Co3+ in these materials.  \n4 Conclusions.  \n Our systematic broad -brush examination and analysis of cobalt ferrite thin films has \nrevealed a great variety of magnetic properties, often quite different to those of the bulk. The \ninfluence of the substrate is evidentl y critical, as are the deposition temperature and oxygen \npressure. Strain engineering is especially rich in CFO, because it works two ways. First is to \ninduce tetragonality in the films, which deforms the strong {100} cubic anisotropy K1c \nassociated high -spin Co2+ in octahedral sites to give a uniaxial anisotropy Ku, which may be \neasy-axis or easy plane, depending on the substrate. Second, a triaxial compressive strain can \nconvert cobalt from a high -spin to a low -spin state, thereby modifying the magnetization, Ms, \nshape anisotropy Ks and magnetocrystalline anisotrop y K1c.  There is therefore a new \nopportunity to modify anisotropy with strain. The example of CFO on STO shows the cobalt \nferrite teetering on an edge, where a slight strain -induced anisotropy change could have a big \neffect. This suggests suggests a  possib ility of using  piezoelectric substrates to switch, based on \ncobalt spin -state transitions and the related anisotropy   20   The next step should be to look carefully for the effects postulated using methods that \nexhibit atomic -scale sensitivity. These include e lectron microscopy with atomic scale \nresolution to evaluate the substrate -induced strain profiles and extended lattice defects, as well \nas spectroscopic methods such as XMCD to determine the cobalt spin and orbital moments,. \nCare will be needed to distingu ish near -surface ions which can be subject to a reduced crystal \nfield from those deeper in the film where the full crystal field will induce a low spin state. \nCo3O4 thin films will be useful reference samples here. Ferromagnetic resonance will be useful \nto evaluate the anisotropy fields and magnetizations independently.  \n         Cobalt ferrite illustrates the man y ways in which a thin film of an oxide containing Co2+ \ncan differ from bulk material, and it offers new opportunities for strain engineering of ma gnetic \nlayers in oxide electronics.  \nAcknowledgements. 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(a), (b) XRD patterns of CFO films on MgO(100) prepared at different substrate \ntemperatures Td in vacuum, and (c) at different pressures at  Td = 600 oC . \n \n \n \n \n \n \n \n20 30 40 50 60 70 80 90 100MgO(004)\n650 oC700 oC\n600 oC\n550 oCIntensity (a.u.)\n2 (degree)MgO500 oC\nMgO(002)\nCFO(008)\n90 92 94 96 98100 102 104 106 108 110MgO(004)\n650 oC700 oC\n600 oC\n550 oCIntensity (a.u.)\n2 (degree)MgO500 oC\nCFO(008)\n90 92 94 96 98100 102 104 106 108 110MgO (004)\nVacuum\n0.009  bar\n0.2 bar\n2 bar\n20 barIntensity (a.u.)\n2(degree)MgO100 bar\nCFO (008)(a) (b) (c)  25  \n \n \n \n \nFig. 3.  Reciprocal space maps about the (113) reflection showing in -plane and out of plane lattice \nparameters for CFO films deposited on MgO (001) at 600 C in (a) vacuum (a) 2 Pa c) 10 Pa and \nthe respe ctive RHEED patterns d), e), f).  \n \n \n \n \n \n \n \n \n 26  \n \n \n \n \n \n \n \n \n \nFig.4. Magnetization of CFO films deposited on MgO (001) in vacuum at 500 and 600 oC.  \n \n \n \n \n \n \n \n \n \n \n-5 -4 -3 -2 -1 0 1 2 3 4 5-400-300-200-1000100200300400\n600o CM (kAm-1)\n0H (T) in-plane\n out-of-plane500o C 27  \n \n \n \n \n \n \n \n \n \nFig.5. Magnetization as a function of substrat e temperature for CFO films deposited on MgO \n(001) in vacuum (solid circles), and in 2 Pa solid squares. Laser fluence was 2.1 Jcm-2, except for \nopen circle (2.7 J Jcm-2). \n \n \n \n \n \n \n \n \n \n500 550 600 650 7000100200300400500\n2 Pa(kAm-1)Vacuum\ns (oC) 28  \n \n \n \n \n \n \n \nFig.6. Magnetization of CFO films deposited on MgO (001) at 600 oC in 2 Pa  and vacuum. The \nlaser fluence was 1.8 Jcm-2.  \n \n \n \n \n \n \n \n \n \n \n \n-5 -4 -3 -2 -1 0 1 2 3 4 5-400-300-200-1000100200300400\nVacuumM (kAm-1)\n0H (T) in-plane\n out-of-planeP(O2) = 2 Pa 29  \n \n \n \n \n \n \n \n \n \nFig. 7. Room temperature magnetization for CFO films deposited on MgO (001) at 600 oC and \ndifferent oxygen pressures. Laser fluence was 2.1 Jcm-2. \n \n \n \n \n \n \n \n \n1E-5 1E-4 1E-3 0.01 0.1 1 100100200300400500Ms (kAm-1) \nP(O2) (Pa) 30  \n \n \n \n \n \n \n \n \n \nFig. 8. Room temperature magnetization of CFO films deposited on six different (100) substrates \nat 600C in vacuum. The graphs are plotted on the same vertical scale and expanded  horizontal \nscales to emphasize the influence of the different substrates.  \n \n \n \n \n \n \n \n \n \n-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14-500-400-300-200-1000100200300400500M (kAm-1)\n0H (T) in-plane\n out-of-planeCFO/MAO\n-5 -4 -3 -2 -1 0 1 2 3 4 5\n0H (T) in-plane\n out-of-planeCFO/LAO\n-5 -4 -3 -2 -1 0 1 2 3 4 5\n0H (T) in-plane\n out-of-planeCFO/STO\n-5 -4 -3 -2 -1 0 1 2 3 4 5-500-400-300-200-1000100200300400500M (kAm-1)\n0H (T) in-plane\n out-of-planeCFO/MgO-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14\n-5 -4 -3 -2 -1 0 1 2 3 4 5\n0H (T) in-plane\n out-of-planeCFO/TiO2\n0H (T) in-plane\n out-of-planeCFO/LSAT 31  \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 9 The reciprocal space map of CFO on MgAl 2O4 (100) grown under optimised \nconditions. MgAl 2O4 imposes a strong in -plane compression on the CoFe 2O4 film, which causes a \ndramatic elongation of the out -of-plane lattice parameter to conserve the volume  of the unit cell.  \n \n \n \n 32  \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 10. Magnetization direction for CFO films grown on MgO  (100)  as a function of film \nthickness. In the thinnest films, interface anisotropy is dominant, but in the thickest ones, strain \nrelaxation eliminates t he tetragonal magnetocrystalline anisotropy, and shape anisotropy then \npredominates.  \n \n \n \n \n \n \n \n \n \n \n1 10 100 1000      In-plane\nShape anisotropy Ks           In-plane\nInterface anisotropy i     Perpendicular\nTetragonal magneto-\ncrystalline anisotropy Ku \nt (nm) 33  \n \n \n \n \n \n \n \n \n \n \n \nFigure 1 1. One electron energy diagrams for Co2+ and Co3+ on tetrahedral or octahedral sites in \nhigh and low spin states.  \n \n \n \n \n \n \n \n \n 34  \n \n \n \n \n \n \n \n \n \n \n      Figure 12. Plot of uniaxial anisotropy vs tetragonality for CFO films on different substrates.  \n \n \n \n-1 0 1 2-1000-5000500\nEasy plane\nMAO LSAT LAO STO \nTiO2 MgO 2 Pa Ku (kJm-3)\nTetragonality (%)MgO Easy axis"    },    {        "title": "1704.04384v1.Thermostat_Influence_on_the_Structural_Development_and_Material_Removal_during_Abrasion_of_Nanocrystalline_Ferrite.pdf",        "content": "Thermostat In\ruence on the Structural Development and\nMaterial Removal during Abrasion of Nanocrystalline\nFerrite\nStefan J. Eder,\u0003,yUlrike Cihak-Bayr,yDavide Bianchi,yGregor Feldbauer,zand Gerhard Betz{\nyAC2T research GmbH, Viktor-Kaplan-Stra\u0019e 2, 2700 Wiener Neustadt, Austria\nzInstitute of Advanced Ceramics, Hamburg University of Technology, Denickestra\u0019e 15/K, 21073 Hamburg,\nGermany\n{Institute of Applied Physics, Vienna University of Technology, Wiedner Hauptstra\u0019e 8{10/134, 1040\nVienna, Austria\nE-mail: stefan.eder@ac2t.at\nPhone: +43 (0)2622 81600 161. Fax: +43 (0)2622 81600 99\nAbstract\nWe consider a nanomachining process of hard, abra-\nsive particles grinding on the rough surface of a poly-\ncrystalline ferritic work piece. Using extensive large-\nscale molecular dynamics (MD) simulations, we show\nthat the mode of thermostatting, i.e., the way that the\nheat generated through deformation and friction is re-\nmoved from the system, has crucial impact on tribo-\nlogical and materials related phenomena. By adopting\nan electron-phonon coupling approach to parametrize\nthe thermostat of the system, thus including the elec-\ntronic contribution to the thermal conductivity of iron,\nwe can reproduce the experimentally measured values\nthat yield realistic temperature gradients in the work\npiece. We compare these results to those obtained by\nassuming the two extreme cases of only phononic heat\nconduction and instantaneous removal of the heat gen-\nerated in the machining interface. Our discussion of\nthe di\u000berences between these three cases reveals that\nalthough the average shear stress is virtually tempera-\nture independent up to a normal pressure of approxi-\nmately 1 GPa, the grain and chip morphology as well\nas most relevant quantities depend heavily on the mode\nof thermostatting beyond a normal pressure of 0.4 GPa.\nThese pronounced di\u000berences can be explained by the\nthermally activated processes that guide the reaction of\nthe Fe lattice to the external mechanical and thermal\nloads caused by nanomachining.\nKeywords\nnanomachining, abrasive wear, polycrystal, molecu-\nlar dynamics, thermostat, heat conductivity, electron-\nphonon couplingNote: This document is the unedited Author's ver-\nsion of a Submitted Work that was subsequently ac-\ncepted for publication in ACS Applied Materials & In-\nterfaces , copyright c\rAmerican Chemical Society af-\nter peer review. To access the \fnal edited and pub-\nlished work see http://pubs.acs.org/doi/full/10.\n1021/acsami.7b01237 .\nPlease cite this work as:\nS. J. Eder, U. Cihak-Bayr, D. Bianchi, G. Feldbauer,\nand G. Betz, \\Thermostat In\ruence on the Structural\nDevelopment and Material Removal during Abrasion of\nNanocrystalline Ferrite,\" ACS Applied Materials & In-\nterfaces , DOI: 10.1021/acsami.7b01237, 2017.\n1 Introduction\nThe response of the near-surface microstructure of a\nwork piece or a system component subjected to abra-\nsive conditions can strongly in\ruence its properties and\nbehavior. This aspect of surface modi\fcation is of high\ninterest to the surface \fnishing industry, where mate-\nrial is deliberately removed from the surface to obtain\na desired roughness or texture, but also for tribological\napplications, where the { usually unintended { wearing\nof surfaces a\u000bects a system's performance and service\nlife. Depending on the relative velocities of the surfaces\nin mechanical contact, the friction and possible plastic\ndeformation occurring in the interface can lead to the\ngeneration of considerable amounts of heat, which has a\nlarge impact on the development of the microstructure\nclose to that interface.\nThe geometrical tolerances for components in various\napplications have increasingly entered into the nano-\nmetric length scale, e.g., in high-gloss \fnishing for opti-\ncal purposes1or in nano/micro-electromechanical sys-\ntems (NEMS/MEMS).2As the mechanisms governing\nmaterial removal and wear, but also those responsible\n1arXiv:1704.04384v1  [cond-mat.mtrl-sci]  14 Apr 2017for changes to the crystal structure have their founda-\ntions in the atomistic nature of the component or work\npiece, it makes sense to consider the respective processes\nfrom a nanoscopic point of view.\nThe shift in tribology towards the nano length scale\ntook place over the last decades and showed that com-\nmonly used macroscopic tribological models as well as\nlaws cannot be readily applied at this scale.3{5Partic-\nularly, nanoscale wear processes6{8are not fully under-\nstood because of their highly complex nature.9Over\nthe last years a high e\u000bort has been put into a bet-\nter understanding of single-asperity contacts, which can\nbe probed experimentally in detail using an atomic\nforce microscope (AFM).10This approach is of high im-\nportance from a fundamental point of view; however,\nmany realistic, nano-technological applications such as\nNEMS/MEMS are dominated by multi-asperity con-\ntacts.11,12In such systems nanowear poses a major lim-\nitation for the lifetime, performance, reliability, or the\noverall usability. Thus, it is imperative to investigate\nsuch multi-asperity contacts exhaustively. A further\npressing issue regarding contacts at the nanoscale is the\ncalculation of the real contact area. Commonly applied\ncontinuum mechanics approaches are hardly applicable\nat this length scale;13therefore, various methods have\nbeen proposed to calculate the contact area based on\natomistic principles, see for example refs 12,14,15.\nAs addition to experimental techniques and to obtain\na more complete picture, classical molecular dynam-\nics (MD) simulations have proven very useful to simu-\nlate nanoscale tribological systems.10Modern comput-\ning power allows tracking the time development of fully\natomistic systems consisting of several million atoms\nwithout the a priori assumption of constitutive equa-\ntions governing the systems' behavior, see refs 16,17\nand the references therein for some examples. One im-\nportant aspect of non-equlilbrium MD (NEMD) sim-\nulations, i.e., where energy is continuously introduced\ninto the system via external constraints such as im-\nposed forces or velocities, is the thermodynamically cor-\nrect removal of the generated heat. Several concepts\nhave been proposed to do this in a manner consistent\nwith some thermodynamic ensemble, such as the Nos\u0013 e-\nHoover,18,19the Berendsen,20or the Langevin ther-\nmostats,21,22to name only a few.\nThe feasibility and quality of a NEMD simulation\nis strongly a\u000bected by the choice of the thermostat as\nwell as its subsequent con\fguration. Particularly, this\nincludes the decisions on which parts of the system\nthe chosen thermostat acts upon and on the coupling\nstrength between system and thermostat. In the fol-\nlowing, three distinct scenarios will be presented. For a\nnanomachining simulation as considered in this work it\nwould, in principle, make sense to apply the thermostat\nonly to a part of the system some distance away from\nthe tribological interface. Such a setup would minimize\nthe interference with the physics of the processes oc-\ncurring close to the surface. This implies that there is\na large heat sink attached to the base of the explicitlyconsidered near-surface region, namely the bulk. Un-\nfortunately, typical interaction potentials modeling the\nbehavior of metals, such as EAM23or Finnis-Sinclair,24\ndo not explicitly include electrons and are therefore not\nable to su\u000eciently reproduce the thermal conductivity\nof a metal. As the modeled systems grow ever larger,\nresulting in larger distances between contact zone and\nheat sink, the grossly overestimated temperature gra-\ndients additionally lead to highly unrealistic absolute\ntemperature di\u000berences. We will henceforth refer to this\nscenario as \\base-thermostatted\".\nAn alternative approach is to thermostat the entire\nwork piece. Most frequently, the thermostat is strongly\ncoupled to the atoms so that heat is almost instanta-\nneously transported away. While this certainly allows\none to maintain good temperature control over the work\npiece, one may completely neglect the thermal in\ruence\nof the modeled mechanical process itself by eliminating\nheat conduction altogether. We will call this scenario\n\\fully thermostatted\".\nWhen it comes to atomistic simulations of systems\nfeaturing considerable heat transfer through metals,\nmany authors do not mention which thermostatting\nmethod they use to control the system's temperature.\nIf they do, they seldom disclose how they parametrized\nthe thermostat or justify why they chose the particular\nparametrization. Without any claim to completeness,\nwe list some examples of otherwise high-quality sim-\nulations that are very likely underthermostatted25{28\ndue to base-thermostatting or possibly overthermostat-\nted29{32due to full thermostatting. Out of these ex-\namples, it seems that only Shiari et al.30are aware of\nsome implications of their simpli\fed thermostatting ef-\nforts and concede that another group reported that ma-\nterial removal is greatly in\ruenced by the thermal con-\nditions in the shear zone.33\nGill34gives a good overview of NEMD as well as mul-\ntiscale modeling of heat conduction in solids. However,\nmost of this extensive review focuses on non-metallic\nmaterials. The presented methods intended for met-\nals are either quasi-static, such as the two-temperature\nmethod mainly used for modeling the laser annealing\nof voids,35,36while others require a multiscale approach\nto the problem via dynamic coarse graining37or featur-\ning coupling schemes to continuum.38The latter apply\ntheir ad hoc technique to an examination of frictional\nheating during sliding by solving the heat equation and\nimposing a thermal conductivity.\nIn this work, we adopt the concept of electron-phonon\ncoupling as laid out in refs 39,40. This does not re-\nquire us to implement a time-consuming multiscale ap-\nproach, but rather assumes that due to their high mo-\nbility within the metal, the electrons can be used as\nan implicit heat sink permeating the substrate. So far,\nthis setup is identical to what we described above as\n\\fully thermostatted\". The crucial point of putting the\nelectron-phonon coupling concept into practice is cor-\nrectly parametrizing the thermostat so that it re\rects\nhow the electrons in the metal interact with the lat-\n2tice vibrations. We will consider the two extreme cases\nof thermostatting only the substrate base (\\base ther-\nmostatting\") and thermostatting the entire substrate\nwith a strongly coupled thermostat (\\full thermostat-\nting\") and compare these two limiting scenarios to re-\nsults obtained with a carefully parametrized thermostat\nthat attempts to re\rect the electron-phonon coupling in\nthe metallic work piece as closely as possible. Addition-\nally, we check if considerably extending the heat treat-\nment period of the initial substrate con\fguration signif-\nicantly changes the results of the abrasion simulations.\nWe discuss how the application of the electron-phonon\ncoupled thermostatting scheme a\u000bects some typically\nevaluated tribological quantities such as friction, mate-\nrial removal/wear, contact area, and surface topogra-\nphy, as well as the microstructural development of the\nwork piece. This constitutes an important step towards\nmaking the results of atomistic non-equilibrium simu-\nlations of materials undergoing plastic deformation and\nsubjected to high temperature gradients more reliable.\n2 Modeling and Simulation De-\ntails\nAll our simulations were carried out using the open-\nsource MD code LAMMPS.41The exact procedure of\nour model construction is described in an earlier work.42\nIt consists of a rough 60 \u000260\u000220 nm3\u000b-Fe polycrystal\nincluding randomly oriented abrasives with Gaussian\nsize-distribution. A representative snapshot of the 3D\nmodel is shown in the top right corner of Fig. 1, along\nwith some diagonally arranged tomographs colored ac-\ncording to the grain orientation. The mean equivalent\ndiameter of the bcc Fe grains in the originally con-\nstructed 3D-periodic 60 \u000260\u000260 nm3cube is 12.7 nm,\ncorresponding to an average grain volume of 1080 nm3.\nDue to the introduction of the surface at z= 20 nm,\nout of the 128 remaining grains, the cleft ones no longer\nhave the shape and size of their original Voronoi cells.\nTherefore, the \fnal overall size distribution of the grains\nfeatures 40 grains with volumes of less than 200 nm3, a\nsize that was not present in the original size distribution\nat all, while 20 grains have sizes between 200 nm3and\n400 nm3, compared with only 7 in the original polycrys-\ntal. There are thus approximately 75 grains in the initial\nsubstrate that almost fully retain their original geome-\ntry. The grains are initially randomly oriented, and the\nsurface topography has a fractal dimension of 2.111, an\nRMS roughness of 0.7 nm, and a lower frequency cut-o\u000b\nproducing a typical lateral roughness feature extent of\n23 nm. The atoms located in the lower 3 \u0017A of the sim-\nulation box are kept rigid to emulate bulk support, and\nthe Fe{Fe interactions within the substrate are governed\nby a Finnis-Sinclair potential.43\nTwo slightly di\u000berent initial con\fgurations were pro-\nduced by subjecting them to two di\u000berent types of heat\ntreatments, abbreviated \\HT1\" and \\HT2\" henceforth,\nwith the temperature curves shown in Fig. 2. In bothheat treatments, a Langevin thermostat21,22controlled\nthe temperature of the entire substrate (abbreviated\n\\full\" in some \fgures) during the ramps, while during\nthe constant temperature intervals only a layer with a\nheight of 0.3 nm at the substrate base was thermostat-\nted (\\base\"). In this way, the annealing process is ef-\n\fcient while allowing the crystal grains to reorganize\nwithout too much thermostat interference. This ba-\nsic Langevin thermostat was parameterized with a time\nconstant of 0.5 ps, corresponding to strong electron-\nphonon coupling in metals.44For a comparison between\nthe substrate microstructures after the two heat treat-\nments, see the selected substrate tomographs in the cen-\nter of Fig. 3. These tomographs are colored according\nto grain orientation as in electron backscatter di\u000brac-\ntion, using the inverse pole \fgure coloring standard.\nThe orientations were calculated using polyhedral tem-\nplate matching45as implemented in OVITO,46and the\ncolor rendition was carried out using the MTEX tool-\nbox47,48for Matlab. The di\u000berences between HT1 and\nHT2 are subtle, even when magni\fed, and best visible in\nor around small near-surface grains, see the comparison\nfor slice #1 shown in the center of Fig. 3. The compar-\nison for slice #2 (right side) gives an idea of the largest\noccurring di\u000berences between the two heat treatments,\nwhere the small purple and orange grains located in the\nupper central part of the section have disappeared in\nfavor of neighboring grains after HT2.\nThe 18 abrasives are modeled as rigid truncated octa-\nhedra with diameters ranging from 6{14 nm, randomly\nrotated and distributed laterally, leading to an e\u000bective\nsurface coverage of 37%. As in ref 49, the interactions\nbetween abrasive particles and the Fe surface are mod-\neled using a Lennard-Jones potential with the parame-\nters\"LJ= 0:125 eV and \u001bLJ= 0:2203 nm. These values\nare similar to refs 25,50 and yield an interaction roughly\none order of magnitude weaker than the one for Fe{Fe,\nwhich means that there will be some adhesion between\nthe abrasives and substrate as well as the wear particles.\nThe abrasives are pulled across the surface at a sliding\nvelocity in xdirection of v(slide) = 80 m/s and an an-\ngle of 6:42\u000ewith thex-axis, so that they re-enter the\nsimulation box at di\u000berent ypositions every time they\npass the periodic box boundaries and therefore never\nfollow exactly in their own grinding marks. The rela-\ntive abrasive positions are locked throughout the simu-\nlations, and abrasive rotation is disabled. Furthermore,\nthe abrasives can change their zposition collectively de-\npending on the topography, similar to a grindstone, but\nnot individually. Because of their rigidity, the abrasive\nparticles themselves are not subject to any wear. The\nnormal pressure \u001bzon the abrasives (de\fned as the total\nforce in\u0000zdirection acting on the abrasive particles di-\nvided by the lateral cross-section of the simulation box,\n3595.4 nm2) is kept constant at values ranging from 0.1\nto 0.9 GPa for a simulation time of 5 ns. During the\ngrinding simulations, the Langevin thermostat acts only\ninydirection, (nearly) perpendicular to the directions\n3Figure 1: The image in the top right is a 3D representation of the entire model during the nanomachining process.\nThe bulk has grain boundary coloring (white on blue), whereas the surface has topographic coloring (blue{yellow{\ngreen{red, from low to high) so that the abraded material is shown in red. Abrasives are gray. Machining e\u000bectively\ntakes place in + xdirection. Computational substrate tomography is performed by decomposing the system into 20\nxzslices normal to the ydirection, one every 3 nm. In this example, the atoms are colored according to the grain\norientation similar to electron backscatter di\u000braction (inverse pole \fgure standard, see legend at the bottom with\nblack point clouds representing the individual grains). The tomographs are \fnally arranged on a 4 \u00025 grid for a\nbetter overview, see Fig. 3.\nFigure 2: Temperature curves for the heat treatments \\HT1\" and \\HT2\" leading to the two initial system con\fgu-\nrations used in this work.\n4Figure 3: Left: Substrate tomographs of the initial con\fguration after heat treatment 2 (HT2), sorted from top\nleft to bottom right. Center: Close-up of exemplary slices #1 and #2 (marked by box in left panel) for comparison\nbetween HT1 and HT2. Abrasives are gray, grains are colored according to orientation, inverse pole \fgure (IPF)\nstandard, see legend on the right.\nof normal pressure and grinding, so as not to overly\ninterfere with these external constraints.\n3 Thermostat Parametrization\nThe electronic contribution to the thermal conductiv-\nity\u0014(T), which dominates in metallic systems, is not\naccounted for in the interaction potentials applied in\nclassical MD. We calculated \u0014(T) obtained from the\nstate-of-the-art Finnis-Sinclair potential for Fe used in\nthis work,43which is based only on the phononic con-\ntribution, for temperatures ranging from 300 to 900 K\nusing the Green-Kubo formulas, which relate the en-\nsemble average of the auto-correlation of the heat \rux q\nto\u0014(T). When comparing the results to the experimen-\ntal TPRC data series,51we found that \u0014(T) calculated\nvia MD correlates well with the experimental values,\nbut that the MD values are, on average, 4.5 times lower\nthan the experimental ones.\nWe will now describe a \\smart\" thermostatting\nscheme, based on refs 39,40, which attempts to re\rect\nthe coupling between electrons and phonons in metals.\nSuch an approach is justi\fed as long as the externally\nimposed velocities lie well below the speed of sound in\nthe work piece, and the mean free path for the electron-\nphonon interactions is short enough to be accommo-\ndated in the work piece along the direction of heat con-\nduction. With our grinding velocity amounting to less\nthan 2% of the speed of sound in iron, the \frst condition\nis easily met. An estimation based on the assumptions\nin ref 52 yields an electron-phonon mean free path of\napproximately 2.5 nm, which is roughly one order of\nmagnitude smaller than our work piece thickness. The\nLangevin thermostat acting on the bulk of the substrate\nis parametrized with a coupling constant corresponding\nto the characteristic time that it takes the lattice vibra-\ntion energy of the ionic system to be transferred to the\nelectron gas, which, due to its high thermal conductiv-\nity, can be considered an implicit heat bath. Based onthe Sommerfeld theory of metals, an estimation for this\nelectron-phonon coupling time \u0015can be expressed as40\n\u0015=2me\u0014EF\n\u0002DT0Lne2kBZ; (1)\nwhereT0is the temperature of the heat bath, \u0002 Dthe\nDebye temperature, Lthe Lorenz number, nthe free-\nelectron density, Zthe valence, \u0014the thermal conduc-\ntivity,EFthe Fermi energy, kBthe Boltzmann con-\nstant, and eandmethe electron charge and mass. If\nwe feed this expression with the data for Fe at the ex-\npected mean substrate temperature of T'310 K, we\nobtain a\u0015of about 0.5 ps.\nOn the other hand, we can roughly estimate the ther-\nmal conductivity of our substrate by assuming that al-\nmost all the frictional energy dissipated in the interface\nmust \fnd its way to the heat bath eventually. We ne-\nglect the heat that \rows from the sliding interface into\nthe wear particles as well as radiation losses from the\nsurface. The former contribution depends on the load\nand amounts to several percent of the total, while the\nlatter are not even included in our MD model, but are\nroughly six orders of magnitude smaller than the fric-\ntion power at the surface temperatures we encounter in\nour simulations. We thus set the one-dimensional heat\n\rux to \b q=\u0016\u001bzv(slide) , where\u0016is the coe\u000ecient of\nfriction,\u001bzis the normal pressure, and v(slide) is the\nsliding speed. While \u001bzandv(slide) are kept constant\nas external constraints, \u0016turns out to maintain a value\nof 1 throughout the simulations discussed in this work\n(see Sec. 4.1 for details), so we may also treat it as a\nconstant. The e\u000bective thermal conductivity measured\nfrom our grinding simulations can then be obtained via\nthe Fourier law\n\bq=\u0000\u0014(T)dT(z)\ndz: (2)\nWe can now search for the optimum value of \u0015start-\ning from the value 0.5 ps estimated using Eq. (1) to\nparametrize our Langevin thermostat for practical use.\nThis is done by performing an abrasion simulation at\n5Figure 4: Temperature pro\fles for three di\u000berent normal pressures and three di\u000berent thermostatting procedures.\nThe inset shows the dependence of the ratio between the experimental thermal conductivity and the value estimated\nfrom the simulations, \u0014exp=\u0014simon the electron-phonon coupling time \u0015. The black boxes represent values obtained\nfrom simulations at 0.9 GPa, and the blue curve is a B-spline interpolant.\nconstant \b qup to the point where it reaches a su\u000e-\nciently linear temperature gradient dT(z)=dzwithin the\nsubstrate close to the machining interface and checking\nhow well that gradient corresponds to the experimen-\ntal thermal conductivity of Fe51at the mean substrate\ntemperature, see Fig. 4 for some associated temperature\npro\fles at various loads and values of \u0015. We calculated\n\u0014(T) for several values of \u0015in this way and found that\nthe best reproduction of the experimental thermal con-\nductivity of Fe can be achieved with \u0015= 3:5 ps through-\nout our desired range of normal pressures, c.f. the inset\nin Fig. 4. Simulations of ion beam mixing and dam-\nage production in Fe53with an implementation of the\nelectron-phonon coupling model of ref 40 also came to\nthe conclusion that the time constant best re\recting the\nexperimental data was several times larger than the one\ncalculated via Eq. (1), arriving at values between 1.6\nand 2.2 ps. Other work dealing with metal ablation by\npicosecond laser pulses36arrives at an electron-phonon\ncoupling time of 7 ps. We can thus conclude that the\nestimate for \u0015calculated via Eq. (1) can only serve\nas a basis for empirically \fnding the electron-phonon\ncoupling time that best re\rects experimentally observ-\nable quantities. Table 1 summarizes the coupling times\nparametrizing the Langevin thermostats acting on the\nbase and the bulk of the substrate to implement the\nthree discussed thermostatting schemes.\nAn additional bene\ft of the electron-phonon coupled\nthermostatting scheme discussed above is that the cool-\ning of the removed matter (e.g., via a coolant \ruid)\nis, at least for all practical purposes, accounted for in\nthis way, i.e., even if the wear debris are already de-\ntached from the substrate surface and would therefore\nno longer be thermally coupled to a base thermostat,\nthey can cool o\u000b by themselves.\nIt should be noted that the theoretically calculated\nmaximum surface temperature occurring due to dryTable 1: Overview of the coupling times for the\nLangevin thermostats acting on the substrate base\n(\u0015base) and on the rest of the Fe atoms ( \u0015bulk).\nthermostatting scheme \u0015base[ps]\u0015bulk[ps]\nbase 0.5 |\nelectron-phonon coupled 0.5 3.5\nfull 0.5 0.5\nsliding54{56can rise considerably higher than the sur-\nface temperatures we observe in our simulations. This\nis because \frstly our simulations last only 5 ns, while\nmacroscopic equilibrium temperatures are reached after\nmuch longer periods of sliding. Secondly, the Langevin\nthermostat representing the heat sink of the work piece\nis placed only 20 nm from the friction interface while\nbeing kept at a constant and rather low temperature\nof 300 K, so that by \fxing the allowed temperature\ngradient in the near-surface region, we also limit the\nmaximum surface temperature.\n4 Results and Discussion\n4.1 Time, load, and temperature depen-\ndence of tribological quantities\nIn Fig. 5, we present the time development of the wear\ndepth, the arithmetic mean height, and the RMS rough-\nness of the surface topography for electron-phonon cou-\npled thermostatting. The wear depth is de\fned as the\ncombined volume of all Fe atoms traveling faster than\n90% of the abrasives' speed, so that they can be consid-\nered attached to the grinding particles,57divided by the\ncross-section of the simulation box Anom= 3595 nm2.\nCompared to our previous work on monocrystalline fer-\n6rite substrates and considerably more regular lateral\ndistributions of abrasive particles,49,58{60the transition\nbetween the running-in and the steady-state regime is\nmuch less pronounced, see the kink in most curves in\nFig. 5 (a) at t\u00190:3 ns. As, by our de\fnition, the sub-\nstrate consists of all atoms moving in + xdirection at\nless than 10% of the grinding speed, the substrate to-\npography does not include the shear zone.57However,\nduring the initial impact of the abrasives on the sub-\nstrate and the brief period following it, which is charac-\nterized by strong oscillations of the contact forces due to\nthe substrate elasticity, a shear zone is formed that can\nmomentarily be up to two orders of magnitude larger\nthan its equilibrium size and may extend up to 10 nm\ninto the substrate. This explains the large deviations\nfrom the general trend during the \frst 0.2 ns in pan-\nels (b) and (c), where the topographic quantities are\nshown.\nThe time evolution of the wear depth exhibits a steady\nincrease for the high pressure simulations, which is re-\n\rected in a steady decrease of the mean substrate height\nzsubst. The low pressure simulations hardly show any re-\nduction ofzsubst and a very small increase of the wear\ndepthhw, restricted to the \frst 0.5 ns. This suggests\nthat up to 0.3 GPa the system does not wear signi\f-\ncantly, but that the reduction of the surface roughness,\nas depicted in Fig. 5 (c), is merely a result of material\ntransfer from the peaks into the valleys, i.e., abrasion of\nthe asperity tips and subsequent re-crystallization onto\nthe substrate. The wear particles are formed at the\nvery beginning for low normal pressures, and although\nindividual particles may change their sizes over time,\nthe combined volume of all the wear particles remains\nfairly constant for the rest of the grinding simulation.\nFor the high pressure cases of 0.9 GPa and 0.75 GPa, the\ntime evolution of Sqis unstable, and the \fnal roughness\napproximately equals the initial value.\nIn Fig. 6, we show exemplary xyprojections of the\nreal atomic contact area, colored according to contact\ntemperature after 5 ns of grinding. As explained in de-\ntail in refs 57,61, this contact area is obtained by estab-\nlishing which substrate atoms are in contact with the\nabrasives or the associated wear particles via a distance\ncriterion, and then multiplying the number of these con-\ntact atoms with a constant per-atom contact area. The\ncontact temperature for each contact atom is calculated\nvia the kinetic energy averaged over a spherical control\nvolume with a radius of 1 nm after subtracting the drift\nvelocity of this control volume, cf. ref 42. The three\nexamples shown in Fig. 6, from left to right, are for\nnormal pressures of 0.3, 0.6, and 0.9 GPa, respectively.\nThe top row shows results for base-thermostatted sub-\nstrates, the center one for electron-phonon coupled ther-\nmostatting, and the bottom row fully thermostatted\nsubstrates. The contact area increases with load inde-\npendently of the chosen thermostatting procedure. In\ndetail, the shape and size of individual patches of the\ncontact area are di\u000berent depending on the chosen ther-\nmostatting. However, its more general characteristicsremain unchanged since these are primarily determined\nby the asperities of the substrate as well as the size and\ndistribution of the abrasive particles. Clearly, the con-\ntact temperature dependence on the normal pressure is\nconsiderable when applying only a base thermostat. For\nthe electron-phonon coupling and the fully thermostat-\nted substrate, increasing the normal pressure does not\nin\ruence the contact temperature much. The main dif-\nference between these two variants is the temperature\nof the hottest spots within the contact area, which is\nabout 100 K higher if electron-phonon coupling is ap-\nplied. Additionally, the local temperature distribution\nwithin the contact area patches is more inhomogeneous\nfor electron-phonon coupling.\nIt seems that the choice of the heat treatment proce-\ndure has little in\ruence on the normal pressure depen-\ndence of some tribological, topographical, and thermo-\ndynamic quantities shown in Fig. 7. The only notice-\nable di\u000berence occurs for the \fnal RMS roughness at\nthe highest load, see Fig. 7 (e), but here the error bars\nare also very large and overlap to such extent that it\nmust be questioned if the di\u000berence between HT1 and\nHT2 is statistically relevant.\nIt is more striking that the pressure dependence of the\nshear stress \u001bxdoes not appear to be in\ruenced by the\nthermostatting procedure at all. This is in agreement\nwith other literature,62where it is reported that the\nsubstrate temperature, varied from 0 K to 500 K, has lit-\ntle e\u000bect on what is there called the \\scratching force\".\nWe therefore carried out additional calculations at a\nconsiderably higher normal pressure of \u001bz= 2:5 GPa\nwith base- and electron-phonon coupled thermostatting.\nThey revealed that under these severe conditions a dif-\nference between the obtained shear stresses emerges.\nWhile the electron-phonon coupled results remain on\nthe straight line suggested in Fig. 7 (a), the results\nfor the base thermostat lie approximately 25% lower.\nThis might be explained by a pronounced softening ef-\nfect of the polycrystal due to the high temperatures\nin the surface zone, where the base-thermostatted sub-\nstrate ground at \u001bz= 2:5 GPa reaches z-averaged tem-\nperatures of 1160 K, while the T-maxima in the con-\ntact spots even rise up to 1350 K. Compared to the\nmuch lower surface temperatures of only 450 K for the\nelectron-phonon coupled substrate, the hotter and thus\nsofter base-thermostatted substrate opposes much less\nresistance to shear, equivalent to reduced work harden-\ning of macroscopic samples at high temperatures.\nAs expected, the wear depth in Fig. 7 (b) and the\narithmetic mean height in Fig. 7 (d) exhibit opposing\ntrends, both showing almost identical behavior for all\nthermostatting procedures up to 0.4 GPa and a split-\nting up into three distinct branches for greater normal\npressures. The fully thermostatted substrates are worn\nless than the electron-phonon coupled ones, which in\nturn wear less than the base-thermostatted ones. The\n\fnal RMS roughness of the surface in Fig. 7 (e) already\nshows some di\u000berences at a normal pressure of 0.4 GPa,\nbut this di\u000berence remains small up to 0.6 GPa. At the\n7Figure 5: Wear height hw(a), arithmetic mean height zsubst (b), and root-mean-square roughness Sq(c) over time\nfor electron-phonon coupled thermostatting. The rainbow-style coloring re\rects the normal pressure (blue is low,\nred is high).\n8Figure 6: Exemplary lateral distribution of the contact area ( xyprojection = top view) and the thermal distribution\nwithin the contact area as a function of normal pressure and thermostatting procedure at t= 5 ns. Top: base-\nthermostatted substrates, center: electron-phonon coupling thermostatted substrates, bottom: fully thermostatted\nsubstrates. Left: \u001bz= 0:3 GPa, center: \u001bz= 0:6 GPa, right: \u001bz= 0:9 GPa.\n9Figure 7: Mean shear stress \u001bx(a), \fnal wear depth hw(b), mean normalized real contact area Ac=AnomwithAnom=\n3595 nm2(c), \fnal arithmetic mean height zsubst (d), \fnal root-mean-square roughness Sq(e), and mean contact\ntemperature Tc(f) over normal pressure \u001bz. Full green circles denote results obtained for electron-phonon coupled\nthermostatting, red upwards triangles for base thermostatting with heat treatment 1 (short), yellow downwards\ntriangles for base thermostatting with heat treatment 2 (long), and blue squares for full thermostatting.\n10highest normal pressure, however, the \\best\" achiev-\nable roughness of the base-thermostatted substrates is\nalmost double that of the ones obtained with the other\ntwo thermostatting procedures. Note that the initial\nRMS roughness is 0.7 nm, which means that grinding\nat normal pressures of 0.75 GPa or more with a base-\nthermostatted substrate does not produce a smoother\nsurface. Whereas the pressure dependence of the \fnal\nroughness is almost linear for full and electron-phonon\ncoupled thermostatting, which is in good agreement\nwith ref 49 (where the substrates were so thin that\nthe thermostatting procedure would not have had much\nin\ruence on the result), the base-thermostatted sys-\ntems show a strongly superlinear pressure-dependence.\nThis might be attributed to the following: in the base-\nthermostatted systems, heat transport from the chips\ntowards the heat sink is only possible via phononic heat\nconduction through the contact regions with the sub-\nstrate. The chips can therefore reach temperatures close\nto 1000 K, making them soft. This, in turn, leads to\ntheir occasional slumping over to the sides of the abra-\nsives, leaving high scratch ridges in the likewise soft-\nened near-surface substrate. A linear increase of the\nRMS roughness with the normal pressure implies that\nthe substrate compliance does not change within the\nsubstrate volume that interacts with the abrasives, even\nat higher loads. Only for the base-thermostatted sub-\nstrate the temperature gradient is steep, and Fig. 7 (e)\nshows that this temperature increase is only relevant\nabove 0.6 GPa, as there is not enough friction energy\nto generate su\u000eciently high temperatures at lower pres-\nsures.\nFigure 7 (c) shows the normalized contact area\nAc=Anomover the normal pressure \u001bz, whereAnom=\n3595 nm2is the lateral cross-section of the simulation\nbox.Ac=Anomcan be considered the degree to which\nthe two counterbodies are in contact. Since this con-\ntact is not necessarily \rat (in contrast to Anom), the di-\nmensionless ratio can in principle exceed 1. For the fully\nthermostatted systems, the normal pressure dependence\nis basically linear, while for the base-thermostatted sys-\ntems it exhibits a distinct kink around \u001bz\u00190:5 GPa,\nseparating the normal pressure dependence into two\nregimes of smaller and larger slope for lower and higher\nnormal pressures, respectively. For the systems with\nelectron-phonon coupled thermostatting, the data lie in\nbetween, but markedly closer to those obtained for the\nfully thermostatted systems, with an apparent change\nin slope between \u001bz\u00190:6\u00000:75 GPa.\nAs can be seen in Fig. 7 (f), it becomes evident that in\nthe base-thermostatted substrates, the average contact\ntemperature increases linearly with \u001bzindependent of\nthe heat treatment procedure, with a temperature in-\ncrease of\u001945 K per 100 MPa. For full and electron-\nphonon coupled thermostatting the contact tempera-\nture remains at a constant 350 K and 410 K, respec-\ntively. The temperature di\u000berence between the heat\nbath and the contact zone may therefore only be a func-tion of the abrasive particle geometry, the sliding veloc-\nity, and the thermostat coupling constant.\nIn Fig. 8 we analyze how the shear stress \u001bxand the\n\fnal wear depth h(end)\nw vary with the normalized con-\ntact areaAc=Anom. In panel (a), the slope corresponds\nto the e\u000bective shear strength \u001cof the system. For the\nfully thermostatted substrates, this quantity is almost\na constant, while for the other two thermostatting pro-\ncedures we observe some more pronounced \rattening of\nthe curve, which re\rects the onset of thermal substrate\nsoftening. Panel (b) shows that the \fnal wear depth can\nessentially be considered a superlinear function of the\ncontact area, almost independent of the contact temper-\nature and therefore also the thermostatting procedure.\nWe can therefore formulate that the contact area in-\ncreases due to thermal softening of the substrate, while\nthe separation of atoms from the substrate is eased at\nhigher temperatures. As this correlation of wear depth\nand contact area depends on the local temperature of\nthe near-surface region immediately beneath the contact\nspot, the same superlinear trend is observed irrespective\nof any thermostatting procedure.\n4.2 Thermostat in\ruence on mi-\ncrostructural development\nThe grinding process changes the microstructure as well\nas the defect structure regardless of thermostatting pro-\ncedure or normal pressure. These changes are more pro-\nnounced at higher normal pressures, and their extent\ndepends critically on the thermostatting procedure. We\ntherefore chose the abrasion simulation carried out at\nthe highest normal pressure (0.9 GPa) to best illustrate\nthe possibly drastic di\u000berences between the three ther-\nmostatting variants, see Fig. 9. For an explanation on\nhow the tomographic slices correspond to the 3D model,\nrecall Fig. 1. The analysis scheme using computational\nsubstrate tomography is explained in detail in ref 42. In\nthe following, individual panels of the substrate tomo-\ngraphs in Figs. 3, 9, and 10 will be referred to by their\nrow number mand their column number n, abbreviated\nas Rm/Cn, cf. the bottom left legend in Fig. 9. The\ndefect structure evolving due to dislocations as well as\nany changes of the grain boundaries can be observed\nvia the centro-symmetry (CS) parameter63panels in\nFig. 9 (a,c,e). The di\u000berentiation between grain growth\nand recrystallization or lattice rotation is possible via\nthe orientation panels in Fig. 9 (b,d,f) with grain col-\noring according to inverse pole \fgure (IPF) standard.\nFor an interpretation of the microstructural changes as\na result of thermally activated processes, we refer to the\ntemperature plots resulting from low (0.3 GPa) and high\npressure simulations (0.9 GPa) in Fig. 10. In general,\nwe observe grain growth, abrasion of entire grains, and\nfull or partial rotation of the lattice structure. Part of\nthe friction energy is stored in dislocations or can lead\nto the formation of new grain boundaries.\nThe base-thermostatted simulation results in the\nmost dramatic changes to the initial polycrystalline\n11Figure 8: Shear stress \u001bxand \fnal wear depth h(end)\nw over the normalized contact area Ac=AnomwithAnom=\n3595 nm2.\nstructure, compare Fig. 3 (a) with Fig. 9 (b). We\nobserve massive grain growth, and only grain bound-\naries pinned to the rigid base via the boundary condi-\ntions do not experience any changes. The grain struc-\nture at the surface is strongly disturbed. As the wear\nparticles cannot cool down by themselves in the base-\nthermostatted simulations, their temperatures rise be-\nyond 900 K, see Fig. 10 (b). Although the surface\ncan cool down somewhat between two passing abra-\nsives, the temperature in the near-surface zone remains\nhigh in the base-thermostatted simulation, never drop-\nping below 600 K. The grain boundary mobility in this\nzone is therefore much higher than in the di\u000berently\nthermostatted simulations. Due to the high temper-\natures, the zone featuring modi\fed grain orientations\nextends to depths of 10{15 nm. The latter can be visual-\nized using IPF standard coloring, which reveals massive\nchanges between initial and \fnal grain orientations and\na slight tendency towards a (111) orientation, though\nthe orientation is too weak to refer to it as a preferred\ntexture evolving during grinding.\nAfter grinding at 0.9 GPa, the \fnal near-surface grain\nsize of the electron-phonon coupled substrate is smaller\nthan that of the base-thermostatted substrate. Only\nbeneath the wear particles, the base-thermostatted sub-\nstrate forms numerous tiny grains down to half the\nsubstrate thickness, whereas electron-phonon coupling\nresults in smaller wear particles and hence a reduced\ngrain-re\fned area, compare R2/C1 in Figs. 9 (a,c).\nThe IPF orientation analysis features pronounced\ncolor shading within the larger grains, which indi-\ncates elastic deformations within the grains that are\nalso referred to as residual stresses of type II, or mi-\ncrostresses.64The fact that these elastic deformations\nare stable in more grains than in the base-thermostatted\nsubstrate can be explained by the much lower tempera-\ntures in the electron-phonon coupled substrate. Besides,\nthe surface can cool down to almost 300 K between two\npassing abrasives, and the wear particles themselves arecool and do not heat up the surface as in the base-\nthermostatted process. At the end of the simulation\nthere is no tendency towards a preferred (111) orienta-\ntion for the electron-phonon coupled substrate.\nWe observe grain growth compared to the initial mi-\ncrostructure for the fully thermostatted simulation as\nwell. The CS parameter analysis of the fully ther-\nmostatted simulation in Fig. 9 (e) shows that small\ngrains have been abraded and grain growth has taken\nplace, but only in the \frst nm beneath the surface. As\nthe temperature of the wear particles and the contact\nzone only reaches 350 K, any temperature rise due to\nfrictional heat is roughly limited to the upper 3 nm\nof the substrate, recall Fig. 4. Since grain growth\nis primarily a thermally activated process, the grain\nboundary mobility is only high enough in this limited\nzone of the \frst nanometers beneath the surface (lo-\ncated atz\u001920 nm). Consequently, the grain sizes\nat the surface of the fully thermostatted substrate are\nsmaller than in the electron-phonon coupled substrate\nand much smaller than in the base-thermostatted sub-\nstrate after 5 ns of grinding, see Fig. 9 (e,f). The grain\norientation analysis in panel (f) reveals that grains grow\nat the expense of small grains at the surface, and that\nlarger grains located in the middle of the substrate are\nstable with respect to their orientation and only grow\ntowards the surface.\nThe fully and the base-thermostatted scenarios re\rect\nthe two extremes of microstructural changes. As the\nevolution of the microstructure is primarily driven by\nthe evolving temperature (when comparing equal pres-\nsure processes), thermostatting strongly changes the\ntemperature levels and gradients at di\u000berent depths of\nthe substrate. Electron-phonon coupled thermostatting\nproduces temperatures between the two bounds of the\nfully and base-thermostatted variants, compare Fig. 10\n(c,d) with (a,b) or (e,f). Electron-phonon coupling leads\nto higher maximum temperature levels in the contact\nthan full thermostatting, but also to elevated tempera-\n12Figure 9: Substrate tomographs after 5 ns of grinding at 0.9 GPa (see legend at bottom left for shorthand references\nin the text). Left: coloring according to centro-symmetry (CS) parameter (perfect lattice = dark blue, lattice defects\n= turquoise, grain boundaries = yellow/orange, surface = red), abrasives are gray. Right: grains colored according to\norientation (inverse pole \fgure (IPF) standard, see legend), abrasives are gray. (a,b): base-thermostatted substrate,\n(c,d): electron-phonon coupled thermostatting, (e,f): fully thermostatted substrate.\n13Figure 10: Substrate tomographs colored according to temperature (see colorbars below, abrasives are gray) after\n5 ns of grinding at 0.3 GPa (left) and 0.9 GPa (right). (a,b): base-thermostatted substrate, (c,d): electron-phonon\ncoupled thermostatting, (e,f): fully thermostatted substrate.\n14ture \felds con\fned to substrate regions directly beneath\nthe contact area when compared to base thermostat-\nting, cf. Fig. 4. Furthermore, the maximum tempera-\nture gradients dT=dz (occurring within the shear zone\naroundz\u001920 nm) are considerably steeper for base\nthermostatting than for the two other thermostatting\nmodes.\nThe main microstructural feature that changes dur-\ning grinding in all simulations is the grain size. Grain\ngrowth occurs in part by grain boundary movement and\nin part by lattice rotation. Occasionally, tiny new grains\nform in both non-base-thermostatted variants, but they\nare not su\u000eciently stable to withstand a passing abra-\nsive.\nFor the di\u000berent thermostatting procedures, the re-\nsulting temperature gradient within the substrate and\nthe shear zone determines the reaction of the mi-\ncrostructure at the respective normal pressure. The\nexternal mechanical loads are fairly equal if the pres-\nsure remains \fxed. For a comparison of the evolving\ntemperature gradients at low normal pressure (0.3 GPa)\nfor base, electron-phonon coupling, and full thermostat-\nting, see Fig. 10 (a,c,e). As the T-gradient in the shear\nzone is very steep for base thermostatting, even this\nlow normal pressure machining process changes the mi-\ncrostructure markedly. The degree to which the mi-\ncrostructure is modi\fed is thus directly de\fned by the\nlevel and extent of the three-dimensional temperature\n\feld.\n5 Conclusion and Outlook\nIn this work we have analyzed the sliding interface be-\ntween a nanocrystalline ferritic work piece and hard,\nabrasive particles during a nanomachining process sim-\nulated with molecular dynamics simulations. We have\nshown that choosing a thermostatting procedure that\ncan mimic the electronic contribution to heat conduc-\ntivity is crucial when large amounts of energy are intro-\nduced to metallic systems in a highly localized fash-\nion. In our application, the resulting near-surface\nmicrostructural development, surface topography, and\nquantities such as the wear depth depend strongly on\nthe chosen thermostatting procedure for normal pres-\nsures exceeding 0.4 GPa, while the total shear stress is\nalmost una\u000bected up to a normal pressure of approxi-\nmately 1 GPa. The correlation between plasti\fcation\nof the substrate and varying normal pressures as well\nas the evolving dislocation structure will be discussed\nin detail in a forthcoming publication by some of the\nauthors.\nAs bene\fcial as the thermostatting approach de-\nscribed in this work is for simulating surface \fnishing\nand wear processes, there are several aspects of it that\ncould be improved further. As the target temperature\nof the thermostat is constant throughout the simula-\ntions, it is currently impossible to reach the practically\nobtained equilibrium temperatures for dry friction withsimulation times restricted to several nanoseconds. A\nsimple workaround could be to set the thermostat tem-\nperature close to the theoretically predicted value at\nthat depth in the work piece, but a proper treatment\nwould have to include a coupling of the base thermostat\nto a far-\feld solution of the temperature depth pro\fle.\nThis could be obtained using a one-dimensional Green's\nfunction approach,54,65assuming an averaged heat \rux\nat the sliding interface and a semi-in\fnite work piece,\nthus placing the bulk temperature boundary condition\n\\far away\" from the heat source. Properly done, this\nwould allow the system to reach a realistic surface tem-\nperature. A second aspect that is not trivial to resolve\nis that the Langevin thermostat somewhat unrealisti-\ncally damps all modes of heat conduction equally, which\nmight be improved by employing a more sophisticated\nthermostat. Finally, since nanoscopic heat conduction\ndepends on the crystal system and orientation, it might\nbe an oversimpli\fcation to assume a constant macro-\nscopic heat conductivity for a 20 nm thick surface layer,\neven if it consists of initially randomly oriented grains.\nA proper treatment of this issue would include a real-\ntime grain orientation analysis and subsequent adaptive\nthermostat parametrization, so any bene\ft gained from\nsuch an approach would have to be carefully balanced\nwith the considerable additional computational cost.\nAlthough the thermal and microstructural trends in\nour results match intuition, the highly desirable experi-\nmental validation is unlikely to be exhaustive. All of\nthe thermostatting scenarios in our work except the\nelectron-phonon coupled ones are di\u000ecult to reproduce\nin an experiment, as it is not trivial to suppress elec-\ntronic heat conduction.66,67While one could simply\nconduct nanomachining trials at higher temperatures,\nany in\ruence of excessive thermal gradients would still\nbe lost. That said, it may be possible to verify the\ntime-development of the near-surface structural changes\noccurring in a nanocrystalline sample due to abrasive\nnanomachining. One aspiring suggestion would be the\ndesign of a tribo-experiment to be analyzed in situ using\nsynchrotron X-ray di\u000braction, similar to what has been\ndone to study the growth of bulk grains in deformed\nmetals.68Before going to such an e\u000bort, it may how-\never be sensible to reconsider the choice of work piece\nmaterial, e.g., an fcc structure with a low tendency for\noxidation while still being technologically relevant.\nThere exists a second path for, at least qualitative,\nvalidation of our simulation results. Aided by ever-\nincreasing computational power, we have recently suc-\nceeded in simulating the tribological response of a con-\nsiderably larger work piece consisting of approximately\n25 million atoms. Due to the initial grain size diameters\nof 30 nm, tribological loading actually leads to disloca-\ntion pile-up and near-surface grain re\fnement, both of\nwhich are observed in macroscopic experiments. This\nbodes well for the future, as it means that phenomena\nrelevant to applied interfaces might be reproduced at\na slightly smaller scale, thus uncovering the underly-\ning mechanisms. Ultimately, MD simulation may well\n15evolve into a crucial component of engineering design\ntools for manufacturing, friction, or wear applications.\nAcknowledgement This work was partly funded by\nthe Austrian COMET-Program (Project K2, XTribol-\nogy, no. 849109) and carried out at the \\Excellence\nCentre of Tribology\". SJE, UC-B, and DB acknowl-\nedge the support of the Province of Nieder osterreich\n(Project \\SaPPS\", WST3-T-8/028-2014). GF acknowl-\nedges the \fnancial support from the German Research\nFoundation (DFG) via SFB 986 \\M3\", project A4. The\nauthors wish to thank M. Moseler, M. M user, J. Be-\nlak, A. Vernes, and G. Vorlaufer for helpful discussions\nabout thermal conductivity, electron-phonon coupling,\nand Green's functions.\nReferences\n(1) Rebeggiani, S.; Ros\u0013 en, B.-G. 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Science 2004 ,305, 229{232.\n18"    },    {        "title": "0707.3112v2.The_Ba2LnFeNb4O15_Tetragonal_Tungsten_Bronze__towards_RT_composite_multiferroics.pdf",        "content": " \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThe Ba 2LnFeNb 4O15 “Tetragonal Tungsten Bronze”:  \ntowards RT composite multiferroics  \n \nM. Jossea,*, O. Bidaultb, F. Roullanda, E. Castela, A. Simona, D. Michaua, R. Von der \nMühlla, O. Nguyena and M. Maglionea  \n \na ICMCB-CNRS, Université Bord eaux 1, 33608 Pessac, France \nb I.C.B., CNRS - Université de Bourgogne, 21078 Dijon, France \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n* To whom correspondence should be addressed: josse@icmcb-bordeaux.cnrs.fr \n 1 \n \n \nAbstract \n \nSeveral Niobium oxides of formula Ba 2LnFeNb 4O15 (Ln = La, Pr, Nd, Sm, Eu, Gd) \nwith the “Tetragonal Tungsten Bronze” (TTB) structure have been synthesised by \nconventional solid-state methods. The Neodymium, Samarium and Europium compounds are \nferroelectric with Curie temperature ra nging from 320 to 440K. The Praseodymium and \nGadolinium compound s behave as relaxors below 170 and 300 K respectively. The \nPraseodymium, Neodymium, Samarium, Europium and Gadolinium compounds exhibit magnetic hysteresis loops at room temperature originating from traces of a barium ferrite \nsecondary phase. The presence of both ferroel ectric and magnetic hysteresis loops at room \ntemperature allows considering these materials as composites multiferroic. Based on crystal-chemical analysis we propose some rela tionships between the introduction of Ln\n3+ ions in the \nTTB framework and the chemical, structural and physical properties of these materials. \n \n 2Introduction \n \nMuch attention has been paid in the recent years to multiferroic materials, and \nparticularly to those presenting both ferroma gnetism and ferroelectricity. Magnetoelectric \nmultiferroics, which will be referred  to as multiferroics in the  following, could be used, for \nexample, for the design of new generation of Random Access Memories (RAMs) in which \nboth the electric and magnetic polarisation are used for data storage. Aside from the \nprocessing of multiferroic thin films and composite s [1], the search for bulk multiferroic is a \ngreat challenge. \n However up to now the candidate multiferroic materials for room temperature applications are very few. A large majority of  the “multiferroic” publications of the recent \nyears deal  with perovskite-related ma terials [2,3]. From this abundant literature, two \nlimitations are widely accepted. First, the electro nic structure of the A an B cations in ABO\n3 \nmultiferroics is hardly compatible with a magn etic ordering at elevated temperatures [4]. \nSecond, a coupling between the ferroelectric a nd magnetic order may be obtained when the \nmaterial exhibit a non standard, e.g. spiral, magnetic structure [5], or a toroidal order [6]. So far, only the Bismuth ferrite BiFeO\n3 has demonstrated room temperature multiferroic \nbehaviour, and yet the complex antiferromagnetic ordering yields to a very small remnant magnetisation [7].  \n To face both these limitations, we decide d to investigate completely different \nnetworks, namely the Tetragonal Tungsten Bron ze (TTB) structural family. At the crystal \nchemistry level, this framework has two main a dvantages. First, the numbe r of cationic sites is \nmuch higher than in perovskites (5 instead of 2) thus enabling extended substitutions opportunities to foster magnetic interactions . Next, TTB compounds are known to display \nincommensurate polar state [8,9], that may favour a coup ling between magnetic and \nferroelectric order. On investigating an extende d family of TTB compounds, this paper is an \nexperimental attempt towards this search for alternative multiferroics.  \nThe “Tetragonal Tungsten Bronze” (TTB) struct ural type is related to the Potassium \ntungstate K 0.475WO 3, the structure of which was el ucidated by Magneli [10]. The \ncharacteristic feature of the TTB  crystal structure is the thr ee types of open channels that \ndevelop within its octahedral fr amework (fig. 1). These features allowed solid state chemists \nto perform a wide range of subs titutions, either in these channels or within the octahedral \nframework itself, which led to the discovery of Ba 2NaNb 5O15 [11], one of the first and most \nsignificant representative  of TTB niobates. The physical properties of Ba 2NaNb 5O15 \n 3(Ferroelectricity, non-linear optic s…) gave rise to extended i nvestigations on TTB niobates, \nand to the discovery of many new ferroelectrics and more recently relaxors [12] compositions.  \nBesides this literature - and forty five years before the “revival of the magnetoelectric \neffect” [7] - Fang & Roth reported, in a short pa per without experimental  data [13], some of \nthe title compositions as ferroelectric and ferri magnetic materials, but did not report further \ninvestigations in later articles. A few years late r, Krainik & al. [14] s ynthesized several rare-\nearth/iron substituted TTB niobates, and found some of them to be ferroelectrics. Schmidt, in \nhis “magnetoelectric classification of materials”  [15], referenced two of these TTB niobates. \nIshihara & al. also reported on the TTB fluoride K 3Fe5F15 as a potential multiferroic [16].  \nOn the basis of these works and of our ow n knowledge of TTB niobates, we decided \nto reinvestigate the Ba 2LnFeNb 4O15 (Ln = La, Pr, Nd, Sm, Eu, Gd) system and started an in-\ndepth study of rare-earth/iron substituted TTB niobates in order to confirm and understand \ntheir multiferroic properties. \nIn this paper, we report th e synthesis and physical-chemical  characterisation of several \nTTB niobates, with emphasis on  their dielectric and magnetic properties. We observed that \nseveral compositions display both ferroelectric and magnetic hysteresis loops at room \ntemperature. X-Ray microprobe experiments re vealed that traces of barium ferrite BaFe 12O19 \nare responsible for room temperature magnetic hysteresis. We also discuss the chemical, \nstructural and physical properties of this family of TTB niobates in terms of crystal-chemical \nbehaviour. \n 4Experiments   \n \nBa2LnFeNb 4O15 (Ln = La, Pr, Nd, Sm, Eu, Gd) compounds have been obtained by \nconventional solid state route, from  stoichiometric mixtures of BaCO 3, Fe 2O3, Nb2O5 and \nLn2O3, all reagents with 99.9% minimal purity grad e. Ceramics were sintered at 1300°C and \nexhibited densities ranging from 90 to 96% of the nominal density of the products. \nX-ray diffraction data were recorded on a Philips XPert pro diffractometer (CuK α1, λ \n= 1.54056 Å) with 10 < 2 θ < 130° and step = 0.008°. Rietveld refinements were performed \nusing the program FULLPROF [17]. Experiment al diffractograms were fitted using a \nThompson-Cox-Hastings function (profile n° 7), without any addi tional constraints. \nMicrostructural characterisations were performed on a JEOL 6560 SEM equipped with \nan EDS analyser.  \nX-Ray Microprobe mappings were obtained from a CAMECA SX-100 apparatus (15 \nkV, 10 µA).  \nDielectric measurements were performe d on a Wayne–Kerr 6425 component analyser \nunder dry helium, using gold electrod es, at frequencies ranging from 102 to 2.105 Hz.  \nFerroelectric hysteresis loops were perfor med using an analogical Sawyer Tower \ncircuit with compensation of stra y capacitances and resistances. \nMagnetic hysteresis loops were obtai ned on a SQUID ma gnetometer (MPMS, \nQuantum Design Inc.). \n 5Results and Discussion \n \nAfter standard solid state processing of  powders and ceramics, X-Ray Diffraction \n(XRD) evidenced the successful stabilisation of  the tetragonal tungsten bronze structure for \nall the compositions . \nHowever the XRD diagrams of the Sm, Eu and Gd compounds displayed a few \nadditional lines consistent with the presen ce of a Fergusonite phase of formula LnNbO 4. \nElectron microscopy analysis confirmed the XRD results. We investigated various elaboration \nprocedures in order to avoid this secondary pha se formation. This allowed us to observe the \nlimited thermal stability of TTB ceramics, whic h start decomposing at about 1360°C with \nformation of Hexagonal Tungsten Bronze (HTB) and Barium Ferrite, both identified during \nSEM experiments. However thanks to improve d synthesis and sintering steps during the \nprocessing, the amount of Fergusonite could be decreased in Sm, Eu and Gd samples. HTB \nand Barium Ferrite secondary phases remained undetectable from XRD and SEM analysis. In \naddition we point out that Fergusonite phases have no particular dielectric and magnetic \nproperties in the investig ated temperature range. \nRietveld refinements were performed us ing an averaged model with a “pseudo-\ntetragonal” symmetry (space group Pba2, n°32).  The refinements converged to satisfying \nagreement factors (Table 1, R B ≈ 6, χ² ≈ 2) and confirmed the stabilisation of the TTB crystal-\nstructure. The cell parameters are reported in table 1, the b parameter is not reported because a \nand b refined values were undifferentiated. The Ri etveld plot for the Sm representative, in \nwhich the SmNbO 4 secondary phase is taken into account, is displayed as an example in \nfigure 2 (corresponding atomic coordinates are ga thered in table 2). One can notice significant \nresidues in the difference pattern, which could not be taken into account by any alternative \nstructural model. There are also  some discrepancies of the B iso atomic displacement factors \nfor the octahedral sites (Nb5+ and Fe3+ ions). These features, and th e undifferentiated values of \na and b orthorhombic cell parameters, suggest that  in these TTBs the crystal structure may be \nmodulated. However the “pseudo-tetragonal” stru ctures, even if they are averaged ones, \nprovide a significant amount of re liable information that allows us to extract the trends \ngoverning the crystal-chemistry of these TTB multiferroics. \nIn these materials the octahedral fram ework is statistically occupied by Fe3+ and Nb5+ \nions while the Ba2+ ions occupy the pentagonal channels. The rare earths occupy the square \nchannels. It should be noted that in the case of the gadolinium repres entative, the Rietveld \n 6refinement shows that Ba and Gd atoms are distributed other both pentagonal and square \ntunnels. \nThe crystal structures observed in the Ba 2LnFeNb 4O15 TTBs are related to the \nintroduction of the rare-earth cations in the square-shaped tunnels (see fig. 1) which normally \ndefine a 12-coordination. The Ln3+ ions are reluctant to adopt su ch an environment, and this \nresults in displacements of the O2- ions defining the surrounding octahedra. The cations \noccupying the octahedral sites are almost unaffected by this distorsion. In particular, although the 4c general position allows any kind of displacement, the corresponding (Nb\n5+, Fe3+) \ncationic sublattice remains almost perfectly te tragonal. The accommodation of the rare earth \nthus induces distortions in th e octahedral framework. If corre lated, these distortions may \ninduce a modulated crystal structure . We note that, even in abse nce of Ln substitutions, the \nTTB network itself is likely to produce modulated struct ures, as in the case of Ba 2NaNb 5O15 \n[8,9].  \nThe influence of the Ln3+ ions on the TTB framework can be illustrated by the \nevolution of the cell volume (tab. 1). The io nic radius of lanthani des ions is known to \ndecrease with increasing atom ic number. Thus the cell volu me of the TTB matrix should \ndecrease accordingly, when going from La to Gd. This trend is observed while going from La \nto Sm representatives, but the following represen tatives, Eu and Gd, po ints out a different \ncrystal chemical behaviour. In particular the increase in the cell volume of the Gadolinium composition  is probably related to the Ba/Gd statis tical distribution ove r both pentagonal and \nsquare channels suggested by th e corresponding Rietveld refinement. Taking into account the \npresence of fergusonite secondary phases in Sm, Eu and Gd TTbs, these observations suggest \nthat the amount of smaller rare-earths that can be accommodated in the TTB framework is \nlimited. This limitation is probably related to the distortion induced by the accommodation of the rare earth, depending on its ionic radius. These statements allow separating rare earth \nsubstituted TTBs into two groups, on crystal-chem ical basis: La-Nd (total or almost total \naccommodation of the rare earths) and Sm-Gd (p artial accommodation of the rare earths and \ndetection of LnNbO\n4 from XRD)). \nFrom Scanning Electron Microscopy, the cer amics are highly densified (compactness \n> 95%) with micronic particles shaped like facete d rods at the surface. EDS analysis indicated \ncationic ratios in agreement with the expected 2:1:1:4 Ba:Ln:Fe:Nb ratio associated with the \nBa2LnFeNb 4O15 composition, although Ln ratio was slightly lower in Sm, Eu and Gd \nsamples. Traces of fergusonite phase were detected in the Pr and Nd samples, suggesting that in both these materials the accommodation of the ra re earth is not complete. Thus SEM results \n 7confirm that the tetragonal tungsten bronze struct ure has been successfully stabilized after the \nintroduction of paramagnetic cations and indicate that the microstructure of the ceramics is \noptimal.  \nDielectric measurements on the La sample did not reveal any ferroelectric transition \ndown to 82K, although the evolution of the dielectric permittivity suggests a possible \ntransition below 80 K, i.e.  beyond our experimental conditions. \nThe Nd, Sm and Eu representativ es are ferroelectric below T C = 323 , 405 and 440 K \nrespectively, as illustrated by figure 3a. This is confirmed by ferroelectric hysteresis loops \nwhich were obtained at room temperature, i.e.  below the transition temperature for the three \nsamples (figure 4).  \nOn the other hand, the Pr and Gd samples are relaxors-ferroelectrics below 170K and \n300K respectively (Fig. 3b). This  means that both the real and imaginary parts of the \ndielectric permittivity display a broad dielec tric maximum whose temperature shifts on \nsweeping the frequency. Because of this rela xor-ferroelectric behaviour, no ferroelectric \nhysteresis loop could be recorded  in these latter materials. \nThe details of the dielectric properties of all samples are gathered in table 2. One can \nnote that the dielectric tran sition temperature and the maxi mum permittivity follow similar \ntrends, although the Eu TTB displays a quite  high permittivity. However the dielectric \npermittivities in Tetragonal Tungsten Bronze ferro electrics can be affected by order/disorder \nphenomenon in the different kind of channels of this framework, and we suspect that this is \nthe case for Eu TTB. The polarisations extracte d from ferroelectric hy steresis loops for the \nNd, Sm and Eu TTBs are al so rather low, as compared to th e usual polarisations observed in \nTTB ferroelectrics (10-20 µC.cm-2). This could be related to a modulated structure (suggested \nby Rietveld refinements) that i nduces misalignments of the local  dipoles and thus reduces the \noverall polarisation. \nThe dielectric studies show that a ferroel ectric behaviour can be obtained at room \ntemperature and above in some of these magn etically substituted TTBs. Rietveld studies \nindicate that the electric dipoles in the Nd, Sm and Eu samples origin ate from an off-center \nposition of the Nb5+ ions within the octahedral sites. Al l these sites display one short and one \nlong axial bond length, in agreement with a stat istical distribution of niobium and iron over \nthe whole octahedral framework. Off-center positions of the Nb5+ ions are also observed in \nthe relaxor Ba 2GdFeNb 4O15, but display inhomogeneous distorti ons in the different octahedral \nsites.  \n 8The evolution of the ferroelectric propertie s in the Nd, Sm and Eu compounds can be \nrationalised from a crystal-chemical point of  view, which could be resumed as follows:  \n- The smaller the rare earth, the greate r the distortion of the TTB framework \n- The greater the distortion of the TTB framework, the greater the permittivity \n- The greater the distortion of the TTB fr amework, the higher the Curie temperature \nThese considerations indicate that the accommodation of the rare earth in the TTB \nframework is the critical parameter impacting the dielectric properties in this family of \nmaterials.  \n After this description of ferroelectric properties we will now focus on the magnetic \nproperties observed in these samples at room temperature.  \nMagnetic hysteresis loops at 300 K have been  obtained from all the samples except the \nLa one (fig 5). The 20 kOersted s magnetisations are comparable, while the coercive fields are \nabout 500-700 Oersteds for Pr, Eu and Gd samples and about 2000 Oe for Nd and Sm \nsamples. It should be noted that all the sample s, except the La one, are strongly attracted to a \nNdFeB magnet at room temperature. \nThe details of the dielectric and magnetic  properties of the studied samples are \ngathered in table 3. It is wo rth mentioning that the room te mperature magnetizations of the \nTTB samples evolve in a similar way as the diel ectric transition temperature. As was observed \nfor the dielectric properties, the variation of the magnetic properties with respect to the rare earth shows that the Ln\n3+ ions play a critical role concerning the magnetic behaviour of our \nTTBs samples. \nThe magnetic study suggests a successful ma gnetic substitution, as  all the samples, \nexcept the La one, exhibit hyst eresis loops at room temperature. The existence of a \nspontaneous magnetisation at room temperature in these samples, however, is surprising \nconsidering that : \n- The Ln3+ ions are not likely to be magneti cally ordered at room temperature \n- The Fe3+ ions are diluted and disordered in the [NbO 6] octahedral framework \nThis latter statement in particular, is not in favour of a long range magnetic order. \nThus we will now discuss the possible links between the crystal-chemistry and the magnetic \nand ferroelectric properties of these niobate s, which may account fo r the origin of the \nmagnetic properties.  \nThe trends in magnetic properties, as well as  the evolution of dielectric properties, can \nbe correlated to the evolution of cell volume.  As shown by figure 6, the molar magnetisation \nof the TTB samples, measured at room temper ature under a magnetic field of 20 kOe, and the \n 9dielectric ordering temperature evolve inversely proportional with  respect to the cell volume. \nThis correlation is reproduc ible and is observed for several sets of samples.  Thus the \ncorrelation between the magnetic behaviour of TTBs and their structural and dielectric \nproperties may support an intrinsic origin for the multiferroic character of the Ba 2LnFeNb 4O15 \nTTBs (Ln = Pr, Nd, Sm, Eu, Gd).  \nHowever X-ray microprobe analysis, initially  performed in order to determine more \nprecisely the rare-earth compos ition in fergusonite-containing sa mples, revealed traces of a \nFe-rich secondary phase. This parasitic phase is  not detected within the La TTB (fig. 7b and \n7a respectively), which also do not exhibit magn etic hysteresis loops at room temperature. \nMoreover the Fe-rich phase cont ains a small amount of barium, suggesting a barium ferrite \nBaFe 12O19 composition, i.e a widely used permanent magnet. Thus the magnetic hysteresis \nloops observed at room temperature in most of the Ba 2LnFeNb 4O15 samples originate from \ntraces of barium ferrite.  \nThe presence of barium ferrite, and its am ount, can be related to the accommodation of \nthe rare earth in the TTB matrix. In the La sample, the La3+ ion is totally accommodated \nwithin the TTB framework, and no fergusonite phase LaNbO 4 appear. To maintain the charge \nbalance of the TTB phase, Fe3+ ions must be totally incorporat ed in the TTB framework of the \nLa sample. In the other samples (Ln = Pr, Nd, Sm, Eu, Gd) the ra re earths are partly \naccommodated, and the excess of lanthanide oxi de reacts with niobium oxide to form \nLnNbO 4 fergusonite. Thus, to maintain electrical neutrality of the TTB framework in these \ncases, iron must be only partly accommodated in the TTB  framework. The excess of iron \noxide reacts with a small proporti on of barium oxide to form barium ferrite related phases. It \nappears that the nature of the rare earth, and particularly its size, i.e. its ability to be \naccommodated within the TTB framework, controls  the appearance of the magnetic barium \nferrite phase in our samples. \nThis latter correlation indicates that the rare earth introduced in the TTB framework,  in \nfine, has a direct influence on it s chemical stability, its crys tal-chemistry and its physical \nproperties. Beyond the correlation shown in fig. 6, one has to take into account the additional \nand highly critical parameter of the chemical stability of these TTB compositions. \nBesides this correlation between chemical stability, crystal-chemistry and physical \nproperties, it is worth mentioning that all the TTB samples (except the La one), display both \nspontaneous polarisation a nd magnetisation and can be considered as composites \nmultiferroics. The particular interest of these multiferroic composites is that the magnetic phase is generated in situ, during the elabor ation of the sample, which should favour the \n 10quality of the interfaces between the ferroel ectric and magnetic part of the composite. \nAdequate chemical substitutions can also modify the amount and properties of this magnetic \nphase. Moreover, four of these composites materials display both ferroelectric and ferromagnetic hysteresis loops at room temperatur e. The particular case  of the Pr and Gd \nsamples should be noted, for they are, belo w 170 and 300 K respectively, two examples of \ncomposite relaxor multiferroics. Last but not l east, this work shows that it is possible to \nintroduce magnetic cations in a ferroelectric framework without  loosing the ferroelectric \nproperties.  \n \n 11Conclusion \n \nThe experiments reported in this paper de monstrate that five over the six studied \nTetragonal Tungsten Bronze samples are composite multiferroics below their \nferroelectric/relaxor Curie temperature. The multiferroic properties of these materials are directly influenced by the natu re of the rare earth accommodat ed in the TTB framework. Thus \nthe Nd, Sm, Eu and Gd representatives are multiferroic at room temperature and above. \nMoreover the Gadolinium and Praseodymium representative are relaxor multiferroics. The Ba\n2LnFeNb 4O15 niobates (Ln = Nd, Sm, Eu, Gd) consti tute a family of room-temperature \ncomposite multiferroics. Both the composite and single-phase approaches are still under \ninvestigation. \nWe would like to point out again our ini tial approach, which was to introduce new \ncrystalline frameworks in the field of multiferroic materials.  We hope that this work will \ncontribute to the widening of the crystal-chemical aspects in multiferroic materials, for such an approach may provide a substantial support to the design of single-phased multiferroics. \n \n \n 12Acknowledgements \n \nThe authors want to thanks L. Raison for the X-Ray microprobe experiments, and E. \nLebraud and S. Pechev for the coll ection of X-Ray diffraction data.  \nThis work is supported by the European Network of Excellence “FAME” \n(www.famenoe.net ) and the STREP MaCoMuFi ( www.macomufi.eu ). \n \n 13References \n \n[1] Zheng, H., Wang, J., Lofland, S.E., Ma, Z., Mohaddes-Ardabilli, L., Zhao, T., Salamanca-\nRiba, L., Shinde, S. R., Ogale, S. B., Bai, F ., Vielhand, D., Jia, Y., Schlom, D. G., Wuttig, M., \nRoytburd, A. &  Ramesh, R. Science (2004), 303, 661-663. \n[2] Kimura, T., Goto, T., Shintani , H., Ishizaka, K., Arima, T. & Tokura, Y. Nature (2003), \n426, 55-58. \n[3] Spaldin, N. A. & Fiebig, M. Science (2005), 309, 391-392. \n[4] N. A. Hill, J. Phys. Chem. B (2000) , 104, 6694-6709 \n[5] Mostovoy, M. Phys. Rev. Lett. (2006), 96, 067601 \n[6] Van Aken, B. B., Rivera J.-P., Schmid H. & Fiebig M. (2007) Nature 449, 702-705 \n[7] Fiebig, M., J. Phys. D.: Appl. Phys., (2005), 38, R123-R152. \n[8] Schneck, J.; Toledano, J. C.; Joffrin, C.; Aubree, J.; Joukoff, B.; Gabelotaud, A. Phys. \nRev. B, (1982), 25(3) , 1766-1785. \n[9] C Filipic, Z Kutnjak, R Lortz, A Torres-Pard o, M Dawber and J F Scott J. Phys.: Condens. \nMatter (2007), 19, 236206 \n[10] Magneli, A., Arkiv foer Kemi, (1949), 1, 213-221. \n[11] Jamieson, P.B.,Abrahams, S.C.,Ber nstein, J.L., J. Chem. Phys., (1969), 50, 4352-4363. \n[12] Ravez, J., Simon, A., C. R. Chimie, (2002), 5, 143-148. \n[13] Fang, P. H., Roth, R. S., J. Appl. Phys., (1960), 31(5) , 278S. \n[14] Krainik, N. N., Isupov, V.  A., Bryzhina, M. F., Agranovska ya, A. I., Kristallografiya, \n(1964), 9(3) , 352-357. \n[15] Schmid, H., Magnetoelectric Interaction Phenomena In Crystals, ed. A.J. Freeman & H. \nSchmidt (Gordon & Breach), (1972), 121-146. \n[16] Ishihara, S., Rivera, J.-P., Kita, E., Ye , Z.-G., Kubel, F., Schmid, H., Ferroelectrics \n(1994), 162(1-4) , 51-61. \n[17] Rodriguez-Carvajal, J., Physica B.(1993), 192, 55-69 \n \n 14Figures \n \n \n \n \n \nFigure 1 \n \n \n 15 \n**  \n \nFigure 2 \n \n \n 16 \n \n \nFigure 3a \n 17 \n \n \n \nFigure 3b \n \n 18 \n \nFigure 4 \n 19 \n \nFigure 5 \n 20 \n \n \n \n \n \nFigure 6 \n 21a \n \nb \n \nFigure 7 \n \n 22Figures captions \n \nFigure 1: Crystal structure of Ba 2LnFeNb 4O15 viewed along the c axis, showing the distorsion \nof the square channel. \n \nFigure 2: Rietveld Plot for Ba2SmFeNb 4O15 (dots: experimental, continuous line: calculated), \ninsert shows SmNbO 4 contributions (*). Upper vertical bars indicate TTB Bragg positions \n(lower bars: SmNbO 4). \n \nFigure 3: a) Dielectric properties of Ba 2LnFeNb 4O15 TTBs, Ln = Nd, Sm, Eu at 10kHz b) Ln \n= Pr, Gd, above vertical scale break: real part of the dielectric constant, below scale break : \nimaginary part of the dielectric constant (for clarity only the rele vant sections of the \nimaginary part are displayed). \n \nFigure 4: Ferroelectric hysteresis loops at r oom temperature for the Nd, Sm and Eu TTBs \n \nFigure 5: Magnetic hysteresis loops at room temperature for the Ba 2LnFeNb 4O15 (except La) \nTTBs \n \nFigure 6: Evolution of unit cell volume, mola r magnetization (RT, H = 20kOe) (divided by \n2.5 for scale purpose) and dielect ric ordering temperature of Ba 2LnFeNb 4O15 TTBs versus \nionic radius of the rare earth. \n \nFigure 7: X-Ray microprobe mappings of a Ba 2LaFeNb 4O15 (a) and Ba 2EuFeNb 4O15 (b) TTB \nsamples (BSE : backscattered electron image of  the mapped zone). White dots in the Eu and \nFe cartography of Ba 2EuFeNb 4O15 correspond to EuNbO 4 and barium ferrite, respectively. \n \n 23 \nLn a (Å) c (Å) V (Å3) RBB Rp Rwp Rexpχ² \nLa 12.514(1) 3.941(1) 617.2 5.74 15.1 13.7 10.35 1.77 \nPr 12.484(1) 3.928(1) 612.3 5.87 15.4 16.6 10.07 2.73 \nNd 12.478(1) 3.925(1) 611.0 6.45 21.9 19.4 12.65 2.39 \nSm 12.458(1) 3.928(1) 609.8 5.83 17.0 13.6 11.76 1.35 \nEu 12.460(1) 3.930(1) 610.2 7.06 19.8 17.0 11.55 2.17 \nGd 12.485(1) 3.936(1) 613.5 6.24 21.7 17.4 13.27 1.74 \n \n \nTable 1 \n 24 \n \nAtom Wickoff occupancy x y z B Biso (Ų) \nBa 4c 1.000 0.1656(2) 0.6768(2) 0.3510(6) 0.77(3) \nSm 2a 1.000 0 0 0.3480(7) 0.81(3) \nNb1 2b 0.800 0 1/2 0.8713(5) 0.45(5) \nFe1 2b 0.200 0 1/2 0.8713(5) 0.45(5) \nNb2 4c 0.800 0.2847(4) 0.4267(3) 0.8555(5) 0.02(2) \nFe2 4c 0.200 0.2847(4) 0.4267(3) 0.8555(5) 0.02(2) \nNb3 4c 0.800 0.5741(3) 0.2878(4) 0.8510(6) 0.02(2) \nFe3 4c 0.200 0.5741(3) 0.2878(4) 0.8510(6) 0.02(2) \nO1 4c 1.000 0.163(2) 0.499(2) 0.753(3) 0.70(13) \nO2 4c 1.000 0.490(2) 0.157(2) 0.850(2) 0.70(13) \nO3 4c 1.000 0.291(1) 0.775(1) 0.813(2) 0.70(13) \nO4 4c 1.000 0.351(1) 0.575(2) 0.733(3) 0.70(13) \nO5 4c 1.000 0.436(1) 0.378(1) 0.885(1) 0.70(13) \nO6 4c 1.000 0.211(1) 0.950(1) 0.358(2) 0.70(13) \nO7 4c 1.000 0.092(1) 0.190(1) 0.403(2) 0.70(13) \nO8 2b 1.000 0 1/2 0.346(3) 0.70(13) \n \nTable 2 \n \n \n 25 \n \n La Pr Nd Sm Eu Gd \nDielectric prop. - Rel. Ferroel. Ferroel. Ferroel. Rel. \nTC or T m (K) - 170 323 405 440 300 \nε’max (1 kHz) 178  186 206 266 731 137 \nPS (µC/cm2) - - 0.53 1.08 0.59 - \nMagnetic prop. (RT) - Hyster. Hyster. Hyst er. Hyster. Hyster. \nHc (Oe) - 540 1860 2150 580 700 \nM(2T) (e.m.u. / mol) 192.8 430.0 741.9 916.2 1004.5 554.3 \n \n \nTable 3 \n \n 26Tables captions \n \nTable 1: Cell parameters (b ≈ a) and volume, agreement factors of the Rietveld \nrefinement. \n \nTable 2: Atomic coordinates and isotropic atomic displacement parameters for \nBa2SmFeNb 4O15\n \nTable 3: Dielectric and magnetic properties of Ba 2LnFeNb 4O15 samples (ε ’max taken at \nTC or T m except for La TTB (80K)) \n \n 27"    },    {        "title": "1201.1629v1.Effect_of_mechanical_stresses_on_the_coercive_force_of_the_heterophase_non_interacting_nanoparticles.pdf",        "content": "Effect of mechanical stresses on the coercive force of the heterophase\nnon-interacting nanoparticles\nLeonid Afremov1and Yury Kirienko2;\u0003\nFar-Eastern Federal University, Vladivostok, Russia\n1afremovl@mail.dvgu.ru,2yury.kirienko@gmail.com,\u0003corresponding author\nKeywords: heterophase particles, mechanical stress, maghemite, cobalt ferrite, elongated nanopar-\nticles, coatings, interfacial exchange interaction.\nAbstract. The theoretical analysis of the effect of uniaxial stress on the magnetization of the system of\nnoninteracting nanoparticles is done by an example of heterophase particles of maghemite, epitaxially\ncoated with cobalt ferrite. It is shown that stretching leads to a decrease in the coercive force Hc, and\ncompression leads to its growth. The residual saturation magnetization Irsof nanoparticles does not\nchange. With increasing of interfacial exchange interaction, coercive force varies nonmonotonically.\nIntroduction\nIt is known that the reduction of size of the particle leads to an intensification of reactivity of the\nmagnetic material. As a result, it is natural to assume that small particles are rather heterophase than\nhomogeneous. Moreover, the formation of neighboring magnetic phases can be caused by processes\nof oxidation or disintegration of the solid solution (see, eg, [1, 2]) occurring in the magnetically or-\ndered grain. Most of the ultrafine magnetic materials of practical interest is two-phase or multiphase\nsingle-domain particles. For example, they are carriers of information in the magnetic memory ele-\nments, also they are widely used in modern biophysics. Numerous experiments with ultradispersed\nmagnetic materials discovered the dependence of such magnetic properties as coercive force, rema-\nnent magnetization and magnetic susceptibility on values and prehistory of the mechanical stresses\napplied to the samples. The inverse problem is of independent interest: to determine the magnetic\nprehistory of the sample from its known mechanical properties. This task is extremely important for\nthe magnetic measurements of fatigue of metal constructions, as well as in the study of the paleointen-\nsity. Owing to the research based on the dependence of magnetic properties on mechanical stresses,\nultradispersed magnets are used, for example, in the sensors of heavy load, in the technology of trans-\nformer cores, and control systems of fatigue of metal structures, in electronic article surveillance and\nmany other technical developments.\nModel\n1. Homogeneously magnetized nanoparticle (phase 1) of volume Vhas the form of ellipsoid with\nelongation1q1, and its long axis oriented along the Oz-axis (see fig. 1).\n2. Nanoparticle contains an uniformly magnetized ellipsoidal inclusion (phase 2) with a volume\nv=\"Vand elongation of q.\n3. The angle between the long axes of the particle and the inclusion is \u000b.\n1elongation — the ratio of the length aof semi-major axis of the ellipsoid to the length bof semi-minor onearXiv:1201.1629v1  [cond-mat.mes-hall]  8 Jan 2012O ZY\nα/tildewideZ\nI(1)\ns\nθ(1)I(2)\ns\nθ(2)\n1\n2H σFigure 1: Illustration of the model of two-phase particle\n4. It is considered that the axes of crystallographic anisotropy of both uniaxial ferromagnets are\nparallel to the long axes of the ellipsoids, and the vectors of spontaneous magnetization of\nphases I(1)\nsand I(2)\nslie in the plane yOz, that contains the long axes of the magnetic phases, and\nmake angles \u0012(1)and\u0012(2)with theOzaxis, respectively.\n5. Both external magnetic field Hand uniaxial mechanical stresses \u001bare applied along the Oz-\naxis.\n6. The volume of nanoparticles exceeds the volume of superparamagnetic transition.\nResults\nThe calculation of the magnetization, held within the framework of the above mentioned model of\ntwo-phase particles — maghemite ( \r-Fe2O3) epitaxially coated with cobalt ferrite ( CoFe 2O4) — is\nshown in fig. 2. It is easy to see that stretching shifts the magnetization curves to lower magnetic\nfieldsH, and the compression leads to the opposite effect. At the same time mechanical stresses\ndo not affect the saturation magnetization, which is determined by the thickness of cobalt coating.\nThese results are determined by the dependence of the critical fields of magnetization reversal on the\nstresses: stretching decreases the critical fields of magnetization reversal and compression increases\nit, and the coercivity of the particles changes consequently.\nCoercivityHcdepends not only on stress but also on the magnitude of the exchange interaction\nthrough the interface Ainand on the relative amount of cobalt coating \u001c= 1\u0000\". WhenAin= 0or\nAin= 3\u000210\u00008erg/cm coercivity of the particles increases monotonically with an increase in the\nrelative volume of coating, whereas for Ain=\u00003\u000210\u00008erg/cm behavior of the Hcis nonmonotonic\n(see fig. 3).\nIn addition, a negative exchange interaction leads to a decrease in the coercive force Hcas com-\npared withAin= 0, and positive — to its increasing. Features of the dependence of the coercive\nforce of the interfacial exchange interaction Ainshown in fig. 4. One can see that the nonmonotonic\nbehaviorHc=Hc(A)is characteristic of nanoparticles with a large ( \u001c= 0:9) or small (\u001c= 0:1)\nthickness of the cobalt coating and is being implemented in both positive and negative values of Ain.\nThe nonmonotonic behavior of the coercive force can be explained by considering the ratio of\nthe exchange interaction Ainand interfacial magnetostatic interaction Ams. WhileAin> Ams, thecoercivity of the system is determined by the critical field of magnetization reversal from a state with\na parallel orientation of the magnetic moments of the phase of nanoparticles to the state in which\none phase (conventionally, the first ) is antiparallel to the external magnetic field Hand the other par-\nallel to it. According to [3], this critical field decreases with increasing of Ain. WhenAin< Ams,\nnanoparticles with parallel magnetic moments of the phases remagnetized by switching of the mag-\nnetic moment of the first phase into a state parallel to the field H. The critical field of magnetization\nreversal in this case should increase with increasing of Ain.\nNoted above nonmonotonic behavior of Hcdid not observed in [4, 5], which is obviously as-\nsociated with a narrower, than in the present study, spectrum of the critical fields of magnetization\nreversal of two-phase particles. A qualitative comparison of the results with similar calculations pre-\nsented in [5–7] shows that, just as in these papers, with the growth of phase CoFe 2O4, coercivity\nincreases up to saturation.\nReferences\n[1] F. Stacey and S. Banerjee, The physical principles of rock magnetism . New-York: Elsevier, 1974.\n[2] A. Gapeev and V . Tselmovich, “The microstructure and domain structure of multiphase oxidized\ntitanomagnetites,” Physics of the earth and planetary interiors , vol. 70, no. 3–4, pp. 243–247,\n1992.\n[3] L. Afremov, Y . Kirienko, and T. Gnitetskaya, “Influence of mechanical stresses on the magnetic\nstate of dual-phase particles,” in Proceedings of the 8th International Conference Problems of\nGeocosmos , (St.-Petersburg), pp. 306–310, 2010.\n[4] G. Sawatzky, F. Van Der Woude, and A. Morrish, “M ¨ossbauer study of several ferrimagnetic\nspinels,” Physical Review , vol. 187, pp. 747–757, 1969.\n[5] J.-S. Yang and C.-R. Chang, “Magnetization curling in elongated heterostructure particles,” Phys.\nRev. B , vol. 49, no. 17, pp. 11877–11885, 1994.\n[6] J.-S. Yang and C.-R. Chang, “The influence of interfacial exchange on the coercivity of acicular\ncoated particle,” Journal of Applied Physics , vol. 69, no. 11, pp. 7756–7761, 1991.\n[7] A. Aharoni, “Magnetization buckling in elongated particles of coated iron oxides,” Journal of\nApplied Physics , vol. 63, no. 9, pp. 4605–4608, 1988.0 1 2 3 4 5 6H,kOe 050100150200250300I,G\nt=0.1\nt=0.5t=0.9\nks=-0.8\n0 1 2 3 4 5 6H,kOe 050100150200250300I,G\nt=0.1\nt=0.5t=0.9\nks=0\n0 2 4 6 8H,kOe 050100150200250300I,G\nt=0.1\nt=0.5t=0.9\nks=0.8Figure 2: The effect of mechanical stresses k\u001b= 3\u0015100\u001b=KAand the relative volume of the cobalt\ncoating\u001c= 1\u0000\"on the magnetization of elongated nanoparticles (elongation q1= 3, constant of\ninterfacial exchange interaction Ain= 0,\u0015100andKAare magnetostriction constant and anisotropy\nconstant, respectively)\n0.0 0.2 0.4 0.6 0.8 1.00.40.50.60.70.80.91.0hc\nks=-0.8ks=0ks=0.8Ain=-3´10-8ergcm\nt\n0.0 0.2 0.4 0.6 0.8 1.00.40.50.60.70.80.91.0hc\nks=-0.8ks=0ks=0.8\nAin=0\nt\n0.0 0.2 0.4 0.6 0.8 1.00.40.60.81.01.2hc\nks=-0.8ks=0ks=0.8Ain=3´10-8ergcm\nt\nFigure 3: Dependence of the relative coercive force hc=Hc=Hc1of elongated nanoparticles on the\nrelative volume of cobalt coating \u001cand on the value of the exchange interaction through the interface\nAinifq1= 3,Hc1= 2947 erg\n-4 -2 2 40.50.60.70.8hc\nks=-0.8\nA\n-4 -2 2 40.60.81.01.2hc\nks=0.8\nA\n-4 -2 0 2 40.60.81.01.2hc\nks=0\nA\nFigure 4: Dependence of the relative coercive force hc=Hc=Hc1of elongated nanoparticles on\nthe value of the exchange interaction through the interface Ainand on the relative volume of cobalt\ncoating\u001c(solid curve corresponds to \u001c= 0:9, dashed curve — to \u001c= 0:5, dot-dashed curve —\n\u001c= 0:1) ifq1= 3,Hc1= 2947 erg"    },    {        "title": "1303.0205v2.Dispersive_Casimir_Pressure_Effect_from_Surface_Plasmon_Quanta_by_Quasi_1D_Metal_Wires_in_Ferrite_Disks_and_The_Josephson_Frequencies_and_Currents.pdf",        "content": "--,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 1 \n 1 \nDispersive Casimir Pressure  Effect from  Surface Plasmon Quanta \nby Quasi 1D Metal Wire s in Ferrite  Disks  and The Josephson \nFrequencies  and Current s \n \nMahmut Obol  \n \n Ferrite s are distinct  material for electromagn etic applications due to  its unique  spin precession. In this pap er, \nCasimir  pressure effect by deploying magnetically tunable surface plasmon quanta in stratified structure of using ferrite \nand metal wires  is presented . Previously , oscillating surface plasmon quanta were successfully included to m odify  first \nreflection and fir st transmission  characteristics . The oscillating surface plasmon quanta in the modified reflection in such \na system, not only  does  resolve  in a typical matter  in metamaterial, but also provide new applications  such as creating \nCasimir pressure effects through the metamaterial composite shown in  this paper.  The Casimir pressure flips from \nattra ctive state to repulsive state is referred to  cause mechanism of radiation from surface plasmon quanta . Both  Casimir \nforce analysis and  the measured data  of radiations  indicate us the system develops quantized states  by electric flux \ninduced by ferromagnetic resonance . Quantum analysis is used to understand  the discrete radiations  spectra  for our  \nexperimental measurement . The discrete radiations  are reproduced by using time dependent  Schrödinger  representation . \nAs result , we find the Josephson frequency  and Josephson current representations at room temperature  and we used \nthem for extrapolating  voltage induced  in excited ferrites . Josephson frequency at X -band is able to differentiate micron \nvolt difference s and it allows us to report the data for voltage induced by ferromagnetic  resonance in ferrite at room \ntemperature. It is understood that the radiation intensity depends on density of final states and excitation probability \nwhen we come to think  the energy matter.  It seems  possible to create as high as 2 0mW microwave power inside \nwaveguide at X -band.   \n \nI. INTRODUCTION  \n The concept of ne gative refractive index material (NIM) was introduced in 1968 by Vese lago [1] and a fabricated \nmetamaterial composite was  reported in 2000 by Smith  et al.  [2]. The study  of NIM has  been  very popular recently  due to its \nassociation  with extraordinary electromagnetic characteristics  including  superlenses [3]. In general, the electromagnetic \nproperties  of a normal material  are defined by permeability µ = µ r - jµi and permittivity ε = ε r – jεi.  In NIM, t he permeability and --,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 2 \n 2 \npermittivity [4], change their sign from positive to negative , that is , - µ = - µr + jµ i and - ε = - εr + jε i. It appears that a normal \nmaterial with a positive absorption  corresponds to  negative absorption in NIM. Although such unusual  appearance in  \nmetamaterials has been understood  by using causality [5] , such negative  absorption  is beyond  the concepts of classical  \nelectrodynamics  [4].  \n Recently, the theoretical concept of a ferrite with metal wires involving  metamaterial composite  was reported by  \nRachford et al. [6] . We applied this  concept  by fabricating  a similar me tamaterial composite of metal wires and ferrites, and were \nable to demonstrate a characteristic measurement in terms of  permea bility and permittivity . We immediately encountered  \napparent negative absorption  phenomena . We introduced surface plasmon quanta  [7] in our meta material composite, which  \nremoves  atypical phenomenon  of negative absorption . The  measured permeability and  permittivity  can be expressed differently \nfrom before  as follows ; - µ = - µr - jµi and - ε = - εr - jεi. These  show both real negativ e permeability and real negative \npermittivity for NIM . The absorption components  of metamaterial  however,  remain the same as in normal material s. It uses no \ncasualty principles throughout  our measurement  process . This normalization comes  with the inclusion of  surface plasmon  quanta  \n[7] in NIM .  \n In this study, a metamaterial composite constructed by met al wires and ferrites disks was  used. Here  ferrites were \nresponsible for creating  negative permeability and the metal wires were responsib le for inducing  negative permittivity  [8]. We \nshall refer to  this metama terial composite of  metal wires and  ferrite disks as MWIFD  in this paper.  The spe cific metal wire  \nconfiguration alone  eliminates the electromagnetic wave propagatio n through MWIFD because of  the plasmonic  metal wire  \nstructure  inside the waveguide [8] . The nullified  electromagnetic wave propagation does get recover ed when  spin precessions are  \ndriven around the metal wires by RF in the presence of an external magnetic field.   Evidentl y, metal wires inside  MWIFD \ncomposite , gain induced transient currents in  the presence  of RF and spin precession s in excited ferrites.   \n To find a solution or a derivation of  Maxwell equation s for the system thus become s quite a complex , if one cannot  \ndistinguish the se currents appropriately .  Although  numerous numerical Maxwell solvers are available, simultaneous dispersive \npermeability and permittivity  characteristics  present  enormous  challenges to using  those simulation codes  in negative refractive \nindex media  [4]. Despite those challenges, finite difference time domain simulation technique s were  successfully implemented in \na ferrite based negative index medium  [6]. Most r ecently, a simultaneous permeability and permittivity measurement of ferrite  \nwas also reported  by [9] . We noted that magnetically excited ferrite s not only  create  dispersive permeability  characteristics  but \nalso create di spersive permittivity  within the ferromagnetic resonance spectrum.  This is attributed to induced electric flux along \nthe magnetization axis which creates  ferromagnetic re sonance in ferrites and  show s an excellent agreement with  recent work  \n[10], where  voltage  enhancement is seen  by uniform ferromagnetic resonance  excitation  in a yttrium iron garnet . Also, i n a \nrecent repor t [11] , O. Mosendz et al. showed  the voltage induced by inverse spin hall effect  and the  voltage  induced were shown --,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 3 \n 3 \nby using  the terms of  AC spin current and DC spin cur rent. Authors  [11] noted that the voltage in duced  at ferromagnetic \nresonance spectrum is proportional to the sample length. The electric flux induced by ferromagnetic resonance [9] is also \nproportional to the sample leng th along its  magnetization axi s and we noted that this finding  also finds very good ag reement with \na recent report  [11].   \n To properly understand  internal wave mechanism  of propagating mode  in MWIFD, this paper introduces a dielectric \nvector potential concept  into the system of MWIFD.  In doing so, the induced electric flux contributions from  spin flow \nprecessions in excited ferrite s in MWIFD  can interact with  metal wires  perfectly  by theoretical consideration . This  extends  \nmagnetostatic limit into levels  of Maxwell equations due to  spins precession  around the metal wires as well as the ma gnetization \naxis of the magnetic medium. This enables us to construct an additional  set of Maxwell equations, which can recover the  \nevanescent surface plasmon mode  by using  spin flow along  RF wave propagation direction in the excited  ferrite disks. The \nrecreated wave propagation in MWIFD composite is defined as  a cross product of two different evanescent modes  in this study . \nNamely Feynman’s notation  [12] was used  to derive  propagation of such surface plasmon s in MWIFD successfully . Then t he \npropagating surface plasmon quant a from evanescent waves were used to derive  Casimir  pressure  effects . Thus, it is natural for \nus to propose  Casimir pressure effects in either dielectric or magnetic mirrors. Obtained  Casimir pressure in this study it does flip \nits signs  from  an attractive st ate to a repulsive state  when its  refractive index  and reflectivity do so. The obtained discrete Casimir \nforces and the obtained discrete radiation spectrum indicate us system develops quantized states  by voltage indu ced by \nferromagnetic resonance. As such we used the Schrödinger representation to derive Josephson frequency and Josephson current  \nrepresentations at room temperature . By using the Josephson frequencies, we are able to extrapolate the voltage induced by \nferromagnetic resonance in YIG, and it implies that we are able to differentiate micron voltage differences in the system.  When \nwe come to think radiation intensity issue s in terms of energy matter, we realized that the density of states is  important.  We \npresent details of this study  in sections below.  \n \nII. THE MWIFD  COMPOSITE  IN WAVEGUIDE  AND DIELECTRIC VECTOR POTENTIAL  \n Ferrite disks t ransversel y magnetized ferrite disk to  a perpendicular direction  of wave propagation in a rectangula r \nwaveguide have  been known for a  long tim e. Normally,  ferrites are used to  understand  Maxwell ’s equations and  equation of \nmotion for spin precessions.  However, a composite of MWIFD inside a waveguide  is difficult to describe with  Maxwell 's \nequation s due to spin precession around the metal wires  (see Fig.1) . In this experiment , we apply an external static magnetic field \nalong  Y axis. T he electromagnetic  wave  TE 10 propagates  along  Z-axis.  Our experimental observation  shows that in this \nconfiguration the inserted metal wires  in ferrite disks  are fully capable of eliminating the  electromagnetic wave propagation  in --,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 4 \n 4 \nwaveguide  in the absence of an  exter nal magnetic field . An appropriate external magnetic field along the Y -axis is ap plied to the \ncomposite perpen dicular to the wave propagation.  This process recovers certain spectrum of  propag ation of  nullified \nelectromagnetic  wave throug h the waveguide.  This is a complex system , because it posses R F electromagnetic waves  and spin \nprecession associated  waves  with additional internal magnetic fields  [13, 14 , 15, and 16]. This study does not specify  the spin \nprecessions associated waves either from nonreciprocal surface wave s or reciprocal bulk wave s from excited ferrite ; one may \nrefer to the report  [14] for detailed analysis of how to differentiate them.     \n We now present expression for  effective permittivity of a surface plasmon state of this  system is shown in equa tion (1) \nand t he expression for  plasm onic [8] resonance frequency as well as  permittivity of nine parallel aligned metal wires in a unit \ncell (Fig.2) is shown in equation (2). The metal wires were considered as a negative permittivity medium using the MWIFD \ncomposite in our formulation p rocess. A multilayered non homogeneous  made up medium is loaded inside waveguide , see fig.1 . \nIn order to continuously use the TE 10 mode for this  configuration inside waveguide, the configuration needs a medium \nhomogenization constraint  as shown below  \n2 2\nmmx\nffx\nnk\nnk . One easily  writes propagation  constants for  ferrite medium and \nplasmonic medium inside waveguide , they are  \n2\n2 2 2cn k km mx mz and \n2\n2 2 2cn k kf zx fz . So, one obtains \n2 222\n0\nf mfm\nsp fz mzn nnn\nck k k\n and this configuration  allow s us to write the evanescent  wave inside waveguide as follows.  \nzjk\nspspe0\n, \nsp spck 0 , and \n1\nf m ffm\nsp\n  (1) \nwhere  \n)/ln(2\n22\n0\n02\n2\ndb ac\nme n\neffmeff\nm\n  \n and \n2\n1\n\nm\nm        (2) \nMoreover,  ferrite disks in such system have  spin flow precession s.  Usually spin flow associated waves  can be replaced b y an \ninternal magnetic field [13 ], \ndh and there is a requirement known as  magnetostatic limit  \n0dh . The magneto statics \nassociated waves must be seen  if ferrites do not have  metal wires in them.  In a recent study for a YIG ferrite [9], we introduced \nan induced electric flux phenomenon due to the ferromagnetic resonance excitation in ferrite. Ferrites are insulator materials, so \nwe do not expect electri c charge and electric polarization (\ndp ) build up to the ferrite surface s due to  the induced electric flux  by \nferromagnetic resonance  in ferrite s. So, the induced electric  flux (\nmd ) [9] satisfies the follows \n0md and \n0dA , --,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 5 \n 5 \nwhere \nd m A d\n and \nd d d m e p e d \n )0 (4  . By introducing a dielectric vector potential  which implies that  it \nalso associates  harmonic oscillation component  in time , so it allows us to write  an induced  magnetic fields which referred  to the \ncontribution s from uniform s pin excitation and spin flow. As result , we have  \ntA\nchd\nd1 , where \ndA stands for dielectric vector \npotential.  \n \nFig. 1 A composite by metal wires and ferrite disks is located at the center position inside a rectangular waveguide . \n \nFig. 2 Metal wires of aluminum foil placed  in array  formation inside  the ferrite  disk. \nSpecifically, dielectric vector potential  itself alone  may represent  current loop  without  source , but it may  be considered as \nsource s of internal fields by spin  flow and induced electric flux  in excited ferrites . Correspondingly, we have an additional  set of \nMaxwell equation s for such expansion  in this study, as  shown below . \nd d d d ecj Acj Atch    1\n                   (3) \nd d hcj e  ]~[\n                                                       (4) \nwhere \n~ , Polder permeability tensor . By a pplying  principles of  rectangular waveguide , we also easily derive  to flow of surface  \nwave  as follows.  \nzjke\n\n,\nfck\n , and \n2 2f               (5) \nNow, w e have two different  evanescent modes  namely as ψ μ and ψ sp in MWIFD.  In early history  of microwave ferrite studies , \ntwo different parallel travelling waves coupling  phenomena were very well studied by H. Suhl [ 15] and J. R. Eshbach [16] for \nparametric ferrite amplifier purposes  although the limitations of spin instabilities near the ferromagnetic resonances kept the \nproblem challenging . Also, the parallel travelled waves coupling s in terms of evanescent waves case was  studied  by S. E. Miller \n[17]. Since we have two different parallel evanescent waves in here, t he successful propagatin g wave from evanescent waves \n--,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 6 \n 6 \ncoupling that may be expressed as follows , the displacement of the surface wave of spin precession around the surface plasmon \nstate that causes fluctuations of surface plasmon states and the vice versa . The mathematical expression kept using Feynman 's \nnotation  [12] for above  statement  and it is written as follows . \n jkzsp\n                                                   (6) \nsp spspjkz \n 0\n                                                 (7) \nUsing equations (6) and (7), we obtain  desired  propagation co nstan t for restored  wave propagation  by coupling  evanescent waves \ninside waveguide , is shown below ,  \n02\nspkk\n and \nsp f sp ncjcj                  (8) \nEquation (8) , stands for a propagation constant of restored  wave through MWIFD composite inside  waveguide  (see Fig.1) , \ndefined as\nspjk . Intuitively, now, the power flow  along  z-axis inside waveguide, which can be expressed as \ndmAAc S22\n.  Because  magnetic vector potential  Am of RF and introduced dielectric vector potential Ad of spin flow \nprecession are orthogonal to each other in nature.  It implies that  cross product of  dielectric  and magnetic vector potentials  are \nalso responsible to  transfer  power  flow by coupled evanescent surface wave s in waveguide .   \n Above analysis shows that one can  change  systems permittivity by using external magnetic field's influence while \nsystem permeability due to change. A pair nickel ferrite  disks with 0.5 mm thickness was used  to cover metal wire of aluminum \nfoil. The t able 1 represents the data summary for nickel ferrite s used for MWIFD composite  inside rectangular waveguide.  \n          Table 1  \na D b 4πM s H0 ∆fferrite  ∆fmetal  \n2.3cm  1.5mm 2.2mm  4kOe 4.5kOe 0.8G Hz 0.01G Hz \n2.3cm  1.5mm  2.2mm  4kOe  5.5kOe  0.8GHz  0.01GHz  \n \nThe magnitudes of transmission coefficients S 21 measured using vector network analyzer is presented in Fig.3  and Fig. 4 . \nAccording to various textbooks convention and technical definition, the S 21 is defined as  \n\n\n)1()2(\n21\nIVVS  , where  V(2)+ = voltage \nmeasured at terminal (2) and V I(1)+ = voltage incident upon terminal (1). The S 21 is usually called \"forward voltage gain\" or \n\"transmission coefficient\" , it is a tr ansmission from right to left through the network.  --,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 7 \n 7 \n In MWFID composite, t he experimental  data support the findings in equation (1) and (5). One cannot imagine the data \nin Fig.3 without accepting the concept the  permittivity  change  under external magnetic fields influence due to spin flow of \nexcited ferrite that sweeps the metal wires.   \n \nFig. 3 Measured S 21 parameters showed correspondence to the positive and negative permeability and permittivity spectra in Fig.3 for 4.5kOe and 5 .5kOe. The \nelimina ted wave  at 0kOe, and the eliminated wave  was almost recovered by m eans of full transmission at highlighted bands  in Fig.3. It implies that the negative \npermittivity in 0kOe becomes positive when the ferrite sees the external fields of 4.5kOe and 5.5kOe.  Addition ally, we measured a point at 1.3kOe field; it is \nconsist ed to the measurements  in [7]. Here, nickel ferrite is used in MWIFD composite.  \nIII. DISCRETE RADIATION  BANDS AND DISPERSIVE CASMIR PRE SURE  FLIPS  \n \n The experimental measurement  in Fig.3 confirmed the relationship between permeability and permittivity in MWIFD.  \nOur analysis  also indicates  that multiple re sonant sates existence in MWIFD . A pair YiG ferrite disks with a thickness ten mil \neach was purchased from Countis Laboratory  to cover m etal wires of aluminum foil. The table 2 represents  data summary for \nMWIFD composite inside rectangular waveg uide, where loading factor was into account to saturated magnetization due to very  \nthin YIG ferrite disks  were used in MWIFD .  Interestingly, the experimentation in Fig.4 showed the double resonant states which \nwere expected by  our analysis .  \nTable 2  \na d b 4πM s H0 ∆fferrite  ∆fmetal  \n2.3cm  2mm  2.2mm  1.2kOe  2.6kOe  0.03GHz  0.01GHz  \n2.3cm  2mm  2.2mm  1.2kOe  2.98kOe  0.03GHz  0.01GHz  \n \nThe measured magnitudes of transmission coefficient S 21 due to resonant permeability and permittivity spectra of MWIFD inside \nthe wav eguide are presented in Fig. 4, respectively. However, i nterpreting  the Fig.4 is  still difficult  within scope of  using \nconv entional knowledge of meta materials and magnetics  alone . It is that has been d ifficult for us to interpret multiple pass bands  \nby using published information  thus far .  \n--,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 8 \n 8 \n \nFig. 4 Measured  S21 parameters showed the  recreated pass  band for positive refractive index and negative refractive index and they fairly linear dependent on \nexternal magnetic fields  of 2.6kOe and 2.98kOe . The eliminated wave was recorded at 0kOe.  The measurement shows that wider pass band may correspond  to \nthe positive refractive index while  the narrower pass band corresponds to the negative refractive index.  The grey  circled region we don’t know them yet.  Here, \nYiG (yttrium iron garnet) is  used in  MWIFD.  We recall this figure in section 4, where Josephson  frequencies are used to interpret the discrete radiation s in here.   \n       The Casimir effect was introduced by H.B. G. Casimir and E. M. Lifshitz almost 60 years ago  [18, 19 ]. The use  of surface  \nplasmon modes  for Casimir forces was described recently by Munday et al.  and Zao et al.  [20, 21 ]. The effects of m acroscopic \nquantum electrodynamics (QED) were also shown  by Buhmann  et al.  [22] to a magneto -electric medium,  where a dispersive \nCasimir force  was shown in terms of  permeability and permittivity  of electromagnetic wave medium .  Recently, a dynamic \nCasimir effect by using Josephson metamaterial was also reported in [23]  by L. Lahteenmaki et al. , and the report showed that \nenergy correlated  photon generation by touching the ground state energy in vacuum.  All those  findings suggest  that there is a \nrepulsive Casimir pressure effect in magneto -electric and metamaterial composites. For a composite o f quasi 1D metal wires in  \nferrite disks  described  here;  multiple reflections  [7] for Figs.1 and 2 were  considered using equation  [9] below.  \n   \n) 1()0(@2 2   dkj djksp sper re r                    (9) \nIn equation (9) , reflection  configuration  includes  oscillating surface p lasmon quanta  in the  first reflection of MWIFD. It \nresembles the  characteristics of partition function of discrete microstates in statistical dynamics. For  such matter of periodic \nmetal lic mirrors  in metamaterialized ferrite composite structure  in Fig.1 and 2 , we share the idea of Lifshitz who extended \nCasimir force into medium [19 ], and the Casimir energy  per unit area  is designed in respect  to Lifshitz  and Milloni  [19, 24] \nexpression s that may now  be written as, \n  \n\n\n\n\n\n djkj\nkkSsp\nsprer kdkd E\n1ln\n)2( 2\n02\n           (10)                                                                \n--,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 9 \n 9 \nWhere\n j , and the k space has o scillating surface plasmon quanta  that is represented by k sp.  By taking a condensed \nvariable  \ndncjsp , where n sp the refracti ve index of propagating surface plasmon  in MWIFD  and d is the width of each \nperiodic metal wires . We  have obtained a Casimir  force for a periodi c metal mirrors that lies between  ferrite insulator s as below . \n      \n\nd\nr er\nn dcF\nspS\n\n03\n421\n16~                                   (11) \nTo test  reasonableness of  equation (11), let us first take a look for two  extreme  ideal cases of the problem (lossless cases, for \nexample) . If reflectivity to be  r = -1 (\nsp ) we have obtained exact  Casimir force  [18] that is for frequency dependent \ndielectric mirror (\nsp ), shown in T able 3 . If reflectivity to be  r = 1 (\nf )we have obtained exact  Casimir force by \nBoyer  [25] and Hushwater  [26] that is for frequency dependent magnetic mirror (\nf ), also shown Table  3. \n                         Table 3 Casimir forces for ideal medium and ideal cases  \n r = -1 r = 1 \nCasimir f orces \n(ideal medium)  \n42\n240 dncF\nspS \n42\n24087\ndncF\nspS\n\n  \nCasimir f orces \n(if ideal medium \nbecomes air)  \n42\n240dcFS \n42\n24087\ndcFS\n\n  \n \nFor both cases, if  refractive index  of medium hovering  close to zero (\n0spn ), then  positive or negati ve Casimir  force \nbecomes enormous  Casimir pressures, which may be observed through the macroscopic systems. In order to work out equation  \n(11) for a dissipative system, one requires  refra ctive index of such system to remain  a measurable quantity throughout due to  \nKramers -Kronig dispersion relationships  [27]. This requirement  prevents unnecessary gains by  such systems in the calculation \nprocess. We use the following transformation  in here  \nz e, which  allows us  to move the equation (11 ) into a comp lex domain,  \nshown  in equation (12)  below . \n                           \ndzzrzz\nnr\ndcF\ncspS1)(ln\n16~3\n42                                   (12) \n2 13\n42\n~ ~)(ln1 11\n16~\nS Scsps\nF Fdzzrzz n dcF\n\n\n\n \n \nWhere  --,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 10 \n 10 \n\n\n\n\n\n\n03\n4 1)(ln\n8~\nzspSnzj\ndcF\n\nrzspSnz\nj dcF\n\n\n\n\n3\n4 2)(ln1\n8~\n\n \nThis is the Casimir pressure e ffect from surface plasmon quanta  in MWIFD  described here , which is able to interpret obtained \nexperimental data in Fig. 4 in terms of discrete radiation  pattern s. One easily finds discrete Casim ir pressures flips  by using \nequation (12 ), where the strict Cauchy integration t echnique  was used  for. The Casimir pressure flips may indicate  instantaneou s \npower release or absorption between quantized  energy state transition s of surface plasmon quanta  in MWI FD. By using usual \nconvention  of resonant states, it usually absorb s S21 transmissions . Pure negative and positive refractive index is achieved  below \nand beyond the resonant states  which  have  the limits  to interpret discrete pass bands in  experimental observations in this  paper . \nWe do expect the repulsive Casmir pressure effec t be further confirmed b y showing  such radiation in other experimental \ndetermination  (other than power ratio measurement, S 21). Such  radiation source s may be applied  as principal radiation source in \nbiological  tissue detection  techniques  as well as  renewable energy techniques.   \n \nIV. JOSEPHSON FREQUENCIES AND CURRENT S \n \n According to the recent  reports  in [9, 10, 11, 28, 29, 30 and 31 ], the ferromagnetic resonances and spin waves in ferrites \nthey offer a new  opportunity for creating electric flux or spin electromotive force by uniform spin excitations . In previous \nsection s, this study presented induced electric flux associated surface wave propagation and the dispersive  Casimir forces for the \ncreation of potential electromagnetic waves radiations.  Both Casimir force analysis and measured discrete radiation spectra in \nfig.4 show that MWIDS system develops quantized states.  Although the metal wires  in ferrite disks is  considered a plasmon \nmatter in terms of negative permittivity medium  for the surface wav e propagation ,  the metal wires is  also the  conductors who \nsees the induced electric flux by ferrite  excitation .  As result , we use d the Schrödinger  representation , where dynamics specified \ntime dependent wave function and it is a convenient approach  to de al with interaction between electrons in metal wires and  \nelectric flux  induced by ferromagnetic resonance  in ferrites.  The Schrödinger equation  is shown here as \n)()( ˆ )(0 t tvh t id t   \n, where  \n0ˆh stands  Hamiltonian  operator for  linear stationary electron  in MWIFD , and \nti\ndq d eve tv 0)(\nstands  electron (\nqe, electron  charge ) volt energy of an electron in metal wire via voltage  induced  by \nferromagnetic res onance in MWIFD . This electron volt energy via voltage  induced by ferromagnetic resonance  in MWIFD  must \nbe as \n) (0\n0 dqve h , so it can be considered a time dependent perturbation in the Schrödinger equation. By using Fermi's \nGolden R ule [32], one obtain s resonance excitation frequencies  (radia tion frequencies) as follows\n0\ndq\nnj ve\n . This simply --,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 11 \n 11 \nrepresents Josephson frequencies in the system which means electrons in the system  (MWIFD)  develop  quantized discrete states \nby electric flux induced by ferromagnetic resonance in ferrite.  According to the data in Fig. 4, we ar e able to calculate the \nvoltages induced for resonant excitations  as follows , 5.7µV at 8.8GHz , 6.7µV at 10.2GHz and 6.8µV at 10.4GHz . The order of \nmagnitude of obtained data very well agrees  with other reports  in [10, 11, and 30 ]. According to the Fermi's Golden Rule, we \nhave the transition probability as follows,\n)(2 20\nnj sp j dqn nj ve w \n , we will use it when  we come to think system 's \nradiation intensity in terms of energy matter . Over there \n)(nj , namely  density of states. Let us make a reasonable  estimate  to \nthe system 's power creation  for a possible transition by \n0\n0 2100\n200 3 ˆdqb dq eea zee  , where \n0ba , Bohr radius. S uppose we \nhave electric flux induced by ferromagnetic resonance as\n1 4 0105 Vm ed and density of states reach ed as \n1110)(nj sp\n(i.e., we know the density of free electron gas  in a good conductor  as\n3 221048.8 cm N  and one finds  electron  density in a n \nexcited  surface plasmon to be as \n1110Nsp ),  then the peak vol tage inside waveguide  about to be  5Volt, which means \nabout a 20mW microwave power flow inside waveguide of  X-band  possible . Apparently , it will be an  enormous power creation \nif we achieve  it.  In near future, we shall discuss density of states  for a study of  radiation intensity matter in a separate paper. \nMore r ecently , Barnes and Meakawa in their report [28] they showed the Berry phase inclusion in the Schrödinger equation  due \nto spin electromotive force by ferrite  and Meakewa also reported a Josephson current induced by ferromagnetic resonance in \nYIG [29]. An ISHE (inv erse spin hall effect) report [30 ] the authors showed pulsed  microwave excitation  that creates greater  \nvoltage signature compare d to the temporal evolution of dir ectly excited spin wave mode in YIG. We also noted that the inverse \nspin hall  by using spin torque dynamics  was also in great interest to amplify microwave signals [31 ]. \n According to those experimental evidences  and findings , we may be able  to write the S chrödinger representation here in \na new form as follows , \n \nt\ndqt\ndqdttve\ni dttve\ni\nt et h et i00\n00)(\n0)()(ˆ )(    , where \nqe electron charge .  One assumes we should  \nbe able to reach a phase condition w here the phase component in above  Schrödinger representation has to be integer value  as \nt\ndqn dttve\n002 )(\n, where  n, integer number and  Planck's constant as\nseV161058.6 . If so, o ne obtains  the \nquantized current flow  in metal wire  as follows,  \n\n\n\n \n\n\n\nLRne RLen\neh\nLen iqhk\nq qhkq\nqq\nld_\n_ 2 , where \nqhkR_ , \nKlitzing resistance and L, inductance . In our system of MWIFD the charge in metal wires has to be conserved , so one obtains  \ninduced charge distribution in metal wires as\ntLR\nqtLR\nldkqh kqh\nene eqq  _ _\n) ( , where \nq ld ne q . The above formulation s --,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 12 \n 12 \npresent  us two interesting  phenomena . One of them  is that  one can achieve quantized current flow at ro om temperature, which \nmay be  understood as Josephson current  [29] at room temperature . For example, i n fig. 4,  unlike  those  resonance radiati on \nspectra , they are also many unusual deep absorption cuts in  S21, one can see those deep  absorption  cuts at  9.9GHz, 11.1GHz and \n11.3GHz  in fig.4 . They may be the frequencies spectra where  we reached the condition s for Josephson currents.  Perhaps, they \nare the frequencies where DC Josephson currents available.  Furthermore, our charge distribution  shows that induced charge on \nmetal wires decays very fast if it is not located on negative inductance spectrum, which may be  an additional  indication why \npulsed microwave excitation better for induced voltage creation  [30].  \n  \nV. CONCLUSION  \n \n Ferrite materi als researches  scientists  reported inverse spin hall effect , spin electromotive force , electric flux induced by \nferromagnetic resonances (FMR)  and every other  voltage in duced by FMR . They all clearly related to the electromotive force , \nwhich can be represented by  E field . We found electric flux induced by ferromagnetic resonance in a ferrite which very much \nfinds agreement with  other report s. In this paper,  the concept of a dielectric  vector potential was also introduced into the system \nof MWIFD,  to generate  a new set of Maxwell equations  to recover  evanescent surface plasmon  due to flow by  bulk or surface \nspin waves  in ferrite . The d ispersive  Casimir  pressure  is derived  by using  surface plasmon quanta which removes atypical \nphenomenon of negative absorption  from a composite of 1D metal wires structure in ferrite disks.  The composite show ed both \nnegative permittivity and permeability s imultaneously  to be a metamaterial . The Casimir pressure flips from attractive state to \nrepulsive state is referred to the  cause of radiation from  surface plasmon quanta  in this paper . Evidently , repulsive  radiation \npressure  from Casimir pressure  flips due to energy state transition s of surface plasmon quanta  in MWIFD , which is understood \nas a cause  mechanism  to interpret  radiation pattern  of measurements  in terms of classical interpretation . Moreover, Schrödinger \nrepresentation was used here to derive Josephson frequency and Josephson current  for the MWIFD system . The Josephson \nfrequency allowed us to extrapolate the  voltage  induced  by ferromagnetic resonance in YIG. We also realized that density of \nstates is important to the radiation intensity in term s of energy matter.  This study  may be beginning of a new quest  in searching \nhigh power radiation sources by  using ferrite based metamaterials although quantization of the electrodynamics i n metamaterial \nmedium is another approach .  \n Author gratefully acknowledges that  expe rimental measurements in this paper used Microwave Materials Laboratory \nfacilities at Northeastern University, Boston MA.   \nREFERENCES  \n[1] V.G.Veselago, \"The electrodynamics of substances with simultaneously negative values of ɛ and µ\" , Sov. Phys. Usp. 10, 509 (1968).  --,  April 7,  2013  \nMahmut Obol (email:mahmut.obol@gmail.com) is with Biomagnomics, MA 02466  Page 13 \n 13 \n \n[2] D.R.Smith, W.J.Padilla, D.C.V ier, S.C.Nemat -Nasser , and S.Schultz, “Composite medium with simultaneously  negative permeability and permittivity” , \nPhys. Rev. Lett. 84, 4184 (2000).  \n[3] J. B. Pendry and D. R. 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Lifshitz, “The theory of molecular attract ive forces between solids \", Sov. Phys. JETP 2 73 (1956).  \n[20] Munday, Capasso and Parsegian, “Measured long range repulsive Casimir –Lifshitz forces” , Nature 457 (2009) 170.  \n[21] R. Zhao , J. Zhou , Th. Koschny , E. N. Economou , and C. M. Soukoulis , “Comparison of chiral metamaterial designs for repulsive Casimir force” , Phys. \nRev. Lett. 103, 103602 (2009).  \n[22] Stefan Yoshi Buhmann, Dirk -Gunnar Welsch, Progress in Quantum Electronic s, Vo. 31, Issue 2, Pages 51 -130 (2007).  \n[23] Pasi Lähteenmäki, G. S. Paraoanu, Juha Hassel, and Pertti J. Hakonen , \"Dynamical Casimir effect in a Josephson metamaterial\", PNAS 2013 110 (11) \n4234 -4238 ; published ahead of print February 12, 2013 , doi: 10.1073/pnas.1212705110 . \n[24] Peter W. Milonni and Mei Li Shin, \"Casimir Forces\", Contemporary Physics, 1992, Volume 33, number 5, pages 313 -332 \n \n[25] Timothy H. 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Karenowska, E. Sai toh, and B. Hillebrands, \n\"Temporal evolution of inverse spin Hall effect voltage in a magnetic insulator -nonmagnetic metal structure \", Appl. Phys. Lett. 99, 182512 (2011) . \n[31] Lin Xue, Chen Wang, Yong -Tao Cui, J. A. Katine, R. A. Buhrman, and D. C. Ralph , \"Conditions for microwave amplification  in magnetic tunnel \njunctions \", APPLIED PHYSICS LETTERS 99, 022505 (2011) . \n[32] L. D. Landau and E. M. Lifshits, \"Quantum Mechanics \", Vol.3, third edition, 1977, Elsevier Science Ltd.  "    },    {        "title": "1505.04136v1.Linear_roto_antiferromagnetic_effect_in_multiferroics__physical_manifestations.pdf",        "content": "Linear ro to-antiferromagnetic effect in  multiferroics: physical manifestations \n \nAnna N. M orozovska1, Victoria V. Khist2, Maya D. Glinchuk3, Venkatraman Gopalan4*, \nEugene A. Eliseev 3†\n1Institute of Physics, National Acad emy of Sciences of Ukraine,  \n41, pr. Nauki, 03028 Kiev, Ukraine \n2 National U nivers ity of  Water Mana gement and Nature Resources Use, \nRivne, 33028, st. Soborna, 11, Ukraine \n3 Institute for Problem s of Mate rials Science, National Academy of Sciences of Ukraine,  \n3, Krjijanovskogo, 03142 Kiev, Ukraine \n4 Departm ent of Materials Science and Engineer ing, Pennsylvania State University, University \nPark, PA 16802, USA \n \nAbstract \nUsing the th eory of symm etry and th e microscop ic model we predicted the possibility of a linear \nroto-an tiferromagnetic ef fect in the perovsk ites with struc tural antiferrod istortive and  \nantiferrom agnetic long-range orde ring and found the necessary condi tions of its occurrence. The  \nmain physical m anifestations of this effect ar e the sm earing of the antiferrom agnetic transition \nand the ju mp of the specific h eat near it. In  the ab sence of extern al fields linea r roto-\nantiferrom agnetic coup ling can in duce a weak antiferrom agnetic ordering above the Neel \ntemperature,  but below the tem perature of antife rrodistortiv e trans ition. Therefore, there is the \npossibility of  observing weak antif errom agnetis m in m ultiferroics such as bis muth f errite \n(BiFeO 3) at tem peratures T>T N, for which the Neel tem peratu re TN is about 645 K, and the \nantiferrodistortive tran sition tem perature is about 1200 K. By quantitative com parison with \nexperim ent we m ade estim ations of  the lin ear ro to-an tiferromagnetic ef fect in the so lid solu tions  \nof multiferroic Bi 1-xRxFeO 3 (R=La, Nd). \n                                                 \n* Corresp onding author: vgopalanpsu@gmail.com  \n† Corresp onding author: euge ne.a.elisee v@gmail.co m\n 1 \n1. Introduction \nMultiferroics, generally  defined as ferroics with  several ty pes of long-range order interacting \nwith each other, are unique m odel system s for funda mental phy sical studies of versatile \ncouplings between the spontaneous  polarization, magnetization, stru ctural and antiferrom agnetic \norder parameters [ 1, 2, 3, 4]. The m ost well-known and im portant  effect for applications of \nmultiferro ics is th e magnetoelectric coupling  between the polarization and magnetization , \nthrough which one can write information by an elec tric field and readout it by a m agnetic field \n[1, 4, 5].  \nGiven the u nique im portance of  multiferroics for wide variety of  applica tions, othe r types \nof coupling s are active ly inve stigated in  anti ferrodistor tive multiferroics in  addition to the \nmagnetoelectric coupling [ 6, 7, 8, 9, 10]. The couplin gs are asso ciated with  the presence of  \nstructural order param eter and its gradient. In  the case of a inhom ogene ous distribution of the \norder parameter, which is inev itable near the surface or in the presence  of developed dom ain \nstructure of ferroelectric, m agnetic  or structural types, there is a coupling between the various \norder parameters and th eir grad ients  [6-10]. Ther efore, according to the theory of symmetry, the \nflexoelectric-antiferrodistor tive and roto-flexo electric coupling  between structural, polar and \nmagnetic order param eters [ 11, 12, 13] can exist in an tiferrodis tortive m ultiferroics in add ition \nroto-m agnetic and roto -electrical coupling [ 14, 15]. The prefix \"roto \" com es from  the word \nrotation and indicates the static rotation of som e atom ic groups with respect to other parts of the  \ncrysta l [14]. In the wor k the term  \"roto -symmetry\" m eans only rotational symm etry of the  \noxygen octahedra MO 6 with respec t to the cube A 8 in antif errodistor tive perovskite s with the \nstructural form ula AMO 3. Oxygen atom s are displaced with  respect to the centers o f the cube \nfaces A 8 in the antiferrodistortiv e phase, the angle or the value of the corresponding \ndisplacem ent is a structural or der param eter (see e.g. [16]). \nCoupling between the various orders param eters can be bilinear, linear-quadratic and \nbiquadratic in the order param eters powers, depending on the exten t to which  the re levan t \nparam eter (or gradient) is proport ional to the physical effect it has generated [1]. Biquadratic \neffects exist for arbitrary symmetry of m ultiferroic [17, 18, 19, 20], but the v alues of the \ncorresponding tensor coupling constants strong ly depend on its shape and size [21]. The  \nappearan ce of non-zero bilinear effects is m aterial -specific,  they are determ ined by the spatial \nmagnetic and roto- symm etry of the m aterial (see  e.g. [4, 7, 14]). Consequently bilinear effects \nare significantly less common, but their physical m anifestations can be much m ore strong and \nnon-trivial, rather than the m anifestations of biquadratic eff ects [4]. Perhaps that is why \nresea rchers are ac tively \"hunting \" for bilinea r coupling ef fects in m ultiferroics. \n 2In this wor k we predicted the po ssibility of  linea r roto -antiferromagnetic coup ling \nexistence in perovskites with antiferrodistortive and antiferrom agnetic ordering and found the \nnecessary conditions of the coupling occurrence. Also we  discuss the m ain physical \nmanifestations of the effect, su ch as the sm earing of antiferro magnetic (AFM) tr ansition, specific \nheat jum p near the transition and weak antif errom agnetism above the Neel temperature. We \nchose m ultiferroic bism uth ferrite (BiFeO 3) as the  model m aterial. \nOur choice of BiFeO 3 is based on th e fact that th e material is  one of  the most prom ising \nmultiferro ic with a relativel y high m agnetoelectric couplin g coefficien t; it reveals  \nantiferrodistortive order at tem peratures below 1200 K; is ferroelectric with a high spontaneous \npolarization below 1100 K and antiferro magnetic below Neel tem perature TN≈(640 – 650) K [4, \n22]. Despite the great am ount of experim ental studies on the B iFeO 3 multiferroic prope rties [ 23, \n4, 5], m any im portant issues rem ain unclear in the sense of understanding of the physical \nmechanis ms responsible for the emergence and m anifestation of these properties [22]. In other \nwords, the theoretica l descrip tion of  BiFeO 3 physical properties is far behind the experim ent. In \nparticular ab initio  calculations, which allow determ ining the param eters of antiferrodistortive \nand antiferrom agnetic subsystem s, magnetoelectric, roto-magnetic and r oto-electric couplings of  \nthe co rrespo nding long -range ord er parameters w ith each o ther in BiFeO 3, are absent to date. On \nthe o ther hand re liable exper imental results,  which ana lyses, as  we will show b elow, allow \nmaking conclusions about the exclusive im porta nce of the roto-type couplings in BiFeO 3. \nBefore pres enting  the problem  statement and orig inal re sults,  let us make som e \ncomments about chosen research methods. As we di scuss the principal possibility of a new kind \nof interaction between two long-range orde r p arameters (antiferrodis tortive and \nantiferromagnetic ) in th e bulk of  multif erroic, in or der to e stablish th e existence of  a par ticular \ninteraction between the orde r param eters and unam biguously define the structure of  \ncorresponding m aterial tensor, one can use the th eory of symmetry, if its  spatial and m agnetic \nsymmetry group is known [14]. F unctional for m of the antife rrodistive-antif errom agnetic \ncoupling co ntribu tion to  the f ree energy and its e ffect on phase trans itions in m ultiferroics can be \nestab lished within th e framework of  the conti nuous m edium  mean-field  Ginzburg-L andau theory \n[11, 13]. However, it is im possible to define the value of the coupl ing strength, i.e.  to calculate \nnon-zero coupling constant for a given m aterial , using phenom enological approach and the \ntheory  of sy mmetry. One can  estim ate the s trength of antiferrodistive- antiferrom agnetic coupling \neither from  first principle quant um mechanic al calculations  within  a specific m icroscopic m odel, \nor from the f itting to ex perim ental d ata. Both of  these app roaches ar e indispensable  to determ ine \nthe coupling  constants and com plement each oth er well, but by them selves they are n ot free from  \ndrawbacks. Most of the first prin ciple calcu lation s (such as carried ou t in the fram ework of DFT)  \ndo not take into  accou nt correctly  inhom ogeneous long-range depolarization  electric field  in \n 3ferroics  and the totality  of the s tructural and magnetic antiferrodistortiv e modes, as well as d o \nnot allow to say anything about the tem peratur e dependence of the coupling constants. However  \nit is possible to extract the coupling constant from  the experim ent sufficiently precisely and \nunam biguously, if the contribution of  other effects is known with in error m argins. \nIn the work we consider step -by-step the m icroscopic picture necessary for the  \noccurrence of linear roto-antife rromagnetic coupling in antiferrodistortive antiferrom agnets wit h \nthe structural form ula AMO 3, establish th e trans formation la w of  linear  roto-an tiferromagnetic  \neffect tensor, find its nonzero co mponents of the theory  of symmetry and estim ate its num erical  \nvalue from the sm earing  of specific heat jum p near AFM transition for bism uth ferrite and its \nsolid so lutions Bi 1-xRxFeO 3 (R=La, Nd). \n \n2. Micros copic model f or bilinea r roto-ant iferromagnetic coupling (LRM) appearance  \nThe antif erromagnetic o rder par ameter of  the two-subla ttice a ntiferrom agnet is an ax ial vecto r L, \nthat is equa l to the dif ference of  magnetiz ation vectors of m agnetic atom s in two equivalent sub-\nlattice s А and В, () 2B AM M L−= ,  и  (see Figure 1a ). The \nantiferrodistortive order param eter is an axial vector, which is the angle of oxygen o ctahedra tilt \n. Below we use the equivalent form  of order param eters defined as the o xygen disp lacem ent \nfrom  symmetric position ∑\n=µ=A\niiB A g\n1S M ∑\n=µ=B\niiB B g\n1S M\nϕ\nϕ=Φ tana , which can be calculated as the product of pseudocubic \nlattice constant  with tangent of angle a ϕ. As a rule, the angle  changes its sig n in \nneighbouring cells, related to  different sublattices А and В, namely ϕ\nΦΦΦ ≡−=B A . The  \ncontribution  of bilin ear roto-m agne tic coupling into the free energy is described  be the following  \nexpression: \n() ()j iij j Bi Aiij\nBj Bi Aj Aiij L\nRM L M M M M g Φχ≡Φ−χ\n≡Φ+Φχ\n=\n2 2            (1a) \nEquation (1a) is invariant under the tim e inversion and the transl ation on the basic vector of \npseudocubic lattice, sin ce magnetiza tion vectors of each sub lattices M j change their signs, and \nthe sublattice A transform s into the sublat tice B under such translation, therefore vector Φ also \nchanges its sign and from  the m acroscopic point of  view nothing changes in the system . Thus the \nnecessa ry condition f or the line ar roto-an tiferromagnetic eff ect appeara nce is the s imultaneou s \nsign change of the vectors com ponents M i and Φj in the neighbouring sublattices А and В. \nOtherwise the corresponding com ponent ijχ of the roto -antif errom agnetic  tensor  is id entic ally \nzero in a h igh tem perature paren t phase, i.e. is  becom es zero everywhere as it follo ws from  the \nfree energy expansion continuity on the ir reducible r epresentation of the parent phase. The sam e \n 4speculations  leads us to the con clusion abo ut im possibility of  the nonzero linear roto -\nferrom agnetic te rm  appearance.  j i ijMΦχ\nLet us underline that the trilin ear roto-antiferrom agnetic coup ling, described by invariants   \n are l k j i ijklLΦΦΦχl kji ijkl LLLΦ χ~, should appear sim ultaneously with the bilinear roto-\nantiferrom agnetic coupling considered above, as well as higher odd order couplings of the type. \nBelow we will concen trate on the study of the bilinear roto-an tiferrom agnetic coupling (1 ) \nphysical m anifestations,  since assu me that the bilin ear effect should d ominate over the higher \norder odd on es under th e sam e other conditions.  \nThe transform ation law of the linear roto-a ntiferrom agnetic effect tensor com ponent s ijχ \nunder the point group symmetry operati ons with the m atrix elem ents, Сij, has the form \n. The transf ormation laws of the order param eters are ()mkjk imtr\nij CCχ −=χ 1 ()f kf k CCΦ =Φ det  и \n. Determ inant det( С) = ±1; the factor tr deno tes either the presence \n(tr = 1) or th e absence ( tr = 0) of the tim e-reversal operati on coupled to the space transfor mation \nС()()p iptr\ni LCC L det1−=\nij. Here the summation is perform ed over the repeating indexes. \n For the m agnetic and spatial symm etry groups corresponding to BiFeO 3 (spatial group is \ncR3, magnetic group is m3 or  −+−\nxzI 23 [24]), nonzero com ponents of  are  ijχ\n3\n333\n223\n11BiFeO BiFeO BiFeOχ≠χ=χ .                                                   (1b) \nNonzero components for EuTiO 3 are . 3\n213\n12EuTiO EuTiOχ−=χ\n \n BiFeO3 \nEuTiO3 \n-S\n+STi\nOEu\n+Φ -Φ ΟBi\nFe\n+Φ -Φ -S +Sах\nА ВВ А\nFigure 1. Schematic illustra tion to the m icroscopic m odel of the bi linear ro to-an tiferrodistor tive \ncoupling in the antif errodistor tive (AFD) phase of antiferrom agnets with the structural form ulae \n 5AМO3. The tilt Φ and  spin S local values should be oppos ite for the neighbouring oxygen \noctahedrons, as shown in the figure f or BiFeO 3 and EuTiO 3.  \n \n3. Physical manifes tations of the b ilinea r roto -antiferro magnetic cou pling \nDescription of an antiferrodist ortive antif erromagnet in the f ramework of the phenom enological \nfree energy approach shows, that the bili near roto-antiferrom agnetic coupling, in the \nmanifestations we a re interes ted in, does not inf luence the behavi our of m agnetic and dielectric  \nsusceptibilities in ex ternal m agnetic or electri c fields, i.e. the coup ling is un related with \nmagnetoelectric effect. Thus we can consider an antif errodistor tive antif erromagnet in th e \nabsence of external fields for the purposes of th e study. For the sake of simplicity we regard that \nboth antiferrodistortive and antif errom agnetic phase transitions  are of the second order. \nThe expres sion for the free energy density  of the uniform antiferrodistortive-\nantiferrom agnet in the absence of  extern al m agnetic and elec tric fields in the isotropic 1D-\napproxim ation has the following form \nL\nRM AFD AFM g g gg ++= ,                                 (2a) \n()\n4 24\n2 LLTgLL\nAFM β+α= ,    ()4 2\n4 2Φβ+Φα=Φ ΦTgAFD ,            (2b) Φχ=L gL\nRM\nTo describe the order param eters saturation behavior at low tem peratures we used the quantum -\ncorrected f ormula for  the coefficient ()() ( )q Q Q Q QT Q TT TT T T coth coth )( − α=α  in Eqs.(2), \nwhich is  valid in  a wid e temperatur e interval including low  and high temperatures[ 25], where \nsubscrip t Q=L  and the tem perature  is equal to Neel temperatu re  of AFM param eter \nappearan ce; subscrip t Q=Φ and the tem perature  is equal to AFD transition tem perature  of \nthe oxygen tilt appearance. At high tem peratures  the f ormulae tr ansforms into the  \nclass ical lim it, qTNT\nqTST\nQT T>>\n()(N LT L TT T −α=α ) and ()()S T TT T −α=αΦ Φ . Further let us solve the \nequation s of state appro ximately in  the assum ption tha t the antiferrod istortive ord er parameter Φ \nappears  at essentiall y higher tem peratures TS than the tem peratu re TN of spontaneous reversible \nantiferrom agnetic order param eter L appearance. This  allows us to m ake a decoupling \napproxim ation on the coupling coefficient χ. This is by the way a typical situation realizing for \ne.g. Bi 1-xRxFeO 3 (R=La, Gd, Nd, x=0 – 0.2) w ith TN≈(635 – 655) K and TS≈1200 K [4, 24], \nEuTiO 3 with TN≈5 K and TS≈280 K [ 26, 27, 28].  \n Equations of state, 0=∂∂ Lg  and 0=Φ∂∂g , contains  built- in fields Φχ and Lχ \ncorrespondingly. As we have shown in the S upplem ent, the built-in fields leads to the \nantiferromagnetic  (AF M) and  antif errodistor tive (AFD) tra nsition  temperatu res shifts, which  are \nquadratic on the param eter χ, nam ely:  \n 6() ()\nT LTN S S N AFM T T T T T\nΦααχ+−−+=2\n2\n21\n21,                                   (3a) \n() ()\nT LTN S S N AFD TT T T T\nΦααχ+−++=2\n2\n21\n21.                                 (3b) \nThe shifts given by expressions (3) are relatively small under the validity of  the strong inequality \n. Under the sim ultaneous validity of the late r inequality and the \ncondition  typical for antiferrodis tortiv e anti ferrom agnets, expressions (3) can be \nsimplified to the f orm, ()2 2χ>>−ααΦ N S T LT TT\nS N T T<<\n( )ST LT N AFM T T TΦααχ−≈ 42 and ()ST LT S AFD T T TΦααχ+≈ 42.  \nBesides the shif t (3), built- in field Φχ leads to the sm earing of t he AFM order param eter \nL above the Neel transition tem perature, i.e. in the param agnetic phase. The s mearing effect \nincreases under the increase of χ value, as  is shown in th e Figure 2а by solid curves. Under the  \nabsence of linear coupling ( χ=0) one has ()L LT L βα±=  at NTT< . Structural order param eter \nis practically independent on χ and equal to ()()Φ Φβα±≈Φ T TT  (see dotted curve in Figure  \n2а). \nUnfortunately, the antiferrom agnetic order param eter L by itself is not directly \nobservable, but som e notion about its behavi or can be obtained from the tem perature \ndependences of neutron scattering and specific h eat, if  the contr ibution  rela ted with the long -\nrange order appearance can be extracted. In particular the analyses  and com parison with  \nexperim ent of the specific heat ch anges 22\ndTgdT CP−=δ  allo ws us to verif y the th eoretica l \npredictions m ade. In the typical case AFD AFM T T<  com pact expres sion of the specific heat \nacquires the for m: \n⎪\n⎩⎪\n⎨⎧\n><⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ Φα+α−=δΦ\n. ,0, ,2 22 2\nAFDAFDL\nP\nTTTTdTd L\ndTd\ndTdTC                             (4) \nAs one can  see from  the Figure 2b-c  the hea t capac ity variation p eculiarity appe ared in the \nvicin ity of  AFM transition, tha t is break a t 0=χ , which becom es sm eared and shifted with χ \nincre ase. A t  and tem peratu res 0=χNTT<  the heat capacity change is as socia ted w ith the  \nappearan ce of AFM order param eter, ()L N LT T T L β−α±≈ , in the im mediate vicinity be low \nthe AFM order phase transition. At 0=χ  and te mperatures S N TT T<<  the param eter 0=L . \nTherefore u nder th e absence of bilinear coup ling between  the sub lattices m agnetization  and \n 7antiferrodistortive tilts, only th e sharp jum p appears on th e specific heat at Neel tem peratu re TN. \nThe jum p value is equal to ()L LT NN\nPT C βα=δ 22.  \nNote that linear m agnetoelectric  effect (if  one e xists in a co ncrete an tiferrom agnet) does \nnot contribute into the specific heat behaviour in the abs ence of external fields,  and so the \nquestion about the contribution of other coupling, biquadratic e.g.  roto-electric, m agnetoelectric \nor roto -magnetic, to the specific heat sm earing  near TN arises. If these or others contributions \nexist how they can be separa ted from  the ones caused by the considered biline ar coupling ? In \norder to ans wer the  ques tion let us es timate the  contribution of the biquad ratic couplings between \ndifferent order param eters and their m ean squi re fluctuations into  the spec ific hea t of \nantiferrodis tortiv e ferroelectric-antiferrom agnet. \nIn orde r to calcu late the contribution one can modify the free energy (2) by add ing the \nferroele ctric contrib ution, ()4 2P PT gp P FE β+α= , and biquadratic couplings, \n. Ferroelectric polarization leads to the occurren ce of additional  \nterm in Eq.(4), tha t is equal to 22 2 2 22LP P L gBQ λ+Φη+Φξ=\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛α−≈δ22P\ndTd\ndTdT CP\nP  and nonzero in ferroelectric phas e at \ntemperatures . FETT<\nRigorously speaking the biquadratic term s can only shift corresponding transition \ntemperature s, but cannot lead to an y sm earing of diffuseness in the tra nsition region. Thus the \nsmearing ef fect is re lated  with the b ilinear term  ΦχL in therm odynam ic limit.  \nUsing Ginzburg-Levanuk approach [ 29] for the estim ation of  the order p arameters m ean \nsquare fluctuations contribution in to the spec ific heat in  the vicinity of AFM phase transition, w e \ninclude the gradien t terms in th e free energy (2 ), which have the sim plest form  in the isotropic \napproxim ation, () ()()2 2 2\n2 2 2L P gL P\ngr ∇γ+∇γ+Φ∇γ=Φ, and the entropy that density near AFM \nphase trans ition is a pproxim ately equal to ()∫γ+−απ≈max\n02 2\n2ln2k\nL AFM LTB AFM\nfl dkk TT kTkg . \nCorresponding expression for the order param eters fluctuations contribution into the specific \nheat chang e near AFM has the form  [29]: \nAFM LLT B fl\nPTTTkC\n−πγα≈δ\n23232\n8                                      (5) \nExpression (5) diverges at  for fi nite  due to the fluctuati ons, which contribution \ndisappears in the lim iting case  (therm odyna mic lim it of Landau theory). Elem entary \nestim ations m ade for the typ ical va lues of  parameters αAFMTT=Lγ\n∞→γL\nLT, βL и γL [30], have shown, that the \n 8smearing effect defined in Eq. (5), is essential only in very narrow vicinity of antiferrom agnetic \nphase transition (for temp eratures interval from fractions of  Kelvin to f ew Kelvin w ide), w hile \nthe experimentally observed range  of specific heat jum p smearing is of order of 20-50 Kelvin. \nReally, a t χ=0 the ratio \nN LT N LL B\nN\nPfl\nP\nTT TTk\nCC\n−απγβ=δδ\n232\n4 becom es less than 0.01 already at \n1>−NTT К. Consideration of χ-effect Barret' s law for tem peratu re dep endence of ()TLα  could \nnot change significantly this es timation. Therefo re, fluctuati on (5) does not m ake a significant \ncontribution  to the sm earing of the s pecific heat  jump, observable in experim ent, approving the \nconclusion that m ost of the sm earing of AFM pha se transition is associ ated with bilinear \ncoupling  only. ΦχL\nBelow we consider m ultiferroic BiFeO 3, which is antif errod istive f erroele ctric – \nantiferromagnet w ith critical tem peratu res TN≈ 645 K, TC≈1100 K and TS≈1200 K [22]. Our \nfitting of  temperature d ependence o f AFM order param eter L in B iFeO 3 obtained from  neutron \nscattering experim ent of Fischer et al [ 31] is shown Figure 2a . The fi tting of temperature \ndependence of specific heat and  its part associated with transi tion to AFM phase is shown in \nFigure 2b-c for the experim ental results of  Kallaev et al [ 32]. Sol id curves fro m Fig.2 \ncorrespond to nonzero value of effective param eter ()ΦΦβαβχ=χT L~=2, 5, 10 SI units, \ndashed curv es correspon d to the case =χ~0. The best f itting w as obtained  for 2 SI units. It is \nclearly seen  that abrup t jumps of order param eter L temperature dependences, corresponding to \nthe calcu lation at =χ~\n=χ~0, is in a satisf actory agreem ent w ith experim ental p oints below  TN. The \nsame situation is for the curve calcu lated at =χ~2 SI units, f or which th e small L exists ab ove the  \nNeel tem perature, d ecreases with tem perature increas es an d tends to  zero at . However  \nthe specific heat featu res observ ed at ST T→\nNTT=  are ev idently b lurred in th e tem perature region \n, at that the sm earing can be satisfactory described only at 2 SI units, but not \nat 0. Therefore we can conclude that the di ffuse \"tail\"  of the antif errom agnetic orde r \nparam eter L (as shown in the Figu re 2а), that exists at no nzero N N T T T 1.1<< =χ~\n=χ~\nχ~ values, pointed out on the \npossibility of  the \"w eak\" im proper antiferrom agnetis m induced by the antiferrodistortive \nstructural ordering. \nHowever nothing definite can be concluded about the χ~ value from  the analyses of the \nexperim ental data shown in the Figure 2а, because it is extrem ely com plex to register relativ ely \nsmall sublattices m agnetization values by neutron s cattering. On the other hand it is evident from \nthe specific heat behavior shown in the Figure 2b-c , that the inequa lity >χ~1 SI units is  \nnecessary for the satisfactory agreem ent with exp eriment. \n 9 \n \n0.20.40.60.8 1 1.21.400.20.40.60.81 Sublatti ce magn etization L/L 0 \n(a) L Φ \nTemperature T/TN χ BiFeO 3 \n0.50.60.70.8 0.9 11.10.320.340.360.380.4\nTemperature T/TN Specif ic heat Cp (J/mol K) \n(b)χ BiFeO 3 \n0 200 400 600  8000246810 Specif ic heat ∆Cp (J/mol K) \nTemperature T (K) (c) χ BiFeO 3 \nTN0 2 4 6 8 1005101520\nTemperature T  (K) Specific Hea t (J/mol K) \nTN \n(d)EuTi O3\n \nFigure 2. (a) Sublattices m agnetization ()()0LTL  and (b) heat cap acity variation as a functio n \nof reduced tem perature NTT . Sym bols are experim ental data for BiFeO 3 from Fischer et al [31] \non neutron scattering and Kallaev  et al [32] for specific heat correspondingly. Curves are \ncalculated by us for e ffective coupling constant =χ~2, 5, 10 SI units (solid curves) and χ = 0 \n(dashed curve); 645=NT  K, 550=LT  K, 100=ΦT  K, 1200=ST  K. Dotted cu rve in  the p lot \n(b) is the AFD order param eter, 0ΦΦ , that is alm ost independent on χ value for chosen \nparam eters. (c) Temperature  depend encies of the anom alous  contribution to the BiFeO 3 specific \nheat. Sym bols are experim ental data from  [32]. Solid curves are calculated at 2, 5, 10 SI  \nunits, dashed curves corresponds to χ = 0. (d) Specific heat variation of EuTiO=χ~\n3 near the AFM \ntransition. Sym bols represent the experim ental data [26-28]. Effective coupling constant χ = 0 \n(dashed cu rve) and 2 SI units (solid curve); =χ~=NT 5.5 K, =ST 285 K. Other param eters of \nEuTiO 3 are listed in the Ta ble 1 in the ref [15]. \n \n 10Heat cap acity variation of EuTiO 3 near the A FM transition  is show n for com parison in \nthe Figure 2d . As one can see from  the plot nonzero χ~ (solid curve ) describes the experim ental \ndata better than χ = 0 (dashed curve). It is worth to unde rline that the sm earing of sublattice \nmagnetization and sp ecific heat for BiFeO 3 and EuTiO 3 shown in Figs.2  looks like the sm earing \nof ferroelectric properties in  external electric field [ 33]. This obviously confirm ed the statem ent \nthat the te rms  and  can b e considered  as built -in fields in the lattices. Φχ Lχ\nIn the n ext section  we show how  the existe nce of  roto-antiferrom agnetic coupling can be  \napproved from  the specific he at behavior in the BiFeO 3-based solid so lutions,  and estim ate th e \neffect value.  \n \n4. Determination of the roto-antiferr omagnetic coupling constant for B i1-xRxFeO 3 solid \nsolutions \nAvailable experim ental results demonstrate noticeable featu res of the tem perature d ependencies \nof the specific heat in Bi 1-xRxFeO 3 (R=La, Nd, x=0 – 0.2) solid solutions [ 34]. The features  \nappears at the tem perature of the antiferrom agnetic phase transition th at is about (640-650) К. \nCorresponding experim ental result s are shown by sym bols in the Figures 3. As one can see from  \nthe figure d ashed curves calcu lated  at 0~=χ  and different com position х do not describe the \nspecific h ear sm earing at tem peratures . Solid cu rves, calcu lated as (2 – 2.5) SI units \nand  K in dependence of х, describ e the sm earing effect adequately, p roving the \nimportance of the bilinear roto-an tiferrom agnetic effect for the unde rstanding of the specific heat \nbehaviour n ear the antif errom agnetic phase tr ansition. The inclus ion of the bilinear ro to-\nantiferrom agnetic effect is neces sary for the quan titativ e description of th e experim ental data.  NTT> =χ~\n)655 645(−=NT\n \n 11 \n 550  600  650  7000.3 0.35  0.4 0.45  0.5 \nx=0.2 \nx=0.1\nx=0.05Bi1-xNdxFeO 3 \nTemperature T (K) (a) \nHeat cap acity Cp (J m ol/K) \n 550  600  650  7000.360.380.40.420.44 Bi1-xLaxFeO3 Heat capacity Cp (J mol/K) \n(b)x=0.1 \nx=0.2 \nx=0 \nTemperature T (K)\n \nFigure 3. Temperature dependence of the specific heat  near AFD phase transition of the solid \nsolution s Bi1-xRxFeO 3 (R=La, Nd, x= 0 – 0.2). Sym bols are experim ental data for Bi 1-xRxFeO 3 \nfrom  Amirov et al [34] for h eat capacity correspondingly. D ashed curves are calculated by us for \ndimensionless coupling constant 0~=χ . Solid curves correspond  to different nonzero =χ~(2 – \n2.5) SI units and  K depending on the  com positio n x,  K, )655 645(−=NT 550=LT 100=ΦT  K, \n K.  1200=ST\n \nNotice that we did not aim ed to dete rmine all m aterial pa rameters of  Bi1-xRxFeO 3 from \nthe f itting of  the specif ic heat varia tion ()TCpδ  and nor malized antiferrom agnetic order \nparam eter ()()0LTL  tem perature dependences, because it  was im possible. O nly the r atio \n()L LTβα 22 can be defined from  specific heat jum p , and the tem perature dependence of the  \nratio N\npCδ\n()LT LTαα  can be determ ined from  the tem perature dependence ()()0LTL . In order to \ndefine the value of αLT one needs the tem peratu re dependence o f the antiferrom agnetic \nsusceptibility that we could not found in litera ture. Despite the difficulty we reached our goal \nand found the effective value of the roto -antiferrodistortive coupling constant, \n()ΦΦβαβχ=χT L~, from the f itting of  exper imental data .  \n \n5. Conclusions \nThe possibility of  the linea r roto -antiferromagnetic ef fect existence  in perovsk ite-\nmultiferroics with the structur al formula A MO3, antiferrodistive and an tiferrom agnetic ordering \nis dem onstra ted. W ithin the f ramework of  the th eory of symm etry and th e microscopic model the  \nnecessary conditions for this eff ect occurrence are the s imultane ous change in the sign of the \ncorresponding com ponents of the elem entary  magnetization vectors in the neighboring \n 12antiferromagnetic  sublattices coup led w ith the change o f the antiferrodis tive displacem ent \ndirection in the neighb oring oxygen octahedron MO 6. Let us underlin e tha t the trilin ear roto -\nantiferrom agnetic coup ling should  appear sim ultaneously with the considered bilin ear ro to-\nantiferrom agnetic coupling, as well as higher odd order coup lings of the type. \nPhysical m anifestations of roto-antif errom agnetic effect is sm earing  of th e \nantiferrom agnetic ph ase tran sition  and the em ergence of s mall \"improper\" antiferrom agnetic \norder parameter L abov e the Neel tem perature and below the tem perature of antiferrodis tortive \ntransition. T he param eter L is induced by the product of the AFD order param eter Φ on th e roto-\nantiferrom agnetic coupling constant χ, at that the term  χΦ acts as effective built-in conjugated \nfield for th e param eter. Therefore,  there is  the possibility to observe weak \"improper\" \nantiferrom agnetism  induced by the stru ctural antiferrod istive ordering Φ above the Neel  \ntemperature.  For exam ple, in bis muth ferrite, for which the antif errom agnetic trans ition \ntemperature is of  the order of (640 -655) K , and the tem perature an tiferrodistor tive transition  goes \nabove 1200 K, the tem perature dependence of L was m easured by neut ron scattering m ethod.  \nHowever, the available experim ental data cannot  say anything definite about the value of a \nsmearing effect, becaus e it is extrem ely difficult to reg ister a sufficiently sm all value of L by \nneutron scattering.  \nRoto-antiferrom agnetic effect al so leads to the s mearing of th e jum p of the specific h eat \nnear the tem perature o f antife rrom agnetic phase transition. By quantitative com parison  with  \nexperim ents we made estim ates of  the roto- linear effect in antif erromagnetic so lid solution s of \nmultiferroic Bi1-xRxFeO 3 and determ ine the optim al value of  the roto-an tiferrom agnetic coupling  \nconstan ts from the f itting to expe rimental d ata. Calculated dependencies de scribe the experim ent \nquite sa tisfactory, thu s proving the im portance of  bilinea r roto-an tiferromagnetic ef fect for the  \nunderstanding of the m echanism s responsible for the tem perature dependence of the specific heat \nbehaviour near the antiferro magnetic phase transition. \n \n 13 \nReferences  \n                                                 \n1 Manfred. Fiebig, \"Reviv al of the m agnetoelectric effect.\" Journal of Physics D: Applied \nPhysics  38, no. 8: R123 (2005). \n2 N. A. Spaldin and M. Fiebig, Science 309, 391-392 (2005). \n3 J.M. Rondinelli, N.A. Spaldin. Advanced Materials.23, 3363–3381 (2011). \n4 A. P., Pyatakov, A. K.Zvezdin, Mag netoelectric and m ultiferroic m edia. Physics-Usp ekhi, \n55(6), 557-581 (2012). \n5 J. F. Scott, \"Data sto rage: Multiferroic m emories.\" Nature m ateria ls 6, no. 4: 256-257 (2007). \n6 V.G. Bar’yakhtar, V.A. L’vov, D.A. Yablonskii. JETP Lett. 37, 673 (1983). \n7 Maxim  Mostovoy, Phys. Re v. Lett. 96, 067601 (2006).  \n8 A. Sparavigna, A. Strigazzi, and A.  Zvezdin, Phys. Rev. B 50, 2953 (1994). \n9 A.P. Pyatakov and A.K.  Zvezdin, E ur. Phys. J. B 71, 419–427 (2009). \n10 B.M. Tanygin. Journal of Magnetism  and Magnetic Materials 323, 616–619 (2011). \n11. E.A. Eliseev, M.D. Glinchuk, V. Khist, V.V.  Skorokhod, R. Blinc, A.N. Morozovska. Linear \nmagnetoelectric coupling and ferroelectricity indu ced by the flexom agnetic effect in ferroics. \nPhys.Rev. B 84, 174112 (2011) \n12. Morozovska, Anna N., Eugene A. Elisee v, Maya D. Glinchuk, Long-Qing Chen, and \nVenkatram an Gopalan. \"Interfacial p olarizati on and pyroelectricity in antif errodisto rtive \nstructures induced by a flexoel ectric effect and ro tostriction. \" Physical Review B 85, no. 9 \n(2012): 094107. \n13. Eugene A. Eliseev, Sergei V. Kalinin, Yijia Gu , Maya D. Glinchuk, Vi ctoria Khist, Albina \nBorisevich, Venkatram an Gopalan, Long-Qing Chen, and Anna N. Morozovska. Universal \nemergence of spatially-m odulated  structures induced by flexo-antiferrodist ortive coupling in \nmultiferro ics. Phys.Rev. B 88, 224105 (2013). \n14 Venkatram an Gopalan, and Daniel  B. Litvin. \" Rotation -reversal symm etries in crystals and \nhanded structures.\" Nature materia ls 10, no. 5: 376-381(2011). \n15 Eugene A. Eliseev, Maya D. Glinchuk, Ve nkatram an Gopa lan, Anna N. Morozovska. \nRotom agnetic couplings influence on the magnetic properties of  antiferrodistortive \nantiferromagnets http://arxiv.org/abs/1409.7108\n16. Hirom oto Uwe, and Tunetaro Sakudo. Physical Review B  13, no. 1: 271 (1976). \n17. M. J. Haun, E. Fur man, T. R. Hale mane and L.  E. Cross, Ferroelectrics 99, 55 (1989) \n18. B. Houchmanzadeh, J. Lajzerowicz a nd E Salje, J. Phys.: Condens. Matter 3, 5163 (1991) \n19. E.V. Balashova and A.K. Tagant sev, Phys. Rev. B 48, 9979 (1993). \n20. A.K. Taga ntsev, E. Courtens and L . Arzel, Phys. Rev. B, 64, 224107 (2001). \n 14                                                                                                                                                             \n21. M.D. Glinchuk, E.A. Eliseev, A.N. Morozovska, R. Blinc. Phys. Rev. B 77, № 2, 024106-1-\n11 (2008) \n22 Gustau Catalan and Jam es F. Scott, Adv. Mater. 21, 1–23 (2009) \n23 G. A. Smolenskii, L. I. Chupis, Sov. Phys. Usp. 25, 475 (1982). \n24 Кадомцева , Звездин , Письма  в ЖЭТФ , 79, 705 (2004) \n25 J. H. Barrett,  Phys. Rev. 86, 118 (1952). \n26 T. Katsufuji and H. Takagi, Phys. Rev. B  64, 054415 (2001). \n27 V. V. Shvartsm an, P. Borisov, W . Kleemann, S. Ka mba, T. Katsufuji. Phys. Rev. B  81, \n064426 (2010). \n28 P. G. Reuve kamp, R. K. Krem er, J. Kohl er, and A. Buss mann-Holder. P hys. Rev. B 90, \n094420 (2014) \n29 B. A. Strukov and A. P. Levanyuk, Ferroelectri c Phenom ena in Crystals (Springer, Berlin, \n1998) \n30 {kB=1.38×10-23 J/K, αTP = 4.366⋅10−7, TN = 650 K, βP = 0.323*10-17, γLP = 10−8 m3/F} \n31 P. Fischer, M. PoIom ska, I. Sosnowska, M.  Szym anski J. Phys. C: Solid St. Phys., 13, 1931-\n40(1980). \n32 С.Н. Каллаев  et al, Теплоемкость  и диэлектрические  свойства мультиферроиков  \nBi1−xGdxFeO3 (x = 0−0.20). Физика  твердого  тела, том 56, вып. 7, 1360-1363 (2014) \n33 Lines, Glass, textbook \n34 Амиров  et al, Bi 1-xRexFeO 3 (Re=La, Nd; x=0-0,2) (2010) \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 15                                                                                                                                                             \nAFD Coupling AFM g g gg \nSupplementary Information \nThe approxim ate expression for the free energy of  the antif errodis tortive-antiferrom agnet in the \nabsence of external m agnetic (H=0) and el ectric (E=0) f ields has the f ollowing form \n,                                 (S.1a) ++=\n()\n4 24\n2 LLTgLL\nAFM β+α=()4 2\n4 2Φβ+Φα=Φ ΦTgAFD ,                 (S.1b) \n2 2\n21Φξ+Φχ= L L gL\nRM j iij Coupling ,                                    (S.1c) \n()() ( )N L L L LT L TT TT T T coth coth )( − α= Temperature-dependen t coefficients are α  and \n() () () ( )S T TT TT T TΦ Φ ΦΦ Φ − α=α coth coth\nL\nRM. Further let us so lve the equ ations of state \napproxim ately in the a ssum ption that the antiferrodistortive order param eter Φ appears a t \nessentially higher temperatures TS than the tem perature TN of s pontaneous reversible \nantiferrom agnetic order param eter L appearance. This  allows us to m ake a decoupling \napproxim ation on the coupling constants ξ and χ. This is b y the way a typica l situation  \nrealizing for e.g. BiFeO 3, EuTiO 3, etc. In a scalar approxim ation equations of state for the order \nparam eter determination  are \n02 3=Φχ+Φξ+β+α=∂∂L L LLgL\nRM L L ,                      (S.2a) \n02 3=χ+Φξ+Φβ+Φα=Φ∂∂\nΦ Φ L LgL\nRM\n() 0.                       (S.2c) \nThe phase transition point of th e long-range order appearance can be found from  the condition of \nlinearized s ystem  =Φχ+−α LTTN LT , () 0=χ+Φ−αΦ L TTS T\n() nonzero determ inant, which \nis () 02=χ−−α−αΦ S T N LT TT TT . Corresponding rigorous and ap proxim ate expressions and \ntheir for tran sition tem peratures beco me \n() ()\nST LTN\nT LTN S S N AFMTT T T T T T\nΦ Φ ααχ−≈ααχ+−−+=4 21\n212 2\n2        (S.3a) \n() ()\nST LTS\nT LTN S S N AFDTT T T T T T\nΦ Φ ααχ+≈ααχ+−++=4 21\n212 2\n2\n()         (S.3b) \nOne can see from  Eqs.( S.3) hat the correction to  transition tem peratures  is proportional to the \nsquire of the coupling constant χ. In the decoupling approxim ation  \n()\nΦΦ\nβα−±≈ΦTTS                                                    (S.4) \n 16                                                                                                                                                             \nLχ\nSand the term   can be neglected in Eq.(S.2b). Then Eq .(S.2a) im mediately transf orms into the  \nequation for L in the som e built-in field , nam ely: Φχ\n( )S L SL\nRM L L L Φχ−=β+Φξ+α3 2\n0>.                                         (S.5a) \n, has t he following form The solution of the equation that  has physical sense at positive ΦχS\n()()()( )\n()χΦ∆βΦξ+α−χΦ∆=\n, 3632 , 2\n32 3 2 3\nSL SL\nRM L STL\n() () ().                               (S.5b) \n( )31\n2 32 3 227 43 9 , ⎟⎠⎞⎜⎝⎛Φχβ+Φξ+αβ+Φχβ=χΦ∆S L SL\nRM L L S L S Here the function . Expression \n(S.5b) m eans, that the AFM phase  transition b ecom es diffuse (s ee Figure 2a ). Heat capacity  \nvariation related with AFD-AFM modes can be cal culated as the secon d derivative of the free \nenergy on temperature T, multiplied b y T, namely in our approxim ations: \n⎪⎩⎪⎨⎧\n><⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ Φα+ΦΦα+α+α−≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ Φα+α−≡⎟\n⎠⎞⎜\n⎝⎛\n∂∂−≡⎟\n⎠⎞⎜\n⎝⎛\n∂∂+∂∂+Φ\nΦ∂∂−≡−=δ\nΦ ΦΦ\n. ,0, ,2 22 2\n2\n22 2\n222 222\nAFDAFDL LLV\nTTTT\ndTd\ndTd\ndTd L\ndTdLdTdL\ndTdTdTd L\ndTd\ndTdTTg\ndTdTTg\ndTdL\nLg\ndTdg\ndTdTdTgdT C\n             (S.6) \n 17"    },    {        "title": "2305.04370v1.Band_gap_and_pseudocapacitance_of_Gd__2_O__3__doped_with_Ni___0_5__Zn___0_5__Fe__2_O__4_.pdf",        "content": "1 \n \nBand gap and pseudocapacitance  of Gd 2O3 doped with Ni 0.5Zn0.5Fe2O4 \nM. Azeem1, Q. Abbas2,3, M. A. Abdelkareem2, A.G. Olabi2,4 \n1Department of Applied Physics and Astronomy, University of Sharjah, 27272, University City, \nSharjah, United Arab Emirates.  \n2Sustainable Energy & Power Systems Research Centre, RISE, University of Sharjah, P.O. Box \n27272, Sharjah, United Arab Emirates  \n3Institute of Engineering and Energy Technologies (IEET), School of Engineering, Computing & \nPhysical Sciences, University of the West of Scotland, Paisley, PA1 2BE, UK  \n4Mechanical Engineering and Design, Aston University, School of Engineering an d Applied \nScience, Aston Triangle, Birmingham, B4 7ET, UK  \nAbstract:  \nHerein, w e present a detailed study of the structural, optical, and electrochemical responses of \nGd 2O3 doped  with nickel zinc ferrite nanoparticles.  Doping of Ni 0.5Zn0.5Fe2O4 nanoparticle s to \nGd 2O3 powder was done through thermal decomposition at 1000 ֯C. The average grain size of the \nmixture was determined to be approximately  95 nm, and phases of cubic Gd 2O3, GdO, and \northorhombic prisms of GdFeO 3 were identified . The focused ion beam energy dispersive X -ray \nspectrum ( FIB-EDX) mapping results clearly show the morphology of the particles with Gd and Fe \nas the dominant elements. The structural data were compared with the  spectroscopic \nmeasurements confirming the formation of multiple phas es of oxides and ferrites. The measured \noptical band gap is significantly redshifted to 1.8 eV and is close  to that of  nitride compounds of \ngadolinium  metal. The m easured  specific capacitance was almost  7 Fg-1 at a current density of \n1 Ag-1, showing  a small drop of 27% when the current density is increased to 10 Ag-1. Cyclic \nvoltammetry (CV) plots of the ferrite doped Gd 2O3 electrode at a scan rate of 5 to 100 mV/s \nindicate the pseudocapacit ive nature of the material.  \n  2 \n \nIntroduction  \nThe Gd3+ ion carries a large spin magnetic moment  (S=7/2)  which makes the gadolinium -based  \ncomposite materials interesting [1].  They have found their applications in a variety of fields  \nincluding  magnetic coolers [2], high -performance  conductors [3], super capacitors  electrodes [4], \noptoelectronics [5], and magnetic resonance im aging(MRI) [6,7] . A very recen t review report has \nhighlighted the spintronic potential of the Gd3+ composites in the ferroic -orde r [ferromagnetic, \nferroelectric , and multiferroic] oxides [8]. Gadolinium ferrite based material systems,  \n(Gd xFe1−x/NiCoO ) [9] and cobalt gadolinium oxide ( Co/Gd 2O3), [10] exhibit  a controllable positive \nexchange bias . Magnetoresistance , coercivity, Curie temperature, magnetic anisotropy , \nsaturated magnetization , and magnetic  remanence  of several other rare earth oxide based \nmaterials systems have also been studied  [11,12] . \nIt appears that most of the attention has been focused on the magnetic properties of rare ea rth \noxide -based heterostructures whereas the optical and electr ochemical  properties have largely \nbeen ignored. We know that the f undamental optical absorption edge  for Gd 2O3 lies in the range \nof 4.9 eV to 5.6 eV [13–15] depending on the preparatory methods, crystallite size and the \nmorphology of the prepared materials, from nanoparticles  [16] to bulk  [13]. These  \nexperimentally determined values differ widely with the theoretical calculations  [17,18] . There \nare very limited attempts to show the practicality  of these materials systems  in devic es. A single \ndomain, crystalline Gd 2O3 film grown on Si by molecular beam epitaxy has shown a potential to \nbe used in super capacitors with a leakage current density of about 10-3 A/cm2[4]. The Gd 2O3 has \naslo been successfully employed as an effective component of high -performance composites in \nnumber of studies  [19,20] . Lanthanide doped  metal oxides  have show n to improve  the specific \ncapacitance . For example, for the case of double perovskites Gd 2NiMnO 6, the measured specific \ncapacitance was ⁓400.46 Fg-1 at current density of 1 Ag-1 [21] while  Gd-doped HfO 2 and \nAluminium doping in Sr 3Co2Fe24O41 [22,23]  are also  found to have potential to be used in the \nsupercapacit or application as well. These values of the capacitance are not very far from the \ntransitio n metal based oxides such as cobalt -doped ZnO quantum dots where the specific \ncapacitance of 697 Fg-1 is observed along with power and energy density of around 1026 W/kg \nand 24Wh/kg, respectively [24]. However, NiCoP/NC electrodes [25] have achieved superb rate -3 \n \ncapability, showing capacitance of 1127 Fg−1 and 873 Fg−1 at curre nt density values of  1 and \n16 Ag−1, respectively, long -term durability (75.5% retention after 8000 cycles) and ultra -high \nenergy density of around 52.5 Wh kg−1.  \nIn this report we make  an attempt to unravel the  fundamental properties of the ferrite doped \nGd 2O3 such as the energy band gap and pseudocapacitance  relevant to the practical applications . \nThe Gd 2O3 was choosen due to it s highly stable nature when compared with other rare earth \nmetal due to its half -filled 4f7 levels  [26].  We shed light on its electronic structure by measuring \nthe optical band gap.  The free carrier density is  estimated t o be  of the order 1026 cm-3 which \ncontributes to the conduction of electrons despite its semiconducting nature. Our study show s \nthat ferrite based Gd 2O3 heterostructures have  great potential for energy -efficient, high density, \nand non -volatile data storage.   \nMaterials and methods  \nThe gadolinium oxide (Gd 2O3) and nickel zinc ferrite (Ni 0.5Zn0.5Fe2O4) nanoparticles were  \npurchased from commercial vendors. The  average size of Ni0.5Zn0.5Fe2O4 nanoparticles was \naround 50 nm. The sample was prepared by mixing 5 g of Gd 2O3 with 1 g of Ni0.5Zn0.5Fe2O4 \nnanoparticles in a pestle and mortar. The color of the mixture was reddish dark brown  which \nchanged to dark gray after t hermal decomposit ion at a temperature of 1000  ֯C for 12 hours . The \nphases of the fabricated  material and their crystal structures were examined by X -ray diffraction \n(XRD) patterns using a Bruker D8 Advance X -ray diffractometer. The Cu Kα radiations of energy \n8.04 keV, corresponding to an x -ray wavelength of 1.5406 Å were used to irradiate the sa mples. \nThe patterns were obtained in a 2θ range of 20° to 85° with a step size of 0.02. The phase analysis \nwas performed by using the software QualX . [27] \nThe Raman spectra were collec ted by using the Renishaw inVia Raman spectromete r. A site of \nthe specimen was exposed to a laser source with a spot size of 50  mm and  a  laser  power of 1%  \n(14 mW) . For the heterogeneous  samples reported here , the Raman signals from the different \ncomposites overlap ped and were  indistinguishable. There fore, up to  45 spectra were collated \nfrom different sites of the same sample with the same laser source, and an avera ge was \ncalculated. The average  spectra were , then,  normalized. The acquisition time for each spectrum 4 \n \nwas 50 s.  The Fourier transform infrared  (FTIR) transmittance spectra were obtained by using \nJASCO FTIR model 6300  type  A spectrometer. Transmittance was measured by setting the \nresolution of the spectrometer to 2 cm-1 in the frequency range of 400 cm-1 to 2000 cm-1. \nThe morphology of the fabricated material was studied by using a  Tescan VEGA XM variable \npressure focused ion beam (FIB) scanning electron microscope (SEM ). The X -ray spectrometer \nembedded in the SEM system  also provides energy dispersive spectra (ED X) for the localized \nchemica l analysis.  The FIB -EDX mapping was done of  the cross -sections of selected regions on \nthe samples.  \n A differential reflectance spectrometer (DRS), model UV -2600i, was used to measure the  diffuse  \nreflectance  from the sample. Approximately 1 mm thick pellets were prepared to measure the \nreflectance  in the wavelength range of 200 nm to 800 nm (corresponding to the photon energy \nrange of 6.2 eV to 1.5 eV) with a data interval of 0.5  nm. Barium sulfate, with an absorption \ncoefficient almost equal to  zero ( and reflectance clo se to 1), was used as a standard to measure  \nrelative reflectance .  \nElectrodes for electrochemical measurements were prepared using carbon in the form of well \ngrinded powder (grind for around 20 mins ). Approximately 80  wt% active carbon material, \n10wt% Cabot carbon black as conductivity enhancer and 10 wt% PVDF powder as binder were \nused for the fabrication of electrodes. The electrode component s were mixed with ethanol for \n2 hours to form a paste. A syringe  is used  to drop the pas te on a 1 cm2 on the carbon sheets . The \ncarbon sheets are cleaned using H 2SO4 and HNO 3. The electrodes  were left to dry in oven at 70  ℃ \nfor 24 hours.  A standard three -electrode electrochemical test cell was utilized  to evaluate the \nelectrochemical characteristics (cyclic voltammetry, galvanostatic charge/discharge , and \nelectrochemical impedance spectroscopy) of synthesized materials using a potentiostat  (BioLogic \nSP-200 Potentiostat, UK). For the three -electrode configuration, prepared materials were utilized \nas a working electrodes . Platinum plate and Hg/HgO were used as a counter and reference \nelectrode, respectively. A 2 M KOH aqueous solution was use d as an electrolyte.  \nResults and discussions  5 \n \nThe XRD pattern of the thermally decomposed mixture of Gd 2O3 doped with Ni 0.5Zn0.5Fe2O4 is \nshown in Figure 1. The average grain size was determined to be approximately 95  nm by using \nScherrer equation . The domi nant phases are of cubic Gd 2O3 with cubic gadolinium monoxide \n(GdO) as minority. The strongest peaks at 2θ ≈ 28.56º, 33.09º, 35.17°, 39.04°, 42.60°, 47.51º, \n52.10°, 54.99° and 56.40º are attributed to Gd 2O3 phases with planes oriented in the [222], [312], \n[411], [332], [413], [440], [611], [514] and [622] directions respectively. Also, there are several \nweak phases of Gd 2O3 above 2θ ≈  60º.  The phases of the cubic Gd 2O3 belong to the space group \nIa- 3 with the lattice parameter a=10.8120  Å. The cell volume is approximately 1263.92 Å3.  \nPhases of gadolinium ferrites (GdFeO 3) were identified at 2θ ≈ 20.22º, 22.95º, 23.17º, 31.87º, \n33.53º, 39.8º, 41.2º, 46.95º, and 48.22º, and 59.15º, corresponding to planes oriented along \n[101], [110], [002], [020], [200], [022], [202], [220], [023] and [204] respectively. Whereas, the \npeaks at 2θ ≈ 32.91º, 53.23º, and 59.53º are because of the diffraction from the planes along \n[112], [131]  and [312] directions, respectively, indicative of positive orthorhombic pyramids. The \ndiffraction angles and the corresponding plane orientations suggest that blocky crystals of  \nGdFeO 3 exists in the form of orthorhombic prisms of the 1st, 2nd, and 3rd order. Most of the phases \nof GdFeO 3, therefore, belong to the orthorhombic system having a 2 -fold axis of rotational \nsymmetry coincident with the c axis of the crystal without the presence of a 3 -fold axis of \nrotational symmetry.  The cell parameters for the orthorhombic crystal system of GdFeO 3 follow \nthe conventional order of a<b<c  with the values 5.3490 Å, 5.6089 Å and 7.6687  Å, respectively. \nThe cell volume is approximately equal to 230.08 Å3. The axis of elongation is along c axis. The \nspace  group  is P b n m.  6 \n \nFigure 1 Experimental XRD pattern. The insets show multiple splitting of the diffracted peaks.  \nFurther, multiple splitting of the diffraction peaks, inset (a -c) Figure 1, indicate the phase \ntransformations. The split between 47.2º and 47.9º, inset (a) Figure 1, shows two phases of \ngadolinium oxide identified as, Gd 2O3 and GdO, diffracted from two parallel planes, [440] for \nGd 2O3 and [220] for GdO. Both of these phases are  also identified between 56.2º and 56.9º, \nshown in inset (b) of Figure 1, reflecting from parallel planes again, [622] and [311] of Gd 2O3 and \nGdO respectively. The cubic GdO belongs to the space group F -4 3 m with the cell parameter \na=5.4000 Å and the cel l volume =157.46 Å3. Inset (c) of Figure 1 shows multiple splitting in \ndiffraction peaks of GdFeO 3. The twin peaks between 59.2º and 59.4º are contributed by the \ndiffraction from [234] and [243] planes. The more intense peaks are from [622] and [204] \ndirec tions. It should be particularly noted that the GdO and GdFeO 3 have metallic character  [18] \nwhereas Gd 2O3 is semiconductor.  \n20 30 40 50 60 70 80 90Relative Intensity (arbitrary units)\n2q\n47.2 47.3 47.4 47.5 47.6 47.7 47.8 47.9Mixture of Gd2O3 and Ni0.5Zn0.5Fe2O4 \nnanoparticles annealed at 1000 ̊C.Relative Intensity (arbitrary units)\n2q ExperimentalGd2O3 [440]\nGdO [220]\n56.0 56.1 56.2 56.3 56.4 56.5 56.6 56.7 56.8Mixture of Gd2O3 and Ni0.5Zn0.5Fe2O4 \nnanoparticles annealed at 1000 ̊C.Relative Intensity (arbitrary units)\n2q Experimental\nGdO [311]Gd2O3 [622]\n58.6 58.8 59.0 59.2 59.4 59.6 59.8 60.0 60.2Mixture of Gd2O3 and Ni0.5Zn0.5Fe2O4 \nnanoparticles annealed at 1000 ̊C.Relative Intensity (arbitrary units)\n2q Experimental GdFeO3 [622]GdFeO3 [234]GdFeO3 [243]\nGdFeO3 [204](a) (b) (c)7 \n \nEvidence of multiple phases of  Gd 2O3 and GdFeO 3 phas es also comes from Raman spectroscopic \nmeasurements  as shown in Figure 2 . Rare  earth sesquioxides , however, are luminescent, and \ntherefore strong non -vibrational luminescent bands quite often hide weak Raman vibrational \nbands. Therefore,  to distinguish the Raman signals from the laser -induced  luminescence, the \nspectra were  collated by usin g two laser sources  of different wave lengths, 488  nm and 785 nm . \nThe bands shared by the spectra obtained by both the laser sources are true Raman bands. The \nfingerprint region for the rare earth alloys lies below 1000 cm-1, [28,29]  however , we attempted  \nto explore the  spectral range of 100 cm-1 to 2500 cm-1. Moreover , the Raman signals are resolved \nby using the Lorentz model . Lorentzian peak function has a bell shape and much wider tails than \nthe Gaussian function . According to the d ispersion theory , the bands of solids can be modeled \nwith damped harmonic oscil lators, which under certain approximations, result in the Lorentz \nprofile .[30] However, t here is always  some uncertainty in the excitation response and the \nlocation of the energy level of the excited state so the duration in which a molecule remains in \nthe excited state influences the width of the Lorentzian line. The Lorentzian peak function is \ndefined by four parameters as follows,  \n \nf(x)=2A\nπ (w\n4(x−xc)2+w2), \nwhere “xc” is the peak position  i.e. the center of the peak , “w” is full width half maximum \n(FWHM)  and “A” is the area under the curve. The height  of the Lorentz peak can be determined \nby,  \nH= 2A\nπw . \nA summary of the fitting parameters of the Lorentzian peak function  along with peak positions  \nis shown in Table 1. The fitting was performed by using a commercial software [31].  8 \n \n \nFigure 2 Raman spectra obtained by laser  sources  of the wavelengths 488 nm (upper panel) \nand 785  nm (bottom panel).  The common  peaks present in both the spectra are true Raman \nbands.  \n \nFigure 3 Flourescence signals induced in Gd 2O3 by the laser source of the wavelength 785 nm.  \n100 200 300 400 500 600 700 800 900 1000 1100 1200 1300\n  Experimental spectrum\n Lorentz fit peaks\n Cumulative Lorentz fitSource laser 785 nm \n Source laser 488 nmIntensity (a.u.)\nRaman shift (cm-1)\n1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500Intensity (a.u.)\nWavenumber (cm-1)9 \n \n \n  10 \n \nTable 1 Fitting parameters for the Lorentzian peak function  \nPlot  Phase  xc w A χ2 R2 \nPeak1  Gd 2O3 120.4 ± 0.2  14.8 ± 1.2  0.6 ± 0.08  \n5.4 x 10-6 0.98  Peak2  Gd 2O3 135.7 ± 0.2  12.6 ± 1.04  0.5± 0.07  \nPeak3  Gd 2O3 236.0 ± 0.8  17.3 ± 2.9  0.4 ± 0.07  \nPeak4  GdFeO 3 274.5 ± 1.0  18.6 ± 4.0  0.4 ± 0.08  \nPeak5  GdFeO 3 304.8 ± 0.7  21.4± 3.1  0.8 ± 0.1  \nPeak6  Gd 2O3 335.6 ± 0.6  30.5 ± 3.1  2.2 ± 0.2  \nPeak7  GdFeO3 358.1 ± 0.1  12.8 ± 0.5  1.9 ± 0.1  \nPeak9  GdFeO 3 541.2 ± 0.6  32.7 ± 1.6  5.0 ± 0.3  \nPeak10  GdFeO3 563.9 ± 0.6 20.0 ± 1.9  1.8 ± 0.2  \nPeak11  GdFeO 3 621.0 ± 0.4  9.0 ± 1.5  0.3 ± 0.1  \nPeak12  GdFeO 3 635.6 ± 0.9  17.4 ± 2.8  0.5 ± 0.08  \nPeak13  GdFeO 3 679.8 ± 0.5  3.4 ± 1.7  0.1 ± 0.02  \nPeak14  GdFeO 3 688.3 ± 0.5  7.3 ± 1.6  0.2 ± 0.03  \nPeak15  GdFeO 3 765.5 ± 0.9  7.4 ± 3.3  0.1 ± 0.06  \nPeak16  GdFeO 3 777.0 ± 0.8  16.3 ± 2.2  0.6 ± 0.09  \n \nThere are 16 Raman bands , shared by the spectra obtained by 488 nm (upper panel of Figure 2) \nand 785 nm (lower panel of Figure 2) laser sources. The bands associated with Gd 2O3 are at about \n120 cm-1, 136 cm-1, 236 cm-1, 336 cm-1, and 564 cm-1 whereas vibrational bands located at about \n275 cm-1, 305 cm-1, 358 cm-1, 541 cm-1, 621 cm-1, 636  cm-1, 680 cm-1, 688 cm-1, 766 cm-1, and \n777 cm-1 belong to GdFeO 3. Our observations are in close agreement with the previous studies \non the similar material s ystems .[32–35] \nFigure 3 shows the spectra in the range of 1300 cm-1 to 2500 cm-1 obtained by 785 nm laser \nsource where very strong fluorescence peaks can be observed . While the abundance of the \nfluorescence signals create difficulties in the  interpretation of the Raman spectrum, i n this case \nwe are able to identify Gd 2O3 by these signals in agr eement with the previous report  [35]. 11 \n \nThe SEM micrographs, Figure 4(a,b), show the morphology of the fabricated materials. It appears \nthat the nanoparticles of an average diameter of around 250 nm form micrometer -sized \naggregates  of GdFeO 3 and Gd 2O3. The representative EDX spectra, shown in Figure 5(a,b), were \ncollated from several randomly chosen sites of the samples confirming  that the Gd and Fe \nelements are the highest in concentration followed by Ni and Zn. However, certain sites were \ncompletely de pleted of the Ni and the Zn atoms as is apparent in Figure 5(b). The FIB -EDX \nmapping was used to analyze selected particles in the mixture, Figure 5(c). It was observed that \nthe bulk of the particles mainly contains Gd (≈71 wt%), oxygen (≈ 20 wt%) and Fe (≈ 7 wt%) as \nconfirmed by the dominant peaks in the EDS spectrum of the particles.  \n  12 \n \n \nFigure 4 (a,b) The SEM micrographs from two different randomly  selected sites. Micro -sized \nclusters formed by the aggregation of particles of the average diameter of 250 nm.  \n \nFigure 5  (a, b) EDX and (c) FIB -EDX of ferrite doped Gd 2O3 nanoparticles.    \n13 \n \nThe FTIR spectrum is shown in Figure 6. In the fingerprint region i.e. below 1 600 cm-1, the \nabsorption band s at 1508  cm-1, 1398 cm-1, and 1388 cm-1, inset (a) Figure 6, are the characteristic \nfeatures of gadolinium oxide. Note that the previous studies sho w only two  absorption  bands \n[36] in this spectral range , as opposed to three in this study,  indicati ve of  two phases of \ngadolinium oxide such as Gd 2O3 and GdO. Figure 6 inset(b)  shows s everal bands in the spectral \nrange of 600 cm-1 to 400 cm-1 ascribed  to metal -oxides. The band at 547 cm-1 is due to the Gd-O \nvibration of Gd 2O3 whereas the bands from 500 cm-1 to 400 cm-1 are due to Fe -O and Gd -O \nvibrations.  \n \nFigure 6  FTIR spectrum showing absorption bands of the mixed phases. Inset (a) and (b) show \nzoomed in views in the spectral ranges 1330 cm-1 to 1450 cm-1 and 400 cm-1 to 600 cm-1. \n  \n2000 1800 1600 1400 1200 1000 800 600 400020406080100\n1460 1440 1420 1400 1380 1360 13406080100\n13851398\n600 580 560 540 520 500 480 460 440 420 4000204060Transmittance (%)\nWavenumber (cm-1)406415424436447463\n480\n491501\n543Transmittance (%)\nWavenumber (cm-1)16171508\n(a)\n(b)14 \n \nDiffuse ref lectance spectrum  and the calculated absorption coefficient is s hown Fi gure 7(a,b)  as \na function of photon energy.  The d iffuse reflected light  emits after penetrating and repeatedly \nrefracting, transmitting and scattering inside the sample.  A monotonic fall in the reflectance \nspectrum , Figure 7(a), is observed above 1.0 eV to  nearly 4.5 eV where it becomes almost flat but \nthen falls again abruptly  at about photon energy value of 5.6 eV. This is a clear indication of \npresence of the optical gap. A rise in the reflectance  below 1.0 eV  signifies  free carriers. The \ndiffuse reflectance is related to the absorption and scattering coefficient by following relationship  \nas put forward by Kublek a-Munk , \nF(R∞)= K\nS=(1−R∞)2\n2R∞, \n \nwhere “F(R∞)” is referred to as the Kubelka -Munk function, and “R∞” is the absolute   \nreflectance of the sample, “K” is the absorption coefficient, and “S” is the scattering coefficient. \nThe scattering coefficient v aries depending on the size and density of the particles  as well as \nrefractive index of the materials. H owever, i t is not dependent on the light frequency  so the \nscattering coefficient  is a constant  in the Kubelka -Munk  model  assuming that refractive index of \nthe absorbing materials is in the range 2.1 – 2.5. [37] The F(R∞) , therefore, is proportional to \nthe absorption coefficient  [38,39] . Considering t he refractive index of Gd 2O3 is about 2.1  [40] and \nthe sample pellets of about 1  mm thickness are assumed to be  of infinite thickness , [41] the \nabsorption coefficient  calculated from  Kubelka -Munk  model , Figure 5(b),  show s an onset of \nabsorption at photon energy ~1.8 eV, which is on average 4.0 eV smaller than the values  [13–\n15,42]  reported for Gd 2O3. The rise in the absorption, however, is not monotonic and becomes \nflat at around 3.8 eV up to 5.6 eV where a second absorption edge lies. A double onset of \nabsorption , at 1.8 eV  and 5.6 eV , indicates the formation of a new valence band due to the \nhybridi zation of the atomic orbitals of the ferrites and Gd3+. It should be noted that the \nfundamental absorption edge at 1.8  eV is only 0.5  eV greater  compared to those of  nitrides \ncompounds of rare  earth  metals  [43–45] which are considered to be the  leading  materials with \npotential in spintronic applications. A significant absorption in the subgap region is indicative of \na high density of electronic charge carriers. The free carrier density can be esitmated  by using the 15 \n \nDrude model  [46] and is found to be of the order 1026 cm-3 which is the characteristic of a heavily \ndoped semiconductor  [47]. In a doped state, the charge car riers form donor and acceptor levels , \nabove the valence and below the conduction bands respectively. These delocalized states are \nresponsible for a metallic like electronic transitions in the subgap region . \n \nFigure 7 (a) Diffuse reflectance spectrum  shows a fall in the reflectance above 1.8 eV(b) Absorption  \ncoefficient calculated by using  the Kublek -Munk model  from the reflectance spectrum indicates \npresence of the energy band gap . \n \nThe specific capacitance,  in units of F.g−1, was calculat ed using the following equation, \nSpecific  capacitance  =I.Δt\nm.ΔV  , \nwhere “ I” is the discharge current, “ Δt”  is the discharge time, “ ΔV” is the operating potential \nwindow, “ m” is the mass loading of the active electrode material. A potential window of 0.30 to \n0.53  V was used during both the galvanostatic charge -discharge (GCD) and cyclic voltammetry \n(CV) segments of experiment.  Figure 8 shows t he GCD measurement  profiles at current densities \nof 1 to 10 Ag-1. Specific capacitance was measured from the discharge curve at all current \ndensities and highest capacitance of 7 Fg-1 was measured at 1 Ag-1. An excellent cyclability is \ndisplayed with modest drop of around 27% in capacitive performance when current density was \nincreased ten times from 1 Ag-1 to 10 Ag-1. The specific capacity values for different current \n16 \n \ndensities are listed in Table 2 . The drop in specific capacitance at higher current densities can be \ncredited to the diffusion effect of electrolyte ions on the surface of active material. At higher \ncurrent densities the inner active sites cannot precede the redox reaction effectively . [48] \nTable 2: Specific capacity values of Gd 2O3at different current densities.  \nCurrent density  \n(A/g)  1 2 4 6 8 10 \nSpecific capacitance  \n(F/g)  7 6 5.48  5.28  5.12  5.10  \n \n0 2 4 6 8 10 12 14 16 180.30.350.40.450.50.55Potential (V) vs Hg/HgO\nTime (s) 1A/g\n 10A/g\n \n \nFigure 8 The GCD profile of ferrite doped Gd 2O3 electrodes at current density from 1 to 10 Ag-1. \n  17 \n \nThe cyclic voltammetry ( CV) plots of ferrite doped Gd 2O3 electrode at the scan rate of 5 mV/s to \n100 mV/s is shown in  Figure 9. The CV  profiles are almost  symmetrical in shape within the applied \npotential window. Well defined oxidation/reduction peaks present in all CV curves represents \nthe pseudocapacitance nature of the tested electrode at all scan rates.  \n0.3 0.35 0.4 0.45 0.5 0.55-1-0.8-0.6-0.4-0.200.20.40.60.81 \n Current density (mA cm-1)\nPotential (V) vs. Hg/HgO 100mV/s\n 80mV/s\n 60mV/s\n 40mV/s\n 20mV/s\n 10mV/s\n 5mV/s\n \nFigure 9 CV profile of Gd 2O3 electrodes at scan rates from 5 to 100 mVs-1. \n \n  18 \n \nElectrochemical impedance spectroscopy (EIS) is a widely used technique to evaluate the \nresistive and capacitive behaviour of any active materials for supercapacitor applications.  Figure \n10(a) shows  a typical complex plane impedance plot of ferrite doped Gd 2O3 electrodes where a \nhigh frequency intercept of real axis provides an equivalent series resistance (ESR) value, which \nis the sum of solution and active materials resistances , at around 0.99  Ω as shown in inset of \nFigure 10(a). The low frequency region of the curves at 45° symbolizes an excellent capacitive \nbehaviour of the electrode material. [49] \n \n \n \n \n \n \n \nFigure 10 (a) EIS Nyquist plots of Gd 2O3 electrodes in frequency range of 10 mHz to 100 kHz \nwith wave amplitude of 5 mV.  (b) Cyclic stability of Gadolinium oxide (inset: initial and final five \ncharge –discharge cycle).  \n \nMaterial stability is another vital aspect of any active material for its potential of real word \napplications. Gadolinium oxide based active material showed excellent long -term durability with \nthe capacity retention of around 90% after 5,000 cycles as show n in Figure 10(b) where inset \nshows first and last five charge discharge cycles. This exceptional retention performance can be \ndue to the robustness nature of this material and good conductivity (low resistance) as shown by \nEIS plot , Figure 10 (a).  \n \n  \n19 \n \nWe finally show the energy density as a function of power density as shown in the Figure 11. The \nspecific energy density of 0.98 W h /kg at the specific power density of 504 W/kg and an energy \ndensity of 0.70 W h /kg was retained at higher power density 5100 W/kg. It can be witnessed \nthat there was slight drop in energy density when power density was increased by over ten folds.  \n \nFigure 11: Ragone plot illustrating energy density as a function of power density of \nGd2O3 doped with Ni 0.5Zn0.5Fe2O4 based super c apacitor.  \n  \n20 \n \nConclusion  \nWe have reported a clear experimental signature of the direct optical energy gap in the ferrite \ndoped Gd 2O3. The nanostructures of the ferrite doped Gd 2O3 with grain size of approximately \n95 nm were prepared by thermal decomposition. The fundamental energy band gap appeared \nto be red shifted at 1.8 eV, as determined by absorption coefficient using the Kubleka -Munk \nmodel . The highest s pecific capacitance was measured to be of 7 Fg-1 at current density of at 1 \nAg-1 which showed a drop of only 27% as the current density was increased. Although, the drop \nin the capacitance is small, the specific capacitance itself, however, is of modest value. A \ncomparative CV curves obtained for different scan rates  illustrates the p esudocapacitve nature \nof the ferrite doped Gd 2O3. The technique s employed in this study show that  Gd 2O3 doped with \n20 wt% concentration of Ni0.5Zn0.5Fe2O4 exhibits an improved electrochemical behaviors  which  \nmay be used for high -performance pseudocapacitor applications. Ferrite doped Gd 2O3 \nheterostructures synthesized with simple approach of thermal decomposition display  superior \nelectrode behaviours  and could be used to fabricate more electroactive electrode  materials.  \nAcknowledgement  The work is supported by the University of Sharjah competitive research \ngrant number 21021430112. Authors are thankful to Anakha Udayan for data acquisition.  \nEthical Compliance  All procedures performed in studies involving human participants were in \naccordance with the ethical standards of the institutional and/or national research committee \nand with the 1964 Helsinki Declaration and its later amendments or comparable ethical \nstandards.  \nConflict of Interest declaration  The authors declare that they have NO affiliations with or \ninvolvement in any organization or entity with any financial interest in the subject matter or \nmaterials discussed in this manuscript.  \n  21 \n \nReferences  \n[1]  Lei X gong, Jockusch S, Turro N J, Tomalia D A and Ottaviani M F 2008 EPR \ncharacterization of gadolinium(III) -containing -PAMAM -dendrimers in the absence and in \nthe presence of paramagnetic probes J. 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Uddin1,* \n1Department of Physics, Chittagong University of Engineering and Technology (CUET), \nChattogram 4349, Bangladesh . \n2Materials Science Division, Atomic Energy Center, Dhaka 1000, Bangladesh . \n3Department of Glass and Ceramic Engineering, Bangladesh University of Engineering and Technology (BUET), \nDhaka 1000, Bangladesh.  \n \n \nABSTRACT  \n \nWe report the synthesi s of Y-substituted Mg -Zn [Mg 0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05)] ferrites  \nusing conventional standard ceramic technique.  The samples  were characteriz ed by X-ray \ndiffraction (XRD) analysis, field emission  scanning electron micros copy (FESEM), FTIR \nspectroscopy , UV–Vis spectroscopy  and quantum design physical properties measurement \nsystem (PPMS).XRD  patterns confirm the single phase cubic spinel structure up to  x = 0.03 and \nappearance of a secondary phase of YFeO 3for higher Y contents.  FESEM images depict the \ndistribution of grains and EDS spectra confirmed the absence of any unwante d element. \nCompletion of solid state reaction and formation of spinel structure has  been revealed  from FTIR \nspectra. The FTIR data along with lattice constant, bulk density and porosity were further used to \ncalculate the stiffness constant ( Cij), elastic constant and Debye temperatures. Mechanical \nstability of all studied compositions is confirmed from Cij using Born stability conditions. \nBrittleness and isotropic nature are also confirmed using  Poisson’s ratio and anisotropy \nconstants , respectively . The enhancement of  dc electrical resistivity  (105Ω-cm to 106Ω-cm) with \nY content is observed . The energy band gap  (increased with Y contents)  is found in good \nagreement with dc electrical res istivity. Ferrimagnetic  to paramagnetic phase change has been  \nobserved from  the field dependent high temperature magnetization curves . The magnetic \nmoments  and saturation magnetization we re found to be decreased with increasing temperature. \nThe Curie temperature (T c) has been measured from temperature  dependent magnetic moment \n(M-T) and initial permeability ( µʹi-T) and found to be  in good agreement with each other. \nDecrease in T c with Y content  is due to redistribution of cations and weakening of the exchange \ncoupling constant . The magnetic phase transition has been analyzed by Arrott plot and found to 2 \n have second order phase transition.  The dc resistivity endorses the prepared ferrites are suitable \nfor high frequency and high temperature magnetic device applications as well.  \nKeywords:  Y-substitute d Mg -Zn ferrites, dc resistivity,  FTIR  spectroscopy , mechanical  \nproperties,  UV-Vis spectroscopy, high temperature hysteresis loop , Curie temperature.  \n \n1. Introduction  \nSince last century,  the spinel type ferrites have come out as one of the leading materials in the \nfield of materials science due t o their vast applications. The p hysics involved in spinel structure \nis still interesting as their properties can easily be tuned, consequently their application s as well . \nIn recent years, new opportunities have been disclosed and numerous directions have been \ndeveloped for research on  these materials  by manipulating the ir physical properties . The long -\nestablished applications of the ferrites (in thin film, bulk or nano form ) are in  transformer \napplications, solar hydrogen production, multi -layer chip inductors, high density magnetic \nrecording, telecommunications, microwave devices, as contrast agents in magnetic resonance \nimaging (MRI), sensors, catalysts etc.  [1-8]. Moreover, the new application of spinel ferrites has \nbeen listed as photocatalysts for photocatalytic degradation of dyes [ 9-12].  \nMg-Zn ferrite is  one of the most  used soft ferrites because of its high electrical resistivity, low \ncost, low dielectric l oss, high mechanical hardness and superior environmental  stability [ 13]. \nMg-Zn ferrites with higher resistivity (106–107 Ω-cm) make  its suitability in high frequency \napplications [13]. Most  of (90%) Mg2+ ions are located on the B-sites and small fraction  (10%)  \noccupies  in the A-sites [ 14, 15]. However, the Zn2+ ions have  a preference to occupy  A-sites in \nthe spinel lattice [ 16, 17]. The Fe3+ ions are distributed on both A- and B-sites [ 18]. Therefore, \nthe physical properties of the Mg -Zn ferrites are determined by the cations distribution over the \nA- and B-sites. The physical properties can be altered by introducing the different metallic ions \nresults the cations distribution modification o n the A- and B-sites. The factors that determine \ncation distributions are ionic radius, charge, site preference and level of substituent ion, methods, \nconditions and sintering temperature for the preparation of ferrites  [19-22].  \nStudy of ferrites usually in most cases has been limited to their structural, electrical and magnetic \nproperties  [9-25]. However,  study of mechanical properties of ferrites has scientific importance 3 \n due to t heir potential applications  in industry and in research field. Elastic constants are closely \nrelated to many important physical properties of solids, for instance acoustic -phonon vibration \nand internal stress, Debye temperature. This type of study provides the information about  \nbonding nature  within the materials . Als o to understand the thermal properties and minimum \nchanges in devices performance with mechanical shock, temperature, vibration, etc. during the \nmanufacturing and testing of ferrite materials  [26] Due to the mentioned interest the study of \nmechanical prope rties have already been considered by researchers  [26-29]. Moreover, the study \nof magnetic properties  of Mg-Zn/other ferrites have also been  usually  limited in range of room \ntemperature to as low as 4 K  [30]. But the most technological uses of these ferrites are  either at \nor above room temperature.  The hysteresis loop, magnetization as well as magnetic moment are \nstrongly dependent on operating temperature . The room temperature measurement of magnetic \nproperties cannot provide exact behavior of magnetic material at higher temperature since the \ntemperature dependent magnetic behavior is highly non -linear. Therefore, to design a magnetic \ndevice using soft ferrites, thermal effects on magnetic properties need to be considered [ 31-32].  \nThis prospect  also motivates the researchers and some reports on the high temperature magnetic \nproperties also available [ 30, 33-34]. Furthermore, the study of dc electrical resistivity provides \nimportant information regarding suitability of materials in high frequency  application. In \naddition, determination of band gap of semiconducting materials is very important for their \npractical application. Therefore , the present paper report s the investigation of mechanical \nproperties, high temperature magnetic properties along with FTIR spectra and UV -Vis \nmeasurement of Mg -Zn ferrites.   \nMg-Zn ferrites have been subjected by a  large number of research group  to modify the different \nproperties using different synthesis routes and conditions [ 12, 14, 35-49], where  some of them \nare dealt with tailoring electrical properties and /or magnetic properties of Mg -Zn ferrites.  The \nmotivation of our present research is to enhance the electrical resistivity of the Mg -Zn ferrites by \nY3+ substitution for Fe3+ with favorable magnetic properties that is  one of the significant \ncharacteristics of ferrites for applications in high frequency devices.  Melagiriyappa  et al . [35] \nhave been studied the electrical properties of Sm substituted Mg -Zn ferrites where increased of \nelectrical resistivity has been  reported. The initial permeability and saturation magnetization (M s) \nare focus and the variation is also being  explained with the variation of Cu contents as well as \ndensity of the samples [ 36]. Mn substituted Mg -Zn ferrites have been investigated by Mohseni  et 4 \n al. [37] in which the structural, vibrational, micro -structural and magnetic properties are \nconsidered. The increase in Ms and decrease in coercivity due to Mn substitution is reported. The \neffect of Cr3+ substitution in the Mg–Zn ferrite has been studied by Haralkar  et al. [42] and Xia \net al. [43]; they have reported on the decrease of Ms with Cr contents.  Mukhtar  et al. [48] have \ninvestigated the Pr substituted Mg -Zn ferrites where t he samples are  characterized by XRD, \nFTIR spectroscope, SEM , EDX and VSM measurements . The decrease in Ms and increase in \ncoercive field ( Hc) has been reported with increase in Pr contents.  The i mprovement in Magnetic \nproperties of Mg -Zn ferrites  by Co substitution ha s also been reported [ 45]. Recently, the effect  \nof Y substitution on the electrical resistivity and magnetic properties of different ferrites has been \nreported [ 34, 50 -56]. But to the best of our knowledge, Y is not considered as a substituent  in the \nMg-Zn ferrites yet.  We have studied the effect of Y substitution on structural properties, \nmorphological study, ac conductivity, dielectric properties, permeability and room temperature \nmagnetic properties of Mg -Zn ferrites and results are available  elsewhere [ 57].  \nTherefore, i n this article, FTIR spectroscopy , mechanical properties, dc resistivity, UV -Vis \nspectroscopy  and temperature dependent magnetic properties of Y -substituted Mg -Zn ferrites \nprepared from nano -sized raw materials  have been presented  for the first time .  \n2. Materials and method s \nA series of Mg 0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) ferrites , in the steps of 0.01,  have been synthesized \nusing analytical grade nano -sized materials by conventional ceramic technique. The use of nano \npowders as raw materials is inspired by successful completion some other projects that can be \nfound elsewhere [58-61]. The detail  of the sample preparation is presented elsewhere [ 57]. The \nsample  preparation method comprises the following o perations as shown in Fig. 1 . The X-ray \ndiffraction (XRD) spectra of the prepared samples  has been  recorded  by a PHILIPS X ’Pert PRO \nX-ray diffractomete r in the range 2θ = 15–700 using Cu  Kα radiation (λ = 1.54 Å).  The \nmicrostructure images  have been taken by the Field Emission S canning Electron Microscope \n(FESEM ) (model: JEOL JSM -7600F) equipped with an  energy dispersive spectrometer (EDS).  A \nPerkinElmer FTIR spectrometer has been used to compute the Fourier Transform Infrared  \n(FTIR) spectra using the  attenuated total reflection  (ATR ) technique in the range  400–1000 cm-1. \nElastic properties were calculated using FTIR data  along with structural parameters . For dc \nelectrical  resistivity measurements, the as -prepared powders were pressed to pellet (diamet er 8.4 5 \n mm, thickness  2.4 mm ) and sintered.  Conductive silver paint was applied  and two probe method \nwas used to measure the dc electrical resistivity. Temperature dependent M vs H curves  and \nmagnetic moment measurement were done  by a Quantum Design PPMS VSM magnetometer.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1. Flow chart of sample preparation method.  \n3. Results and discussion  \n3.1 Phase analysis  \nFig. 2 depicts t he XRD  patterns of Mg0.5Zn0.5YxFe2-xO4(0 ≤ x ≤ 0.05)  ferrites . The peaks \nare indexed for different atomic planes and compared with reported peaks of different ferrites \n[34, 50 -56]. The XRD patterns reveal synthesis of spinel ferrites in which the single phase noted  \nup to x=0.03 . A secondary  phase is appeared  at 2=33.2 ° for higher Y contents . This extra phase \n(marked as red  circle) results from YFeO 3 because of  highly reactiv e tendency  of Fe3+ ions with \nY3+ ions [ 62]. This type of secondary phase is also  shown  in case of Y substitution in  other \nferrites  [55-56]. The structural parameters have been calculated using  the XRD data and can be \nfound  elsewhere [ 57].  Oxide of raw \nmaterials (Fe 2O3, \nMgO, ZnO and \nY2O3) Weighing by \ndifferent mole  \nPercentage  \n  \nDry mixing \nby Milling ,  \n6 hrs  \n \nPressing to \ndesired shape  \n  \nMilling  \n2.5 hrs  \n Pre-sintering  \n(850 °C, 4 hrs)  \nSintering  \n(1250 °C, 4 hrs)  \n  \nFinish ed \nProduct  6 \n  \n \n \n \n \n \n \n \n \n \n \nFig. 2. The XRD pattern s of Mg0.5Zn0.5YxFe2-xO4(0 ≤ x ≤ 0.05).  \n \n3.2 Microstructure study  \nThe physical properties, especially the magnetic properties , are sensitive to the  microstructure  of \nferrites and by manipulation of microstructure the properties can also be tailored.  Fig. 3 \nillustrates  the FESEM images of Mg 0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05)  where considerable  \nmodification in microstructu re is observed due to the Y substitution.  The ED S spectra for each \ncomposition are presented in Fig. 4(a -f).  The  qualitative analysis of EDS spectra  endorses  the \nexistence  of Mg, Zn, Fe, Y and O ions and nonattendance  of any undesired  element.  The \nquantitative analysis of EDS spectra provides the ratio of metal cations and anions present in the \nsamples which are also calculated [Table 1] and found in good agreement with other reported \nresults  [14].  \nTable 1 Cations -anions % (At) and cations -anions ratio of Mg 0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05).  \nY contents \n(x) Grain size \n(µm)  cations (At %)  anions (At %)  Ratio of cation: \nanion  \n0.00 1.22 40.03  59.97  2.802:4.197  \n2.847:4.152  \n2.820:4.179  \n2.946:4.053  \n3.355:3.775  \n2.688:4.312  0.01 3.65 40.68  59.32  \n0.02 3.75 40.29  59.71  \n0.03 1.28 42.09  57.91  \n0.04 1.07 47.94  53.94  \n0.05 1.87 38.40  61.60  \n20 30 40 50 60 70\n  Intensity (arb. u.)x = 0.05x = 0.04x = 0.03x = 0.02x = 0.01\n2 (degree)x = 0.00(220)(311)\n(222)(400) (422)(551) (440)7 \n  \n \n \n \n \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. 3 . FESEM images of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05)  [57]. \n \n(b) x=0.01  \n (a) x=0.00  \n(d) x=0.03  \n (c) x=0.02  \n \n(e) x=0.04  \n (f) x=0.05  \n 8 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n3.3. FTIR spectroscopy  \nFTIR spectr a of ferrites generally exhibit two major peaks for the M-O vibrations  at A- and B- \nsub-lattice  [63]. Fig.  5 illustrates  the FTIR spectra of the considered ferrite compositions  in \nwhich two major peaks are noted within the  expected  range of frequency : 365 to 1100 cm-1. \nTable 2 shows  the peaks  point s for the studied compositions , the completion of solid state \nreaction has been confirmed from the position of the band within the expected range of \nfrequency for  ferrites with spinel structure  [64]. The bands at higher frequency (ν1 = 560 cm-1)  \n \n \nFig. 4 . EDS spectra of Mg0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) . \nx=0.00  x=0.01  \nx=0.02  x=0.03  \nx=0.04  x=0.05  9 \n correspond  to the stretching vibrations of the Metal–Oxygen  bunch  in the A-sites and at lower \nfrequency (ν2 = 396-402 cm-1) are allocated  for the stretching vibration  of the Metal –Oxygen \nbunch in the B-sites, respectively  [65]. The band frequency ν1 remains constant for all \ncomposition . The band frequency ν 2 slightly shifts to lower frequency side for x = 0.04 and \nhigher frequency side at x = 0.05, this shift in peak position is might be related  to the appearance \nof impurity phase. These measurements  (XRD, FESEM and FTIR)  confirmed the  successful \nincorporation  of Y into Mg-Zn ferrites.   \n \nTable 2 \nFTIR absorption bands for  Mg0.5Zn0.5YxFe2-xO4. \nY \n(x) ν1  \n(cm-1) ν2 \n(cm-1) FcT×105(dy\nnes/cm) FcO×105(d\nynes/cm) Kav×105\n \n(dynes /cm) \n0 560 400 2.29 1.17 1.73 \n0.01 560 400 2.29 1.17 1.73 \n0.02 560 400 2.29  1.17 1.73 \n0.03 560 400 2.29  1.17 1.73 \n0.04 560 396 2.29  1.14 1.72 \n0.05 560 402 2.29  1.18 1.74 \n \n \n \n \n \n \n \n \n \n3.4 Mechanical  properties   \nThe stability and bonding characteristics among the adjacent atomic plane can be understood  \nfrom the stiffness constants. For cubic crystal system , there are three stiffness constants: C11, C12 \nand C44. The FTIR spectroscopy data can be used to determine  the stiffness constant , elastic \nconstants and Debye temperature etc.  [66, 67]. The equations used to calculate C11 is given by  \n[68]𝐶11=𝐾𝑎𝑣𝑎 , where Kav is the  average of tetrahedral and octahedral force constant calculated \nWave number (cm-1)400 600 800 1000Transmittance (%)\n30603060025503060030600306090\nx=0.00x=0.01x=0.02x=0.03x=0.04x=0.05\n45\n67\n8910\n11Fig. 5. FTIR spectra of Mg0.5Zn0.5YxFe2-\nxO4(0 ≤ x ≤ 0.05) . \n 10 \n [Table 2] from FTIR spectra using the following equation  [69]:𝐹𝑐=4𝜋2𝑐2𝜈2𝑚, where , c, υ and \nm are the speed of light in free space, vibrational frequency  and the reduced mass of Fe3+ and O2- \nions, respectively. The  equivalent reduced mass is about  2.061 ×10-23 g and a is the lattice \nconstant.  The relation [ 68]: 𝐶12=𝜍𝐶11\n1−𝜍 is used to estimate the  C12, where σ is the Poisson’s ratio  \n[70]: 𝜍=0.324 ×[1− 1.043 ×𝑃 ], P is the pore fraction.  The C44 is calculated from the \nrelation:  𝐶44=𝐶11−𝐶12\n2. The obtain ed values  of Cij [Table 3] confirmed  that the ferrites \ncompositions under consideration satisfy the Born criteria for mechanical stability  [71, 72]: \n𝐶11−𝐶12>0; 𝐶11+2𝐶12>0; 𝐶44>0.Therefore, the synthesized compositions are \nmechanically stable.  The stiffness constant  C11 represent s the elasticity of length and provide s \nthe stiffness of materials  along the (100) direction while the constants C12 and C44 born the \nelasticity in shape . The variation in stiffness constants with Y contents is shown in Fig. 6 (a).  \n \n \n \n \n \n \n   \n \nFig. 6.  The variation of (a) stiffness constants ( Cij) and elastic moduli with Y contents.  \nThe values of C11 and C12 increase  up to x = 0.02 and then C12 gradually decreases while C11 \ndecreases up to x = 0.04 and increases further . The improvement in C11 and C12 endorse the good \nsolubility of Y ions into B-sites and strengthen  of the inter -atomic bonding . While the decline  \ntrend  in C11 and C12 curves indicates  the decrease in crystallization process . Microstructure \nimages [Fig. 3] also ratify  the crystallization of prepared samples. In this case , the variation in \nC11 and C12 and bulk density [Table 3] may be attributed from the variation of grains size that \ncan be observed from Fig. 3.  [73]. The equations  [70]:𝐸= 𝐶11−𝐶12 (𝐶11+2𝐶12)\n(𝐶11+𝐶12), 𝐵=1(𝐶11+2𝐶12)\n3, \n0.00 0.01 0.02 0.03 0.04 0.0530405060708090C12 and C44 (GPa)\nY contents (x)190195200205210215220\nC11\nC12\nC11 (GPa)C44(a)\n0.00 0.01 0.02 0.03 0.04 0.057580859095100\nE\nG\nY contents (x) B and G (GPa)B(b)\n180190200210\nE (GPa)11 \n 𝐺=𝐸\n2(𝜍+1) are used to calculate the Young’s modulus ( E), Bulk modulus ( B) and shear modulus \n(G) and presented in Table 3. Fig. 6 (b) represent s the variation in different elastic moduli [ B, G \nand E] with Y contents ( x). The bulk modulus provide s the value of required resistance for \nvolume deformation,  an increase is observed up to x = 0.02 due to improvement in \nmicrostructure and then decreases  gradually. This variation might be related to the variation in \nbulk density as well as the grains size  (x) [73]. The shear and Young’s modulus are noted to \ndecrease for x = 0.01 and then increase gradually with  the variation of x. The value of E provides \nthe stiffness of solids where the high value of E indicates the  stiffer  solids.  The values of E and G \nof substituted compositions are lower than that of the parent one [ x= 0.00]. The change in inter-\natomic bonding is responsible for the variation in elastic moduli [ 26]. Although, the variation of \nelastic moduli [Fig. 6] is shown as a function of Y contents in magnified scale, but if we have a \nlook on the values presented  in Table 3, an insignificant variation is observ ed. Table 3 shows the \nPoisson’s ratio [ σ] for al l composition s. The Poisson’s ratio is extremely important in materials \napplication point of view which determine s the ductile behavior or brittleness of materials with a \ndecisive  factor  of 0.26  [74, 75]. A value of σ less than 0.26 indicates the brittleness of materials \notherwise ductile behavior of materials. Like other ceramics materials, brittleness of studied \ncompositions  is confirmed  and the values of σ for all compositions lie in the expected range  from \n-1 to 0.5, confirming excellent  elastic behavior with the theory of elasticity  [76]. The anisotropy \nconstant A (A= 2C44/(C11-C12)) [77-78] also calculated [Table 3] and found to have the value of 1  \n(one)  for all composition, indicating isotropic nature of cubic ferrites compositions.  \nThe Debye temperature ( ΘD), the most fundamental property  of solids, used to resolve many of \ncharacteristics thermal properties ; determine  the lower temperature region and higher \ntemperature region for solids.  Moreover, the ΘD also provides the information regarding bonding \nstrength. The higher value of ΘD is associated with stronger bonds in solids. T he average sound \nvelocity can be used to  calculate ΘD by the following equation [79]: ΘD=hkB   3n4π  NAρ/\nM 13 𝑣m, where the symbols signify the usual meanings  [75]. The average  sound velocity  is \ncalculated by the equation  [79]:𝑣m= 13  1𝑣l3+2𝑣s3    −13 , where vl \n[𝑣l=  3B+4G 3ρ  12 ] and vs [𝑣s= Gρ  12 ] are the longitudinal and transverse sound \nvelocities  in an isotropic material.  12 \n  \n \n \n \n \n \nFig. 7. The variation of sound velocities and Debye temperature with Y contents  in the \ncomposition of Mg 0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) . \nTable 3 \nThe lattice constant ( a), bulk density (ρb), porosity ( P), average force constant ( Kav), stiffness \nconstant ( Cij) elastic moduli ( E, B, G ), Poisson’s ratio ( σ), anisotropic constant ( A), sound \nvelocities ( vl, vs, vm) and Debye temperature ( ΘD). \nParameters  x =0.00 x =0.01 x =0.02 x =0.03 x =0.04 x =0.05 \na (Å) 8.4309 8.4327 8.4366 8.4350 8.4269 8.4288 \nρb(kg/m3) 4006  4172  4197  4166  4119  4084  \nP (%) 18.08  14.76  14.25  15.06  16.38  17.17  \nKav. (N/m)  173.40  173.40  174.58  173.40  172.24  173.99  \nC11 (GPa)  205.55  205.5 9 206.86  205.38  204.01  206.14  \nC12 (GPa)  32.19  42.10  43.99  41.15  36.89  34.93  \nC44 (GPa)  86.68  81.70  81.43  82.11  83.56  85.60  \nE (GPa)  196.83  191.18  191.43  191.64  192.71  196.01  \nB (GPa)  89.98  96.57  98.28  95.89  92.60  92.00  \nG (GPa)  86.68  81.70  81.43  82.11  83.56  85.60  \nσ 0.13 0.17 0.17 0.16 0.15 0.14 \nA 1.0 1.0 1.0 1.0 1.0 1.0 \nv1 (m/s)  6918.9  6742.7  6740.1  6748.3  6777.8  6851.1  \nvs (m/s)  4365.4  4091.0  4063.0  4109.9  4194.3  4278.7  \nvm (m/s)  4803.9  4520.7  4492.6  4539.8  4625.3  4713.8  \nΘD (K) 607.43  579.12  576.38  580.70  589.12  598.39  \n \nThe calculated sound velocities mentioned are  presented in Table  3 along with the calculated ΘD. \nIt is observed [Table 3] that the transverse sound velocit y is lower than the longitudinal  sound \nvelocity. Energy is transferred from particle to particle by vibration of particle during \n0.00 0.01 0.02 0.03 0.04 0.054.24.95.66.37.0\nvsvlvl, vs, vm (103 m/s)\nY contents (x)vm\n570580590600610\nD\nD (K)13 \n propagation of wave through a medium . During the propagation of transverse wave, the vibration \nof particle is at right angles  with the direction of wave transmission , caused a higher energy \nrequirement to vibrate the adjacent  particle.  As a result the energy of the wave reduced and \nhence the velocity of transverse wave  is lower than the longitudinal  wave and is roughly  half of \nlongitudinal  wave velocity  [80]. Fig. 7 illustrates  the variation of ΘD with Y contents in which it \nseen that the ΘD decreases with Y contents up to x = 0.02 and then increases with Y contents. \nThe similarity in the variation of sound velocities and ΘD is observed  similar to other ferrites [ 26, \n73, 76]. The variation of ΘD roughly follows the variation of  rigidity  modulus  [76] as can be \nshown in Fig. 6 (b) . \n3.5 dc resistivity  \nHigh resistivity of ferrites is essential for most electronic application and therefore, the study of \ndc electrical resistivity is important  for prediction of their application in electronic and high \nfrequency devices. The room temperature  dc electrical resistivity  (𝜌𝑅𝑇) of Mg 0.5Zn0.5YxFe2-xO4 (0 \n≤ x ≤ 0.05) is estimated  by the following equation:  𝜌𝑅𝑇=𝑅𝐴\n𝐿, where, R, A and L  is resistance , \narea and thickness , respectively  of the pellet.  The value of 𝜌𝑅𝑇 is noted  to increase d due to Y \nsubstitution unlike for  x = 0.0 1 (Fig. 8 (a)) , can be explained by  Verwey mechanism based on  \nelectrons exchange involving  the ions of the element with different valence states positioned  \noctahedral (B-site) [81-82]. In the present case , the Fe2+ and Fe3+ is responsible for the \nconduction because of the hopping tendency between them  [83, 84] and the Fe2+ ions are  the \nproduct of sintering process [ 42]. The value of 𝜌𝑅𝑇 is found to be  0.45 × 105 Ω-cm for x = 0  and \n0.40 × 105 Ω-cm, 1.57  × 106 Ω-cm, 1.78  × 106 Ω-cm, 2.19  × 106 Ω-cm and 1.43  × 106 Ω-cm for x \n= 0.01, 0.02, 0.03, 0.04 and 0.05, respectively. Fig. 8 (a) clearly indicates the increase in \nresistivity for substituted composition (except for x = 0.01) where the resistivity decreases \nslightly. The ac resistivity was found to decrease for x = 0.01 [ 57]. The Y ions replace Fe3+ ions \nat B-sites. [50], simultaneously s mall fraction  of Zn ions forfeit  balanced  by forcing  some Fe3+ to \nmove at the A-sites.  The Fe ions have higher electronic valence than Zn ions; hence the charge \ncarriers (here, metallic vacancy) have been increased by balancing the electrical charge, results  \ndecrease in resistivity. Two mechanisms might be involved  in increas ing resistivity , firstly, the \nenhancement in resistivity due to Y substitution is attributed by forming the stable bonds \n(electric) of  Y3+ and Fe2+, causes localization of Fe2+, limiting the Verwey mechanism  [81], 14 \n hence upturn the resistivity. Secondly, the preference of Y3+ ions to reside in  the B-sites, lead to a \nreplacement of Fe3+ ions at  B-sites, results a reduction of Fe2+ formation. Since, Y3+ ions do not \ntake place in the conduction due to its stable oxidation state and electrons hopping takes place \nbetween Fe3+ and Fe2+, therefore, the degree of electrons change ( Fe2+↔ Fe3+) is reduced and \nconsequently the resistivity is increased. The resistivity of materials greatly depends on its \nmicrostructure: the porosity and the distribution of grains. The lowering of resistivity for x = 0.05 \ncontents may be explained using the micros tructure [Fig. 3(f)] of substituted compositions and \nthe improved connectivity between the grains [ 85]. From Fig. 3 (d -e), it is seen that the grains \ndistribution are more homogeneous for x = 0.05 than that for x = 0.03 and 0.04; and expected to \nhave the i mproved connectivity between the grains. Therefore, a slight decrease in resistivity is \nalso expected.  \n \n \n \n \n \n \n \n \n \n \n \nFig. 8. (a) variation dc resistivity with Y contents , and (b) plot of [ F(R ∞)hν]2 against hν of \nMg 0.5Zn0.5YxFe2-xO4 (0≤x≤0.05)  ferrites . \n \n3.6 Diffuse reflectance results   \nTo know  the optical properties of Y -substitute d Mg -Zn ferrites, the UV –Vis diffuse reflectance \nstudy  has been  performed . The optical reflectance data can be used to calculate the band gap \nusing the equation 𝐹 𝑅∞ =𝐴 ℎ𝜈−𝐸2 𝑛\nℎ𝜈, where,  F(R ∞)=(1-R)2/2R ∞, is the  Kubelka -Munk Function \n[86]. Therefore, the plot of [ F(R ∞)hν]2 against  photon energy (hν) gives the way of calculating \nthe energy band gap Eg (eV), by extrapolati ng the linear part to the x-axis ( energy axis ). The \n0.00 0.01 0.02 0.03 0.04 0.050.00.51.01.52.02.5\n Y contents (x)\n    (-cm 106)(a)\n1 2 3 4 5 6 7 \n \nh (eV)x=0.00Eg= 2.56 eVEg= 2.59 eV\nx=0.01 \n Eg= 2.71 eV\nx=0.02 \n Eg= 2.73 eV\nx=0.03\n [F(R)h]2Eg= 2.78 eV\nx=0.04 \n Eg= 2.58 eV\nx=0.05  \n (b)15 \n band gap is found to be increased due to the Y  substitution that is consistent with the Y -\nsubstituted Co ferrites [ 53]. The variation  in band gap indicates the change in structural \nparameters attributed from the change in Y contents [ 53]. The band gap  enhancement  is \nassociated with the variation of dc resistivity [Fig. 8 (b)].  \n3.7 Magnetic properties  \n3.7.1 High temperature  magnetic properties  mapping  \nThe magnetization vs applied magnetic field (upper part of hysteresis loops ) at different \nmeasuring  temperatures , as shown in Fig. 9 (a-f), exhibiting typical magnetic nature  at elevated \ntemperature, revealing the existence of magnetic ordered structure near Tc (in section 3. 7.3). \nBeyond  Tc, destruction of magnetic order  is also observed i.e., transition from ferrimagnetic  \n(below T c) to paramagnetic (above T c) nature. It is [Fig. 9 (a -f) also expected  that the coercivity \n(Hc) and remanent  magnetization  (Mr) decrease  with increasing temperature and  tend to be zero \nnear Tc. It is very important that the ferrimagnetic ordered structured remain s up to T c indica ting \nthe suitability of these materials to use them  at higher temperature.  \n  \n \n \n \n \n \n \n \n \n \n \n \n \n0 10k 20k 30k 40k010203040M (emu/g)\nApplied field (Oe) 37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 C(b)\n0 10k 20k 30k 40k01020304050(a)\nApplied field, H (Oe)M (emu/g) 37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 C16 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 9 (a-f) The M vs H plot  of Mg 0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) m easured  at different  \ntemperatures.  \nThe value of M s for each composition at different selected  temperatures is  obtained by the \nfollowing: at first the M s vs 1/H curves are  plotted (figures not shown here)  and are extrapolated \nto zero i.e., 1/H = 0  [53]. The obtained M s values  are plott ed in Fig. 10 (a) for different Y \ncontents at different temperature . The variation of M s with Y contents has been explained \nelsewhere [ 57]. A decrease in the M s with increasing temperature is observed due to more \nrandomness of magnetic moments that attributed from the increase in thermal energy. A change \nin site occupancy of cation  may also induce due to thermal fluctuation;  results reduced M s at \nhigher temperatu res. \n \n0 10k 20k 30k 40k010203040 37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 C Applied field, H (Oe)M (emu/g)(c)\n0 10k 20k 30k 40k01020304050  37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 CApplied field, H (Oe)M (emu/g)(d)\n0 10k 20k 30k 40k05101520253035M (emu/g)\nApplied Field, H (Oe) 37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 C(e)\n0 10k 20k 30k 40k010203040\n(f) 37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 C\nApplied Field, H (Oe)M (emu/g)17 \n  \n \n \n \n \n \n \n \n  \n  \n \n \n  \n \nFig. 10. Variation of (a) saturation magnetization at different measuring temperatures , (b) \ntemperature dependent magnetic moment and (c) dM/dT vs temperature graph for different Y \ncontents.  \n \nIt is interesting to note that the width of the magnetic transition, inferred from the magnitude of \nthe FWHM of the dM/dT vs T curve [Fig. 10 (c)] and the magnetic ordering temperature (taken \nat the peak of the dM/dT versus T graph ) show features strongly dependent on the level of Y \nsubstitutions. For the Y substituted compounds  [Fig. 10 (c)] , the transition width does not show \nany systematic variation with Y content. The Y -free compound is characterized by a very large \ntransition width. The transition temp erature lies within a narrow range of 341 ± 4 K. This implies \nthat the average mean -field exchange energy among the spins remains insensitive to the Y \ncontent. It should be noted that, substitutional disorder always introduces local strains inside the \n0.000.150.0000.0370.0000.0390.0000.0630.0000.0430.0000.066\n327 227 127x = 0.00 \n \nTemperature (C)\n \n27x = 0.01 \n x = 0.02 \n x = 0.03\n M (emu)x = 0.04 \n x = 0.05  \n (b)\n0.00 0.01 0.02 0.03 0.04 0.0530405060(a)\n 27 C\n 37 C\n 47 C)\n 57 C\n 65 CMs (emu/g)\nY contents \nnge interactions inside these samples.   \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 5.2.29 dM/dT vs temperature graph for Y substituted compositions . \n360 420 480 540 600-0.0030.000Y=0.02\n dM/dT \nT (K)\n360 420 480 540 600-0.0030.000Y = 0.03\n dM/dT \nT (K)\n360 420 480 540 600-0.0030.000\nY = 0.04\n dM/dT \nT (K)\n360 420 480 540 600-0.006-0.0030.000\nY = 0.05\n dM/dT \nT (K)\n360 420 480 540 600-0.0030.000Y = 0.00\n dM/dT \nT (K)\n360 420 480 540 600-0.0030.000Y = 0.01\n dM/dT \nT (K)18 \n comp ounds. These local distortio ns and i nhomogeneities lead to a distribution in the exchange \nenergies. Within the mean -field approximation, the Curie temperature is directly proportional to \nthe exchange energy [ 87]. Structure specific corrections often lower the theoretically predicted \ntransition temperature, but this link between transition temperature and exchange energy remains \nvalid. Within this framework, width of the magnetic transition is directly related to the \ndistribution and homogeneity of atoms ins ide the solid. Therefore, different transition widths in \nthe Y substituted compounds reflect the distribution and inhomogeneity of the exchange \ninteractions inside these samples.   \n \n3.7.2 Curie temperature   \nThe magnetic moment and initial permeability and h ence the magnetic properties  are very \nsensitive to temperature . Both exhibit an  abrupt dropping near  the transition point from \nferrimagnetic to paramagnetic state.  Therefore, the Curie temperature (T c) has been  measured by \napplying the two basic characteristics: (i) the samples are  subjected to an applied magnetic field \nof 100 0 Oe and the magnetic moment is  calculated at different temperature. The heating rate is \nkept slow to avoid the nucleation and particle growth at elevated temperatures. The v ariation of \nmagnetic moment with temperature is shown in Fig .10 (b) where the  magnetic moment is found \nto decrease  with increasing tempera ture as expected and tends to  zero at different measuring \ntemperature for different compositions. The T c value is  then calculated from the first derivative \nof magnetic moment against  temperature graph , Fig. 10 (c). Moreover, the T c is also calculated \nfrom the t emperature dependen t initial permeability, µi plot as shown in Fig .11 (a). A peak, also \nknown as Hopkinson peak, is observed from the graph where the value of initial permeability \ndropped sharply . The characteristic temperature at which the peak observed is known as the T c \nwhere the spin magnetic moments are completely disordered  and the conversion from \nferrimagnetic  to paramagnetic materials is taken place. The obtained T c is found to be decreased \nwith the Y substation. The value of T c depends on the cation distribution on A- and B-site and the \nexchange coupling constant  [53]. The coupling constant due to exchange interaction between \nions in sub -lattice ( A and B) is much stronger than the coupling constant due to exchange \ninteraction among the ions in the same sub -lattice sites ( A/B). The substitution of Y ions in the \nspinel str ucture leads to increase in the bond lengths and distance between ions presents  at the A- \nand B–sites that  results a decrease in exchange constants and hence  the T c values [ 53]. The T c 19 \n value for the Mg 0.5Zn0.5Fe2O4 (x = 0.00) is obtained from the M-T graph and µʹi-T graph  and \nfound to be 77°C and 75°C, respectively  which  is higher than that of (Tc ~47°C ) reported by K hot \net al. [88] and Hashim  et al. [44] and lower than that  of (Tc ~97°C ) obtained by Mazen  et al. \n[89]. \n \n \n \n \n \n \n \n \n  \n \nFig. 11. (a) The temperature  dependent initial permeability, and (b) the variation of Curie \ntemperature with Y contents.  \n \n3.7.3 Phase transition  \nIn section 3. 7.1, the ferr imagnetic to paramagnetic phase transition has been observed and the \ntransitions temperatures hav e also been obtained  (section 3 .7.2). The  determination of the nature \nof phase transition (first or second order)  is also significant . The Arrott plot ( M2 (emu/g)2 against \nH/M (Oe.g/emu))  has been used  to see the nature of phase transition . Banerjee  [90] has proposed \nthe criteria for first and second order transition. The positive slope of Arrott plot  indicates the \nsecond order phase transition otherwise the transition is first order . The curves intercept at M2 \n(Y-axis) indicate the  presence of spontaneous magnetization  while the curve passing through the \norigin measure the T c [91]. Fig. 1 2 (a-f) illustrates that the second order phase transition has been \noccurred in the prepared samples.  It is also observed that the spontaneous magnetization is \npresent up to  ≤ 75 °C for x = 0.00, ≤  69 °C for x = 0.01, ≤  67 °C for x = 0.02, ≤  65 °C for x = \n0.03 and ≤  63 °C for x = 0.04 and 0.05  and there after the curves (after T c for each compositions ) \nare straight upwards from H/M axis ( x-axis) which indicates the paramagnetic behavior . \n40 60 80 100200300400500600700\n x=0.00\n x=0.01\n x=0.02\n x=0.03\n x=0.04\n x=0.05 \n i\nTemperature (C)(a)\n0.00 0.01 0.02 0.03 0.04 0.0560657075\n  Tc (C)\nY contents From -T\n From M-T(b)20 \n  \n \n \n \n \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 2. The Arrott plot  of Mg 0.5Zn0.5YxFe2-xO4 (0≤ x ≤0.05) ferrites . \n \n0 1k 2k 3k 4k 5k051015202530  37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 CH/M (Oe.g/emu)M2  (emu/g )2(a) \n01k 2k 3k 4k 5k 6k024681012141618\nH/M (Oe.g/emu)(b) M2  (emu/g )2 37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 C\n0 1k 2k 3k 4k 5k05101520 37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 CH/M (Oe.g/emu)M2  102 (emu/g )2(c)\n0 1k 2k 3k 4k 5k0510152025  37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 CH/M (Oe.g/emu)M2  102 (emu/g)2(d)\n01k2k3k4k5k6k7k024681012M2  (emu/g )2\nH/M (Oe.g/emu) 37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 C(e)\n01k 2k 3k 4k 5k 6k0246810121416\nH/M (Oe.g/emu)M2  (emu/g )2(f) 37 C\n 47 C\n 52 C\n 57 C\n 62 C\n 67 C\n 69 C\n 71 C\n 73 C\n 75 C\n 77 C\n 79 C\n 81 C\n 83 C\n 85 C\n 87 C\n 92 C\n 97 C\n 102 C\n 127 C\n 147 C\n 167 C\n 187 C21 \n 4. Conclusions  \nThe Yttrium -substituted Mg -Zn [Mg 0.5Zn0.5YxFe2-xO4 (0 ≤ x ≤ 0.05) ] ferrites , with spinel \nstructure as confirmed from  the XRD and FTIR studies , have been synthesized using ceramic  \ntechnique.  The calculated stiffness constant revealed the mechanical stability  of the \ncompositions . The bulk modulus is increased for substituted compositions  up to x = 0.2 and \ndecreased thereafter.  The shear modulus and Young’s modulus are found to be decreased at x = \n0.01 and then increased with further increase in Y contents . The values of Poisson’s ratio (0.13 \nto 0.17) indicating  the brittleness character of studied compositions.  The value of anisotropic \nconstants ( A = 1) be a sign of isotropic nature of studied spinel ferrites. The variation  of ΘD with \nincreasing  Y contents is associated with the variation  in rigidity  of the compositions . The dc \nresistivity increases from 0.45×106 Ω-cm (for x = 0.00)  to 1.57 ×106 Ω-cm, 1.78×106 Ω-cm, \n2.19×106 Ω-cm and 1.43×106 Ω-cm for x =0.02, 0.03, 0.04 and 0.05, respectively. The obtained  \noptical energy band gap are found to be 2.56 eV (for x = 0.00) to 2. 59 eV, 2.71 eV, 2.73 eV, 2.78 \neV and 2. 58 eV for x = 0.01, 0.02, 0.03, 0.04 and 0.05, respectively.  Both the ferrimagnetic and \nparamagnetic nature have been confirmed from  temperatures dependent magnetization curves . \nVery close  values of Tc obtained from the M-T and µʹi-T is noticed  which is decreased due to the \nweakening of exchange coupling constant  with Y substitution . The second order phase transition \n(ferrimagnetic to paramagnetic ) has been occurred.  The prepared ferrites are suitable for high \nfrequency application as well as at temperature higher than room temperature (up to T c). \nAcknowledgments  \n \nThe authors are grateful to the Directorate of Research and Extension, Chittagong \nUniversity of Engineering and Technology, Chattogram -4349, Bangladesh under grant \nnumber CUET DRE (CUET/DRE/2016 -2017/PHY/005) for arranging the financial support \nfor this work. We are thankful for the laboratory support of the Materials Science Division, \nAtomic Energy Center, Dhaka -1000, Bangladesh.  \nReferences  \n1. A. Goldman, Modern Ferrites Technology, 2nd ed. (Springer, New York, USA, 2006).  \n2. M. A. Ahmed, H. E. Hassan, M. M. Eltabey, K. Latka, T. R. Tatarchuk, Phys . B: Phys . Cond . Mater . 530 \n(2018) 195.  \n3. B. R. Babu, T. Tatarchuk, Mater. Chem. Phys. 207 (2018) 534.  \n4. T. Tatarchuk, M. Bououdina, W. Macyk, O. Shyichuk, N. Paliychuk, I. Yaremiy, B. AlNajar, M. Pacia, \nNanoscale res. lett. 12(2017) 1.  22 \n 5. 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